Modern Learning Theories and Mathematics Education - Robert Siegler
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.
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
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
“ 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:
<ul><li>A basic issue in many modern learning theories involves how knowledge is represented </li></ul><ul><li>In mathematical cognition, this issue involves the underlying representation of numerical magnitudes (Dehaene, 1997; Gelman & Gallistel, 2001; Case & Okamoto, 1996) </li></ul><ul><li>Empirical research indicates that linear representations linking number symbols with their magnitudes are crucial for a variety of important mathematics learning outcomes </li></ul>Theoretical Background: The Centrality of Numerical Magnitude Representations
Number Presented Number Presented Number Presented Progression from Log to Linear Representation — 0-100 Range (Siegler & Booth, 2004)
Progression from Log to Linear Representation — 0-1,000 Range (Siegler & Opfer, 2003) Sixth Graders Number Presented R 2 lin = .97 Number Presented Median Estimate Second Graders R 2 log = .95
<ul><li>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). </li></ul>The Centrality of Numerical Magnitude Representations
Correlations Among Linearity of Magnitude Representations on Three Estimation Tasks (Booth & Siegler, 2006) ** p < .01; * p < .05 Grade Task Measurement Numerosity 2 nd Number line .65** .55* Measurement .54** 4 th Number line .84** .70** Measurement .60**
Correlations Between Linearity of Estimation and Math Achievement (Booth & Siegler, 2006) Estimation Task Grade ** p < .01; * p < .05 Number Line Measurement Numerosity 2 nd .53** .62** .48** 4 th .47* .54** .35
Issue in Mathematics Education: Low-Income Children Lag Behind in Mathematical Proficiency Even Before They Enter School <ul><li>1. Children vary greatly in mathematical knowledge when they enter school </li></ul><ul><li>2. Numerical knowledge of kindergartners from low-income families trails far behind that of peers from higher-income families (ECLS, 2001) </li></ul>
<ul><li>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) </li></ul><ul><li>4. Large, early, SES related differences become even more pronounced as children progress through school </li></ul>
Applying Theory to Educational Problem Might inadequate representations of numerical magnitudes underlie low-income children’s poor numerical performance?
<ul><li>Counting experience is likely helpful and necessary, but insufficient </li></ul><ul><li>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) </li></ul>Applied Goal Raised New Theoretical Question: What Leads Anyone to Form Initial Linear Representation?
<ul><li>Board games might play a crucial role in forming linear representations of numerical magnitudes </li></ul><ul><li>Designed to promote interactions between parents and peers </li></ul><ul><li>Also provides rich experiences with numbers </li></ul>Playing Board Games
<ul><li>The greater the number a token reaches, the greater the </li></ul><ul><ul><li>Distance that the child has moved the token </li></ul></ul><ul><ul><li>Number of discrete hand movements the child has made </li></ul></ul><ul><ul><li>Number of number names the child has spoken </li></ul></ul><ul><ul><li>Time spent by the child playing the game </li></ul></ul><ul><li>Thus, playing number board games provides visuo-spatial, kinesthetic, auditory, and temporal cues to links between number symbols and their magnitudes </li></ul>Key Properties of Board Games Like Chutes and Ladders
<ul><li>Goal was to investigate whether playing the number board game: </li></ul><ul><ul><li>Improves a broad range of numerical skills and concepts </li></ul></ul><ul><ul><li>Produces gains that remain stable over time </li></ul></ul>Effects of Playing the Number Board Game (Ramani & Siegler, 2008)
<ul><li>Participants : 129 4- and 5-year-olds from Head Start classrooms (mean age = 4.8), 52% African-American </li></ul><ul><li>Experimental Conditions : </li></ul><ul><ul><li>Number Board Game (N = 69) </li></ul></ul><ul><ul><li>Color Board Game (N = 60) </li></ul></ul><ul><li>Design : Pretest, 4 training sessions, posttest, 9 week follow-up. </li></ul>Methods
<ul><li>Training Procedure : </li></ul><ul><ul><li>Children play a total of 20 games over 4 sessions in a 2 week period, 15-20 minutes/session </li></ul></ul><ul><ul><li>Child spins spinner, gets 1 or 2, says while moving token (e.g.) “5, 6” or “blue, red” </li></ul></ul><ul><ul><li>Feedback and help if needed </li></ul></ul><ul><li>Measures : </li></ul><ul><ul><li>0-10 Number Line Estimation </li></ul></ul><ul><ul><li>1-9 Numerical Magnitude Comparison </li></ul></ul><ul><ul><li>1-10 Counting </li></ul></ul><ul><ul><li>1-10 Numeral Identification </li></ul></ul>
Numerical Magnitude Comparison M % Correct * * * p < .001
Counting M Mean Counts Without Error * * * p < .001
Number Line Estimation: Linearity of Individual Children’s Estimates M Mean R 2 lin * * * p < .001
Numeral Identification M * * % Correct * p < .001
Percent Correct Addition Answers (Siegler & Ramani, 2009) M % Correct * p < .05 *
Experiment 2: Game Playing Outside the Lab <ul><li>Participants : 145 4- and 5-year-olds </li></ul><ul><ul><li>115 children from prior Head Start sample </li></ul></ul><ul><ul><li>30 children recruited from middle- to upper- middle income families </li></ul></ul><ul><li>Procedure : Informal activities questionnaire </li></ul><ul><ul><li>Board games, card games, & video games played outside of school </li></ul></ul><ul><ul><li>Name the different games </li></ul></ul>
Percent of Children Who Had Played Each Type of Game M * * % of Children * * p < .01
Correlations Between Game Playing Experience and Numerical Knowledge Among Head Start Children M ** p < .01; * p < .05 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
Correlations Between Playing Chutes and Ladders and Numerical Knowledge M * p < .05 Number Line Linearity Magnitude Comparison Counting Numeral Identification Played Chutes & Ladders .20* .18 .19* .24*
<ul><li>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. </li></ul>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.”
<ul><li>Conclusion 9 from NMAP: </li></ul><ul><li>“ 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.” </li></ul>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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