Shaping the river: Contexts of mathematical development and their implications for assessment and standard-setting. .

1. Shaping the river:Contexts of mathematical development and their implications for assessment and standard-setting.. Christopher CorreaKevin F. Miller University of Michigan

2. Overview • Quality control inspectors? • Getting from standards and assessment to learning • Historical perspectives • Poor mathematical performance in the U.S. is a comparatively recent phenomena • Sources of differences in mathematical performance between children in China and the U.S. • Symbol structure • Educational practices • The case of fractions • Implications for assessment & standards setting

3. The uses of assessment • “Quality control inspectors”? • How might assessment information lead to increased learning? • Interaction between • Kinds of information provided • Models of learning • Instructional practices

4. Overview Quality control inspectors? Getting from standards and assessment to learning Historical perspectives Poor mathematical performance in the U.S. is a comparatively recent phenomena Sources of differences in mathematical performance between children in China and the U.S. Symbol structure Educational practices The case of fractions Implications for assessment & standards setting

5. Historical perspectives: “A calculating people” • “When a country has a national sport as part of its culture—that is, contests that almost all members find interesting to participate in or watch—the team representing that country tends to be strong, even if its population is small…I think that school mathematics is a national (in the sense used above) intellectual pursuit in Asian countries, but not in the United States. Whereas students cannot escape from it in the former, in the latter they are told (at least implicitly) that, if they are poor at or dislike mathematics, they are free to seek achievement in some other area. • Hatano, G. (1990). Toward the cultural psychology of mathematical cognition. Comment on Stevenson, H. W., Lee, S-Y. (1990). Contexts of achievement. Monographs of the Society for Research in Child Development, 55 (1-2, Serial No. 221), 108-115.

6. These things can change… • America in the 1830s • “We are a traveling and a calculating people,” said Ohio booster James Hall in his book Statistics of the West, with evident satisfaction. “Arithmetic I presume comes by instinct among this guessing, reckoning, expecting and calculating people,” said English traveler Thomas Hamilton, with evident distaste. Each man attached a different value to the idea, but both agreed that Americans in the 1830s had some sort of innate reckoning skill that set them apart from Europeans. • Cohen, P. C. (1982). A calculating people: The spread of numeracy in early America. Chicago: U. of Chicago. (pp. 3-4)

7. Geary, D. C., Salthouse, T. A., Chen, G-P, & Fan, L. (1996). Are East Asian Versus American Differences in Arithmetical Ability a Recent Phenomenon? Developmental Psychology, 32, 254-262 • Younger (20 year old) and Older (late 65-75 year old) adults • Matched for health, education • Given Arithmetic and basic cognitive tasks

8. Addition performance • Addition problems like 19 + 8 + 27 = ? • # correctly solved in 2 minutes • Interaction between age and culture

9. Subtraction performance • Subtraction problems like 35-9 = ? • # correctly solved in 2 minutes • Interaction between age and culture

10. Perceptual speed – No culture effect • Perceptual speed task • Identifying whether number strings are the same or not • 179359821 • 178539821 • # correctly solved in 3 minutes • ETS kit of factor-referenced tests (Ekstrom et al., 1976)

11. Mental rotation – no culture effect • Rotation of figures in 3-dimensional space • # solved in 3 minutes • From ETS kit of factor-referenced tests (Ekstrom et al., 1976)

12. Conclusions • Differences are • Large • Specific • Probably recent in origin • Where might they come from? • Differences in what children learn • Differences in educational environment • Differences in teaching processes

13. Overview Quality control inspectors? Getting from standards and assessment to learning Historical perspectives Poor mathematical performance in the U.S. is a comparatively recent phenomena Sources of differences in mathematical performance between children in China and the U.S. Symbol structure Educational practices The case of fractions Implications for assessment & standards setting

14. Language and Learning to Count • Children need to learn a system of number names as they learn to count • Not a trivial task

15. Number names in Chinese & English - Part ICounting to Ten • Both languages share an unpredictable list • No way to induce “five” from “one, two, three, four” • Linguistically, learning to count to ten should be of equal difficulty in both languages

16. Number names in Chinese & English - Part IIFrom Ten to Twenty • Chinese has a clear base-ten structure • similar to Arabic numerals: 11 = “10…1” • English lacks clear evidence of base-ten structure • Names for 11 and 12 not marked as compounds with 10. • Larger teens names follow German system of unit+digits name, unlike larger two-digit number names • compare “fourteen” and “twenty-four”

17. Number names in Chinese & English - Part IIIAbove Twenty • Both languages share a similar structure • similar to Arabic numerals: 37 = “3x10 + 7” • For Chinese, this extends previous system • For English, it represents a new way of naming numbers

18. A longitudinal view

19. Learning difficulties reflect language structure ..and they don’t stop here!

20. The Panda’s snack • Language affects only some aspects of early number knowledge • No language difference for counting-principle errors such as double-counting • Mastering number list and understanding numerosity not the same • Producing sets of n items • No language difference

21. Continuing effects • Learning Arabic numerals involves a mapping from verbal number names • Teens continue to cause problems *

22. Summary • Early mathematical development is a mix of language-dependent and universal factors • Sensitivity to symbol structure begins very early • Base-ten concepts and “teens” are problematic for speakers of English • Foundation for later mathematics • Other sources

23. Overview Quality control inspectors? Getting from standards and assessment to learning Historical perspectives Poor mathematical performance in the U.S. is a comparatively recent phenomena Sources of differences in mathematical performance between children in China and the U.S. Symbol structure The case of fractions Educational practices Implications for assessment & standards setting

24. Interaction between symbol structure and classroom processes – Rational numbers • English terms somewhat opaque • Numerator • Denominator • Chinese terms more transparent • 分子 (“fraction child”) • 分母 (“fraction mother”) *****

25. Transparent fraction terms – does it make a difference? • Not entirely clear • Miura et al. (1999) vs. • Paik & Mix (2003) • Children in both countries likely to pick “numerator + denominator” foil • Language support can only take you so far… 1/3 = ? return

26. Student thinking – components of knowledge • In addition to content domains, the most recent TIMSS student assessment also considers different types of thinking: • Knowing facts, procedures, and concepts • What students need to know • Applying knowledge and conceptual understanding • Applying what the student knows to solve routine problems or answer questions • Reasoning • Solving for unfamiliar situations and multi-step problems

27. Thinking about Fractions • Knowing • Applying • Reasoning

28. Thinking about Fractions • Knowing • Applying • Reasoning

29. Thinking about Fractions • Knowing • Applying • Reasoning

30. Fourth-grade students in many high-achieving countries are proficient in problem solving. Fourth-grade student achievement in Japan, TIMSS 2003

31. Fourth-grade student achievement in Netherlands, TIMSS 2003

32. U.S. fourth-grade students are more proficient at knowing procedures and knowledge than applying knowledge to solve problems. Fourth-grade student achievement, TIMSS 2003

33. …though this changes over time. Eighth-grade student achievement, TIMSS 2003

34. Teacher Knowledge • Teachers may be more comfortable facilitating complex problem solving if they have a strong understanding of mathematical concepts

35. Teacher Knowledge in the U.S. • Zhou et al. (2006) compare pedagogical content knowledge of U.S. and Chinese teachers

36. Why? • K-12 experiences • Teacher Training • Few content specialists • Changes in workforce • Few opportunities for learning

37. Cultural Beliefs and Scripts • Beliefs about learning • Can you get smarter? • Americans more likely to believe that intelligence is fixed and that success is due to ability rather than effort • Can be maladaptive • Teaching & Assessment practices can reinforce this idea

38. Desire similar outcome • achieving good scores, doing “well” Different motivation for pursuing this outcome Self-Theories of IntelligenceDweck, 1999 Incremental Theorists Entity Theorists • Intelligence is fixed • Trait largely determined by nature • Intelligence is malleable • Quality that can be increased through nurture • Learning goals • seeking to develop ability • Performance goals • seeking to validate ability as good relative to others “When I take a course in school, it is very important for me to validate that I am smarter than other students.” “In school I am always seeking opportunities to develop new skills and acquire new knowledge.”

39. Beliefs about Teaching We used multiple correspondence analysis to represent individual teachers and their responses to the interview question, “How do students best learn mathematics?”

40. Beliefs about Teaching …and find within-group similarities

41. Practice • “I think that drill is very important… I think that there is just no substitute for going over it and over it and over it until it’s very firmly entrenched in their minds.” (Teacher U4-2) • With some students, there can be something said for a lot of rote practice at the very beginning of a concept… there are some students who need 40 problems of it just to get the pattern of what they're doing. (Teacher U4-3)

42. Remembering and Understanding • Attributions for student errors by Chinese & American teachers and college students • Remembering vs.Understanding

43. Scripts for teaching • Customary teaching practices that are informed by a culture’s underlying values and assumptions of what teaching should be like.

44. National Scripts

45. National Scripts

46. National Scripts

47. Identifying a Script for Chinese Lesson Introductions Data Participants include 19 fourth- and fifth- grade teachers from six schools in Beijing, China Videotaped Classroom Observations Lesson introductions are defined as “what happens between the beginning of class and the final solution of the first novel problem.” Interviews

48. A Proposed Script for Lesson Introductions in China Teacher Reviews Old Knowledge (whole-class) Students Develop Methods and Solutions (small groups) Students Share Results (whole-class)

49. Teacher Reviews Old Knowledge (whole-class) • Students Develop Methods and Solutions (small groups) • Students Share Results (whole-class)

50. I. Teacher reviews old knowledge • Video example