1. Modern Learning Theories and Mathematics Education: Bidirectional Contributions, Bidirectional Challenges 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 investigators funded by IES, most of my pre-IES research was theoretical (Definition: “Without any likely application”) IES 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. I. Contributions to Mathematics Education from Applying Modern Learning Theories II. Contributions to Modern Learning Theories from Mathematics Education Applications III. Challenges to Modern Learning Theories from Mathematics Education Applications IV. Challenges to Mathematics Education from Modern Learning Theories Bidirectional Contributions, Bidirectional Challenges

5. I. Contributions to Mathematics Education from Applying Modern Learning Theories

6. “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:

7. 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

8. The Number Line Task 0 100 71

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

10. 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

11. 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

12. Correlations Among Linearity of Magnitude Representations on Three Estimation Tasks (Booth & Siegler, 2006) GradeTask Measuremen t Numerosit y 2 nd Number line.65**.55* Measureme nt.54** 4 th Number line.84**.70** Measureme nt.60** **p <.01; *p <.05

13. Correlations Between Linearity of Estimation and Math Achievement (Booth & Siegler, 2006) Number LineMeasurementNumerosity 2 nd.53**.62**.48** 4 th.47*.54**.35 Estimation TaskGrade **p <.01; *p <.05

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

15. 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)

16. 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

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

18. 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?

19. 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

20. Chutes and Ladders

21. 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

22. Number Board Game

23. Color Board Game

24. 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)

25. 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

26. 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

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

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

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

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

31. Percent Correct Addition Answers (Siegler & Ramani, in press) M % Correct *p <.05 *

32. II.Contributions to Modern Learning Theories from Mathematics Education Applications

33. Point to need for single theory to integrate learning of concepts, procedures, facts, and problem solving Demonstrate need to identify everyday experiences that build conceptual understanding Theoretical Contributions of Number Game Application NMAP Conclusion: “10. The curriculum must simultaneously develop conceptual understanding, computational fluency, and problem solving skills... These capabilities are mutually supportive, each facilitating learning of the others.”

34. Illustrate need to identify central conceptual structures (Case & Okamoto, 1996) Raise question of what other specific activities contribute to numerical magnitude representations: Counting objects in row Addition via counting fingers Conversation about numerical properties Other games (e.g., war) Suggest that inadequate fraction magnitude representations partially due to lack of experiences that indicate correlational structure (1/3 + 1/3 = 2/6)

35. III.Challenges to Modern Learning Theories from Mathematics Education Applications

36. NMAP Executive Summary, p. 32: “There are many gaps in current understanding of how children learn algebra and the preparation that is needed before they enter algebra.” Considerable high quality research is available regarding math learning in preschool and first few grades, but far less on later math learning. Theories and empirical studies need to address learning of pre- algebra, algebra, and geometry. Virtue of theory-based applications: Open up theories; help avoid trap of “more and more about less and less.”

37. Conclusion 12 from NMAP: “Difficulty with fractions (including decimals and percentages) is pervasive and is a major obstacle to further progress in mathematics, including algebra.” Remarkable agreement among NMAP members and algebra teachers on importance of fractions for learning algebra. But no evidence. Need for robust measures of moderately general knowledge structures, such as understanding of fractions, so can investigate these relations. Such robust measures require better theory of what’s central to (e.g.) understanding fractions.

38. IV.Challenges to Mathematics Education from Modern Learning Theories

39. 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.”

40. 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.” IES has generously supported research on learning principles and on 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.

42. Score Trend in NAEP Mathematics Average Scores, 1973 - 2008