1. 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

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 9/24/2009PME-NA 20092

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 9/24/2009PME-NA 20093

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 9/24/2009PME-NA 20094

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 9/24/2009PME-NA 20095

6. Strategies for estimation Exact Calculation26 X 42 = 1092 Round one 30 x 42 = 1260, or 26 x 40 =1040 Round both30 x 40 = 1200 Truncate 2  x 4 or 8, and add two zeros to make 800 9/24/2009PME-NA 20096

7. Appropriateness: Proximity Proximity: Which strategy will provide the closest estimate to the actual number? 9/24/2009PME-NA 20097 Exact Calculation26 X 42 = 1092 Round one26 x 40 =1040% Deviation = 5% Round both30 x 40 = 1200% Deviation = 10% Truncate 2  x 4 or 8, and add two zeros to make 800% Deviation = 27%

8. False illusion of proximity Exact Calculation29 X 31 = 899 Round one31 x 30 =930% Deviation = 3% Round both30 x 30 = 900% Deviation = 0.1% Truncate 3  x 3 or 9, and add two zeros to make 900% Deviation = 0.1% 9/24/2009PME-NA 20098 The more you round, the greater the error? In some cases, round both or trunc provides a more proximal estimate than round one

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 9/24/2009PME-NA 20099

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) 9/24/2009PME-NA 200910

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 9/24/2009PME-NA 200911

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 9/24/2009PME-NA 200912

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 9/24/2009PME-NA 200913

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) 9/24/2009PME-NA 200914

15. Compare packet 9/24/2009PME-NA 200915

16. Sequential packet 9/24/2009PME-NA 200916 page next page next

17. Assessment Assessment was similar for both studies Individual pretest and posttest Procedural knowledge Flexibility Conceptual knowledge 9/24/2009PME-NA 200917

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 9/24/2009PME-NA 200918

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 9/24/2009PME-NA 200919

20. Conceptual knowledge Core concepts related to estimation –Definitions of estimation –Acceptance of multiple strategies of estimation and multiple values of estimates 9/24/2009PME-NA 200920

21. Results Prior knowledge at pretest Impact of prior knowledge on flexibility 9/24/2009PME-NA 200921

22. Prior knowledge at pretest Study 1 students did well at pretest Study 2 students significantly lower 9/24/2009PME-NA 200922 Study 1 Study 2 Procedural.78.30 Flexibility.73.46 Conceptual.48.35

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 9/24/2009PME-NA 200923

24. Gains pre/post Students in Studies 1 and 2 made comparable gains, despite differences in their pretest knowledge 9/24/2009PME-NA 200924 Knowledge TypeStudy 1Study 2 Pretest Posttest Pretest Posttest Procedural.78.85.30.68 Flexibility.73.89.46.68 Conceptual.48.56.35.47

25. However.... On flexibility subscales, the differences between Studies 1 and 2 begin to emerge 9/24/2009PME-NA 200925 Knowledge TypeStudy 1Study 2 Pretest Posttest Pretest Posttest Flexibility.73.89.46.68 Multiple ways.75.92.24.62 Closeness.76.86.56.69 Ease.62.89.58.72

26. 1. Multiple ways subscale 9/24/2009PME-NA 200926 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

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 9/24/2009PME-NA 200927

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) 9/24/2009PME-NA 200928

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 9/24/2009PME-NA 200929

30. Flexibility use at posttest Use of multiple strategies Choice of appropriate strategies 9/24/2009PME-NA 200930

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 9/24/2009PME-NA 200931

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 9/24/2009PME-NA 200932

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 9/24/2009PME-NA 200933

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 9/24/2009PME-NA 200934

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? 9/24/2009PME-NA 200935

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 9/24/2009PME-NA 200936

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 9/24/2009PME-NA 200937

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 9/24/2009PME-NA 200938

39. Thanks! 9/24/2009PME-NA 200939 Jon R. Star Harvard University Jon_Star@harvard.edu Bethany Rittle-Johnson Vanderbilt University Bethany.Rittle-Johnson@vanderbilt.edu 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.