1. 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.

2. 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

3. 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

4. “ 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:

6. The Number Line Task 0 100 71

7. Number Presented Number Presented Number Presented Progression from Log to Linear Representation — 0-100 Range (Siegler & Booth, 2004)

8. 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

10. 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**

11. 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

12. Causal Evidence: External Magnitude Representations and Arithmetic Learning (Booth & Siegler, 2008) ** p < .01 ** % Correct Sums

15. Applying Theory to Educational Problem Might inadequate representations of numerical magnitudes underlie low-income children’s poor numerical performance?

18. Chutes and Ladders

20. Number Board Game

21. Color Board Game

25. Numerical Magnitude Comparison M % Correct * * * p < .001

26. Counting M Mean Counts Without Error * * * p < .001

27. Number Line Estimation: Linearity of Individual Children’s Estimates M Mean R 2 lin * * * p < .001

28. Numeral Identification M * * % Correct * p < .001

29. Percent Correct Addition Answers (Siegler & Ramani, 2009) M % Correct * p < .05 *

31. Percent of Children Who Had Played Each Type of Game M * * % of Children * * p < .01

32. 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

33. 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*