With Cognitive Foundations for Improving Mathematical Learning,weclosea five-volume journey through the evolution and ea

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*Table of contents : Cover......Page 1Cognitive Foundations for ImprovingMathematical Learning......Page 2Copyright......Page 5Contributors......Page 6Foreword: Cognitive Foundations for Improving Mathematical Learning......Page 8Chapters That Deal With Possible Cognitive Foundations for Numeracy......Page 9Chapter That Deals With Home and Parental Influences......Page 10Chapters That Deal With Interventions and Their Effects......Page 11References......Page 13Preface......Page 16Introduction......Page 18Brief History of Mathematics Intervention Studies......Page 19Design Factors......Page 20Diminishing Intervention Impacts Across Time (Fadeout)......Page 22Transfer......Page 24What Are the Cognitive Foundations for Improving Mathematics Learning?......Page 25Fluid Intelligence......Page 27Executive Function and Working Memory......Page 28Implications......Page 30Domain-General Interventions......Page 31Domain-Specific Components......Page 32Computer-Based Training of Number Line Judgments......Page 33Training to Enhance Subitizing Speed......Page 35Training Finger Differentiation: Its Impact on Arithmetic Ability......Page 37Using Electrical Brain Stimulation to Improve Numerical and Arithmetic Processing......Page 38Parental Influences on Child Cognition and Mathematical Learning......Page 39A Brief History of Research on Parental Influences......Page 40Research on Children's Home Numeracy Environment......Page 41Conclusion......Page 42References......Page 278Introduction......Page 54The Uruguayan Context......Page 55Classroom Geometric and Arithmetic Abilities......Page 59Approximate Number Abilities......Page 60The Present Intervention Study......Page 62Teachers Responses to the Software......Page 63Design of the Current Study......Page 64Participants......Page 65Intervention Games......Page 68Results......Page 69IQ and Repeater Status by SES Quintile......Page 70Pre-Intervention Arithmetic by Grade and Repeater Status......Page 71ANS and Arithmetic Ability......Page 73Pre- to Postintervention Improvement......Page 74Conclusions and Future Directions......Page 79References......Page 82Introduction......Page 85Integrated Theory of Numerical Development......Page 86Numerical Magnitude Understanding in Early Childhood......Page 91Play and Games in Mathematics Development......Page 92Playing Traditional Games to Promote Numerical Knowledge......Page 93Computer and Tablet Games......Page 97Preschool Programs Using Games and Play......Page 99Conclusions and Future Directions......Page 100References......Page 101Introduction......Page 107Training Studies Using ``The Number RaceÂ´Â´......Page 108Nonsymbolic vs. Symbolic Training......Page 111Brief ANS Training......Page 112Long-Term ANS Training......Page 114Mechanisms Behind Long-Term ANS Training Improvements......Page 115Conclusions and Future Directions......Page 118References......Page 119Introduction......Page 123Number Talk......Page 124Questionnaire Studies......Page 125Observational Studies in the Lab......Page 126Naturalistic Home Observations......Page 127Experimental Studies......Page 130Experiments in the Lab......Page 131Experiments in the Field......Page 132Spatial Talk......Page 133Summary: Math Talk......Page 135Gesture: An Additional Support for Children's Math Learning......Page 136Counting Gestures......Page 137Cardinal Number Gestures......Page 138Gesture and Arithmetic......Page 140Summary: Gesture......Page 141Parental Math Attitudes and Beliefs: Intergenerational Findings......Page 142Intergenerational Effects of Math Anxiety......Page 143Other Negative Attitudes Toward Math......Page 144Conclusions and Future Directions......Page 146References......Page 149Introduction......Page 159The Curriculum Research Framework (CRF)......Page 160Category I: A Priori Foundations......Page 161Phase 2. Subject Matter A Priori Foundation......Page 162Phase 4. Structure According to Specific Learning Model and Learning Trajectory......Page 163Phase 6. Formative Research: Small Group......Page 178Phase 7. Formative Research: Single Classroom......Page 179Phase 8. Formative Research: Multiple Classrooms......Page 180Phase 10. Summative Research: Large Scale......Page 181Conclusions and Future Directions......Page 182References......Page 183Connections Between Early Mathematics Development and General Language......Page 190Early Connections Between Mathematics and Literacy Skills......Page 191What is Content-Specific Mathematical Language?......Page 193Correlational and Experimental Evidence on the Relations Between Mathematical Language and Mathematics Performance......Page 194Interventions to Improve Mathematical Language......Page 197Mathematical Language and Numeracy Instruction......Page 200Spatial Language......Page 201Developing Methods for Mathematical Language Instruction......Page 202References......Page 203Introduction......Page 209Early Numeracy Skills are Important for Future......Page 210Identifying Children at Risk for Mathematical Learning Difficulties......Page 211Early Numeracy Interventions for Low-Performing Children......Page 215Studies With ThinkMath Intervention Programs......Page 219Conclusions and Future Directions......Page 222Acknowledgments......Page 356The SES-Related Gap in Children's Early Mathematical Knowledge......Page 229Potential for Early Curricular Intervention to Reduce the Math Gap......Page 231The Pre-K Mathematics Intervention......Page 232Responsiveness of Low-Performing Children......Page 234First Approach: Tutorial Interventions in Mathematics and Attention......Page 235Math Screening Measure......Page 237Study Design......Page 238The Math Intervention: Pre-K Mathematics Tutorial......Page 239The Attention Intervention......Page 240Measures and Assessment Procedures......Page 241Is the Tutorial-Based Math Intervention Effective?......Page 242Does Attention Training Have a Facilitative Effect on Math Outcomes?......Page 243Second Approach: Intensification by Providing 2 Years of Tier 1 Math Intervention......Page 244Child Sample......Page 245The Math Intervention: Pre-Pre-K Mathematics and Pre-K Mathematics......Page 246Measures and Assessment Procedures......Page 248Does This 2-Year Intervention Improve the Mathematical Knowledge of Very Low-Performing Children?......Page 249Effectiveness of the First Intervention Approach......Page 250Comparison of the Two Intervention Approaches for Very Low-Performing Children......Page 251Conclusion 1. Most Low-Performing Children Respond to Intensified Support in Mathematics......Page 252Conclusion 3. Public Preschool Programs Should Provide High-Quality, Intensive Math Support, But New Policies and Resources .........Page 253References......Page 254Analogy and Analogical Reasoning in Mathematics......Page 260Where Does the Analogy "Numbers are Points on the Line" Come From?......Page 264Tapping Into Students Understanding of the Dense Ordering of Rational Numbers......Page 265Could the Number Line Support Students Understanding of Density?......Page 267Does the Number Line Have an Effect on Students Reasoning About Density?......Page 268Is Density More Accessible to Students in a Geometrical Rather Than in an Arithmetical Context?......Page 270Using the "Numbers are Points" Analogy and the ``Rubber Line" Bridging Analogy......Page 274Conclusions and Future Directions......Page 27711The Role of Visual Representations in Mathematical Word Problems......Page 282Students Natural Use of Visual Representations in Mathematics......Page 283Embedding Visual Representations in Text......Page 289Teaching Students to Create or Complete Diagrams as They Solve Problems......Page 291Integrating Visual Representations With Text......Page 294Schema-Based Instruction (SBI): Integrated Instruction in Word Problems and Visual Representations......Page 297Conclusions and Future Directions......Page 302References......Page 30312The Role of Cognitive Processes in Treating Mathematics Learning Difficulties......Page 308Why Embed Cognitive Process Training within Direct Skills Intervention?......Page 310Why Focus on Word Problems?......Page 311Conceptual Model for Linking Language Comprehension and Our Approach to Explicit Skills Word-Problem Intervention......Page 312Study Overview......Page 315Preliminary Results......Page 316Allocating Varying Forms of Explicit Skills Intervention to Subgroups of Learners With Different Cognitive Profiles......Page 319Why Focus Specifically on Fraction Magnitude Comparisons and Word Problems?......Page 320Why Incorporate a Self-Explaining or Word-Problem Instructional Component Into the Multicomponent Fractions Intervention?......Page 321Does Supported Self-Explaining Compensate for Limitations in Cognitive Processes?......Page 322Study Overview......Page 325Results......Page 327Summary and Future Directions......Page 329References......Page 330Introduction......Page 334Patterns of Effects Across Time and Theories of Children's Mathematical Development......Page 335Measurement-Based Explanations......Page 336Cognitive Processing Explanations......Page 340Implications for the Study of Children's Mathematical Development......Page 350Targeting At-Risk Children......Page 352Targeting Advanced Skills in Older Children......Page 353Complementary Follow-Through Interventions......Page 354The Possibility of Different Effects of Improving Early Math Intervention at Scale?......Page 355References......Page 357C......Page 360D......Page 361G......Page 362L......Page 363N......Page 364P......Page 365S......Page 366W......Page 367Z......Page 368Back Cover......Page 369*

Mathematical Cognition and Learning

Cognitive Foundations for Improving Mathematical Learning

Mathematical Cognition and Learning Series Editors Daniel B. Berch David C. Geary Kathleen Mann Koepke

VOLUME 5 Cognitive Foundations for Improving Mathematical Learning Volume Editors David C. Geary Daniel B. Berch Kathleen Mann Koepke

Mathematical Cognition and Learning

Cognitive Foundations for Improving Mathematical Learning Edited by

David C. Geary Psychological Sciences University of Missouri Columbia, MO, United States

Daniel B. Berch Curry School of Education University of Virginia Charlottesville, VA, United States

Kathleen Mann Koepke Eunice Kennedy Shriver National Institute of Child Health and Human Development (NICHD) National Institutes of Health (NIH) Bethesda, MD, United States

Academic Press is an imprint of Elsevier 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States 525 B Street, Suite 1650, San Diego, CA 92101, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 125 London Wall, London, EC2Y 5AS, United Kingdom Copyright © 2019 Elsevier Inc. All rights reserved. Disclaimer Notice The following were written by a U.S. Government employee within the scope of her official duties and, as such, shall remain in the public domain: Preface and Chapter 1. The views expressed in this book are those of the authors and do not necessarily represent those of the National Institutes of Health (NIH), the Eunice Kennedy Shriver National Institute of Child Health and Human Development (NICHD), or the U.S. Department of Health and Human Services (DHHS). No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-815952-1 ISSN: 2214-2568 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Nikki Levy Acquisition Editor: Emily Ekle Editorial Project Manager: Barbara Makinster Production Project Manager: Anusha Sambamoorthy Cover Designer: Matthew Limbert Typeset by SPi Global, India

Contributors Pirjo Aunio, Special Needs Education, Faculty of Educational Sciences, University of Helsinki, Helsinki, Finland Drew Bailey, School of Education, University of California, Irvine, Irvine, CA, United States Daniel B. Berch, Curry School of Education, University of Virginia, Charlottesville, VA, United States Talia Berkowitz, Department of Psychology, University of Chicago, Chicago, IL, United States Douglas H. Clements, Morgridge College of Education, Marsico Institute, University of Denver, Denver, CO, United States Caitlin Craddock, Department of Special Education, Vanderbilt University, Nashville, TN, United States Emily N. Daubert, Department of Human Development and Quantitative Methodology, University of Maryland, College Park, MD, United States Dinorah De Leo´n, Centro de Investigacio´n Ba´sica en Psicologı´a, Montevideo, Uruguay Lydia DeFlorio, College of Education, University of Nevada, Reno, Reno, NV, United States Dahiana Fitipalde, Centro de Investigacio´n Ba´sica en Psicologı´a, Montevideo, Uruguay Douglas Fuchs, Department of Special Education, Vanderbilt University, Nashville, TN, United States Lynn S. Fuchs, Department of Special Education, Vanderbilt University, Nashville, TN, United States David C. Geary, Psychological Sciences, University of Missouri, Columbia, MO, United States Dominic J. Gibson, Department of Psychology, University of Chicago, Chicago, IL, United States Justin Halberda, Johns Hopkins University, Baltimore, MD, United States Asha K. Jitendra, University of California, Riverside, Graduate School of Education, Riverside, CA, United States Yemimah King, Department of Human Development and Family Studies, Purdue University, West Lafayette, IN, United States xi

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Alice Klein, Center for Early Learning, STEM Program, WestEd, San Francisco, CA, United States Josh Langfus, Johns Hopkins University, Baltimore, MD, United States Susan C. Levine, Department of Psychology; Department of Comparative Human Development; Committee on Education, University of Chicago, Chicago, IL, United States Melissa E. Libertus, Department of Psychology and Learning Research and Development Center, University of Pittsburgh, Pittsburgh, PA, United States Alejandro Maiche, Centro de Investigacio´n Ba´sica en Psicologı´a, Montevideo, Uruguay ´ lvaro Mailhos, Centro de Investigacio´n Ba´sica en Psicologı´a, Montevideo, Uruguay A Amelia S. Malone, Department of Special Education, Vanderbilt University, Nashville, TN, United States Kathleen Mann Koepke, National Institutes of Health, Eunice Kennedy Shriver National Institute of Child Health & Human Development, Bethesda, MD, United States Amy R. Napoli, Department of Human Development and Family Studies, Purdue University, West Lafayette, IN, United States David J. Purpura, Department of Human Development and Family Studies, Purdue University, West Lafayette, IN, United States Geetha B. Ramani, Department of Human Development and Quantitative Methodology, University of Maryland, College Park, MD, United States Julie Sarama, Morgridge College of Education, Marsico Institute, University of Denver, Denver, CO, United States Nicole R. Scalise, Department of Human Development and Quantitative Methodology, University of Maryland, College Park, MD, United States Pamela M. Seethaler, Department of Special Education, Vanderbilt University, Nashville, TN, United States Prentice Starkey, Center for Early Learning, STEM Program, WestEd, San Francisco, CA, United States Xenia Vamvakoussi, Department of Early Childhood Education, University of Ioannina, Ioannina, Greece John Woodward, School of Education, University of Puget Sound, Tacoma, WA, United States

Foreword: Cognitive Foundations for Improving Mathematical Learning

Ann Dowker Department of Experimental Psychology, Oxford University, Oxford, United Kingdom

In recent years, there has been increased interest in early mathematical development. This has included both interest in the domain-general and domain-specific underpinnings of early mathematical development (Chu, van Marle, & Geary, 2015; Hannula, Lepola, & Lehtinen, 2010; Hohol, Cipora, Willmes, & Nuerk, 2017), and interest in early interventions to ameliorate and where possible prevent mathematical difficulties (Butterworth, Varma, & Laurillard, 2011; Clements & Sarama, 2011; Cohen Kadosh, Dowker, Heine, Kaufmann, & Kucian, 2013; Dowker, 2017; Fuchs, Fuchs, & Compton, 2012). On the whole, these areas have tended to be studied and discussed separately, though a better understanding of the foundations of early numerical abilities could make it much easier to develop suitable interventions. This book makes a very important contribution to our knowledge and understanding by bringing together and integrating the topics of cognitive foundations and interventions in mathematics and reviewing recent work in these areas. In the introductory chapter, the editors David Geary, Daniel Berch, and Kathleen Mann Koepke explore the key issues of the book. They discuss both the domain-general and domain-specific factors that contribute to mathematical development and mathematical difficulties, and how different types of interventions have addressed these. Domain-general factors include fluid intelligence and executive function and working memory. There have been numerous attempts to train children in domain-general abilities, especially working memory, and a few individual results have appeared potentially promising (e.g., Klingberg, 2010). However, as the editors and authors and others point out, there is little consistent evidence that training in domain-general abilities transfers to tasks that are dissimilar to those on which children have been trained (Melby-Verlag, Redick, & Hulme, 2016; Sala & Gobet, 2017). In discussing the chapters, xiii

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I will group them not chronologically but thematically as follows: (1) those that propose and discuss cognitive foundations of numeracy; (2) a chapter that discusses informal influences on children’s numeracy development, specifically parental input; (3) those that describe and discuss the development and evaluation of intervention programs; and (4) discussions of current problems with interventions and how evaluations of their effectiveness might be improved. Because of the integrative nature of the book, many of the chapters include elements of more than one of these, but I will group the chapters according to what appears to be the main focus.

CHAPTERS THAT DEAL WITH POSSIBLE COGNITIVE FOUNDATIONS FOR NUMERACY One domain-specific ability that has been proposed to be an important foundation for numeracy is the Approximate Number System (Dehaene, 1997). Melissa Libertus discusses the relationship between the Approximate Number System (ANS) and mathematical performance. As she points out, studies suggest a correlation between nonsymbolic magnitude estimation and arithmetic (e.g., Mazzocco, Feigenson, & Halberda, 2011), but it tends to be less strong than the correlations between symbolic magnitude estimation and arithmetic (e.g., De Smedt, Noe¨l, Gilmore, & Ansari, 2013). Moreover, training in symbolic number comparison is more likely than training in nonsymbolic number comparison to lead to improvement in arithmetic (Dillon, Kannan, Dean, Spelke, & Dufio, 2017; Honore & Noel, 2016). The author proposes that the ANS may affect mathematical development not so much directly as through increasing attention to and engagement with mathematical material. This may even have intergenerational effects: recent studies (Braham & Libertus, 2017) suggest that parents’ ANS acuity correlates with the amount of mathematical talk that they use when playing with their children, and with their children’s mathematical abilities. With regard to domain-general factors, several authors discuss the importance of both language and visual-spatial representation to arithmetical cognition and learning. David Purpura, Amy Napoli, and Yemimah King discuss the importance of developing mathematical language. Studies have shown that both language and print knowledge in general are related to informal numeracy and numeral knowledge (Austin, Blevins-Knabe, Ota, Rowe, & Lindauer, 2011; LeFevre et al., 2010; Purpura & Ganley, 2014; Purpura, Hume, Sims, & Lonigan, 2011). Purpura, Napoli, Wehrspann, and Gold (2017) found that instruction in mathematical language through dialogic reading improved children’s general numeracy skills, but not their general vocabulary. Analyses showed that it was largely instruction in quantitative language that improved numeracy skills; instruction in spatial language did not have such an effect. However, visual and spatial representations are also important to mathematical learning. Asha Jitendra and John Woodward have studied primary

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school children of various ages, and have focused on the ways in which visual representations may facilitate word problem solving. They review their own and others’ studies, both of how the use of diagrams in text may improve performance, and of how children may be encouraged to develop and use their own visual representations. Xenia Vamvakoussi discusses the importance in early numerical development of another domain-general ability: analogical reasoning, whereby children may make sense of unfamiliar situations by extrapolating from familiar ones. The author points out that young children frequently use analogical reasoning spontaneously, but often fail to use analogies generated by others such as teachers. For example, children and even adolescents may fail to make productive use of the analogy “numbers are points on the line.” The author and colleagues found that secondary school pupils’ use of this analogy was improved by explicit instruction as to its implications, and by the use of a bridging analogy of a rubber number line. Lynn Fuchs, Douglas Fuchs, Amelia Malone, Pamela Seethaler, and Caitlin Craddock discuss the importance of understanding and using cognitive, linguistic, and social-emotional processes in treating mathematics learning difficulties. The authors discuss two ways in which this can be applied to interventions. One is to include cognitive, linguistic and/or social-emotional processes within direct skills interventions. For example, Fuchs, Fuchs, Craddock, Seethaler, and Geary (2017) incorporated language comprehension training in an intervention to improve primary school children’s word problem solving. Preliminary results have suggested that this adds to the effectiveness of the intervention. The other is to allocate different forms of direct skills interventions to groups of pupils with different profiles of cognitive, linguistic and/or social-emotional processes. For example, Fuchs et al. (2016) found that children with poor working memory and/or poor reasoning abilities benefitted more from a fractions intervention if it included verbal explanations than if it involved word problem training, whereas the reverse was true of children with better working memory and reasoning.

CHAPTER THAT DEALS WITH HOME AND PARENTAL INFLUENCES Susan Levine, Dominic Gibson, and Talia Berkowitz discuss some parental influences on young children’s numeracy development. These include parents’ use of numerical language, their use of number gestures (a sometimes neglected topic), and the intergenerational transmission of attitudes toward mathematics. Negative parental attitudes toward mathematics, especially mathematics anxiety, have been found to have adverse effects on children’s mathematical attainment. The authors suggest that interventions that involve parents in helping their children with mathematics (see for example the Pre-K Mathematics curriculum by Klein and Starkey, discussed later) may help to break this cycle, and discuss some interventions that they are themselves developing.

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CHAPTERS THAT DEAL WITH INTERVENTIONS AND THEIR EFFECTS Pirjo Aunio discusses general principles of assessment and intervention. Aunio and R€as€anen (2016) had developed a model of foundational abilities in arithmetic, with four main groups of crucial abilities: (1) symbolic and nonsymbolic number sense; (2) understanding mathematical relations; (3) counting skills; and (4) basic skills in arithmetic. The author emphasizes the importance of identifying children at risk for learning difficulties in mathematics as early as possible. The chapter proposes the use of Curriculum-Based Measurement: the combination of conventional achievement testing and the use of time series (that permit scores for the same individual to be compared at different time points). Reliable assessment may be used to guide suitable interventions. The author discusses in particular the development and use of the “ThinkMath” materials, which target numerical relational skills, counting skills, and simple arithmetical word problems. Alice Klein, Prentice Starkey, and Lydia DeFlorio studied the effects of interventions on American preschool children from disadvantaged backgrounds. The researchers had developed a curricular intervention, Pre-K Mathematics, for children from low-income homes, who were thus at risk of low achievement at school. The intervention includes small-group activities run by teachers using activities with concrete manipulatives to support development of numerical abilities, arithmetic problem solving, spatial abilities and geometrical reasoning, pattern knowledge and informal measurement. It also includes suggestions for parents for games and activities to pursue with children at home. Several studies have shown that the curriculum has positive effects on children’s mathematical performance, as measured on the CMA and TEMA tests, at the time of entering kindergarten, with significant improvements from their performance at entry to prekindergarten. However, those who are in the lowest quartile for mathematical performance at the time of entering prekindergarten seem to benefit little from the intervention. These researchers also examined the effect of giving 541 children from this group either of two additional tutorial interventions. One intervention involved individual tutoring sessions in mathematical tasks related to the overall Pre-K mathematics curriculum. The other combined such tutoring with attention-training tutorials. The children in these individualized intervention groups improved significantly more than those who received just the curricular intervention. The inclusion of attention training did not lead to greater improvement than the mathematics training on its own, again indicating that domain-specific interventions may be more successful than domain-general interventions in improving mathematics performance. The researchers tested yet another form of intervention: giving the curricular intervention without additional individualized tutoring, but administered over a 2-year period starting at age 3, rather than a 1-year period starting at

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age 4. This was found to improve outcomes significantly in comparison with children who received only one year of intervention. Thus, it appears that children who appear relatively resistant to a curricular intervention that benefits most children can be helped by intensification of the intervention, either through adding individualized activities or increasing the time devoted to the curricular intervention. Geetha Ramani, Emily Daubert, and Nicole Scalise discuss the role of play and games in building up preschoolers’ numerical abilities. For example, they report the work of Geetha Ramani and Robert Siegler (Ramani & Siegler, 2008; Siegler & Ramani, 2008) on the use of board games with nursery school and kindergarten children from disadvantaged backgrounds. They proposed that SES differences in early numeracy may reflect different levels of prior experience with informal numerical activities, such as numerical board games. Most middle-class children will have at least some early experience with traditional board games, such as Chutes and Ladders, which involve counting steps on numbered squares, whereas children from disadvantaged backgrounds often do not. The authors found that preschoolers from low-income families performed worse on tests of numerical magnitude estimation than peers from better-off backgrounds. However, playing a simple linear numerical board game for four 15-minute sessions eliminated the differences in numerical estimation proficiency. Playing games that substituted colors for numbers did not improve numerical performance, while a circular numerical board game improved number recognition but not numerical estimation. These findings have been replicated with groups of children in the UK (Whyte & Bull, 2008) and Sweden (Elofsson, Gustafson, Samuelsson, & Tr€aff, 2016). Other number games have also been shown to improve preschoolers’ number understanding, such as dominoes (Brankaer, Ghesquie`re, & De Smedt, 2015) and numerical computer games (Maertens, De Smedt, Sasanguie, Elen, & Reynvoet, 2016; Schacter & Io, 2016; Toll & Van Luit, 2013). While most of the chapters in this book relate to interventions for preschool and kindergarten children, interventions for early primary school are also described. Josh Langfus, Alexandro Maiche, Dinorah De Leon, Dahiana Fitipalde, Alvaro Mailhos, and Justin Halberda studied 386 second- and third-grade children in Uruguay. The researchers created a software application comprising simple mathematical games involving magnitude comparisons for approximate numbers of objects, time durations, and surface areas. The games included novel characters such as “monsters,” visual rewards for good performance, such as “stars” and online tracking. The children in eight Montevideo classrooms played the games over a period of one month. They were compared with children in eight similar classrooms in the same schools undergoing business-as-usual mathematics instruction. All children in both groups had access to tablets. At the end of the month, all children showed significant improvement in tests of IQ, Vocabulary, Geometry, and especially Arithmetic Ability and Approximate Number System Ability. Overall, the intervention and business-as-usual children did not differ significantly, which may be in part due to the fact that they all

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experienced positive effects from being in schools, which were taking part in the project, and receiving regular visits from the researchers. There was some suggestion that the intervention had a positive effect on children, who were in the lowest SES group and/or had repeated a grade.

CHAPTERS THAT DEAL WITH PROBLEMS WITH INTERVENTIONS AND THEIR EVALUATION, AND SUGGEST IMPROVEMENTS Julie Sarama and Douglas Clements point out some important weaknesses in the research often carried out on curriculum development. For one thing, it often involves only product evaluation. Researchers may investigate the effectiveness of an intervention or classroom program in comparison with business-as-usual teaching or with other interventions; but it is often not used for the foundations of development of the interventions in the first place. The authors describe their own “Building Blocks” curriculum, which was informed by evidence at all stages of development and evaluation, especially as regards learning trajectories in different components of numerical abilities and how these are influenced by interventions. This chapter focuses in particular on learning trajectories in subitizing. Drew Bailey discusses reasons why early academic interventions often show significant effects on school readiness, but these effects often fade out during the years after the end of treatment. This may be in part because of the componential nature of arithmetical ability (see Dowker, 2008, 2015): interventions that target components that are important in early schooling may not necessarily affect or transfer to other components that are more important later on. Also, the instruments (tests) used may differ over the years, both in content and in scaling. Another possibility is that the effects of early intervention fade out due to forgetting. Bailey argues that it is important to have more long-term follow-up studies if we are to devise interventions with better long-term impact.

CONCLUSION To conclude, this book brings together many important and up-to-date theories and findings that will stimulate much further research on the cognitive foundations of numeracy and their applications to interventions to prevent or ameliorate mathematical difficulties. It will also facilitate the development of such interventions and help to ensure that they are truly evidence-based.

REFERENCES Aunio, P., & R€as€anen, P. (2016). Core numerical skills for learning mathematics in children aged five to eight years—a working model for educators. European Early Childhood Education Research Journal, 24(5), 684–704. Austin, A. M. B., Blevins-Knabe, B., Ota, C., Rowe, T., & Lindauer, S. L. K. (2011). Mediators of preschoolers’ early mathematics concepts. Early Child Development and Care, 181, 1181–1198.

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Braham, E. J., & Libertus, M. E. (2017). Intergenerational associations in numerical approximation and mathematical abilities. Developmental Science, 20, e12436. Brankaer, C., Ghesquie`re, P., & De Smedt, B. (2015). The effect of a numerical domino game on numerical magnitude processing in children with mild intellectual disabilities. Mind, Brain, and Education, 9, 29–39. Butterworth, B., Varma, S., & Laurillard, D. (2011). Dyscalculia: from brain to education. Science, 332, 1049–1053. Chu, F. W., van Marle, K., & Geary, D. C. (2015). Early numerical foundations of young children’s mathematical development. Journal of Experimental Child Psychology, 132, 1–8. Clements, D. H., & Sarama, J. (2011). Early childhood mathematics intervention. Science, 333, 968–970. Cohen Kadosh, R., Dowker, A., Heine, A., Kaufmann, L., & Kucian, K. (2013). Interventions for improving numerical abilities: present and future. Trends in Neuroscience and Education, 2, 85–93. Dehaene, S. (1997). The number sense: How the mind creates mathematics. Oxford: Oxford University Press. De Smedt, B., Noe¨l, M.-P., Gilmore, C., & Ansari, D. (2013). How do symbolic and non-symbolic numerical magnitude processing relate to individual differences in children’s mathematical skills. A review of evidence from brain and behavior. Trends in Neuroscience and Education, 2, 48–55. Dillon, M. R., Kannan, H., Dean, J. T., Spelke, E., & Dufio, E. (2017). Cognitive science in the field: a preschool intervention durably enhances intuitive but not formal mathematics. Science, 357, 47–55. Dowker, A. (2008). Individual differences in numerical abilities in preschoolers. Developmental Science, 11, 650–654. Dowker, A. (2015). Individual differences in arithmetical abilities: the componential nature of arithmetic. In R. Cohen Kadosh & A. Dowker (Eds.), Oxford handbook of mathematical cognition (pp. 878–894). Oxford: Oxford University Press. Dowker, A. (2017). Interventions for primary school children with difficulties in mathematics. In J. Sarama, D. Clements, C. Germeroth, & C. l. Day-Hess (Eds.), Advances in Child Development and Behavior, Vol. 53: The Development of Early Mathematics Education (pp. 255–287). New York, NY: Elsevier. Elofsson, J., Gustafson, S., Samuelsson, J., & Tr€aff, U. (2016). Playing number board games supports 5-year-old children’s early mathematical development. The Journal of Mathematical Behavior, 43, 134–147. Fuchs, L. S., Fuchs, D., & Compton, D. L. (2012). The early prevention of mathematics difficulty: its power and limitations. Journal of Learning Disabilities, 45, 257–269. Fuchs, L. S., Fuchs, D., Craddock, C., Seethaler, P. M., & Geary, D. C. (2017). In Embedding word-problem specific language comprehension within word-problem intervention Paper presented at the 2017 math cognition conference. Fuchs, L. S., Malone, A., Schumacher, R. F., Namkung, J. M., Hamlett, C. L., Jordan, N. C., et al. (2016). Supported self-explaining during fraction intervention. Journal of Educational Psychology, 108, 493–508. Hannula, M., Lepola, J., & Lehtinen, E. (2010). Spontaneous focusing on numerosity as a domainspecific predictor of arithmetical skills. Journal of Experimental Child Psychology, 107, 394–406. Hohol, M., Cipora, K., Willmes, K., & Nuerk, H.-C. (2017). Bringing back the balance: domain general processes are also important in numerical cognition. Frontiers in Psychology, 8, 499.

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Honore, N., & Noel, M. P. (2016). Improving preschoolers’ arithmetic through number magnitude training: the impact of non-symbolic and symbolic training. PLoS ONE, 11, e0166685. Klingberg, T. (2010). Training and plasticity of working memory. Trends in Cognitive Sciences, 14(7), 317–324. LeFevre, J.-A., Fast, L., Skwarchuk, S.-L., Smith-Chant, B. L., Bisanz, J., Kamawar, D., et al. (2010). Pathways to mathematics: longitudinal predictors of performance. Child Development, 81, 1753–1767. Maertens, B., De Smedt, B., Sasanguie, D., Elen, J., & Reynvoet, B. (2016). Enhancing arithmetic in pre-schoolers with comparison or number line estimation training: does it matter? Learning and Instruction, 46, 1–11. Mazzocco, M. M. M., Feigenson, L., & Halberda, J. (2011). Pre-schoolers’ precision of the approximate number system predicts later school maths performance. PLoS ONE, 6, e23749. Melby-Verlag, M., Redick, C., & Hulme, C. (2016). Working memory training does not improve performance on measures of intelligence or other measures of “far transfer”: evidence from a meta-analytic review. Perspectives on Psychological Science, 11, 512–534. Purpura, D. J., & Ganley, C. M. (2014). Working memory and language: skill-specific or domaingeneral relations to mathematics. Journal of Experimental Child Psychology, 122, 104–121. Purpura, D. J., Hume, L. E., Sims, D. M., & Lonigan, C. J. (2011). Early literacy and early numeracy: the value of including early literacy skills in the prediction of numeracy development. Journal of Experimental Child Psychology, 110, 647–658. Purpura, D. J., Napoli, A. R., Wehrspann, E. A., & Gold, Z. S. (2017). Causal connections between mathematical language and mathematical knowledge: a dialogic reading intervention. Journal of Research on Educational Effectiveness, 10, 116–137. Ramani, G. B., & Siegler, R. S. (2008). Promoting broad and stable improvements in low-income children’s numerical knowledge through playing number board games. Child Development, 79, 375–394. Sala, G., & Gobet, F. (2017). Working memory training in typically developing children: a metaanalysis of the available evidence. Developmental Psychology, 53, 671–685. Schacter, J., & Io, B. (2016). Improving low-income preschoolers mathematics achievement with Math Shelf, a preschool tablet computer curriculum. Computers in Human Behaviour, 55, 223–239. Siegler, R. S., & Ramani, G. B. (2008). Playing board games promotes low-income children’s numerical development. Developmental Science, Special Issue on Mathematical Cognition, 11, 655–661. Toll, S. W., & Van Luit, J. E. (2013). Accelerating the early numeracy development of kindergartners with limited working memory skills through remedial education. Research in Developmental Disabilities, 34, 745–755. Whyte, J. C., & Bull, R. (2008). Number games, magnitude representation, and basic number skills in preschoolers. Developmental Psychology, 44, 588–596.

Preface With Cognitive Foundations for Improving Mathematical Learning, we close a five-volume journey through the evolution and early development of number competencies (Volume 1); the brain and genetic foundations for these and more complex abilities (Volume 2); the cognitive bases for the learning of evolutionarily novel mathematics, from arithmetic to trigonometry (Volume 3); the influences of language and culture on mathematical learning and cognition (Volume 4); and now formal and informal instructional approaches for improving children’s mathematics learning (Volume 5). With this final volume, we see clear links between topics and discoveries covered in previous volumes and the development of intervention approaches. These include interventions focused on the relation between our evolved number sense and children’s early math learning; the influence of home, family, and mathematical language on early math learning; and the integration of cognitive science approaches to mathematical learning into educational interventions. These represent an exciting step forward and a much needed conciliation between basic research in mathematical cognition and mathematics education interventions that we hope will continue well beyond the publication of this volume. As with the previous volumes, we anticipate this one will be of interest to researchers, graduate students, and undergraduates specializing in cognitive development, cognitive neuroscience, educational psychology, early childhood education, special education, and many other disciplines. More so than the four other volumes, this final volume should be particularly interesting for researchers and practitioners in both mathematics and preschool education. The chapters cover a wide range of approaches and topics. The approaches range from small-scale game-based interventions focused on improving preschoolers’ number knowledge to large-scale randomized controlled studies focused on older and academically at-risk children. And, the topics range from fundamental number knowledge, to mathematical language, to use of analogy and visuospatial representations to facilitate mathematical learning. The final chapter addresses the vexing but critically important issue of intervention fade-out effects and the factors that might be underlying these effects. The chapters herein provide cutting-edge descriptions of the latest approaches to our understanding of the informal (e.g., at home, with parents) and formal (i.e., in educational settings) interventions that can be used to improve young children’s and older students’ mathematical learning, as well

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as discussion of potential shortcomings of these approaches and the steps that need to be taken to improve them. This volume could be used as a textbook for several kinds of courses taught in psychology (e.g., educational psychology) and education (e.g., mathematics education, instruction and learning, early childhood education). In the introduction, we provide an overview of methodological issues and the domain-general and domain-specific cognitive foundations that can influence the effectiveness of various intervention approaches, as well as a brief foray into newly emerging interventions (e.g., brain stimulation) and parental influences on mathematical learning. The early chapters in the volume focus on young children and describe different types of interventions from board games to tablet-based games to large-scale randomized controlled studies, all focused on better preparing children for formal mathematical learning at school entry or as an adjunct to their initial formal learning. The chapters also cover informal influences on children’s early learning, including the role of mathematical language (e.g., understanding more versus less) and the ways in which activities with parents can boost children’s emerging understanding of number. The later chapters focus on older students’ mathematics learning and the integration of insights from cognitive science into intervention design, including the use of visuospatial representations to facilitate students’ ability to solve mathematical word problems, analogical reasoning to facilitate students’ conceptual understanding of number, and how students’ cognitive strengths and weaknesses influence their responsiveness to different intervention approaches. As noted, the volume closes with a sobering but thoughtful discussion of intervention fadeout and provides insights into how we might better address this issue. We thank the Child Development and Behavior Branch of the Eunice Kennedy Shriver National Institute of Child Health and Human Development, NIH for the primary funding of the conference on which this volume is based and Dr. Joan McLaughlin, Commissioner of the National Center for Special Education Research (NCSER), U.S. Department of Education for supplementary funding of the meeting. We are also grateful to Dr. Camilla Benbow, Dean of the Peabody School of Education and Human Development at Vanderbilt University for hosting the conference and to Dr. Lynn Fuchs for her significant contributions to the planning for and successful execution of virtually every phase of this meeting. Finally, we are indebted to the administrative staff of Peabody College for ensuring that the requisite conference facilities were in place and that the meeting ran smoothly and efficiently. David C. Geary Daniel B. Berch Kathleen Mann Koepke

Chapter 1

Introduction: Cognitive Foundations for Improving Mathematical Learning David C. Geary*, Daniel B. Berch† and Kathleen Mann Koepke‡ *

Psychological Sciences, University of Missouri, Columbia, MO, United States Curry School of Education, University of Virginia, Charlottesville, VA, United States ‡ National Institutes of Health, Eunice Kennedy Shriver National Institute of Child Health & Human Development, Bethesda, MD, United States †

INTRODUCTION Children’s acquisition of some level of competence with mathematics has never been as important as it is today. Mathematical knowledge and skills expand educational and employment opportunities and allow one to better navigate the increasingly number-imbued realities of life in the modern world. As the educational and economic benefits to mathematical knowledge increase, so do the costs of falling behind. Reducing these costs by improving the mathematical development of at-risk children will yield substantive real-world benefits to these individuals and to the communities in which they will reside as adults. As detailed by the authors in this volume, there are many ways to potentially reduce these costs, from increasing parents’ engagement in numeracyrelated activities with their preschool children to formal interventions to better prepare children for success when they enter school to interventions that target core mathematical competencies (e.g., fractions). These interventions provide some very promising initial steps toward the development of systematic approaches toward ensuring that all children will eventually have the mathematical knowledge needed for functioning in educational, employment, and dayto-day contexts, but much remains to be learned. The chapters in this volume largely provide reviews of specific interventions or classes of intervention, including issues surrounding fadeout (loss of intervention gains over time). In our introductory chapter, we step back and begin with an overview of intervention work more generally, including

Mathematical Cognition and Learning, Vol. 5. https://doi.org/10.1016/B978-0-12-815952-1.00001-3 Copyright © 2019 Elsevier Inc. All rights reserved.

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discussion of methodological issues and the cognitive mechanisms that support mathematics learning as related to intervention research. The authors of the chapters within this volume have done an excellent job of overviewing current intervention approaches but were not able to cover all of them. Thus we discuss some of the approaches that were not touched upon in these chapters, to provide the reader with a more complete synopsis of the efforts that are being made to improve children’s mathematical learning. We close with a brief discussion on how parents can influence their children’s mathematical development and suggestions for improving research in this area.

BRIEF HISTORY OF MATHEMATICS INTERVENTION STUDIES For several millennia, there has been discussion and oftentimes vociferous debate regarding how to best educate youth. The debate reflects a general tension between a teacher-directed classical education with set standards to child-directed Romantic approaches, following Rousseau (1979/1762). The tension between these approaches has fueled much of the controversy in modern education, including the “reading wars” and “math wars” (Klein, 2007; Loveless, 2001). Resolving these tensions is more consequential than your typical “academic feud,” as the resolution or resolutions will influence how and what children are taught in school and are likely to be most keenly felt by those students in the most need of well-vetted educational interventions. Project Follow Through was designed, at least in part, to provide an experimental (actually quasi-experimental) evaluation of the basic tenets of these broad educational approaches and was the largest educational intervention that has ever been conducted (Stebbins, 1977). The project was conducted from 1968 to 1977 across several cohorts of children at risk of poor educational outcomes. In total, about 200,000 children across 180 communities throughout the United States participated in a kindergarten through third grade intervention. Each participating school partnered with a university, research institute, or some other group to develop and implement the four-year intervention grounded in differing educational principles and academic foci. The principles reflected the basic teaching philosophy and broadly included direct instruction, cognitive approaches (e.g., focus on problem solving), and social/affective approaches focusing on, for instance, self-efficacy. The educational goals ranged from a focus on mastery of basic skills to more complex problem solving and conceptual understanding. Relative to Title I comparison groups and with respect to performance on standardized achievement tests, children in the direct instruction programs with a focus on basic skills and mastery had better outcomes on basic skills, problem solving, and had a better academic self-concept than did students in the other programs. There was some fadeout (below) over time, but many of these gains were maintained throughout the elementary school years (Meyer, 1984; Meyer, Gersten, & Gutkin, 1983).

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Of course, the results were roundly criticized (House, Glass, McLean, & Walker, 1978) and eventually suppressed in favor of more Romantic approaches to education. The “reading wars” and “math wars” were to follow in the succeeding decades, and in turn were followed by the National Reading Panel (2000), National Mathematics Advisory Panel (2008), and the founding of the Institute of Education Sciences in 2002. The combination, though routinely critiqued and debated (e.g., Benbow & Faulkner, 2008; Boaler, 2008), resulted in more rigorous and scientific standards for evaluating educational programs generally and interventions for at-risk children. Many of the intervention programs described in this volume are the direct result of these changes and all of them represent empirically and scientifically grounded approaches for evaluating and ultimately developing interventions that will be effective.

METHODOLOGICAL ISSUES IN INTERVENTION RESEARCH As was just noted, there is a long history of debate over how students’ mathematical competencies should be assessed—qualitatively or quantitatively— and even deeper divisions about how children learn (e.g., discovery learning versus direct instruction), what should be emphasized during instruction (e.g., concepts versus procedures), and the types of interventions (e.g., randomized controlled trials) that provide the most credible evidence for informing educational practice (e.g., Brickman & Reich, 2015; Cai et al., 2018; Cobb, Yackel, & Wood, 1992; Donaldson, Christie, & Mark, 2015; Geary, 1994). Although we acknowledge that qualitative work can be informative, our focus is on quantitative approaches and research grounded in cognitive psychology, and issues that can compromise the usefulness of these methods for informing educational practice. We describe a few of these most basic issues.

Design Factors The implementation of effective educational policy at the national or state level and the implementation of instructional programs that consistently promote students’ achievement are dependent on the availability of high-quality educational research, including research on interventions (National Research Council, 2002; Pianta, Barnett, Burchinal, & Thornburg, 2009). Researchers must adhere to basic scientific principles in the design of such studies and be alert to potential threats to their validity. The same threats to validity that are of concern to researchers can also be used by consumers (e.g., teachers, policy makers) to evaluate the quality of any particular study, and determine whether and to what extent teaching practices might be adjusted based on these results (Willingham, 2012). We note a few issues regarding the design of educational intervention studies and potential threats to their validity (Gersten et al., 2005).

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The What Works Clearing House (2017) provides guidelines for evaluating the quality of intervention research and how this quality influences the strength of the inferences that can be drawn from these studies; the National Mathematics Advisory Panel (2008) adopted very similar standards in their reviews of research related to children’s mathematics learning. The best evidence for the effectiveness of interventions comes from randomized controlled trials (RCTs) and quasi-experimental studies. In the former, students (or students clustered within groups, such as classrooms or schools) are randomly assigned to intervention or one of several control groups. Random assignment should ensure that factors that can influence the intervention (e.g., motivation) will be evenly distributed across the intervention and control groups and thus will not bias the results of the study. Quasi-experimental studies, such as Project Follow Through, differ from RCTs in that there is no randomization. Common threats to the validity of both RTCs and quasi-experimental studies include nonequivalence of groups at pretreatment (e.g., groups differ in prerequisite skills) and attrition during the intervention. There are various statistical techniques to adjust for these issues, but the extent to which these improve the quality of the results depends on other factors, such as whether attrition was similar or not in the intervention and control groups. Quasi-experimental studies come with an additional risk of unmeasured differences between the intervention and control groups. For instance, the intervention group may be composed of students who volunteered for the study, and the control group composed of students who declined to participate. Even if the groups are similar on some dimension (e.g., achievement test scores), this does not ensure they are equivalent on other dimensions (e.g., motivation). As a result, group differences at the end of the intervention could be due to the intervention or some unmeasured differences between the groups; the same issue could occur in RCTs, if there are significant numbers of students, teachers, or schools that decline to participate (Bell, Olsen, Orr, & Stuart, 2016). The What Works Clearing House (2017) also suggests that sample sizes need to be large enough to detect a .25 standard deviation intervention effect with a power of .80. In other words, there needs to be enough participants to ensure detection of a modest intervention effect at least 80% of the time. Even when these basic issues are addressed, Gersten et al. (2005) note additional factors that could undermine the validity or at least the usefulness (e.g., translating into classroom practices) of RCTs and quasi-experimental studies. These include clear and detailed descriptions of both the content and process of the intervention (e.g., one-on-one, small group, 10 sessions, etc.) and the content and process of the instruction received by the comparison group. The latter often includes students in business-as-usual classrooms; and clear description of what they receive in these classrooms and how it differs from the intervention is critical to the interpretation of the results of such studies (e.g., Fuchs, Geary, et al., 2013, Fuchs, Schumacher, et al., 2013). The fidelity with which the intervention was implemented, as we discuss in the

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next section, is essential to determining its effectiveness. The content, reliability, and validity of the measures used to assess the intervention are also critical and help to determine whether and the extent to which there are educational effects beyond the specific content covered in the intervention, as we discuss in “Transfer” section.

Fidelity of Implementation A well-designed intervention, one focusing on enhancing students’ conceptual understanding or procedural fluency in a specific area of mathematics is of course critical to the development of effective instructional approaches (National Research Council, 2004). The proper evaluation of the efficacy of lab studies and effectiveness of field studies requires that the program is implemented in ways the developer intended. The expectations for the fidelity of interventions in medicine are well established and have gained considerable traction the past two decades in education, although opportunities for improvement remain (Swanson, Wanzek, Haring, Ciullo, & McCulley, 2013). O’Donnell (2008) notes five basic criteria for ensuring fidelity of an intervention: (a) adherence—whether the components of the intervention are being delivered as designed; (b) duration—the number, length, or frequency of sessions implemented; (c) quality of delivery—the manner in which the implementer delivers the program using the techniques, processes, or methods prescribed; (d) participant responsiveness—the extent to which participants are engaged by and involved in the activities and content of the program; and (e) program differentiation—whether critical features that distinguish the program from the comparison condition are present or absent during implementation (O’Donnell, 2008, p. 34)

Demonstrating positive student gains based on interventions that have been implemented with fidelity in lab settings and in restricted educational settings (e.g., intervention implementation monitored by researchers) is just the first step. The results still leave unanswered the question of whether the intervention can be scaled up to large numbers of classrooms and implemented with fidelity without researcher supports (Cai et al., 2017). In the scaling up of interventions to real-world classrooms there are many additional threats to fidelity of implementation, including differences in teacher preparation and autonomy, characteristics of students within each classroom, and general school characteristics (e.g., level of absenteeism), among other factors (McDonald, Keesler, Kauffman, & Schneider, 2006).

Diminishing Intervention Impacts Across Time (Fadeout) There are many interventions designed to improve children’s academic skills and most of them in fact do so, that is, there are demonstrable gains

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in achievement at the end of such interventions relative to peers who did not participate in the intervention (Barnett, 2011). Unfortunately, nearly all of the effects of these interventions and especially those focused on specific academic skills during the preschool or elementary school years fadeout over time (Bailey, Duncan, Odgers, & Yu, 2017; Puma et al., 2012; Watts, Duncan, Clements, & Sarama, 2018). It is not that all of the gains these children achieved fade, but rather the children who did not receive the intervention tend to catch up within a few years (e.g., Clements, Sarama, Wolfe, & Spitler, 2013). Not all studies show a complete fadeout of early intervention effects, and for some interventions there are often other benefits that are not reflected in achievement scores (e.g., increase in high school graduation rate; Deming, 2009). Even if there are (sometimes) modest longer term gains to achievement and (also modest) gains in noncognitive areas, the fade-out effects are sobering and indicate a need to rethink the approach to intervention (Bailey, this volume; Fuchs et al., this volume). Bailey et al. (2017) provide a thoughtful discussion of why fade-out effects occur and what might be done to enhance the effectiveness of interventions. Their first point is that interventions designed to build academic skills (e.g., phonemic awareness or counting) that children will be exposed to during normal schooling, whether or not they participate in an intervention, will per force result in some degree of fadeout. This is because children who were not in the intervention will eventually—often the following academic year—be instructed on these same skills and thus will catch up to some extent or completely to the children who participated in the intervention. Skills that are malleable and where the timing of their acquisition is important for subsequent learning or other critical outcomes (e.g., high school graduation) may be an exception. Cortes and Goodman’s (2014) study of the effectiveness of algebra doubledosing—students at risk for failing algebra I enroll in two algebra courses in 9th grade (standard algebra, and algebra with supports)—provides an example. The intervention students receive the same content as other students, but achieving some level of competence in this content has potentially outsized consequences. Passing algebra is critical to later mathematics learning and preparation for the labor market (Rivera-Batiz, 1992) and is also a high school graduation requirement in many states. The double-dose of algebra in 9th grade resulted in modest but significant gains in algebra grades as well as later grades in geometry and trigonometry and, at least for some students, an increase in graduation rates. Bailey et al. (2017) also highlight the importance of interventions that seek to change students’ beliefs about learning; changing beliefs that mathematics takes talent that you either have or do not to a belief about the importance of hard work and persistence is one example (Blackwell, Trzesniewski, & Dweck, 2007). Their final point is that the long-term effectiveness of interventions may depend on children’s subsequent placement in sustaining environments, such as the quality of the schools they attend after they complete the intervention (Currie & Thomas, 2000).

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Transfer Transfer effects refer to the relation between training or an intervention in one domain on performance in similar (near transfer) or seemingly very different (far transfer) domains. The relation between working memory training—such as practice at tracking and retaining information—on similar working memory tasks (near transfer) or on fluid intelligence or mathematics achievement (far transfer) illustrates different types of transfer. Studies of this type span more than a century, with an early conclusion that: Improvement in any single mental function rarely brings about equal improvements in any other function, no matter how similar … [the] spread of practice [transfer] occurs only where identical elements are concerned in the influencing and influenced function. (Thorndike & Woodworth, 1901, p. 250)

The early evidence thus suggested that transfer can occur but only to the extent to which the trained competencies are embedded within the transfer task. In theory, this could occur for domain-general abilities (below), whereby training to improve working memory might transfer to fluid intelligence or to academic competencies to the extent the latter are dependent on working memory. At this time, the potential for working memory training to transfer to fluid intelligence or to academic competencies is vigorously debated (e.g., Au et al., 2015; Harrison et al., 2013), and little if any far transfer likely occurs (Melby-Lerva˚g, Redick, & Hulme, 2016). Melby-Lerva˚g et al. confirmed that working memory training often improves performance on near transfer tasks, but there was very little consistent support for transfer to other competencies (including mathematics), when appropriate controls are used (e.g., an active versus passive control group; see also Sala & Gobet, 2017). In total, these results are consistent with Thorndike and Woodworth’s (1901) early conclusion. The educational implications are that transfer effects will largely occur within the same academic domain and that the strength of transfer will vary directly with the degree of overlap between the training and transfer skills, and this does appear to be the case for mathematics (e.g., Fuchs et al., 2014; Fuchs, Malone, Schumacher, Namkung, & Wang, 2017; Gersten et al., 2008). An example is provided by Fuchs et al.’s (2017) interventions to improve children’s fraction competencies. A nice feature of these interventions is their grounding in Siegler, Thompson, and Schneider’s (2011) integrated theory of number development; specifically, focusing on children’s understanding of fraction magnitudes. The intervention involved activities designed to focus students on the relative magnitudes of fractions, such as identifying the larger of two fractions, ordering a series of fractions from smallest to largest, and placing fractions on a number line. The fractions instruction for the business-as-usual control group focused on part-whole relations (e.g., fractional proportion of shaded areas of a rectangle; see Fuchs, Geary, et al., 2013, Fuchs, Schumacher, et al., 2013).

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Relative to the control group, the interventions significantly improved students’ performance on tasks that were similar to the intervention (near transfer), specifically, accuracy at placing fractions on a number line (Fuchs et al., 2017). The students also showed substantive gains in competence at procedurally solving fractions problems and on released fractions items from the National Assessment of Educational Progress. The two latter measures are not far transfer, but they are at least intermediate, as they showed that improving students’ conceptual understanding of fractions transferred to their ability to solve fractions problems. Using contrasting interventions, Fuchs et al. (2017) identified how different components of the intervention interacted with students’ domaingeneral abilities (below). For instance, students with low working memory capacity especially benefitted from explaining their reasoning behind judgments about fractions magnitudes; this explanation component was not necessary for students with adequate working memory capacity. They also found that instruction on how to solve fractions word problems that involved multiplication transferred to addition word problems, but instruction on addition did not transfer to multiplication. The implication is that the most effective use of instruction time (in terms of promoting transfer) is to focus on multiplication.

WHAT ARE THE COGNITIVE FOUNDATIONS FOR IMPROVING MATHEMATICS LEARNING? A combination of domain-general abilities, such as working memory, and domain-specific knowledge contribute to the growth of academic competencies but their relative importance is not fully understood, including whether their contributions change over time or with students’ level of expertise (Ferrer & McArdle, 2004; Gustafsson & Undheim, 1992; Von Aster & Shalev, 2007). Cattell’s (1987) investment theory provides one influential example of the mix of traits that promote academic development. He focused on importance of fluid intelligence (abstract problem solving), in combination with interests and personality, in the development of crystallized intelligence, including domain-specific academic knowledge (see also Ackerman & Beier, 2006). Intelligence influences the ease of learning new information, but interests and personality (e.g., conscientiousness; Poropat, 2009) determine how and where these abilities are invested. Other researchers have demonstrated that current levels of domain-specific knowledge influence the ease of making further gains in this knowledge (Thorsen, Gustafsson, & Cliffordson, 2014; Tricot & Sweller, 2013). As with other academic domains, the relative contributions of domaingeneral abilities and domain-specific knowledge to subsequent mathematics achievement are not fully understood and may vary across grade, level of student knowledge, and mathematical content (Bailey, Watts, Littlefield, & Geary, 2014; Friso-van den Bos, van der Ven, Kroesbergen, & van Luit, 2013; Fuchs, Geary, Fuchs, Compton, & Hamlett, 2016; Geary, 2011;

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FIG. 1 Changes in the relative contribution of overall domain-general abilities, including intelligence and working memory, and domain-specific knowledge (e.g., recall of basic facts, number and fractions knowledge) on mathematics achievement in the subsequent grade. (By Geary, D. C., Nicholas, A., Li, Y., & Sun, J. Journal of Educational Psychology, 109, 689. Copyright 2017, by American Psychological Association. Reprinted with permission.)

Siegler et al., 2012; Watts et al., 2015). Two recent longitudinal studies indicated that domain-general abilities are important throughout students’ mathematical development, and that the relative importance of domain-specific knowledge increases across successive grades (Geary, Nicholas, Li, & Sun, 2017; Lee & Bull, 2016). The basic pattern is shown in Fig. 1, whereby the combined effect of domain-general abilities, including working memory and intelligence, ranges between .45 and .55 from 2nd to 8th grade, inclusive; a 1 standard deviation increase in domain-general abilities is associated with about a .5 standard deviation increase in math achievement in the following grade, controlling prior math skills. In contrast, prior-grade math competencies on subsequent math achievement are much smaller than the influence of domaingeneral abilities in the early grades, but become as important as domain-general abilities by middle school. A more complete understanding of the grade-to-grade contributions of domain-general and domain-specific effects on subsequent mathematics achievement and changes in the relative magnitude of these effects will facilitate our understanding of children’s mathematical development generally, and provide insight into when, where, and how to target interventions to

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improve this development. In the following sections, we briefly overview the core domain-general competencies that contribute to mathematics achievement (and academic achievement more generally) and our current understanding of the effectiveness of domain-general and domain-specific interventions.

Domain-General Components Intelligence, executive function, and working memory are the most consistently studied domain-general predictors of mathematical development and achievement. The relative importance of these individual abilities may vary with age, students’ level of expertise in mathematics, and mathematical content.

Fluid Intelligence Cattell’s and Horn’s fluid intelligence indexes people’s ability to identify the underlying rules or concepts in novel problem-solving domains (Cattell, 1963; Horn, 1968). As Cattell stated, “Fluid general ability … shows more in tests requiring adaptation to new situations, where crystallized skills [domain-specific knowledge] are of no particular advantage” (Cattell, 1963, p. 3). Mathematics is an evolutionarily novel domain and thus fluid intelligence should be a significant contributor to individual differences in the ease of learning newly introduced mathematical concepts and particularly important as mathematics becomes increasingly abstract in later grades (Geary, 1995, 2005). In theory, however, the relative importance of fluid intelligence will decline once the concept is understood, but with the continuous introduction of new and more abstract concepts in the standard mathematics curriculum, fluid intelligence will remain important. Most studies of the relation between intelligence and mathematics achievement have used a composite measure (e.g., standardized IQ test) that technically does not provide a direct assessment of fluid intelligence but will be highly correlated with it (Walberg, 1984). One of the largest of these studies included 70,000 students and found that intelligence measured at age 11 years was highly correlated (r ¼ .77) with standardized mathematics achievement scores 5 years later (Deary, Strand, Smith, & Fernandes, 2007). In an analysis of almost 5000 children and adolescents, Taub and colleagues separated the contributions of other factors, such as crystallized intelligence, from fluid abilities and found the latter predicts mathematics achievement, albeit the strength of the relation was somewhat lower than that found by Deary et al. (b ¼ .37 to .75) and varied across age (Taub, Keith, Floyd, & McGrew, 2008). Similar to the pattern shown in Fig. 1, contributions of crystallized knowledge and fluid intelligence to the prediction of adolescents’ mathematics achievement were about the same, but crystallized knowledge was not important for younger children. Although their measure of crystallized knowledge included many areas, not just mathematics, it should be a reasonable proxy for mathematics knowledge.

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This is not to say that domain-specific knowledge is not particularly important for children’s early mathematics learning; it depends on the mathematics being assessed. Indeed, a similar overall pattern emerges for the age at which preschool children learn their first mathematical concept—the cardinal value or quantities represented by number words—and for their mathematics achievement generally; a combination of prerequisite domain-specific knowledge (e.g., knowing the list of count words) and intelligence is important (Geary & vanMarle, 2016; Geary, vanMarle, Chu, Hoard, & Nugent, in press). The age at which children learned this mathematical concept and again fluid intelligence independently predict these children’s school-entry number knowledge and mathematics achievement 3 years later (Geary, vanMarle, Chu, Hoard, et al., in press, Geary, vanMarle, Chu, Rouder, et al., 2018). The pattern across all of these studies is consistent with Cattell’s (1987) investment theory; “… this year’s crystallized ability level is a function of last year’s fluid ability level—and last year’s interest in school work” (Cattell, 1987, p. 139). In other words, strong fluid abilities accelerate the learning of mathematics, assuming sufficient motivation and engagement with the material, and this domain-specific foundation along with fluid abilities jointly contribute to future learning. Environmental factors such as the quantitative activities at home and quality of curricula material will be important for exposing children to the appropriate mathematics content (LeFevre et al., 2009; Ramani & Siegler, 2008).

Executive Function and Working Memory The constructs and associated measures of executive function and working memory come from the neuropsychological and cognitive traditions, respectively, but capture many of the same underlying competencies, as we detail later. Performance on measures of fluid intelligence and working memory are at least moderately correlated (Ackerman, Beier, & Boyle, 2002), but appear to assess independent competencies (Embretson, 1995; Jurden, 1995). Our perspective is that the key aspects of working memory as related to mathematics involve attentional control and the ability to update information represented in working memory (Bull & Lee, 2014; Iuculano, Moro, & Butterworth, 2011), as contrasted with the logical problem solving and ease of concept learning captured by fluid intelligence. Whatever the specifics, both types of measures independently predict academic success (Clark, Pritchard, & Woodward, 2010; Taub et al., 2008). The relations between executive function, working memory, and children’s mathematics achievement are assessed somewhat differently for preschool- and school-age children. In the preschool studies, the focus is on a single measure of executive function and performance on such measures consistently predict later mathematics achievement (Blair & Razza, 2007; Bull, Espy, & Wiebe, 2008; Clark et al., 2010). At these ages, executive function captures the ability to

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maintain attentional focus on the task at hand and to inhibit prepotent responding, that is, inhibit a previously used response that is correct in one context but is not correct in the current context. Toward the end of the preschool years, these functions differentiate into the central executive, phonological loop, and visuospatial sketchpad components of working memory (Baddeley & Hitch, 1974; Wiebe, Espy, & Charak, 2008). The central executive in turn captures the core abilities of executive function (Allan & Lonigan, 2011) and can be decomposed into the ability to suppress prepotent responses, shift attention between tasks (shifting), and explicitly monitor and update information (updating) represented in the phonological loop or visuospatial sketchpad (Miyake et al., 2000). More recently, the term “working memory” has been used to represent the updating component, and this is how we use working memory hereafter. The studies of school-age children have consistently revealed a relation between one or several components of Baddeley and Hitch’s (1974) system and mathematics achievement (DeStefano & LeFevre, 2004; Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007; McLean & Hitch, 1999; Swanson & Sachse-Lee, 2001). The most consistent finding is that the higher the capacity of the central executive the better the performance on measures of mathematics achievement and cognition (Bull et al., 2008; Mazzocco & Kover, 2007; Passolunghi, Vercelloni, & Schadee, 2007). This relation holds across the entire continuum of mathematics achievement and is a core deficit of children with learning difficulties in mathematics (Geary, 2004). In a review of this literature, Bull and Lee (2014) concluded that working memory (the updating component) is particularly important for mathematics learning and commonly found to be associated with mathematical learning difficulties (Geary, Hoard, Nugent, & Bailey, 2012). The strength of the relation between mathematics achievement and the phonological loop and visuospatial sketchpad is reduced and sometimes eliminated when performance on working memory and intelligence measures are controlled (Fuchs et al., 2010a, 2010b; Geary, Hoard, Nugent, & Bailey, 2013; Holmes & Adams, 2006). The relations that are found tend to be restricted to specific aspects of mathematics (Andersson, 2010; Bull et al., 2008; Geary et al., 2007; Krajewski & Schneider, 2009). The phonological loop contributes to the encoding and processing of number words and numerals and to processes that involve them (e.g., counting “six, seven, eight” to solve 6 + 2; Geary, 1993; Krajewski & Schneider, 2009; Logie & Baddeley, 1987). The phonological loop may also contribute to skill at solving mathematical word problems, presumably because phonological skills contribute to the comprehension of the reading-related components of word problems (Swanson & Sachse-Lee, 2001). The visuospatial sketchpad also contributes to the solving of word problems (Johnson, 1984), and some other aspects of mathematics, such as visualizing and representing quantities on the number line (Gunderson, Ramirez, Beilock, &

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Levine, 2012). The former effect is likely related to ease of constructing diagrams to represent quantitative relations in the word problems. Overall, however, the strength of the relations between visuospatial memory and mathematics performance tends to get larger across grades, potentially due to changes in the mathematical content covered in higher grades (Li & Geary, 2013, 2017; Mix & Cheng, 2011). There are also different components of visuospatial competencies and some of these may contribute to different aspects of mathematical development (Raghubar, Barnes, & Hecht, 2010). Kytt€al€a and Lehto (2008) found that the relation between visuospatial ability and high school students’ mathematics performance, controlling for fluid intelligence, varied with whether the visuospatial task required simultaneous (maze memory), sequential (e.g., block recall), or active (3-dimensional mental rotation) processing. Memory for simultaneously presented visuospatial information predicted overall mathematics achievement, whereas sequential processing made unique contributions to performance on written word problems and active processing to geometry. Follow-up studies are needed to confirm the specificity of these relations, given the small number of these types of studies. Although performance on measures of executive function and the central executive requires attentional focus and inhibition of distractors, these measures do not fully capture children’s attentive behavior in classroom settings. In-class attentive behavior is rated by teachers and includes items such as, “Gives close attention to detail and avoids careless mistakes” (Swanson et al., 2012). These measures predict mathematics achievement above and beyond the contributions of executive function, the central executive, and fluid intelligence (Fuchs et al., 2006; Geary et al., 2013).

Implications These types of findings and those reviewed in the next section imply that interventions that are simply focused on addressing deficits in mathematical knowledge will not be sufficient for many children, if they do not also include features that address domain-general deficits. Fuchs et al.’s (2016) approach of integrating domain-specific instruction with supports that help to compensate for any such deficits may be useful in the design of any such intervention (see also Fuchs et al., this volume). In a structured intervention focused on improving elementary student’s number knowledge, Fuchs, Geary, et al. (2013), Fuchs, Schumacher, et al. (2013) found that children with below average fluid abilities gained more if the intervention included speeded practice of the instruction material at the end of each session. Children with above-average abilities also gained from this practice, but the relative gains were larger for the below-average students. More generally, meta-analyses conducted by the National Mathematics Advisory Panel (2008)—and consistent with the results from Project Follow Through (Stebbins, 1977)— indicated that students with difficulties in mathematics learning benefit from

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explicit, teacher-directed instruction (Gersten et al., 2008) that may help to compensate for domain-general deficits.

Domain-General Interventions As we discussed previously, the majority of studies investigating the role of domain-general factors on mathematical learning and development have focused primarily on fluid intelligence, executive function, and working memory. This being said, it is important to point out that researchers have also begun to study a much wider range of domain-general processes that may also influence mathematical cognition and its development. In fact, a recent special issue of the Journal of Numerical Cognition was devoted primarily to the topic of domain-general factors that contribute to numerical and arithmetic processing. In the introduction to this special issue, Knops, Nuerk, and Gobel (2017) reviewed the surprisingly wide range of domain-general factors examined in this collection of articles, including (in addition to intelligence, executive function, attention, and working memory): general language, mathematical language, ordinality (nonnumerical), perceptual speed, self-regulation, social power, and creativity, among others. Although we will not discuss the findings that explored the extent to which, if any, these factors are associated with mathematical skills, it is worth focusing here on the two studies in this collection that investigated the impact of domain-general interventions. Honore and Noe¨l’s (2017) study examined whether training visuospatial working memory in 5- to 6-year-olds would not only enhance working memory skills (i.e., visuospatial and/or verbal working memory) but also transfer to numerical abilities; specifically, whether training resulted in improvement in counting and comparison (numerical and collection) skills, number line judgments, and addition. The Cogmed visuospatial working memory training program was administered to these children for 20 min per each school day over 5 weeks. This computerized program is adaptive, in that the difficulty level (i.e., number of items to be remembered) is adjusted on a trial-by-trial basis, thereby taxing working memory capacity to its limit. A control group received the same Cogmed training program except with no adaptive feature, that is, they only had to remember the same, low level number of items they began with—one or two, and were trained for only 10 min per school day. Interestingly, although the children in the experimental condition exhibited an improvement in visuospatial working memory, there was no effect on verbal working memory. More importantly, even though there was a small improvement in Arabic number comparison, this failed to be sustained 10 weeks following the training (which was also the case for the visuospatial working memory effect). Furthermore, the training did not even show an immediate enhancement of counting, collections comparison, or addition.

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The other intervention study in the special issue (Ramani, Jaeggi, Daubert, & Buschkuehl, 2017) is summarized in the chapter by Ramani et al. (this volume). Briefly, although there was some evidence that the numerical magnitude knowledge of Kindergartners was enhanced by playing an adaptive, visuospatial working memory training game on a tablet computer, the only children who exhibited this improvement were those with initially lower, numerical magnitude knowledge. Furthermore, domain-general training was not as effective as domain-specific training with numerical magnitudes. According to Knops et al. (2017), the outcomes of these two studies suggest that significant associations between domain-general factors and numerical or arithmetic skills that have been found in prediction studies do not “easily translate” into improved mathematics learning in intervention studies. With respect to the current volume, we direct the reader to some of the domain-general interventions that were based on one or two of the other factors discussed by Knops et al. (2017) in their special issue. Namely, both the role of mathematical language in mathematics learning and its influence via an instructional intervention are described by Purpura et al. (this volume). Relatedly, the chapter by Fuchs et al. (this volume) examines the effect of embedding language instruction (both general language and mathematical language) within a wordproblem-solving intervention. In both of these cases, the interventions are focused on competencies and knowledge that are directly usable when solving many types of mathematics problems, as contrasted with more general abilities, such as visuospatial ability. Additionally, the chapter by Vamvakoussi (this volume) examines the value of incorporating analogies in mathematics instruction as analogical reasoning can be viewed as a domain-general cognitive process (Alvarez et al., 2017; Forbus, Gentner, Markman, & Ferguson, 2010). Again, the use of analogies is embedded directly in mathematics instruction, and not the focus of an independent intervention to improve general analogical reasoning abilities.

Domain-Specific Components In their introduction to the special issue discussed previously, Knops et al. (2017) also examined in detail the nature of the domain-specific factors that were investigated in their collection of articles, as well as various meanings that have been attributed to this construct. In their analysis, they point out that this term has been applied to several different components of mathematical cognition studies, including alleged domain-specific tasks, behaviors, representations, and processing mechanisms. Rather than exploring these finer distinctions here, we prefer to point out that the domain-specific numerical and mathematical representations, processing mechanisms, and skills investigated to date include, but are not limited to: approximate and exact number judgments; spatial-numerical associations; counting skills, addition and subtraction operations; numerical estimation; fractions knowledge (conceptual and procedural) and arithmetic; and algebraic as well as geometrical reasoning.

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Domain-Specific Interventions Researchers have developed a rather diverse set of interventions for improving domain-specific components of math skills ranging from basic numerical representations and processing mechanisms to higher order mathematical concepts and operations. Examples of such interventions are discussed in many of the following chapters, with the majority focusing on improving the numerical and arithmetic skills of preschool- and early school-age children. In this section, we have chosen to describe a computer-based program for training number line estimations by children with mathematical learning disabilities and their typically achieving peers, as it provides an interesting intervention that targets a numerical skill that is generally considered a critical foundation for subsequent learning of both whole number and rational number arithmetic.

Computer-Based Training of Number Line Judgments As several of the authors in the present volume describe (Aunio, this volume; Langfus et al., this volume; Ramani et al., this volume; Sarama & Clements, this volume), a growing number of computer-based games have been developed for improving basic numerical and arithmetic skills in children. In this section, we describe a software program that focuses primarily on training number line judgments, known as Rescue Calcularis. Development of this program was originally based on neuropsychological and neuroimaging findings from studies of children with developmental dyscalculia (DD) and designed specifically for improving their construction of and access to a mental number line (Kucian et al., 2011). In the first published study making use of this software, 8- to 10-year-old children with DD and an age-matched control group of typically developing (TD) children were shown an Arabic digit, an addition or subtraction problem, or different numbers of dots, and were tasked with positioning the solution to each problem (i.e., a number) on a number line using a joystick. These problems were administered within the context of saving one’s home planet with the use of a spaceship. Following 5 weeks of training (15 min a day, 5 days a week), children in both groups showed improved performance on both the number line and arithmetic tests. Furthermore, use of a brain imaging technique—(fMRI, functional magnetic resonance imaging)—revealed reduced recruitment of relevant brain regions (i.e., bilateral parietal) supporting number line processing, which the authors attributed to an automatization of the cognitive processes required for solving number line problems. In a more recent study employing the Rescue Calcularis game, Michels, O’Gorman, and Kucian (2018) administered number line training to 9.5year-old children with developmental dyscalculia (DD) over a 5-week period, as well as to TD children to whom the DD participants were matched on age, gender, and handedness. Furthermore, all children were pre- and posttested on a numerical ordinality judgment task during which fMRI and functional

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connectivity data were collected. The children with developmental dyscalculia exhibited abnormally high hyperconnectivity in frontal, parietal, temporal, and visual regions prior to the training; in other words, they were engaging too many brain regions when attempting to order numerals. The numerical and arithmetic performance of the DD group as well as their number line skills improved significantly following training, but only the latter reached the same level of accuracy as that of the TD group. And of particular interest is that not only did the functional hyperconnectivity of the DD children decline to the point of disappearing, but also a classification analysis of the posttest fMRI data (controlling for age and IQ) could not distinguish between the DD and TD groups. These outcomes led the authors to conclude that number line training normalized the DD children’s prior, aberrant neuronal activity and efficiency and generated widespread changes across the distributed brain regions involved in the various stages of information processing needed for them to make correct numerical order judgments. A recent offshoot of the Rescue Calcularis game is Calcularis—a computerbased training program that like its forerunner focuses on the learning of basic numerical skills, arithmetic operations, and number line estimation (Rauscher et al., 2016). Although this game retains the core features of “Rescue Calcularis”—primarily emphasizing the construction of and access to the mental number line—it includes a more comprehensive training of math skills and is also more flexibly adaptive to each child’s learning profile and knowledge state. The program is based upon the robust theoretical principles of both the Triple-Code model (Dehaene, 1992) and a four-step developmental model (Von Aster & Shalev, 2007). Briefly, the two areas of the training program cover: (1) transcoding between alternative representations of number and principles of numerical understanding such as cardinality and ordinality, with the component games ordered hierarchically; and (2) the concepts underlying arithmetic operations and their automatization. Seven- to ten-year-old typically developing children were randomly assigned to a Calcularis training group who received 6–8 weeks of daily (5 days per week) training at home, a computerbased, spelling training control group, or an untrained wait-list control group. The Calcularis group showed significantly greater improvements in number line estimation and subtraction (but not in addition) as compared to both control conditions, with medium to large effect sizes. As just described, these number line training programs have yielded some important findings. That being said, replications and extensions of such studies are certainly needed before larger scale interventions making use of these programs are warranted. Indeed, a recent critical review and analysis of the numerical training literature by Szucs and Myers (2017) focusing on improving both approximate and exact components of number sense reveal a sizable number of problems with this research. Several of the 10 studies reviewed were considered to have been poorly designed (e.g., poor choice of control condition activities); lacked sufficient power; used inadequate statistical

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procedures; failed to follow up the most crucial alternative hypotheses of the findings that either had been or could be raised; and there was a severe shortage of critical comments pertaining to the methods, findings, and conclusions of other relevant studies cited. Szucs and Myers conclude by recommending specific research designs, methods, procedures, and statistical analyses; thorough reporting of this information in published articles that should avoid post hoc theorizing and making convoluted arguments when explaining findings; and overcoming citation biases by taking a more critical stance when evaluating evidence from relevant prior studies. This rather devastating critique is certainly worthy of serious attention by researchers planning future training studies.

Domain-Relevant Interventions In addition to domain-general and domain-specific interventions for improving math skills, we suggest another category that is worthy of consideration: domain-relevant interventions. This term represents approaches aimed at enhancing numerical and arithmetic processing indirectly via either training of nonquantitative but still pertinent skills or activation of domain-general cognitive processes that are more relevant for improving numerical judgments than nonquantitative ones (Karmiloff-Smith, 2015; Knops et al., 2017). Examples that we cover here include: (a) providing alerting cues that engage attention toward stimulus features that can enhance the speed of executing enumeration judgments; (b) training nonquantitative, finger differentiation skills that have been shown to be associated with arithmetic abilities; and (c) delivering noninvasive, mild electrical stimulation to cortical brain regions that have been reliably implicated in numerical processing.

Training to Enhance Subitizing Speed Subitizing refers to the fast and accurate enumeration of small sets of elements (e.g., dots), typically within a range of 1–4 items (Mandler & Shebo, 1982; Trick & Pylyshyn, 1994; see also Sarama & Clements, this volume, for a more detailed examination of subitizing in young children). Although subitizing was historically considered to be a preattentive process, more recent research has shown that if attentional engagement in this enumeration task is reduced (e.g., by concurrent tasks designed to capture attentional resources), the subitizing process is impaired. This finding led Gliksman, Weinbach, and Henik (2016) to ask whether inducing an increase in attentional engagement during an enumeration task could lead to improved performance. Alertness denotes the attentional system’s state of general readiness to react to a given sensory stimulus that regulates attentional processing to that stimulus as well as the initiation of a response to it (Petersen & Posner, 2012; Wiegand, Petersen, Bundesen, & Habekost, 2017). And as Gliksman et al. point out, researchers can manipulate alerting by presenting

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brief warning cues, usually an auditory tone or beep that induces a state of high arousal. Importantly, these warning cues are considered neutral in that they provide no information concerning either the identity of the ensuing target stimulus or its spatial location. Furthermore, subitizing has been shown to be based in part on pattern recognition (Ashkenazi, Mark-Zigdon, & Henik, 2013; Mandler & Shebo, 1982), where quantities within the subitizing range (e.g., three dots) can be identified as familiar whole figures comprised of smaller elements, for example, the spatial configuration of three dots may form a triangle. Taking such evidence into account, Glicksman et al. also proposed that alerting could facilitate the subitizing process by enhancing global perceptual processing of geometric shapes such as a triangle and square composed, respectively, of three and four elements in a set; in contrast to a random arrangement of say, seven dots where the individual dots in the array are likely to be processed locally (Navon, 1977). Furthermore, these authors noted that strong evidence already exists from other kinds of tasks (e.g., face recognition; letter identification) indicating that alerting cues can induce a global processing bias. The participants in their first experiment were required to say aloud the number of dots presented in random arrays, consisting of those considered to be within the subitizing range (1–4 dots) and those within the small estimation range (5–9 dots). An alerting cue (auditory tone) appeared briefly before the display of the target array on half of the trials. Although alerting cues facilitated the response times for correctly enumerating the quantities within the subitizing range, they had no such effect on the small estimation range. The authors hypothesized that the facilitative effect of alerting cues on the subitizing process may be attributable to the enhancement of global processing, a process that was previously associated with both alerting and subitizing. To test this hypothesis, they carried out a second experiment with college students where the procedure was basically the same as that of Experiment 1, except that the arrays containing quantities within the small estimation range were presented in canonical arrangements creating hierarchical shapes— global figures made up of dots such as 6 dots in two vertical, parallel rows of 3 each, as on a die. They reasoned that if alerting regulates the subitizing process by enhancing global processing, then it should likewise improve performance even in the small estimation range, if the quantities are arranged in a canonical array. In other words, small local elements should create a larger global element that would enable participants to apprehend all the displayed dots at the same time rather than having to count each of them in succession. Indeed, this is exactly what happened. Furthermore, Gliksman et al. suggested that future research may be able to investigate whether extant training procedures known to increase alertness and global processing could also enhance enumeration in special populations such as those with developmental dyscalculia.

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Training Finger Differentiation: Its Impact on Arithmetic Ability In the chapter concerning early math development in the home environment, Levine et al. (this volume) discuss numerous studies concerning the role of gesture in general and finger pointing and counting in particular as these may contribute to the early development of numerical and arithmetic skills. One of the interesting findings they mention is an association between children’s math skills and the ability to discriminate between their fingers. More specifically, research on this topic begins with measuring the ability to distinguish between one’s own fingers when they are touched with a physical object with no visual cues available. This is skill is formally known as finger gnosis. To date, several studies have reported significant correlations between finger gnosis and numerical as well as arithmetic abilities. Fayol, Barrouillet, and Marinthe (1998) found that a measure of finger gnosis administered at age 5 predicted numerical ability up to 3 years later—even better than a measure of general intelligence. Similarly, Noel (2005) found that finger gnosis in first grade predicted the ability to map numerals and their associated magnitudes (but not reading ability) in second grade. Likewise, studies carried out by Penner-Wilger and Anderson (2013) and PennerWilger et al. (2009) reported associations between finger gnosis and several kinds of numerical tasks including magnitude comparisons and number line estimation. In contrast, several studies have failed to find any such associations (Long et al., 2016; Newman, 2016; Soylu, Raymond, Gutierrez, & Newman, 2017). Soylu, Lester, and Newman (2018) suggest that evidence demonstrating the codevelopment of finger gnosis, counting, and arithmetic skills might explain some of the conflicting findings on the relation between finger gnosis and arithmetic performance. Soylu, Lester, and Newman (2018) also provide a detailed analysis of three purported mechanisms underlying these associations: the localization model—close proximity of brain regions (in the parietal cortex) associated with both finger and number processing; the functional model—a functional relation between fingers and number processing that can be attributed to a developmental association originating in early finger counting experiences; and an evolutionary model—finger sensorimotor neural circuits originally evolved for visually guiding fine motor movements are reused to accommodate the representation and processing of numerical quantities. In the first attempt to test the functional hypothesis (and the only one to do so prior to 2016), Gracia-Bafalluy and Noel, 2008 administered 8 weeks of finger differentiation training to a group of first graders with poor finger gnosis skills. The results showed greater strengthening of finger gnosis for this group as compared with another group of first graders who were trained on story comprehension and a third group of children with initially high finger gnosis skills who spent 8 weeks engaging in business as usual. Furthermore, arithmetic abilities and ordinality judgments also improved significantly.

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However, Fischer (2010) criticized this study, arguing that the purported effects of finger differentiation training may be at least partially attributable to a statistical artifact: regression toward the mean. Testing a larger sample and employing much better controls, Wasner, Nuerk, Martignon, Roesch, and Moeller (2016) found that finger gnosis was a unique albeit weak predictor (accounting for only 1%–2% of the variance) of children’s preliminary arithmetic skills at the onset of first grade, independent of the effects of age, gender, general cognitive ability, short-term memory, and antecedents of numerical competencies. Nevertheless, they concluded that their findings are of some theoretical import.

Using Electrical Brain Stimulation to Improve Numerical and Arithmetic Processing Over the past decade, there has been a growing effort to examine whether weak (low intensity), transcranial electrical stimulation (tES) of the brain can be used for enhancing cognition (Sarkar & Cohen Kadosh, 2016a, 2016b). Indeed, enough studies have been published to have permitted several meta-analyses to be carried out (Simonsmeiera, Grabner, Heina, Krenza, & Schneidera, 2018). Nevertheless, most researchers working in this area would agree that this kind of research is still in its early stages, and that its use for improving numerical cognition in particular is even newer. Sarkar and Cohen Kadosh (2016a, 2016b) have provided a thorough review of such studies, examining the principles underlying this technology and the findings concerning its uses for enhancing numerical and arithmetic processing and skills. Here, we briefly describe two studies that demonstrate some of the potential value of these approaches. First, it should be noted that the two most frequently used forms of tES are transcranial direct current stimulation (tDCS) and transcranial random noise stimulation (tRNS). Basically, tDCS delivers a constant current through two electrodes attached to the scalp that produces anodal stimulation where it enters the brain (the anode), and cathodal stimulation where it leaves the brain and reenters the electrode (the cathode). The more recently applied form, tRNS, generates randomly assigned amplitudes of current that fluctuate between positive and negative amplitudes, producing “noise” in the targeted brain region, exerting excitatory effects at both electrodes. Furthermore, the rationale for selecting what are considered reliable target regions for brain stimulation is based on evidence drawn from neuroimaging research (including meta-analyses) that denote the locations where cortical activation has taken place during certain kinds of tasks or events. As Sarkar and Cohen Kadosh (2016a) note the parietal lobes and their subregions constitute the two areas of brain that have been implicated in numerical tasks most consistently across a substantial number of studies. Effects of brain stimulation on numerosity discrimination. Cappelletti, Barth, Fregni, Spelke, and Pascual-Leone (2007) administered tRNS bilaterally

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to the parietal lobes of healthy adults coupled with intensive cognitive training on a standard numerosity discrimination task—560 trials an hour a day for five successive days to determine whether precision of the approximate number system (ANS) could be improved. This condition was compared with three other conditions: cognitive training without brain stimulation, stimulation with no cognitive training, and cognitive training coupled with stimulation bilaterally to the motor cortex (i.e., a control brain region). The participants in the parietal stimulation plus training group exhibited significantly more improvement than did the participants in all of the other three conditions, and lasted for a longer period of time—up to 16 weeks following training. Furthermore, the improved ANS precision of the combined experimental condition exhibited transfer to other quantity judgment tasks (i.e., spatial and temporal discriminations). According to Sarkar and Cohen Kadosh (2016b), parietal stimulation facilitated the cortical plasticity associated with learning, inducing both the initial improvement in numerical discrimination and its maintenance over time. Effects of brain stimulation on arithmetic skills. Snowball et al. (2013) carried out a brain stimulation study with adults in which arithmetic skills was trained over 5 consecutive days using two types of problems: (1) drills requiring recall of answers to problems previously presented, and (2) problems requiring calculation by manipulating numerical operands in accord with a specific algorithm. Control tasks were also administered. The experimental group’s cognitive training was accompanied by brain stimulation (tRNS) applied bilaterally to the dorsolateral prefrontal cortex (a region implicated in the acquisition of arithmetic), while a control group received sham stimulation during cognitive training. (As Sarkar & Cohen Kadosh, 2016b describe it, sham stimulation consists of applying a current of the same strength and intensity as the treatment condition, but for a significantly shorter duration, e.g., 30 s as compared with 5 min—a typically effective duration, in order to rule out placebo effects.) The authors found that not only did brain stimulation coupled with cognitive training improve performance on both the drill and calculation tasks, but also that these improvements were maintained for a period of 6 months after training.

PARENTAL INFLUENCES ON CHILD COGNITION AND MATHEMATICAL LEARNING Much of what children learn occurs outside of formal educational settings. This learning includes human universals, such as language, that do not in fact require formal schooling, as well as knowledge that has been viewed by many as the purview of formal education, including symbolic mathematics. We now understand that parents can significantly influence children’s initial learning of symbolic mathematics and their readiness for mathematics learning in school. In the following, we provide overviews of parental influences on children’s learning in general, as related to numeracy, and associated methodological issues.

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A Brief History of Research on Parental Influences Until recent decades, little was known about mathematical learning prior to formal instruction, and even less about parental and home influences on this learning. This is surprising because from a broader evolutionary perspective, we would expect important early influences on children’s learning following from the significant expansion of brain size during hominid evolution, an associated increase in the length of the human developmental period, and almost certainly an increase in the amount of information children must learn to prepare them for adulthood (Bogin, 1999; Foley & Lee, 1991; Geary & Bjorklund, 2000; Wynn, 2002). These shifts were later accompanied by the earliest indications of symbolic representations during our evolutionary history (Street, Navarrete, Reader, & Laland, 2017), and eventually those related to number (Geary, Berch, & Mann Koepke, 2015). Such changes provided the foundation for culture and its cross-generational “evolution” (Richerson & Boyd, 2005), and placed a premium on children’s ability to learn (Geary, 2007). The earliest examples of cultural transfer of knowledge no doubt began through children’s observation and imitation of adults (Lancy, 2014; Tramacere, Pievani, & Ferrari, 2017) and likely included play with others, including older children, with imitation and emulation at its core (Riede, Johannsen, Hogberg, Nowell, & Lombard, 2018). Thus it is perhaps surprising that the major and best known theories of cognitive development make only modest reference to parental influences. Despite Piaget’s strong interest in evolutionary theories, his own theory of cognitive development placed little emphasis on the role of imitation in learning or to the direct influence parents (or any other adult) may have on that development. Instead, the developmental progression through Piaget’s four biologically preordained stages of development required that parents only be aware of the child’s incomplete knowledge and then provide an environment that would enable growth of that knowledge. In Piaget’s theory, the child is the active agent moving through the environment, constructing new knowledge as she passes through the cognitive development stages. In contrast, Vygotsky placed much greater emphasis on the social nature of the coconstruction of knowledge and “zones of proximal development” (Vygotsky, 1978, pp. 84–91), highlighting the role of others, including parents in cognitive development. However, in Vygotsky’s theory, parents are adjuvants to cognitive development; they have little direct effect and are not discussed at length. Only Bandura’s (1986) social learning theory affords parents a substantial and direct role in their child’s cognitive development (see also Rosenthal & Zimmerman, 1978). Here, observation and imitation are paramount to cognitive development and parents as “models” have a pivotal role in shaping their child’s development through both reinforcing behaviors and modeling. These theories focus our attention on the importance of children’s social context for their early learning but have not been thoroughly assessed with respect to children’s mathematical development (NMAP, 2008).

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Fortunately, more recent investigators have focused on the specifics of how parents can influence their children’s early mathematical cognition (e.g., Aunio, Tapola, Mononen, & Niemivirta, 2016; Casey et al., 2018; Kleemans, Segers, & Verhoeven, 2016; Skwarchuk, Sowinski, & LeFevre, 2014). These studies are a crucial step in the right direction, but span very different populations and use diverse assessments, so it is difficult to generalize from them. Further, the studies do not include all the parental variables that may be relevant or of interest; these would include parental math anxiety (del Rio, Susperreguy, Strasser, & Salinas, 2017), parental expectations of child aptitude or achievement (Lefevre, Clarke, & Stringer, 2002), parental valuation of math learning (Di Paola, 2016), or parental beliefs about the aptitude for or importance of mathematics for boys and girls (del Rio et al., 2017; Stoet, Bailey, Moore, & Geary, 2016). Nevertheless, to date, the most consistent parental effects are seen in parental numeracy expectancy and parental engagement.

Research on Children’s Home Numeracy Environment As mentioned, we have achieved important insights into the characteristics of children’s home numeracy environment that are associated with their mathematical development and later mathematics achievement (e.g., BlevinsKnabe & Austin, 2016; Elliott & Bachman, 2018; Purpura et al., this volume; Ramani et al., this volume). These characteristics include parental socioeconomic status (SES; Anders et al., 2012; Ginsburg & Russell, 1981); parental education (Anders et al., 2012; Purpura & Reid, 2016); literacy activities (reading) (Anders et al., 2012; Deng, Silinskas, Wei, & Georgiou, 2015); parental number vocabulary (Skwarchuk et al., 2014), frequency, and variety of number words and number-related activities that parents use when engaged with their children (Cankaya & LeFevre, 2016; Casey et al., 2018; Kleemans, Peeters, Segers, & Verhoeven, 2012; Kluczniok, Lehrl, Kuger, & Rossbach, 2013; LeFevre et al., 2009; LeFevre, Polyzoi, Skwarchuk, Fast, & Sowinski, 2010; Zippert & Ramani, 2017; see also Pupura et al., this volume); and home enrichment with numeracy-related materials (e.g., board games) for children to explore (Skwarchuk, Vandermaas-Peeler, & LeFevre, 2016; see also Ramani et al., this volume). As an example, Casey et al. (2018) found that parents’ number talk with their 3-year-old children and especially their bringing attention to number sets (e.g., this is “three” diamonds, ♦♦♦) were predictive of their later mathematics achievement in kindergarten and first grade, controlling a host of child and parental characteristics (e.g., intelligence). Other studies have shown that preschoolers’ frequent engagement in board games that involve numbers and counting is associated with later number knowledge and mathematics achievement (Benavides-Varela et al., 2016; Ramani & Siegler, 2008; see also Ramani et al., this volume). Napoli and Purpura (2018) found that parental

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report of engagement in number-related activities with their children (e.g., counting, pointing out and naming numerals) was correlated with children’s later number knowledge (see also LeFevre et al., 2009). Still, much remains to be determined, despite significant progress over the last several decades.

Methodological Issues The internal consistency and contributions of the previously cited studies are strong, but there are many issues surrounding the definition of numeracy activities and their measurement that remain to be resolved before the consistency of across-study findings can be fully evaluated (see Elliott & Bachman, 2018). For example, many studies use a single category of “number-related activities,” while other studies break this into distinct categories of “formal activities” and “informal activities.” Another inconsistency across studies and perhaps even within some studies is the identification of whether or not there are siblings within the home. Sibling play may increase the time spent in informal number-related activities but decrease the quality of that activity time. Likewise, few studies currently report on the use of full-time or parttime out-of-the-home care; this daycare may include enriched or impoverished, but otherwise unreported number-related activities and serve as an important confound to study results. Many, if not most, parental and home environment studies rely heavily on parental report. Although checklists can add consistency to this data collection, several concerns remain around self-report reliability, as well as accuracy for delayed recall of frequencies and past events. A few investigators have attempted to overcome concerns of self-report data by conducting naturalistic observations in the home or laboratory-based observations of parentchild interactions. Although these measures can be less biased, more reliable, and eliminate issues around recall of activities, such assessments also add to the cost of the study, may add time and scheduling burdens to the participants and the investigator, and may elicit a Hawthorne effect during periods of direct observation. Finally, the digital world is rapidly approaching saturation levels, yet few studies currently report or include the digital connectivity of homes and the use of electronic devices to engage children in number-related activities (see Ramani et al., this volume). How parents are or will be using digital devices as part their repertoire of number-related activities will be an important measure to include in future research.

CONCLUSION As outlined in the first four volumes of this series, research on children’s mathematical knowledge and its development has blossomed over the past three decades. We now have a much firmer grasp on infants’ and young children’s intuitive sense of number, the evolved systems that support them, and

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how these systems interact with early experiences to help children acquire their first understanding of symbolic mathematics (Volume 1). The development of functional and structural brain imaging techniques has led to insights into the neural systems that support people’s intuitive understanding of number and that support mathematics learning, including how the brain changes in response to targeted interventions (Volume 2). Advances in the cognitive sciences have led to insights into human learning generally, and the application of these insights to mathematics has resulted in a much deeper understanding of the cognitive mechanisms that can impede or facilitate mathematics learning inside and outside of the classroom (Volume 3). Important insights have also been gained regarding the wider influences of culture (e.g., market economic, mathematics curriculum) and language (e.g., structure of number words) on children’s mathematical development (Volume 4). In this final volume, we focus more directly on educational practices. The latter of course includes the standard curriculum that will be well known to readers of this series, but our focus is on the specifics of how informal parental activities can foster children’s learning of mathematics, and on the formal interventions that are needed to support the development of children who are at risk for long-term difficulties in learning mathematics. Much has been learned, but much remains to be done. The reduction of intervention fadeout and the development of interventions for students with domain-general (e.g., working memory) deficits are significant issues that remain to be tackled. Although these are daunting issues, we are confident that advances in the coming decades will make substantive inroads into addressing them.

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Chapter 2

The Effects of SES, Grade-Repeating, and IQ in a Game-Based Approximate Math Intervention Josh Langfus*, Alejandro Maiche†, Dinorah De Leo´n†, Dahiana Fitipalde†, A´lvaro Mailhos† and Justin Halberda* * †

Johns Hopkins University, Baltimore, MD, United States Centro de Investigacio´n Ba´sica en Psicologı´a, Montevideo, Uruguay

INTRODUCTION What tools can be used to close the gap between low- and high-achieving students? While there may be many ways to intervene—for example, teacher training, improving facilities, updating curricula—a great deal of attention has focused on the potential benefits of using technology as a teaching tool in the classroom. Nonprofit organizations like One Laptop Per Child (Trucano, 2011) and the World Computer Exchange (2016) have partnered with schools around the world to give students access to computers, tablets, and the internet. To the extent that digital literacy will be a valuable skill for tomorrow’s workforce, providing early access to technology might be a useful investment. Less clear is the role that technology can play in bolstering instruction in traditional subject areas such as science, math, and reading. Some exploratory work has examined the effectiveness of technology-based classroom interventions in math education, with results suggesting benefits of training that focuses on specific skills, for example, manipulating decimals (Zhang, Trussell, Gallegos, & Asam, 2015) and multiplication and division (Pilli & Aksu, 2013). In contrast, other recent work has explored the effectiveness of nontechnology interventions in locations with restricted access to computers, though it is not clear that improvement on these intervention activities transfers to other math abilities (Dillon, Kannan, Dean, Spelke, & Duflo, 2017). In the context of recent interest in technology’s role in education, we present some initial findings from an ongoing classroom technology intervention in Mathematical Cognition and Learning, Vol. 5. https://doi.org/10.1016/B978-0-12-815952-1.00002-5 Copyright © 2019 Elsevier Inc. All rights reserved.

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elementary schools in and surrounding Montevideo, Uruguay—the country’s capital and largest city. This effort is unique because Uruguay stands out as one of the first countries in the world to commit to providing public access to information and communications technology (ICT) on a nation-wide scale. Not only has the Uruguayan government invested in establishing free internet connections in schools and public places across the country, but also since 2008 it has partnered with the nonprofit organization One Laptop Per Child to provide every school-aged child with their own computer to be used in the classroom and at home. The joint initiative is called Plan Ceibal (“Conectividad Educativa de Informa´tica Ba´sica para el Aprendizaje en Lı´nea” or “Educational Connectivity/Basic Computing for Online Learning”), and over the last decade, the program has expanded from its start in the nation’s primary schools to include every student in Uruguay from preschool through elementary and middle school. The dramatic increase in access to ICT has created a need for content on these platforms that educators can use in the classroom. Our team has partnered with a dedicated group of teachers at public elementary schools in the Montevideo area to create educational software for the tablets and evaluate its effectiveness. The present study is the latest from this ongoing project; a previous study engaged 503 first-graders in a number estimation task (Odic et al., 2016), and in the work presented here we introduced 386s- and third-graders from a wide range of backgrounds to three tablet-based magnitude training games. We used a pretest, intervention, posttest design, along with a Business-As-Usual (BAU) Control group, to examine the effectiveness of playing these games on several measures, including a standardized test of school math ability (speeded arithmetic). The context of Uruguay also affords us the opportunity to look at whether such interventions helps some students more than others, e.g., as a function of Socioeconomic Status (SES) and Grade-Repeating status. As we will discuss in more detail later, our project includes students from schools across the SES spectrum, and therefore allows us to study the relationships between SES and measures of achievement and cognitive abilities, as well as how SES might moderate the impact of our intervention software. In addition to SES, the large proportion of students in Uruguayan schools that repeat a grade allowed us to examine whether our intervention was particularly helpful for these students, and more generally to ask what kinds of students benefit most from our technology-based intervention. Our initial hope was that students who are most vulnerable or disadvantaged might stand to gain the most from our intervention games which focus on simple, intuitive, “core” magnitude discrimination. To explore how the current study may begin to answer these questions, it is important to understand the context of schools in Uruguay.

THE URUGUAYAN CONTEXT One hope of the educational community in Uruguay is that providing equivalent technology to all children may help to bridge the gaps between schools in

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high- and low-SES communities. As mentioned previously, the schools in Uruguay provide a unique test case for exploring the potential of technological interventions, not only because of the willingness of teachers and students to engage with technology, but also because this technology is accessible to schools in communities across the socioeconomic spectrum. Table 1 summarizes the number of students per school per SES quintile included in the current study. The Uruguayan National Public Education Administration (ANEP) uses a variety of factors to classify each school’s SES context: household education level, socioeconomic indicators (e.g., percentage of households with access to potable water), and social integration (e.g., head of household’s employment status and percentage of household children attending school). Using an aggregate of these factors, ANEP classifies schools into quintiles (with level 1 as the lowest). Uruguay’s national government tracks a wide range of data about the nation’s public education system, including enrollment and completion rates across many demographics, and these data highlight differences across socioeconomic contexts. A 2017 report from the National Institute of Education Assessment (INEEd) showed that, while access to education has improved over the last 10 years, striking disparities still exist across levels of SES in terms of enrollment, retention, and grade-repetition. These disparities can be seen from the beginning of enrollment in school through graduation. For example, between 2013 and 2015, preschoolers (age 3) in the least advantaged schools were less likely to meet minimum standards for required attendance (at least 141 days in the year) compared to students at the most advantaged schools; 28% of students in SES Quintile 5 did not meet this standard, compared to 46% in Quintile 1 (INEEd, 2017). Though overall attendance increases, and the gap narrows as students progress to higher grades, the data show a consistent 10% difference in meeting the standard for minimum attendance between the highest and lowest SES schools from first grade through sixth grade (INEEd, 2017). Socioeconomic differences are also apparent in the number of students who repeat a grade, which can occur for several reasons including lack of attendance or poor performance. Already in the earliest grades there are differences between higher and lower SES Quintiles with respect to children being on-target to progress through school at the appropriate age. For example, at 7 years of age only 4% of students in Quintile 5 schools had repeated or were repeating a grade, while the figure jumps to 12% for children in Quintile 1 schools, according to country-wide data (INEEd, 2017). And, most dramatically, by the time students graduated from secondary school, 83% of students in the least advantaged schools had repeated at least one grade, compared to 24% in the most advantaged programs (INEEd, 2017). Arguably the most striking differences across SES manifest in terms of the number of students who leave education at either the primary-, middle-, or high-school level. While Uruguay has made significant strides in ensuring that nearly every student, regardless of SES, completes primary education, only

TABLE 1 Condition Assignment by School With SES Quintile Information (1 5 Low, 5 5 High) School

A

B

C

D

E

F

G

H

SES Quintile

1

1

2

2

3

4

5

5

Classrooms

a

b

a

a

b

a

b

a

b

a

b

a

b

a

b

c

n

17

18

20

27

31

20

22

26

27

24

27

29

28

26

19

27

X

X

X

X

X

X

X

X

X

X

Grade

Condition

2 3

X

BAU control

X

Game intervention

X

X X

X

X X

X X

X X

X

X X

X X

X

Note that for all schools except for B, every school had 2 year-matched classrooms—one for each condition (Intervention and Control).

X X

X

X

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50% of least advantaged students graduate from middle school, compared to 95% of students from Quintile 5 schools, and the trend continues through the end of high school, where the gap is 15% vs. 71% (INEEd, 2017). In terms of the effect of SES on access to education, and foreshadowing some of our own result, data from our study show the same patterns as the INEEd report. On both the pre- and postintervention evaluation days of the current study, more students in the lower SES schools were absent compared to students at higher SES schools (see Table 2). Students at poorer schools are also more likely to have repeated a grade; within our sample of 386 children, the percentage of students whose birthdays fell after the legal cutoff date for their grade level was 5.4% at Quintile 5 schools compared to 41.8% at Quintile 1 schools (see Table 3)—a rate that is even more dramatic than what INEEd has reported for the national percentages in Quintile 1 (INEEd, 2017). While these disparities are disheartening, the fact that the schools who have partnered with us for this project seem to represent a microcosm

TABLE 2 Attendance Data by SES Quintile for the Testing Days Quintile

1

2

3

4

5

Overall

N

55

100

53

49

129

386

% Present both

53

40

55

90

77

62

% Absent pre

18

26

15

8

12

16

% Absent post

22

26

25

2

9

16

% Absent both

7

8

6

0

3

5

Total %

100

100

100

100

100

100

Number of students enrolled in each Quintile is given at the top, and percent of students who were present for both, one, or neither test are given below. Students in lower SES classrooms (Quintiles 1, 2, and 3) were overall less likely to be in school testing days.

TABLE 3 Number of Students per SES Quintile Counted as Repeaters Quintile

Repeaters

Total Students

Percent Repeating

1

23

55

41.8

2

27

100

27.0

3

9

53

17.0

4

4

49

8.2

5

7

129

5.4

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Cognitive Foundations for Improving Mathematical Learning

of the issues affecting the education system as a whole in Uruguay allows this and future studies to identify interventions that stand a greater chance of scaling up to meet those broader challenges.

BACKGROUND FOR THE CURRENT STUDY Socioeconomic Status in Education Children from low SES households typically underperform their middle- and high-SES peers in school achievement (Sirin, 2005; Valle-Lisboa et al., 2016). These differences may be the result of a range of factors associated with SES (Wilkinson & Pickett, 2010). Performance differences have been observed in many specific domains, including language development (Hart & Risley, 1975; Hoff, 2003), IQ (Turkheimer, Haley, Waldron, D’Onofrio, & Gottesman, 2003), cognitive ability (Larson, Russ, Nelson, Olson, & Halfon, 2015), spatial knowledge (Levine, Vasilyeva, Lourenco, Newcombe, & Huttenlocher, 2005; Verdine, Irwin, Golinkoff, & Hirsh-Pasek, 2014), as well as mathematics (Klibanoff, Levine, Huttenlocher, Vasilyeva, & Hedges, 2006). Based on this literature, we expect to find differences in the initial cognitive and curricular abilities of children as a function of SES. Given these differences in baseline abilities, we might expect that SES differences will differentially affect the impact of our training games and test-retest improvement.

Classroom Geometric and Arithmetic Abilities As with other content areas, math achievement varies as a function of SES in the United States (Sirin, 2005), Uruguay (Valle-Lisboa et al., 2016) and other South American countries (INEEd, 2017) for all ages. These differences can be tracked across a variety of domains, e.g., intuitive number sense, spatial reasoning. In the present work, we focused on assessing two domains of math achievement: arithmetic and geometry. Our measure of geometry was used primarily as a pilot and consisted of an assessment developed by the Spelke lab and, in a collaboration, translated into Spanish by one of our authors (A.M.) and used with permission (E. Spelke, personal communication, June 23, 2017). A typical question on this measure showed students several shapes and asked them, for example, to circle all that had sides of equal length. Another question in this pilot task assessed students’ understanding of symmetry by asking whether a given shape was the same on either side of a dotted line. This measure will serve as a control to contrast with our main measure of interest: arithmetic ability. To measure arithmetic ability, we used a version of the math fluency subtest of the Woodcock-Johnson Tests of Achievement (Woodcock, McGrew, & Mather, 2001/2007) adapted for use in Spanish-speaking contexts: the Baterı´a III Woodcock-Mun˜oz (Woodcock et al., 2001/2007) (see Fig. 1). The test measures the number of addition, subtraction, and single-digit multiplication problems children can complete in 3 min. It is a reliable measure of arithmetic achievement and has been widely administered in many countries.

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FIG. 1 An example page from the Woodcock-Mun˜oz speeded arithmetic assessment. Students had 3 min to complete as many problems as possible.

Because norming data for this test do not exist for Uruguay, here we report the raw scores (i.e., the number of problems completed correctly), age-normalized scores (normed to our sample), and regressions controlled for age, geometry, IQ, and vocabulary where appropriate. We expect that our children will differ in math ability (geometry and arithmetic) as a function of SES. We will also look at how this relates to IQ and ANS ability and Repeater status, and how these factors may influence the effectiveness of the intervention.

Approximate Number Abilities Our magnitude training intervention approach was inspired by work in developmental psychology that has focused on the relationship between classroom

44

Cognitive Foundations for Improving Mathematical Learning

mathematics and an intuitive number sense mediated by the approximate number system (ANS: Halberda & Feigenson, 2008). The ANS is a core knowledge system (Feigenson, Dehaene, & Spelke, 2004) that is present in newborns (Izard, Sann, Spelke, & Streri, 2009) and improves over the course of development (Halberda & Feigenson, 2008; Halberda, Ly, Wilmer, Naiman, & Germine, 2012; Odic, Libertus, Feigenson, & Halberda, 2013). The system supports people’s intuitive understanding of the approximate quantity of collections of items or events and supports the ability to add and subtract these quantities. Such a system might be specific to number, or the ANS may be part of a more general magnitude system that includes, e.g., surface area, time, and length (Lourenco, Bonny, Fernandez, & Rao, 2012; Sokolowski, Fias, Ononye, & Ansari, 2017; Walsh, 2003). Because we want to remain open to the possibility of a generalized magnitude system, we created mini-games for surface area, time, and number. However, in previous work we have found that each of these dimensions uniquely correlates with school math ability, suggesting some independence among these dimensions (Odic et al., 2016). Our own proposal is that the shared aspects of these representations may derive from the computations that allow one to compare magnitudes (e.g., to determine that one duration is longer than another, or that one size is larger than another), rather than from shared representations of quantity (Odic, Pietroski, Hunter, & Lidz, 2013). For the current paper, we are interested in improvements in this or any other parts of these systems, and in the possibility that training in magnitude games can transfer to speeded arithmetic performance. A critical component of these ongoing debates is whether ANS precision plays a causal role in shaping formal math abilities. Intervention studies provide a tool to assess this hypothesis by intervening to improve ANS precision and observing if this leads to an improvement in formal math ability. A handful of research projects have looked at classroom interventions using computerized training programs to improve the precision of cognitive abilities, such as the ANS, that are associated with school math performance (DeWind & Brannon, 2012; Hyde, Khanum, & Spelke, 2014). There have been mixed successes, and more research is needed to understand what kinds of interventions are most successful and for which students. One possibility is that these types of ANS interventions may be particularly helpful in fostering foundational understandings of basic concepts or in situations where a child is struggling to attain proficiency with early skills. In the present work, our measure of ANS ability will be the number of problems solved in a timed paper-and-pencil dot comparison task (Fig. 2). This is a pilot version of a task that we are actively developing which aims to measure ANS ability in a group- or individual-administered, paper-andpencil format (Mailhos et al., 2018). We expect to find differences in ANS ability across SES in pretest, and we also expect to replicate results showing a correlation between ANS ability and arithmetic ability (DeWind & Brannon, 2012; Hyde et al., 2014; Odic et al., 2016; Zosh, Verdine, Halberda, HirshPasek, & Golinkoff, 2018). Lastly, we will also look at whether SES,

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FIG. 2 Sample page from modified dots discrimination task. Students were instructed to consider each pair of squares and place an X in the box with more dots in it. They were given 3 min and told to begin at the top left and continue down each column until time ran out.

Grade-Repeating, and IQ affect the relationships between ANS ability, arithmetic ability, and intervention transfer.

THE PRESENT INTERVENTION STUDY Our primary goal in partnering with communities in Montevideo was to create a resource that could improve students’ math performance and that students could use in the classroom and at home. To do this, we aimed to develop an app that students would be motivated to engage with over an extended period of time and that trained approximate magnitude comparison over various domains.

46

Cognitive Foundations for Improving Mathematical Learning

FIG. 3 (A) The tablet children used in the intervention and sample images from trials of each of the three training games: ANS (B), area (C), and time (D).

To make the intervention engaging for students, we developed a completely novel app that included extensive original artwork and characters (e.g., “monsters”), rewards for good performance (e.g., “stars”), and online tracking of gameplay. The app included three mini-games (see Fig. 3B–D), each focusing on one of the three abilities that have been discussed in the literature as being relevant to general magnitude representation: time, area, number. We sought to improve students’ math abilities through training on these magnitude comparison tasks over the course of a 1-month intervention.

Teachers’ Responses to the Software Successfully translating from the lab to the classroom requires building bridges between researchers and educators, and our project cannot succeed without the work of teachers invested in partnering with us for the long term. Not only are teachers responsible for implementing the intervention in their classrooms, but their feedback is our primary means of understanding how students interact with the app. Nearly all the teachers whose classrooms participated in the current study were also with us for a separate pilot phase, and their feedback was particularly valuable in helping us to understand how to motivate the children to continue engaging with the games. For example, teachers suggested that students be able to personalize their game sessions by selecting an image as an avatar. They also suggested introducing a progress-tracking system into the game that would allow students to see how they were doing and compare their performance with their peers. We implemented these ideas in the version of the

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game for the current study, and we heard from teachers that these changes improved student engagement. At the end of the intervention, the teachers met with our team in Uruguay and offered final comments on the intervention, summarized by our team here: El juego en t erminos generales, fue evaluado positivamente por parte de las maestras, la mayorı´a de ellas transmitieron el entusiasmo de los nin˜os por jugar. La incorporacio´n de las estrellas como premio fue de gran motivacio´n para los nin˜os, por ejemplo se veı´a que comparaban su progreso en los distintos minijuegos con sus compan˜eros a la vez que los motivaba a seguir jugando. La tema´tica del juego fue divertida y muy aceptada por los nin˜os, con respecto a esto se observo´ que cada nin˜o tenı´a un monstruo preferido, incluso nos compartieron dibujos de los monstruos. Para finalizar, luego de terminada la intervencio´n los nin˜os nos preguntaban si ı´bamos a volver con las tablets y el juego. The game, in general, was seen very positively by the teachers, the majority of whom communicated the children’s enthusiasm for playing the game. The incorporation of the stars as a reward system was a great motivator for the kids, for example they saw students comparing progress in the different mini-games with their friends which motivated them to continue playing. The theme of the game was fun and very well accepted by the children, and with respect to this it was observed that each child had a favorite monster, and they shared with us drawings of these monsters. Finally, after the end of the intervention, the children asked us if we were going to use the tablets and play the games again

These comments suggest that students enjoyed engaging with the training app and teachers were easily able to incorporate it into their normal lessons. These are key components of a classroom-based intervention.

Design of the Current Study As mentioned previously, the study implemented a pretest, intervention, posttest design with a Business-As-Usual (BAU) Control group. We assessed formal Arithmetic Abilities (i.e., the number of single- and double-digit addition, subtraction and multiplication problems students could complete in 3 min), IQ [Raven’s Progressive Matrices (Raven, Raven, & Court, 1998)], Approximate Number Abilities (the ability to rapidly determine the larger of two approximate quantities on our paper assessment), Vocabulary, and Geometry Ability. In our analyses, we focus on the diversity of our Uruguayan school sample in terms of SES, Grade-Repeating, IQ, and Arithmetic Ability, and we present improvement scores from pre- to posttest for both our Game Intervention group and our BAU Control group. Figure 4 illustrates the study’s overall design. The boxes on the left and right represent the pre- and postintervention evaluations, respectively. All students participated in these evaluations. Classrooms at each of the participating schools were randomly assigned to either the Game

48

Cognitive Foundations for Improving Mathematical Learning

FIG. 4 Study design. Both groups (Intervention and Control) participated in pre- and postintervention evaluations of cognitive abilities. The intervention group was encouraged to play the three tablet games during the 5-week intervention period, during which the control group received Business-as-Usual instruction.

Intervention condition or the BAU Control condition for the five-week intervention period. During that time, students in the Game Intervention group interacted with the intervention software which consisted of three magnitude training games, each focusing on a different ability: area discrimination, time discrimination, and approximate number discrimination. Note that, consistent with Uruguayan school policy, students in the BAU Control condition also had access to tablets during this time as part of their normal curriculum, but they did not have access to the games we developed for the intervention.

Methods Participants Sixteen classrooms in eight public elementary schools in and around Montevideo participated in the study, including 386 s- and third-graders overall. Half of the classrooms were assigned to the intervention condition and the training game was installed on tablet computers provided by Plan Ceibal. During the 5-week intervention period, students in this group were encouraged approximately once per week by their teachers to play the game at school for a little while and at home as much as they liked. Otherwise, children in the Game Intervention group received Business-As-Usual (BAU) math instruction. The other eight classrooms comprised a BAU Control group and received normal math instruction. These students had access to tablets but not the training game and, as is usual for schools in Uruguay, use of the tablets was left to the discretion of the classroom teacher. For this study, it was important to us that the teachers in each group felt that they could instruct as normal, along with the small addition of access to the game for the Game Intervention group. In Fig. 5 we see a map of Montevideo and the surrounding areas where our schools were located. Each school is lettered and the metrics for each school are summarized in Table 1. In this study, SES information was only available

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FIG. 5 Locations of our schools. From Google Maps image used with in accordance with Google’s fair-use guidelines: https://www.google.com/permissions/geoguidelines/attr-guide/.

at the school level (as mentioned previously). In all but two schools, each intervention classroom was matched with a same-year (second- or thirdgrade) control classroom at the same school. All children spoke Spanish as their first language and all tasks were administered in Spanish. The posttest evaluations were administered by a researcher blind to the experimental conditions of the classrooms (J.L.). Of the 386 students enrolled in classrooms that participated in the study, 19 students (5%) did not attend class on both pre- and postintervention testing days, and some students (overall 32%) were in attendance on one of these days but not both. As mentioned earlier, attendance at school depends to an extent on SES Quintile, and this is reflected in which students tended to be present for our pre- and posttest measures, as summarized in Table 2. Additionally, descriptive statistics about our sample (e.g., age ranges by condition and Grade) can be found in Table 4. Analyses were conducted using data from all students who completed the relevant measures for each statistical test. Evaluation of repeater status. Our analyses considered the interaction between Repeater Status on cognitive abilities (such as IQ) and achievement (e.g., arithmetic ability). We considered Repeaters to be any children who had already repeated or were currently repeating a grade, and we determined

TABLE 4 Descriptive Statistics for Students Included in Analyses Class

Condition

Mean Age (mo)

SD Age

Range (mo)

Females

Total

Second grade

Control

95.24

6.91

85.90–117.10

59

128

Second grade

Intervention

94.22

6.65

85.50–117.40

72

127

Third grade

Control

107.17

7.90

97.73–127.63

31

61

Third grade

Intervention

106.55

6.63

97.73–127.50

35

70

Overall

Control intervention

99.13

9.14

85.90–127.63

90

189

98.79

8.92

85.50–127.50

107

197

All ages are reported here as-of the pretest date before the 5-week intervention period.

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this by comparing each student’s birthday with the state-mandated cutoff date for their grade. By law, Uruguayan schools enforce strict cutoff dates for matriculation; it is illegal for parents to choose to hold a child back and start them later in school, a practice that occurs in some other countries like the United States. While our records did not indicate for each child the reason why they were a year older than their peers, the teachers reported that nearly all of these older children had been asked to repeat a grade because of poor performance. In our dataset, we had roughly the same percentage of Repeaters in Grades 2 and 3 (24% and 27%, respectively).

Materials During the 5-week intervention period, students engaged with the training app using touchscreen tablets specifically designed for children, which they had access to throughout the training period. As mentioned previously, the training app consisted of three mini-games, each designed to probe a distinct modality of magnitude discrimination: time, area, and approximate number. The games could be played without a connection to the internet. Because use of the app was left to the discretion of the teachers in school, and the children themselves while at home, the number of times each child played each game was free to vary; however, in practice, children played each game multiple times over the course of the intervention; see Engagement with the intervention in “Results” section. The cognitive and achievement tests were administered to all students in the study before and after the intervention period by trained researchers following a written protocol. As mentioned previously, we measured ANS Ability, Geometric Ability, Arithmetic Ability, Vocabulary size, and IQ (see Fig. 4). We also tracked SES at a school level using government-reported data (ANEP [Administracio´n Nacional de Educacio´n Primaria], 2012). Intervention Games Upon starting the intervention software, students saw the app’s home page and could select one of the three mini-games by tapping on that game’s icon. In each mini-game, students saw 12 trials in which the ratios of magnitudes (i.e., durations of time, numbers of items, and surface areas) were systematically varied, and they were asked to tap on the larger magnitude. Students received feedback after each trial indicating whether they made a correct response, and the games responded to three consecutive incorrect responses by repeating the game’s instructions. After 12 trials the game ended, and a gold star appeared beside the number of correct answers made in that block. Tapping the screen returned the student to the app’s home page, where the total number of correct responses across all blocks was displayed over the icon for each game. Time discrimination. In the time discrimination game, students had to decide which of two sounds had a longer duration. The screen showed two monsters: a green one on the left and a purple one on the right (see Fig. 3D). Students tapped the screen to start a trial in which each monster took

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Cognitive Foundations for Improving Mathematical Learning

its turn to make a singing-like sound for a specific amount of time. To provide additional visual cues, the monsters’ mouths moved, and they moved their hands to cover their mouths for 1 s after completing their sound. Students tapped the monster that they thought sang for longer and received feedback based on whether they answered correctly. After correct answers, the monsters would laugh and smile; incorrect responses elicited a “No, no!” sound while the monsters shook their heads. The singing sound varied from trial to trial to make the game more interesting. Area discrimination. In the area discrimination game, students had to decide which of two monsters blew the largest “bubble” (see Fig. 3C). As in the time discrimination game, students saw two monsters: a blue one on the right and a yellow one on the left. After tapping the screen to begin the trial, they saw a multicolor circle grow in front of one of the monsters. The circle expanded briefly and stayed at a constant size for a brief interval before appearing to “pop” and disappear. Next, a similar bubble appeared in front of the other monster. The expansion and disappearing of the bubbles was accompanied by “blowing” and “popping” sound effects. After both bubbles disappeared, students tapped the monster they thought blew the biggest bubble. After correct responses, the monsters would laugh and dance to music, while incorrect responses caused them to shake their heads and say “No, no!.” Approximate number discrimination. In the number discrimination game, students were asked to tap the monster that “sneezed” the most “germs” (see Fig. 3B). As in the other games, they saw two monsters on the screen—one on the left and the other on the right. At the start of the trial, the first monster scrunched up its face and made a sneezing sound as a large sneeze-cloud with a certain number of “germs” appeared under its nose. After less than a second, the cloud of germs corresponding to the first monster disappeared and the second monster took its turn to sneeze a cloud of germs. The germs consisted of a discrete number of bounded shapes with different patterns and textures across trials. On some of the trials, the shapes were of equal size, and on other trials the size of the germs varied. On half of the trials, the total area of the germ shapes was equal between monsters (viz., the total area was controlled across numbers), and in the other half of trials the size of each germ and the total area varied (viz., the total area did not consistently correlate with the number of germs). These controls ensured that both individual germ size and total germ area were poor predictors of number in our stimuli. As in the other two games, students received auditory and visual feedback for the responses in the form of the monsters laughing and dancing for correct responses or shaking their heads (i.e., for incorrect responses).

Results Our most important measures were Arithmetic Ability (Baterı´a III WoodcockMun˜oz), ANS Ability (paper/pencil ANS task), SES (as reported by ANEP),

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Repeater Status (determined by students’ birthdates), and IQ (Raven’s Progressive Matrices). We also include the brief geometry and vocabulary tests as pilot control measures. We first investigated pretest differences in Arithmetic Ability, ANS Ability, and IQ as a function of SES, Grade, and Repeater Status (i.e., Repeater vs. Non-Repeater). In our analysis of pretest abilities, we grouped both Game Intervention and BAU Control children together because, at this point, no children had received an intervention.

IQ and Repeater Status by SES Quintile Independent from our interests in math intervention, factors like IQ, graderepeating, and SES may have interesting interdependencies of their own. We first asked whether children’s IQ might contribute to determining which children did and did not repeat a grade as a function of SES. In Fig. 6 each dot represents a child’s age-normalized percentile score on the Raven’s Progressive Matrices (IQ) assessment, arranged by SES Quintile and jittered along the x-axis to reveal all children. Filled-in and open dots correspond to Repeaters and Non-Repeaters, respectively. The filled and open gray dots indicate means for each group, with bars representing standard error. Because Raven’s scores are age normalized to each student’s birthdate, the scores in Fig. 6 can be compared across Grades 2 and 3, and across Repeaters and Non-Repeaters, that is, even though Repeaters were older than their classmates they are not favored on this measure.

Repeater distribution within IQ by SES Quintile Repeaters vs. nonrepeaters 100

IQ

75

50

25

0 Quintile 1

Quintile 2

Quintile 3

Quintile 4

Quintile 5

FIG. 6 Repeater Status compared with IQ within each SES Quintile. Filled-in dots indicate students who repeated a grade; open circles indicate students who never repeated a grade. Dark gray dots indicate group means for Repeaters (filled) and Non-Repeaters (open), with bars indicating standard error of the mean.

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Cognitive Foundations for Improving Mathematical Learning

The first thing we notice is that every Quintile contains the full range of IQ scores (i.e., we see scores from approximately 0–100 in each Quintile). Second, there is a small effect of SES on IQ in both Repeaters and Non-Repeaters, with higher SES corresponding to somewhat higher IQ scores on average, though this trend is perhaps minor compared to the broad similarity in spread that we see in this measure (0–100). This difference in measured IQ as a function of SES replicates patterns previously demonstrated in the literature (Turkheimer et al., 2003). One of the most dramatic patterns in Fig. 6 is that the proportion of Repeaters is much higher in the lower quintiles compared to the higher quintiles. The effect of SES on the percentage of students who were repeating a grade is so dramatic that the difference between SES Quintile 1 (disadvantaged) and SES Quintile 5 (advantaged) was nearly eightfold in our sample: 41.8% of students in SES Quintile 1 had or were repeating a grade while only 5.4% of students in SES Quintile 5 had repeated or were repeating a grade. This trend is noticeable in Table 4, but in Fig. 6 we can see the pattern at the level of the child by looking at the ratio of filled to open circles as a function of SES. Finally, in Fig. 6, we see that the mean IQ tends to be lower for Repeaters compared to Non-Repeaters. This result is important for considering how to effectively help these struggling students. Further, it is noteworthy that Repeaters tend to cluster at the bottom of the IQ scale in Quintiles 1 and 2, whereas the mean IQ of Repeaters was similar to Non-Repeaters in two of the three upper Quintiles (and note that in the one upper Quintile where this was not true, Quintile 4, there were only four Repeater children and so a larger sample may be required before concluding that this Quintile would show a significant difference). Interestingly, while lower IQ is linked with repeating a grade in SES Quintiles 1 and 2, it is less linked in the more advantaged schools (SES Quintiles 3 and 5). Thus while it is true that many factors may impact performance on IQ tests like the one we have used here, our data support the inference that SES serves as a protective factor against repeating a grade. One possible explanation is that lower IQ students who might otherwise repeat a grade may receive more support in higher SES schools compared to students in disadvantaged schools and therefore be able to follow the standard trajectory. Conversely, students in low SES schools appear to need a higher IQ to avoid having to repeat a grade. This interpretation would be consistent with recent work showing that SES may moderate the impact of environmental factors that affect school performance. More specifically, higher SES mitigates the potential negative effects of factors such as low IQ (Tucker-Drob & Bates, 2016; Turkheimer et al., 2003).

Pre-Intervention Arithmetic by Grade and Repeater Status Next, we considered Pre-Intervention Arithmetic Ability. Fig. 7 shows the performance for four groups of children across SES and Grade (i.e., Grade 2 Non-Repeaters, Grade 2 Repeaters, Grade 3 Non-Repeaters, and Grade 3

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Mean pretest arithmetic ability

Pretest arithmetic ability across SES Quintiles

Grade Third Second

30

Repeater status Nonrepeater Repeater

20

Sample size 10 30 50 70

10

0 1

2

3

4

5

SES Quintile

FIG. 7 Pretest Woodcock-Mun˜oz Arithmetic Ability score by SES quintile separated by Grade (second and third) and Repeater Status (Repeater vs. Non-Repeater). For this figure, four separate planned regressions were run (one for each group). The effect of SES on performance is apparent in each group (all n 35), indicated by the upward slope of the lines as SES Quintile increases (all P < .01). The top pair of lines represents third-graders, while bottom pair represents secondgraders (reflecting the positive difference that one additional year of school has on speeded arithmetic performance); dashed lines through triangles and solid lines through circles correspond to Repeaters and Non-Repeaters, respectively (these lines are regression lines based on the raw data from each child in the group). The size of each triangle and circle is proportional the sample size in that category. Symbols within each Quintile have been slightly jittered horizontally to prevent overlap.

Repeaters) in our assessment of Arithmetic Ability. If SES affects Arithmetic Ability, we should see a positive trend across SES with increasing Arithmetic Ability scores as SES Quintile increases. In Fig. 7, we see this pattern of higher scores for children in higher SES schools across Grade and Repeater Status as revealed in a regression of Arithmetic Ability by SES [F(1,314) ¼ 38.81, P < .001, R2 ¼ .11)]. This reveals that SES impacts arithmetic ability and that children in higher SES Quintiles do better than their peers in less advantaged schools, as found in other studies (Goldin et al., 2014; Klibanoff et al., 2006; Odic et al., 2016; Valle-Lisboa et al., 2016; Wilkinson & Pickett, 2010; Zosh et al., 2018). Considering Repeater Status in Fig. 7, note that despite Repeaters being a year older than their Non-Repeater classmates they nevertheless performed below the level of the Non-Repeater children on the pretest of Arithmetic Ability as revealed by planned t-tests within each Grade (Second Graders: t(72.53) ¼ 3.47, P < .001; Third Graders t(49.68) ¼ 3.35, P < .01). This highlights the importance of considering Repeater Status as a factor influencing students’ performance.

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Cognitive Foundations for Improving Mathematical Learning

ANS and Arithmetic Ability Next, we considered the relationship between Arithmetic Ability and ANS Ability in our sample. There is a continuing debate concerning whether ANS abilities are related to school mathematical abilities (Bugden & Ansari, 2016; Clayton, Gilmore, & Inglis, 2015; DeWind & Brannon, 2012; Fazio, Bailey, Thompson, & Siegler, 2014; Gilmore et al., 2013; Odic et al., 2016; Xenidou-Dervou, Molenaar, Ansari, van der Schoot, & van Lieshout, 2017). We used the preintervention scores from all children to determine whether Arithmetic Ability is correlated with ANS Ability. The top plot in Fig. 8 shows the simple correlation ANS ability relates to arithmetic ability

Arithmetic ability

60

y = .38x + 9.1 R2 = .19 P < .001

40

20

0 0

20

40

60

ANS ability ANS ability relates to arithmetic ability Controlling for IQ, vocabulary, geometry, and age

Arithmetic residuals

40 y = .2x + 9 × 10–16 r 2 = .07 20

P < .001

0

–20 –20 SES Quintile

0 ANS residuals 1

2

3

20 4

40 5

FIG. 8 Correlation and partial correlation (controlling for IQ, Vocabulary, Geometry Ability, and Age) between ANS ability and arithmetic ability. The top figure shows the simple correlation between the measures; the bottom figure shows the partial correlation. Each symbol corresponds to a child, and the shapes indicate the child’s SES Quintile. Both regressions are significant and each SES Quintile contributes to these effects.

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between these measures while the bottom shows the partial correlation of Arithmetic Ability and ANS Ability controlling for IQ, Vocabulary, Geometry Ability, and Age. The colors correspond to the SES Quintiles of students’ schools. The first thing we notice is that the relationship is significantly positive in both cases (P < .001) suggesting a relationship between ANS and Arithmetic Ability, even when controlling for other factors (correlation: F(1,304)¼ 70.07, P < .001, R2 ¼ .19; partial correlation: F(1,304)¼ 21.23, P < .001, r2 ¼ .065). An additional age-normalized regression (not shown) between ANS Ability and Arithmetic Ability was also significant (this regression is one way of creating agenormalized standard scores for these tasks): F(1,316)¼ 67, P < .001, r2 ¼ .17. Second, we see that the colored dots are evenly spread throughout the trend, indicating that no single SES Quintile is driving the effect. Indeed, individual regressions were also performed for each SES Quintile separately and the regression slopes were all positive suggesting that all SES Quintiles contribute to this effect. Therefore contrary to some claims in the literature (Bugden & Ansari, 2016; Clayton et al., 2015; Gilmore et al., 2013; Xenidou-Dervou et al., 2017) and consistent with others (DeWind & Brannon, 2012; Fazio et al., 2014; Odic et al., 2016) our data show a link between ANS Ability and Arithmetic Ability (across SES Quintiles and controlling for many relevant measures).

Engagement With the Intervention Thus far we have seen noteworthy effects of SES and Repeater-Status on several measures in our sample. We also found that ANS Ability correlates with Arithmetic Ability while controlling for many factors. Before turning to consider how performance changed from pre- to posttest, we first report children’s engagement with the intervention games. Recall that children in the Game Intervention group (n ¼ 197) had access to three discrimination games on their tablets for 5 weeks and they were free to play them as much or as little as they liked while at home. Fig. 9 shows children’s total number of games played over the course of the 5 weeks. The total number of games played suggests that children liked the games and did volunteer to play them. Because each discrimination game lasted approximately 3 min per play, this intervention can be considered a brief amount of exposure to the intervention games over the course of the 5 weeks. Pre- to Postintervention Improvement We turn now to considering how performance changed from pre- to posttest. Fig. 10 shows the percentage change from pretest to posttest performance (viz., performance 5 weeks later), collapsing across SES Quintiles, Grade, and Repeater-Status, for both Game Intervention and BAU Control children on each of our measures: Arithmetic Ability, ANS Ability, Vocabulary, Geometry, IQ. We saw significant gains for all groups of children on all tasks as shown by planned t-tests for each bar in Fig. 10 (all Ps < .05 except

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Cognitive Foundations for Improving Mathematical Learning

Student engagement by game Mean (SE) number of plays T % of Tablets played N times

0.06

(T) Time

10 (0.6)

(N) Number

16 (1.8)

(A) Area

26 (2.4)

N

0.04

0.02

A

0.00 0

50

100

150

Number of plays

Control Intervention Control Intervention Control

ANS

Intervention

Vocabulary

Control

Geometry

Intervention

Arithmetic

FIG. 9 Distribution of number of plays per child per game over the course of the 5-week intervention for all children in the Game Intervention group. The graphs are density distributions and show the proportion of children for each number of plays. Most children played each game around 10 times while the more active players pulled the mean number of plays higher for some games.

IQ

Intervention Control 0.00

0.25

0.50 Percent change - pre to post

0.75

1.00

FIG. 10 Percent improvement on each assessment for both BAU Control and Game Intervention children. Both groups showed significant improvements from pre- to posttest on each of our measures (all P < .05 except Geometry, Intervention group P ¼ .09), with the greatest improvement seen in the math assessments of Arithmetic Ability and ANS ability.

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Geometry Ability in the Intervention group: P ¼ .09). It is also noteworthy that we saw the greatest gains for our two math assessments: Arithmetic Ability and ANS Ability. Considering possible differences between the gains seen for BAU Control children and those seen for Game Intervention children, Fig. 10 does not reveal noteworthy improvements of the Game Intervention children above and beyond the BAU Control children. This is perhaps because of the large overall improvements we saw in all children. We next look at improvement as a function of SES Quintile and Repeater-Status as these factors may modulate who does and who does not improve. We computed z-transformed change scores for both Arithmetic Ability and ANS Ability by taking each child’s change in number of problems answered correctly from pre- to posttest and dividing by the SD of that child’s classroom scores on the pretest: Postchild Prechild zdiff ¼ SDðPrechild Þ Recall that both our Arithmetic Ability assessment (Baterı´a III WoodcockMun˜oz) and ANS Ability assessment were timed assessments which measured the number of problems correctly solved during 3 min. Dividing the difference from pre- to posttest by the SD of the pretest scores for each child’s classroom is what makes this a z-normalized change score, and it has the consequence that each child’s change score is roughly normalized across our factors of interest. That is, a z-normalized change score from a Grade 2 child in the Game Intervention group of the 5th SES Quintile will be comparable to the z-normalized change score from a Grade 3 child in the BAU Control group of the 1st SES Quintile. In both cases, the z-normalized change score indicates how much the child improved relative to the variability in the pretest scores of their immediate peers. Arithmetic ability. In Fig. 11, the left side shows the z-normalized change scores in Arithmetic Ability for Non-Repeater children and the right side shows the z-normalized change scores for Repeater children. The first pattern to notice is that all of the mean change scores are above 0 (the level of no change), indicating that all groups of children improved from pretest to posttest. This duplicates the pattern of bars in Fig. 10. The change scores across groups seems to hover around a value of 1. This means that children, in general, improved about 1 SD above the mean of their classroom group from pretest to posttest. Next, on the left side we see that for no SES Quintile did the Game Intervention children improve significantly more than the BAU Control children, with the exception of trends in the 1st and 3rd SES Quintiles. Being conservative, we can say that this intervention did not show noteworthy improvement over the test-retest improvement in the matched BAU Control classrooms for children who did not repeat a grade.

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Cognitive Foundations for Improving Mathematical Learning

Change in arithmetic ability by quintile and condition in Repeaters and Non-Repeaters

Classroom-normalized arithmetic change score

Non-Repeaters

Repeaters

Condition

2

Control Intervention

Grade Second

1

Third

1

2

3

4

5 1 SES Quintile

2

3

4

5

FIG. 11 Change in arithmetic ability by quintile and condition (BAU control vs. game intervention) for Repeaters (right) and Non-Repeaters (left).

Turning now to children who were or had already repeated a grade (right side), we see two noteworthy standouts for z-normalized change scores: the 3rd SES Quintile Game Intervention group and the 4th SES Quintile Game Intervention group. Planned t-tests compared the Game Intervention children to the BAU Control children in each SES Quintile. The test for the 3rd SES Quintile Repeaters (Game Intervention, M ¼ 2.55, SD ¼ 0.600; BAU Control M ¼ 0.575, SD ¼ 0.588) revealed that the Game Intervention children showed improvement significantly greater than the BAU Control children [t(5.998) ¼ 4.693, P < .01]. The children in the 4th SES Quintile Game Intervention group are showing higher z-normalized change scores than most other groups; however, we did not have a child in the 4th SES Quintile BAU Control group who was a Repeater, so we cannot carry out the same t-test on this group. These groups suggest that the game intervention was more effective than simply test-retest improvement in at least some SES Quintiles for children who had or were repeating a grade and that there may be trends for lower SES children who did not repeat a grade (e.g., SES Quintiles 1 and perhaps 3), but these trends were insufficient in our sample to overcome the already large improvements seen in the BAU Control children and must be taken as merely suggestive. As one indication that these improvements may generalize—given greater training or more effective yoking of pre- and posttesting to minimize the large test–retest improvements we saw here in the BAU children—we looked at whether the groups who showed noteworthy gains were different from their peers (e.g., examining whether their gains were an epiphenomenal result of their having extremely low pretest scores, which made large gains easier to

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Change in ANS ability by quintile and condition in Repeaters and Non-Repeaters Non-Repeaters

Repeaters

Classroom-normalized ANS change score

10.0

7.5

Condition Control Intervention

5.0 Grade Second Third

2.5

0.0 1

2

3

4

5

1

2

3

4

5

SES Quintile

FIG. 12 Change in ANS ability by SES quintile and condition (BAU control vs. game intervention) for repeaters (right) and non-repeaters (left).

achieve). In both pretest ANS Ability and pretest Arithmetic Ability, children who improved more after the Game Intervention were no different than their peers—suggesting that these improvements did not merely result from aberrantly low pretest scores for these children. ANS ability. Next, in Fig. 12, we consider the z-normalized change scores for performance on the ANS Ability assessment. Here again, we computed the change in number of correct answers from pre- to posttest and divided by the SD of scores within each child’s classroom. In Fig. 12 we see these scores separated by intervention condition and Repeater Status. The first pattern to notice is that all of the mean change scores are above 0 (the level of no change), indicating that all groups of children improved from pretest to posttest. This again duplicates the pattern of bars in Fig. 10. The change scores across groups seem to hover around a value of 1.75. This means that children, in general, improved about 1.75 SDs above the mean of their classroom group from pretest to posttest. On the left side we see that Game Intervention children in SES Quintile 5 improved significantly more than their peers in the BAU Control group as revealed by a planned t-test [t(91.212) ¼ 4.553), P < .001]. However, children in SES Quintiles 3 and 4 showed the opposite trend with BAU Control children tending to show more improvement then the Game Intervention children [Q3: t(20.019) ¼ 1.890, P ¼ .07; Q4: t(35.900) ¼ 1.445, P ¼ .16]. Thus for children who were not repeating a grade, on the left side of Fig. 12, we cannot say that there was any systematic benefit of the Game Intervention above and beyond did the BAU Control children. All children improved on this assessment.

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Cognitive Foundations for Improving Mathematical Learning

Considering now the children who had or were repeating a grade, on the right side of Fig. 12, we see a trend where Game Intervention children improved more than the BAU Control children in SES Quintile 3 (p ¼ .25). However, none of these trends attained significance, perhaps due to the large variability from these small sample sizes.

Summary of Pre- to Postintervention Improvement Across all measures, there is noteworthy improvement from pretest to posttest for both the Game Intervention and BAU Control children, most likely due to test-retest improvement as well as the positive effects of having a special visitor come to the classroom and engage children in special tests—note that the tester for the posttest (J.L.), who was blind to the experimental condition for each classroom, was a particularly special visitor as he is male (atypical in education settings in Uruguay), an American, and a Spanish speaker with an American-Iberian accent. This may have contributed to better focus and overall performance on posttest versus pretest (which was administered by female Uruguayan graduate students (D.L. & D.F.). While this overall improvement made it difficult to see improvement in our Game Intervention children above and beyond the improvement in our BAU Control children, and in spite of the small sample sizes in some of these groups, we did see some significant results and trends suggesting that there may be a positive influence of the game-based intervention training. But for certain, the strongest results in the current sample concern the effects of SES and Repeater-Status on Arithmetic Ability, along with the differential links between IQ and these factors, and the relationship between Arithmetic Ability and ANS Ability in our sample. Continued work to tailor interventions to the community is necessary.

CONCLUSIONS AND FUTURE DIRECTIONS Researchers and educators have a shared interest in understanding the factors that influence students’ educational outcomes, and our work is consistent with previous findings that the socioeconomic status of a child’s community is related to their performance in the classroom. Even before our intervention started, students in lower SES schools scored lower on measures of math achievement compared to students at more advantaged schools. We also saw that lower SES students are more likely to repeat a grade compared to higher SES students, and that having a lower IQ has a disproportionately negative impact on the likelihood of repeating a grade for lower SES children. One reason for this may be that lower IQ students at higher SES schools are better supported, either within the classroom or at home, compared to similarly scoring students at lower SES schools. An important future direction may be investigating the specific ways that more affluent schools succeed in supporting students at risk for repeating a grade so that these methods

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can be implemented at lower SES schools. Certainly availability of resources has a role to play here, but interventions such as the one presented in this study suggest that there are possibilities for reducing the gap between highand low-achieving students using existing infrastructure. One method for enriching the educational experience of vulnerable students may be through technology. The Uruguayan educational system provides a unique opportunity to investigate this question, given the country’s investment in educational technology in schools that span the SES spectrum. While not conclusive at this stage, our results are consistent with the idea that the tablet can serve as an important vehicle for intervention and that it may be useful for improving math performance for students who are the most disadvantaged. What is more certain is that students seemed to show across-the-board improvement in a number of areas when special adults visited their classrooms and evaluated their performance. To the extent that lower SES students feel different societal expectations in terms of their academic potential, providing special attention to these students may prove an important tool in helping them reach the same levels of achievement as their more affluent peers. The design of the current study had some limitations which should be addressed in future interventions. A more fine-grained way to measure the impact of the game would have been to look at dosage effects; however, this stage of the project included no standardization of the amount of time that the students interacted with the intervention media. In part, this was by design; the project is a collaboration with the teachers who agreed to use the game in their classrooms. By giving them the freedom to implement the game in the ways they saw fit, we stood to learn from their feedback about how best to integrate the intervention game with the normal classroom curricula. Future deployments of this intervention may include lesson planning guides based on suggestions from teachers in the current study. While the effects of SES and Repeater Status are central to our interpretation of this study’s results, we were unable to balance our sample of students by age across SES Quintiles because classrooms in the study were included based on their willingness to participate. For the most part, only one grade level was represented within each SES Quintile, so it made comparisons across grade levels difficult given the large effect of Grade on our outcome measures (viz., Arithmetic Ability score). Similarly, we would have liked to have been able to examine in more detail the effect of repeating a grade, but the number of Repeater students was impossible to control in the sample. Future studies might focus on schools with high levels of repetition to examine the effects of the intervention in these groups. Having a more complete sample would allow us to identify which students are most helped by our intervention; however, this would depend on the availability of classrooms willing to participate in the study. This study’s main goal was practical: to create a tool for teachers to use in the classroom that could improve students’ math performance. However, these results also speak to important scientific questions about the relationship

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between underlying cognitive abilities and classroom math performance. One of the strongest results in this study was the evidence for a relationship between Arithmetic Ability and ANS Ability. We found that ANS Ability related to Arithmetic Ability across all SES Quintiles controlling for IQ, Vocabulary, Geometry Ability, and Age. The inclusion of these control measures was important because there is a continuing debate concerning whether ANS abilities are related to school mathematical abilities when controlled for other factors (Bugden & Ansari, 2016; DeWind & Brannon, 2012; Fazio et al., 2014; Gilmore et al., 2013; Odic et al., 2016; Xenidou-Dervou et al., 2017). Contrary to some claims in the literature (Bugden & Ansari, 2016; Clayton et al., 2015; Gilmore et al., 2013; Xenidou-Dervou et al., 2017) and consistent with others (Halberda et al., 2012; Odic et al., 2016) our data show a consistent link between ANS Ability and Arithmetic Ability across SES Quintiles. Considering the importance of intervention research, to the extent that magnitude training can improve formal math ability, this suggests a causal link between intuitive number sense and children’s formal understanding of math. Our data are consistent with other findings that have shown such a link (DeWind & Brannon, 2012; Odic et al., 2016; Odic, Libertus, et al., 2013), but further work is needed to explore more fine-grained questions about training type and specific formal skills. The intervention in this study trained children on three different magnitude tasks (area, time, and approximate number). A future study might look at the correlation between amount of student interaction with each of these games and amount of improvement in various achievement domains (e.g., addition/subtraction, geometry, time estimation). For example, if the partial correlation between ANS training and formal math ability were significant when accounting for approximate area training, this would suggest a privileged connection between the ANS system and symbolic math, as found by Lourenco et al. (2012) for adults. This would align with previous work that has shown a privileged relationship between ANS and symbolic math compared to symbolic math and time discrimination (Odic et al., 2016). It may also be the case that different magnitude training tasks are more strongly linked to performance in various outcome domains; for instance, we might observe privileged relationships between area training and formal geometry as well as between ANS training and formal arithmetic. These are important scientific questions about the foundations of human numerical understanding that could be addressed in future studies. Understanding underlying mechanisms is key to progress in many areas of science. When we succeed in applying this understanding to improving the lives of people, our work becomes even more thrilling. The current project reveals that there are children who will benefit from interventions to improve math ability, and that these improvements may vary as a function of IQ and SES. It also shows that there is a link between ANS Ability and Arithmetic Ability across SES Quintiles. Nevertheless, our work also highlights the

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challenges of developing an intervention that will bridge the gap between game-based improvements and gains in math understanding. Through our partnership with teachers and students, we aim to continue exploring how equal access to technology and purpose-built software can improve educational outcomes, and ultimately help to create a more equitable society.

REFERENCES ANEP [Administracio´n Nacional de Educacio´n Primaria]. (2012). Relevamiento de contexto sociocultural de escuelas de educacio´n primaria [Survey of primary school’s socio-cultural context]. Montevideo: ANEP. Obtenido de http://www.anep.edu.uy/anep/index.php/codicenpublicaciones/category/117-publicacionesdivision-de-investigacion-y-estadistica-educativa? download¼1132:relevamiento-de-contextosociocultural-2010. Bugden, S., & Ansari, D. (2016). Probing the nature of deficits in the ‘Approximate Number System’ in children with presistent developmental dyscalculia. Developmental Science, 19(5), 817–833. Clayton, S., Gilmore, C., & Inglis, M. (2015). Dot comparison stimuli are not all alike: the effect of different visual controls on ANS measurement. Acta Psychologica, 19(5), 177–184. DeWind, N. K., & Brannon, E. M. (2012). Malleability of the approximate number system: effects of feedback and training. Frontiers in Human Neuroscience, 6, 1–10. Dillon, M. R., Kannan, H., Dean, J. T., Spelke, E. S., & Duflo, E. (2017). Cognitive science in the field: a preschool intervention durably enhances intuitive but not formal mathematics. Science, 357(6346), 47–55. Fazio, L. K., Bailey, D. H., Thompson, C. A., & Siegler, R. S. (2014). Relations of different types of numerical magnitude representations to each other and to mathematics achievement. Journal of Experimental Child Psychology, 123, 53–72. Feigenson, L., Dehaene, S., & Spelke, L. (2004). Core systems of number. Trends in Cognitive Science, 8(7), 307–314. Gilmore, C., Attridge, N., Clayton, S., Cragg, L., Johnson, S., Marlow, N., et al. (2013). Individual differences in inhibitory control, not non-verbal number acuity, correlate with mathematics achievement. PLoS ONE, 8(6). Goldin, A. P., Hermida, M. J., Shalom, D. E., Costa, M. E., Lopez-Roesenfeld, M., Segretin, M. S., et al. (2014). Far transfer to language and math of a short software-based gaming intervention. Proceedings of the National Academy of Sciences of the United States of America, 111(17), 6443–6448. Halberda, J., & Feigenson, L. (2008). Developmental change in the acuity of the “number sense”: the approximate number system in 3-, 4-, 5-, and 6-year-olds and adults. Developmental Psychology, 44(5), 1457–1465. Halberda, J., Ly, R., Wilmer, J. B., Naiman, D. Q., & Germine, L. (2012). Number sense across the lifespan as revealed by a massive Internet-based sample. Proceedings of the National Academy of Sciences of the United States of America, 109(28), 11116–11120. Hart, B., & Risley, T. R. (1975). Incidental teaching of language in the preschool. Journal of Applied Behavior Analysis, 8(4), 411–420. Hoff, E. (2003). The specificity of environmental influence: socioeconomic status affects early vocabulary development via maternal speech. Child Development, 74(5), 1368–1378. Hyde, D. C., Khanum, S., & Spelke, E. S. (2014). Brief non-symbolic, approximate number practice enhances exact symbolic arithmetic in children. Cognition, 131(1), 92–107.

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INEEd. (2017). Informe sobre el estado de la educacio´n en Uruguay: 2015–2016. Montevideo: INEEd. Izard, V., Sann, C., Spelke, E., & Streri, A. (2009). Newborn infants perceive abstract numbers. Proceedings of the National Academy of Sciences of the United States of America, 106(25), 10382–10385. Klibanoff, R. S., Levine, S. C., Huttenlocher, J., Vasilyeva, M., & Hedges, L. V. (2006). Preschool children’s mathematical knowledge: the effect of teacher “math talk.” Developmental Psychology, 42(1), 59–69. Larson, K., Russ, S. A., Nelson, B. B., Olson, L. M., & Halfon, N. (2015). Cognitive ability at kindergarten entry and socioeconomic status. Pediatrics, 135(2), 440–448. Levine, S. C., Vasilyeva, M., Lourenco, S. F., Newcombe, N. S., & Huttenlocher, J. (2005). Socioeconomic status modifies the sex difference in spatial skill. Psychological Science, 16(11), 841–845. Lourenco, S. F., Bonny, J. W., Fernandez, E. P., & Rao, S. (2012). Nonsymbolic number and cumulative area representations contribute shared and unique variance to symbolic math competence. Proceedings of the National Academy of Sciences of the United States of America, 109(46), 18737–18742. ´ ., Maiche, A., Langfus, J., De Leon, D., Fitipalde, D., Sa´nchez, I., & Halberda, J. Mailhos, A (2018). Towards a paper-and-pencil ANS assessment tool. (in prep.). Odic, D., Lisboa, J. V., Eisinger, R., Gonzalez, M. O., Maiche, A., & Halberda, J. (2016). Approximate number and approximate time discrimination each correlate with school math abilities in young children. Acta Psychologica, 163, 17–26. Odic, D., Libertus, M., Feigenson, L., & Halberda, J. (2013). Developmental change in the acuity of approximate number and area representations. Developmental Psychology, 49(6), 1103. Odic, D., Pietroski, P., Hunter, T., & Lidz, J. (2013). Young children’s understanding of “more” and discrimination of number and surface area. Journal of Experimental Psychology: Learning, Memory, and Cognition, 39(2), 451–461. Pilli, O., & Aksu, M. (2013). The effects of computer-assisted instruction on the achievement, attitudes and retention of fourth grade mathematics students in North Cyprus. Computers & Education, 62, 62–71. Raven, J., Raven, J. C., & Court, J. H. (1998). Manual for Raven’s progressive matrices and vocabulary scales. San Antonio, TX: Pearson. Sirin, S. R. (2005). Socioeconomic status and academic achievement: a meta-analytic review of research. Review of Educational Research, 75(3), 417–453. Sokolowski, H. M., Fias, W., Ononye, C. B., & Ansari, D. (2017). Are numbers grounded in a general magnitude processing system? A functional neuroimaging meta-analysis. Neuropsychologia, 105, 50–69. Trucano, M. (2011). What’s next for plan ceibal in Uruguay? EduTech: A World Bank Blog on ICT use in Education. 10 de June de. Obtenido de: http://blogs.worldbank.org/edutech/planceibal2. Tucker-Drob, E. M., & Bates, T. C. (2016). Large cross-national differences in gene x socioeconomic status interaction on intelligence. Psychological Science, 27(2), 138–149. Turkheimer, E., Haley, A., Waldron, M., D’Onofrio, B., & Gottesman, I. I. (2003). Socioeconomic status modifies heritability of IQ in young children. Psychological Science, 14(6), 623–628. ´ . E., Mailhos, A ´ ., Luzardo, M., Halberda, J., & Maiche, A. (2016). Valle-Lisboa, J., Cabana, A Cognitive abilities that mediate SES’s effect on elementary mathematics learning: the Uruguayan tablet-based intervention. Prospects: Comparative Journal of Curriculum, Learning, and Assessment, 46(2), 301–315.

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Verdine, B. N., Irwin, C. M., Golinkoff, R. M., & Hirsh-Pasek, K. (2014). Contributions of executive function and spatial skills to preschool mathematics achievement. Journal of Experimental Child Psychology, 126, 37–51. Walsh, V. (2003). A theory of magnitude: common cortical metrics of time, space and quantity. Trends in Cognitive Science, 7(11), 483–488. Wilkinson, R., & Pickett, K. (2010). The spirit level: Why equality is better for everyone. London: Penguin. Woodcock, R. W., McGrew, K. S., & Mather, N. (2001/2007). Woodcock-Johnson III tests of cognitive abilities. Rolling Meadows, IL: Riverside. World Computer Exchange. (2016). WCE FY16 annual report. Hull, MA: World Computer Exchange. Xenidou-Dervou, I., Molenaar, D., Ansari, D., van der Schoot, E., & van Lieshout, C. (2017). Nonsymbolic and symbolic magnitude comparison skills as longitudinal predictors of mathematical achievement. Learning and Instruction, 50, 1–13. Zhang, M., Trussell, R. P., Gallegos, B., & Asam, R. R. (2015). Using math apps for improving student learning: an exploratory study in an inclusive fourth grade classroom. TechTrends, 59(2), 32–40. Zosh, J., Verdine, B., Halberda, J., Hirsh-Pasek, K., & Golinkoff, R. (2018). Preschoolers’ approximate number system varies by socio-economic status. (in prep.).

Chapter 3

Role of Play and Games in Building Children’s Foundational Numerical Knowledge Geetha B. Ramani, Emily N. Daubert and Nicole R. Scalise Department of Human Development and Quantitative Methodology, University of Maryland, College Park, MD, United States

INTRODUCTION Adults’ mathematical knowledge is important for daily functioning, career advancement, and life success (National Research Council, 2001). Laying the foundation for learning this vital knowledge begins early. During early childhood, children acquire a range of numeracy skills that are critical for later mathematical achievement. For example, children’s numerical understanding from the start of kindergarten through the middle of 1st grade is strongly related to mathematics achievement at the end of 1st grade through 3rd grade (Geary et al., 2018; Jordan, Kaplan, Locuniak, & Ramineni, 2007). In a large longitudinal study, children’s numerical knowledge at age 4 predicted their mathematical knowledge at age 15, even after controlling for various factors, such as IQ, socioeconomic stats (SES), and reading comprehension abilities (Watts, Duncan, Siegler, & Davis-Kean, 2014). Moreover, children’s numerical knowledge at age 7 was related to their SES at age 42, again after controlling for factors such as IQ, childhood SES, academic motivation, and reading achievement (Ritchie & Bates, 2013). Given the importance of early numerical knowledge for later achievement, a strong emphasis has been placed on ensuring high-quality mathematical instruction in early childhood classrooms. A key goal of mathematics instruction in these early childhood classrooms is the use of developmentally appropriate practices so that mathematics is purposeful and meaningful for young children (Baroody, 2000). One way to achieve this goal is through play and games. While many mathematics skills need to be explicitly taught, Mathematical Cognition and Learning, Vol. 5. https://doi.org/10.1016/B978-0-12-815952-1.00003-7 Copyright © 2019 Elsevier Inc. All rights reserved.

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playful activities and games in early education curricula can supplement this instruction by providing children with opportunities to practice their early mathematics skills in engaging and meaningful ways. In this chapter, we will argue for the benefits of using age-appropriate play and games in early childhood classrooms to promote children’s numerical knowledge. We will focus on one area of numerical knowledge, specifically numerical magnitude knowledge, by presenting theory and research that underscores the importance of this area In the following section, we present specific examples of games that have been found to promote early numerical magnitude knowledge. Our focus is on both traditional games as well as math-related computer and tablet games to improve numerical knowledge in early childhood. Overall, we will argue for the importance of focusing on theoretically driven design of games to promote young children’s mathematical knowledge.

UNDERSTANDING OF NUMERICAL MAGNITUDES Early childhood is critical for establishing the foundational numerical knowledge that will serve as the basis for children’s future mathematics achievement. One key aspect of early mathematical knowledge that has roots in infancy is children’s understanding of numerical magnitudes or the relative quantities represented by numbers in their various forms. Numerical magnitude knowledge allows us to determine which of two bushes has more berries and to know whether $10 or $25 per hour is a higher payment for our efforts. Numerical magnitude information is also embedded in much of mathematics. For example, understanding numerical magnitudes can help students recognize implausible arithmetic solutions, such as why 10 plus 2 is unlikely to be equal to 102. Individual differences in numerical magnitude knowledge, assessed with tasks such as comparing sets of objects, identifying which of two numerals is more, and estimating the position of a number on a bounded number line, predict mathematics achievement among children and adults (Schneider et al., 2016). People’s numerical knowledge grows throughout the lifespan, typically in conjunction with formal education (Siegler, 2016). The present section reviews an integrated theory of numerical magnitude development and highlights the characteristics of numerical magnitude knowledge in early childhood.

Integrated Theory of Numerical Development From infancy through adulthood, humans are aware of and can discriminate between numerical magnitudes. The integrated theory of numerical development asserts that numerical knowledge across the lifespan can be characterized as an increasing understanding of numerical magnitude information represented by the mental number line (Siegler, 2016; Siegler, Thompson, & Schneider, 2011). The mental number line is a spatial-numerical association shared by humans and some animals, with most members of Western cultures

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representing number in a horizontal line moving from left (smaller quantities) to right (larger quantities; Dehaene, 2011). The theory outlines four conceptually distinct but temporally overlapping trends in magnitude understanding, including discriminating between nonsymbolic sets of objects, linking nonsymbolic quantities to small whole numbers, expanding to magnitudes of larger whole numbers, and then expanding to magnitudes of negative and rational numbers (Siegler, 2016). While each trend centers on a different type of magnitude knowledge, magnitude knowledge is typically assessed with comparison or estimation tasks across trends (Fig. 1). Magnitude comparison tasks involve comparing two or more quantities to determine which is the most, using nonsymbolic quantities (e.g., □□□ vs. □□□□□□) or symbolic numbers (e.g., 3 vs. 5). Depending on the age and skill level of participants, magnitude knowledge is then operationalized as accuracy across trials, average reaction time, or the most difficult ratio that a participant can consistently discriminate. Magnitude estimation tasks involve placing a nonsymbolic quantity or symbolic number in the correct position on a number line with labeled end points.

FIG. 1 Magnitude knowledge tasks. This figure illustrates commonly used magnitude knowledge tasks for each of the four types of numerical magnitude understanding (A–D), in magnitude comparison formats (left side) and number line estimation formats (right side).

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Consistent with the characteristics of the mental number line, number line estimation tasks use a horizontal line labeled with a smaller quantity on the left endpoint and a larger quantity on the right endpoint. Participants’ magnitude knowledge on these types of magnitude estimation tasks is operationalized as the discrepancy between where they positioned the target number and its actual location (percent absolute error), or the degree to which their estimates follow a linear distribution (such that the distance between estimates of 2 and 3 is equivalently spaced to the distance between estimates of 8 and 9). The first trend of numerical magnitude development involves increasingly precise representations of nonsymbolic magnitude. From birth, humans are sensitive to large differences in nonsymbolic numerical magnitude (e.g., 8 objects vs. 16 objects), often attributed to the evolved approximate number sense (ANS; Feigenson, Dehaene, & Spelke, 2004; Izard, Sann, Spelke, & Streri, 2009). As early as the first few days of life, infants possess the ability to discriminate between visual displays of one and three objects (Antell & Keating, 1983; Starkey & Cooper, 1980). However, the ability to discriminate between sets of objects depends on the ratio of the number of objects in each of the sets. Izard et al. (2009) found that newborns showed a behavioral preference (i.e., longer looking times) when a number of objects displayed that matched the number of sounds played simultaneously, as compared to trials where the number of objects and sounds did not match. The newborns showed this preference for ratios of 1:3 (e.g., 4 objects vs. 12 sounds) but did not differ in their looking time behavior for more difficult ratios such as 1:2 (e.g., 4 objects vs. 8 sounds), demonstrating ratio-dependent processing that is found in both children and adults. By the time infants are 6 months old, they can discriminate between sets that differ for number, space, and duration with 2:1 ratios, and by 9 months, infants can discriminate between sets with 3:2 ratios (Brannon, Lutz, & Cordes, 2006; Cordes & Brannon, 2008; Lipton & Spelke, 2003; vanMarle & Wynn, 2006; Xu & Spelke, 2000). The developmental trend continues, with 3-year-olds being able to discriminate 4:3 ratios, 6-year-olds able to discriminate 6:5 ratios, and adults able to discriminate 8:7 ratios (Bonny & Lourenco, 2013; Halberda & Feigenson, 2008; Holloway & Ansari, 2008). Studies of children and adults that employ a comparison task (Fig. 1, set a) often characterize their ANS acuity with a Weber fraction, representing the most difficult ratio of stimuli that an individual can accurately discriminate (Halberda & Feigenson, 2008). Individual differences in nonsymbolic magnitude knowledge, measured by Weber fractions or overall task accuracy, predict concurrent and later mathematics achievement (e.g., Batchelor, Inglis, & Gilmore, 2015; Bonny & Lourenco, 2013; Halberda, Ly, Wilmer, Naiman, & Germine, 2012; Libertus, Feigenson, & Halberda, 2011). Although the relation between nonsymbolic magnitude and math achievement is highly contested, several meta-analyses suggest there is a significant, albeit small, average effect

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across childhood and adulthood (r 0.20; Chen & Li, 2014; Fazio, Bailey, Thompson, & Siegler, 2014; Schneider et al., 2016). The second trend of numerical magnitude development is the process of linking nonsymbolic representations to symbolic representations of small whole numbers (Siegler, 2016). Children begin to recognize and attach meaning to symbolic number words (e.g., one, two, three) and written numerical symbols (e.g., 1, 2, 3) in early childhood. Learning number words is central to young children’s understanding of the counting system (Wynn, 1992). Children begin to use number words to recite the count sequence between the ages of 2 and 3 years (Gelman & Gallistel, 1978), although they may not initially adhere to the standard sequence (i.e., using the words “two, four, five” to count three items). Beginning at age 2 or 3 years, children require about a year of experience counting before they begin to relate specific number words to their corresponding numerosities (Wynn, 1992). By the time they are 3 and 4 years old, children typically know the correct count sequence for numbers one through ten (Fuson, 1988; Siegler & Robinson, 1982). By age five, most children have mastered the main counting principles. For example, children’s understanding of cardinality improves, which link a symbolic referent (e.g., number word) to a nonsymbolic quantity (e.g., tokens). However, children may make some errors such as believing items must be counted consecutively from one contiguous item to the next (adjacency). Despite these errors, children are highly proficient counters by the time they reach kindergarten (Geary, 2006). Although these counting skills are critical for numerical magnitude understanding, there is also evidence that they are not sufficient for learning about the magnitudes of numbers. Specifically, children can count in a number range nearly a year before they acquire an understanding of the magnitudes of those numbers (Le Le Corre & Carey, 2007). Some theorists have suggested children’s symbolic magnitude representations, such as written numerals and number words, are mapped directly onto their nonsymbolic magnitude representations (e.g., Dehaene, 2001). Indeed, both show ratio-dependent performance in magnitude comparison tasks and overlapping regions of neural activity (Nieder & Dehaene, 2009; Reynvoet & Sasanguie, 2016). However, the integrated theory of numerical development aligns with an alternative theoretical perspective that suggests children may learn the meaning of small numbers (up to 10) by associating them with nonsymbolic representations, including their own fingers (Siegler, 2016; Sullivan & Barner, 2014). In other words, having something external to represent quantity, including fingers, helps children to learning the meaning of numbers. Once children have linked small symbolic numbers to nonsymbolic representations, they may then abstract to gain meaning of larger symbolic numbers that could not be precisely represented by approximate nonsymbolic representations (e.g., 100 and 101). Variability in preschool and kindergarten children’s symbolic magnitude knowledge for small numbers (1–10) assessed

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through symbolic magnitude comparison and number line estimation tasks predicts their math achievement in later elementary school (Booth & Siegler, 2006; Kolkman, Kroesbergen, & Leseman, 2013; Sasanguie, De Smedt, Defever, & Reynvoet, 2012). After acquiring an understanding of small symbolic numbers, the third stage in magnitude development is an understanding of larger symbolic numbers. Children’s initial performance on 0–10 number line estimation tasks follows a logarithmic pattern, where smaller numbers are overly spread across the left side of the number line and larger numbers are condensed near the right side of the number line (Siegler, 2016). By ages 5 and 6 on 0–10 number line estimation tasks, children can accurately produce a linear distribution of their estimates, where estimated magnitudes increase linearly with actual magnitude. However, it is this age range children show the same logarithmic distribution on 0–100 number line estimation tasks (Berteletti, Lucangeli, Piazza, Dehaene, & Zorzi, 2010; Kim & Opfer, 2017; Laski & Siegler, 2007). The fourth and final stage of numerical magnitude development is an understanding of rational numbers, including decimals, fractions, ratios, and rates (Siegler, 2016). Although rational number knowledge predicts mathematics achievement and is key to more advanced mathematics and scientific work, many children and adults have a limited understanding of rational magnitudes (Siegler, 2016). Several principles of rational numbers contrast starkly with principles of whole numbers: they do not have singular predecessors or successors, they can be represented by infinitely many symbols (e.g., 0.5 ¼ ½ ¼ 4/8), and may decrease with multiplication or increase with division (see also Vamvakoussi, this volume). These departures from the principles of whole numbers are likely part of the reason why students struggle to develop an understanding of rational number magnitudes. There is some evidence that whole number magnitude understanding relates to later knowledge of fraction magnitudes (Bailey, Siegler, & Geary, 2014). In addition, understanding the relative magnitude of rational numbers may be what allows students to be successful in arithmetic with rational numbers; for example, students who practiced comparing and ordering fractions of varying magnitudes saw greater improvement in their conceptual understanding and arithmetic performance than students who practiced creating part-whole representations of fractions (Fuchs et al., 2013). Thus as with nonsymbolic and symbolic whole numbers, a deeper understanding of the magnitude of rational numbers may underlie numerical development and aid in mathematics performance broadly. In sum, the integrated theory of numerical development describes a progressive broadening of numerical magnitude understanding from infancy to adulthood that is in turn related to math achievement. Each of the four trends of developing magnitude knowledge is a candidate for interventions to promote mathematics learning, although different types of magnitude are more and less relevant for certain age groups.

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Numerical Magnitude Understanding in Early Childhood As the main focus of this chapter is on early childhood, two of the four trends in the integrated theory of numerical development framework are most relevant to this period. In preschool, children are refining their nonsymbolic magnitude knowledge, able to reliably discriminate between sets of objects at a 3:4 ratio around 3 years old and ratios of 5:6 by 6 years old (Halberda & Feigenson, 2008). Children are also mastering the verbal count sequence, recognizing written Arabic numerals, and learning to connect symbolic number representations like number words and numerals to quantities of nonsymbolic objects. On the basis of the integrated theory of numerical development, nonsymbolic magnitude representations do not causally lead to all symbolic number knowledge; however, young children may be attaching meaning to their newly acquiring number symbols by associating them with known nonsymbolic representations (Siegler, 2016; Sullivan & Barner, 2014; vanMarle et al., 2018). This theoretical orientation implies that refining the nonsymbolic magnitude representation system in preschool is perhaps less critical for later math achievement than helping children link their nonsymbolic and symbolic number representations. Indeed, preschoolers’ performance on cardinality tasks strongly predicts later math achievement (Geary & vanMarle, 2016). Fluency with small symbolic numerical magnitudes may support the development of simple arithmetic skills, as well as later fluency with larger numerical whole and rational number. Young children demonstrate arithmetic understanding on nonverbal addition tasks as early as 2 years of age (Starkey, 1992). By the time they are three, many children can solve simple nonsymbolic problems of small numbers (2 + 1), and 4-year-olds can solve addition as well as subtraction problems (3–1; Vilette, 2002). Symbolic numerical magnitude knowledge can help children with their symbolic arithmetic performance by signaling implausible answers (e.g., that the sum of 2 + 4 could not be less than 4). Thus by the time children reach 4 to 5 years of age, they are using a range of numerical knowledge skills including counting, numeral identification, and numerical magnitude knowledge for more advanced mathematics like arithmetic. The central reason to focus on the development of numerical knowledge in early childhood period is because of the evidence linking these early quantitative skills to later mathematics achievement. For example, 6-year-olds’ understanding of symbolic number magnitude predicted their fraction magnitudes and fractions arithmetic—traditionally difficult mathematical concepts—at age 13 (Bailey et al., 2014). These relations held even after controlling for various factors, such as children’s executive function, IQ, race, gender, and SES. Using longitudinal data, multiple studies have found similar relations between early magnitude understanding and later fraction magnitude and arithmetic knowledge (Hecht, Close, & Santisi, 2003; Hecht & Vagi, 2010; Jordan et al., 2013; Vukovic et al., 2014).

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Strikingly, children’s accuracy and linearity of estimates on the number line estimation task is one of the strongest predictors of concurrent and future mathematical achievement (Booth & Siegler, 2006; Geary, 2011; Geary, Hoard, Nugent, & Byrd-Craven, 2008; Laski & Siegler, 2007; Sasanguie et al., 2012; Sella, Tressoldi, Lucangeli, & Zorzi, 2016; Siegler & Booth, 2004) and arithmetic performance (Ashcraft & Moore, 2012; Gunderson, Ramirez, Beilock, & Levine, 2012; LeFevre et al., 2013). These relations are present even after statistically controlling for a range of related variables, including IQ, working memory, nonsymbolic numerical knowledge, reading achievement, race, ethnicity, and parental income (Bailey et al., 2014; Booth & Siegler, 2006, 2008; Cowan & Powell, 2014; Fazio et al., 2014; Geary et al., 2008). Overall, these studies provide strong empirical support for promoting mathematical knowledge in young children, specifically numerical magnitude knowledge.

PLAY AND GAMES IN MATHEMATICS DEVELOPMENT Numerous theoretical perspectives argue for using play and games as developmentally appropriate methods to build young children’s mathematical development. This section will broadly discuss some of these perspectives, as well as the advantages of using play and games, specifically for mathematics learning. Classic developmental theories each include arguments for the benefits of play and games for children’s cognitive growth. For example, Piaget (1962) argued that play can provide children with opportunities to utilize and practice their existing skills. It is through play and interactions with concrete materials in their environment that children can build their knowledge. From a sociocultural perspective, Vygotsky (1976, 1978) also emphasized the value of play for cognitive growth, suggesting that play provides children with experience using cultural tools, such as language and symbols, creating and following specific rules, and moving from use of just concrete objects to thinking in more abstract and symbolic ways. These experiences can promote growth in children’s cognitive abilities, such as attention, memory, and problem-solving. Combining Piaget and Vygotsky’s theories can help to provide balanced and developmentally appropriate ways for teaching mathematics, if done judiciously (Fuson, 2009). Concrete materials can be used to help children make meanings of cultural symbols, such as numbers. For example, using number lines in classrooms as a tool for children can provide a visual representation of the order and magnitudes of numbers. Then using these materials in playful contexts could help children learn about numbers in developmentally appropriate ways. Contemporary theorists and researchers have also discussed the importance of play, specifically in the area of mathematics. As previously discussed, young children possess a range of numerical knowledge prior to

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entering school. This early knowledge has been theorized to develop through early playful experiences that can help build a foundation for later, more complex mathematics (Ginsburg, 2006). The process through which children build mathematical knowledge through everyday experiences is termed “mathematization” (Sarama & Clements, 2009). This can occur through engaging in mathematical-related behavior during play. For example, when playing with collections of items, such as animal figures, children can sort them by their common characteristics, and then realize that the collections provide a way to compare the quantities of the groups, thereby using mathematical reasoning. A specific type of play that can be useful for promoting the development of children’s mathematical knowledge is games, which provide engaging, enjoyable, and structured experiences while helping children to reach targeted learning goals. Specifically, games are socially interactive and capitalize on children’s interest (Hassinger-Das et al., 2017). Games provide an ideal context for playful learning, which involves learning approaches where children are actively involved, engaged with meaningful materials, and interacting with others (HirshPasek, Golinkoff, & Eyer, 2003; Resnick, 1999). Specifically, for math, there are numerous advantages for using games as teaching tools. According to Davies (1995), a few of these advantages are that games provide meaningful contexts to apply and use mathematical skills. Games also allow children with varying abilities in mathematics to interact and potentially learn from one another. Playing games can also increase children’s motivation and build a positive attitude toward mathematics. Thus there are numerous advantages for using play and games as a means for promoting children’s numerical knowledge.

Improving Children’s Numerical Magnitude Knowledge Through Games and Play There is strong theoretical grounding for using play and games for promoting children’s development in the area of mathematics. As an educational tool, however, it is important for games to match the educational objectives and areas that are being targeted in mathematics lessons (Aldridge & Badham, 1993). In this section, we describe empirical work on play and games that have been found to improve children’s early numerical magnitude knowledge. We will argue for the importance of using games that align with this specific area of numerical knowledge.

Playing Traditional Games to Promote Numerical Knowledge Board games have been used for promoting young children’s numerical magnitude knowledge. Siegler and Booth (2004) hypothesized that board games designed with linearly arranged, consecutively numbered spaces of equal sizes can provide multiple cues to both the order of numbers and the numbers’

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magnitudes. Theoretically, such a game provides a physical representation of the hypothesized mental number line upon which numerical magnitude is built (Dehaene, 2011). One example of such a game is Chutes and Ladders. During their play, children experience the visual cues to the order of numbers, auditory cues when saying the numbers on the spaces, and kinesthetic cues when moving the game pieces. Such redundant, multisensory cues are helpful for learning numerical relations even in infancy (Jordan, Suanda, & Brannon, 2008). Building upon this theoretical analysis, a series of studies tested the benefits of playing simple, linear number board games. The studies focused primarily on preschool children from lower-income backgrounds because the numerical understanding of children from low-income households typically lags behind that of children from middle- and upper-income households by up to 8 months (Ramani & Siegler, 2011; Starkey, Klein, & Wakeley, 2004). Therefore young children from lower-income backgrounds may especially benefit from targeted magnitude interventions. In the studies, preschool children from Head Start classrooms played one-on-one with an experimenter either a linear numerical board game with squares numbered from 1 to 10 or an identical version of the game except the squares varied in colors (Fig. 2). Children in these studies played one of the two games for four 15–20 min sessions. Children who played the number board game considerably improved their performance on a numerical line estimation task of the same number range, whereas children who played the color version of the game did not (Siegler & Ramani, 2008). In a subsequent study, children who played the linear numerical board game improved their performance on the number line estimation task, as well as their counting skills from 1 to 10, their numeral identification of the numerals 1–10, and their comparison of the magnitudes of numerals. Further, these improvements were stable over a 9-week period of not having played the board game (Ramani & Siegler, 2008). Simple linear board games can also be scaled up for use in early childhood classrooms. After a brief training, assistant teachers from the Head Start classrooms supervised a small group of children from the classroom while they played the game. Children who played the number board game improved their numerical knowledge. The assistant teachers also adapted the amount and type of feedback they provided during the game. For example, they used less instruction and modeling on how to complete the turns in the last game playing session than in the first game playing session (Ramani, Siegler, & Hitti, 2012). Thus playing the linear number board game can have broad and stable benefits for children’s numerical knowledge in classrooms. The design of games is important when considering its potential benefits for young children’s numerical knowledge. For example, according to the representational mapping hypothesis (Siegler & Ramani, 2009), the more transparent the mapping between materials and the desired internal representation,

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FIG. 2 Linear number board game and linear color board game (Ramani & Siegler, 2008; Siegler & Ramani, 2008).

the greater the learning of these representations. To test this hypothesis, Siegler and Ramani assigned children to play either the linear board game or a circular board game with numbers in the same 1–10 range. It was predicted that the linear game would be more effective than the circular one at increasing numerical magnitude knowledge because the linear board is easier to translate into the mental number line. As predicted, playing the linear board game for roughly an hour increased low-income preschoolers’ numerical magnitude knowledge, whereas playing the circular board game did not. However, playing both games improved children’s numeral identification skills because of the similar practice both groups received in this area. A third group of children who had practice with counting and numeral naming tasks, but not in a game context, did not show improvements on any measure of numerical knowledge. The latter results suggest even though the children engaged with these number activities for the same length of time as children who played the board game, there was no change in their numerical knowledge suggesting the game context was

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important. In further support of the importance of the design of the linear game, preschoolers who played the linear board game learned more from subsequent training on simple arithmetic problems than children in the other two conditions. These results suggest that playing the linear board game helped to produce a linear representation of numbers, which makes it easier to solve the arithmetic problems. Several other empirical studies on board games have found similar improvements in children’s numerical knowledge. For example, the benefits of playing number board games on children’s numerical knowledge have been replicated with preschool-age children in Scotland (Whyte & Bull, 2008) and Sweden (Elofsson, Gustafson, Samuelsson, & Tr€aff, 2016). Further, board games have been used with Kindergarten children. Specifically, they played a board game with the numbers 0–100 arranged in a 10 x 10 matrix for four sessions with an experimenter (Laski & Siegler, 2014). It was hypothesized that having children count the numbers in the spaces would lead to greater encoding of the numerals. They found that only children who played using the counting-on procedures (e.g., being on the space numbered 5 and counting-on by saying 6, 7, 8 while moving the game piece) improved their performance on a number line estimation task, whereas others who played the same game by counting from one while moving their game piece did not show any significant improvements. Overall, these studies suggest that playing board games can promote children’s numerical knowledge and specific aspects of the game design can influence these benefits. In addition to board games, there are other games and playful materials that can provide children with numerical information and practice to promote their numerical magnitude knowledge. For example, traditional numerical playing cards provide simultaneous exposure to symbolic and nonsymbolic magnitude representations, in the form of Arabic numerals and sets of objects (e.g., hearts, diamonds). As shown through board games, exposing children to materials with multiple redundant cues to numbers and their magnitudes can help build their numerical knowledge (Ramani & Siegler, 2008; Siegler & Booth, 2004). Providing children with both the symbolic and nonsymbolic magnitude representations will allow them to use either type of information when playing with the cards. To capitalize on the dual representation of numerical information on the design of cards, Scalise, Daubert, and Ramani (2017) randomly assigned preschool-age children from Head Start classrooms to play one of two games with numerical cards that ranged from 1 to 10. Specifically, children played either a numerical magnitude comparison (commonly known as “War”) or a numerical matching card game (commonly known as “Memory”). The children played for four 15–20 min sessions over a 3-week period. The different games using the same cards allowed for testing how specific card games that vary in the type and amount of numerical knowledge they require and use to

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complete the game can influence children’s learning. As predicted, the magnitude comparison card game led to significant improvements in children’s symbolic magnitude comparison skills, whereas both games led to improvements in children’s counting and numeral identification skills. These results suggest a brief intervention using games with materials that combine nonsymbolic and symbolic magnitude representations can support low-income preschoolers’ numerical magnitude knowledge. Similar to cards, other number-based games can include materials with nonsymbolic and symbolic numerical information. For example, Brankaer, Ghesquie`re, and De Smedt (2015) randomly assigned 8-year-olds with mild intellectual disabilities to play one of two games with dominoes for eight 15-min sessions. One game included dominoes with the Arabic numeral on one side and the corresponding number in dots on the other side, whereas the other game included dominoes with colors and pictures. Both games involved playing a matching game with the dominoes. They found that although there were no differences in accuracy, children who played the numerical domino game became faster on a symbolic magnitude comparison task compared to children who played the color domino game. Together, these studies show that including multiple representations of the magnitude of numbers in game materials are important features of promoting young children’s numerical knowledge. Materials that use redundant cues, such as the spatial information in linear number board games, as well as the symbols and objects in cards and dominoes can help support the development of numerical magnitude knowledge. Further, playful, interactive settings where children take an active role in the game can be an important context for their learning.

Computer and Tablet Games In addition to board and card games, there are numerous technology-based games that have been shown to be effective at improving young children’s numerical knowledge. Preschool- and kindergarten-age children from lowerincome backgrounds played an adaptive computer, number-based game that involved performing numerical magnitude comparisons using dots, numbers, or arithmetic problems. Playing a number-based game compared to playing a reading game improved the children’s numerical magnitude knowledge (Wilson, Dehaene, Dubois, & Fayol, 2009). A more recent study has replicated these results with similar-age children from middle-income backgrounds (Sella et al., 2016). In another study, Toll and Van Luit (2013) randomly assigned kindergarten-age children to play one of two adaptive computer games for eight 25-min sessions; one game focused on comparing the quantities of numbers, and the other focused on learning procedural and conceptual counting knowledge. Compared to a control group, children

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in both numerical conditions improved their counting and arithmetic skills after the intervention and 6 months later. Thus playing well-designed and validated computer-based games can benefit young children’s numerical knowledge. Recently, there has been a rapid growth in tablet computer technology. Given the portability and lower cost of tablet computers compared to desktops computers, tablet computers have become more accessible for use in schools and homes. Further, the touchscreen technology of tablets makes their use extremely intuitive, requiring little instruction so they can be used by a wide audience (Dube & McEwen, 2015). A recent study tested the benefits of playing tablet computers for improving kindergarten children’s numerical knowledge. The games targeted improving either children’s domain-specific skills, their numerical magnitude knowledge, or their underlying domain-general skills, specifically working memory. Geary and Hoard’s (2005) framework of mathematical development suggests that underlying cognitive systems, including the amount of information one can hold and manipulate in memory, directly supports complex mathematical activities. Working memory predicts children’s mathematics performance (Frisovan den Bos, van der Ven, Kroesbergen, & van Luit, 2013) and is likely involved in their process of identifying, encoding, and manipulating numerical magnitude information. Thus promoting children’s working memory skills in addition to their specific numerical skills could lead to greater improvements in their mathematical knowledge. For example, working memory plays a substantial role in the performance of numerical magnitude knowledge tasks (Kolkman, Hoijtink, Kroesbergen, & Leseman, 2013) and is important for solving arithmetic problems (Geary, Hoard, Byrd-Craven, & DeSoto, 2004). In the study, the tablet game that targeted improving children’s domainspecific knowledge of numerical magnitude understanding was similar to a traditional board game. Adapted from Laski and Siegler (2014), the game used a 10 x 10 array with the number from 1 to 100 in each square. The other tablet game targeted improving children’s domain-general knowledge, specifically their working memory. Children were shown a series of the same characters in various colors which were presented upside-down or right-side up. Children had to touch one of the corresponding buttons in the bottom corners of the screen to indicate the orientation of the character. They then had to recall the sequence of colors of the characters. Playing both tablet games improved children’s numerical magnitude knowledge compared to children who instead received the ongoing classroom instruction. After playing one of the two tablet games for 10 sessions that each lasted about 10–15 min, children’s number line estimation performance improved suggesting that both domain-specific and domain-general interventions can facilitate mathematical learning (Ramani, Jaeggi, Daubert, & Buschkuehl, 2017). Other studies are also beginning to show the benefits of touchscreen technology for children’s mathematics skills. Specifically, children played one of three tablet games for 6 sessions of approximately 10 min over a period of

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3 weeks. In the number line estimation game, children were given a number (both symbolic and nonsymbolic quantities) and were asked to place their vehicle on the corresponding place on a number line. In the magnitude comparison game, children were shown two quantities (both symbolic and nonsymbolic) and were asked to tap the larger quantity. The active control game was a memory matching game. Children received auditory and visual feedback in all three games. They found that playing the number line estimation game improved children’s number line performance with the numbers 1 to 10; however, playing both the number line game and the magnitude comparison game improved performance on an arithmetic task (Maertens, De Smedt, Sasanguie, Elen, & Reynvoet, 2016). Overall, the results demonstrate the potential of using a rapidly growing technology to promote young children’s numerical knowledge.

Preschool Programs Using Games and Play One benefit of using games in early childhood classrooms is that they can be integrated with instruction and other games to build children’s numerical knowledge. For example, Passolunghi and Costa (2016) examined the benefits of having children play a board game as well as seven other number-related games targeted at improving different categories of numerical knowledge. These games included a finger counting game, a game requiring children to identify which character had more of an item, and a game involving connecting quantities with a numeral. Other children were randomly assigned to play games targeted toward improving children’s working memory. Children played the games for 1 h twice a week for 5 weeks. They found that number games improved children’s numerical knowledge, whereas the working memory games improved children’s working memory, as well as their numerical knowledge. Another such intervention targeted the understanding of whole numbers, number operation, and number relations of kindergarten-age children from lower-income backgrounds (Dyson, Jordan, & Glutting, 2013). The 8-week intervention provided children with 30-min small group sessions 3 days a week. The program included 11 number games in the intervention, such as board game, counting games, and numerical comparison games. Children who played the number games made larger gains on the number sense measures than children who did not play the games. Importantly, these improvements were still found 6 weeks after the intervention. Similar to traditional games and computer games, tablet-based games can also be used as a part of a larger curriculum. Schacter and Jo (2016) evaluated the benefits of Math Shelf, a preschool mathematics curriculum for tablets with over 100 games that target foundational numerical knowledge, such as subitizing and counting, as well as more advanced concepts, such as cardinal principle, place values, and comparing the magnitudes of numbers. Children from predominantly lower-income backgrounds played games appropriate for their skill level for 10 min twice a week for 15 weeks. Compared to children who

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participated in the standard classroom activities, children who played the Math Shelf games performed significantly higher on a variety of measures of numerical knowledge. They also found that students with the lower numerical knowledge at the beginning of the intervention showed the greatest gains from playing the games. Together, these studies suggest that both traditional games and technology-based games can be used with other instructional materials to maximize their benefits. Several well-known preschool programs targeted at improving mathematics skills have taken this approach by including many playful activities. Many of these programs integrate informal learning activities and play with ageappropriate, explicit classroom instruction. One such curriculum is Building Blocks (Clements & Sarama, 2007, 2008), which includes whole classroom math-related activities with small group activities, and computer games. Preschoolers from low-income backgrounds given the Building Blocks curriculum made much greater progress than a control group in number, geometry, measurement, and recognition of patterns. In Pre-K Mathematics (Klein, Starkey, Clements, Sarama, & Iyer, 2008), children engage in small group activities and games. One critical aspect of the program is that these activities and games focus on using concrete manipulatives. In large randomized control trials, prekindergarten children from lower-income backgrounds showed greater growth in their mathematical knowledge when their teachers used Pre-K Mathematics in the classroom as opposed to the standard mathematics curriculum.

CONCLUSIONS AND FUTURE DIRECTIONS Overall, there is strong theoretical and empirical support for the benefits of playing games on children’s numerical knowledge. Activities that target promoting children’s numerical magnitude knowledge are advantageous, given the importance of this knowledge for children’s later mathematical achievement. Both traditional games and computer games that are designed to focus children’s attention on the numerical information in the materials seem most effective in promoting children’s numerical magnitude knowledge. This can include focusing their attention through the materials used in the game, instructions on how to play the game, or the practice with numbers children have while playing the game. In this chapter, we focused on numerical magnitude knowledge, but there are other related foundational areas of numeracy, such as cardinality and counting principles, where children could likely benefit from play-based activities. Future research that continues to identify how various aspects of games influence children’s learning of foundational numerical knowledge could be particularly important. This could help to determine which games should be used to target the learning of specific skills or mathematical concepts, or how the games can be used in conjunction with other instructional materials. Future

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research in this area could also address the common fade-out effects found with early mathematics interventions (Bailey, Duncan, Watts, Clements, & Sarama, 2018). Specifically, longer term follow-up studies are needed to assess the long-term benefits of the math-related games, and if there are fade-out effects, to develop booster interventions that reduce could these effects. Research that continues to investigate the benefits of games will help in the development of age-appropriate practices so that mathematics instruction in early childhood classrooms is purposeful and meaningful for young children.

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Ramani, G. B., & Siegler, R. S. (2011). Reducing the gap in numerical knowledge between low- and middle-income preschoolers. Journal of Applied Developmental Psychology, 32, 146–159. https://doi.org/10.1016/j.appdev.2011.02.005. Ramani, G. B., Siegler, R. S., & Hitti, A. (2012). Taking it to the classroom: number board games as a small group learning activity. Journal of Educational Psychology, 104(3), 661–672. Resnick, L. B. (1999). Making America smarter. Education Week, 38–40. Reynvoet, B., & Sasanguie, D. (2016). The symbol grounding problem revisited: a thorough evaluation of the ANS mapping account and the proposal of an alternative account based on symbolsymbol associations. Frontiers in Psychology, 7(1581), 1–11. https://doi.org/10.3389/fpsyg. 2016.01581. Ritchie, S. J., & Bates, T. C. (2013). Enduring links from childhood mathematics and reading achievement to adult socioeconomic status. Psychological Science, 24(7), 1301–1308. https://doi.org/10.1177/0956797612466268. Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: learning trajectories for young children. New York, NY: Routledge. Sasanguie, D., De Smedt, B., Defever, E., & Reynvoet, B. (2012). Association between basic numerical abilities and mathematics achievement. British Journal of Developmental Psychology, 30, 344–357. Scalise, N., Daubert, N. A., & Ramani, G. B. (2017). Narrowing the early mathematics gap: a play-based intervention to promote head start preschoolers’ number skills. Journal of Numerical Cognition, 3(3), 559–581. https://doi.org/10.5964/jnc.v3i3.72. Schacter, J., & Jo, B. (2016). Improving low-income preschoolers mathematics achievement with Math Shelf, a preschool tablet computer curriculum. Computers in Human Behavior, 55, 223–229. Schneider, M., Beeres, K., Coban, L., Merz, S., Schmidt, S. S., Stricker, J., et al. (2016). Associations of non-symbolic and symbolic numerical magnitude processing with mathematical competence: a meta-analysis. Developmental Science, 20, e12372. https://doi.org/10.1111/ desc.12372. Sella, F., Tressoldi, P., Lucangeli, D., & Zorzi, M. (2016). Training numerical skills with the adaptive videogame “the number race”: a randomized controlled trial on preschoolers. Trends in Neuroscience and Education, 5, 20–29. https://doi.org/10.1016/j.tine.2016.02.002. Siegler, R. S. (2016). Magnitude knowledge: the common core of numerical development. Developmental Science, 19, 341–361. https://doi.org/10.1111/desc.12395. Siegler, R. S., & Booth, J. (2004). Development of numerical estimation in young children. Child Development, 75, 428–444. Siegler, R. S., & Ramani, G. B. (2008). Playing board games promotes low-income children’s numerical development. Developmental Science, 11, 655–661. Special Issue on Mathematical Cognition. Siegler, R. S., & Ramani, G. B. (2009). Playing linear number board games – but not circular ones – improves low-income preschoolers’ numerical understanding. Journal of Educational Psychology, 101, 545–560. Siegler, R. S., & Robinson, M. (1982). The development of numerical understandings. Advances in Child Development and Behavior, 16, 241–312. Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62, 273–296. Starkey, P. (1992). The early development of numerical reasoning. Cognition, 43(2), 93–126. Starkey, P., & Cooper, R. G. (1980). Perception of numbers by human infants. Science, 210(4473), 1033–1035.

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Chapter 4

Understanding the Link Between the Approximate Number System and Math Abilities Melissa E. Libertus Department of Psychology and Learning Research and Development Center, University of Pittsburgh, Pittsburgh, PA, United States

INTRODUCTION Imagine you are at the grocery store to get a few items that you forgot for dinner. As you reach the checkout, you scan the lanes trying to figure out which will be the fastest. Most likely you will look at the other shoppers’ carts and compare which lane has the fewest shoppers with the fewest items without exactly counting how many items each shopper is purchasing. You can complete this comparison process quickly and effortlessly and within seconds decide where you will line up. Quick estimation and comparison processes such as the one described in the previous scenario rely on our approximate number system (ANS). The ANS allows us to form approximate representations of quantities without the need to count or understand symbolic labels for quantities (Dehaene, 1997). Since these representations are inherently imprecise and this imprecision increases as the quantities increase, our ability to discriminate between quantities is determined by their ratio. For example, it is easier to determine that a shopper with 15 items in his cart has more items than a shopper with only 10 items compared to determining that a shopper with 45 items has more than a shopper with 40 items. While ratio-dependent discrimination is present in children and adults at all ages (Halberda & Feigenson, 2008; Xu & Spelke, 2000), individuals differ in the precision of their ANS representations (Halberda, Ly, Willmer, Naiman, & Germine, 2012; Halberda, Mazzocco, & Feigenson, 2008; Libertus & Brannon, 2010). Even though the ANS can be useful under certain circumstances such as quickly comparing which checkout lane will move the fastest, it does not afford Mathematical Cognition and Learning, Vol. 5. https://doi.org/10.1016/B978-0-12-815952-1.00004-9 Copyright © 2019 Elsevier Inc. All rights reserved.

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precise number representation that may be needed to figure out how much our items will cost together. As educated humans, we have acquired symbolic representations such as number words and Arabic numerals to represent exact quantities (Fuson, 1988; Gelman & Gallistel, 1978; Wynn, 1992). Importantly, these exact representations allow us to perform precise arithmetic and even learn more complex math such as trigonometry, algebra, and calculus. Despite the fact that the ANS only offers imprecise numerical representations, it seems to be linked to people’s symbolic math abilities. Three recent meta-analyses (Chen & Li, 2014; Fazio, Bailey, Thompson, & Siegler, 2014; Schneider et al., 2016) have shown that across a large number of correlational studies, there is a small but significant association such that people with more precise ANS representations tend to perform better on a variety of math assessments. This association appears to be stronger in children than adults (Fazio et al., 2014; Inglis, Attridge, Batchelor, & Gilmore, 2011) and for math assessments that rely more heavily on the processing of quantities instead of the application of rote memorized procedures (Schneider et al., 2016). While these meta-analyses support the existence of a link between the ANS and math, they are based on correlational findings, which leave open the possibility that other factors may explain the observed associations. For example, some researchers have argued that tasks that are designed to measure the ANS and math both require visuospatial processing and that positive correlations may be explained by this shared need (Allik & Tuulmets, 1991; Dakin, Tibber, Greenwood, Kingdom, & Morgan, 2011; Durgin, 1995; Gebuis, Cohen Kadosh, & Gevers, 2016; Ginsburg & Nicholls, 1988; Leibovich, Katzin, Harel, & Henik, 2016; Tibber et al., 2013; Tibber, Greenwood, & Dakin, 2012; Tokita & Ishiguchi, 2010; Vos, van Oeffelen, Tibosch, & Allik, 1988; Zhou, Wei, Zhang, Cui, & Chen, 2015). Other researchers have argued that the shared reliance on executive functions, especially inhibitory control, may explain the observed correlations (Fuhs & McNeil, 2013; Gilmore et al., 2013) although the evidence for this explanation is mixed (Keller & Libertus, 2015).

TRAINING STUDIES Training studies are one way to circumvent the limitations of correlational studies and to examine whether the ANS and math are causally linked. If participants are randomly assigned to different conditions, some of which may train the ANS and others may not, any subsequent improvements in math ability in participants who completed ANS-related training conditions are most likely due to the reliance on the ANS during training.

Training Studies Using “The Number Race” Early training studies often combined several different number processing tasks that tapped into the ANS and other basic number processing skills. For example, “The Number Race” is an adaptive computer game that was

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initially designed to improve math abilities in children with dyscalculia (Wilson, Dehaene, et al., 2006). It adapts to each child’s individual performance to minimize failure and maintain an appropriate level of difficulty. Children perform number comparisons and simple arithmetic using a mix of both nonsymbolic and symbolic numerical information. Speed is emphasized by asking children to perform the comparisons quicker than a virtual opponent. Additionally, choosing the larger quantity is incentivized by having children advance their game token on a board by the number of items in the chosen set. At the same time, this step is intended to help children train the connection between numbers and space. In the first evaluation of “The Number Race,” a sample of nine dyscalculic children played the game 4 days a week for 5 weeks (Wilson, Revkin, Cohen, Cohen, & Dehaene, 2006). Each training session lasted about 30 min. Comparisons between pre- and posttraining performance revealed significant increases in subitizing speed (i.e., the speed in which children enumerated small quantities) and subtraction accuracy after training. However, the lack of a control group limits the conclusions that can be drawn from this study. Three follow-up studies were designed to remedy this shortcoming. In the first follow-up study, Wilson, Dehaene, Dubois, and Fayol (2009) used a cross-over design to train 4- to 6-year-olds from low socioeconomic status households with “The Number Race” and a commercially available reading software as a control training. Half of the children completed six sessions of “The Number Race” first and then four sessions of the reading software, the other half first completed four sessions of the reading software and then six sessions of “The Number Race.” Each session lasted about 20 min. Unlike in the earlier study, Wilson and colleagues only found significant improvements in symbolic number comparison (i.e., identifying the larger of two Arabic numerals) following training on “The Number Race.” No transfer to addition skills was observed. In the second follow-up study, R€as€anen, Salminen, Wilson, Aunio, and Dehaene (2009) divided a sample of 6-year-olds into three different groups: Half of the children were assigned to the passive control group, that is, they only completed the pre- and postassessments and participated in regular classroom instruction in the interim. Half of the other children completed daily 10- to 15-min sessions of “The Number Race” for a total of 3 weeks. The remaining children completed “Graphogame-Math,” another computerized number game that emphasizes the correspondence between small quantities and their labels. For example, children hear a number word such as “three” and have to select the correct match from 2 to 5 visually presented options. Choices include dot patterns, number symbols, and in the end symbolic addition (e.g., “1 + 2”) and subtraction problems (e.g., “4 1”). Children in both training conditions showed significant improvements in symbolic number comparisons as compared with the children in the passive control condition. However, similar to the findings by Wilson et al. (2009), no transfer to improvements in addition and subtraction skills were observed.

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In the third follow-up study, Sella, Tressoldi, Lucangeli, and Zorzi (2016) used “The Number Race” in a randomized controlled trial with 5-year-old Italian preschoolers. Half of the children completed the Italian version of “The Number Race,” the other half played a computerized game to foster drawing skills. Both groups completed 10 weeks of training, two sessions per week for about 20 min each. Children in “The Number Race” training showed significantly more precise mapping of numbers onto a number line (ranging from 1 to 10 and from 1 to 20) as well as better mental calculation skills after training compared to the children in the control group. While these previous studies focused on preschool-aged children and those that just started formal education, Kuhn and Holling (2014) designed a computerized number sense training similar to “The Number Race” for 3rd and 4th graders. This training consisted of two number line tasks and a number comparison task. In the number line tasks children had to indicate the position of a single number, the result of a computation, or multiple numbers on number lines that ranged from 0 to 20 or 0 to 100. In the number comparison task, children had to select the largest number out of two or four whereby numbers were shown as nonsymbolic arrays of shapes, numerals, or symbolic calculations (e.g., “8 + 6”). Children completed two out of the three tasks each day for 10 min each for a total of 15 days (3 weeks of school). Children in the number sense training showed significantly greater gains on a standardized math assessment compared to a passive control group but not compared to a working memory training group. However, children in the number sense training group improved mainly in arithmetic, while children in the working memory training group improved in word problem solving. These findings corroborate that a computerized training targeting basic number processing such as mapping numbers onto space and comparing numbers may improve some aspects of children’s math abilities. These findings further suggest that such gains can be accomplished in older, typically achieving children (average age 9 years). To disentangle the role of number comparison versus number line tasks for training-related improvements, Maertens, De Smedt, Sasanguie, Elen, and Reynvoet (2016) designed two separate training games and tested them in a sample of 5-year-old children. In the number comparison training, children saw nonsymbolic and symbolic stimuli as well as a combination of both types of stimuli and had to choose the numerically larger of the two. In the number line training, children saw a line anchored either by nonsymbolic or symbolic stimuli and had to place a number on the line relative to the two anchors. Maertens and colleagues compared these number training conditions to a memory-game control group and a passive control group. Children in the three active training groups completed six sessions of their respective games, each lasting for about 10 min, over a period of 3 weeks. Children in both number training conditions but not the control conditions improved in arithmetic (e.g., symbolic problems such as “6 + 3 ¼ …” and word problems such as

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“Here you see two red balloons and three blue balloons. How many balloons are there together?”). These findings suggest that comparison skills and the mappings of numbers onto space are linked to math abilities. In sum, findings from training studies using “The Number Race” and similar training games that required symbolic and nonsymbolic number comparisons as well as mappings of numbers onto space are mixed and largely inconsistent. While performance on the trained tasks generally improves, transfer to math more generally is less reliable. However, variations in the samples (ages, socioeconomic status, initial math abilities), duration and timing of the intervention, the nature of the control, as well as the pre- and posttraining assessments make it difficult to draw firm conclusions about the effectiveness of these kinds of training approaches.

Nonsymbolic vs. Symbolic Training All of the previously reviewed training studies used both nonsymbolic and symbolic training stimuli. Thus it is unclear whether the observed improvements in some of the math-related tasks are due to the training with nonsymbolic or symbolic representations or the need to connect the two. To disentangle whether training on nonsymbolic stimuli differs from training on symbolic stimuli, Obersteiner, Reiss, and Ufer (2013) trained four groups of first graders on slightly modified versions of “The Number Race.” In the “Approximate Version” of the game, children performed nonsymbolic number comparisons as well as estimations in which they had to judge which of two symbolic estimates was nearest to the correct answer. In contrast, in the “Exact Version” of the game, children had to judge the exact number of dots in the nonsymbolic stimuli. One group of children was trained on the Approximate and the Exact Versions of the computer game, another group was trained only on the Approximate Version, and a third only on the Exact Version. The fourth group of children was trained on a computer game designed to improve language skills. All students completed 10 training sessions, 30 min each over a period of 4 weeks. Children who completed only the Approximate or the Exact Version, but not a combination of both or the language training, performed higher on a subsequent arithmetic test. While these findings suggest that training involving nonsymbolic numerical stimuli can lead to improvements in math, the “Approximate Version” required children to also process number symbols in the estimation portion of the game. Thus there were again some symbolic representations, which make it impossible to ascertain that this training was purely targeting the ANS. To avoid these confounds, Honore and Noel (2016) created two parallel training conditions, one that used entirely nonsymbolic number stimuli and another that only used symbolic number stimuli. They compared these training conditions along with a storybook reading control condition in a sample of 5-year-olds. In both nonsymbolic and symbolic number training conditions,

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children had to complete a comparison task and a number line task using the respective stimuli. In the comparison task, children were shown three arrays of dots or three Arabic numerals and three animals in various sizes. Children were instructed to give the smallest number to the smallest animal, the medium number to the medium-sized animal, and the largest number to the largest animal. In the number line task children were shown a line bounded by two arrays of dots or two numerals (1 and 10 or 1 and 20) and asked to indicate where a third number presented either nonsymbolically or symbolically would be with respect to the two anchors. All children completed ten 30-min sessions of their assigned training condition, 15 min per task during each session. While the nonsymbolic training condition yielded significant improvements in nonsymbolic number comparison and nonsymbolic number line performance, only the symbolic training yielded improvements in arithmetic. The findings from these two studies thus suggest that the link between symbolic number representations and math may be stronger than the link between the ANS and math. However, they show that the ANS is malleable and leave open the possibility that improvements in math may occur with more or different forms of nonsymbolic training. These conclusions are in line with those by Dillon, Kannan, Dean, Spelke, and Duflo (2017) who found that nonsymbolic numerical and geometrical training implemented in a large number of preschools in India led to long-term gains in nonsymbolic skills but not school math. Dillon and colleagues suggest that a more effective intervention may need to align the nonsymbolic content more clearly with the school math curriculum and/or that the training should occur at the same time that children receive formal instruction in school math so that children may form connections between nonsymbolic training content and school math.

Training Specific Aspects of Nonsymbolic Number Processing To more clearly isolate specific aspects of approximate number processing for math, a number of recent training studies have focused on single nonsymbolic tasks during training. Some of these studies used brief training durations to assess their effects on math abilities within the same session, while others looked at the effects of long-term training.

Brief ANS Training Hyde, Khanum, and Spelke (2014) used brief training on approximate numerical addition, approximate number comparison, line length addition, and brightness comparison to address whether approximate number training was more effective at improving arithmetic performance in first graders than nonnumerical addition or nonnumerical comparison training. In the approximate numerical addition training, children saw two arrays of dots successively moving behind an occluder and they had to indicate whether the sum was more or less numerous than a third comparison array (Fig. 1).

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Training tasks a. Numerical addition

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FIG. 1 Schematic depiction of training tasks used by Hyde et al. (2014). Stimulus events are organized horizontally from start (top) to finish (bottom). The numbers indicate the duration of presentation. The horizontal arrows indicate stimulus movement. The vertical arrows indicate the following event. Reprinted with permission.

In the approximate numerical comparison training, children saw one array of dots move behind an occluder and they had to indicate whether it was more or less numerous than a comparison array. In the line length addition training, children saw two line segments successively move behind an occluder and they had to indicate whether the sum was taller or shorter than a third line segment. Finally, in the brightness comparison training, children saw an oval of a certain brightness disappear behind an occluder and they had to indicate whether it was more or less bright than a comparison circle. Hyde and colleagues found that approximate numerical addition and approximate number comparison training but not line length addition or brightness comparison improved first graders’ symbolic addition performance immediately after the training. This transfer was specific to math as they did not find subsequent improvements in children’s reading performance. In a follow-up study, Khanum, Hanif, Spelke, Berteletti, and Hyde (2016) replicated the finding that approximate numerical addition but not line length addition improves symbolic addition in Pakistani first graders. Instead of comparing approximate number training to training conditions using other types of stimuli, Wang, Odic, Halberda, and Feigenson (2016) made use of the fact that children perform significantly better on nonsymbolic number comparisons when they start out with the easiest comparisons and gradually receive more difficult comparisons than when they begin with the

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hardest comparisons and receive gradually easier ones (Odic, Hock, & Halberda, 2014). In this study, Wang et al. (2016) asked 5-year-old children to first complete a nonsymbolic number comparison task starting either with the easiest comparisons and gradually getting harder or vice versa. Immediately following these two different training conditions, children completed a math assessment tapping into counting, number comparison, calculation, and numeral literacy. Children who first completed the easiest number comparisons performed significantly better on the nonsymbolic number comparison task as well as on a subsequent math assessment compared to children who first completed the hardest number comparisons and these effects were medium sized. Importantly, these improvements did not extend to a vocabulary assessment supporting the idea that engagement of the ANS leads to changes in math, but not vocabulary. While these brief training studies suggest that engagement of the ANS can lead to immediate improvements in math performance, the underlying mechanisms for these improvements are unclear. Hyde et al. (2014) suggest that “symbolic arithmetic draws on at least some overlapping cognitive and/or neural structures used to represent approximate number” (p. 104). Activating these overlapping structures during the brief training session may serve as a “warm-up” before performing math. However, this account cannot fully explain the results of Wang et al. (2016) because in their study both training conditions engaged the ANS. Instead, these results could be explained by training-related differences in “children’s internal sense of their ability to reason about quantities,” which may have led those children who completed the easiest comparisons first “to think of themselves as mathematically competent, thereby leading to more perseverance in subsequent symbolic math problems” (Wang et al., 2016, p. 94). Regardless, it is likely that these brief training interventions do not exert strong, long-lasting effects on children’s math abilities even though this assertion remains to be experimentally validated. Long-term training studies are more likely to yield these benefits.

Long-Term ANS Training A set of recent ANS training studies sought to determine if long-term training of the ANS can lead to changes in adults’ math abilities. To this end, Park and Brannon (2013) trained a group of adults on a nonsymbolic arithmetic task for ten sessions every day, each lasting about 25 min. Similar to the training task used in Hyde et al.’s (2014) approximate addition training, participants saw an array of dots move behind an occluder and a second array was added (addition) or a subset moved out from behind the occluder (subtraction). Participants then either saw a third array and had to indicate whether the sum or difference of the first two arrays was more or less than the third array or they saw two arrays and had to indicate which contained the sum or difference of the first two arrays. Park and Brannon (2013) found that this nonsymbolic arithmetic training led to significant improvements in a symbolic arithmetic

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assessment compared to a passive test-retest control and an active general worldknowledge training. A follow-up study (Park & Brannon, 2014) replicated the improvements in symbolic arithmetic following nonsymbolic arithmetic training while failing to find such improvements following a nonsymbolic number comparison training similar to the approximate number comparison training used by Hyde et al. (2014). The authors argue that the manipulation of nonsymbolic number representations may be a critical component of this training, which relates to the manipulations necessary to successfully solve symbolic arithmetic. Recent efforts have expanded on these initial studies to examine whether young children benefit in similar ways from long-term ANS training. Park, Bermudez, Roberts, and Brannon (2016) trained 3- to 5-year-old children on a nonsymbolic addition and subtraction computer game similar to the one used with adults (Park & Brannon, 2013, 2014). Children in the ANS training group completed ten sessions, 12 min each, over a period of 2–3 weeks while children in the control group completed a computerized memory game for the same period of time. They found significantly greater improvement on a standardized math assessment for those children who completed the nonsymbolic arithmetic training compared to those who completed the memory training. Building on these findings, Libertus, Odic, Feigenson, and Halberda (2018) showed that 5 weeks of nonsymbolic number comparison training in young children also leads to greater improvements on a standardized math assessment compared to a phoneme awareness training. In both training conditions, children were trained every other day for 15 min for a total of 5 weeks. Interestingly, even though the nonsymbolic number comparison training relied entirely on visual stimuli (as did all other training studies reviewed before), children in the number comparison training group also performed significantly better on an auditory approximate number comparison task after training. In this task, they heard two sequences of rapidly presented tones, one high pitched and one low pitched, and had to indicate which of the two sequences contained more tones. This training-related transfer suggests that the visual nonsymbolic number comparison training does not merely improve children’s ability to inhibit irrelevant perceptual cues that are specific to the visual stimuli (e.g., total surface area, density, or convex hull), but instead sharpens their ANS representations that are independent of the format in which stimuli are presented (i.e., dot arrays in the visual modality or tone sequences in the auditory modality).

MECHANISMS BEHIND LONG-TERM ANS TRAINING IMPROVEMENTS While these long-term training studies suggest that ANS training can lead to improvements in math performance in both adults and children, the underlying mechanisms for these improvements are unclear. Hyde, Berteletti, and Mou (2016) suggested two possible mechanisms: the Overlapping

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Operations Hypothesis and the Representational Overlap Hypothesis. According to the Overlapping Operations Hypothesis, the shared reliance on similar mental operations (e.g., addition and subtraction) may explain why training on nonsymbolic number formats may lead to improvements in math abilities (see Park & Brannon, 2014, for a similar explanation). This account fits well with training studies that used nonsymbolic arithmetic and showed subsequent improvements in symbolic arithmetic (Hyde et al., 2014; Park & Brannon, 2013, 2014), but it cannot explain why nonsymbolic number comparison training yields improvements in symbolic arithmetic (Hyde et al., 2014) or math abilities more broadly (Libertus et al., 2018; Wang et al., 2016). Instead, these findings may be better explained by the Representational Overlap Hypothesis, which purports that symbolic number representations rely to some extent on ANS representations. However, this account cannot explain why nonsymbolic arithmetic but not nonsymbolic number comparison training in adults yields improvements in symbolic arithmetic (Park & Brannon, 2014). While these previous hypotheses focused on training-related changes to mental processes closely related to the training content, another possible explanation is that long-term ANS training leads to greater engagement with, interest in, or attention to math-related information outside of the training context, which in turn may improve math abilities. For example, children who complete ANS training repeatedly over the course of several weeks may become more interested in their math lessons at school or engage in more math-related activities such as number games outside of school. They might also pay more attention when other people in their environment talk about numbers and math such as a parent talking about the costs of groceries while shopping. Preliminary support for the viability of this explanation comes from correlational studies examining the role of math-related input for children’s math abilities. Direct observations typically quantify math input in terms of the frequency of math talk, that is, math-related utterances such as “we need three spoons of sugar” or “there are more than ten people on the bus.” Previous studies examining the amount of parental math talk that young children are exposed to have found positive associations with children’s math ability (Casey et al., 2018; Elliott, Braham, & Libertus, 2017; Gunderson & Levine, 2011; Levine, Suriyakham, Rowe, Huttenlocher, & Gunderson, 2010; Susperreguy & DavisKean, 2016; see also Levine et al., this volume). The more parents talked about numbers and math with their children during everyday activities or play in the lab, the greater the children’s counting skills and performance on standardized math assessments. Similarly, the frequency of parent-reported math activities in the home has also been linked to children’s math achievement (Anders et al., 2012; Blevins-Knabe & Musun-Miller, 1996; Kleemans, Peeters, Segers, & Verhoeven, 2012; Missall, Hojnoski, Caskie, & Repasky, 2015; Skwarchuk, 2009). For example, home activities, such as counting, relate to children’s understanding of numerical symbols (LeFevre et al., 2009) and activities

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such as playing board games and puzzles predict children’s performance on nonsymbolic math tasks and children’s developing fluency with basic number skills (Skwarchuk, Sowinski, & LeFevre, 2014; Vandermaas-Peeler, Nelson, Bumpass, & Sassine, 2009; see also Ramani et al., this volume). While variability in parents’ math-related input has been observed in a number of different studies, it is unclear why parents differ in the provision of math-related input. On the one hand, it seems that if parents perceive early math learning to be important, they report greater engagement in math activities such as playing board games with numbers or measuring ingredients while cooking than if they perceive math to be less important (BlevinsKnabe, Berghout Austin, Musun, Eddy, & Jones, 2000; Missall et al., 2015; Musun-Miller & Blevins-Knabe, 1998; Sonnenschein et al., 2012). On the other hand, parents who report feeling more comfortable teaching math or report having better math skills report greater engagement in math-related activities with their children (Blevins-Knabe et al., 2000; Missall et al., 2015). One of the shortcomings of these previous studies is that they used parental report to measure parents’ math skills and math-related input. To test whether similar relations between parents’ cognitive abilities, their provision of math-related input, and children’s math abilities could be observed when parents and children were tested directly we completed a series of studies. First, we found that parents with more precise ANS representations also had children with more precise ANS representations (Braham & Libertus, 2017; Navarro, Braham, & Libertus, in press). Second, parents’ ANS precision was a significant predictor of children’s ability to solve math word problems, but not written calculation or speeded mental arithmetic problems (Braham & Libertus, 2017). Thus we hypothesized that parents with more precise ANS representations may talk more about math with their children during everyday activities which may provide the children with more learning opportunities to apply math in everyday contexts. In a follow-up study, we provided evidence that parents’ ANS indeed relates to their provision of math-related input (Elliott et al., 2017). In this study, mothers were observed while they played with their children in the lab for 10 min. We quantified the amount of math talk that occurred during these 10 min by counting how often mothers used a number word. We found that mothers’ ANS precision as well as their self-reported math ability were significant predictors of how often they used number words greater than ten. In turn, the more frequently mothers talked about numbers greater than ten, the better their children tended to perform on a standardized math assessment. Thus individual differences in ANS precision can be linked to attention to and engagement with math-related information in everyday contexts, which in the case of greater parental conversation about these topics, leads to more opportunities for their children to learn math. Obviously, more research is needed to determine whether children may prompt their parents to provide content or whether parents initiate the

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math-related input themselves. Regardless, linking the math input findings to the previous ANS training studies raises the intriguing possibility that ANS training may heighten people’s interest in or attention to numbers and other math-related information outside of the training which in turn may increase their opportunities to improve their math abilities. For example, engaging in ANS training may lead children to seek out more math-related activities alone or with their parents and this greater math-related input may lead to improvements in math ability. We recently showed that experimentally manipulating children’s exposure to math talk indeed results in greater subsequent focusing on number in an unrelated context (Braham, Libertus, & McCrink, in press). In this study, we asked preschool-aged children and their parents to play together in a children’s museum exhibit using either a numerical prompt or a nonnumerical prompt. Before and after playing with their parents, children completed assessments to measure their tendency to spontaneously focus on number when imitating the actions of an experimenter. After playing with their parents, children whose parents received the numerical prompt showed greater spontaneous focus on number compared to children whose parents received the nonnumerical prompt. These findings suggest that when parents provide their children with mathrelated input, it increases children’s later spontaneous attention to numerical information, which in turn has been shown to predict children’s math abilities (Hannula, Lepola, & Lehtinen, 2010; Hannula-Sormunen, Lehtinen, & R€as€anen, 2015).

CONCLUSIONS AND FUTURE DIRECTIONS In sum, previous correlational and training studies provide compelling support for the link between the ANS and math. However, attempts to explain the mechanisms behind this link via reliance on shared mental operations or shared representations have not been successful at integrating all of the findings across studies. One alternative explanation raised here is that ANS training may lead to greater engagement in math-related activities or heightened attention to math-related learning opportunities, which in turn improve math abilities. The initial findings presented in this chapter at least suggest that this idea is tenable, but further work is needed to directly test this hypothesis. For example, future long-term ANS training studies could assess participants’ interest in math-related activities and information before, during, and after the training to test whether ANS training leads to increased interest. Relatedly, future studies could vary whether children are receiving concurrent instruction in math while completing ANS training and what math concepts children are learning. These studies would provide important information for potential applications of ANS training in the classroom but also elucidate which aspects of math are most closely linked to the ANS.

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REFERENCES Allik, J., & Tuulmets, T. (1991). Occupancy model of perceived numerosity. Perception & Psychophysics, 49(4), 303–314. Anders, Y., Rossbach, H.-G., Weinert, S., Ebert, S., Kuger, S., Lehrl, S., et al. (2012). Home and preschool learning environments and their relations to the development of early numeracy skills. Early Childhood Research Quarterly, 27, 231–244. Blevins-Knabe, B., Berghout Austin, A., Musun, L., Eddy, A., & Jones, R. M. (2000). Family home care providers’ and parents’ beliefs and practices concerning mathematics with young children. Early Child Development and Care, 165(1), 41–58. Blevins-Knabe, B., & Musun-Miller, L. (1996). Number use at home by children and their parents and its relationship to early mathematical performance. Early Development and Parenting, 5(1), 35–45. Braham, E. J., & Libertus, M. E. (2017). Intergenerational associations in numerical approximation and mathematical abilities. Developmental Science, 20(5), e12436. Braham, E. J., Libertus, M. E., & McCrink, K. (in press). Improving young children’s spontaneous focus on number through guided parent-child interactions at a children’s museum. Developmental Psychology. Casey, B. M., Lombardi, C. M., Thomson, D., Nguyen, H. N., Paz, M., Theriault, C. A., & Dearing, E. (2018). Maternal support of children’s early numerical concept learning predicts preschool and first-grade math achievement. Child Development, 89(1), 156–173. Chen, Q., & Li, J. (2014). Association between individual differences in non-symbolic number acuity and math performance: a meta-analysis. Acta Psychologica, 148, 163–172. Dakin, S. C., Tibber, M. S., Greenwood, J. A., Kingdom, F. A., & Morgan, M. J. (2011). A common visual metric for approximate number and density. Proceedings of the National Academy of Sciences of the United States of America, 108(49), 19552–19557. https://doi. org/10.1073/pnas.1113195108. Dehaene, S. (1997). The number sense: How the mind creates mathematics. New York: Oxford University Press. Dillon, M. R., Kannan, H., Dean, J. T., Spelke, E., & Duflo, E. (2017). Cognitive science in the field: a preschool intervention durably enhances intuitive but not formal mathematics. Science, 357, 47–55. Durgin, F. H. (1995). Texture density adaptation and the perceived numerosity and distribution of texture. Journal of Experimental Psychology: Human Perception and Performance, 21(1), 149–169. Elliott, L. E., Braham, E. J., & Libertus, M. E. (2017). Understanding sources of individual variability in parents’ number talk with young children. Journal of Experimental Child Psychology, 159, 1–15. Fazio, L. K., Bailey, D. H., Thompson, C. A., & Siegler, R. S. (2014). Relations of different types of numerical magnitude representations to each other and to mathematics achievement. Journal of Experimental Child Psychology, 123, 53–72. Fuhs, M. W., & McNeil, N. M. (2013). ANS acuity and mathematics ability in preschoolers from low-income homes: contributions of inhibitory control. Developmental Science, 16(1), 136–148. https://doi.org/10.1111/desc.12013. Fuson, K. C. (1988). Children’s counting and concepts of number. New York: Springer-Verlag. Gebuis, T., Cohen Kadosh, R., & Gevers, W. (2016). Sensory-integration system rather than approximate number system underlies numerosity processing: a critical review. Acta Psychologica, 171, 17–35.

104 Cognitive Foundations for Improving Mathematical Learning Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge: Harvard University Press. Gilmore, C., Attridge, N., Clayton, S., Cragg, L., Johnson, S., Marlow, N., et al. (2013). Individual difference in inhibitory control, not non-verbal number acuity, correlate with mathematics achievement. PLoS ONE, 8(6), e67374. Ginsburg, N., & Nicholls, A. (1988). Perceived numerosity as a function of item size. Perceptual and Motor Skills, 67, 656–658. Gunderson, E. A., & Levine, S. C. (2011). Some types of parent number talk count more than others: relations between parents’ input and children’s cardinal-number knowledge. Developmental Science, 14(5). https://doi.org/10.1111/j.1467-7687.2011.01050.x 21884318. Halberda, J., & Feigenson, L. (2008). Developmental change in the acuity of the ‘number sense’: the approximate number system in 3-, 4-, 5-, and 6-year-olds and adults. Developmental Psychology, 44(5), 1457–1465. Halberda, J., Ly, R., Willmer, J., Naiman, D., & Germine, L. (2012). Number sense across the lifespan as revealed by a massive internet-based sample. Proceedings of the National Academy of Sciences of the United States of America, 109(28), 11116–11120. https://doi.org/10.1073/ pnas.1200196109. Halberda, J., Mazzocco, M. M., & Feigenson, L. (2008). Individual differences in non-verbal number acuity correlate with maths achievement. Nature, 455(7213), 665–668. Hannula, M. M., Lepola, J., & Lehtinen, E. (2010). Spontaneous focusing on numerosity as a domain-specific predictor of arithmetical skills. Journal of Experimental Child Psychology, 107(4), 394–406. https://doi.org/10.1016/j.jecp.2010.06.004. Hannula-Sormunen, M. M., Lehtinen, E., & R€as€anen, P. (2015). Preschool children’s spontaneous focusing on numerosity, subitizing, and counting skills as predictors of their mathematical performance seven years later at school. Mathematical Thinking and Learning, 17(2–3), 155–177. Honore, N., & Noel, M. P. (2016). Improving preschoolers’ arithmetic through number magnitude training: the impact of non-symbolic and symbolic training. PLoS ONE, 11(11), e0166685. Hyde, D. C., Berteletti, I., & Mou, Y. (2016). Approximate numerical abilities and mathematics: insight from correlational and experimental training studies. In M. Cappelletti & W. Fias (Eds.), Vol. 227. Progress in brain research (pp. 335–351). Amsterdam: Elsevier. Hyde, D. C., Khanum, S., & Spelke, E. S. (2014). Brief non-symbolic, approximate number practice enhances subsequent exact symbolic arithmetic in children. Cognition, 131, 92–107. Inglis, M., Attridge, N., Batchelor, S., & Gilmore, C. (2011). Non-verbal number acuity correlates with symbolic mathematics achievement: but only in children. Psychonomic Bulletin & Review. https://doi.org/10.3758/s13423-011-0154-1. Keller, L., & Libertus, M. E. (2015). Inhibitory control may not explain the link between approximation and math abilities in kindergarteners from middle class families. Frontiers in Psychology, 6, 685. https://doi.org/10.3389/fpsyg.2015.00685. Khanum, S., Hanif, R., Spelke, E., Berteletti, I., & Hyde, D. C. (2016). Effects of non-symbolic approximate number practice on symbolic numerical abilities in Pakistani children. PLoS ONE, 11(10), e0164436. Kleemans, T., Peeters, M., Segers, E., & Verhoeven, L. (2012). Child and home predictors of early numeracy skills in kindergarten. Early Childhood Research Quarterly, 27, 471–477. Kuhn, J.-T., & Holling, H. (2014). Number sense or working memory? The effect of two computer-based trainings on mathematical skills in elementary school. Advances in Cognitive Psychology, 10(2), 59–67. LeFevre, J. A., Skwarchuk, S. L., Smith-Chant, B. L., Fast, L., Kamawar, D., & Bisanz, J. (2009). Home numeracy experiences and children’s math performance in the early school years. Canadian Journal of Behavioural Science, 41(2), 55–66.

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Leibovich, T., Katzin, N., Harel, M., & Henik, A. (2016). From ‘sense of number’ to ‘sense of magnitude’—the role of continuous magnitudes in numerical cognition. Behavioral and Brain Sciences, 40, 1–62. Levine, S. C., Suriyakham, L. W., Rowe, M. L., Huttenlocher, J., & Gunderson, E. A. (2010). What counts in the development of young children’s number knowledge? Developmental Psychology, 46(5), 1309–1319. https://doi.org/10.1037/a0019671 20822240. Libertus, M. E., & Brannon, E. M. (2010). Stable individual differences in number discrimination in infancy. Developmental Science, 13(6), 900–906. https://doi.org/10.1111/ j.1467-7687.2009.00948.x. Libertus, M. E., Odic, D., Feigenson, L., & Halberda, J. (2018). Visual training of approximate number sense improves auditory number sense and school math ability. [in prep]. Maertens, B., De Smedt, B., Sasanguie, D., Elen, J., & Reynvoet, B. (2016). Enhancing arithmetic in pre-schoolers with comparison or number line estimation training: does it matter? Learning and Instruction, 46, 1–11. Missall, K., Hojnoski, R. L., Caskie, G. I., & Repasky, P. (2015). Home numeracy environments of preschoolers: examining relations among mathematical activities, parent mathematical beliefs, and early mathematical skills. Early Education and Development, 26(3), 356–376. Musun-Miller, L., & Blevins-Knabe, B. (1998). Adults’ beliefs about children and mathematics: how important is it and how do children learn about it. Early Development and Parenting, 7(4), 191–202. Navarro, M., Braham, E. J., & Libertus, M. E. (in press). Intergenerational associations of the approximate number system in toddlers and their parents. British Journal of Developmental Psychology. Obersteiner, A., Reiss, K., & Ufer, S. (2013). How training on exact or approximate mental representations of number can enhance first-grade students’ basic number processing and arithmetic skills. Learning and Instruction, 23, 125–135. Odic, D., Hock, H., & Halberda, J. (2014). Hysteresis affects approximate number discrimination in young children. Journal of Experimental Psychology: General, 143(1), 255–265. Park, J., Bermudez, V., Roberts, R. C., & Brannon, E. M. (2016). Non-symbolic approximate arithmetic training improves math performance in preschoolers. Journal of Experimental Child Psychology, 152, 278–293. Park, J., & Brannon, E. M. (2013). Training the approximate number system improves math proficiency. Psychological Science, 24(10), 2013–2019. Park, J., & Brannon, E. M. (2014). Improving arithmetic performance with number sense training: an investigation of underlying mechanism. Cognition, 133(1), 188–200. R€as€anen, P., Salminen, J., Wilson, A. J., Aunio, P., & Dehaene, S. (2009). Computer-assisted intervention for children with low numeracy skills. Cognitive Development, 24, 450–472. Schneider, M., Beeres, K., Coban, L., Merz, S., Schmidt, S. S., Stricker, J., et al. (2016). Associations of non-symbolic and symbolic numerical magnitude processing with mathematical competence: a meta-analysis. Developmental Science, 20(3), e12372. https://doi.org/ 10.1111/desc.12372. Sella, F., Tressoldi, P., Lucangeli, D., & Zorzi, M. (2016). Training numerical skills with the adaptive videogame “the number race”: a randomized controlled trial on preschoolers. Trends in Neuroscience and Education, 5, 20–29. Skwarchuk, S. L. (2009). How do parents support preschoolers’ numeracy learning experiences at home? Early Childhood Education Journal, 37, 189–197. Skwarchuk, S. L., Sowinski, C., & LeFevre, J. A. (2014). Formal and informal home learning activities in relation to children’s early numeracy and literacy skills: the development of a home numeracy model. Journal of Experimental Child Psychology, 121, 63–84.

106 Cognitive Foundations for Improving Mathematical Learning Sonnenschein, S., Galindo, C., Metzger, S. R., Thompson, J. A., Huang, H. C., & Lewis, H. (2012). Parents’ beliefs about children’s math development and children’s participation in math activities. Child Development Research. 2012, Article ID 851657, https://doi.org/10.1155/ 2012/851657. Susperreguy, M. I., & Davis-Kean, P. E. (2016). Maternal math talk in the home and math skills in preschool children. Early Education and Development, 27(6), 841–857. Tibber, M. S., Greenwood, J. A., & Dakin, S. C. (2012). Number and density discrimination rely on a common metric: similar psychophysical effects of size, contrast, and divided attention. Journal of Vision, 12(6), 8. https://doi.org/10.1167/12.6.8. Tibber, M. S., Manasseh, G. S. L., Clarke, R. C., Gagin, G., Swanbeck, S. N., Butterworth, B., et al. (2013). Sensitivity to numerosity is not a unique visuospatial psychophysical predictor of mathematical ability. Vision Research, 89, 1–9. Tokita, M., & Ishiguchi, A. (2010). Effects of element features on discrimination of relative numerosity: comparison of search symmetry and asymmetry pairs. Psychological Research, 74(1), 99–109. https://doi.org/10.1007/s00426-008-0183-1. Vandermaas-Peeler, M., Nelson, J., Bumpass, C., & Sassine, B. (2009). Numeracy-related exchanges in joint storybook reading and play. International Journal of Early Years Education, 17(1). https://doi.org/10.1080/09669760802699910. Vos, P. G., van Oeffelen, M. P., Tibosch, H. J., & Allik, J. (1988). Interactions between area and numerosity. Psychological Research, 50(3), 148–154. Wang, J., Odic, D., Halberda, J., & Feigenson, L. (2016). Changing the precision of preschoolers’ approximate number system representations changes their symbolic math performance. Journal of Experimental Child Psychology, 147, 82–99. Wilson, A. J., Dehaene, S., Dubois, O., & Fayol, M. (2009). Effects of an adaptive game intervention on accessing number sense in low-socioeconomic-status kindergarten children. Mind, Brain and Education, 3(4), 224–234. Wilson, A. J., Dehaene, S., Pinel, P., Revkin, S. K., Cohen, L., & Cohen, D. (2006). Principles underlying the design of “the number race”, an adaptive computer game for remediation of dyscalculia. Behavioral and Brain Functions, 2, 19. https://doi.org/10.1186/1744-9081-2-19. Wilson, A. J., Revkin, S. K., Cohen, D., Cohen, L., & Dehaene, S. (2006). An open trial assessment of the number race, an adaptive computer game for remediation of dyscalculia. Behavioral and Brain Functions, 2(1), 20. Wynn, K. (1992). Children’s acquisition of the number words and the counting system. Cognitive Psychology, 24, 220–251. Xu, F., & Spelke, E. S. (2000). Large number discrimination in 6-month-old infants. Cognition, 74(1), B1–B11. Zhou, X., Wei, W., Zhang, Y., Cui, J., & Chen, C. (2015). Visual perception can account for the close relation between numerosity processing and computational fluency. Frontiers in Psychology, 6, 1364.

Chapter 5

Mathematical Development in the Early Home Environment Susan C. Levine*,†,‡, Dominic J. Gibson* and Talia Berkowitz* *

Department of Psychology, University of Chicago, Chicago, IL, United States Department of Comparative Human Development, University of Chicago, Chicago, IL, United States ‡ Committee on Education, University of Chicago, Chicago, IL, United States †

INTRODUCTION Parent support of children’s early math learning tends to take a back seat to other aspects of children’s development—notably early language and literacy skills. Indeed, parents of preschoolers report that they emphasize language and literacy more than mathematics in the home environment, and they also report that they believe this should be the case in preschool settings (e.g., Cannon & Ginsburg, 2008; LeFevre et al., 2009). While parent support of children’s early math development should not supplant their support of language/ literacy development, we argue that early math development is important and consequential, and that family involvement is integral to children’s development in this domain. We base this argument on a few key findings. First, the math support parents provide to young children, notably the math language and activities they engage in with their children, is related to children’s foundational math skills as they enter school (e.g., Gunderson & Levine, 2011; Levine, Ratliff, Cannon, & Huttenlocher, 2012; Levine, Suriyakham, Rowe, Huttenlocher, & Gunderson, 2010; Pruden, Levine, & Huttenlocher, 2012). Second, the level of children’s math knowledge at kindergarten entry predicts their long-term academic success not only in math, but also in reading (e.g., Claessens & Engel, 2013; Duncan et al., 2007), although the processes that link early skills to later skills are not fully understood (Bailey, this volume; Bailey, Duncan, Watts, Clements, & Sarama, 2018). At a societal level, these findings are gaining attention because of projections for steep growth in STEM-related (science, technology, engineering, and mathematics) jobs in the coming decades and the central role of mathematics

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in broad aspects of STEM achievement (e.g., U.S. Department of Education, 2016). Issues of equity are also paramount, because many children from lower socioeconomic backgrounds enter school behind in math, which places them at a disadvantage for the STEM employment opportunities that are expected to grow during their lifetime. Beyond considerations of STEM workforce needs, and associated equity issues, mathematics knowledge is helpful to making good decisions, both related to one’s personal life (e.g., health, finances, etc.) and to being an informed citizen who can understand data that bear on a wide range of societal issues (e.g., Reyna, Nelson, Han, & Dieckmann, 2009). We argue that providing a home environment that supports early math development is an attainable goal for everyone, because adults engage in mathematical thinking every day—when they shop, cook, measure, make plans, think about time, plan routes, assemble, and build things. Thus there are ample opportunities for parents and caregivers to engage children in mathematical thinking. In this chapter, we primarily examine the home math environment through the lens of early “math talk,” both numerical and spatial, not only because of its relation to early math skill, but because of evidence that this kind of talk is malleable in terms of quantity and quality (e.g., Gibson, Gunderson, & Levine, under review; Vandermaas-Peeler, Boomgarden, Finn, & Pittard, 2012). In addressing this important kind of early math support, we not only consider the math talk parents provide but also the cospeech gestures that commonly accompany math talk, asking if this makes a difference to children’s mathematical learning and if so, why. In addition, we consider how the attitudes that parents have about math might influence their math talk. We know from prior research that math anxious individuals tend to avoid math themselves, and we have preliminary evidence that math anxious parents not only engage in less math talk with their young children (Eason, Nelson, Dearing, & Levine, 2017) but that their math interactions tend to be less effective (Herts et al., 2017; Maloney, Ramirez, Gunderson, Levine, & Beilock, 2015). In considering the relation of math talk in the home environment to young children’s math development, we organize the chapter around the following the three questions: (1) To what extent is parent math talk in the early home environment related to children’s math outcomes, and is there evidence that this relation is causal? (2) What is the role of gesture in supporting math learning, and why might this be another important facet of children’s math input? (3) Are parent math attitudes and beliefs related to math in the home environment and by extension, to children’s math achievement?

PARENT MATH LANGUAGE AND ASSOCIATED OUTCOMES Number Talk When parents are asked about early math supports, they are highly likely to think about counting and other number-related activities (e.g., Anderson, 1997;

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Cannon & Ginsburg, 2008). Indeed, both questionnaire and observational studies of the early home environment reveal that number-related activities and talk are a common form of parent-child math engagement. Although these methods have differing strengths and weaknesses, and provide different kinds of data, the results that have emerged are largely convergent, supporting a relation between parents’ number-relevant talk and activities, and children’s number knowledge.

Questionnaire Studies While parent questionnaires have the advantage of capturing general levels and types of number engagement, they are limited by parents’ memories of what actually occurred, and may be subject to demand characteristics, particularly if the respondents know the research is focused on early math. In addition, because questionnaires require the recall of past events, they cannot provide detailed information about the quantity and quality of number talk or accompanying co-speech gestures that may be important to children’s number learning. Despite these limitations, questionnaire studies reveal a positive correlation between parents’ reported engagement in home numeracy activities and children’s number knowledge (e.g., Blevins-Knabe & Musun-Miller, 1998; LeFevre et al., 2009; LeFevre, Clarke, & Stringer, 2002; Manolitsis, Georgiou, & Tziraki, 2013; Niklas & Schneider, 2014; Skwarchuk, Sowinski, & LeFevre, 2014). Early questionnaire studies mainly focused on parent reports of formal math activities that involved directly teaching children to count, to read and write numerals, or to do simple calculations. These studies largely showed a positive relation between the frequency of these activities and children’s math skills (e.g., Blevins-Knabe & Musun-Miller, 1998; LeFevre et al., 2002). More recently, LeFevre and colleagues drew a distinction between direct math activities where the purpose was to teach about number (e.g., counting, numeral recognition, simple arithmetic) and indirect math activities, where there is the potential to learn about number but this is incidental (LeFevre et al., 2009). They asked how these different categories of activities, direct and indirect, related to kindergarten-2nd graders’ number knowledge (number, addition, and subtraction) and math fluency (latency to correct responses on addition calculations). In a parent questionnaire study, they found that, controlling for a range of background factors that related to children’s math scores, the frequency of indirect math experiences was significantly related to both math knowledge and math fluency, whereas the frequency of direct math experiences was significantly related only to math fluency. In a related study, Skwarchuk et al. (2014) found that parent reports about the frequency of kindergarteners’ direct numeracy activities were related to children’s symbolic number skills whereas their reports about the frequency of indirect math activities were related to children’s nonsymbolic arithmetic. In contrast to these findings, a questionnaire study carried out with the parents of 3- to 8-year-olds found that a general home math factor (but not a direct

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math activities factor, an indirect math activities factor, or a spatial activities factor) was significantly related to children’s math skills, as reported by parents (accounting for 8% of the variance), controlling for parents’ approximate number sense (ANS) acuity, their math anxiety, and household income (Purpura, Napoli, Wehrspann, & Gold, 2017). Considered together, these studies provide some evidence for differential relations of direct and indirect math activities to children’s math skills, but there are inconsistencies across studies, perhaps due to different age groups and measures. It is also possible that the distinction between indirect and direct math activities is not tapping the key factors that are important to children’s math skills. Instead, it may be that the math content that is taught, and the ways it is being taught during direct and indirect activities, rather than the formality or directness of the math activities per se, is driving the relations to child math outcomes. Observing parent-child math interactions during direct and indirect math activities opens the door to probing this issue more deeply.

Observational Studies Like studies utilizing questionnaires, observational studies have both advantages and disadvantages. On the plus side, they allow researchers to measure actual parent supports of young children’s math rather than relying on parent reports of such behaviors. Further, they provide detailed information about what was actually said and what accompanying supports were used, such as the use of gestures and manipulatives, thus providing more detailed information about the qualitative as well as quantitative aspects of math interactions. However, by their very nature, observational studies are time limited, capturing only a small slice of life. Thus there is a risk that the math that occurs during the observations might not be representative of the math support the child actually receives. Moreover, as is the case in questionnaire studies, if parents are told that the researchers are interested in math, there are likely demand characteristics that lead to more math talk than is typical. Yet, even with these limitations, observational studies, like questionnaire studies, show that parents’ number talk is related to children’s number knowledge, and that these relations hold controlling for many variables that could provide alternative explanations for these relations, such as parent education, overall parent talk, child math knowledge, and family SES (e.g., Casey et al., 2018; Levine et al., 2010; Ramani, Rowe, Eason, & Leech, 2015). Observational Studies in the Lab In observational studies, researchers commonly videotape parent-child dyads playing with a standard set of toys for a fixed amount of time and code the number talk provided by the parent. In one study of this kind, Casey et al. (2018) utilized videotapes of 140 mothers interacting with their 36-month-old children who participated in the NICHD Study of Early Child

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and Youth Development from the Boston site. They coded maternal number talk during interactions with two of the boxes in the 3-Boxes task (NICHD Early Child Care Research Network, 1999). These two boxes contained a cash register and dress up clothes (Box 2) and Duplo blocks of various sizes (Box 3), both of which afforded the opportunity for number talk. Mothers were simply asked to have their child play with the toys in each of the boxes, reducing the chance that demand characteristics could influence parent math talk. The researchers coded parent labeling of numerals, labeling the cardinal value of sets, and engaging in one-to-one counting. Controlling for variables including child gender, child IQ, maternal education, income to needs ratio, as well as other variables that might be associated with number talk, results showed that labeling the cardinal value of sets, but not the other categories of number talk, predicted children’s performance on the Woodcock-Johnson Applied Problems subtest when children were 4.5 years old and in 1st grade. Using a similar approach, Ramani et al. (2015) examined parent number talk with 3- to 5-year-old children during a 15-min observation session where dyads were asked to play with a book, puzzle, and board game. All parent number talk was coded and categorized into the following categories: “foundational” number talk (i.e., counting, numerical identification) and “more advanced” number talk (i.e., cardinality, ordinal relations, and arithmetic). Parent talk about foundational number concepts was not related to children’s math skills, either foundational (e.g., counting, numeral identification) or more advanced (counting principles, enumeration and cardinality, number line estimation, and magnitude comparison), possibly because many children had already learned the foundational concepts and input focused on these concepts was not helpful to children’s learning of more advanced number concepts. In contrast, parent use of more advanced number talk was related to children’s advanced number knowledge, controlling for child age.

Naturalistic Home Observations In other observational studies, researchers have observed parents and children engaging in naturalistic interactions and have then coded number talk that occurred, typically videotaping but occasionally audiotaping the interactions. We took this approach in a longitudinal language project examining the relation of parent language input to children’s language and cognitive development. The families were recruited to represent the demographics of the greater Chicago area with the requirement that all participating families reported that English was the language spoken in the home. In this study, we visited parent-child dyads in their homes every 4 months, beginning when children were 14months of age and continuing until they were 30months of age. During each visit, the researcher videotaped the parent-child dyads for approximately 90 min, instructing the parents to go about their normal activities. At certain visits, particular aspects of children’s language and cognitive skills were assessed, including their math knowledge.

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Levine et al. (2010) found substantial variation in the amount of number talk parents engaged in during five 90-min sessions with their 14- to 30-month-old children (Mean parent number tokens (use of numbers between 1 and 10) ¼ 90.8 (sd ¼ 61.3); range: 4–257). In other words, there was about a 60-fold difference between the children who heard the fewest number words and those who heard the most number words. Extrapolating, this range amounts to a substantial input gap between a child hearing as few as 1500 number words to as many as 100,000 number words in a year, assuming 8-h waking days. The majority of these parent number words involved labeling the cardinal value of sets ((e.g., “You have three bears!”) (50%) or counting (32%)—either counting objects (three trucks: “One (points to first truck), two (point to second truck, three”) or reciting the count list without present objects, with the former being more common. Further, parent number words were most frequent for smaller numbers. As is shown by Fig. 1, number tokens fall off steeply from 1 to 10, with 76.0% of all parent number tokens consisting of 1–3, a pattern that has been reported by others (Ramscar, Dye, Popick, & O’Donnell-McCarthy, 2011). As is the case for many aspects of language input to young children, number talk was associated with socioeconomic status, as assessed by a composite of education and income (r ¼ .30, P < .05). Moreover, recent analyses show that the SES difference was significant for cardinal number talk (r ¼ .49, P < .01) but not for counting (r ¼ .06, P ¼ .75). We also asked whether parents engage in more number talk with boys than girls but did not find significant differences. The main question Levine and colleagues asked (Levine et al., 2010) was whether amount of parent number talk as assessed by their number word tokens while interacting with their child during five sessions from 14 to 30 months, predicted children’s number knowledge. Using multiple regression, they found that variation in parent number talk predicted children’s cardinal number knowledge at 46 months of age as assessed on the Point-to-X task

FIG. 1 Cumulative production of each numerosity 1–10, by parents and children. (From Levine, Huttenlocher, Taylor, & Langrock, 1999; reproduced with the permission of the publisher.)

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(where children see two groups of objects and are asked to point to the group with x objects), and that this relation held controlling for family socioeconomic status, child number tokens, and parent other word tokens (see Fig. 2). In a follow-up study, Gunderson and Levine (2011) coded qualitative aspects of the same parent-child interactions coded by Levine et al. (2010). In a multiple regression analysis they found that parents’ counting or labeling the cardinal value of visible sets when children were 14–30 months of age was significantly related to children’s cardinal number knowledge at 46 months whereas this was not the case for counting and cardinal utterances that did not refer to visible sets (e.g., practice reciting the count sequence). Further, despite the fact that larger number words were relatively rare, parents’ counting or labeling visible sets from 4 to 10 was significantly related to children’s understanding of the cardinal meanings of number words at 46 months of age whereas parents’ counting or labeling set sizes in the subitizable range (1–3) was not. In Table 1 we provide two examples that illustrate the difference between a number interaction that uses number words to refer to present objects (Example 1) versus one that does not (Example 2). Although the first exchange does not contain as many number words as the second, the child has more opportunity to link the count sequence to the cardinality of the set in this interaction than in the second. Other studies have examined parent number talk during daily routines such as meal time. Susperreguy and Davis-Kean (2016) coded 4-h of talk that occurred during breakfast time and dinner time, because parent-child conversations are particularly rich during these times (e.g., Tabors, Beals, & Weizman, 2001). Parent number talk was highly variable across the 40 families in the study, ranging from 4 to 195 instances. As in the Levine et al. (2010) study, the most common type of parent math talk involved labeling cardinal values of sets (42%). Counting, ordinal numbers, and naming digits each accounted

Child cardinality knowledge

18 16 14 12 10 8 6 4 2 0 0

1

2 3 4 Parent number talk (log)

5

6

FIG. 2 Scatterplot displaying the relation between parent cumulative number word token (log) when child age was between 14 and 30 months and child cardinal number knowledge at 46 months (n ¼ 44). (From Levine et al., 2010; reproduced with the permission of the publisher.)

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TABLE 1 Examples of Types of Math Talk Parents Engage in with their Young Children Example 1: 30 months old boy; Father and son looking at photo album F: I think that was his fourth birthday so there’s four candles, see? One, two, three, four. (father points to each candle in turn). Example 2: 26 month old girl; Mother repeatedly tossing child in air M: One, two, three, whee! C: Again! M: Again? M: How many more times? C: Four? M: Four? Ok. M: One, two, three! C: Again! ….(continues until the child has been thrown in the air four times, each time with the count “One, two, three” accompanying each throw, sometimes with a whee!).

for an additional 10% of parent number talk. Further, they found that parent number talk was related to children’s math knowledge, as assessed by their scores on the Test of Early Mathematics Ability (TEMA; Ginsburg & Baroody, 2003), administered 1 year after the naturalistic observations took place.

Experimental Studies As summarized previously, questionnaire and observational studies provide evidence that parent number talk and children’s number knowledge are positively related. However, these studies do not provide evidence of a causal relation even when they control for other variables that could account for this relation. One could argue, for example, that children who have more math knowledge or are more interested in math may elicit or initiate number talk with their parents, although Anderson (1997) reports that parents are typically the ones to incorporate math into activities in the home. It is also possible that parents with higher math ability not only talk more about number, but also have children who are predisposed to learn math with greater ease (e.g., Elliott, Braham, & Libertus, 2017). A growing set of experimental studies are providing evidence of a causal link between parent number talk and activities and children’s number knowledge. These studies show that it is possible to enhance the number talk parents provide to young children and that doing so leads to gains in children’s number knowledge. Vandermaas-Peeler and colleagues conducted two studies in which

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parents were instructed to support their children’s number learning either in the context of playing a board game (Vandermaas-Peeler, Ferretti, & Loving, 2012) or making a recipe (Vandermaas-Peeler, Boomgarden, et al., 2012). In both of these studies, the parents in the numeracy awareness intervention group provided more number talk and more advanced number talk than parents who were not given number talk instructions (i.e., the control group). For example, in the cooking context the parents in the intervention group were more likely to engage the child in addition and subtraction, moving beyond the kinds of counting activities that they were prompted to engage in. In both numeracy awareness interventions groups, children were asked more numeracy questions and likely because of this more frequently provided numerical information. In the cooking study, children’s learning was assessed by administering the TEMA-2 immediately following the activity. The intervention group did not score higher than the control group on this test, likely because it takes more input than was provided in a single activity to develop the math concepts that are tapped on this test, such as cardinal number knowledge. Nonetheless, both the cooking and the board game interventions resulted in significant enhancements in the quality and quantity of parent number support, suggesting that such interventions could result in changes in children’s math knowledge over time.

Experiments in the Lab Experimental studies have also shown that certain activities and kinds of talk lead to gains in children’s number knowledge (see also Libertus, this volume; Ramani et al., this volume). For example, researchers have shown that when 4-year-olds from low socioeconomic backgrounds played a linear number board game, they showed gains in number line representations and on comparing number magnitudes, relative to control groups. Those who played the linear board game also showed greater gains than control children in simple arithmetic, after they were provided with a lesson on this topic (e.g., Ramani & Siegler, 2008; Siegler & Ramani, 2009). In terms of the number talk provided, Laski and Siegler (2014) found that a critical component of the linear board game intervention is that it requires counting on, for example, when the child spun a “two” and was on the fifth space, they counted “six, seven” rather than “one, two.” Mix, Sandhofer, Moore, and Russell (2012) conducted a training study, manipulating the context in which children heard new number words. Once a week for 6 weeks, children heard experimenters either only count sets, only label the cardinal values of sets, label the cardinal values of sets and then immediately count each object in the set, or alternate between counting and labeling cardinal values at each session. At the end of the training period, only those children who had received the combined counting and cardinal labeling training showed significant improvements in their understanding of the cardinal principle. This finding indicates that hearing an adult both count and label the

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cardinal value of sets helps children learn a foundational math concept and does so more effectively than providing children with counting or set size labels alone or even alternating between these two kinds of input. Unfortunately, this kind of input, counting and labeling set sizes, occurs relatively rarely in naturalistic interactions (Casey et al., 2018; Levine et al., unpublished data). The results of two other training studies suggest that young children learn more about number when object sets are not distracting. That is, perceptual richness of familiar objects may make them overly attractive and focus children’s attention on the objects themselves rather than on their role as members of a set that is being counted (Petersen & McNeil, 2013). Consistent with this finding, young children’s understanding of cardinality improved when they were trained with pictures of objects but not when they were trained with the objects themselves (Petersen et al., 2014). Together, these studies demonstrate that small differences in both the type of input and the manipulatives involved in various activities can lead to differences in children’s learning and performance.

Experiments in the Field Building on findings showing that families are more likely to embrace their role in supporting children’s language and literacy than to embrace their role in supporting young children’s math (e.g., Cannon & Ginsburg, 2008; Skwarchuk, 2009), several research teams have examined whether interventions that involve sharing number books with young children at home can increase their number knowledge (Gibson et al., under review; Purpura et al., 2017). In a parent-delivered book intervention study, Gibson et al. assessed children’s understanding of cardinal number using the Give-N task (Wynn, 1990) and randomized them into groups that received one of three book types: number books that focused on 1–3, number books that focused on 4–6, or control adjective books. Parents were asked to read the books to their child as many times a week as possible over the course of a month. At the end of the month, children were once again tested on their understanding of number words. As predicted, children who received the adjective book did not make significant gains in their number knowledge over the month long period. However, children in the number book conditions progressed about one knower level, on average. For instance, some children understood the quantities represented by ‘one’ and ‘two’ at the beginning of the month, and understood ‘one,’ ‘two,’ and ‘three’ at the end of the month. Further, children who only understood the number 1, or the numbers 1 and 2 at pretest learned from the small (1–3) but not the large (4–6) number books, whereas children who also understood the numbers 3 and 4 at pretest learned from both the small and large number books. These findings suggest that at the beginning of number word learning, children benefit from having the input tailored to their level of

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knowledge whereas once children understand three or four number words, they can benefit from exposure to a wider range of number input. This difference suggests that the knowledge children are working on when they are just beginning to learn the cardinal meaning of number words—learning the meaning of the next number word—is qualitatively different from the knowledge they are working on when they are three- and four-knowers—learning the cardinal principle. Even though parents tend to tailor their number input to young children’s number knowledge (Saxe, Guberman, & Gearhart, 1987), some parents appear to do this in a more effective way than others, opening the door for taking this factor into account in the design of interventions.

Spatial Talk While there is no doubt that numerical skills are central to children’s mathematical development, there is evidence that spatial skills contribute to this development. There is accumulating evidence from developmental studies that spatial abilities are correlated with mathematical abilities (e.g., Casey et al., 2018; Gunderson, Ramirez, Beilock & Levine, 2012; Li & Geary, 2013, 2017; Mix et al., 2016, 2017), as well as evidence from studies of adults that spatial abilities are broadly predictive of STEM achievement and career paths, even controlling for verbal and mathematical abilities (e.g., Wai, Lubinski, & Benbow, 2009). There is some evidence for a causal relation, as several studies have shown that training a spatial skill—for example, mental rotation—leads to gains in numerical aspects of math— for example solving equivalence problems (e.g., Cheng & Mix, 2014), although not all studies find this relation (e.g., Hawes, Moss, Caswell, & Poliszczuk, 2015). As was the case for parent number talk, observational studies have revealed marked variation in parent spatial talk to young children. In our longitudinal language study, we coded and analyzed three categories of spatial words that parents and children use to describe objects, which we dubbed “what spatial words,” including nine observation sessions that occurred every 4 months between child ages 14 and 46 months. The categories we coded were talk about shapes (e.g., circle, square triangle), sizes (e.g., tall, short, wide, narrow), and spatial features (curved, straight, corner, etc.). Although “where spatial words” (e.g., in, on, under, in front of, above, etc.) are frequently used to talk about locations, we did not include these words because their use was very highly correlated with overall parent talk, and thus it is difficult to examine how these types of words relate to children’s spatial thinking. Over nine 90-min observations (total 13.5 h), parents averaged 167 spatial words referring to these categories, with a range of 5–525 words. Assuming 8-h days, we estimated that a child at the high end of input would hear over 113,000 “what spatial words” in a year and a child at the low end would only hear only about 1100 of these words, more than a 100-fold difference.

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FIG. 3 (A) Sample item from the Spatial Transformation task (Levine et al., 1999; Pruden et al., 2012). Children were asked to point to the shape the pieces would make if they were put together. (B) Sample item from the Spatial Analogies test. Children were asked to select the one picture that “goes best” with the target picture. (From Huttenlocher & Levine’s (1990) Primary Test of Cognitive Skills, Monterey, CA: CTB/McGraw-Hill Companies, Inc. Reproduced with permission of McGraw-Hill Companies, Inc.)

Moreover, as for parents’ number talk, parents use of spatial words when talking to their children was related to children’s spatial tokens, controlling for parent other-word tokens. Further, parent spatial language was related to children’s spatial skills on nonverbal spatial tests given at 54 months age, including a mental rotation test and a spatial analogies test (see Fig. 3), relations that were mediated by children’s spatial language production (Pruden et al., 2012). Other studies also have provided evidence that spatial language is related to young children’s spatial thinking. For example, Verdine et al. (2014) assessed 3-year-old children’s spatial thinking using a spatial assembly task. Parents of the participating children were asked to check off which of 14 spatial words they used with their children. Findings showed that parents’ report of their use of particular words—between, below, above, and near—predicted children’s performance on the spatial assembly task. Moreover, the relations between the spatial words parents checked off and children’s performance on particular assembly items were quite specific. For example, two of the spatial words, above and below, were significantly related to children’s vertical location scores, but not to their other spatial skills. Verdine et al. hypothesized that this was because knowledge of these words was particularly helpful in determining the correct level of a piece in the construction of an object, perhaps by providing children with another way to encode the spatial relationship

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present in the model or by focusing their attention on the dimension and actions that were relevant to achieving a match to the model. The relation of spatial language to spatial thinking has led researchers to ask whether parent spatial talk to young children is more likely to occur in certain contexts than others. One study found more parent spatial talk to their 3- to 5-year-olds during block play than during non-construction activities (Ferrara, Hirsh-Pasek, Newcombe, Golinkoff, & Lam, 2011). Moreover, parent and child spatial language was more frequent when parent-child dyads engaged in guided block play that involved following step-by-step instructions to assemble a structure than during either free play with blocks or play with an already assembled structure. In the longitudinal language study described earlier, Levine and colleagues examined the frequency of parent-child spatial activities such as block play and puzzle play with preschool boys and girls. Between 26 and 46 months of age, about half of the dyads (51%) engaged in puzzle play, with no sex difference in the number of boys and girls who played with puzzles at least once over the six sessions that were coded (Levine et al., 2012). Nonetheless, parents tended to play with more complex puzzles with boys than girls, and parent spatial language was more frequent with more complex puzzles, resulting in boys hearing more parent spatial language than girls during puzzle play. Further, in this same database, parent-child block play was more common for boys than girls between 26 and 46 months of age, although not between 14 and 22 months (Petersen & Levine, in preparation). Given the Ferrara et al. (2011) findings, greater engagement of boys than girls in block play could result in them hearing more parent spatial language than girls. In fact, Pruden and Levine (2017) found that boys in our longitudinal language study use more spatial language than girls during naturalistic interactions with their parents between 14 and 46 months of age, although this was not the case for other aspects of language. This sex difference was fully mediated by a difference in parent spatial language use. This finding raises the possibility that differences in parent spatial talk to boys and girls could contribute to the male advantage on certain spatial tasks (e.g., Linn & Petersen, 1985; Moore & Johnson, 2008; Quinn & Liben, 2008; Uttal et al., 2012) perhaps by calling their attention to spatial information in the world and/or by reducing cognitive load while they are solving spatial problems.

Summary: Math Talk Considered together, the take-home message from studies of parents’ math talk with their children is that both their number and spatial talk are related to children’s talk about these domains and the development of children’s knowledge and skills in these domains. Parents’ early number talk is related to children’s cardinal number knowledge, controlling for a variety of other alternative factors that could account for this relation. Further, parents’ spatial talk is related to children’s spatial thinking as tapped by nonverbal

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tasks such as those assessing mental transformation skill and block construction. Importantly, experimental studies show that it is possible to enhance parents’ math conversations with young children, and that this results in enhanced child performance on outcome measures, providing evidence of a causal link between the parents’ math input and children’s math learning. A remaining question is whether these interventions lead to enduring achievement gains and/or greater interest in mathematics. There is some indication that this may be the case for number talk, based on the few studies with longer term follow-up tests, but even these only follow-up a few years later. For example, Geary and colleagues found that children’s early acquisition of the cardinal principle is associated with better school entry number knowledge, controlling for other factors (Geary et al., 2018).

GESTURE: AN ADDITIONAL SUPPORT FOR CHILDREN’S MATH LEARNING One of the most common features of early math talk between parents and young children is the frequent use of gestures (Fuson, 1988; Goldin-Meadow, Levine, & Jacobs, 2014; Suriyakham, 2007). For instance, pointing and fingercounting are some of the most frequent strategies used by children during counting and arithmetic activities (Fuson, 1982, 1988; Geary & Burlingham-Dubree, 1989; Gelman & Gallistel, 1978; Graham, 1999; Saxe, 1977; Siegler & Shrager, 1984). As children get older, they continue to rely on their hands when they solve more complicated mathematical problems such as when explaining their solutions to mathematical equivalence problems (e.g., “3 + 5 + 7 ¼ 8 + 7”; Perry, Church, & Goldin-Meadow, 1988). Likewise, parents and teachers frequently gesture when counting sets or talking about arithmetic (Flevares & Perry, 2001; Goldin-Meadow, Kim, & Singer, 1999; Suriyakham, 2007). Accordingly, many theories suggest that number gestures play a role in the development of verbal number knowledge and counting principles (e.g., Butterworth, 1999, 2005; Fuson, 1982, 1988; Gelman & Gallistel, 1978; Gracia-Bafalluy & Noe¨l, 2008). However, despite the ubiquity of number gestures, there is surprisingly little agreement concerning the precise role (or roles) that gestures play in number development. This partly stems from the fact that unlike in the case of parent number talk there is not yet a substantial body of research linking parents’ number gestures to children’s number knowledge and academic math performance more broadly. Nevertheless, research into how children and parents gesture about number and what function children’s own gestures play in their developing understanding of number and mathematics can provide clues into how input involving number gestures may benefit children. Therefore in this section we focus on how parents and children use number gestures as well as highlight some initial evidence that input involving gestures may have a positive impact on children’s mathematical development.

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Counting Gestures Many of the proposed functions of number gestures revolve around children’s comprehension and implementation of the counting principles (Alibali & DiRusso, 1999; Di Luca & Pesenti, 2008; Fuson, 1988; Gelman & Gallistel, 1978; Graham, 1999; Potter & Levy, 1968; Saxe, 1977; Saxe & Kaplan, 1981). Counting gestures, such as pointing to objects while counting and raising fingers while reciting the count list, are used by children as young as the age of two (Fuson, 1982; Gelman & Gallistel, 1978). As children age and improve in their ability to accurately count, their use of counting gestures appears to increase. For instance, Suriyakham (2007) found that 30-month-olds produce more counting utterances without pointing gestures than with counting gestures, but by 38 months children count with pointing gestures as frequently as they count without them. Similarly, Saxe (1977) found that three-year-olds either do not gesture or fail to coordinate their points and verbal counting when tasked with reproducing sets containing 4–9 items. By the age of four, children almost always pointed while counting in this context and did so correctly on the majority of trials. It has been hypothesized that these pointing gestures play a functional role in the development and implementation of the counting principles. For instance, children may use counting gestures to keep track of the number words while reciting the count list and to help maintain the stable order of the numbers in the count list (Fuson, 1982; Wiese, 2003). Pointing while counting also appears to be particularly useful for understanding and implementing the one-to-one correspondence principle by helping children map a single number word to each item being counted (Alibali & DiRusso, 1999). In fact, 3- to 4-year-old children who watched an experimenter count believed a count was incorrect if the experimenter pointed to one object twice even if the experimenter used the correct order and number of number words when counting (Briars & Siegler, 1984). Moreover, Saxe (1977) observed that 4-year-olds almost always pointed during successful implementation of oneto-one correspondence during counting. Researchers have also highlighted the connection between number processing and finger representation as evidence that children’s fingers play a role in the development of counting (Domahs, Moeller, Huber, Willmes, & Nuerk, 2010; Gerstmann, 1940; Moeller et al., 2012). Notably, children’s ability to represent their fingers mentally and discriminate between their fingers is correlated with math ability (Fayol, Barrouillet, & Marinthe, 1998; Noe¨l, 2005). Number gestures have also been linked to numerical cognition by studies demonstrating common neuroanatomical substrates of finger representation and number processing (Pesenti, Thioux, Seron, & Volder, 2000; Piazza, Mechelli, Butterworth, & Price, 2002; Pinel, Piazza, Le Bihan, & Dehaene, 2004).

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Cardinal Number Gestures Cardinal number gestures (e.g., holding up five fingers to indicate 5) represent another significant portion of children’s early number input and developing ability to communicate about numbers. Although the specific combinations of fingers that make up canonical number gestures often differ between cultures, the use of conventionalized cardinal number gestures is common across cultures (Bender & Beller, 2012). Cardinal number gestures are particularly interesting since they can be used in combination with or in place of number words to label specific set sizes. However, they also are iconic representations of number in that they represent quantities through one-to-one correspondence between fingers in a gesture and items in a set. There is some evidence to suggest that learning iconic representations may be easier than learning arbitrary symbol-meaning mappings, particularly around the age of two when children begin to learn about numbers (Namy, Campbell, & Tomasello, 2004). Accordingly, researchers have argued that number gestures and other iconic number systems are easier for children to learn than the purely symbolic systems, like number words (Wiese, 2003) and may therefore serve as a bridge between nonsymbolic and symbolic representations of number (Gunderson, Spaepen, Gibson, Goldin-Meadow, & Levine, 2015). Some have argued that children become proficient with number gestures only after becoming proficient with number words and therefore that number gestures cannot support the acquisition of number words. For instance, in one study, Nicoladis, Pika, and Marentette (2010) found that 4- to 5-year-olds (but not 2- to 3-year-olds) were more accurate when labeling sets using verbal number labels (number words) than cardinal number gestures, with the greatest difference in accuracy coming on larger sets (6, 7, 8, and 9). They concluded that children’s knowledge of number words exceeds their knowledge of number gestures, and thus that there was no evidence to suggest children use their knowledge of number gestures to learn number words. Crollen, Seron, and Noe¨l (2011) interpreted these findings to mean that number gestures do not precede the acquisition of number words and thus that the acquisition of number words is not rooted in children’s acquisition of cardinal number gestures. Importantly, Nicoladis et al.’s (2010) finding that 4- to 5-year-olds have great proficiency with number words than number gestures may be due to children at this age already knowing the cardinal principle. Having knowledge of the cardinal principle would mean that these children were likely well practiced using number words and at or near ceiling on tasks involving number words. Consistent with this interpretation, we found that subset knowers (children who had not yet learned the cardinal principle) were significantly more accurate when labeling the number of items in a set using gesture compared to speech (Gunderson et al., 2015). This effect appeared to be strongest for numbers immediately above children’s knower level (i.e., numbers which

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children had not yet mastered in speech). In contrast, cardinal principle knowers showed either no difference or a slight advantage for labeling sets in speech compared to gesture depending on the set size. Subset knowers also showed an advantage when labeling larger sets (above four) using number gestures. Specifically, subset knowers made larger cardinal gestures when labeling larger sets sizes (i.e., 10) than when labeling smaller set sizes (i.e., 5). In contrast, subset knowers’ verbal responses did not increase with the size of the set. Another interesting facet of Gunderson et al.’s (2015) findings is children’s simultaneous use of verbal labels and gestures, which often did not match. In such cases, children’s gestures were much more accurate than their speech. Previous studies outside the domain of early number development have found that such gesture-speech mismatches are predictive of subsequent language (Iverson & Goldin-Meadow, 2005) and conceptual development (Church & Goldin-Meadow, 1986; Perry et al., 1988). Therefore in a follow-up training study, Gibson et al. (under review) looked at whether children who produced such mismatches in the early stages of number development (one- and two-knowers) were more likely to learn a new number word than those who did not make such mismatches. They found that indeed children who mismatched and received rich number input during training (counting, cardinal labeling of the set size, and comparison of adjacent set sizes) were more likely to learn a new number word than both those who did not mismatch and those who received sparse number input during training (just counting). This finding suggests that number gestures in combination with speech are at least predictive of children’s subsequent learning and perhaps play a functional role in this learning. Analyses of parents’ and children’s use of number gestures at home provides additional evidence that number gestures may be a pathway for positively impacting children’s verbal number knowledge. Suriyakham (2007) found that children whose parents frequently gestured when counting and labeling sets said more number words and performed better on early measures of cardinality such as the Point-to-X task than children whose parents gestured less frequently when counting and labeling sets. Together, these findings show that prior to fully comprehending number words, children are more accurate when labeling both small and large sets using gestures compared to speech (Gunderson et al., 2015). Moreover, how children gesture about numbers in combination with their speech can help reveal whether children are on the cusp of learning a new number word (Gibson et al., under review). Although more experimental research is necessary to establish a causal link between parents’ gestures and children’s subsequent number knowledge, there is some evidence that these are related (Suriyakham, 2007). These findings open the door to the possibility that parents’ number gestures and children’s knowledge of cardinal number gestures could play a supportive role in children’s acquisition of number words and

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could be leveraged to promote verbal number knowledge. Again, this is important given the relation between children’s early cardinal number knowledge and their later math skills (Geary et al., 2018).

Gesture and Arithmetic Gestures continue to be revealing and useful as children progress to solving simple arithmetic problems. For instance, Levine, Jordan, and Huttenlocher (1992) asked 4.5- to 7-year-olds to solve basic addition and subtraction problems that were presented in the form of word problems, number fact problems, and nonverbal problems. The latter were presented with sets of disks that were either added to form a larger hidden set or were subtracted to form a smaller hidden set. Children below the age of 5.5 years rarely used their fingers when attempting to solve the arithmetic problems. However, the older children used their fingers in much the same way that they would use manipulatives. In fact, when manipulatives were not available to children, as was the case on word problems and number fact problems, children used their fingers more than when manipulatives were available, as was the case on the nonverbal problems, suggesting that they viewed fingers and manipulatives as substitutable. Specifically, children may use their fingers as a working memory aid during problem solving (Geary, 1990). Furthermore, children were more accurate in solving the verbal arithmetic problems when they used their fingers than when they did not, a finding that mirrors that of Jordan, Kaplan, Ramineni, and Locuniak (2008) for kindergarten age children. Children from different socioeconomic backgrounds have been shown to differ in their finger use during calculations. For instance, in kindergarten, middle-income children are more likely to use their fingers to perform arithmetic than low-income children, but by first grade, the reverse is true: low-income children are more likely to use their fingers than middle-income children ( Jordan, Levine, & Huttenlocher, 1994). Likewise, middle-SES children’s use of fingers during arithmetic decreases between kindergarten and second grade while low-SES children’s use of fingers increases over this period ( Jordan et al., 2008). Importantly, Jordan et al. (2008) found that finger use in kindergarten was a strong positive predictor of calculation abilities in second grade. However, finger use in second grade was a negative predictor of children’s calculation abilities. These findings suggest that gestures may serve as a helpful tool for children’s introduction to arithmetical operations but may become a crutch as children are required to move to more advanced strategies. In support of this idea, Siegler and Jenkins (1989) have argued that children’s use of their fingers when solving problems evolves over the course of mathematical development. For example, initially children use a “count all” strategy (holding up fingers for each addend of an addition problem and then counting all of the raised fingers) before moving to a “count on” strategy (counting on from the first addend, e.g., holding up two fingers and then

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counting “three, four, five, and so on” for the second addend). This progression suggests that children’s gestures can reveal how their strategies for arithmetic are changing and becoming more advanced and may even help support that developmental change. For instance, Siegler and Shrager (1984) suggested that early arithmetic strategies such as finger and verbal counting might help children form problem-answer associations that enable them to progress to more advanced retrieval-based problem solving. As children progress to more advanced mathematical concepts, such as mathematical equivalence, gestures continue to play an important role in how children think and communicate about math problems (McNeil et al., 2006). Likewise, adults frequently gesture when explaining difficult math problems to children (Crowder & Newman, 1993; Flevares & Perry, 2001; Neill, 1991; Roth & Welzel, 2001; Zukow-Goldring, Romo, & Duncan, 1994). In a study of one-on-one instruction with 9- and 10-year-old children, teachers communicated 40% of the problem solving strategies they taught through gesture (Goldin-Meadow et al., 1999). Importantly, there is some evidence to suggest that children benefit more from instruction that involves gesture than instruction that does not involve gestures (Church, Ayman-Nolley, & Mahootian, 2004; Valenzeno, Alibali, & Klatzky, 2003). In fact, gestures may be especially effective in getting children to generalize to new types of problems when learning abstract mathematical concepts (Novack, Congdon, Hemani-Lopez, & Goldin-Meadow, 2014). As an inherently spatial form of communication, co-speech gesture appears to be helpful in supporting children’s spatial language and spatial skill as well. Parents vary widely in how often their spatial utterances are accompanied by gestures, ranging from 0% to 44% of their spatial utterances in one study (Cartmill, Pruden, Levine, & Goldin-Meadow, 2010). Furthermore, young children’s spatial language is more highly predicted by parents’ spatial language that is accompanied by gesture than by parents’ spatial language that is unaccompanied by gesture (Cartmill et al., 2010). Additionally, Young and colleagues have shown in an experiment, that spatial language that is accompanied by gesture is more effective in supporting children’s puzzle ability than spatial language not accompanied by co-speech gesture or by nonspatial language, either with or without co-speech gesture (Young, Cartmill, Levine, & Goldin-Meadow, 2014).

Summary: Gesture In sum, young children frequently gesture when communicating about number and other aspects of mathematics (e.g., Jordan et al., 2008; Pruden et al., 2012). Children’s gestures can reveal a lot about what they know and how they are thinking about math problems, enabling parents and teachers to target input to children’s current stage of development (e.g., Gibson et al., under review; Jordan et al., 2008; Perry et al., 1988). Moreover, children’s own

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gestures may help scaffold their early verbal mathematical knowledge. Similarly, parents and teachers use gesture to aid in communicating abstract mathematical ideas (e.g., Crowder & Newman, 1993; Flevares & Perry, 2001; Neill, 1991) and there is some evidence to suggest that such uses of gesture can help children better understand numbers, arithmetic, and spatial thinking (Novack et al., 2014; Suriyakham, 2007; Young et al., 2014).

PARENTAL MATH ATTITUDES AND BELIEFS: INTERGENERATIONAL FINDINGS As discussed previously, parents vary greatly in the quantity and quality of math talk they engage in with their children. Even when controlling for factors such as income level and parental education, we see variation in parents’ home numeracy practices (e.g., Levine et al., 2010; Susperreguy & Davis-Kean, 2016). It is important to understand the factors that might undermine parents’ engagement in high-quality math talk with their children, because little or low-quality talk can influence children’s mathematical development (e.g., Hyde, Else-Quest, Alibali, Knuth, & Romberg, 2006; Maloney et al., 2015). One such factor is parents’ math anxiety, or the fear and apprehension around doing math (Hembree, 1990; Richardson & Suinn, 1972). Math anxiety is a global phenomenon, experienced by both children and adults (Foley et al., 2017; Harari, Vukovic, & Bailey, 2013; Ramirez, Gunderson, Levine, & Beilock, 2013), and is associated with worse performance on standard math achievement tests (e.g., Ashcraft, 2002; Hembree, 1990; Ramirez et al., 2013). This underperformance in math may then continue to spiral downward as math anxiety and poor performance feed off of one another in a vicious cycle (Gunderson, Park, Maloney, Beilock, & Levine, 2017). One reason math anxious individuals may exhibit poorer performance on math tasks is that the cognitive resources (e.g., working memory) that are normally used when solving math problems are depleted by the fears and negative emotions they experience in these contexts (Beilock & Carr, 2005; Beilock, Schaeffer, & Rozek, 2017; Foley et al., 2017; Hopko, Ashcraft, Gute, Ruggiero, & Lewis, 1998). In support of this idea, in both children (Ramirez et al., 2013) and adults (Beilock & Carr, 2005), high (but not low) working memory individuals show performance deficits in high pressure situations because the worries they experience intrude into their working memory and thereby reduce the resources they otherwise could deploy to solve these problems. Spatial ability, which is related to math achievement, is also vulnerable to the effects of anxiety. Studies with adults have found that anxiety related to spatial tasks, such as a females fear of confirming gender stereotypes about spatial abilities, hinders performance on those tasks (Campbell & Collaer, 2009; Lawton, 1994; McGlone & Aronson, 2006). As with math anxiety, this

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relation starts early in schooling. Female students with high working memory who experience spatial anxiety exhibit reduced mental rotation ability compared to their high working memory peers who are not anxious about spatial tasks (Ramirez, Gunderson, Levine & Beilock, 2012).

Intergenerational Effects of Math Anxiety These findings provide a sense of the impact that math and spatial anxiety have on the lives of those who experience these anxieties, but work from our lab has also found an intergenerational impact of math anxiety, with adults’ math anxiety leading to lower math achievement for children. In one study, we assessed the math anxiety of 17 first- and second-grade female teachers, and then looked at the math achievement and beliefs about math of the 117 students in their classrooms (Beilock, Gunderson, Ramirez, & Levine, 2010). We found that while there was no relation between students’ math achievement and their teachers’ level of math anxiety at the beginning of the year, girls’ math achievement was negatively correlated with teachers’ math anxiety at the end of the year. The relation of teacher math anxiety to girls’ math achievement was mediated by girls’ gender ability beliefs—female students with math anxious teachers came to have relatively increased acceptance of traditional gender norms in school (i.e., girls are better at reading, boys are better at math), which in turn impacted their math performance.Although the relationship of teacher math anxiety to boys’ math achievement was not significant in the Beilock et al. (2010) study, in a much larger study, teacher math anxiety was negatively associated with both girls’ and boys’ math learning over the school year (Schaeffer et al., under review). And it is not only teachers’ math anxiety that is negatively related to children’s math achievement, but also parents’ math anxiety (Berkowitz et al., 2015; Casad, Hale, & Wachs, 2015; Maloney et al., 2015). When highly math anxious parents report frequently helping their children with math homework, their children tend to learn less math over the year compared to children of high math anxious parents who help with homework less and to children of low math anxious parents, even when controlling for parent and child math ability (Maloney et al., 2015). In fact, we start to see negative effects of math anxiety on parent-child math interactions even during the preschool years. Using the same longitudinal database as the Levine et al. (2010) study described previously, we examined whether parent math anxiety was related to the quantity of number talk they engaged in with their children (Berkowitz, Gibson, Monahan, & Levine, 2017). Since math anxious individuals tend to avoid math, we reasoned that they might also avoid talking about math topics with their children. We found a relation between parent number talk when children were young (14–30 months old)

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and parent math anxiety, as assessed using the Short Mathematics Anxiety Rating Scale (Alexander & Martray, 1989) for higher, but not lower SES families. Among the higher SES parents, even the most basic types of math talk (e.g., cardinal labeling of sets) were lower in math anxious parents. One reason we might only have seen the relation of parent math anxiety and early number talk among higher SES families is that there is less math talk in lower SES families overall, regardless of level of math anxiety. Another recent study in our lab with 50 parent-child dyads from largely middle income backgrounds found preliminary evidence that math anxious parents engage in less number talk with their preschool children than less math anxious parents (Berkowitz, 2018; Eason et al., 2017). Additionally, we showed that differences in the number prompts (e.g., “How many are there?” and “Can you count the plates?”) parents produced was a greater predictor of child number talk than parents’ number statements. While we did not find conclusive evidence in this sample that the difference in types of parent number talk was related to parents’ math anxiety, future work is planned to explore this possibility. While less work has explored the intergenerational impacts of spatial anxiety, we have found evidence that adult’s spatial anxiety may impact children’s performance on spatial tasks. Gunderson, Ramirez, Beilock, and Levine (2013) looked at teachers’ spatial anxiety and measured the spatial skills of the students in their classrooms. They found that at the end of the year, student spatial skills were negatively related to teachers’ spatial anxiety, even after accounting for other variables such as teachers’ math anxiety. This may be due to teachers with high levels of spatial anxiety avoiding introducing spatial activities into the classroom, or presenting spatial activities in a less effective manner, though the exact mechanisms that account for this relation are not yet fully understood. Since spatial skills are an important component of math and other STEM areas, further study of the negative impacts of adults’ spatial anxiety on children’s spatial ability is an important area for future research.

Other Negative Attitudes Toward Math Looking beyond math anxiety, parental attitudes about math can influence their children’s achievement. Children of parents who frequently express their negative attitudes toward math (e.g., “I am not good at math”) have both decreased interest and lower achievement in math, even when controlling for parents’ education (Hyde et al., 2006). In part this is because parents’ negative attitudes toward math may lead them to have low expectations for their children’s math achievement or to fail to impart the value of math to their children during everyday interactions (Harackiewicz, Smith, & Priniski, 2016; Rozek, Svoboda, Harackiewicz, Hulleman, & Hyde, 2017).

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Furthermore, differences in parents’ expectations for children’s math achievement predict children’s math achievement as early as the age of five (DeFlorio & Beliakoff, 2015). It may be that children pick up on their parents’ attitudes and beliefs about math, and when parents model positive attitudes about math in their own behaviors, their children are more likely to adopt and act on those values (Simpkins, Fredricks, & Eccles, 2012). Parents’ process praise (“You did a great job!”) rather than person praise (“You’re so smart”) has also been found to relate to children’s growth mind-sets, which in turn has been found to relate to their math achievement (Cimpian, Arce, Markman, & Dweck, 2007; Gunderson, Gripshover, et al., 2013; Gunderson et al., 2018; Skwarchuk et al., 2014). These findings underline the importance of developing interventions that combat the detrimental impact of parents’ negative math attitudes on their children’s math achievement. In one such effort, we carried out a longitudinal study assessing the effectiveness of a math app, Bedtime Math, in boosting the math achievement of children of math anxious parents (Berkowitz et al., 2015). As mentioned previously, in other studies we observed a negative relationship between the math anxiety of important adults in children’s lives (i.e., parents and teachers) and students’ performance in mathematics in early elementary school (e.g., Beilock et al., 2010; Maloney et al., 2015). Therefore we hypothesized that helping parents and children engage in high-quality math interactions could help change the nature of this relation between parent math anxiety and student math achievement. The app contained nightly passages, along with five to six math-related questions for parents and children to solve together. As such, the app not only provided parents with an opportunity for meaningful math interactions with their child, but also helped model how math can be incorporated into children’s many interests via engaging discussions that could spill over into daily activities and routines. We found that the children of high math anxious parents showed greater math gains over the school year when they were randomized into the math group compared to a reading control group (see Fig. 4). Moreover, the achievement of this group, unlike that of the children of high math anxious parents who were in the control group, was similar to that of children whose parents were low in math anxiety. Interestingly, the math app interactions were of higher quality for the dyads where the parent was not math anxious than for the dyads with a math anxious parent (Herts et al., 2017). Nonetheless, the children of the math anxious parents benefited most from this intervention, suggesting that the interactions that took place were of sufficient quality to support math gains. Importantly, we find that children of math anxious parents sustain these gains (compared to children of math anxious parents in the control group) through at least the end of 3rd grade, even though app use dramatically decreases after the first year. One potential mechanism for the sustained effect, supported by

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FIG. 4 Students’ W-scores on the WJ Applied Problems Subtest in Spring of 1st grade. Students with high math anxious parents who were assigned to the math group performed as well as students of low math anxious parents in both the math and the reading (control) groups. In contrast, students of high math anxious parents in the reading group performed worse than all the other groups on the math assessment.

a mediation analysis, is a change in the math anxious parents’ expectations and values for their children’s math achievement (Schaeffer, Rozek, Berkowitz, Levine, & Beilock, in press).

Summary: Parental Math Attitudes and Beliefs To recap, individuals’ negative attitudes around math, including their math anxiety, are associated with poorer performance on math-related tasks in both adults and children. Additionally, there is evidence that parents’ negative math attitudes and beliefs have adverse implications for their children’s performance and achievement in mathematics. The quantity and quality of parents’ math interactions with their children appear to be lower when parents are math anxious. However, existing studies indicate that interventions supporting parent-child math interactions can make a meaningful difference in the math achievement of children with math anxious parents.

CONCLUSIONS AND FUTURE DIRECTIONS Our review highlights several important messages about the early home math environment and children’s early math learning. First, it shows that the home math environments young children experience vary widely, and that these variations predict children’s early mathematical knowledge. Second, it shows that specific contexts may support particular kinds of math knowledge. For example, indirect math activities such as games tend to support nonsymbolic arithmetic whereas more direct math activities such as numeral recognition

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and counting tend to support symbolic number knowledge, although these differences may be related to the kinds of math learning experiences provided in these contexts. That is, games that involve counting and learning to recognize numerals may result in the same kinds of learning as more direct math activities. Third, our review shows that it is important to tune home math talk to the level of children’s understanding and thus, for parents to be aware of children’s knowledge levels. Although Saxe et al. (1987) have shown that parents are sensitive to children’s math knowledge levels, their intuitions about the kinds of inputs that are most helpful in enhancing children’s knowledge are likely to benefit from evidence-based tips. Fourth, children’s gestures can provide information about their readiness to learn a new number concept and particular kinds of cospeech gestures can be an effective way for parents to help children understand challenging math concepts. Fifth, both the math supports that parents provide and children’s math outcomes appear to be linked to parents’ attitudes about math, notably parents’ math anxiety but also other attitudes they have about math. Finally, and most importantly, several studies show that parent math talk is malleable and that it is possible to increase the quantity and quality of this kind of talk through various means (e.g., nudges to support number learning during cooking, games, number books, math apps such as Bedtime Math), which in turn can lead to gains in children’s math language and math knowledge. There are many outstanding basic and applied research questions involving early math development. To highlight just a few, we know little about the long-term effects of enhancing the early home math environment because most intervention studies only test immediate math gains or at most, gains that are present a few years postintervention (see Bailey, this volume). Second, we need to learn more about the kinds of early math experiences that are most effective in supporting not only children’s math knowledge, but also their math attitudes. We also need to probe how negative math attitudes, such as math anxiety, get started in the first place and how to prevent these attitudes from taking hold and contributing to the vicious cycle of low math achievement and negative math attitudes (Gunderson et al., 2017). There are also important questions about how to scale up knowledge about effective home math supports. One important factor in successful scale up efforts is to develop messages and interventions that are not only evidence based but that also resonate with cultural practices of families. For example, a recent set of family discussions suggested that both Latino and Chinese families resonate more toward math supports being talked about as educational activities rather than games (ZenoMath, 2017). Additionally, Marta Civil’s work brings the funds of knowledge framework to math, which recognizes and harnesses the strengths that families bring to the task of effectively supporting children’s mathematics education, rather than approaching families through the lens of a deficit model. Her work shows that families can effectively support children’s math learning and also demonstrates how parents can serve as

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resources for other parents, and how this can help build a community focused on this important goal (e.g., Civil, 1998; Civil, Bratton, & Quintos, 2005). Fortunately, work with families and community partners to support children’s early math learning is gaining traction and is having positive results. One promising approach is to create communities that embrace the importance of early math and that teach parents how they can support early math (Blumenstock, 2016). These efforts are likely to be more effective when they involve school-home partnerships that enlist parents in supporting curricular goals. Use of bi-generational interventions that involve teaching math to Head Start students and at the same time teaching their families how they can support their children’s math learning in the home environment has been found to be an effective way to increase young children’s math knowledge (e.g., Starkey & Klein, 2000). Community partnerships can also help with supporting children’s math learning and reaching families with messages about how they can support their children’s math learning. A study with the National Museum of Math in New York City where researchers and museum professionals codeveloped tip cards that could support parents’ math conversations with their children exemplifies work with community partners. In this study, families randomized into the tip card group gave higher ratings of enjoyment and learning at the museum than those in the control group (Herts et al., 2018). Another approach involves incorporating parent voices into the design of interventions. In our lab, we have held focus groups to help us incorporate parent voices in developing math messages that resonate and in developing effective math interventions. Our effort to develop effective math messages involved convening focus groups with two groups of African American parents. In two successive focus groups, we were able to iteratively refine messages based on what we learned from parents. We learned that parents preferred messages with specific information that could guide their math supports, messages about including math in everyday activities (but not the message that they did not need to make time for math because of this), and messages that math is more than numbers, but again wanted specific information about other important aspects of math. Additionally, parents liked the comparison of math to reading, which highlighted that parent support is important for both domains and that learning in these domains go hand in hand (Levine, Bradley, & Mix, under review). Technology offers potentially powerful ways to deliver high-quality math activities and math tips to families. This can be done through apps, websites, and texts that can remind families to engage their children in math activities and help them do this effectively. With the increasing spread of smart devices, this opportunity is increasing and becoming ubiquitous. However, it is important to note that not all apps are created equal, and despite the 1000s of “educational” apps that are available, researchers are only starting to develop frameworks for evaluating the potential impact of various apps (Hirsh-Pasek et al., 2015). As shown by our research, an app that encourages parents and

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children to do math together was effective in supporting the math learning of the children of math anxious parents. This app may be more beneficial than math apps that involve the child alone, an important question for future research, particularly because most education apps are designed for children to use on their own and because children are spending increasing amounts of time in front of screens, reducing time for face-to-face interactions (e.g., Common Sense Media, 2017). While all of these approaches and tools can be useful resources to enhance children’s math experiences and to spread positive messages around math, they will have the greatest impact if they are developed in partnership with the parents who will be using them and with multiple sectors of community partners. Creating multisector coalitions that involve families, schools, businesses, health care providers, community organizations, and philanthropists will help ensure that sharing math with young children becomes as much of a focus as sharing books and reading.

ACKNOWLEDGMENTS We thank the National Science Foundation Spatial Intelligence and Learning Center (Grants SBE-1041707 and SBE-0541957 to Susan C. Levine), the National Science Foundation Science of Learning Collaborative Network Grant #1540741 to Susan C. Levine, and the HeisingSimons Foundation Development and Research in Early Mathematics Education (DREME) Network (support to Susan C. Levine). Research reported in this publication was also supported by the Eunice Kennedy Shriver National Institute of Child Health & Human Development of the National Institutes of Health under Award Number P01HD040605.

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140 Cognitive Foundations for Improving Mathematical Learning Potter, M. C., & Levy, E. I. (1968). Spatial enumeration without counting. Child Development, 39, 265–272. https://doi.org/10.2307/1127377. Pruden, S. M., & Levine, S. C. (2017). Parents’ spatial language mediates a sex difference in preschoolers’ spatial-language use. Psychological Science, 28(1), 1583–1596. https://doi.org/ 10.1177/0956797617711968. Pruden, S. M., Levine, S. C., & Huttenlocher, J. (2012). Children’s spatial thinking: does talk about the spatial world matter? Developmental Science, 14(6), 1417–1430. https://doi.org/ 10.1111/j.1467-7687.2011.01088. Purpura, D. J., Napoli, A. R., Wehrspann, E. A., & Gold, Z. S. (2017). Causal connections between mathematical language and mathematical knowledge: a dialogic reading intervention. Journal of Research on Educational Effectiveness, 10(1), 116–137. https://doi.org/10.1080/ 19345747.2016.1204639. Quinn, P. C., & Liben, L. S. (2008). A sex difference in mental rotation in young infants. Psychological Science, 19(11), 1067–1070. https://doi.org/10.1111/j.1467-9280.2008.02201.x. Ramani, G. B., Rowe, M. R., Eason, S., & Leech, K. (2015). Math talk during informal learning activities in head start families. Cognitive Development, 35, 15–33. https://doi.org/10.1016/ j.cogdev.2014.11.002. Ramani, G. B., & Siegler, R. S. (2008). Promoting broad and stable improvements in low-income children’s numerical knowledge through playing number board games. Child Development, 79(2), 375–394. https://doi.org/10.1111/j.1467-8624.2007.01131.x. Ramirez, G., Gunderson, E. A., Levine, S. C., & Beilock, S. L. (2012). Spatial anxiety relates to spatial abilities as a function of working memory in children. The Quarterly Journal of Experimental Psychology, 65(3), 474–487. https://doi.org/10.1080/17470218.2011.616214. Ramirez, G., Gunderson, E. A., Levine, S. C., & Beilock, S. L. (2013). Math anxiety, working memory and math achievement in early elementary school. Journal of Cognition and Development, 14(2), 187–202. https://doi.org/10.1080/15248372.2012.664593. Ramscar, M., Dye, M., Popick, H. M., & O’Donnell-McCarthy, F. (2011). The enigma of number: why children find the meanings of even small number words hard to learn and how we can help them do better. PLoS One, 6(7), e22501. https://doi.org/10.1371/journal. pone.0022501. Reyna, V. F., Nelson, W., Han, P., & Dieckmann, N. F. (2009). How numeracy influences risk comprehension and medical decision making. Psychological Bulletin, 135, 943–973. Richardson, F. C., & Suinn, R. M. (1972). The mathematics anxiety rating scale: psychometric data. Journal of Counseling Psychology, 19(6), 551–554. https://doi.org/10.1037/h0033456. Roth, W.-M., & Welzel, M. (2001). From activity to gestures and scientific language. Journal of Research in Science Teaching, 38(1), 103–136. https://doi.org/10.1002/1098-2736(200101) 38:13.0.CO;2-G. Rozek, C. S., Svoboda, R. C., Harackiewicz, J. M., Hulleman, C. S., & Hyde, J. S. (2017). Utility-value intervention with parents increases students’ STEM preparation and career pursuit. Proceedings of the National Academy of Sciences of the United States of America, 114, 909–914. https://doi.org/10.1073/pnas.1607386114. Saxe, G. B. (1977). A developmental analysis of notational counting. Child Development, 48, 1512–1520. https://doi.org/10.2307/1128514. Saxe, G. B., Guberman, S. R., & Gearhart, M. (1987). Social processes in early number development. Monographs of the Society of Research in Child Development, 52(2), I–162. Serial N. 216. https://doi.org/10.2307/1166071. Saxe, G. B., & Kaplan, R. (1981). Gesture in early counting: a developmental analysis. Perceptual and Motor Skills, 53(3), 851–854. https://doi.org/10.2466/pms.1981.53.3.851.

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Schaeffer, M. W., Rozek, C. S., Berkowitz, T., & Levine, S. C., & Beilock, S. L. (in press). Disassociating the relation between parents’ math anxiety and children’s math achievement: Long-term effects of a math app intervention. Journal of Experimental Psychology: General. Schaeffer, M. W., Rozek, C. S., Maloney, E. A., Berkowitz, T., Levine, S. C., & Beilock, S. L. (under review). Math anxious elementary school teachers undermine students’ math learning. Siegler, R., & Shrager, S. (1984). Strategy choice in addition and subtraction: how do children know what to do. In Origins of cognitive skills (pp. 229–294). Hillsdale, NJ: Lawrence Erlbaum Associates. Siegler, R. S., & Jenkins, E. (1989). How children discover new strategies. Hillsdale, NJ: Lawrence Erlbaum Associates. Siegler, R. S., & Ramani, G. B. (2009). Playing linear number board games—but not circular ones—improves low income preschoolers’ numerical understanding. Journal of Experimental Psychology, 101(3), 545–560. https://doi.org/10.1037/a0014239. Simpkins, S. D., Fredricks, J. A., & Eccles, J. S. (2012). Charting the Eccles’ expectancy-value model from mothers’ beliefs in childhood to youths’ activities in adolescence. Developmental Psychology, 48(4), 1019–1032. https://doi.org/10.1037/a0027468. Skwarchuk, S. L. (2009). How do parents support children’s preschool numeracy experiences at home? Early Childhood Education Journal, 37(3), 189–197. https://doi.org/10.1007/s10643009-0340-1. Skwarchuk, S. L., Sowinski, C., & LeFevre, J. A. (2014). Formal and informal home learning activities in relation to children’s early numeracy and literacy skills: the development of a home numeracy model. Journal of Experimental Child Psychology, 121, 63–84. https://doi.org/10.1016/j.jecp.2013.11.006. Starkey, P., & Klein, A. (2000). Fostering parental support for children’s mathematical development: an intervention with head Start families. Early Education and Development, 11(5), 659–680. https://doi.org/10.1207/s1556693eed1105 7. Suriyakham, L. W. (2007). Input effects on the development of the cardinality principle: Does gesture count? [Doctoral dissertation]. University of Chicago. Susperreguy, M. I., & Davis-Kean, P. E. (2016). Maternal math talk in the home and math skills in preschool children. Early Education and Development, 27(6), 841–857. https://doi.org/ 10.1080/10409289.2016.1148480. Tabors, P. O., Beals, D. E., & Weizman, Z. O. (2001). “You know what oxygen is?” Learning new words at home. In D. K. Dickinson & P. O. Tabors (Eds.), Beginning literacy with language: Young children learning at home and school (pp. 93–110). Baltimore: Paul H. Brookes. U.S. Department of Education (2016). STEM 2026: A vision for innovation in STEM education. Washington, DC: Office of Innovation and Improvement. Uttal, D. H., Meadow, N. G., Tipton, E., Hand, L. L., Alden, A. R., Warren, C., et al. (2012). The malleability of spatial skills: a meta-analysis of training studies. Psychological Bulletin, 139(2), 352–402. https://doi.org/10.1037/a0028446. Valenzeno, L., Alibali, M. W., & Klatzky, R. (2003). Teachers’ gestures facilitate students’ learning: a lesson in symmetry. Contemporary Educational Psychology, 28(2), 187–204. https://doi.org/10.1016/S0361-476X(02)00007-3. Vandermaas-Peeler, M., Boomgarden, E., Finn, L., & Pittard, C. (2012). Parental support of numeracy during a cooking activity with four-year-olds. International Journal of Early Years Education, 20(1), 78–93. https://doi.org/10.1080/09669760.2012.663237. Vandermaas-Peeler, M., Ferretti, L., & Loving, S. (2012). Playing the lady bug game: parent guidance of young children’s numeracy activities. Early Child Development and Care, 182(10), 1289–1307. https://doi.org/10.1080/03004430.2011.609617.

142 Cognitive Foundations for Improving Mathematical Learning Verdine, B. N., Golinkoff, R. M., Hirsh-Pasek, K., Newcombe, N. S., Filipowicz, A. T., & Chang, A. (2014). Deconstructing building blocks: preschoolers’ spatial assembly performance relates to early mathematical skills. Child Development, 85(3), 1062–1076. https:// doi.org/10.1111/cdev.12165. Wai, J., Lubinski, D., & Benbow, C. P. (2009). Spatial ability for STEM domains: aligning over 50 years of cumulative psychological knowledge solidifies its importance. Journal of Educational Psychology, 101(4), 817. https://doi.org/10.1037/a0016127. Wiese, H. (2003). Numbers, language, and the human mind. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511486562. Wynn, K. (1990). Children’s understanding of counting. Cognition, 36(2), 155–193. https://doi. org/10.1016/0010-0277(90)90003-3. Young, C., Cartmill, E., Levine, S. C., & Goldin-Meadow, S. (2014). In Gesture and speech input are interlocking pieces: the development of children’s jigsaw puzzle assembly ability. Proceedings of the annual meeting of the cognitive science society, 36. Retrieved from: https://escholarship.org/uc/item/3h2198h1. ZenoMath. (2017). Summary report of 6 discussion groups developed by Zeno and Purple Group. 15 November 2017. Zukow-Goldring, P., Romo, L., & Duncan, K. R. (1994). Gestures speak louder than words: achieving consensus in Latino classrooms. In A. Alvarez & P. del Rio (Eds.), vol. 4. Education as cultural construction: Exploration in socio-cultural studies (pp. 227–239). Madrid, Spain: Fundacio Infancia y Aprendizage.

Chapter 6

From Cognition to Curriculum to Scale☆ Julie Sarama and Douglas H. Clements Morgridge College of Education, Marsico Institute, University of Denver, Denver, CO, United States

INTRODUCTION Most developers and publishers claim that their curricula are based on research, but few explicate their claims. In this chapter, we briefly assess the state of affairs regarding “research-based curricula” and present a model to mitigate weaknesses in the field that is based on coordinated interdisciplinary research ranging from cognitive science to scale-up. We describe an example in early mathematics. Finding a curriculum that does not claim a research basis is difficult, but these claims are often vacuous, citing theories or empirical results vaguely (Clements, 2007, 2008; Clements & Sarama, 2013; Kinzie, Whittaker, McGuire, Lee, & Virginia, 2015). For example, they often cite research evidence relevant to the beginning or end of the curriculum development process. That is, at the beginning, “research-based” often indicates asserting that the curriculum was built upon broad theoretical frameworks or, with little specificity, “research on students’ thinking.” Such a research-to-practice model alone is inadequate, because it includes a one-way translation of research results to principles to instructional designs and therefore is often

☆ This research was supported by the Institute of Education Sciences, U.S. Department of Education through Grants R305A120813, R305K05157, and R305A110188. The opinions expressed are those of the authors and do not represent views of the U.S. Department of Education. Although the research is concerned with theoretical issues, not particular curricula, a small component of the intervention used in this research have been published by the authors and their collaborators on the project, who thus could have a vested interest in the results. Researchers from an independent institution oversaw the research design, data collection, and analysis and confirmed findings and procedures. The authors wish to express appreciation to the school districts, teachers, and students who participated in this research.

Mathematical Cognition and Learning, Vol. 5. https://doi.org/10.1016/B978-0-12-815952-1.00006-2 Copyright © 2019 Elsevier Inc. All rights reserved.

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flawed in its presumptions, insensitive to changing goals in the content area, and unable to contribute to a revision of the theory and knowledge on which it is built (Clements, 2007). At the other end, research validated may mean that effectiveness of the finished curriculum was evaluated. Not only does this leave out critical stages of a scientific research and development process (Battista & Clements, 2000; Clements & Battista, 2000; Doabler et al., 2014), but the research designs are often weak (Munter, Cobb, & Shekell, 2016). In the area of early mathematics, for example, of 78 elementary school programs evaluated, less than 10% had valid evidence of effectiveness and four of those had only “potentially positive” effects on achievement (Doabler et al., 2014). Why this state of affairs continues is explained by the confluence of many factors, such as instrumentalist views of mathematics and knowledge acquisition as simple transmission, a skepticism or rejection of mathematics curricula in the earliest years, lack of standards for curriculum development, a bias against design sciences, such as curriculum development in particular, in academe, and limited involvement of and communication between, relevant parties (for elaboration, see Clements & Sarama, 2013). This is not to say that there have been no viable attempts to build valid research-based curricula. There are many (for lists of examples, see Clements, 2008; Day-Hess & Clements, 2017). However, they remain relatively small in number and frequently do not explicate the methods and findings of the development process. To address these weaknesses, close the gap between research and practice, and increase the impact of research on the field (Cai et al., 2017), we need scientific approaches to the conceptualization, design, creation, implementation, and scale-up of curricula that are not just “based on” or “validated by” research but that were constructed, refined, and evaluated with a comprehensive program of research and development (Clements, 2007, 2008; Clements & Sarama, 2013).

THE CURRICULUM RESEARCH FRAMEWORK (CRF) Based on a review of research and expert practice (Clements, 2008), we constructed and tested a framework for the construct of research-based curricula. The goal was to promote a valid scientific curriculum development program that addresses two basic questions—about effects and conditions—in three domains: practice, policy, and theory. For example, a curriculum development program should address not only the practical question of whether the curriculum is effective in helping children achieve specific learning goals, but also the conditions under which it is effective. Theoretically, the research program should also address why it is effective and why certain sets of conditions decrease or increase the curriculum’s effectiveness.

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We developed the Curriculum Research Framework (CRF, Clements, 2007), which identifies three categories and ten phases of research and development, along with methods appropriate for each. A core feature of the CRF is that it is grounded in coordinated interdisciplinary research ranging from cognitive science to early childhood and mathematics education to implementation science to scale-up (the final scale-up phrase is complex and has its own elaborated model, Sarama & Clements, 2013). Each phase must yield positive results to proceed to the next. This process can reveal weaknesses that have to be addressed and reevaluated (or the project halted, saving resources before large-scale evaluations are conducted, most likely yielding little to no benefits). This approach has higher validity than others for the same reason: Construct validity tests are more frequent and more trustworthy. For example, if research on students’ thinking and learning in the goal domain is not carefully reviewed or conducted, it is considerably less likely that later phases of development (curricula, professional development, implementation, etc.) will be successful.

The CRF and Early Mathematics We first implemented the CRF in the field of early mathematics, given its importance (Clements & Sarama, 2014; Sarama & Clements, 2009) and the low use of mathematics curricula in the earliest years of schooling in the United States. For example, US teachers tend to use emerging curricula, whereas those in China use mathematics-specific curricula (Li, Chi, DeBey, & Baroody, 2015).>

The CRF Enacted As stated, the CRF includes ten phases for asserting that a curriculum is based on research, which can be ordered by the chronology of typical curriculum development, although they are cyclic or recursive (Clements, 2007, 2008; Clements & Sarama, 2013). In the remainder of this section, we briefly describe each phase and then illustrate how we enacted that phase in the Building Blocks research and development project, a NSF-funded early childhood mathematics research and curriculum development project that was the first to be based on the CRF.

Category I: A Priori Foundations The first category is that of a priori foundations. Here, the nature of the phase is a focused version of the research-to-practice model. That is, the extant research is reviewed and implications for the nascent curriculum development effort drawn. The questions asked regard what is already known that can be

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applied to an anticipated curriculum, concerning psychology, education, systemic change, and so forth in general (phase 1), the specific subject matter content, including the role it would play in students’ knowledge development (phase 2), and pedagogy, including the effectiveness of certain types of activities (phase 3). A general guideline across these evaluation phases is that equity issues (Confrey & Lachance, 2000) be considered. For example, considerable thought should be given to the students who are envisioned as users and who participate in field tests; a convenience sample is often inappropriate. Systemic sociocultural issues should be considered as well (Tate, 1997). For Building Blocks, we used research on and conducted all field tests with two populations: Children from low-resource communities and children with special needs.

Phase 1. General A Priori Foundation Developers review broad philosophies, theories, and empirical results on learning and teaching. Based on theory and research on early childhood learning and teaching (Clements & Sarama, 2007a), we determined that Building Blocks’ basic approach would be finding the mathematics in, and developing mathematics from, children’s activity. That is, we wanted to “mathematize” everyday activities, such as puzzles, songs, moving, and building. For example, teachers might help children mathematize moving their bodies in many ways. Children might count their steps as they walk. They might also move in patterns: step, step, hop; step, step, hop…. They might do both, counting as they walk, “one, two, three, four, five six, ….” These examples show that mathematizing means representing and elaborating everyday activities mathematically. Children create models of everyday situations with mathematical objects, such as numbers and shapes; mathematical actions, such as counting or transforming shapes; and their structural relationships—and use those understandings to solve problems. They learn to increasingly see the world through mathematical lenses. Phase 2. Subject Matter A Priori Foundation Developers review research and consult with experts to identify topics that make a substantive contribution to children’s mathematical development, are generative in children’s development of future mathematical understanding, and are interesting to children. We determined the topics that fit those criteria by considering what mathematics is culturally valued (e.g., standards from domain-specific organizations and states) and empirical research on what constituted the core ideas and skill areas of mathematics for young children (Clements, Sarama, & DiBiase, 2004). We then organized for the development of learning trajectories in the domains of number (subitizing, counting, sequencing, arithmetic), geometry (matching, naming, building, and combining shapes), patterning, and measurement.

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Phase 3. Pedagogical A Priori Foundation Developers review empirical findings on making activities educationally effective—motivating and efficacious—to create general guidelines for the generation of activities. As an example, research using computer software with young children (Clements & Swaminathan, 1995; Sarama & Clements, in press) showed that preschoolers can use computers effectively and that software can be made more effective by employing animation, children’s voices, and clear feedback. Although such software is only a small component of the Building Blocks curriculum, it makes a significant contribution, because research was used in its development, giving the developers information on how to make the software targeted and effective. Another issue that should be considered is for whom the curriculum is intended (e.g., sophisticated reform-oriented teacher, reform-oriented reform teacher, traditional teacher). Is it intended to be “ahead of where the teacher is” or fit the teachers’ current practice (Martin A. Simon, personal communication, May 28, 2002)? We planned that Building Blocks would be considerably “ahead” of the teachers because most had little preparation in mathematics education. However, to reduce the unfamiliarity for them, we also connected all aspects of the curriculum to typical early childhood educational practice whenever this was consistent with (e.g., traditional scheduling) and especially when it strengthened (e.g., an emphasis on child development and processes, not just products) the mathematics.

Category II: Learning Model and Learning Trajectory Within the second category is the most extensive and intensive development phase, in which developers’ structure activities in accordance with theoretically and empirically based models of children’s thinking. This phase involves the creation of research-based learning trajectories—One for every major topic. For this paper, we focus on just one of the many topics from the Building Blocks curriculum, subitizing, or the quick recognition of a number of items in a set without counting, from the Latin “to arrive suddenly.”

Phase 4. Structure According to Specific Learning Model and Learning Trajectory The question is how the curriculum can be constructed to be consistent with, and build upon, students’ thinking and learning, which are posited to have characteristics and developmental courses that are not arbitrary, and therefore not equally amenable to various instructional approaches or curricular routes (this is based on our arching theory of hierarchic interactionalism, to which space constraints allow only short references, but see Sarama & Clements, 2009). What distinguishes phase 4 from phase 3, which concerns pedagogical

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a prior foundations, is not only the focus on the child’s learning, rather than teaching strategies alone, but also on the iterative nature of its application. Fig. 1 diagrams the influence of research that creates a chain of development around this core component of learning trajectories (Confrey, Gianopulos, McGowan, Shah, & Belcher, 2017). We first discuss the three rounded rectangles that represent three fields of research that support the development of each initial (or “Hypothetical”) Learning Trajectory. Mathematical progressions contribute to the identification of goals. We posit that worthwhile goals are based on the big ideas of mathematics: those that are mathematically central and coherent, consistent with children’s thinking, and generative of future learning (i.e., they are part of a coherent mathematical progression, Clements & Conference Working Group, 2004; van Marle et al., 2018). The fundamental importance of cardinal understanding of whole numbers needs no justification and children’s first cardinal meanings for number words may be labels for small sets of subitized objects, even if they determined the labels by counting (Fuson, 1992b; Slusser & Sarnecka, 2011; Steffe, Thompson, & Richards, 1982). Subitizing introduces basic ideas of cardinality—”how many,” ideas of “more” and “less,” ideas of parts and wholes and their relationships, beginning arithmetic, and, in general, ideas of quantity. Developed well, these are related, forming webs of connected ideas that are the building blocks of mathematics through elementary, middle, and high school and beyond. Finally, the Common Core State Standards explicitly describes subitizing: “Students choose, combine, and apply effective strategies for answering quantitative questions, including quickly recognizing the cardinalities of small sets of objects” (National Governor’s Association Center for Best Practices & Council of Chief State School Officers, 2010, p. 9). Psychological research. Psychologically oriented research, from cognitive science, developmental psychology, education, and so forth, was particularly critical and extensive for the topic of subitizing. Indeed, we note that this research has shown that innate competencies that underlie subitizing are the first quantification mechanisms. For example, children as young as 6 months of age and probably younger are sensitive to number. They habituate to 1 vs. 2 or 3 objects (Antell & Keating, 1983; P. Starkey, Spelke, & Gelman, 1990). For example, shown repeated sets of 3, they eventually “get used to” the number, even as color, size, and arrangements change, and become more attentive only when a set with a different number such as 2 is shown. This indicates that infants are sensitive to small numerosities of a set of items before they are taught number words, counting, or finger patterns. Such research contributes, from a psychological perspective, to the importance of the goal of subitizing. However, the main contribution of psychologically oriented research for all topics is to the creation of the developmental progression of the learning trajectories. Given the importance of this to the creation of a learning trajectory, we review the research in some detail to illustrate that creation

Mathematical progressions

Curriculum

Learning trajectories

(abstracted from research)

Psychological research Mathematics education research

1. Mathematical goal 2. Developmental progression 3. Instructional activities

Assessments

LTs Validated

FIG. 1 The logic model for the Phase 4 of the Curriculum Research Framework.

Professional development

Scale up communication

TRIAD scale up model enacted

Child engagement and achievement

Persistence of child achievement

Sustainability of fidelity of implementation

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(for comprehensive reviews, see Butterworth, 2010; Clements, Sarama, & Mac Donald, 2017; Hannula, Lepola, & Lehtinen, 2010; Sarama & Clements, 2009). We begin with research on the nature of subitizing. Subitizing was initially defined in psychology (Kaufman, Lord, Reese, & Volkmann, 1949). Subitizing activity was differentiated from estimation as a unique form of visual number discrimination characterized by speed, exactness, and degree of confidence. For example, individuals identified sets of five or fewer objects quickly (10) full-day sessions of training in regular meetings and frequent coaching. Training included all three components of each learning trajectory, the goal, the developmental progression, and the

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instructional activities and strategies (as in Fig. 3). To understand the goal, teachers study the mathematical content; examples for subitizing include the concept of cardinality and partitioning numbers. A key instructional use of learning trajectories is in formative assessment along the developmental progression. We worked with teachers to study the developmental progression for subitizing, analyze multiple video segments illustrating each level and discuss the mental “actions on objects” that constitute the defining cognitive components of each level; order tasks corresponding to those levels; and practice diagnosis in teams, with a couple of teachers exemplifying behaviors of children at different levels, and one teacher identifying the level of each (we used an online application; an update to it can be seen at learningtrajectories.org). Further, teachers need training in understanding, administering, and especially using data from new assessment strategies (Foorman, Santi, & Berger, 2007). TRIAD training focuses mainly on the curriculum-embedded assessment of Building Blocks’ Small Group Record Sheets. Formative assessment requires more than identifying children’s levels of thinking. Teachers must select and modify instructional activities and strategies that are appropriate and effective for each level. To learn about instructional tasks and strategies, teachers practice the curriculum’s activities, but also analyze them to establish and justify their connection to a particular level of the developmental progression (as in Fig. 3). Across all forms of professional development, the focus is on children’s thinking and learning. Conversations about children learning serves as way to address implementation issues. Although early mathematics is often an uncomfortable topic for early childhood educators, the newness of the learning trajectories for all participants helps establish a sense of shared learning and community. Each session in the last third of professional development includes scheduled time to discuss “learning stories” (Perry, Dockett, & Harley, 2007). Teachers show their record keeping on small group record sheets, and sometimes videos, and discuss their use of learning trajectories in teaching children, including challenges, questions, and successes. These discussions promote peer learning and collaboration and also motivate peers to solve implementation difficulties.

CONCLUSIONS AND FUTURE DIRECTIONS We described our Curriculum Research Framework and illustrated its instantiation with the Building Blocks curriculum, focusing on the goal of subitizing. As diagrammed in Fig. 1, we believe the success it had stemmed from the complementary research bases, and especially the establishment of a set of cognitively grounded learning trajectories that contributed to all aspects of the projects. Future research should critically evaluate the veracity of these beliefs. Future research and development might also evaluate the CRF’s implementation with other grade levels and other topics (see Doabler et al., 2014;

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Kinzie et al., 2015). Just as important, our research designs could not identify which components of the CRF and TRIAD models and their instantiations are critical. Such research would be theoretically and practically useful. The specific contribution of the learning trajectories per se needs to be disentangled and identified. In a present project, we are addressing this issue. In an IES-funded project entitled, “Evaluating the Efficacy of Learning Trajectories in Early Mathematics”, we are testing the efficacy of learning trajectories in a series of eight randomized clinical trials testing different aspects of LTs. These experiments will determine whether LTs are more efficacious than other approaches in supporting young children’s learning. On a practical side, with funding from the Heising-Simons Foundation and the Bill and Melinda Gates Foundation, we are developing a technology-based tool for teachers and teacher trainers that extends a resource we created for the TRIAD evaluation. The Learning and Teaching with Learning Trajectories tool, or LT2, is a new, free resource for early mathematics (see www. LearningTrajectories.org). LT2 provides learning-trajectories-based math resources for teachers, caregivers, and parents. LT2 runs on all technological platforms, addresses new ages—birth to age 8 years—and including new alignments with standards and assessments, as well as new software for children from—their everyday activities, including art, stories, puzzles, and games.

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Chapter 7

Development of Mathematical Language in Preschool and Its Role in Learning Numeracy Skills David J. Purpura, Amy R. Napoli and Yemimah King Department of Human Development and Family Studies, Purdue University, West Lafayette, IN, United States

INTRODUCTION Young children’s acquisition of mathematics skills is tightly linked to the development of other academic and cognitive skills during the early years. In particular, a growing body of evidence highlights the important role that language skills—especially mathematical language (e.g., words and concepts such as many, most, few, fewest)—play in young children’s acquisition of early mathematical knowledge. In this chapter we (1) discuss the connections between general language and mathematics development, (2) define mathematical language and distinguish it from mathematics talk, (3) present the correlational and experimental evidence supporting the role mathematical language plays in early numeracy acquisition and existing mechanisms for intervention, and finally (4) discuss remaining unanswered questions about the mechanisms underlying this relation and future directions regarding this line of research.

CONNECTIONS BETWEEN EARLY MATHEMATICS DEVELOPMENT AND GENERAL LANGUAGE Mathematics proficiency is an academic and economic gatekeeper that provides a foundation for achievement and career skills (Geary, 1994; Jordan, Hanich, & Uberti, 2003; National Mathematics Advisory Panel [NMAP], 2008). Children with early mathematics deficits typically develop their skills at a slower rate than their more advanced peers (Aunola, Leskinen, Lerkkanen, & Nurmi, 2004) and experience lifelong academic and career difficulties (Adelman, 1999; Mathematical Cognition and Learning, Vol. 5. https://doi.org/10.1016/B978-0-12-815952-1.00007-4 Copyright © 2019 Elsevier Inc. All rights reserved.

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Duncan et al., 2007; Evan, Gray, & Olchefske, 2006; McGregor, 1994). Although early mathematics skills form the foundation for the acquisition of later mathematics skills, these early skills do not develop in isolation from other academic and cognitive domains—particularly language and literacy skills (LeFevre et al., 2010; Purpura, Hume, Sims, & Lonigan, 2011). For example, LeFevre et al. (2010) indicated that of the three main pathways by which early mathematics skills develop (quantitative, visual spatial, and linguistic), the strongest and most stable pathway was the linguistic pathway which was broadly defined to include both language and phonological skills. Furthermore, children who have difficulty in one domain have a high likelihood of experiencing difficulties in the other area (Barberisi, Katusic, Colligan, Weaver, & Jacobsen, 2005).

Difficulties in Both Domains Among the approximately 5%–8% of children who are diagnosed with a mathematics learning disability, a large proportion also meet criteria for a reading disability (Geary, 2004). Notably, children with both mathematics and reading difficulties experience more significant difficulties in mathematics than children who only have either mathematics difficulties or reading difficulties alone (Hanich, Jordan, Kaplan, & Dick, 2001; Jordan & Hanich, 2000; Jordan & Levine, 2009; Mann Koepke & Miller, 2013; Willcutt et al., 2013). Donlan (2007) suggests that there may be an interaction between linguistic and visuospatial neural systems during development, resulting in children with specific language impairments also having difficulties in mathematics. Moreover, verbal skills are necessary for children to succeed in calculation tasks that require knowledge of number words (Donlan, 2007; Jordan, Huttenlocher, & Levine, 1994; Lewis, Hitch, & Walker, 1994). For example, language comprehension is a necessary component of word problems, and children who struggle with both mathematics and reading tend to have particular difficulties with story problems (Fuchs et al., this volume; Jordan & Levine, 2009; Powell, Fuchs, Fuchs, Cirino, & Fletcher, 2009). Ultimately, children who tend to have co-occurring mathematics/reading difficulties are at an increased risk for experiencing persistent difficulties in mathematics ( Jordan & Levine, 2009; Mann Koepke & Miller, 2013), which suggests that deficits in language and literacy skills may either lead to difficulties in mathematics or exacerbate existing difficulties, above and beyond any domain-general deficits (e.g., working memory) that may slow learning in both areas.

Early Connections Between Mathematics and Literacy Skills Though a large body of evidence indicates that there are strong relations between mathematics and literacy skills in elementary school (Hooper, Roberts, Sideris, Burchinal, & Zeisel, 2010; Romano, Babchishin, Pagani, & Kohen, 2010) it is also evident that this relation begins early in life (Davidse, De Jong, & Bus, 2014). Notably, Spelke (2003) suggests that early language skills help children

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refine their concepts of basic number. Specifically, by building and applying the constructs of language such as grammatical rules to the underlying approximate number system, children are able to intuit specific quantities, their relations, and their positioning relative to other quantities (Spelke, 2017). Even at the prereading level, emergent literacy skills such as phonological awareness (Krajewski, Schneider, & Niedling, 2008), language (Hooper et al., 2010; LeFevre et al., 2010), and print knowledge (Geary & van Marle, 2016; Neumann, Hood, Ford, & Neumann, 2013; Piasta, Purpura, & Wagner, 2010) are individually predictive of mathematics skills. Yet, though each of these three components of emergent literacy skills has been found to be independently related to mathematics performance, little work focused on identifying if the relations were unique and specific to each component, or if some components were acting as a proxy measure for the other components of literacy development. To begin to address this issue, Purpura et al. (2011) assessed 69 preschool students on a broad measure of mathematics skills and on measures of phonological awareness, language, and print knowledge. These children were then assessed 1 year later on three mathematics measures (two broad measures that included formal and informal skills and one that just included formal skills). The results indicated that language and print knowledge each accounted for between 3% and 9% of the unique variance in predicting the two broader measures of mathematics, and phonological awareness (the core deficit in reading difficulties) was not a significant predictor of mathematics skills. None of the emergent literacy domains were significant predictors of formal-symbolic mathematics skills. These findings indicate that emergent literacy skills—specifically language and print knowledge—have a specific relation with early informal mathematics skills but not with formal skills. Although Purpura et al. (2011) and others (Austin, Blevins-Knabe, Ota, Rowe, & Lindauer, 2011; LeFevre et al., 2010) have shown that both language and print knowledge in general are related to informal numeracy and numeral knowledge and that expressive vocabulary is a significant predictor of almost all individual aspects of early numeracy (Purpura & Ganley, 2014), the distinct manner in which these two literacy components are related to specific aspects of early mathematical performance had not previously been evaluated. Austin and colleagues indicated that, though early language and print knowledge were related to early mathematics performance, it was likely in a developmental manner where children first obtain an understanding of receptive language and letter awareness before obtaining an understanding of early math concepts. To further evaluate this theory, Purpura and Napoli (2015) conducted a study to determine the distinct relation of vocabulary skills and print knowledge to components of early mathematics skills. To do this, 180 preschool children were assessed on a battery of targeted early mathematics and emergent literacy skills. It was found that the relation between language and numeral knowledge is fully mediated by informal numeracy

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skills (e.g., those mathematics skills that do not require the use of Arabic numerals) and the relation between informal numeracy skills and numeral knowledge skills is partially mediated by print knowledge. These findings indicate that language is likely to be more strongly connected to mathematical development early on and print-based skills may subsequently be more strongly connected. In all, this series of studies suggest that the association between language skills and mathematics development is particularly strong in the early years.

Language Interventions and Mathematics Skills Although research suggests that language skills are related to a broad array of early mathematics skills, interventions that focus only on language development do not appear to improve mathematics skills ( Jordan, Glutting, Dyson, Hassinger-Das, & Irwin, 2012). In a randomized control study, Jordan et al. (2012) tested the effectiveness of an 8-week-long intervention to build number sense knowledge in kindergartners at risk for mathematical deficits. Children in this study were randomly assigned to a number sense intervention group, a language intervention group, or a business-as-usual control group. The number sense intervention condition focused on number recognition, number sequencing, verbal subitizing, counting on fingers, number list activities, written number activities, number sets, problem solving and operations, and linear number board game. The language intervention condition focused on teaching 43 vocabulary words (e.g., general vocabulary and quantitative vocabulary) from eight different storybooks. The results showed that children in the number sense intervention condition had significantly higher scores on mathematics assessments by the end of the intervention than children in the language intervention and control groups. Further, the language intervention group did not significantly differ from the control group on posttest assessments of mathematics skills. This study suggests that interventions that focus on general vocabulary are not sufficient for improving children’s mathematics skills. Rather, it may be more effective to teach math-related vocabulary along with number sense concepts. It may also be more effective to focus only on mathematicsspecific vocabulary (i.e., mathematical language) rather than including general vocabulary that is unrelated to mathematics because the language related to mathematics is highly content specific (Harmon, Hedrick, & Wood, 2005).

WHAT IS CONTENT-SPECIFIC MATHEMATICAL LANGUAGE? In contrast to mathematics knowledge (or numeracy; the ability to work with exact numbers such as counting, comparing, adding), mathematical language is a child’s understanding of the key words and concepts used in early math. Specifically, mathematical language consists of terms that are used to describe quantity or spatial relations (e.g., more, fewer, near, below). Mathematical language is

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different from “number talk,” which usually relates to mathematical input in domains such as counting, cardinality, and calculation (Boonen, Kolkman, & Kroesbergen, 2011; Klibanoff, Levine, Vasilyeva, & Hedges, 2006). Unlike number talk, mathematical language does not include references to specific numbers or direct teaching of mathematics skills, and is related to children’s knowledge of specific terms rather than their production of mathematics-focused language. Two specific aspects of mathematical language have been identified as potentially important for early mathematical learning: quantitative language (Barner, Chow, & Yang, 2009) and spatial language (Ramani, Zippert, Schweitzer, & Pan, 2014). Quantitative language includes terms such as “more than,” “less than,” “many,” and “fewer.” Understanding these terms allows children to make and describe comparisons between groups or numbers. For example, knowing that the term “more” can mean an increase in quantity (“give me more”) or can be used in comparative statements (“five is more than two”). This knowledge may allow children to refine their understanding of quantity more precisely as approximate number words seem to develop prior to children’s acquisition of more exact cardinal number knowledge (e.g., children may understand that “five” refers to a quantity but not know the exact quantity; Gunderson, Spaepen, & Levine, 2015). Spatial language consists of words that describe the dimension of objects, locations and directions, and relations between objects, including words such as near and above (Cannon, Levine, & Huttenlocher, 2007). Understanding these spatial terms may allow children to talk about relations between physical objects and between numbers on a number line as well as develop spatial skills that have been found to be important for mathematics development (Mix & Cheng, 2012). For example, spatial words such as “before” and “after” are often connected to the number sequence (e.g., “five comes after four”). Children’s production of spatial language is positively related to their performance on spatial problem-solving tasks (Pruden, Levine, & Huttenlocher, 2011), as well as joint block play (Ramani et al., 2014). A list of the words typically considered to comprise these two types of mathematical language are presented in Table 1.

CORRELATIONAL AND EXPERIMENTAL EVIDENCE ON THE RELATIONS BETWEEN MATHEMATICAL LANGUAGE AND MATHEMATICS PERFORMANCE Mathematical language is one of the strongest predictors of mathematics development during the preschool (Purpura & Logan, 2015) and early elementary school years (Toll & Van Luit, 2014a). Notably, in Purpura and Logan’s study, the numeracy skills, mathematical language, executive functioning, and rapid automatized naming of 114 preschoolers were assessed in the fall and spring of the school year. Using both mixed-effect regressions, it was found that mathematical language was the strongest predictor of

TABLE 1 Quantitative and Spatial Language Words Used in Prior Studies Quantitative Terms

Spatial Terms

A couple

Above

A lot

After

Add

Around

Big, bigger, biggest

Away

Different

Back

Enough

Behind

Few, fewer, fewest

Before

Less than

Below

Least

Between

Many

Bottom

Minus

Close, closer, closest

More, Most

Down

Same

End

Several

Far, farther, farthest

Similar

First

Small, smaller, smallest

Forward

Some

Front

Subtract

High, higher, highest

Take away

In Into Inside Last Long, longer, longest Middle Near On Out, outside Over Through Top Toward Under

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numeracy development (i.e., set comparison, numeral comparison, one-to-one correspondence, number order, identifying numerals, ordinality, and number combinations) during preschool. When utilizing quantile regression which allows for examining predictors of the dependent variable across the spectrum of ability, mathematical language was found to be the best predictor at the 50th and 75th percentiles of ability. Furthermore, mathematical language has also been found to be more proximal to mathematics performance than are general language skills (Purpura & Reid, 2016; Toll & Van Luit, 2014b). Even though general language often predicts mathematics performance (LeFevre et al., 2010; Purpura et al., 2011), when mathematical language is also included in the model, general language is no longer a specific predictor—this is likely because general language acts as a proxy for more precise domain-specific language, including mathematical language. Young children who do not understand specific aspects of mathematical language such as comparative words (e.g., more) struggle to acquire early mathematics skills such as cardinal number knowledge (Barner et al., 2009), which can lead to later difficulties in mathematics development. In fact, in a recent study, Purpura, Day, Napoli, and Hart (2017) found that among a wide range of academic and cognitive skills assessed at the beginning of preschool (e.g., mathematical language, numeracy skills, vocabulary, executive functioning, approximate number system, print knowledge, general vocabulary, and phonological awareness), mathematical language was the strongest and most consistent classifier of the children who would perform the lowest on a numeracy measure at the end of preschool—it was even a better classifier than beginning of preschool numeracy performance. These findings suggest that children who enter preschool with lower mathematical language skills are the most likely to struggle to acquire numeracy skills during the preschool years. This was likely because children with lower mathematical language skills may not have had access to the mathematical instruction occurring in classrooms because they did not understand the language used to discuss these concepts, whereas other children who entered preschool low on numeracy skills but not low on mathematical language knowledge may have been able to benefit more from instruction. Children’s interactions with adults are central to the acquisition of early numeracy skills and mathematical language. The more mathematical talk and mathematical language that children hear at school and at home, the more likely it is that they have higher mathematical skills. For instance, studies have shown that interactions that include mathematical language, both with caregivers and teachers, predict children’s mathematics outcomes (Chard et al., 2008; Clements & Sarama, 2011; Gunderson & Levine, 2011; Levine, Suriyakham, Rowe, Huttenlocher, & Gunderson, 2010; Ramani, Rowe, Eason, & Leech, 2015). It has been postulated that mathematical languagerich interactions are important in developing children’s early mathematics skills (Clements, Baroody, & Sarama, 2013) and children from families with

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low SES are particularly at risk for delayed acquisition of mathematical language because they experience significantly less language-rich environments than their middle and higher SES peers (Hart & Risely, 1995). This language deficit is especially apparent in their exposure to mathematical language (Levine et al., 2010) and manifests into large differences (d ¼ 0.80) in knowledge of mathematical language concepts between children from families with low SES and middle/high SES by the time they reach preschool (Purpura & Reid, 2016). Thus children from families of lower SES are, from early ages, less likely to have acquired mathematical language skills.

Interventions to Improve Mathematical Language A strong emphasis on mathematical language is believed to be a critical component of the success of early mathematics instruction (NCTM, 2006), and researchers have emphasized the importance of including a mathematical language component in general mathematics instruction (Chard et al., 2008; Clements & Sarama, 2011). However, to date only a few intervention studies have focused on directly improving children’s mathematical language skills (Hassinger-Das, Jordan, & Dyson, 2015; Jennings, Jennings, Richey, & Dixon-Krauss, 1992; Powell & Driver, 2015), and these studies have provided strong evidence that children’s mathematical language can be improved through intervention. For example, Hassinger-Das et al. (2015) randomly assigned kindergarten children to a number sense plus mathematical language intervention, a number sense only intervention, or a business-as-usual control group. They found that children in the mathematical language group outperformed the other groups on immediate posttest measures of mathematical language, but not general mathematics measures. Yet, at a delayed posttest, the mathematical language group showed significant improvement on mathematical skills in comparison to the other conditions. These findings suggest that although some benefits of mathematical language instruction may not have been immediately found, children may have been provided with access to the content of instruction by gaining a better understanding of the language components of mathematics which allowed them to further develop their mathematics skills. Importantly, and similar to research conducted by Jennings et al. (1992) and Powell and Driver (2015), the mathematical language intervention targeted both mathematical knowledge and numeracy skills. Though these studies provide evidence of the malleability of mathematical language skills, they cannot be used to disentangle the effects of mathematical language specifically, without the additive component of mathematical knowledge. Critically, despite prior correlational evidence that mathematical language is predictive of mathematical development, these correlational and intervention studies cannot be used to make causal assertions regarding this relation. To address the issue of whether or not mathematical language is causally related to numeracy development, Purpura, Napoli, Wehrspann, and Gold (2017) conducted an intervention study that provided mathematical language

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instruction with no additional numeracy instruction. Forty-seven preschool children from Head Start centers were randomly assigned to a storybook reading intervention or a business-as-usual control group. Children were pre- and posttested on measures of numeracy, mathematical language, and general vocabulary. The mathematical language storybook intervention was administered for approximately 15 min per day, 2–3 days per week, for 7 weeks (M sessions ¼ 14.5). For the first 6 weeks, interventionists read one book per week to small groups of 2–3 children at a time. The foci of books alternated weekly between quantitative and spatial language, though both quantitative and spatial terms were highlighted when possible. During the seventh week, children were allowed to choose one of the six books for each session to review content and to ensure prolonged engagement. An eighth week was used to make up sessions with students who missed earlier sessions. A key aspect of this intervention was that it was designed to include only mathematical language terms (e.g., more, less, near, far) and not numeracy skill content (e.g., number names, counting, addition) to assess the causal relation between mathematical language and numeracy skills. Over the course of the intervention, children were exposed to 56 different mathematical language terms either as part of the story or as questions asked by the interventionists. Dialogic reading (Arnold & Whitehurst, 1994; Lonigan, Anthony, Bloomfield, Dyer, & Samwel, 1999) was used as the primary instructional framework of this intervention in order to promote children’s own use of mathematical language and more fully engage them with the content. The utility of dialogic reading for improving children’s language skills is likely derived from the emphasis on eliciting verbal responses and descriptions from children (Lever & Senechal, 2011). It was hypothesized that this framework would be effective for improving mathematical language as mathematical curricula rich in interaction are effective for improving both mathematics and language skills (Sarama, Lange, Clements, & Wolfe, 2012; see also Hassinger-Das et al., 2015). After the intervention was completed, the results of the study indicated that the intervention had significant effects on both mathematical language skills (Hedge’s g ¼ 0.42) and numeracy skills (Hedge’s g ¼ 0.32), but not general vocabulary. These findings indicate that promoting children’s mathematical language skills also positively affects their general numeracy skills (e.g., a measure that broadly assesses children’s skills in one-to-one correspondence, numeral identification, set comparison, numeral comparison, number order, ordinal number knowledge, story problems, and formal addition). It is likely that exposure to mathematical language terms in an interactive context where children repeatedly heard the words and were also encouraged to use the words themselves allowed children to form an understanding of the terms. As such, it appears that mathematical language may causally underlie numeracy development. Despite the positive findings from the Purpura, Napoli, et al. (2017) intervention study, it is unclear which aspect of mathematical language—quantitative or spatial—was driving the positive transfer to numeracy skills. Our work only

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examined effects of the intervention on the broad measure of mathematical language. However, this question can begin to be answered by examining the intervention’s impact on the quantitative and spatial mathematical language terms separately. When the mathematical language measure was separated into quantitative (six items) and spatial domains (10 items), these new exploratory analyses indicated that the effects on mathematical language were primarily found for quantitative mathematical language (Hedge’s g ¼ 0.80; Fig. 1) and not spatial language (Hedge’s g ¼ 0.20; Fig. 2). These findings suggest that instruction on

FIG. 1 Pre- and posttest mathematical language scores by condition for the quantitative mathematical language items.

FIG. 2 Pre- and posttest mathematical language scores by condition for the spatial mathematical language items.

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quantitative mathematical language, but not spatial mathematical language, underlies the early development of numeracy skills. Furthermore, by examining the pretest and posttest scores of each group in Figs. 1 and 2, it can be seen that for spatial mathematical language, both groups improve by about two points from pre- to posttest. In contrast, for quantitative mathematical language, only the intervention group improved, suggesting that children may not have acquired these quantitative terms over the course of normal schooling. Ultimately, it is likely that quantitative language affects numeracy development because it provides approximate benchmarks of quantitative relations (Gunderson et al., 2015) by which children can then develop more exact knowledge of specific numbers and relations about numbers (Shusterman, Slusser, Halberda, & Odic, 2016). However, given that both quantitative and spatial language were taught in an integrative manner, the more specific claim of whether it is instruction in quantitative language or spatial language that underlies this relation cannot yet be answered fully. A number of critical questions regarding impacts of mathematical language instruction and mechanisms by which it improves children’s mathematics skills need to be evaluated through future research.

UNANSWERED QUESTIONS There is a growing body of evidence that suggests mathematical language may be a key component in young children’s acquisition of early mathematics skills. Not only is it one of the strongest predictors of numeracy development during preschool and elementary school (Purpura & Logan, 2015; Toll & Van Luit, 2014b) and the strongest classifier of later risk status for mathematics difficulties during preschool (Purpura, Day, et al., 2017), but it also appears to causally contribute to numeracy development (Purpura, Napoli, et al., 2017). However, four critical gaps need to be addressed to better understand how mathematical language is related to mathematical skills over time and provide appropriate instruction in this area, including: (1) Investigating the way mathematical language instruction interacts with mathematics content instruction, (2) examining the relation between different aspects of mathematical language (e.g., more, fewer versus before, after) to different aspects of mathematics (e.g., numeracy versus geometry), (3) determining how to best assess mathematical language skills across the years, and (4) developing methods to provide effective mathematical language instruction. Following, we detail these key issues and future directions.

Mathematical Language and Numeracy Instruction It seems evident that quantitative mathematical language underlies early numeracy development; however, the nature of the relation and how these two constructs develop needs to be better understood. Notably, most common references to providing rich mathematical language environments encourage

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“mathematics instruction to be rich in mathematical language” (Clements et al., 2013; NCTM, 2006). Understanding mathematical language terms facilitates broader conceptual knowledge of numeracy (Dunston & Tyminski, 2013) which ultimately should lead to greater gains in numeracy skills (e.g., enumerating quantities, understanding the relations between quantities or numerals, and using informal and formal arithmetic operations). Likely, this is because understanding the quantitative terms provides an approximate understanding of numbers, which then enables children to more readily learn numeracy concepts with direct engagement of numeracy instruction. Suggestions that embedding mathematical language into mathematical knowledge instruction may be more beneficial than mathematical knowledge instruction alone have been partially supported by empirical evidence with elementary-age children (Fuchs et al., this volume; Hassinger-Das et al., 2015; Powell & Driver, 2015). Both of these studies with elementary students compared the combination of mathematical language and numeracy instruction, numeracy instruction alone, and a control (note that though the two studies had different mathematical content focus, the type of comparison groups was the same). Hassinger-Das et al. found that the combined intervention led to significantly improved numeracy skills compared to the numeracy intervention alone only at the delayed posttest, whereas, Powell and Driver found no significant differences between the conditions on addition fluency skills. The findings from existing studies are limited in applicability to younger children for three reasons. First, the specific numeracy content is different across grades, and early informal numeracy skills (e.g., counting, comparison) appear to be more language based than formal skills (e.g., addition, subtraction; Purpura & Napoli, 2015) that were targeted in these studies. Second, the findings from the Powell and Driver (2015) study are based on a measure of addition fluency. If mathematical language broadens conceptual knowledge of skills, the gains may not be manifest in a timed measure at immediate posttest. Third, neither Power and Driver nor Hassinger-Das et al. (2015) contrasted the interventions with a mathematical language only condition which has been shown to be sufficient for the development of early numeracy skills (Purpura, Napoli, et al., 2017). Importantly, is quantitative mathematical language necessary for the development of numeracy skills? Is it simply sufficient? Or, is it complementary where instruction of quantitative language enhances subsequent numeracy instruction? Ultimately, delineating the way in which quantitative language and numeracy instruction jointly or independently affect numeracy development is a necessary step in developing methods for maximizing the effectiveness of early numeracy instruction.

Spatial Language Quantitative language appears to drive the relation between mathematical language and early numeracy skills, but the role of spatial language in the development of mathematical skills is an important area for future research.

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Understanding spatial terms may allow children to talk about relations between physical objects and how to manipulate shapes and objects, as well as develop spatial skills that have been found to be important for mathematical development (Mix & Cheng, 2012). For example, many spatial words such as “near” and “above” are often used in instruction related to early geometry skills. Though spatial language is unlikely to be directly related to numeracy skills, it may be related to more proximal mathematics skills such as geometry (Casey et al., 2015) and spatial skills (Borriello & Liben, 2017; Miller, Vlach, & Simmering, 2017).

Mathematical Language Measures As researchers continue to investigate the role that mathematical language plays in mathematics development, there is a need to develop valid and reliable ways of assessing the understanding of mathematical language (Raghubar & Barnes, 2017). At the present time, no published standardized measure for mathematical language exists. Some work has been done to develop a measure of mathematical language for children in preschool (Purpura & Logan, 2015) and first grade (Powell & Nelson, 2017). However, significant work needs to be conducted to link these assessment frameworks as well as expand them beyond existing grades. Without the development of a uniform measure across these ages, it will be difficult to evaluate if mathematical language plays a continuous role in early mathematics development, or if this role is more foundational in nature.

Developing Methods for Mathematical Language Instruction Developing methods to enhance young children’s math skills before early difficulties become long-standing problems, particularly for children from families of lower SES, is important for enabling academic success. Despite the substantial evidence supporting math language as a critical foundation for early math skills development and the general consensus in major policy documents that math language should be incorporated into preschool math instruction (Clements et al., 2013; National Council of Teachers of Mathematics [NCTM], 2006; Purpura, Napoli, et al., 2017), there are limited available and empirically supported resources for preschool teachers to effectively implement math language instruction in the classroom. Though the concept of math language instruction may outwardly appear relatively simple, merely encouraging teachers to use more math language is not sufficient for improving children’s skills because there is significant variability in the quality of how these language concepts are presented and using a large range of math language may be negatively related to math skill acquisition because it may detract from more targeted language and concepts (Boonen et al., 2011; Lansdell, 1999). Even within effective empirically supported early mathematics curricula such as Building Blocks (Clements & Sarama, 2013) there is little specific structured emphasis on mathematical language concepts built into the lessons.

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Though teachers are instructed to “…develop children’s language and vocabulary by describing what individual children are doing…” in this curriculum, there is little direction for exactly how to do this language-enhanced instruction. There are a few lessons that target comparative skills, but they are mainly limited to concepts of “more.” Parents and teachers often do not have the tools or training to provide such targeted instruction (Ginsburg, Lee, & Boyd, 2008). As a consequence, there is a critical need to develop and evaluate easy-to-implement methods for teachers to provide math language instruction to their preschool students in an engaging manner.

CONCLUSIONS AND FUTURE DIRECTIONS Mathematical language during the preschool years is a critical component of early mathematics development. Children who acquire more advanced quantitative language skills also develop their numeracy skills in tandem. Further research is needed to examine how mathematical language instruction can be combined with mathematical knowledge instruction to enhance children’s early learning opportunities, as well as to identify which components, or combination of components, are most important for building early mathematical knowledge. Subsequently, building on that knowledge to develop effective instructional methods in this area is needed to enable teachers to maximize children’s learning of these concepts during preschool. Ultimately, the support of a causal relation between mathematical language and general mathematics knowledge begins to fill a critical gap in the research literature and complements and enhances prior correlational evidence, but more work is needed to better understand the underlying mechanisms driving these connections and how best to support young children’s learning of these terms and concepts.

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190 Cognitive Foundations for Improving Mathematical Learning Geary, D. C., & van Marle, K. (2016). Young children’s core symbolic and non-symbolic quantitative knowledge in the prediction of later mathematics achievement. Developmental Psychology, 52, 2130–2144. https://doi.org/10.1037/dev0000214. Ginsburg, H. P., Lee, J. S., & Boyd, J. S. (2008). Mathematics education for young children: what it is and how to promote it. Social Policy Report, 22, 3–23. Gunderson, E. A., & Levine, S. C. (2011). Some types of parent number talk count more than others: relations between parents’ input and children’s cardinal-number knowledge. Developmental Science, 14, 1021–1032. https://doi.org/10.1111/j.1467-7687.2011.01050.x. Gunderson, E. A., Spaepen, E., & Levine, S. C. (2015). Approximate number word knowledge before the cardinal principle. Journal of Experimental Child Psychology, 130, 35–55. https://doi.org/10.1016/j.jecp.2014.09.008. Hanich, L. B., Jordan, N. C., Kaplan, D., & Dick, J. (2001). Performance across different areas of mathematical cognition in children with learning difficulties. Journal of Educational Psychology, 93, 615–626. https://doi.org/10.1037/0022-0663.93.3.615. Harmon, J. M., Hedrick, W. B., & Wood, K. D. (2005). Research on vocabulary instruction in the content areas: implications for struggling readers. Reading and Writing Quarterly, 21, 261–280. https://doi.org/10.1080/10573560590949377. Hart, B., & Risley, T. R. (1995). Meaningful differences in the everyday experience of young American children. Baltimore, MD: Paul H. Brookes Publishing Co. Hassinger-Das, B., Jordan, N. C., & Dyson, N. (2015). Reading stories to learn math: mathematics vocabulary instruction for children with early numeracy difficulties. The Elementary School Journal, 116, 242–264. https://doi.org/10.1086/683986. Hooper, S. R., Roberts, J., Sideris, J., Burchinal, M., & Zeisel, S. (2010). Longitudinal predictors of reading and mathematics trajectories through middle school for African American versus Caucasian students across two samples. Developmental Psychology, 46, 1018–1029. https:// doi.org/10.1037/a0018877. Jennings, C. M., Jennings, J. E., Richey, J., & Dixon-Krauss, L. (1992). Increasing interest and achievement in mathematics through children’s literature. Early Childhood Research Quarterly, 7, 263–276. https://doi.org/10.1016/0885-2006(92)90008-M. Jordan, N. C., Glutting, J., Dyson, N., Hassinger-Das, B., & Irwin, C. (2012). Building kindergartners’ number sense: a randomized controlled study. Journal of Educational Psychology, 104, 647–660. https://doi.org/10.1037/a0029018. Jordan, N. C., & Hanich, L. B. (2000). Mathematical thinking in second-grade children with different forms of LD. Journal of Learning Disabilities, 33, 567–578. https://doi.org/ 10.1177/002221940003300605. Jordan, N. C., Hanich, L. B., & Uberti, H. Z. (2003). Mathematical thinking and learning difficulties. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise. Mahwah, NJ: Lawrence Earlbaum Associates Publishers. Jordan, N. C., Huttenlocher, J., & Levine, S. C. (1994). Assessing early arithmetic abilities: effects of verbal and nonverbal response types on the calculation performance of middleand-low income children. Learning and Individual Differences, 6, 413–432. https://doi.org/ 10.1016/1041-6080(94)90003-5. Jordan, N. C., & Levine, S. C. (2009). Socioeconomic variation, number competence, and mathematics learning difficulties in young children. Developmental Disabilities Research Reviews, 15, 60–68. https://doi.org/10.1002/ddrr.46. Klibanoff, R. S., Levine, S. C., Vasilyeva, M., & Hedges, L. V. (2006). Preschool children’s mathematical knowledge: the effect of teacher “mathematics talk” Developmental Psychology, 42, 59–69. https://doi.org/10.1037/0012-1649.42.1.59.

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Krajewski, K., Schneider, W., & Niedling, G. (2008). On the importance of working memory, intelligence, phonological awareness, and early quantity-number-competencies for the successful transition from kindergarten to elementary school. Psychologie in Erziehung und Unterricht, 55, 100–113. Lansdell, J. M. (1999). Introducing young children to mathematical concepts: problems with ‘new’ terminology. Educational Studies, 25, 327–333. https://doi.org/10.1080/03055699997837. LeFevre, J.-A., Fast, L., Skwarchuk, S.-L., Smith-Chant, B. L., Bisanz, J., Kamawar, D., et al. (2010). Pathways to mathematics: longitudinal predictors of performance. Child Development, 81, 1753–1767. https://doi.org/10.1111/j.1467-8624.2010.01508.x. Lever, R., & Senechal, M. (2011). Discussing stories: on how a dialogic reading intervention improves kindergartners’ oral narrative construction. Journal of Experimental Child Psychology, 108, 1–24. https://doi.org/10.1016/j.jecp.2010.07.002. Levine, S. C., Suriyakham, L. W., Rowe, M. L., Huttenlocher, J., & Gunderson, E. A. (2010). What counts in the development of young children’s number knowledge? Developmental Psychology, 46, 1309–1319. https://doi.org/10.1037/a0019671. Lewis, C., Hitch, G. J., & Walker, P. (1994). The prevalence of specific arithmetic difficulties and specific reading difficulties in 9- to 10-year-old boys and girls. Journal of Child Psychology and Psychiatry, 35, 283–292. https://doi.org/10.1111/j.1469-7610.1994.tb01162.x. Lonigan, C. J., Anthony, J. L., Bloomfield, B. G., Dyer, S. M., & Samwel, C. S. (1999). Effects of two shared-reading interventions on emergent literacy skills of at-risk preschoolers. Journal of Early Intervention, 22, 306–322. https://doi.org/10.1177/105381519902200406. Mann Koepke, K., & Miller, B. (2013). At the intersection of mathematics and reading disabilities: introduction to the special issue. Journal of Learning Disabilities, 46, 483–489. https://doi.org/10.1177/0022219413498200. McGregor, E. (1994). Economic development and public education: strategies and standards. Educational Policy, 8, 252–271. https://doi.org/10.1177/0895904894008003001. Miller, H. E., Vlach, H. A., & Simmering, V. R. (2017). Producing spatial words is not enough: understanding the relation between language and spatial cognition. Child Development., 88, 1966–1982. Advance online publication. https://doi.org/10.1111/cdev.12664. Mix, K. S., & Cheng, Y. L. (2012). The relation between space and math: developmental and educational implications. Advances in Child Development and Behavior, 42, 197–243. https://doi.org/10.1016/B978-0-12-394388-0.00006-X. National Council of Teachers of Mathematics (2006). Curriculum focal points for prekindergarten through grade 8 mathematics. Reston, VA: National Council of Teachers of Mathematics. Retrieved from https://www2.bc.edu/solomon-friedberg/mt190/nctm-focal-points.pdf. National Mathematics Advisory Panel (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. US Department of Education. Neumann, M. M., Hood, M., Ford, R. M., & Neumann, D. L. (2013). Letter and numeral identification: their relationship with early literacy and numeracy skills. European Early Childhood Research Journal, 21, 489–501. https://doi.org/10.1080/1350293X.2013.845438. Piasta, S. B., Purpura, D. J., & Wagner, R. (2010). Fostering alphabet knowledge development: a comparison of two instructional approaches. Reading and Writing, 23, 607–626. https://doi. org/10.1007/s11145-009-9174-x. Powell, S. R., & Driver, M. K. (2015). The influence of mathematics vocabulary instruction embedded within addition tutoring for first-grade students with mathematics difficulty. Learning Disability Quarterly, 38, 221–233. https://doi.org/10.1177/0731948714564574. Powell, S. R., Fuchs, L. S., Fuchs, D., Cirino, P. T., & Fletcher, J. M. (2009). Do word-problem features differentially affect problem difficulty as a function of students’ mathematics

192 Cognitive Foundations for Improving Mathematical Learning difficulty with and without reading difficulty? Journal of Learning Disabilities, 42, 99–110. https://doi.org/10.1177/0022219408326211. Powell, S. R., & Nelson, G. (2017). An investigation of the mathematics-vocabulary knowledge of first-grade students. The Elementary School Journal, 117, 664–686. https://doi.org/ 10.1086/691604. Pruden, S. M., Levine, S. C., & Huttenlocher, J. (2011). Children’s spatial thinking: does talk about the spatial world matter? Developmental Science, 14, 1417–1430. https://doi.org/ 10.1111/j.1467-7687.2011.01088.x. Purpura, D. J., Day, E. A., Napoli, A. R., & Hart, S. A. (2017). Identifying domain-general and domain-specific predictors of low mathematics performance: a classification and regression tree analysis. Journal of Numerical Cognition, 3, 365–399. https://doi.org/10.5964/jnc. v.3i2.53. Purpura, D. J., & Ganley, C. M. (2014). Working memory and language: skill-specific or domaingeneral relations to mathematics? Journal of Experimental Child Psychology, 122, 104–121. https://doi.org/10.1016/j.jecp.2013.12.009. Purpura, D. J., Hume, L. E., Sims, D. M., & Lonigan, C. J. (2011). Early literacy and early numeracy: the value of including early literacy skills in the prediction of numeracy development. Journal of Experimental Child Psychology, 110, 647–658. https://doi.org/10.1016/j.jecp.2011.07.004. Purpura, D. J., & Logan, J. A. R. (2015). The nonlinear relations of the approximate number system and mathematical language to early mathematics development. Developmental Psychology, 51, 1717–1724. https://doi.org/10.1037/dev0000055. Purpura, D. J., & Napoli, A. R. (2015). Early numeracy and literacy: untangling the relation between specific components. Mathematical Thinking and Learning, 17, 197–218. https:// doi.org/10.1080/10986065.2015.1016817. Purpura, D. J., Napoli, A. R., Wehrspann, E. A., & Gold, Z. S. (2017). Causal connections between mathematical language and mathematical knowledge: a dialogic reading intervention. Journal of Research on Educational Effectiveness, 10, 116–137. https://doi.org/ 10.1080/19345747.2016.1204639. Purpura, D. J., & Reid, E. E. (2016). Mathematics and language: individual and group differences in mathematical language skills in young children. Early Childhood Research Quarterly, 36, 259–268. https://doi.org/10.1016/j.ecresq.2015.12.020. Raghubar, K. P., & Barnes, M. A. (2017). Early numeracy skills in preschool-aged children: a review of neurocognitive findings and implications for assessment and intervention. The Clinical Neuropsychologist, 31, 329–351. https://doi.org/10.1080/13854046.2016.1259387. Ramani, G. B., Rowe, M. L., Eason, S. H., & Leech, K. A. (2015). Math talk during informal learning activities in head start families. Cognitive Development, 35, 15–33. https://doi.org/ 10.1016/j.cogdev.2014.11.002. Ramani, G. B., Zippert, E., Schweitzer, S., & Pan, S. (2014). Preschool children’s joint block building during a guided play activity. Journal of Applied Developmental Psychology, 35, 326–336. https://doi.org/10.1016/j.appdev.2014.05.005. Romano, E., Babchishin, L., Pagani, L. S., & Kohen, D. (2010). School readiness and later achievement: replication and extension using a nationwide Canadian survey. Developmental Psychology, 46, 995–1007. https://doi.org/10.1037/a0018880. Sarama, J., Lange, A. A., Clements, D. H., & Wolfe, C. B. (2012). The impacts of an early mathematics curriculum on oral language and literacy. Early Childhood Research Quarterly, 27, 489–502. https://doi.org/10.1016/j.ecresq.2011.12.002.

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Shusterman, A., Slusser, E., Halberda, J., & Odic, D. (2016). Acquisition of cardinal principle coincides with improvement in approximate number system acuity in preschoolers. PLoS One. 11(4), e0153072. https://doi.org/10.1371/journal.pone.0153072. Spelke, E. S. (2003). What makes us smart? Core knowledge and natural language. In D. Gentner & S. Goldin-Meadow (Eds.), Language in mind (pp. 277–311). Cambridge, MA: MIT Press. Spelke, E. S. (2017). Core knowledge, language, and number. Language Learning and Development, 13, 147–170. https://doi.org/10.1080/15475441.2016.1263572. Toll, S. W. M., & Van Luit, J. E. H. (2014a). Explaining numeracy development in weak performing kindergartners. Journal of Experimental Child Psychology, 124, 97–111. https:// doi.org/10.1016/j.jecp.2014.02.001. Toll, S. W. M., & Van Luit, J. E. H. (2014b). The developmental relationship between language and low early numeracy skills throughout kindergarten. Exceptional Children, 81, 64–78. https://doi.org/10.1177/0014402914532233. Willcutt, E. G., Petrill, S. A., Wu, S., Boada, R., DeFries, J. C., Olson, R. K., et al. (2013). Comorbidity between reading disability and mathematics disability: concurrent psychopathology, functional impairment, and neuropsychological functioning. Journal of Learning Disabilities, 46, 500–516.

Chapter 8

Early Numeracy Skills Learning and Learning Difficulties— Evidence-based Assessment and Interventions Pirjo Aunio Special Needs Education, Faculty of Educational Sciences, University of Helsinki, Helsinki, Finland

INTRODUCTION Children’s mathematical competencies provide a foundation for later occupational opportunities and for coping with the complexities of life as an adult (Hakkarainen, Holopainen, & Savolainen, 2013; Korhonen, Linnanm€aki, & Aunio, 2014). Early numeracy skills are the footing for children’s mathematics learning at school, as these predict later achievement in mathematics (Geary et al., 2017; Jordan, Glutting, & Ramineni, 2010). Low performance in early numeracy skills often leads to learning difficulties in mathematics. There are several different terms used to describe learning difficulties in mathematics, such as low performance in mathematics, learning difficulties in mathematics, mathematical learning disability, dyscalculia, mathematics disorder, and many more. These various terms reflect differences in the mathematics achievement cutoff scores used to define groups and different origins of the problems, ranging from neurological dysfunctions to inappropriate opportunities to learn and practice mathematical skills (e.g., low socioeconomic status of the child’s family; Ansari, 2015; Mazzocco, 2009). Geary (2013) suggests that children who score at or below the 10th percentile on standardized mathematics achievement tests for at least two consecutive academic years are categorized as having an MLD (Mathematical learning disability). He further suggests that all children scoring between the 11th and 25th percentiles, inclusive, across two consecutive years are classed as LA (Low Achievers). This is a useful approach for education, but it depends on well-developed assessment scales that are accessible to educators. Teachers Mathematical Cognition and Learning, Vol. 5. https://doi.org/10.1016/B978-0-12-815952-1.00008-6 Copyright © 2019 Elsevier Inc. All rights reserved.

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not only have to identify children with mathematics difficulties, but they have to adjust their instruction to support the learning of these children, preferably using evidence-based intervention programs if they are available. Finnish educational authorities have invested in efforts over the last 10 years to boost teachers’ levels of knowledge of individual learning differences in early reading and mathematical skills. The emphasis has been on the early identification of learning difficulties and early intervention. At the same time, there has been a nation-wide initiative toward the Responsiveness to Intervention model and general (Tier 1), intensified (Tier 2) and special educational support (Tier 3) in the national education system (National Core Curriculum for Basic Education, 2014/2016). Along with these changes, the National Ministry of Education and Culture has financially supported researchers to produce evidence-based knowledge for educators to use in the classroom and to produce assessment tools and intervention programs for children with mathematical and reading difficulties. Our research groups in Niilo M€aki Instituutti (University of Jyv€askyl€a) and University of Helsinki have designed two Web services for educators, namely, LukiMat (www.lukimat.fi) and ThinkMath (https://thinkmathglobal. com). LukiMat has three parts: reading, mathematics, and assessment of learning. All parts focus on basic skills learning for 5- to 10-year-olds. Together these services provide educators with evidence-based knowledge about learning and learning difficulties in reading and mathematics, in addition to assessment scales and computerized intervention programs (e.g., Graphogame, the Number Race). ThinkMath provides educators with knowledge about how to efficiently teach children with learning difficulties and hands-on intervention materials to be used with 5- to 8-year-olds, who have problems with learning early mathematical skills. In this chapter, I will describe our research on the evidence-based knowledge, assessment, and intervention tools for teachers.

EARLY NUMERACY SKILLS ARE IMPORTANT FOR FUTURE Early numeracy provides the foundation for later math learning (Aunio & R€as€anen, 2016; Merkley & Ansari, 2016). Critical early numeracy skills include number line estimation and differences in numerical magnitudes (LeFevre et al., 2010; Merkley & Ansari, 2016; Muldoon, Towse, Simms, Perra, & Menzies, 2013), recognition and naming of number symbols (G€obel, Watson, Lerva˚g, & Hulme, 2014; Pinto, Bigozzi, Tarchi, Vezzani, & Accorti Gamannossi, 2016), numerical relational and counting skills (Aunio & Niemivirta, 2010; Purpura & Reid, 2016), basic addition and subtraction skills and early arithmetical word problem-solving skills (Jordan, Kaplan, Ola´h, & Locuniak, 2006). In their recent study, Geary et al. (2017) suggest that all of these skills are first dependent on children’s understanding of the cardinal value of number words and numerals by the time they are 4 years old. Low performance in early numeracy skills is a risk factor for later mathematical learning difficulties (Jordan et al., 2006; Morgan, Farkas, & Wu, 2009)

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and can be observed during children’s average classroom activities. In the classroom, these children have not yet memorized the count sequence, make errors when counting collections of objects, have difficulties comparing the magnitudes of numbers, and have difficulties with basic addition and subtraction (Aunio & Niemivirta, 2010; Desoete, Stock, Schepense, Baeyens, & Roeyers, 2009; Jordan et al., 2006). These differences exist before formal schooling (Aunio, Hautam€aki, Sajaniemi, & Van Luit, 2009) and without intervention (Missall, Mercer, Martinez, & Casebeer, 2012). It is important that researchers communicate with educators in an accessible way. We developed the core factor model of mathematics skills for 5- to 8-year-olds with this in mind (Aunio & R€as€anen, 2016). It aims to be a working model for the educators by presenting them with an overview of the most important skills (i.e., core skills) that develop in early childhood and in the early grades. The model also highlights individual differences in early mathematical skills development. This model was based on a systematic literature review of longitudinal studies of mathematical development in this age range. We also analyzed the assessment batteries designed for identifying children with potential learning difficulties in mathematics. We were able to categorize skills into four main groups of numerical factors that are most crucial to the development of mathematical skills: (1) symbolic and nonsymbolic number sense, (2) understanding mathematical relations, (3) counting skills, and (4) basic skills in arithmetic (Table 1; Aunio & R€as€anen, 2016). Based on this model we have produced assessment and intervention materials and published them in LukiMat and ThinkMath services.

IDENTIFYING CHILDREN AT RISK FOR MATHEMATICAL LEARNING DIFFICULTIES One of the practical challenges in education is to identify at-risk children as early as possible. To do so requires the use of valid and reliable assessment scales. There have been discussions about what is the most efficient way to identify children at risk for learning difficulties in mathematics. One possibility is to use Curriculum-based Measurement (CBM) framework as it combines two well-known educational assessment traditions. The first is the conventional achievement testing paradigm, with documentation to support reliability and validity and with normative frameworks to compare a child’s performance with national norms. The second is behavioral measurement’s time series displays that permit scores for the same individual to be compared at different time points (see Fuchs, 2017). For instance, Purpura, Reid, Eiland, and Baroody (2015) argued for brief CBM measurements, such as Child Math Assessment—Abbreviated (e.g., Starkey, Klein, & Wakeley, 2004), Brief Research-based Early Mathematics Assessment (Weiland et al., 2012), and Preschool Early Numeracy Skills Screener—Brief version (Purpura et al., 2015). The benefit of a short assessment is ease of use, but other measures are needed to follow at-risk children’s learning

198 Cognitive Foundations for Improving Mathematical Learning

TABLE 1 Core Numerical Skills for Learning Mathematics in Children aged 5–8 Years. Core Numerical Factor

Core Numerical Skills Included in Factors

(1) Symbolic and nonsymbolic number sense

Understanding magnitude embedded in the number word sequence Subitising

(2) Understanding mathematical relational skills

One-to-one correspondence related to counting skills Basic arithmetic principles (incl. Associativity, commutativity, inversion) Base-10 arithmetic strategies (e.g., place value understanding)

(3) Counting skills

Knowledge of number symbols Number (word) sequence skills Enumeration skills

(4) Basic skills in arithmetic

Addition skills Subtraction skills

and to inform instruction and intervention. The monitoring tools can be done several times during the school year and they can help establish whether and to what extent children have acquired key skills (Lei, Wu, DiPerna, & Morgan, 2009). Lei et al. (2009) found that early mathematics CBM tasks are reliable and have potential as screening and progress monitoring tools. However, Laracy, Hojnoski, and Dever (2016) reported positive but low predictive validity with CBM measure (IGDIs-EN) and recommend that none of these measures should be used alone to determine risk of later poor performance. Their results also suggest caution in selecting cutoff scores for instructional or intervention purposes, which is also critical in studies focusing on mathematical learning difficulties (Geary, 2013). Fuchs (2004) describes two ways of making CBM. One is a curriculum sampling approach, which is tightly linked to curriculum, and requires different assessments for each school year. The second one is the Robust indicator method, which relies more on indicators representing the core competence in mathematics and having good predictive power relative to later mathematics learning. Our approach follows the Robust indicator method of designing CBM. We have developed screening tools for teachers to help identify children who are at risk for learning difficulties. The skills measured originate

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in the Core Factor model (Aunio & R€as€anen, 2016) and are predictive of later mathematics learning. We have concentrated on developing measurements that include several tasks for each of the four factors. This means that our approach is not a brief measurement (i.e., which could be done in 10 min with children), nor is it a pure progress monitoring tool, which can be repeated several times during a school term (Purpura et al., 2015). Instead, we designed scales that include several skills, work as screeners and include norms to support the decision making. We have devised two sets of screening instruments, one already available for Finnish educators and the other is being designed for Norwegian and South African educators. The first one, LukiMat scales, focuses on core skill factors: symbolic and nonsymbolic number sense, understanding mathematical relations, counting skills, and basic skills in arithmetic. They are designed to be used with kindergarten, first- and second-grade children. Teachers conduct the 30–45-min assessments which can be done individually or in groups. There are three scales that are administered three times a year: in the beginning of the school year (August–September), in the middle of the school year (November–December), and in the end of the school year (April–May). Hellstrand, Korhonen, R€as€anen, Linnanm€aki, and Aunio (in revision) studied the psychometric features of the LukiMat scales with 1139 children (552 boys) from kindergarten (n ¼ 361), first grade (n ¼ 321), and second grade (n ¼ 457) and found that early numeracy skills were divided into four reliable (a ¼ 0.91–0.95) factors (symbolic and nonsymbolic number sense, understanding mathematical relations, counting skills, and basic skills in arithmetic) that were all important in all three age groups (Table 2). The findings suggest that a focus on four core numerical skills will be a good working model for educators and researchers in structuring their educational assessments and for supporting instruction and research on children at risk for learning difficulties. The assessment batteries with teacher’s manuals, children’s materials, and norms are published in the LukiMat web service. We have provided norms for three measurements points in kindergarten, first and second grade. Educators have reported that the assessment is easy to use with groups of children and the results help them design their teaching and educational support (Tier 1 and Tier 2). The second series of assessment batteries, the ThinkMath-math scales, were primarily developed to measure the effects of early numeracy interventions in Finland (Mononen & Aunio, 2014, 2016). We have also begun developing these for use in South Africa (Aunio et al., in revision; Aunio, Mononen, orm€anen, 2016) and Norway (Lopez-Pedersen, Mononen, Ragpot, & T€ Korhonen, and Aunio, in preparation). The ThinkMath scale includes tasks that assess numerical relational skills, counting skills, and simple arithmetical word problem-solving skills. The scales are administered by teachers to groups of students and take about 30–45 min to complete.

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TABLE 2 Core Numerical Skills and Tasks in LukiMat Assessment Batteries Skills Measured

Content of Tasks

Symbolic and nonsymbolic number sense

Comparing magnitudes, approximate counting

Understanding mathematical relational skills

Seriation, comparison, classification, one-to-one correspondence Basic arithmetic principles: additive composition, commutativity, associativity, inversion Understanding mathematical symbols Understanding place-value and baseten system

Counting skills

Counting up or back, skip, count from given number Number identification, recognition, and writing Counting numerosity of a set, counting part of a whole

Basic skills in arithmetic

Addition and subtraction Verbal story problems; symbols

Applying understanding of place-value and base-ten system in counting (several skills measured)

Counting sums of money

In South Africa, we used the kindergarten scale in first grade, as the level of performance was estimated to be low. The correlation between ThinkMath total score and Listening Comprehension scale was statistically significant (r ¼ 0.52, P < .01) indicating that language and early numeracy performance were strongly related (see also Fuchs et al., this volume). We further investigated language effects by dividing children based on whether English was their primary language (n ¼ 79) or English as a second language (ESL; n ¼ 188). We found clear differences between the groups, indicating that it is not wise to use the same norms for all children. Fig. 1 (and Table 3) illustrates the differences between English as a home language and ESL children in five age groups. The preliminary results from our Norwegian study revealed the same three factors as were found in South Africa, namely, numerical relational skills, counting skills, and simple arithmetical word

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FIG. 1 South Africa ThinkMath total score in age groups and in ESL and non-ESL group.

problems. Again, there was a significant correlation with listening comprehension total score and ThinkMath total score (r ¼ 0.30, P < .01). To sum up it is clear that three clusters of skills are important for children’s mathematical development from kindergarten to second grade: numerical relational skills (comparison concepts with quantities and comparison of numbers), counting skills (number sequences forwards and backwards with missing number, and number word–quantity–number symbol relations), and simple arithmetical word problems (verbal addition and subtraction problems). With good screening scales that assess each of these three skills, it is possible for educators to identify children at risk for learning difficulties in mathematics. However, we definitely need more longitudinal studies to demonstrate how core skills develop and relate to later mathematical skills learning. There are also other future challenges, for instance, how to develop evidence-based short screening and deeper monitoring scales so that all educators will have access to them. In addition, it is important to further investigate the relation between language and mathematical performance (Fuchs et al., this volume; Pupura et al., this volume; Purpura, Hume, Sims, & Lonigan, 2011).

EARLY NUMERACY INTERVENTIONS FOR LOW-PERFORMING CHILDREN Meta-analyses of group-based interventions for children with mathematical learning difficulties show that explicit instructional procedures are more effective than other instructional approaches (Baker, Gersten, & Lee, 2002; Dennis et al., 2016; Gersten et al., 2009; Kroesbergen & Van Luit, 2003; Mononen, Aunio, Koponen, & Aro, 2014; Swanson, Hoskyn, & Lee, 1999). Explicit interventions include, for instance, breaking the instruction logical sequences, clearly presenting subject matter, guided practice, independent

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FIG. 3 Possible explanations for fadeout: forgetting. Note: For forgetting to fully account for fadeout, an indirect effect of the treatment on skill at time 2 must be offset by an indirect effect of the treatment on forgetting (SEM). Panel A shows an example in which the treatment group undergoes a net skill loss between the posttest and follow-up. This is less likely in early childhood than the case illustrated in Panel B, in which both groups experience net skill gain, but the treatment group experiences more forgetting than the control group. In the underlying SEM, Tx denotes treatment (1¼ treatment, 0 ¼ control).

gain (Panel B), the control group experiences a higher net skill gain in the posttreatment period. For forgetting to fully account for fadeout, an indirect effect of the treatment on skill at the follow-up assessment must be offset by an indirect effect of the treatment on forgetting (SEM). However, this explanation overlooks important theoretical and empirical considerations. Children probably do not stop practicing the math they learn from an effective early math intervention, because in the early years, math is hierarchically organized: For example, children practice addition as they learn multiplication (Lemaire & Siegler, 1995). Further, children learn a full standard deviation worth of material from kindergarten to grade 1, but math intervention impacts on vertically scaled achievement tests rarely approach that magnitude. Thus children are likely to be learning content from early math interventions that they will be rehearsing regularly in the following year. In long-term postintervention assessments, children who received a successful

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intervention do not perform worse at long-term follow-up assessments than they did on the posttest in the absolute sense; they merely learn less during the time since the end of the intervention than children in the control group (Clements et al., 2013; Elango et al., 2015). In other words, results resemble Fig. 3, Panel B much more than Panel A. Thus fadeout is often reasonably reframed in the context of early interventions as “catchup.”1 But while forgetting is unlikely to be the primary cognitive process underlying fadeout in early math interventions, it may still play a role: Children who receive an effective math intervention could, in principle, experience the same amount of learning as the control group following the end of treatment with more forgetting. My colleagues and I are beginning to look into this possibility and hope to report estimates of the role forgetting plays in fadeout in the near future. Teaching to the Test: If interventions target test content in ways that do not also build fundamental supporting skills, gains may be short-lived: Fadeout is a predictable consequence of teaching to the test. Children’s learning from early math interventions tends to be shallow and superficial, and thus unlikely to persist or transfer to more advanced skills.

It is not difficult to understand why teaching to the test implies fadeout. If children gain only the most superficial understanding of the content to which they are exposed, it is unlikely that this understanding will transfer to more advanced knowledge in the future. Indeed, the “hollowness” of test score gains has been invoked for decades as an explanation for fadeout in experiments that attempt to increase children’s general cognitive ability (Jensen, 1998; but see Protzko, 2016). A schematic depiction of the teaching to the test explanation, along with a SEM representation, appears in Fig. 4. Simply, there is an effect on test scores (Panel A, SEM), but this effect does reflect true changes in the underlying latent skill (Panel B, SEM). Test-specific effects, in this example, do not transfer to later test scores (SEM). Is teaching to the test a fair characterization of existing early mathematics interventions? The question deserves a full treatment, including a systematic review of observer reports of early math interventions and of psychometric analyses of the domains in which children learn most when receiving such interventions compared to business-as-usual control conditions. However, based on my unsystematic evaluation of the evidence, including discussions with early math interventionists, reading reports, and reanalyzing data, I am skeptical that this hypothesis accounts for much of the fadeout we observe. Developers of these interventions are often leading researchers in math cognition, development, and education, and they monitor the fidelity with 1. All catchup will result in fadeout, but fadeout is theoretically possible without catchup, in cases in which the treatment group experiences a net skill loss and the control group exhibits no improvement.

330 Cognitive Foundations for Improving Mathematical Learning

Treatment Control

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FIG. 4 Possible explanations for fadeout: teaching to the test. Note: For teaching to the test to fully account for fadeout, there must be an impact on test-specific variance at the posttest (Panel A, SEM), but no impact of the treatment on children’s true score on some skill (Panel B, SEM). Because test-specific variance in this example does not transfer across testing occasions, there is no effect of the treatment on test score at the follow-up (SEM). In the underlying SEM, Tx denotes treatment (1 ¼ treatment, 0 ¼ control).

which their interventions are implemented. Their descriptions of their math interventions often highlight the extent to which they target children’s conceptual understanding of number (e.g., Clarke et al., 2016), and their interventions show impacts on a diverse set of tests at the end of treatment (Clarke et al., 2016; Smith et al., 2013). Furthermore, interventions often occur in the same or similar contexts as those in which children receive their typical math instruction and extend the dosage of instruction by a substantial amount. All of these suggest to me that early math interventions are providing children with knowledge and understanding no more shallow and superficial than that which they would develop under business-as-usual conditions. However, to reiterate, this is not a systematic review of the extent to which early math interventions provide children with deeper or shallower understanding than they would otherwise receive, and doing so would make a useful contribution to the literature. Constraining Content: This hypothesis posits that children are not exposed to more advanced content than they learned during an effective intervention, leading to fadeout:

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After the conclusion of an effective early math intervention, students return to an instructional environment where teachers tend to target lower-achieving students and instruction does not build on the knowledge they received in the intervention, which leads to stagnation in the treatment group and catchup by the control group.

This explanation for fadeout is intuitively appealing, perhaps because it implies a simple solution: teach more difficult material (or, when it can be done effectively, differentiate instruction). Consistent with this explanation are (1) observations that kindergarten teachers report spending a substantial proportion of math instruction time on content that most of their students already know (Engel, Claessens, & Finch, 2013), and (2) the ubiquitous pattern of “catchup” described previously, where fadeout occurs as the treatment group’s posttreatment trajectory slows while the control group’s does not. Further, it seems highly likely under some conditions. For example, in an extreme case in which children would never encounter any content more advanced than the content to which they had been exposed following an effective early math intervention, these children would only make progress by spontaneously generating it. However, as noted previously, children learn very quickly in the early school years (when learning is expressed in standard deviation units on a vertically scaled achievement test); they learn more in the year following the intervention than the posttest impact of the most effective early childhood interventions of which I am aware. Thus the extreme scenario in which children in the treatment group are not exposed to new content in the year following an effective early intervention is not realistic. Still, it remains possible that the content to which children are exposed following an effective early intervention contributes to fadeout. My colleagues and I used the posttest matching approach described previously to test this possibility (Bailey et al., 2016). If the constraining content hypothesis fully explained fadeout, children in the treatment group and posttest-matched higher-achieving children in the control group should have similar trajectories in the following year (Fig. 5, Panel B and SEM), while the unmatched lowerachieving control children who were not matched to children in the treatment group catch up to both groups. We found that, consistent with Fig. 6, Panel B and contrary to predictions of the constraining content hypothesis, children in the treatment group fell behind their posttest-matched peers from the control group in the following year at approximately the same rate that the control group caught up to the treatment group. This pattern held for higher- and lower-achieving children in the treatment group, consistent with the possibility that preexisting differences between children in the treatment group and higher-achieving posttest-matched controls continued to favor the latter group following the conclusion of the intervention.

332 Cognitive Foundations for Improving Mathematical Learning

Treatment Control

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FIG. 5 Possible explanations for fadeout: constraining content. Note: Panel A shows the standard catchup pattern accompanying fadeout of early math intervention effects. For constraining content to fully account for fadeout, children in the treatment group and similarly high achieving children in the control group must be equally constrained by the low level of content to which they are exposed in the period following the end of the intervention (denoted by the gray line in Panel B). The effect of skill at time 1 on skill at time 2 is thus constrained at the higher end of skill at time 1 for both groups, indicated by the negative effect of skill at time 12 on skill at time 2 (SEM). In the underlying SEM, Tx denotes treatment (1¼ treatment, 0 ¼ control). Figure adapted from Bailey et al. (2016).

Modest Transfer: The Modest Transfer explanation posits that causal effects of earlier skills on later learning are real positive smaller than many assume, so that changes to early skills lead to diminishing effects on more advanced skills: Early math skill boosts are insufficient to generate long-term effects on children’s much later math achievement. Catchup occurs when transfer from basic skills to more sophisticated skills is modest and development under counterfactual conditions is rapid.

The explanation for fadeout I find most consistent with the current literature, which has survived several tests since my colleagues and I first described it (Bailey, Watts, Littlefield, & Geary, 2014), is that although children’s early math skills are necessary for their later math learning, they are not sufficient

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FIG. 6 Possible explanations for fadeout: modest transfer. Note: Panel A shows the standard catchup pattern accompanying fadeout of early math intervention effects. If modest transfer accounts for fadeout, children in the treatment group and similarly high achieving children in the control group will diverge following the end of treatment (Panel B) at the same rate as the treatment effect in Panel A decays. In this case, the predicted effect of skill at time 1 on skill at time 2 is smaller in the treatment group than in the control group, indicated by the higher slope in the control group in the posttreatment interval in the graphs and the negative treatment by skill interaction in the SEM. In the underlying SEM, Tx denotes treatment (1 ¼ treatment, 0 ¼ control). Figure adapted from Bailey et al. (2016).

to produce substantial long-term effects on children’s math achievement. Before outlining evidence for this hypothesis, I note why I provide it. Several math cognition researchers have expressed to me that the minimal transfer idea is overly pessimistic and that advancing it will do harm, either to the field or to children who would otherwise benefit from early math intervention. My hope is that these ideas will benefit the field and children in one of two ways. First, the strongest way to falsify this hypothesis would be to design an early math intervention that shows a large initial impact with an impact of about the same magnitude several years later. I do not think or hope that this chapter will convince all readers of this hypothesis, and I think it would be a mistake to conclude that designing better early math interventions is no longer a good use of anyone’s time. Second, if the hypothesis continues to survive attempts at falsification, I hope it will help motivate math cognition researchers to

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pursue research on complementary approaches to improving children’s academic outcomes (see Fuchs et al., this volume), which I will focus on later in this chapter. The idea that transfer of learning would be modest in math development is counterintuitive. As described previously under Teaching to the Test, basic skills are frequently reemployed in service of learning and performance of more advanced skills. Consistent with this idea, children’s math skills around school entry are robust statistical predictors of their much later mathematics achievement, statistically controlling for a wide range of cognitive and contextual covariates (e.g., Aunola, Leskinen, Lerkkanen, & Nurmi, 2004; Duncan et al., 2007; Geary, Hoard, Nugent, & Bailey, 2013; Jordan, Kaplan, Ramineni, & Locuniak, 2009; Siegler et al., 2012; Watts, Duncan, Siegler, & Davis-Kean, 2014). How can one reconcile these findings with evidence of fadeout from experimental studies? One possibility is that the set of statistical controls employed in these studies is insufficient to capture all of the factors influencing children’s math learning throughout their development. If these omitted variables are positively related to both early and later math achievement, a failure to control for them would lead us to overestimate the causal effect of changes in children’s early math achievement on their much later math achievement. Consistent with this hypothesis, we found in two longitudinal datasets that factors influencing children’s math learning throughout development accounted for more of the long-term stability in children’s math achievement scores during schooling, and that commonly used control variables failed to account for between 1/3 and 1/2 of the variance in these factors (Bailey et al., 2014). To rule out the hypothesis that preexisting math knowledge per se constituted much of the residual variation in this stable unmeasured factor, we tested whether an effective early math intervention acted on children’s math achievement via (1) a factor with similar influences on children’s math achievement throughout development, (2) an autoregressive process, where children’s math achievement at time t affects their math achievement at time t + 1, which in turn affects their math achievement at time t + 2, or some combination of these pathways. Only hypothesis 2 was supported (Watts et al., 2017). Accounting for the unmeasured persistent variation in children’s academic achievement, in these two studies, we estimate an effect of children’s prior achievement of approximately .35 SD following a standard deviation boost in children’s math achievement in the previous year, and .12 (¼.35*.35) SD following a standard deviation boost in children’s math achievement two years ago. In other words, a 1 SD gain in math achievement is 1st grade is predicted to result in a .35 SD gain in 2nd grade, but only a .12 SD gain in third grade. This pattern is close to what has been reported in experimental studies that boosted children’s math achievement near school entry and followed children for at least a year subsequent to the end of treatment

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(Bailey et al., 2018). It is important to note that the highly significant .35 SD effect size is fully consistent with the hypothesis that important math transfer take place from one year to the next, but because longer-run transfer estimates are a product of their .35 SD annual building blocks, they rapidly diminish in size. Transfer matters, but math achievement in a given year is the product of many other influences as well. Because these models reconcile robust relations between early and later math achievement with observed patterns of fadeout, it is not clear that explanations specific to early math interventions (relative to other factors leading to specific changes in children’s math achievement) are necessary. To the extent that this pattern of diminishing impacts reflects overlap between the items assessed in consecutive years, transfer of learning contributes even less to persistence. Still, these findings do not rule out the possibility of the other explanations proposed previously. Indeed, some of the ideas discussed previously are likely complementary explanations for modest transfer (e.g., the heterogeneity of math achievement across development implies that early gains will not necessarily lead to later gains; forgetting and constraining content could impede transfer). So far, this explanation is more technical than conceptual. How might one make sense of the possibility that boosts in children’s early math skills would not lead to lasting advantages relative to children’s business-as-usual educational experiences, despite the strong evidence for the importance of early math skills for later math learning and performance? Two important factors require consideration: (1) Although early skills are necessary for more advanced math learning and performance, they are not sufficient. For example, addition fact fluency is not sufficient for learning multiplication. To know how to multiply, one must at least have some familiarity with the multiplication symbol and a procedure for multiplication. (2) As noted previously, in the early school years, children in the control group who receive business-as-usual math instruction progress quickly (this point applies to pre-k and the early school years and may vary across grades, a possibility I will discuss later). Taken together, these factors limit the potential for large, long-lasting transfer effects. Hypothetically, imagine there are three sequential math skills, S1, S2, and S3, and that learning each skill is a necessary but not sufficient condition for learning the subsequent skill. Receiving effective instruction on a skill increases the probability of learning the next skill before a peer with the same socioeconomic status, reading achievement, IQ, and subsequent schooling by one half. However, importantly, in the time it takes the child to learn the more advanced skill, peers also master the previous skill. In this case, a child who receives an effective intervention, which changes her from not knowing S1 to knowing S1, will see her probability of learning S2 before an otherwise equal peer increase from .5 (an equal chance of either student learning S2 first) to .5 + (50% * .5), or .75. If she learns S2 before her peer, she has a probability of .75 of learning S3 before her peer. Therefore the

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probability of learning S3 before her peer is the probability of learning S2 first and S3 first (.752), plus the probability of learning S2 second but still learning S3 first (.252), or .625. As the number of skills in the sequence increases, the initial boost in the probability that the student treated on S1 will outperform her matched peer will decay by one half, approaching an asymptote of .5 (equally likely to learn; see Fig. 7). Obviously, this is an oversimplified scenario. For example, there is not a single route to learning a specific skill, and it is possible that more basic skills continue to influence more advanced skill learning after intermediate skills are learned. However, because of the rapid learning rates for control-group children discussed previously, the assumption that catchup occurs on the more basic skill during the period in which transfer occurs may not differ substantially from what happens. Consistent with this simplified model, approximately exponential decay of treatment effects (rapid decline after the end of treatment followed by more modest declines thereafter) has been observed in math and other early academic interventions (Bailey et al., 2018; Li et al., 2017). Some plausible objections to the ubiquity of the Modest Transfer explanation remain. Perhaps skills that have been (or can be) targeted by early math interventions do not develop quickly under counterfactual conditions and

0.8

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might affect children’s math learning throughout development. Further, the hypothesis that treatment effects regularly approach an asymptote of exactly zero, rather than some small but positive value that the average early intervention study is not powered to detect, deserves further empirical attention. Some meta-analytic estimates of long-term impacts of early childhood educational interventions (which include more than just math instruction, but often show a similar pattern of declining impacts after the end of treatment) approach an asymptote of approximately .05 SD (Li et al., 2017); to what extent this small but positive estimate reflects cognitive skill building effects, long-term effects of environmental enrichment via other processes, publication bias in reported outcomes, or sampling error, is not clear. However, while an effect of .05 may strike the reader as quite small, it is important to bear in mind that the average initial effect in these studies was just over .20 SD; to the extent that the long-term asymptote scales with the size of the initial effect (i.e., if 1/4 of the initial effect is regularly maintained years later), substantial long-term effects may be possible. The relative influences of mediating and moderating cognitive and educational processes across children’s long-term mathematical development are not well understood and deserve additional empirical attention.

IMPLICATIONS FOR THE STUDY OF CHILDREN’S MATHEMATICAL DEVELOPMENT Understanding fadeout has important implications for the study of children’s mathematical development. I have reviewed these implications elsewhere (Bailey, 2018), but summarize the main points here. First, the disappointing but important regularity of fadeout conflicts with common understandings of mathematical development. Specifically, as reviewed previously, analyses of nonexperimental longitudinal datasets have yielded estimates of the effects of early math achievement on much later math achievement that are higher than most corresponding estimates from experimental studies (Bailey et al., 2018). Fortunately, relevant high quality experimental and quasi-experimental studies exist. Resolving the discrepancy between experimental and nonexperimental findings in research on children’s mathematical development in a robust and replicable way is a worthy endeavor for some math cognition and educational researchers (especially those doing nonexperimental research) to pursue. In short, theories about the long-term effects of math knowledge on other math knowledge must, at the very least, be consistent with regularities in the existing literature. For this to happen, researchers conducting nonexperimental work on cognitive development need to provide clear and thoughtful theoretical interpretations of their results. For example, consider the empirical regularity that children’s earlier math achievement is a robust statistical predictor of their much later math achievement, controlling for a variety of child- and familylevel covariates. If we estimate a partial correlation of, say, .4 between

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children’s grade 1 and grade 5 math achievement, what is the proper interpretation of this estimate? Is it the causal effect of a 1 SD boost in children’s grade 1 math achievement on their grade 5 math achievement which, given the nonexperimental source of variation, may still suffer from lingering bias? Or is it only useful for prediction, perhaps indicating that achievement scores are a useful way to identify first graders who are at risk for low grade 5 math achievement, because of many causal factors in addition to their low grade 1 math achievement (e.g., working memory constraints)? The parameter underlying the .4 estimate is rarely discussed in method and results sections. On one hand, it is admirable that researchers do not wish to overstep their data. On the other hand, the scientific and educational value of this work is constrained by the extent to which it can inform theories that will make specific predictions about the effects of changes to children’s grade 1 math achievement in the real world (Borsboom, 2013). Making clear the assumptions of the model and the interpretation of the parameter of interest exemplify what Meehl (1990) described as “precision in the derivation chain.” This is not to argue that researchers ought to naı¨vely proclaim their results to be unbiased causal estimates. On the contrary, the goal is to use prior knowledge yielded from causally informative designs along with a series of robustness checks and falsification tests based on this knowledge to gauge, perhaps imprecisely, the likely magnitude and direction of lingering biases. The approach, known as “coherent pattern matching,” is detailed in Shadish, Cook, and Campbell (2002). For example, if our models yield similar estimates of effects of early math achievement on reading achievement, or on math achievement a year later and math achievement 5 years later, these might be signs of model misspecification. Some of the models I described in the section on the Modest Transfer explanation appear to recover the long-run observed effects of experimentally induced changes in early math achievement on children’s math achievement several years later (Bailey et al., 2018), but this should be seen as only one of possibly many models that are consistent with the fadeout data. Alternative specifications, replications, extensions, and robustness checks of these methods in the context of children’s mathematical development would make useful contributions to the literature. Being able to predict the long-term effects of changes in one math skill on others reasonably accurately, and to understand how these vary over time (Geary, Nicholas, Li, & Sun, 2017), would be even more useful. Finally, more practically, math cognition theory and real-world practice will benefit from larger field experiments that target specific skills and include long-term follow-up assessments after the intervention. My discussion of plausible explanations of fadeout was largely informed by a handful of RCTs of math interventions in pre-k or the early school years that assessed children at least a year after the end of treatment. This work, along with field studies in developing countries (Barner et al., 2016; Dillon et al., 2017), provides a rich combination of reasonably large sample size and well-measured

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constructs, along with a source of exogenous variation not present in nonexperimental longitudinal studies. Generating models that fit well in nonexperimental longitudinal datasets, replicate across datasets, and reliably reproduce the patterns of treatment effects observed in experimental datasets would allow for a better integration of cognitive developmental theory, longitudinal data analysis, and educational practice.

How can Researchers of Mathematical Cognition Help Produce Long-Lasting Effects? In reanalyses of experimental and nonexperimental longitudinal datasets, and in the design of early math interventions, it is worth considering whether some types of target skills produce more persistent effects than other skills. My colleagues and I have hypothesized that these skills share at least three important characteristics (Bailey, Duncan, Odgers, & Yu, 2017): First, and most obvious, is that the skill must be malleable by the early intervention of interest. Long-term impacts in the absence of any short-term impacts (sometimes called “sleeper effects”) are intuitively unlikely. Second, the skill must be fundamental for academic success. Ideally, the skill will be something that is used in the learning or performance of more advanced math. Third, skills most likely to produce persistent effects are those that do not develop quickly under counterfactual conditions. This is especially important for interventions targeting early achievement skills because, as discussed earlier, young children learn malleable and fundamental skills quickly—even in the absence of some prior intervention. Unfortunately, even assuming its accuracy, this framework also does not clearly identify the early skills meeting all three conditions (which Bailey et al. call “trifecta” skills) that might be targeted via early intervention. Schools ought to be promoting the learning of all fundamental and malleable skills, but doing so makes these skills develop more rapidly in the absence of intervention. Further, skills that do not develop readily under counterfactual conditions but may be fundamental for academic success (e.g., solving systems of equations) may not be readily malleable (i.e., teachable) for young children. Following, I highlight some possible cases in which all three criteria might be met. I caution that this is speculative, and again, hope that in attempting to falsify these ideas, the field will learn some practically useful principles.

Targeting At-Risk Children Knowledge of the characteristics of children at risk for persistently low mathematics achievement may allow us to target children who under business-asusual conditions would take several years to learn as much as they would learn from a successful early math intervention (e.g., Berch & Mazzocco, 2007; National Mathematics Advisory Panel, 2008). This has been an appealing

340 Cognitive Foundations for Improving Mathematical Learning

approach for interventionists, who often disproportionately target at-risk children. The long-term effects of interventions aimed at at-risk children may vary substantially depending on the targeted skill and age of intervention. I propose that a useful metric worth reporting in math intervention research is the ratio of the treatment effect on the targeted skill to growth in the control group during the same year. Based on the framework described previously, interventions with ratios substantially higher than 1 would be predicted to produce the longest lasting effects.

Targeting Advanced Skills in Older Children Another way to approach this would be to target older children. As noted previously, children progress far more slowly in middle and high school than in elementary school (Hill et al., 2008). This is not purely a psychometric artifact. For example, numerous children and adults struggle to understand fractions many years after they are introduced during schooling (NMAP, 2008; Schneider & Siegler, 2010). Children’s knowledge of fractions is fundamental to their ability to learn more advanced math, and it can be changed in children at risk for persistently low math achievement (Fuchs et al., 2013). I am not aware of a study that has examined the effects of an effective fraction intervention many years later, but based on the framework I outlined previously, this seems like a promising direction. Math cognition randomized controlled trials with multiyear follow-up intervals for older children are not common. However, two relevant and important examples come from large causally informative regression discontinuity designs, in which children who scored below a set threshold on a math achievement placement test were required to take two periods of math instead of one. In Chicago, children who performed below a set threshold on a math test were required to take a second period of algebra in ninth grade. Children who scored just below this threshold showed persistent benefits several years later relative to children who scored just above the threshold (Cortes, Goodman, & Nomi, 2015). Most notably, the effect of being double dosed on high school graduation was 12 percentage points (the graduation rate at the cutoff was 58%). This is certainly an important study for helping to understand the factors that might improve at-risk children’s educational outcomes. However, the pattern of impacts of the program implies that math cognition may not have been the primary active ingredient. In grade 10, children took a standardized math test, and the effect of being double dosed was a nonsignificant .09 SD. In grade 11, there was a statistically significant impact on students’ ACT Math scores (.18 SD), but oddly, it was smaller than the impact on their ACT Verbal scores (.27 SD). Elsewhere, we hypothesized that this effect might tell us more about the important effects of receiving credit for a high school algebra class on students’ persistence in high school (Bailey et al., 2017); being double dosed increased children’s chances of receiving a C or better

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in algebra by 12 percentage points. The extent to which an even more effective algebra intervention would lead to persistent effects on children’s real world outcomes, and whether this knowledge influences children’s later outcomes directly, rather than simply through course completion, is not clear, but are important relevant questions for math interventionists. Taylor (2014) evaluated a similar program in Miami that assigned sixth graders who performed below a set threshold on a math test to take a second period of math in sixth grade. The effect on grade six math achievement was .16–.18 SD, which was reduced to approximately one-third of this size by the end of grade eight. Impacts in high school were null. Perhaps the difference in the persistence of these two interventions was attributable to the systemic advantage afforded to the Chicago children who would have failed high school algebra if not for the intervention. In contrast, perhaps the link between middle school math performance and educational attainment is less mechanistic. The possible impacts of interventions on “staying on track” in educational settings warrant consideration by interventionists and by developmentalists interpreting the long-term effects of interventions (Bailey et al., 2017). The end-of-treatment effects in both of these studies were much smaller than some obtained by early math interventions, offsetting any potential benefit of targeting children with slower learning under counterfactual conditions. The promise of the approach of targeting older children depends on the inevitability of this pattern. Given the strong focus of math cognition research on younger children (for a review of the current state of the field, see Alcock et al., 2016), perhaps a somewhat increased focus on older children’s learning and cognition would be sensible.

Complementary Follow-Through Interventions Several interventionists have told me that they do not intend their early math interventions to be inoculations against later low achievement, and that children must receive better instruction following the conclusion of effective early interventions for effects to be sustained. Based on the patterns of effects observed following the end of effective early math interventions, this seems like a reasonable stance. Before considering what an effective complementary follow-through intervention might look like, it is important to consider what it means for an intervention to be complementary. In economics, complementarity refers to the condition under which the causal effect of one variable (early math intervention, in this case) is larger for people who also receive some other treatment (subsequent schooling environment, in this case). To establish that a second intervention has sustained the effect of the first intervention, one would ideally rerandomize children from both the treatment and control groups from an RCT of an early math intervention into a treatment or control group for a second intervention. A positive interaction between receiving the

342 Cognitive Foundations for Improving Mathematical Learning

first intervention and the second intervention (i.e., whereby the effect of the earlier intervention is more persistent for children who received the second intervention than for children who did not) would be evidence of complementarity, which would mitigate fadeout. I have not seen this design used in published RCTs of early math programs, but identifying complementary interventions would be a very useful step toward raising the long-term achievement of children at risk for persistently low math achievement with targeted intervention. Proof of concept for the idea of complementary interventions comes from recent work that estimates the effect of attending Head Start, the effect of attending a school receiving increased funding, and the interaction between these variables ( Johnson & Jackson, 2017). Findings suggest positive effects of both on adult income, but larger effects of Head Start attendance for children who subsequently attended schools that received more funding. This work is clever and causally informative, but our ability to draw conclusions about complementary math cognition interventions is limited, because the independent variables of interest (spending) are so distally related to math instruction. Importantly, complementarity is a hypothesis, not a law. The opposite pattern of substitutability is also possible. For example, a study of Danish children found positive effects on educational attainment for children who received pre-k and positive effects of receiving a nurse home visiting program in infancy, but the effect of pre-k disappeared almost completely for the children who had received the nurse home visiting program, meaning that the programs were nearly perfect substitutes for each other (Rossinust, 2016). Slater & W€ What would an effective complementary follow-through intervention look like? As reviewed in the summary of the constraining content hypothesis previously, I am skeptical as to whether merely teaching more advanced content would disproportionately benefit children who received an effective early intervention (except in the unrealistic case that the early intervention was so effective that there was almost no overlap in achievement between the groups by the end of treatment). Jenkins et al. (2018) found that the long-term effects of receiving an effective pre-k math intervention did not interact with classroom quality in kindergarten and first grade. Perhaps complementary intervention is most likely when early and later interventions are more directly aligned in pedagogy, terminology, instructional materials, and teacher knowledge. Of course, this is also speculative.

The Possibility of Different Effects of Improving Early Math Intervention at Scale? Finally, while the constraining content hypothesis has limited support under existing conditions, it is likely that at some level of scale and effective implementation, effective early intervention could push up the level of later instruction for children across the elementary school years. For example, changes in county level spending for early childhood education programs in North Carolina

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across years were associated with changes in children’s math and reading scores at age 11 (Dodge, Bai, Ladd, & Muschkin, 2017). Importantly, effects were positive even for children who were not eligible for participation in these programs, consistent with the possibility of “spillover” effects on nonparticipants. Perhaps implementing a universal program for young children that improved their school-entry math skills would allow instruction in later grades to change. Of course, to the extent this would become economically and practically feasible to implement, math cognition researchers would likely favor such a program. In the meantime, perhaps the impacts of one-year math interventions assigned at the school or district level on children’s achievement many years later would be a theoretically and practically useful set of interventions to study.

CONCLUSIONS AND FUTURE DIRECTIONS Explanations for fadeout, their implications for research practices, and their implications for intervention are conceptually intertwined. However, while interventionists are concerned with designing treatments that will produce persistent effects in the field, lab research on mathematical cognition rarely addresses the problem of persistence. This is a shame, because math cognition researchers possess important knowledge about learning, transfer, and memory, along with (although not usually expressed in these terms) the malleability, fundamentality, and development under counterfactual conditions of aspects of children’s math knowledge. All of these ideas feature prominently into explanations for fadeout and implications for practice. The purpose of this chapter is to attempt to introduce math cognition researchers to the regularity of fadeout in field studies that follow children for a long time following the conclusion of an effective early math intervention and to get them thinking about reasons why, what this means for their research, and whether they might be able to improve math intervention by combining this kind of information with their knowledge of developmental changes in mathematical cognition and learning. In the short-term, thoughtful theoretical interpretation of nonexperimental research findings, comparison of these findings to experimental findings when possible will help us make more accurate predictions about the long-term effects of changes to early math skills on important outcomes. Combining these theories with tests involving variation in targeted skills, child baseline achievement and age, and complementary interventions, would further develop these theories and yield practically important information as well.

ACKNOWLEDGMENTS I am grateful to the Eunice Kennedy Shriver National Institute of Child Health & Human Development of the National Institutes of Health under award number P01-HD065704. I also thank Dan Berch, Greg Duncan, Dave Geary, and Kathy Mann Koepke for helpful comments on a previous draft.

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Index

Note: Page numbers followed by “f ” indicate figures, and “t” indicate tables.

A Academic feud, 2 “Accumulator” perspective, 150–151 “Actions-on-objects” theory, 150 Adolescents’ Mathematics achievement, 10 Adults’ Mathematical knowledge, 69 Ainsworth, S., 280–281 Aldous, C., 269–270, 289–290 Alexander, P. A., 248 Algebra, 6 Analogical reasoning, 247–250 Analogy, 247–250 numbers are points on the number line, 251–252, 254 Anderson, A., 114 Anderson, M. L., 20 Approximate number discrimination game, 52 Approximate number system (ANS), 43–45, 61, 64, 91–92 arithmetic ability and, 56–57 link with Math, 92 symbolic Math abilities, 92 training studies, 92–99 brief ANS training, 96–98 long-term ANS training, 98–99 nonsymbolic number processing, specific aspects of, 96–99 nonsymbolic vs. symbolic training, 95–96 using “The Number Race”, 92–95 Arabic numerals, 91–92 Area discrimination game, 52 Arithmetic, 92–93, 124–125 Arithmetic abilities, 59 and ANS, 56–57 classroom geometric and, 42–43 Arithmetic ability assessment, 59 Arithmetic performance, 96–97 Arithmetic test, 95 Arithmetic training, 98–99 Assessment Record Sheets (ARS), 233 At-risk children, 1, 3 At-risk students, 302

Attention training, 229 Aunio, P., 93, 199, 206–207

B Baddeley, A. D., 12 Bailey, D. H., 6, 209 Bandura, A., 23 Baroody, A. J., 197–198 Baxter, J., 271 Beilock, S. L., 128 Beitzel, B., 280–281 Bempeni, M., 256 Berteletti, I., 96–97, 99–100 Boonen, A., 283, 285–286 Booth, J., 77–78, 276, 280–281 Bootstrapping process, 248–249 Brankaer, C., 81 Brannon, E. M., 98–99 Bridging analogies, 250, 264–265 Building Blocks curriculum, 84, 146–147 Bull, R., 12 Business-As-Usual (BAU) Control group, 38, 47–48, 60–62

C Campbell, D. T., 338 Campbell, K., 274–275, 281–282 Cantor-Dedekind axiom, 251–252 Cardinal number gestures, 122–124 Cardinal number knowledge, 113 Casey, B. M., 24–25, 110–111 Cattell, R. B., 8, 10–11 Child cognition, parental influences, 22–25 children’s home numeracy environment, 24–25 history of research, 23–24 methodological issues, 25 Child Math Assessment (CMA), 227–228, 234 Child Math Assessment scores, 221f

347

348 Children auditory and visual feedback, 82–83 early Mathematical knowledge SES-related gap in, 215–217 foundational numerical knowledge early numerical knowledge, 69–70 integrated theory, 70–74 mathematics development, play and games in, 76–84 numerical magnitudes knowledge, understanding, 70–76 numerical magnitudes, understanding, 70–76 home numeracy environment, 24–25 mathematical competencies, 195 mathematical development, 9–10 cognitive processing explanations, 327–337 explanations of fadeout, 323–337 implications for, 337–343 instrumentation hypothesis, 324 measurement-based explanations, 323–327 time and theories, 322–337 math learning gestures, 120–126 support for, 120–126 number learning, 109 numerical magnitude knowledge computer and tablet games, 81–83 improving through games and play, 77–83 reasoning skills, 205–206 spatial thinking, 118–119 working memory skills, 82 Chutes and Ladders, 77–78 Cimen, E., 276 Clarified analogy, 252 Clement, J., 250, 261–262, 264–265 Cogmed visuospatial working memory training program, 14 Cognitive development, subitizing, 153–154 Cognitive intervention, 222 Cognitive load theory, 283–284 Cognitive processes, 296–297 implications for practice, 316 limitations in, 309–311 Cognitive processing explanations, 8–14, 327–337 Cognitive science, 145 Cognitive skill building effects, 336–337 Cognitive training explicit skills intervention, 297–306 instruction on language comprehension, 299 word problems, 298–299

Index

Cohen Kadosh, R., 21–22 Coherent pattern matching, 338 Collis, K., 274 Common Core State Standards for Mathematics (CCSS-M), 270 Compensatory moderator effect, 310, 311f Comprehension issues, 282–283 Computer technology, 161–162 Conceptual metaphors, 248 Conceptual subitizer, 162–163 Conceptual subitizing, 155 Condition effect, 304 Content-specific mathematical language, 178–179 Cook, T. D., 338 Core numerical skills, 198t, 200t Costa, H. M., 83 Counting gestures, 121 Creative Curriculum, 217 Crollen, V., 122 Crowley, K., 308–309 Cummins, D. D., 301 Curriculum-based Measurement (CBM) framework, 197–199 Curriculum development, 143–144 Curriculum research framework (CRF), 144–166 core feature of, 145 evaluation, 162–166 formative research, multiple classrooms, 164–165 formative research, single classroom, 163–164 formative research, small group, 162–163 market research, 162 summative research, large scale, 165–166 summative research, small scale, 165 learning model, 147–162 learning trajectory, 147–162 a priori foundations, 145–147 general a priori foundation, 146 pedagogical a priori foundation, 147 subject matter a priori foundation, 146 specific learning model, 147–162

D Daubert, N. A., 80–81 Davies, B., 77 Davis-Kean, P. E., 113–114 Day, E. A., 181 Dean, J. T., 96 Deary, I. J., 10

349

Index

Dehaene, S., 93, 269 De Smedt, B., 81, 94–95 Developmental dyscalculia (DD), 16 Dever, B. V., 197–198 Dewolf, T., 276 Diagram literacy, 275, 280 Dialogic reading, 183 Diezmann, C., 276–278 Dillon, M. R., 96 DISC strategy, 288 Domain-general components, 10–14 executive function, 11–13 fluid intelligence, 10–11 implications, 13–14 working memory, 11–13 Domain-general interventions, 14–15 Domain-relevant interventions, 18–22 arithmetic skills, brain stimulation effects, 22 numerical and arithmetic processing, electrical brain stimulation, 21–22 numerosity discrimination, brain stimulation effects, 21–22 subitizing speed, enhancing, 18–19 training finger differentiation, arithmetic ability, 20–21 Domain-specific components, 15 Domain-specific interventions, 16–18 number line judgments, computer-based training, 16–18 Donlan, C., 176 Driver, M. K., 182, 186 Dubois, O., 93 Duflo, E., 96 Dumas, D., 248

E Early Child and Youth Development, 110–111 Early childhood. See also Child cognition classrooms, 69–70 math instruction, 323 numerical magnitude understanding in, 75–76 Early curricular intervention, pre-k mathematics intervention, 218–220 Early mathematics development, 107, 175–178 Early math learning, 107 Early number talk, 119–120 Early numeracy intervention, low-performing children, 201–208 Early numeracy skills, 196–197 low performance, 196–197 Early training studies, 92–93

Educational community, Uruguay, 38–39 Educational intervention studies, 204–205t Educationally oriented research, 156 Eiland, M. D., 197–198 Elen, J., 94–95 Elia, I., 277–278 Elicited self-explaining, 308 End-of-treatment effects, 341 English, L., 276–278 Euclidean line, 251, 254–255 Executive function, 11–13. See also Working memory Experimental studies, 114–115 in field, 116–117 in lab, 115–116 Explicit interventions, 201–203 Explicit skills intervention cognitive profiles, 306–316 fraction magnitude comparisons, 307–308 fractions at fourth grade, 307 multicomponent fractions intervention, 308–309 word problems, 307–308

F Fagnant, A., 278–280 Fang, H., 279, 283–284 Fayol, M., 93 Feigenson, L., 97–99 Ferrara, K., 119 Fidelity assessment, 233 Field notes, 272 Finger gnosis, 20 Finger representation, 121 Finger sensorimotor neural circuits, 20 First-grade number knowledge intervention, 295 First intervention approach, 236–237 Fischer, J.-P., 20–21 Fitzhugh, J. I., 150 Fluid intelligence, 10–11 Fractions Faceoff!, 312 Fuchs, L. S., 7–8, 13–15, 198–199, 295, 298–299, 307–310, 312, 316 Fyfe, E. R., 308–309

G Gagatsis, A., 277–278 Game Intervention children, 57–62 Game Intervention condition, 47–48 Game Intervention group, 57 Games and play, preschool programs, 83–84

350

Index

Geary, D. C., 13–14, 82, 195–196, 295, 298 General language, 175–178 Gersten, R., 4–5 Gestures, 120–126 cardinal number, 122–124 counting, 121 Ghesquie`re, P., 81 Gibson, D., 116–117, 123 Gliksman, Y., 18–19 Gobel, S., 14 Grade and repeater status, 54–55 Grade-Repeating status, 38 GraphoGame Math, 93, 208 Greeno, J. G., 301 Gunderson, E. A., 113, 123, 128

Intervention approaches, 1–2 Intervention classrooms, 271–272 Intervention games, 51–52 approximate number discrimination, 52 area discrimination, 52 time discrimination, 51–52 Intervention research methodological issues, 3–8 design factors, 3–5 implementation, fidelity of, 5 intervention impacts, diminishing, 5–6 transfer effects, 7–8 Investment theory, 8, 11 IQ, 53–54. See also Intelligence Izard, V., 72

H

J

Halberda, J., 97–99 Hanif, R., 96–97 Hart, S. A., 181 Hassinger-Das, B., 182, 185–186 Hatano, G., 250 Hawthorne effect, 25 Head Start classrooms, 78, 182–183, 342 Hegarty, M., 276, 281–282 Hellstrand, H., 199 Henik, A., 18–19 Hitch, G. J., 12 Hoard, M. K., 82 Hojnoski, R. L., 197–198 Holling, H., 94 Honore, N., 14, 95–96 Horn, J. L., 10 Hulme, C., 297–298 Human cognition, 249. See also Executive function; Fluid intelligence; Working memory Huttenlocher, J., 124 Hyde, D. C., 96–100

I Inductive reasoning, 248 INEEd report, 41–42 Informal mathematical knowledge, 218–219 Informal numeracy skills, 177–178 Information and communications technology (ICT), 37–38 Instructional intervention, 15 Integrated theory, 70–75 Intelligence, 8. See also Fluid intelligence Intergenerational findings, 126–130

Jablansky, S., 248 Jenkins, E., 124–125 Jenkins, J. M., 342 Jennings, C. M., 182 Jo, B., 83–84 Jordan, N. C., 124, 178 Journal of Numerical Cognition, 14

K Kannan, H., 96 Kaplan, D., 124 Khanum, S., 96–97 Kindergarten, 73, 230–231 children, 80 Kintsch, W., 301 Klahr, D., 150 Klein, T., 288–289 Knops, A., 14–15 Koedinger, K., 276, 280–281 Korhonen, J., 199 Kozhevnikov, M., 276, 281–282 Kucian, K., 16–17 Kuhn, J.-T., 94 Kytt€al€a, M., 12–13

L Lakoff, G., 248 Language comprehension, 299–304, 306 Language interventions, 178 Language therapy, 298–299 Laracy, S. D., 197–198 Learning trajectories, formative evaluation and validation, 162 Lee, K., 12

351

Index

LeFevre, J. A., 109 Lehto, J. E., 12–13 Lei, P.-W., 197–198 Lester, F., 20 Levine, S. C., 112–114, 119, 124, 127–128 Li, W., 322 Libertus, M. E., 99 Linear color board game, 79f Linear number board game, 79f Line diagrams, 280 Linnanm€aki, K., 199 Literacy skills, 176–178 Locuniak, M. N., 124 Loehr, A. M., 308–309 Logan, G. D., 161 Low-income children, 216 Low-performing children intensified support in mathematics, 238–239 residual subgroup of, 239 Lucangeli, D., 94 LukiMat scales, 196–197, 199 LukiMat web service, 208

M MacDonald, B. L., 155, 161 Maertens, B., 94–95 Magnitude knowledge tasks, 71f Malone, A., 307–310, 312, 316 Marentette, P., 122 Martignon, L., 21 Materials, intervention, 51 Math abilities, 94, 98–99 Math anxiety, 109–110, 126 intergenerational effects of, 127–128 Math+Attention condition, 224–227 Math Cognition Conference, 323 Math cognition randomized controlled trials, 340 Mathematical cognition, 321, 338–339 long-lasting effects, 339–343 at-risk children, targeting, 339–340 complementary follow-through interventions, 341–342 early math intervention at scale, 342–343 older children, targeting advanced skills, 340–341 skills, characteristics, 339 Mathematical development, 126, 163–164 Mathematical language instruction, developing methods, 187–188 interventions to improve, 182–185

mathematics performance and, 179–185 measures, 187 and numeracy instruction, 185–186 unanswered questions, 185–188 Mathematical learning child cognition, parental influences, 22–25 difficulties, 197–201 Mathematical learning disability (MLD), 176, 195–196 Mathematical progressions, 148 Mathematical talk, 181–182 Mathematics development, play and games in, 76–84 Mathematics intervention studies, history of, 2–3 Mathematics learning cognitive foundations for, 8–22 domain-general components, 10–14 domain-general interventions, 14–15 domain-relevant interventions, 18–22 domain-specific components, 15 domain-specific interventions, 16–18 Mathematics, literacy skills, connections, 176–178 Mathematics proficiency, 175–176 Mathematics skills, 175, 178 Math Only condition, 224–225 Math Shelf, 83–84 Math talk, 100–102, 108, 119–120 Math wars, 2–3 Meehl, P. E., 337–338 Melby-Lerva˚g, M., 7, 297–298 “Memory”, 80–81 Mental number line, 70–72 Michels, L., 16–17 Miller, M. R., 308–309 Mix, K. S., 115–116 Modest Transfer, 336–337 Modest Transfer explanation, 338 Moeller, K., 21 Mononen, R., 206–207 Moore, J. A., 115–116 Mou, Y., 99–100 Myers, T., 17–18

N Napoli, A. R., 24–25, 177–178, 181–185 Nathan, M. J., 301 National Assessment of Educational Progress, 8 National Institute of Education Assessment (INEEd), 39

352 National Mathematics Advisory Panel (NMAP), 4, 13–14, 307 NCTM standards, 271 Negative attitudes, math, 128–130 Newman, S., 20 Nicoladis, E., 122–123 Noe¨l, M. P., 14, 20, 95–96, 122 Non-Repeaters, 54 Nonsymbolic arithmetic, 98–100 Nonsymbolic number comparisons, 95–100 “No successor” principle, 260 Nuerk, H.-C., 14, 21 Number gestures, 121, 123 Number knowledge, 112–114 Number knowledge intervention, 303–304, 306 Number line, 254–264 students’ interpretations of, 254–255 students’ reasoning about density, 255–257 students’ understanding of density, arithmetical and geometrical context, 257–261 “The Number Race”, 92–95, 208 “Numbers are points” analogy, 261–264 Numbers are points on the line, 254 Number sense training, 94 Number talk, 108–117 experimental studies, 114–117 naturalistic home observations, 111–114 observational studies, 110–111 questionnaire studies, 109–110 Number words, 91–92, 156 Numeracy instruction, 185–186 Numeracy skills, 183–185 Numerical magnitudes knowledge, understanding, 70–76 development, integrated theory, 70–74 Nu´n˜ez, R., 248

O Obersteiner, A., 95 Object files, 151 Object individuation, 152 Observational studies, 110 in lab, 110–111 Odic, D., 97–99 O’Donnell, C. L., 5 O’Gorman, R., 16–17 One Laptop Per Child, 37–38 Oral language comprehension, 298 Oral language instruction, 298 Overlapping Operations Hypothesis, 99–100

Index

P Pantziara, M., 277–278 Parental math attitudes, 126–130 Parental math beliefs, 126–130 Parent-child Math engagement, 108–109 Parent-delivered book intervention, 116–117 Parent Feedback Forms (PFF), 234 Parent math language number talk, 108–117 outcomes, 108–120 spatial talk, 117–119 Parent number talk, 111, 114–115, 117–118 Parent-reported math activities, 100–101 Parents, 22–25, 101, 116–117 cognitive abilities, 101 Math anxiety, 126 Math skills, 101 spatial language, 118 support, 107 Park, J., 98–99 Participants, 48–51 Passolunghi, M. C., 83 Pedagogical content knowledge, 220 Penner-Wilger, M., 20 Perceptual subitizing, 153–154 Performance assessments, 272 Perry Preschool program, 321 Piaget, J., 76 Pika, S., 122 Pinel, P., 269 Plan Ceibal, 37–38, 48 Po´lya, G., 248, 252, 264 Powell, S. R., 182, 186 Pre-intervention arithmetic ability, 54–55 Pre-K Mathematics, 84, 232–235 Pre-K Mathematics intervention, 218–221 low-performing children, responsiveness, 220–221 Pre-K Mathematics Tutorial (PKMT), 221–222, 225–226, 229–230 Pre-Pre-K Mathematics, 232–235 Prerequisite ability moderator effect, 311f Preschool, 75 Preschool-age children, 80 Preschool programs, mathematics skills, 84 Preschool teachers, 216–217 inadequate preparation of, 217 Pre- to postintervention improvement, 57–62 Pruden, S. M., 119 Psychological research, 148

Index

Public preschool programs, 239–240 Purpura, D. J., 15, 24–25, 177–178, 181–185, 197–198

Q Quantitative language, 179, 186–187 Quasi-experimental studies, 4 Quintile, 53–54

R Ramani, G. B., 80–81, 111, 307–308 Ramineni, C., 124 Ramirez, G., 128 Randomized controlled trials (RCTs), 4 R€as€anen, P., 93, 199 Ratio-dependent discrimination, 91 Rational number knowledge, 74 Rational numbers betweenness property of, 252 dense ordering of, 252–253 infinity of intermediates, 252 Reading comprehension skills, 282 Reading wars, 2–3 Recognition of number, 154–155 Reid, E., 197–198 Reiss, K., 95 Relational reasoning, 248 Repeater children, 54 Repeater status, 53–54, 63 evaluation, 49–51 Representational mapping hypothesis, 78–80 Representational overlap hypothesis, 99–100 “Rescue Calcularis” game, 16–17 Research-based curricula, 143–144 Research-to-practice model, 145–146 Reuter, T., 278, 280 Reynvoet, B., 94–95 Richland, L. E., 250 Rittle-Johnson, B., 308–309 Roesch, S., 21 Rousseau, J.-J., 2 “Rubber line” bridging analogy, 261–264 Russell, C., 115–116

S Sakakibara, T., 250 Salminen, J., 93 Sandhofer, C. M., 115–116 Sarkar, A., 21–22 Sasanguie, D., 94–95

353 Saxe, G. B., 121, 130–131 SBI curriculum, 285–287, 289 Scalise, N., 80–81 Schacter, J., 83–84 Schema-based instruction (SBI), 284–289, 302, 315 Schema-based intervention, 300–301 Schneider, M., 7 School-age children, 12 Schumacher, R. F., 13–14, 307–308, 316 Science, technology, engineering, and mathematics (STEM), 107–108, 117, 128 Scientific curriculum development program, 144 Second intervention approach, 237 Self-efficacy, 2 Sella, F., 94 Seron, X., 122 Shadish, W. R., 338 Short Mathematics Anxiety Rating Scale, 127–128 Shrager, S., 124–125 Siegler, R. S., 7, 77–78, 124–125, 289–290, 307–309 Singer, L. M., 248 Skill learning impacts, hypothetical pattern, 336f Skwarchuk, S. L., 109–110 Sleeper effects, 339 Smith, C. L., 248–249 Snapshots activity, 162–163 Social learning theory, 23 Socioeconomic status (SES), 38–41, 63, 181–182 in education, 42 IQ, 53–54 repeater status, 53–54 Soylu, F., 20 Spatial ability, 126–127 Spatial anxiety, 128 Spatial language, 119, 125, 179, 186–187 Spatial talk, 117–119 Spelke, E. S., 96–97, 269 “Spillover” effects, 342–343 Split-attention effect, 284 Spontaneous self-explaining, 308 Staley, R., 280–281 Stanescu, R., 269 Structural equation model (SEM), 324–325 Students’ ACT Math scores, 340–341 Subitizing, 153–154 Subset knowers, 122–123 Suriyakham, L. W., 121, 123

354 Susperreguy, M. I., 113–114 Symbolic number comparisons, 95 Symbolic numerical magnitude knowledge, 75 Szucs, D., 17–18

T Tape diagrams, 276 Tax working memory, 309–310 Taylor, E., 341 Teacher math anxiety, 127 Teachers, instructional goal, 219–220 Teachers’ responses, software, 46–47 Teaching to the Test, 334 TEMA-3, 227–228, 231, 235 Terwel, J., 278–279 ThinkMath intervention programs, 196–197, 199, 203, 205–208 Thompson, C. A., 7 Thorndike, E. L., 7 Tier 1 Math interventions child sample, 231–232 intensification, 230–236 mathematical knowledge, economically disadvantaged backgrounds, 235 math intervention, 232–234 measures and assessment procedures, 234 participants, 231–232 Pre-K Mathematics, 232–234 Pre-Pre-K Mathematics, 232–234 study design, 232 very low-performing children, mathematical knowledge, 235–236 Time discrimination game, 51–52 Toll, S. W., 81–82 Training studies, 92–99 brief ANS training, 96–98 long-term ANS training, 98–99 nonsymbolic number processing, specific aspects of, 96–99 nonsymbolic vs. symbolic training, 95–96 using “The Number Race”, 92–95 Transcranial direct current stimulation (tDCS), 21 Transcranial random noise stimulation (tRNS), 21–22 Transfer, 7–8, 336–338 Tressoldi, P., 94 TRIAD model, 165–166 Tsivkin, S., 269 Tutorial-based math intervention, 228–229 Tutorial interventions attention intervention, 226–227

Index

in mathematics and attention, 221–230 math intervention, 225–226 measures and assessment procedures, 227–228 study design and participants, 223–225 child sample, 224 math screening measure, 223–224 randomized experiment, 224–225 Tweney, R. D., 269 2-digit calculations, 295–296

U Ufer, S., 95 Unresponsiveness, 296 Uruguayan educational system, 63 Uruguayan National Public Education Administration (ANEP), 38–39

V Vamvakoussi, X., 15, 255–256 Vandermaas-Peeler, M., 114–115 Van Dooren, W., 276 Van Luit, J. E., 81–82 Verbal number knowledge, 123–124 Verdine, B. N., 118–119 Verschaffel, L., 276 Very low-performing children. See also Mathematical learning disability (MLD) intervention approaches, 237–238 mathematical knowledge, 235–236 Vigilance game, 227 Visual representations embedding in text, 276–278 integrating with text, 281–284 schema-based instruction, 284–289 students’ natural use of, 270–275 teaching students, diagrams creation, 278–281 Visuospatial processing, 92 Visuospatial sketchpad, 12–13 Visuospatial working memory, 14 Vlassis, J., 278–280 Vygotsky, L. S., 23, 76

W Wang, J., 97–98 “War”, 80–81 Wasner, M., 21 Watson, J., 274 Weber fractions, 72–73

355

Index

Wehrspann, E. A., 182–183 Weinbach, N., 18–19 Westerholm, A., 207–208 What Works Clearing House, 4 Wilkins, J. L. M., 155 Wilson, A. J., 93 Woodcock-Johnson Tests of Achievement, 42–43 Woodcock-Mun˜oz Arithmetic Ability score, 55f Woodward, J., 271 Woodworth, R. S., 7 Word-problem intervention, 299–302

Word-problem language, 303–305 Word problems, 298–299 integrated instruction in, 284–289 Working memory, 11–13. See also Executive function capacity, 314 training, 7 World Computer Exchange, 37

Z Zbrodoff, N. J., 161 Zorzi, M., 94