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Modern Learning Theories and Mathematics
Education
The research reported here was supported by the Institute of Education Sciences, U.S. Department of Education, through Grant R305H050035 to Carnegie Mellon University. The opinions expressed are those of the author and do not represent views of the Institute or the U.S. Department of Education.
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Like many developmental psychologists, most of my early research was theoretical (Definition: “Without any likely application”)
Over time, my kids (and granting agencies) motivated me to think harder about ways in which the research could be applied to important educational problems without sacrificing rigor
One outcome has been my current research applying theories of numerical cognition to improving low-income preschoolers’ mathematical understanding
A Little Personal Background
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Another outcome has been to increase my interest in broader issues of application, i.e., educational policy issues
This growing interest in applications led me to abandon my traditional “just say no” policy regarding commissions and panels and accept appointment to the National Mathematics Advisory Panel (NMAP). Main role was in learning processes group
The present talk combines perspectives gained from doing the applied research and from participating in the learning processes group of NMAP
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“9. Encouraging results have been obtained for a variety of instructional programs developed to improve the mathematical knowledge of preschoolers and kindergartners, especially those from low-income backgrounds. There are effective techniques – derived from scientific research on learning – that could be put to work in the classroom today to improve children’s mathematical knowledge.”
“14. Children’s goals and beliefs about learning are related to their mathematics performance. . . When children believe that their efforts to learn make them ‘smarter,’ they show greater persistence in mathematics learning.”
Conclusions of NMAP:
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A basic issue in many modern learning theories involves how knowledge is represented
In mathematical cognition, this issue involves the underlying representation of numerical magnitudes (Dehaene, 1997; Gelman & Gallistel, 2001; Case & Okamoto, 1996)
Empirical research indicates that linear representations linking number symbols with their magnitudes are crucial for a variety of important mathematics learning outcomes
Theoretical Background: The Centrality of Numerical Magnitude Representations
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The Number Line Task
0 100
71
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NumberPresented
NumberPresented
NumberPresented
Progression from Log to Linear Representation — 0-100 Range
(Siegler & Booth, 2004)
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Progression from Log to Linear Representation — 0-1,000 Range
(Siegler & Opfer, 2003)
Sixth Graders
Number Presented
R2lin = .97
Number Presented
Med
ian
Est
imat
e
Second Graders
R2log = .95
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Linearity of magnitude representations correlates positively and quite strongly across varied estimation tasks, numerical magnitude comparison, arithmetic, and math achievement tests (Booth & Siegler, 2006; 2008; Geary, et al., 2007; Ramani & Siegler, 2008; Whyte & Bull, 2008).
The Centrality of Numerical Magnitude Representations
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Correlations Among Linearity of Magnitude Representations on Three Estimation Tasks
(Booth & Siegler, 2006)
Grade Task Measurement Numerosity
2nd Number line .65** .55*
Measurement .54**
4th Number line .84** .70**
Measurement .60**
**p < .01; *p < .05
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Correlations Between Linearity of Estimation and Math Achievement
(Booth & Siegler, 2006)
Number Line Measurement Numerosity
2nd .53** .62** .48**
4th .47* .54** .35
Estimation TaskGrade
**p < .01; *p < .05
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Causal Evidence: External Magnitude Representations and Arithmetic Learning
(Booth & Siegler, 2008)
0
25
50
Pretest Posttest% C
orre
ct A
rith
met
ic A
nsw
ers
Feedback + Magnitude RepresentationFeedback **p < .01
**
% C
orre
ct S
ums
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Issue in Mathematics Education: Low-Income Children Lag Behind in
Mathematical Proficiency Even Before They Enter School
1. Children vary greatly in mathematical knowledge when they enter school
2. Numerical knowledge of kindergartners from low-income families trails far behind that of peers from higher-income families (ECLS, 2001)
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3. Kindergartners’ numerical knowledge strongly predicts later mathematical achievement — through elementary, middle, and high school (Duncan, et al., 2007; Jordan et al., 2009; Stevenson & Newman, 1986)
4. Large, early, SES related differences become even more pronounced as children progress through school
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Applying Theory to Educational Problem
Might inadequate representations of numerical magnitudes underlie low-income children’s poor numerical
performance?
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• Counting experience is likely helpful and necessary, but insufficient
• Children can count in a numerical range more than a year before they can generate a linear representation of numerical magnitudes in that range (Condry & Spelke, 2008; LeCorre & Carey, 2007; Schaffer et al., 1974)
Applied Goal Raised New Theoretical Question: What Leads Anyone to Form
Initial Linear Representation?
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• Board games might play a crucial role in forming linear representations of numerical magnitudes
• Designed to promote interactions between parents and peers
• Also provides rich experiences with numbers
Playing Board Games
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Chutes and Ladders
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The greater the number a token reaches, the greater the• Distance that the child has moved the token • Number of discrete hand movements the child
has made• Number of number names the child has spoken• Time spent by the child playing the game
Thus, playing number board games provides visuo-spatial, kinesthetic, auditory, and temporal cues to links between number symbols and their magnitudes
Key Properties of Board Games Like Chutes and Ladders
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Number Board Game
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Color Board Game
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Goal was to investigate whether playing the number board game:
• Improves a broad range of numerical skills and concepts
• Produces gains that remain stable over time
Effects of Playing the Number Board Game
(Ramani & Siegler, 2008)
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Participants: 129 4- and 5-year-olds from Head Start classrooms (mean age = 4.8), 52% African-American
Experimental Conditions:• Number Board Game (N = 69)• Color Board Game (N = 60)
Design: Pretest, 4 training sessions, posttest, 9 week follow-up.
Methods
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Training Procedure:• Children play a total of 20 games over 4 sessions
in a 2 week period, 15-20 minutes/session• Child spins spinner, gets 1 or 2, says while
moving token (e.g.) “5, 6” or “blue, red”• Feedback and help if needed
Measures:
• 0-10 Number Line Estimation
• 1-9 Numerical Magnitude Comparison
• 1-10 Counting
• 1-10 Numeral Identification
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50
60
70
80
90
100
Pretest Posttest Follow-up
Number Board Game Color Board Game
Numerical Magnitude ComparisonM
% C
orr e
ct **
*p < .001
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6
7
8
9
10
Pretest Posttest Follow-up
Number Board Game Color Board Game
CountingM
Mea
n C
ount
s W
itho
ut E
rror **
*p < .001
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0
10
20
30
40
50
Pretest Posttest Follow-up
Number Board Game Color Board Game
Number Line Estimation:Linearity of Individual Children’s
EstimatesM
Mea
n R
2 lin *
*
*p < .001
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50
60
70
80
90
100
Pretest Posttest Follow-up
Number Board Game Color Board Game
Numeral IdentificationM
**
% C
orr e
ct
*p < .001
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0
10
20
30
40
50
Pretest PosttestLinear Board GameCircular Board GameControl Numerical Activities
Percent Correct Addition Answers(Siegler & Ramani, 2009)M
% C
orre
ct
*p < .05
*
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Experiment 2: Game Playing Outside the Lab
Participants: 145 4- and 5-year-olds• 115 children from prior Head Start sample• 30 children recruited from middle- to upper-
middle income familiesProcedure: Informal activities questionnaire
• Board games, card games, & video games played outside of school
• Name the different games
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10
30
50
70
90
Board Games Card Games Video Games
Head Start Children Middle-Income Children
Percent of Children Who Had Played Each Type of Game
M
*
*%
of
Chi
ldr e
n*
*p < .01
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Correlations Between Game Playing Experience and Numerical Knowledge Among Head Start Children
M
Number Line Linearity
Magnitude Comparison
Counting
Numeral Identification
No. of Board Games .38** .26** .20* .25**
No. of Card Games .18 .28** .11 .13
No. of Video Games .21* .02 .00 .07
**p < .01; *p < .05
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Correlations Between Playing Chutes and Ladders and Numerical Knowledge
M
Number Line Linearity
Magnitude Comparison
Counting
Numeral Identification
Played Chutes & Ladders .20* .18 .19* .24*
*p < .05
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Young students in East Asia and some European countries spend more time on math, encounter more challenging and conceptually richer curricula, and learn more. No reason why we can’t do the same. Belief that young children aren’t ready to learn relatively advanced concepts contradicts both national and international data.
Conclusion 15 from NMAP:“Teachers and developers of instructional materials sometimes assume that children need to be a certain age to learn certain mathematical ideas. However, a major research finding is that what is developmentally appropriate is largely contingent on prior opportunities to learn. Claims that children of particular ages cannot learn certain content because they are too young have consistently been shown to be wrong.”
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Conclusion 9 from NMAP:“There are effective techniques — derived from scientific research on learning — that could be put to work in the classroom today to improve children’s mathematical knowledge.”
Funding agencies have generously supported research on learning principles and on small scale programs that implement these principles. As always, we need more research, but some of the research is now sufficiently advanced for broad implementation, at least on an experimental basis. The challenge for the field of mathematics education is how to use the programs and principles to improve educational practice.
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