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El. knyga: Cognitive Foundations for Improving Mathematical Learning

Edited by (University of Virginia, Charlottesville, VA, USA), Edited by (National Institute of Child Health and Human Development, NIH, Rockville, MD, USA), Edited by (University of Missouri, Columbia, MO, USA)

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Cognitive Foundations for Improving Mathematical Learning, Fifth Edition, the latest release in the Mathematical Cognition and Learning series, focuses on informal learning environments and other parental influences on numerical cognitive development and formal instructional interventions for improving mathematics learning and performance. Chapters cover the use of numerical play and games for improving foundational number knowledge, school math performance, the link between early math abilities and the approximate number system, and how families can help improve the early development of math skills.

In addition, the book examines learning trajectories in early mathematics, the role of mathematical language in acquiring numeracy skills, evidence-based assessments of early math skills, approaches for intensifying early mathematics interventions, the use of analogies in mathematics instruction, schema-based diagrams for teaching ratios and proportions, the role of cognitive processes in treating mathematical learning difficulties, and more.

  • Identifies the relative influence of school and family on math learning
  • Discusses the efficacy of numerical play for improvement in math
  • Features learning trajectories in math
  • Examines the role of math language in numeracy skills
  • Includes assessments of math skills
  • Explores the role of cognition in treating math-based learning difficulties
Contributors xi
Foreword: Cognitive Foundations for Improving Mathematical Learning xiii
Ann Dowker
Preface xxi
1 Introduction: Cognitive Foundations for Improving Mathematical Learning
1(36)
David C. Geary
Daniel B. Berch
Kathleen Mann Koepke
Introduction
1(1)
Brief History of Mathematics Intervention Studies
2(1)
Methodological Issues in Intervention Research
3(1)
Design Factors
3(2)
Fidelity of Implementation
5(1)
Diminishing Intervention Impacts Across Time (Fadeout)
5(2)
Transfer
7(1)
What Are the Cognitive Foundations for Improving Mathematics Learning?
8(2)
Domain-General Components
10(4)
Domain-General Interventions
14(1)
Domain-Specific Components
15(1)
Domain-Specific Interventions
16(2)
Domain-Relevant Interventions
18(4)
Parental Influences on Child Cognition and Mathematical Learning
22(1)
A Brief History of Research on Parental Influences
23(1)
Research on Children's Home Numeracy Environment
24(1)
Methodological Issues
25(1)
Conclusion
25(1)
References
26(11)
2 The Effects of SES, Grade-Repeating, and IQ in a Game-Based Approximate Math Intervention
37(32)
Josh Langfus
Alejandro Maiche
Dinorah De Leon
Dahiana Fitipalde
Alvaro Mailhos
Justin Halberda
Introduction
37(1)
The Uruguayan Context
38(4)
Background for the Current Study
42(1)
Socioeconomic Status in Education
42(1)
Classroom Geometric and Arithmetic Abilities
42(1)
Approximate Number Abilities
43(2)
The Present Intervention Study
45(1)
Teachers' Responses to the Software
46(1)
Design of the Current Study
47(1)
Methods
48(4)
Results
52(10)
Conclusions and Future Directions
62(3)
References
65(4)
3 Role of Play and Games in Building Children's Foundational Numerical Knowledge
69(22)
Geetha B. Ramani
Emily N. Daubert
Nicole R. Scalise
Introduction
69(1)
Understanding of Numerical Magnitudes
70(1)
Integrated Theory of Numerical Development
70(5)
Numerical Magnitude Understanding in Early Childhood
75(1)
Play and Games in Mathematics Development
76(1)
Improving Children's Numerical Magnitude Knowledge Through Games and Play
77(6)
Preschool Programs Using Games and Play
83(1)
Conclusions and Future Directions
84(1)
References
85(6)
4 Understanding the Link Between the Approximate Number System and Math Abilities
91(16)
Melissa E. Libertus
Introduction
91(1)
Training Studies
92(1)
Training Studies Using "The Number Race"
92(3)
Nonsymbolic vs. Symbolic Training
95(1)
Training Specific Aspects of Nonsymbolic Number Processing
96(3)
Mechanisms Behind Long-Term ANS Training Improvements
99(3)
Conclusions and Future Directions
102(1)
References
103(4)
5 Mathematical Development in the Early Home Environment
107(36)
Susan C. Levine
Dominic J. Gibson
Talia Berkowitz
Introduction
107(1)
Parent Math Language and Associated Outcomes
108(1)
Number Talk
108(9)
Spatial Talk
117(2)
Summary: Math Talk
119(1)
Gesture: An Additional Support for Children's Math Learning
120(1)
Counting Gestures
121(1)
Cardinal Number Gestures
122(2)
Gesture and Arithmetic
124(1)
Summary: Gesture
125(1)
Parental Math Attitudes and Beliefs: Intergenerational Findings
126(1)
Intergenerational Effects of Math Anxiety
127(1)
Other Negative Attitudes Toward Math
128(2)
Summary: Parental Math Attitudes and Beliefs
130(1)
Conclusions and Future Directions
130(3)
Acknowledgments
133(1)
References
133(10)
6 From Cognition to Curriculum to Scale
143(32)
Julie Sarama
Douglas H. Clements
Introduction
143(1)
The Curriculum Research Framework (CRF)
144(1)
The CRF and Early Mathematics
145(1)
The CRF Enacted
145(1)
Category I A Priori Foundations
145(2)
Category II Learning Model and Learning Trajectory
147(15)
Category III Evaluation
162(4)
Conclusions and Future Directions
166(1)
References
167(8)
7 Development of Mathematical Language in Preschool and Its Role in Learning Numeracy Skills
175(20)
David J. Purpura
Amy R. Napoli
Yemimah King
Introduction
175(1)
Connections Between Early Mathematics Development and General Language
175(1)
Difficulties in Both Domains
176(1)
Early Connections Between Mathematics and Literacy Skills
176(2)
Language Interventions and Mathematics Skills
178(1)
What is Content-Specific Mathematical Language?
178(1)
Correlational and Experimental Evidence on the Relations Between Mathematical Language and Mathematics Performance
179(3)
Interventions to Improve Mathematical Language
182(3)
Unanswered Questions
185(1)
Mathematical Language and Numeracy Instruction
185(1)
Spatial Language
186(1)
Mathematical Language Measures
187(1)
Developing Methods for Mathematical Language Instruction
187(1)
Conclusions and Future Directions
188(1)
References
188(7)
8 Early Numeracy Skills Learning and Learning Difficulties---Evidence-based Assessment and Interventions
195(20)
Pirjo Aunio
Introduction
195(1)
Early Numeracy Skills are Important for Future
196(1)
Identifying Children at Risk for Mathematical Learning Difficulties
197(4)
Early Numeracy Interventions for Low-Performing Children
201(4)
Studies With ThinkMath Intervention Programs
205(3)
Conclusions and Future Directions
208(1)
References
209(6)
9 Improving the Mathematical Knowledge of At-Risk Preschool Children: Two Approaches to Intensifying Early Math Intervention
215(32)
Alice Klein
Prentice Starkey
Lydia DeFlorio
The SES-Related Gap in Children's Early Mathematical Knowledge
215(2)
Potential for Early Curricular Intervention to Reduce the Math Cap
217(1)
The Pre-K Mathematics Intervention
218(2)
Effectiveness of the Pre-K Mathematics Intervention
220(1)
First Approach: Tutorial Interventions in Mathematics and Attention
221(2)
Study Design and Participants
223(2)
The Math Intervention: Pre-K Mathematics Tutorial
225(1)
The Attention Intervention
226(1)
Measures and Assessment Procedures
227(1)
Findings
228(2)
Summary of Findings From Tutorial Intervention Study
230(1)
Second Approach: Intensification by Providing 2 Years of Tier 1 Math Intervention
230(1)
Study Design and Participants
231(1)
The Math Intervention: Pre-Pre-K Mathematics and Pre-K Mathematics
232(2)
Measures and Assessment Procedures
234(1)
Findings
235(1)
Summary of Findings
236(1)
Effectiveness of the First Intervention Approach
236(1)
Effectiveness of the Second Intervention Approach
237(1)
Comparison of the Two Intervention Approaches for Very Low-Performing Children
237(1)
Conclusions and Future Directions
238(1)
Conclusion 1 Most Low-Performing Children Respond to Intensified Support in Mathematics
238(1)
Conclusion 2 A Residual Subgroup of Low-Performing Children Showed Low Response to Intensified Math Support; Research is Needed to Understand Why
239(1)
Conclusion 3 Public Preschool Programs Should Provide High-Quality, Intensive Math Support, But New Policies and Resources Are Needed
239(1)
Acknowledgments
240(1)
References
240(7)
10 The Use of Analogies in Mathematics Instruction: Affordances and Challenges
247(22)
Xenia Vamvakoussi
Analogy and Analogical Reasoning in Mathematics
247(4)
Where Does the Analogy "Numbers are Points on the Line" Come From?
251(1)
Tapping Into Students' Understanding of the Dense Ordering of Rational Numbers
252(2)
Could the Number Line Support Students' Understanding of Density?
254(1)
Does the Number Line Have an Effect on Students' Reasoning About Density?
255(2)
Is Density More Accessible to Students in a Geometrical Rather Than in an Arithmetical Context?
257(4)
Using the "Numbers are Points" Analogy and the "Rubber Line" Bridging Analogy
261(3)
Conclusions and Future Directions
264(1)
References
265(4)
11 The Role of Visual Representations in Mathematical Word Problems
269(26)
Asha K. Jitendra
John Woodward
Students' Natural Use of Visual Representations in Mathematics
270(6)
Embedding Visual Representations in Text
276(2)
Teaching Students to Create or Complete Diagrams as They Solve Problems
278(3)
Integrating Visual Representations With Text
281(3)
Schema-Based Instruction (SBI): Integrated Instruction in Word Problems and Visual Representations
284(5)
Conclusions and Future Directions
289(1)
References
290(5)
12 The Role of Cognitive Processes in Treating Mathematics Learning Difficulties
295(26)
Lynn S. Fuchs
Douglas Fuchs
Amelia S. Malone
Pamela M. Seethaler
Caitlin Craddock
Embedding Cognitive Training Within Explicit Skills Intervention to Facilitate Transfer to Academic Performance
297(1)
Study Context
297(2)
Conceptual Model for Linking Language Comprehension and Our Approach to Explicit Skills Word-Problem Intervention
299(3)
Study Overview
302(1)
Preliminary Results
303(3)
Some Tentative Conclusions
306(1)
Allocating Varying Forms of Explicit Skills Intervention to Subgroups of Learners With Different Cognitive Profiles
306(1)
Study Context
307(2)
Does Supported Self-Explaining Compensate for Limitations in Cognitive Processes?
309(3)
Study Overview
312(2)
Results
314(2)
Implications for Practice
316(1)
Summary and Future Directions
316(1)
References
317(4)
13 Explanations and Implications of Diminishing Intervention Impacts Across Time
321(26)
Drew Bailey
Introduction
321(1)
Patterns of Effects Across Time and Theories of Children's Mathematical Development
322(1)
Possible Explanations of Fadeout
323(14)
Implications for the Study of Children's Mathematical Development
337(2)
How can Researchers of Mathematical Cognition Help Produce Long-Lasting Effects?
339(4)
Conclusions and Future Directions
343(1)
Acknowledgments
343(1)
References
344(3)
Index 347
David C. Geary is a cognitive developmental and evolutionary psychologist at the University of Missouri. He has wide ranging interests but his primary areas of research and scholarly work are childrens mathematical cognition and learning and Darwins sexual selection as largely but not solely related to human sex differences.

Professor Geary directed a 10-year longitudinal study of childrens mathematical development from kindergarten to ninth grade, with a focus on identifying the core deficits underlying learning disabilities and persistent low achievement in mathematics. The study was funded by the National Institutes of Health (US), including through a MERIT award to professor Geary. One result has been the identification of the school-entry number knowledge that predicts economically-relevant mathematical competencies in adolescence. As a follow-up, professor Geary is directing a second longitudinal study, funded by the National Science Foundation (US), to identify the preschool quantitative competencies that predict this school-entry number knowledge. Professor Geary has also published conceptual and theoretical articles on individual differences in childrens mathematical learning, as well as a book published by the American Psychological Association, Childrens mathematical development (1994); recently translated into Korean. Professor Geary has also contributed to applied and policy related work on this topic, serving, for instance, on the Presidents National Mathematics Advisory Panel, and chairing its learning processes task group.

Professor Gearys interests in evolution are reflected in two of his other books published by the American Psychological Association, The origin of mind: Evolution of brain, cognition, and general intelligence (2005), and Male, female: The evolution of human sex differences (1998, 2010 second edition). The corresponding empirical work ranges from the study of changes in brain volume during hominid evolution to human mate choices to hormonal responses to simulated (video game) competition. Professor Gearys current interests in this area follow from several of his collaborative studies on the effects of prenatal toxin exposure on sex differences in cognition and behavior in mice. Specifically, traits related to Darwins sexual selection are often exaggerated relative to other traits. These would include, for example, the bright plumage of the males of many species of bird that in turn is a good indicator of their behavioral and genetic health. These traits are particularly sensitive to environmental disruption, even in healthy individuals. Professor Gearys in progress book, The evolution of vulnerability, is focused on these traits in humans and how they can be used to identify at-risk populations and individuals.

Daniel B. Berch is Professor of Educational Psychology and Applied Developmental Science at the University of Virginias Curry School of Education. Prior to this position, he was Associate Dean for Research and Faculty Development at the Curry School. Before coming to the University of Virginia, Professor Berch served as Associate Chief of the Child Development and Behavior Branch at the National Institute of Child Health and Human Development, NIH. His previous federal service included a year spent as a Senior Research Associate at the U. S. Department of Education, advising the Assistant Secretary for Educational Research and Improvement. Professor Berch is a cognitive developmental psychologist with interests ranging from the development of numerical cognition and mathematical learning disabilities (MLD) to evolutionary perspectives on education. He has published articles on childrens magnitude representations, the development of number sense, and the role of working memory in MLD. He is senior editor of the book, Why is math so hard for some children? The nature and origins of mathematical learning difficulties and disabilities (co-edited by Michele Mazzocco).

Among other honors, he received the NIH Award of Merit, was elected Fellow of the American Psychological Associations Division of Experimental Psychology, served as an ex officio member of the U.S. Department of Educations National Mathematics Advisory Panel commissioned by President George W. Bush, was elected to the Evolution Institutes Scientific Advisory Board (and chairs its Education Subcommittee), was appointed to the AIM Academys Research Advisory Board, and served as a member of the Professional Advisory Board of the National Center for Learning Disabilities for six years. During the past several years, Professor Berch has been working on the implications of evolutionary theory for educational research and practice, publishing a book chapter on instructing evolved minds, serving as one of the Evolution Institutes primary organizers of a 2013 conference on evolutionary perspectives in educational research funded by the American Educational Research Association, and is co-author (with David Geary) of an article entitled Evolutionary Approaches to Understanding Childrens Academic Achievement” to be published in Wileys forthcoming online reference work, Emerging Trends in the Social and Behavioral Sciences.

Kathleen Mann Koepke, Ph.D., is Director of the Math and Science Cognition and Learning, Development & Disorders Program in the Child Devlopment and Behavior Branch (CDBB) of the Eunice Kennedy Shriver National Institute of Child Health and Human Development (NICHD), National Institutes of Health (NIH), Rockville, MD (kmk@nih.gov).

Dr. Mann Koepke oversees a research program focused on developing and supporting research and training initiatives to increase knowledge relevant to the development of math and science cogniton, reasoning, knowledge, and abilities, both in animals and in humans from birth through all years of formal education in diverse learners with and without disabilities. This research focus recently lead to her serving as co-Guest Editor of a special journal issue regarding the co-occurrence of math and reading disabilities (Mann Koepke, K and Miller, B. (Eds.) At the Intersection of Math & Reading Disabilities. Journal of Learning Disabilities. 2013: 46(6)).

She is a lifecourse developmental cognitive neuroscientist/psychologist with a passion to serve the cognitively challenged and/or disabled via promoting new and innovative basic research and theoretically-grounded evidence-based intervention strategies to maximuze function. Dr. Mann Koepke has served in the Division of Extramural Programs across the National Institute on Aging (NIA), the National Institute of Nursing Research (NINR), and now NICHD, overseeing research on cognitive and neurological development and disorders, including neurobiological, behavioral and caregiving research foci. She has served on numerous federal and national committees aimed at advancing research and services for young learners and persons with cognitive and/or physical disability or differences, co-authoring numerous calls for new research to close significant gaps, as well as peer-reviewed publications. Prior to coming to NIH, she was faculty in Neurology (Psychology) at Washington University-St. Louis School of Medicine where she managed the universitys Alzheimers Disease Research Center (supported in part by grant P50AG05681), and served as Director of its Education Core and its Rural Outreach Satellite. While there, expanding on her enthusiasm for the use of newly developed technologies as tools for cognition, she developed and regularly contributed to the first-ever freely available web-based online educational support system for anyone interested in Alzheimers disease (AD) and for formal and informal dementia patient caregivers; the cite has garnered numerous national and international awards. This early online educational service has been used & replicated around the globe as a model for online disease/disabilty-focused educational support service.