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Jon R. Star Harvard University Bethany Rittle-Johnson Vanderbilt University

The Role of Prior Knowledge in the Development of Strategy Flexibility: The Case of Computational Estimation. Jon R. Star Harvard University Bethany Rittle-Johnson Vanderbilt University. Thanks to…. Funding from the US Department of Education Thanks to schools in Michigan and Tennessee

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Jon R. Star Harvard University Bethany Rittle-Johnson Vanderbilt University

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  1. The Role of Prior Knowledge in the Development of Strategy Flexibility: The Case of Computational Estimation Jon R. StarHarvard University Bethany Rittle-JohnsonVanderbilt University

  2. Thanks to… • Funding from the US Department of Education • Thanks to schools in Michigan and Tennessee • Holly Harris for help with data collection and analysis • Research assistants at Michigan State University, Vanderbilt University, and Harvard University for help with all aspects of this work PME-NA 2009

  3. Plan for this talk • Background • Flexibility • Computational estimation • Role of prior knowledge • Results of two studies that helped us explore prior knowledge and its impact on flexibility PME-NA 2009

  4. Flexibility • Knowledge of multiple strategies • Use of efficient strategies, including the ability to select the most appropriate strategy for a given problem and a given problem-solving goal PME-NA 2009

  5. Computational estimation • Widely studied in 80’s and early 90’s • Less so in recent years • Process of mentally generating an approximate answer for a given arithmetic problem (Rubenstein, 1985) • Distinct from “mental computation,” which means finding the exact answer PME-NA 2009

  6. Strategies for estimation PME-NA 2009

  7. Appropriateness: Proximity • Proximity: Which strategy will provide the closest estimate to the actual number? PME-NA 2009

  8. False illusion of proximity • The more you round, the greater the error? In some cases, round both or trunc provides a more proximal estimate than round one PME-NA 2009

  9. Appropriateness: Ease • How quickly the estimate can be generated • Generalizations can be made based on studies on middle school students (Star & Rittle-Johnson, 2009) • Truncation is an easier (faster) strategy than round both for multiplicands greater than or equal to 20 • Round one is easier to implement than round both where one multiplicand is near ten PME-NA 2009

  10. Prior knowledge and flexibility • Students’ prior knowledge may have impact on their gains in flexibility • Learners need initial familiarity with one strategy before they can become flexible in the use of multiple strategies (Rittle-Johnson, Star, & Durkin, in press) PME-NA 2009

  11. Research questions • What is the role of students’ prior knowledge of estimation strategies in the effectiveness of interventions designed to promote flexibility knowledge? • What is the role of prior knowledge on flexibility of use? • Multiple strategies • Appropriate strategies PME-NA 2009

  12. Participants • Study 1: • 65 fifth graders in an urban, private school • Fluent users of the round both strategy at pretest • Study 2: • 157 fifth and sixth graders in a small, rural school • Began study with moderate to low prior knowledge of strategies for estimation PME-NA 2009

  13. Design • Pretest/Intervention/Post-test design • Intervention occurred in partner work during math classes • Random assignment of pairs to condition • Both conditions present in all classrooms • Class periods began with a short period (10 minutes) of instruction • Students studied worked examples with partner and also solved practice problems on own PME-NA 2009

  14. Intervention • Interventions in Study 1 and 2 were similar • One week intervention focused on comparison of worked examples presented side-by-side versus sequential study of the examples • Focused on the three estimation strategies discussed earlier • (Effect of the intervention is not the focus of this paper) PME-NA 2009

  15. Compare packet PME-NA 2009

  16. Sequential packet next page next next page PME-NA 2009

  17. Assessment • Assessment was similar for both studies • Individual pretest and posttest • Procedural knowledge • Flexibility • Conceptual knowledge PME-NA 2009

  18. Procedural knowledge • How to estimate, using both whole-number multiplication problems and transfer problems • Mental • Estimate 32 x 17 mentally and quickly • Familiar • Estimate 12 x 24 and 113 x 27 • Transfer • Estimate 1.19 x 2.39 and 102 ÷ 27 PME-NA 2009

  19. Flexibility • Knowledge • Multiple strategies (“Multiple ways”) • Recognize and evaluate ease of use (“Ease”) • Recognize and evaluate closeness of estimate (“Closeness”) • Use of strategies • Coded students’ strategies on familiar whole-number multiplication problems PME-NA 2009

  20. Conceptual knowledge • Core concepts related to estimation • Definitions of estimation • Acceptance of multiple strategies of estimation and multiple values of estimates PME-NA 2009

  21. Results • Prior knowledge at pretest • Impact of prior knowledge on flexibility PME-NA 2009

  22. Prior knowledge at pretest • Study 1 students did well at pretest • Study 2 students significantly lower PME-NA 2009

  23. Strategies on pretest PK items • Study 1: Almost all students began with considerable fluency with the round both strategy • 92% used round both on at least one problem • 38% also familiar with round one • Study 2: Students used round both much less frequently at pretest • 49% used round both on any problem at pretest • 11% used round one at pretest PME-NA 2009

  24. Gains pre/post • Students in Studies 1 and 2 made comparable gains, despite differences in their pretest knowledge PME-NA 2009

  25. However.... • On flexibility subscales, the differences between Studies 1 and 2 begin to emerge PME-NA 2009

  26. 1. Multiple ways subscale • Assessed students’ knowledge of multiple strategies for generating estimates • Are students able to generate an estimate for a problem in at least two ways? • Study 1 students’ subscale score rose from 75% to 92%, while Study 2 students’ scores improved much more dramatically, from 24% to 63%. • However, Study 1 students were almost at ceiling at pretest in their knowledge of multiple strategies PME-NA 2009

  27. 2. Ease subscale • Assessed students’ knowledge of which estimation strategies were easiest to compute • Study 1 students’ subscale scores grew from 62% to 89% while Study 2 students’ gains were slightly less, from 58% to 72%. • Students in both studies made comparable gains in recognizing the relative ease of truncation strategy, but Study 1 students made greater gains in recognizing relative ease of round one PME-NA 2009

  28. 3. Evaluation of proximal strategies • Assessed students’ knowledge of which strategies yielded proximal (i.e. closer) estimates • Similar gains from both studies, 76% to 86% (Study 1) and 56% to 69% (Study 2) PME-NA 2009

  29. Overall flexibility results • Students in Study 2 made the greatest gains in their knowledge of multiple strategies • Began with little knowledge of strategies other than round both • Study 2 students gained appreciation of the relative ease of trunc over round both • Study 1 students showed superior performance on all subscales and greater gains on identifying strategies for ease of computation PME-NA 2009

  30. Flexibility use at posttest • Use of multiple strategies • Choice of appropriate strategies PME-NA 2009

  31. Use of multiple strategies • Study 1 students increased their use of round both and round one, while the use of trunc decreased • Study 2 students increased their use of all three estimation strategies • Study 1 students were more likely to use multiple strategies on the posttest • 53% in Study 1 used at least two of the three strategies as compared to 29% in Study 2 PME-NA 2009

  32. Choice of appropriate strategies • Coded whether students switched to a more appropriate strategy • For problems where round one is a more appropriate strategy than round both, how many students switch from round both to round one? • Only considered students who showed fluency with round both at pretest • 25% of Study 1 and only 5% of Study 2 switched PME-NA 2009

  33. Choice of appropriate strategies • For problems where trunc is a more appropriate strategy than round both, how many students switch from round both to trunc? • Only considered students who showed fluency with round both at pretest • 3% of Study 1 switched from round both to trunc, while 19% of Study 2 switched PME-NA 2009

  34. Discussion • Prior knowledge mattered! • Students with significant fluency in the round both strategy (Study 1) show greater gains in flexibility than students who were substantially less fluent (Study 2) • Also superior in flexibility use by using greater diversity of strategies as well as more frequent selection of the most appropriate strategy PME-NA 2009

  35. Discussion • Prior knowledge didn’t matter?! • Students with lower prior knowledge (Study 2) made greater gains in their knowledge of multiple strategies and comparable gains in learning relative merits of trunc strategy in terms of ease and proximity • Students in Study 2 were also more likely to switch from round both to trunc • A choice to optimize for ease? PME-NA 2009

  36. Role of prior knowledge?! • A possible explanation is that students with high prior knowledge in Study 1 switched strategies to get a more proximal estimate • Already had an easy-to-execute strategy with round both, but were interested in switching if this meant getting a closer estimate • Students in Study 2 with lower prior knowledge switched for greater ease of implementation • Attracted by an easy-to-compute strategy with truncate PME-NA 2009

  37. Implications • Assessment of flexibility and interventions designed to promote flexibility should include both knowledge and use • Does prior knowledge help or hinder learning? • Students with higher prior knowledge may be reluctant to adopt new strategies, except under certain conditions (e.g., new goal of proximity) • Students with minimal prior knowledge may be overloaded with multiple strategies but may be attracted to strategies that are easy to implement PME-NA 2009

  38. In conclusion... • Prior knowledge plays an important but complex and nuanced role in the development of strategy flexibility • Flexibility can and should be an instructional goal for all students • Students’ prior knowledge may promote or hinder students’ knowledge of multiple strategies and their ability to select the most appropriate strategy for a given problem PME-NA 2009

  39. Thanks! Star, J.R., Rittle-Johnson, B., Lynch, K., & Perova, N. (in press). The role of prior knowledge and comparison in the development of strategy flexibility: The case of computational estimation. ZDM - The International Journal on Mathematics Education. Jon R. Star Harvard UniversityJon_Star@harvard.edu Bethany Rittle-JohnsonVanderbilt University Bethany.Rittle-Johnson@vanderbilt.edu PME-NA 2009

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