Professor Sarah Lubienski Fulbright Scholar at the

1. Promoting Equity through Problem Solving: Results from Two Decades of Mathematics Instructional Reform in the United States Professor Sarah Lubienski Fulbright Scholar at the Centre for the Advancement of Science Teaching and Learning (CASTeL) Dublin City University On sabbatical leave from the Department of Curriculum and Instruction, University of Illinois

2. Introduction My Background • Past work • Studies of math teaching, learning and equity • Large-scale analyses of national data (only basic analyses discussed here)

4. Introduction My Background • Past work • Studies of math teaching, learning and equity • Large-scale analyses of national data (only basic analyses discussed here) Current situation - Fulbright DCU Sept-Dec, 2010 Current research • U.S. reform led by the National Council of Teachers of Mathematics (NCTM) • Project Maths in Ireland • Equity

5. Anyon (1980) found social class differences in how U.S. students were being educated: Upper-class schools: • Education for executives • Emphasis on creativity, problem-solving, critical thinking Working-class schools: • Training for obedient workers • Emphasis on basic skills, following instructions, obeying authority

6. NCTM Reforms In response to concerns about equity and the changing needs of society, the U.S. National Council of Teachers of Mathematics released: 1) Curriculum and Evaluation Standards (1989). 2) Teaching Standards (1991) 3) Assessment Standards (1995) 4) Principles and Standards for School Mathematics (2000)

7. NCTM Reforms Less: Teacher lecture Memorization of rules with little understanding Repetitive drill More: Student understanding of concepts Mathematical reasoning, communication Learning through problem solving Balance among 5 curricular strands (Geometry, Msmt., Algebra/Functions, Number, Probability/Stats), “Mathematical power for all”

8. Project Maths • Emphasis on: • Student understanding of concepts • Mathematical reasoning, communication • Real world examples & applications • Problem solving skills • Statistics/Probability, Geometry/Trigonometry, Number, Algebra, Functions.

9. Project Maths versus NCTM Standards • Similar goals • Similar rhetoric/rationale--competing internationally, changing workforce needs • Some differences: • Secondary school focus of Project Maths • Syllabus “levels” (foundation, ordinary, higher) • Implementation plan • NCTM as teacher organization has no power, no money, no role in the government • Lack of coherence in U.S. reforms – professional development, assessments, curriculum frameworks

10. NCTM Implementation • Despite NCTM’s lack of authority & resources, the Standards have had a remarkably large impact on: • Textbooks – directly and indirectly • State standards • Assessments • Teacher education and professional development • Still -- changing teacher practice and consistently detecting effects with students has been difficult. “Surface changes” easier to obtain than substantive change.

11. What has happened to U.S. student achievement in mathematics since the first NCTM Standards? Studies of problem-centered textbooks—when used as intended—have been promising: Increase in conceptual understanding Similar levels of procedural fluency Advantages compounded over time (less summer loss) (Riordan & Noyce, 2001; Schoenfeld, 2002; Senk & Thompson, 2003, Tarr, et al, 2008)

12. What has happened to U.S. student achievement in mathematics since the first NCTM Standards? Our “National Test”: National Assessment of Educational Progress (NAEP) Administered sporadically at 4th, 8th, and 12th grades Main NAEP (vs. Long-Term-Trend NAEP) Main NAEP Results 1990-2009 Achievement significantly increased at 4th & 8th grades. Because of recent framework change, 12th grade trends are less clear.

13. NAEP Mathematics Achievement by Grade, 1990-2009 s.d.≈ 33 pts

14. Backlash Despite Successes • Critics rightly charge that most research has been conducted by pro-reformers

15. Backlash Despite Successes • Critics rightly charge that most research has been conducted by pro-reformers • NCTM’s initial “Decreased Emphasis” list – PR problem

16. NCTM’s Increased & Decreased Emphases (A few examples from list for Grades 9-12)

17. Backlash Despite Successes • Critics rightly charge that most research has been conducted by pro-reformers • NCTM’s initial “Decreased Emphasis” list – PR problem • Initial drafts of some NCTM-aligned curricula may not have emphasized symbolic procedures enough (revisions were made – e.g., Core Plus)

18. Backlash Despite Successes • Critics rightly charge that most research has been conducted by pro-reformers • NCTM’s initial “Decreased Emphasis” list – PR problem • Initial drafts of some NCTM-aligned curricula may not have emphasized symbolic procedures enough (revisions were made – e.g., Core Plus) • “Mathematically Correct” organization fanning backlash flame with mathematicians and parents

19. Backlash Despite Successes • Critics rightly charge that most research has been conducted by pro-reformers • NCTM’s initial “Decreased Emphasis” list – PR problem • Initial drafts of some NCTM-aligned curricula may not have emphasized symbolic procedures enough (revisions were made – e.g., Core Plus) • “Mathematically Correct” organization fanning backlash flame with mathematicians and parents • Long-Term Trend NAEP did not show same increases as Main NAEP

20. Long Term Trend NAEP ResultsMathematics – Age 9, 13 & 17From NAEP 2008 Long Term Trend report, p.29 s.d.≈ 33 pts

21. The Promise of the NCTM Reforms We are convinced that if students are exposed to the kinds of experiences outlined in the Standards, they will gain mathematical power. This term denotes an individual's abilities to explore, conjecture, and reason logically…and the development of personal self-confidence. National Council of Teachers of Mathematics Curriculum and Evaluation Standards, 1989, p. 5

22. Sounds good…Do all benefit as intended?

23. NAEP Mathematics Achievement by Race/Ethnicity, 1990-2009, Grade 4

24. NAEP Mathematics Achievement by Race/Ethnicity, 1990-2009, Grade 4 & 8

25. Percentage of students agreeing "Learning math is mostly memorizing facts” (2000) Take a guess…

26. Percentage of students agreeing "Learning math is mostly memorizing facts” (2000)

27. Percentage of students agreeing "Learning math is mostly memorizing facts” (2000)

28. Pat has 3 fish bowls. There are 4 plants and 5 fish in each bowl. Which gives the total number of fish? • 3 + 5 • 3 X 4 • 3 X 5 • 3 + 4 + 5

29. Pat has 3 fish bowls. There are 4 plants and 5 fish in each bowl. Which gives the total number of fish? • 3 + 5 • 3 X 4 • 3 X 5 • 3 + 4 + 5 67% White & Asian Correct 40% Other Black, Hispanic, Am. Ind. 22% White students 43% Black students For more item-level analyses, see: Lubienski, S. T., & Crockett, M. (2007). NAEP mathematics achievement and race/ethnicity. In P. Kloosterman and F. Lester (Eds.) Results from the Ninth Mathematics Assessment of NAEP.

30. Girls’ and Boys’ Math Achievement • K-12 math achievement differences are small – usually about .1 s.d. • Differences in confidence and attitudes toward math tend to be larger than achievement disparities. • 56% of 4th grade boys vs. 43% of girls agree “I am good at math.”

31. 1990–2009 NAEP Scores by Gender, Grade 4

32. If reform-oriented instruction is implemented across all schools, will SES- and race-related achievement gaps narrow?

33. One Study that Points to Challenges: Do lower- and higher-SES students react to a "reformed" curriculum and pedagogy differently? What Led Me To This? • Writing and piloting new middle-school mathematics curriculum • My own working-class background

34. Instruction in this classroom • Connected Mathematics Project Trial Materials • Launch-Explore-Summarize model of teaching through problem solving

35. Two Themes 1) Authority/direction 2) Contextualization/abstraction The first theme arose in relation to students’ experiences with the open nature of the problems and whole-class discussions. The second theme arose specifically in relation to the contextualized nature of the “real world” problems.

36. Comparing Traditional Math withProblem-Centered MathLower-SES Students Sue: I like the other way better…Some of the questions in the books I don’t understand at all, they are really confusing… Lynn: I think I’m better with number problems than with problem solving like we're doing. Nick: In 5th grade when we had those math books I did better than I am now…[The old books gave] specific directions… Dawn: I don’t like this math book because it doesn’t explain EXACTLY!

37. Higher-SES students Rebecca: This year were doing stuff that I like...Before, we just sat there with 100’s of problems on a page... Guinevere: It's a lot easier [with the new curriculum]... I guess our family's, just, we are word problem kind of people. Christopher: Before we just learned about how to do problems and not how it has to do with real life. . . Sometimes the directions are unclear, but I just go and do what I think it's trying to say... I just try my best to figure out the directions, so If I get it wrong it's just because of directions and not because I did the problem wrong.

38. Higher-SES Guinevere: Yes, because I need to get my point across. Samantha: Yes, because I want other people to understand my ideas. I like arguing. Benjamin: Yes, because I like to let people know what I'm thinking. "Do you participate much in class discussions? Why or why not?" (From final survey)

39. Higher-SES Guinevere: Yes, because I need to get my point across. Samantha: Yes, because I want other people to understand my ideas. I like arguing. Benjamin: Yes, because I like to let people know what I'm thinking. Lower-SES Rose: Yes. If I know what I'm talking about. But if I'm confused I just listen. Sue: Sometimes, only if I know I've got the right answer. Dawn: No because I don't like to be wrong in front of a whole group. Carl: No, because I always feel awkward. "Do you participate much in class discussions? Why or why not?" (From final survey)

40. Do you learn from whole-class discussions?Lower-SES Students Sue: I learn better from just like the teacher instead of the whole group…When everyone is there they give their opinions and stuff it may not be right, and I mix those two up, and it just confuses me.

41. Higher-SES Students Rebecca: Yeah, I think it helps me learn more things instead of just like doing it on your own, I can know everybody’s opinions and take it into consideration. Do you find it confusing when you have all those different opinions out? Not really, some of em aren’t true, and some of em are, and I can figure out which ones are true and which ones aren’t and stuff.

42. Abstraction from Contextualized Problems Two brief examples • The Pizza Sharing Problem -- Intended to teach division interpretation of fractions • Popcorn Box Problem -- Intended to teach unit pricing ideas

43. Pizza Problem Imagine that you entered a pizza parlor and saw pizzas being served at two tables of your friends, one table with 10 people and the other with 8 people.

44. Approaches to the Pizza Problem

45. Popcorn Box ProblemWhich is the better buy?

47. Working Class Jobs traditionally involve: Obedience, conformity. In child-rearing, parents tend to: Emphasize obedience to authority, rules. Show/tell how to solve problems. Encourage communicating and reasoning in a more contextualized manner. Middle Class Jobs traditionally involve: Creativity, autonomy. In child-rearing, parents tend to: Emphasize reasoning, intellectual curiosity. Guide problem-solving with questions. Encourage decontextualized uses of language and reasoning. Class Cultural Differences (An Oversimplified, Dangerous Chart)

48. So is rote learning better for low-SES students? No. This would be a repeat of past problems.

49. Other Research on Reform-Oriented Instruction • Again, most evidence suggests that reform-oriented instruction helps U.S. students’ problem solving skills & conceptual understanding (e.g., Senk & Thompson, 2003; Riordan & Noyce, 2001). • A study in Brazil found that reform-oriented math instruction increased scores for both rich and poor, but widened the gap between them (Franco, Sztajn & Ortigao, 2007). • Research in England suggests that working-class students can benefit from reform-oriented math instruction (Boaler, 2002), but contextualized problems can pose unexpected difficulties for working-class students (Cooper & Dunne, 2000).

50. Will similar patterns occur in Ireland? I don’t know. The point is that schools must pay attention to equity when implementing instructional reforms.