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CHAPTERSIXThere Is Nothing So Practical asa Good TheoryRobert S. Siegler, Lisa K. Fazio and Aryn PykeContents1. Introduction 1722. Development of Mathematical Knowledge 1732.1 Early mathematical abilities 1732.2 Development of numerical knowledge in low- and174middle-income populations2.3 Differences in exposure to math of preschoolers from different 175economic backgrounds3. Effects of Large-Scale Math Curricula 1754. Theoretical Background of a Targeted Intervention 1785. From Theory to Practice: Developing a Board Game Intervention 1835.1 Assessing the benefits of the numerical board game 1846. Conclusions 189Acknowledgments 192References 192AbstractChildren from low-income families begin school with less mathematicalknowledge than peers from middle-income backgrounds. This discrepancyhas long-term consequences: Children who start behind usually staybehind. Effective interventions have therefore been sought to improve themathematical knowledge of preschoolers from impoverished backgrounds.Some curriculum-based interventions have met with impressive success;two disadvantages of such interventions, however, are that they are quitecostly in terms of the time and resources they require and their multifacetedlessons make it impossible to determine why they work. In this chapter, wedescribe a theoretical analysis that motivated the development of a simple,brief, and inexpensive intervention that involved playing a linear numberboard game. Roughly an hour of playing this game produced improvementsin numerical magnitude comparison, number line estimation, counting,numeral identification, and ability to learn novel arithmetic problems bypreschoolers from low-income backgrounds. The gains were greater thanthose produced by playing a parallel nonnumerical game or engaging inPsychologyof Learningand Motivation, Volume 55ISSN 0079-7421, DOI 10.1016/B978-0-12-387691-1.X0001-4# 2011 Elsevier Inc.All rights reserved.171
172 Robert S. Siegler et al.other numerical activities. Detailed analyses of learning patterns indicatedthat this theoretically motivated, game-based intervention exercises mostof its effects by enhancing and refining children’s representations of numericalmagnitudes.1. INTRODUCTIONWhen children enter kindergarten, they already differ greatly inmathematical knowledge. These early differences have large and lastingconsequences. Initial knowledge predicts subsequent success in manydomains (Bransford, Brown, & Cocking, 1999), but the relation betweenearly math knowledge and subsequent math achievement is particularlystrong and persistent (e.g., roughly twice as strong as the relation betweenearly and later reading achievement; Duncan et. al, 2007). Limited mathematicalknowledge has large, deleterious economic and occupationalconsequences in adulthood (McCloskey, 2007; Rivera-Batiz, 1992).Children from low-income families are particularly at risk for difficultieslearning mathematics. They tend to begin school with less numericalknowledge than their classmates from middle- or high-income households,and over the course of schooling, they fall steadily further behind(e.g., N. C. Jordan, Kaplan, Olah, & Locuniak, 2006). Again, similarpatterns are present in other areas such as reading, but the differences inmathematical knowledge tend to be greater (Case, Griffin, & Kelly, 1999).Recognition of these problems has led a number of researchers todevise programs intended to improve the mathematical knowledge ofpreschoolers from low-income families and thus to reduce the gapbetween them and children from more affluent backgrounds. However,these large-scale interventions are costly, both financially and in the timeneeded to train teachers to implement them. Because these interventionsare so multifaceted, they also yield limited information regarding how toimprove future interventions and future theories of numerical cognition.In this chapter, we briefly describe the early development of numericalknowledge, differences between the numerical knowledge of preschoolersfrom low-income and middle-income backgrounds, and three broad-basedcurricular interventions that have been found to improve that knowledge.We then present a theoretical analysis that suggests that underdevelopedknowledge of numerical magnitudes is a key source of the mathematicaldifficulties of children from low-income backgrounds. Next, we describean intervention, based on the theoretical analysis, designed to improveyoung children’s knowledge of numerical magnitudes. Finally, we presentevidence regarding the effectiveness of the intervention and discuss generalimplications of the findings.
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Series EditorBRIAN H. ROSSBeckman I
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viContents3. Science of Multimedia
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xContributorsHenry L. Roediger, III
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xiiPrefaceand there has been much e
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xivPrefaceInterventions for improvi
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32 Henry L. Roediger et al.Benefit
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34 Henry L. Roediger et al.Hasher,
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36 Henry L. Roediger et al.Son, L.
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38 John Sweller1. INTRODUCTIONCogni
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40 John Sweller‘‘baby-talk.’
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42 John Swellerof cognitive or meta
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44 John SwellerTable 1Natural Infor
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46 John Swellerconfigurations. A ch
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48 John Swellerreproduction results
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50 John Swellerto mutation, without
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52 John Swellerusing a complex or,
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54 John SwellerThe human cognitive
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56 John Swellerepigenetic system ha
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58 John SwellerWe can determine lev
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60 John Swellerthan memorizing the
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62 John SwellerTable 2EffectVariabi
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64 John Swelleroccurs when students
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66 John Sweller3.3.3. The Split-Att
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68 John Swellermerely restates the
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70 John SwellerThe expertise revers
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72 John Swellerobtained a reverse m
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74 John Swelleractivities that othe
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76 John SwellerRenkl, A. (2005). Th
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78 Richard E. Mayercognitive theory
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80 Richard E. Mayerpictures beginni
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82 Richard E. MayerThe dualchannelp
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84 Richard E. Mayerlearner selects
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86 Richard E. MayerRosenthal, Rosno
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88 Richard E. MayerGenerative proce
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90 Richard E. Mayer4.2. Evidence-ba
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92 Richard E. Mayersituation could
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94 Richard E. Mayer(Moreno & Mayer,
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96 Richard E. Mayer4.3.1. Segmentin
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98 Richard E. Mayerpresenting the w
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100 Richard E. MayerThe multimedia
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102 Richard E. Mayergame in industr
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104 Richard E. MayerClark, R. C., &
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106 Richard E. MayerMayer, R. E., &
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108 Richard E. MayerSweller, J. (19
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110 Timothy J. Nokes and Daniel M.
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Incorporating Motivation into a The
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The Power of Comparison in Learning
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CHAPTEREIGHTThe Ubiquitous Patterns
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The Ubiquitous Patterns of Incorrec
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CHAPTERNINEConceptual Problem Solvi
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Conceptual Problem Solving in Physi
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IndexAAcademic performancedrive the
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Index 301Correct method comparison,
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Index 303LLabor in vain, 28Language
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Index 305OOntological categories of
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Index 307WWeb-delivered multimedia
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CONTENTS OF RECENT VOLUMESVolume 40
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Contents of Recent Volumes 311Under
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Contents of Recent Volumes 313Volum
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