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There Is Nothing So Practical as a Good Theory 183Individual differences in the quality of magnitude representations canbe detected from early ages. For example, kindergartners who are lessaccurate at number line estimation obtain lower achievement test scores atthe end of their kindergarten year (Siegler & Booth, 2004) and firstgraders who have been identified as having math difficulties are lessaccurate at number line estimation than classmates without math difficulties(Geary et al., 2007). These findings, together with the strong relationsbetween the quality of magnitude representations and many other aspectsof early mathematical understanding, suggested that targeting an interventionto improve magnitude representations might yield far-reachinggains in mathematical knowledge.5. FROM THEORY TO PRACTICE: DEVELOPING A BOARD GAMEINTERVENTIONWhat experiences might facilitate a child’s development of a linearrepresentation of numerical magnitude? Counting experience likely contributes,but such experience appears insufficient: Children can oftencount perfectly in a numerical range at least a year before they even knowthe ordering of numerical magnitudes in that range, much less possess anequal interval scale of the magnitudes (Le Corre et al., 2006; Ramani &Siegler, 2008; Schaeffer et al., 1974). Thus, young children may ‘‘lack anunderstanding of the concrete significance of the verbal [counting] string’’(Petitto, 1990, p. 70).If counting is insufficient, what other experiences might scaffold a linearrepresentation of numerical magnitudes? Experiences that provide redundantcues to numerical magnitudes can be especially effective in helpingchildren represent and discriminate the magnitudes (e.g., K. E. Jordan et al.,2008). One activity that provides such redundant cues is playing linear,numerical board games—that is, board games with linearly arranged, consecutivelynumbered, equal-sized spaces, such as the first row of Chutes andLadders. As noted by Siegler and Booth (2004), the greater the number inthe current square, the greater (a) the distance the child has moved thetoken, (b) the number of discrete moves the child has made, (c) the numberof number words the child has said and heard, and (d) the amount of timesince the game began. Consistent with the view that these cues to numericalmagnitude are useful, Ramani and Siegler (2008) found that preschoolerswho reported having played Chutes and Ladders produced number lineestimates that were more accurate and more closely followed a linearfunction than peers who reported never having played the game.Of course, correlation is not causation. Therefore, to experimentallytest the benefits of playing such games, Siegler and Ramani (2008) devised
184 Robert S. Siegler et al.a number board game similar to the first row of Chutes and Ladders (therow with the numbers 1–10.) As shown in Figure 3, this linear physicalrepresentation transparently reflects the ratio characteristics of the numbersystem. Just as 6 is twice as large as 3, the position of the ‘‘6’’square is twiceas far from the start position as the position of the ‘‘3’’square; it takes twiceas much time to reach the ‘‘6’’ square; and the child needs to say twice asmany number words and make twice as many hand movements with thetoken to reach ‘‘6.’’ In combination, the auditory, visuospatial, kinesthetic,and temporal cues to the magnitudes of the numbers 1–10 wereexpected to provide a robust, multimodal foundation for a linear representationof numerical magnitudes in that range.Children who played the linear board game were required to name thenumber in each square that they traversed during the game; thus, if theywere on the ‘‘3’’ square and spun ‘‘2,’’ they needed to count ‘‘4, 5.’’ Thismeant that the child needed to encode the number in the square, somethingthat preschoolers do not do automatically (Berch, Foley, Hill, &Ryan, 1999) but that seems crucial for correlating the number’s identitywith its magnitude.5.1. Assessing the benefits of the numerical board gameSiegler and Ramani (2008) provided the first test of the effects of playingthe linear numerical board game. They randomly assigned 36 preschoolersfrom low-income backgrounds to play one of the two games depictedin Figure 3—the number board game or the color board game. Each gameboard featured 10 horizontally arranged, equal-sized squares of variouscolors. The color version and the number version of the game werealmost identical, the only difference being that in the number game,the numbers 1–10 were listed consecutively from left to right, one numberper square. On each turn, the child spun a spinner and moved his orher token the indicated number of squares (one or two). For each squaretraversed, the child was required to state its color (in the color game) or its[(Figure_3)TD$FIG]Figure 3 Game boards for the linear number game (left) and the color game (right).Siegler, R. S. (2009). Improving the numerical understanding of children from lowincomefamilies. Child Development Perspectives, 3, 118–124.
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Series EditorBRIAN H. ROSSBeckman I
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Academic Press is an imprint of Els
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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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2 Henry L. Roediger et al.delayed t
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22 Henry L. Roediger et al.9. BENEF
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30 Henry L. Roediger et al.of some
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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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Page 148 and 149: Incorporating Motivation into a The
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Page 152 and 153: CHAPTERFIVEOn the Interplay of Emot
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Page 186 and 187: CHAPTERSIXThere Is Nothing So Pract
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Page 214 and 215: CHAPTERSEVENThe Power of Comparison
Page 216 and 217: The Power of Comparison in Learning
Page 242 and 243: CHAPTEREIGHTThe Ubiquitous Patterns
Page 244 and 245: The Ubiquitous Patterns of Incorrec
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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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