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Contrasting Cases in Mathematics Lessons Support Procedural Flexibility and Conceptual Knowledge

Contrasting Cases in Mathematics Lessons Support Procedural Flexibility and Conceptual Knowledge. Jon R. Star Harvard University Bethany Rittle-Johnson Vanderbilt University.

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Contrasting Cases in Mathematics Lessons Support Procedural Flexibility and Conceptual Knowledge

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  1. Contrasting Cases in Mathematics Lessons Support Procedural Flexibility and Conceptual Knowledge Jon R. Star Harvard University Bethany Rittle-Johnson Vanderbilt University EARLI Invited Symposium: Construction of (elementary) mathematical knowledge: New conceptual and methodological developments, Budapest, August 29, 2007

  2. Acknowledgements • Funded by a grant from the United States Department of Education • Thanks to research assistants at Michigan State University and Vanderbilt University: • Kosze Lee, Kuo-Liang Chang, Howard Glasser, Andrea Francis, Tharanga Wijetunge, Holly Harris, Jen Samson, Anna Krueger, Heena Ali, Sallie Baxter, Amy Goodman, Adam Porter, and John Murphy

  3. Comparison • Is a fundamental learning mechanism • Lots of evidence from cognitive science • Identifying similarities and differences in multiple examples appears to be a critical pathway to flexible, transferable knowledge • Mostly laboratory studies • Not done with school-age children or in mathematics (Gentner, Loewenstein, & Thompson, 2003; Kurtz, Miao, & Gentner, 2001; Loewenstein & Gentner, 2001; Namy & Gentner, 2002; Oakes & Ribar, 2005; Schwartz & Bransford, 1998)

  4. Central tenet of math reforms • Students benefit from sharing and comparing of solution methods • “nearly axiomatic,” “with broad general endorsement” (Silver et al., 2005) • Noted feature of ‘expert’ math instruction • Present 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; Stigler & Hiebert, 1999)

  5. “Contrasting Cases” Project • Experimental studies on comparison in academic domains and settings largely absent • Goal of present work • Investigate whether comparison can support learning and transfer, flexibility, and conceptual knowledge • Experimental studies in real-life classrooms • Computational estimation (10-12 year olds) • Algebra equation solving (13-14 year olds)

  6. Why algebra? • Area of weakness for US students; critical gatekeeper course • Particular focus: Linear equation solving • Multiple strategies for solving equations • Some are better than others • Students tend to memorize only one method • Goal: Know multiple strategies and choose the most appropriate ones for a given problem or circumstance

  7. Solving 3(x + 1) = 15 Strategy #2: 3(x + 1) = 15 x + 1 = 5 x = 4 Strategy #1: 3(x + 1) = 15 3x + 3 = 15 3x = 12 x = 4

  8. Similarly, 3(x + 1) + 2(x + 1) = 10 Strategy #1: 3(x + 1) + 2(x + 1) = 10 3x + 3 + 2x + 2 = 10 5x + 5 = 10 5x = 5 x = 1 Strategy #2: 3(x + 1) + 2(x + 1) = 10 5(x + 1) = 10 x + 1 = 2 x = 1

  9. Why estimation? • Widely studied in 1980’s and 1990’s; less so now • Viewed as a critical part of mathematical proficiency • Many ways to estimate • Good estimators know multiple strategies and can choose the most appropriate ones for a given problem or circumstance

  10. Multi-digit multiplication • Estimate 13 x 44 • “Round both” to the nearest 10: 10 * 40 • “Round one” to the nearest 10: 10 * 44 • “Truncate”: 1█ * 4█and add 2 zeroes • Choosing an optimal strategy requires balancing • Simplicity - ease of computing • Proximity - close “enough” to exact answer

  11. Flexibility is key in both domains • Students need to know a variety of strategies and to be able to choose the most appropriate ones for a given problem or circumstance • In other words, students need to be flexible problem solvers • Does comparison help students to become more flexible?

  12. Intervention • Comparison condition • compare and contrast alternative solution methods • Sequential condition • study same solution methods sequentially

  13. Comparison condition

  14. next page next page next page Sequential condition

  15. Outcomes of interest • Procedural knowledge • Conceptual knowledge • Flexibility

  16. Procedural knowledge • Familiar: Ability to solve problems similar to those seen in intervention • Transfer: Ability to solve problems that are somewhat different than those in intervention

  17. Conceptual knowledge • Knowledge of concepts

  18. Flexibility • Ability to generate, recognize, and evaluate multiple solution methods for the same problem • “Independent” measure • Multiple choice and short answer assessment • Direct measure • Strategies on procedural knowledge items (e.g., Beishuizen, van Putten, & van Mulken, 1997; Blöte, Klein, & Beishuizen, 2000; Blöte, Van der Burg, & Klein, 2001; Star & Seifert, 2006; Rittle-Johnson & Star, 2007)

  19. Flexibility items (independent measure)

  20. Flexibility items (independent measure)

  21. Method • Algebra: 70 7th grade students (age 13-14)* • Estimation: 158 5th-6th grade students (age 10-12) • Pretest - Intervention (3 class periods) - Posttest • Replaced lessons in textbook • Intervention occurred in partner work during math classes • Random assignment of pairs to condition • Students studied worked examples with partner and also solved practice problems on own *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, 99(3), 561-574.

  22. Results • Procedural knowledge • Flexibility • Independent measure • Strategy use • Conceptual knowledge

  23. Procedural knowledge Students in the comparison condition made greater gains in procedural knowledge.

  24. Flexibility (independent measure) Students in the comparison condition made greater gains in flexibility.

  25. Flexibility in strategy use(algebra) Strategies used on procedural knowledge items:

  26. Conceptual knowledge Comparison and sequential students achieved similar and modest gains in conceptual knowledge.

  27. Overall • Comparing alternative solution methods rather than studying them sequentially • Helped students move beyond rigid adherence to a single strategy to more adaptive and flexible use of multiple methods • Improved ability to solve problems correctly

  28. Next steps • What kinds of comparison are most beneficial? • Comparing problem types • Comparing solution methods • Comparing isomorphs • Improving measures of conceptual knowledge

  29. Thanks! You can download this presentation and other related papers and talks at http://gseacademic.harvard.edu/~starjo Jon Star Jon_Star@Harvard.edu Bethany Rittle-Johnson Bethany.Rittle-Johnson@vanderbilt.edu

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