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It Pays to Compare: Effectively Using Comparison to Support Student Learning of Algebra

It Pays to Compare: Effectively Using Comparison to Support Student Learning of Algebra. Bethany Rittle-Johnson Jon Star. Our approach to improving students’ mathematics learning. Identify instructional practices used in exemplary and typical classrooms

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It Pays to Compare: Effectively Using Comparison to Support Student Learning of Algebra

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  1. It Pays to Compare: Effectively Using Comparison to SupportStudent Learning of Algebra Bethany Rittle-Johnson Jon Star

  2. Our approach to improving students’ mathematics learning • Identify instructional practices used in exemplary and typical classrooms • Use cognitive science literature to focus on practices most likely to help student learning • Experimentally evaluate impact of the instructional practice on student learning and develop instructional guidelines IES Conference 2008

  3. Potential of comparison • Mathematics Education: Central tenet of reform efforts; used by teachers • Cognitive Science: A fundamental learning mechanism IES Conference 2008

  4. Central tenet of math reforms • Students benefit from sharing and comparing solution methods • “nearly axiomatic”, “with broad general endorsement” (Silver et al., 2005) • Noted feature of ‘expert’ math instruction • Present in classrooms in high performing countries such as Japan and Hong Kong (Ball, 1993; Fraivillig, Murphy, & Fuson, 1999; Huffred-Ackles, Fuson, & Sherin Gamoran, 2004; Lampert, 1990; Silver et al., 2005; NCTM, 1989, 2000; Richland et al 2007; Stigler & Hiebert, 1999) IES Conference 2008

  5. Used in some Algebra textbooks Sobel, M.A., Maletsky, E. M., Lerner, N., & Cohen, L.S. (1985) Algebra One, Harper and Row Inc.   IES Conference 2008

  6. But does comparison improve student learning? • No evidence that comparison improves student learning in mathematics • Cognitive science research suggests that it should… IES Conference 2008

  7. Comparison in cognitive science “The simple, ubiquitous act of comparing two things is often highly informative to human learners…. Comparison is a general learning process that can promote deep relational learning and the development of theory-level explanations” (Gentner, 2005, pp. 247, 251)

  8. Fundamental learning mechanism • Lots of evidence from cognitive science • Identifying similarities and differences in multiple examples is an important pathway to flexible, transferable knowledge • Mostly laboratory studies • Rarely done with school-age children or in mathematics (Catrambone & Holyoak, 1989; Gentner, Loewenstein, & Thompson, 2003; Gick & Holyoak, 1983; Kurtz, Miao, & Gentner, 2001; Loewenstein & Gentner, 2001; Namy & Gentner, 2002; Oakes & Ribar, 2005; Schwartz & Bransford, 1998) IES Conference 2008

  9. Does comparison support math learning? • Goal of our IES grant • Investigate whether comparison can support conceptual and procedural knowledge of equation solving (and estimation) • Explore what types of comparison are most effective • Experimental studies in intact classrooms IES Conference 2008

  10. Why equation solving? • Often students’ first exposure to abstraction and symbolism of mathematics • Area of weakness for US students (Blume & Heckman, 1997; Schmidt et al., 1999) • According to NCTM and National Math Panel Report, linear equation solving should be a focal point of math instruction in middle school • Although real-world contexts and informal solution methods are powerful for simple problems, equations and equation solving are more effective for complex problems (Koedinger, Alibali & Nathan, 2008) IES Conference 2008

  11. Multiple methods for solving equations Method #1: 3(x + 1) = 15 3x + 3 = 15 3x = 12 x = 4 • Some are better than others • Students tend to memorize only one method • Example: Solving 3(x + 1) = 15 Method #2: 3(x + 1) = 15 x + 1 = 5 x = 4 IES Conference 2008

  12. Study 1 • Research question: Does comparing solution methods improve equation solving knowledge? • Research design: Randomly assigned to: • Comparison condition • Compare and contrast alternative solution methods • Sequential condition • Study same solution methods sequentially Rittle-Johnson, B. & Star, J.R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology. IES Conference 2008

  13. Translation to the classroom • Students study and explain worked examples with a partner • Based on core findings in cognitive science -- the advantages of: • Worked examples (e.g. Sweller, 1988) • Generating explanations (e.g. Chi et al, 1989; Rittle-Johnson, 2006) • Peer collaboration (e.g. Fuchs & Fuchs, 2000) IES Conference 2008

  14. Comparison condition IES Conference 2008

  15. next page next page next page Sequential condition IES Conference 2008

  16. Predicted outcomes • Students in comparison condition will make greater gains in: • Procedural knowledge, including success on novel problems • Procedural flexibility (e.g. use more efficient methods; evaluate when to use a procedure) • Conceptual knowledge (e.g. equivalence) IES Conference 2008

  17. Study 1 Method • Participants: 70 7th-grade students and their math teacher • Design: • Pretest - Intervention - Posttest • Replaced 2 lessons in textbook • Intervention occurred in partner work during 2 1/2 math classes • Intervention: • Randomly assigned to Compare or Sequential condition • Studied worked examples with partner • Solved practice problems on own IES Conference 2008

  18. Procedural knowledge assessment • Equation Solving • Intervention: 1/3 (x + 1) = 15 • Posttest Familiar: -1/4 (x – 3) = 10 • Posttest Novel: 0.25 (t + 3) = 0.5 IES Conference 2008

  19. Procedural flexibility • Use of more efficient solution methods on procedural knowledge assessment • Knowledge of multiple methods • Solve each equation in two different ways • Evaluate methods: Looking at the problem shown above, do you think that this way of starting to do this problem is a good idea? An ok step to make? Circle your answer below and explain your reasoning. IES Conference 2008

  20. Conceptual knowledge assessment IES Conference 2008

  21. Gains in procedural knowledge F(1, 31) =4.49, p < .05 IES Conference 2008

  22. Flexible use of procedures • Comparison students more likely to use more efficient method and somewhat less likely to use the conventional method Solution Method at Posttest (Proportion of problems) ~p = .06; * p < .05 IES Conference 2008

  23. Gains in flexible knowledge of procedures IES Conference 2008 F(1,31) = 7.73, p < .01

  24. Gains in conceptual knowledge No Difference IES Conference 2008

  25. Summary of Study 1 • Comparing alternative solution methods is more effective than sequential sharing of multiple methods • Improves procedural transfer and flexibility • In mathematics, in classrooms IES Conference 2008

  26. Comparison can help: Now what? • Replicated findings for fifth graders learning computational estimation • Goal: Develop guidelines for using comparison to support mathematics learning • Starting Point: Standard classroom practices IES Conference 2008

  27. What are teachers doing? • US teachers use comparison in 8th grade math lessons (average 4 per lesson) • Types of comparisons (used with equal frequency): • Compare two similar problems with same basic solution • Compare two moderately similar problems or solutions • Compare a problem to a mathematical rule or principle • Compare a problem to a non-mathematical situation (Richland , Holyoak & Stigler, 2004 analysis of TIMSS videos) IES Conference 2008

  28. What are teachers doing? • May not be using comparison well • Teachers, rather than students, initiate comparisons and make links between examples • When they present multiple solutions, rarely provide support for or discuss comparisons • Don’t know which types of comparison support learning • e.g. Comparisons to contexts from different domains rarely support learning in laboratory studies. (Richland , Holyoak & Stigler, 2004; Richaland, Zur & Holyoak, 2007; Chazan & Ball, 1999) IES Conference 2008

  29. What about Algebra I textbooks? • Compare two similar problems with same basic solution method (Equivalent Equations) Bellman,A.E., Bragg,S.C., Charles, R.I., Hall,B., Handlin, W.G., & Kennedy, D. (2007) Algebra 1, Pearson Education Inc, Pearson Prentice Hall IES Conference 2008

  30. Algebra I textbooks • Compare problems with different structures (Different Problem Types) Hollowell, K.A., Ellis, W., & Schultz, J.E. (1997). HRW Algebra. Holt, Rinehart, & Winston.  IES Conference 2008

  31. Algebra I textbooks • Compare different solution methods to same problem (Solution Methods) Sobel, M.A., Maletsky, E. M., Lerner, N., & Cohen, L.S. (1985) Algebra One, Harper and Row Inc.   IES Conference 2008 31

  32. Comparison in Algebra 1 textbooks Informal analysis of 10 Algebra I textbooks - chapter on multi-step linear equations IES Conference 2008

  33. What should be compared? • Variety of comparisons are being used in math classrooms • What are benefits and drawbacks to different types of comparisons? • Study 1 confirms that comparing solution methods aids learning, as suggested by expert teaching practices • Cognitive science literature suggests that comparing two problems solved with the same solution method should benefit learning IES Conference 2008

  34. Study 2 • Research question: What are the relative merits of comparing solution methods vs. comparing problems? • Research design: Randomly assigned to: • Compare solution methods • Compare problems that: • Are very similar (Equivalent) • Have different problem features (Different problem types) IES Conference 2008

  35. Types of comparison Solution Methods (one problem solved in 2 ways) Problem Types (2 different problems, solved in same way) Equivalent (two similar problems, solved in same way) IES Conference 2008

  36. Study 2 Method • Participants: 161 7th & 8th grade students from 3 schools (more diverse sample) • Design: • Pretest - Intervention - Posttest - 2 week Retention • Replaced 3 lessons in textbook • Randomly assigned to • Compare Solution Methods • Compare Problem Types • Compare Equivalent • Intervention occurred in partner work • Assessment adapted from Study 1 IES Conference 2008

  37. Conceptual knowledge results * F (2, 153) = 5.76, p = .004, 2 = .07 IES Conference 2008

  38. Procedural knowledge results No differences, even on novel problem types IES Conference 2008

  39. Flexible use of procedures * F (2, 153) = 4.96, p = .008, 2 = .06 IES Conference 2008

  40. Flexible knowledge of procedures * F (2, 153) = 5.01, p = .008, 2 = .07 IES Conference 2008

  41. Summary • Across studies, Comparing Solution Methods often supported the largest gains in conceptual knowledge, procedural knowledge and procedural flexibility • Supported attention to multiple methods and their relative efficiency, which both predicted learning IES Conference 2008

  42. Guidelines for using comparison • Provide a written record of examples • Leverage current use of worked examples in textbooks • Contrast important dimensions in the examples, such as problem features or solution methods • Contrasting correct and incorrect solution methods can help too (Kelley Durkin, IES pre-doc research) • Have students compare a familiar method to an unfamiliar method • Invite comparisons by using common labels and prompting for specific comparisons, including efficiency of the methods • Be sure students, not just teachers, are comparing and explaining • Incorporate some direct instruction IES Conference 2008

  43. What’s next? • Teacher Professional Development for using comparison in Algebra I courses • Type of comparison matched to prior knowledge and sequencing different types of comparison

  44. Acknowledgements • For slides, papers or more information, contact: b.rittle-johnson@vanderbilt.edu • Funded by a grant from the Institute for Education Sciences, US Department of Education • Thanks to research assistants at Vanderbilt: • Holly Harris, Shanelle Chambers, Jennifer Samson, Anna Krueger, Heena Ali, Kelley Durkin, Kelly Cashen, Calie Traver, Sallie Baxter, Amy Goodman, Adam Porter, John Murphy, Rose Vick, Alexander Kmicikewycz, Jacquelyn Beckley and Jacquelyn Jones • And at Michigan State: • Kosze Lee, Kuo-Liang Chang, Howard Glasser, Andrea Francis, Tharanga Wijetunge, Beste Gucler, and Mustafa Demir • And at Harvard: • Martina Olzog, Jennifer Rabb, Christine Yang, Nira Gautam, Natasha Perova, and Theodora Chang IES Conference 2008

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