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Investigating Student Thinking about Estimation: What Makes a Good Estimate?

Investigating Student Thinking about Estimation: What Makes a Good Estimate?. Jon R. Star Kosze Lee, Kuo-Liang Chang Tharanga Wijetunge Michigan State University Bethany Rittle-Johnson Vanderbilt University. Acknowledgements .

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Investigating Student Thinking about Estimation: What Makes a Good Estimate?

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  1. Investigating Student Thinking about Estimation: What Makes a Good Estimate? Jon R. Star Kosze Lee, Kuo-Liang Chang Tharanga Wijetunge Michigan State University Bethany Rittle-Johnson Vanderbilt University

  2. Acknowledgements • Funded by a grant from the Institute for Education Sciences, US Department of Education, to Michigan State University • Thanks also to Howard Glasser (Michigan State) and to Holly A. Harris and Jennifer Samson (Vanderbilt) AERA Presentation, Chicago

  3. Computational Estimation • Widely studied in 1980’s and 1990’s • Still viewed as a critical part of mathematical proficiency • We know a lot about what makes a good estimator • We don’t know as much about how students think about the processes and products of estimation (Case & Sowder, 1990; Reys, Bestgen, Rybolt, & Wyatt, 1980; Lindquist, 1989; Lindquist, Carpenter, Silver, & Matthews, 1983; National Research Council, 2001) AERA Presentation, Chicago

  4. Computational Estimation • Widely studied in 1980’s and 1990’s • Still viewed as a critical part of mathematical proficiency • We know a lot about what makes a good estimator • We don’t know as much about how students think about the processes and products of estimation (Case & Sowder, 1990; Reys, Bestgen, Rybolt, & Wyatt, 1980; Lindquist, 1989; Lindquist, Carpenter, Silver, & Matthews, 1983; National Research Council, 2001) AERA Presentation, Chicago

  5. Computational Estimation • Widely studied in 1980’s and 1990’s • Still viewed as a critical part of mathematical proficiency • We know a lot about what makes a good estimator • We don’t know as much about how students think about the processes and products of estimation (Case & Sowder, 1990; Reys, Bestgen, Rybolt, & Wyatt, 1980; Lindquist, 1989; Lindquist, Carpenter, Silver, & Matthews, 1983; National Research Council, 2001) AERA Presentation, Chicago

  6. Computational Estimation • Widely studied in 1980’s and 1990’s • Still viewed as a critical part of mathematical proficiency • We know a lot about what makes a good estimator • We don’t know as much about how students think about the processes and products of estimation (Case & Sowder, 1990; Reys, Bestgen, Rybolt, & Wyatt, 1980; Lindquist, 1989; Lindquist, Carpenter, Silver, & Matthews, 1983; National Research Council, 2001) AERA Presentation, Chicago

  7. What Makes an Estimate Good? • Simplicity • Good estimates are easy to compute • For example, • 11 x 31 • An easy way to estimate is to round both numbers to the nearest 10 • 10 x 30 = 300 (LeFevre, GreenHam & Waheed, 1993; Reys & Bestgen, 1981) AERA Presentation, Chicago

  8. What Makes an Estimate Good? • Proximity • Good estimates are close to exact answer • For example • 11 x 57 • By rounding only the 11 to the nearest 10, we get a close estimate • 10 x 57 = 570, which is only 57 (or 9%) from the exact answer of 627 (LeFevre, GreenHam & Waheed, 1993; Reys & Bestgen, 1981) AERA Presentation, Chicago

  9. What Makes an Estimate Good? • Simplicity and proximity seem very straightforward features of estimates • Complex relationships between: • the problems one is estimating • the strategies one uses • whether an estimate is easy and/or close to the exact value AERA Presentation, Chicago

  10. What Makes an Estimate Good? • Simplicity and proximity seem very straightforward features of estimates • Complex relationships between: • the problems one is estimating • the strategies one uses • whether an estimate is easy and/or close to the exact value AERA Presentation, Chicago

  11. What Makes an Estimate Good? • Simplicity and proximity seem very straightforward features of estimates • Complex relationships between: • the problems one is estimating • the strategies one uses • whether an estimate is easy and/or close to the exact value AERA Presentation, Chicago

  12. What Makes an Estimate Good? • Simplicity and proximity seem very straightforward features of estimates • Complex relationships between: • the problems one is estimating • the strategies one uses • whether an estimate is easy and/or close to the exact value AERA Presentation, Chicago

  13. What Makes an Estimate Good? • Simplicity and proximity seem very straightforward features of estimates • Complex relationships between: • the problems one is estimating • the strategies one uses • whether an estimate is easy and/or close to the exact value AERA Presentation, Chicago

  14. For example • Which yields a closer estimate, rounding one number to the nearest ten or rounding both numbers to the nearest ten? Round One number Round Two numbers AERA Presentation, Chicago

  15. For example • Intuition: Round one yields a closer estimate • 13 x 44 (exact answer 572) • Round one: • 10 x 44 = 440, which is 132 (23%) off • Round two: • 10 x 40 = 400, which is 172 (30%) off AERA Presentation, Chicago

  16. For example • But it depends on the problem! • 13 x 48 (exact answer 624) • Round one: • 10 x 48 = 480, which is 144 (23%) off • Round two: • 10 x 50 = 500, which is 124 (20%) off AERA Presentation, Chicago

  17. Purpose of study AERA Presentation, Chicago

  18. Purpose of study • Investigate students’ difficulties with estimation • Investigate students’ thinking about what makes an estimate good AERA Presentation, Chicago

  19. Purpose of study • Investigate students’ difficulties with estimation • Investigate students’ thinking about what makes an estimate good AERA Presentation, Chicago

  20. Method • Part of a larger study • 55 6th graders • Private middle school in US South • Worked on packets of problems in pairs • 2 days of problem solving • Partners interactions audio-taped AERA Presentation, Chicago

  21. Method • Part of a larger study • 55 6th graders • Private middle school in US South • Worked on packets of problems in pairs • 2 days of problem solving • Partners interactions audio-taped AERA Presentation, Chicago

  22. Method • Part of a larger study • 55 6th graders • Private middle school in US South • Worked on packets of problems in pairs • 2 days of problem solving • Partners interactions audio-taped AERA Presentation, Chicago

  23. Method • Part of a larger study • 55 6th graders • Private middle school in US South • Worked on packets of problems in pairs • 2 days of problem solving • Partners interactions audio-taped AERA Presentation, Chicago

  24. Method • Part of a larger study • 55 6th graders • Private middle school in US South • Worked on packets of problems in pairs • 2 days of problem solving • Partners interactions audio-taped AERA Presentation, Chicago

  25. Method • Part of a larger study • 55 6th graders • Private middle school in US South • Worked on packets of problems in pairs • 2 days of problem solving • Partners interactions audio-taped AERA Presentation, Chicago

  26. Materials • Worked examples with questions • Independent practice AERA Presentation, Chicago

  27. Materials • Worked examples with questions • Independent practice AERA Presentation, Chicago

  28. Sample of a worked example given 3. How is Allie’s way similar to Claire’s way? 4a. Use Allie’s way to estimate 21 * 43. 4b. Use Claire's way to estimate 21 * 43. 4c. What do you notice about these estimates? AERA Presentation, Chicago

  29. Analysis • Listened to audio with attention to students’ perceptions of good estimates AERA Presentation, Chicago

  30. Results • Students refer to simplicity and proximity in various ways when thinking about what makes an estimation good • Simplicity/Easiness: 4 ways • Proximity/Closeness: 2 ways AERA Presentation, Chicago

  31. What makes an estimation “Easy”?The first way • Compute “in your head” and not on paper AERA Presentation, Chicago

  32. Example: Compute in your head • One student said: “You can't really do [Catherine’s way] in your head, you'll get confused what number you're on. So Marquan's way is easier.” AERA Presentation, Chicago

  33. What makes an estimation “Easy”?The second way • Compute “in your head” and not on paper • Time spent in using a strategy AERA Presentation, Chicago

  34. Example: Time spent • One student pointed that a method is harder: “It’s going to take longer” • Another student argued: “I think Jenny's way is easiest on this one. I know it's not as quick.” AERA Presentation, Chicago

  35. What makes an estimation “Easy”?The third way • Compute “in your head” and not on paper • Time spent in using a strategy • Using particular strategies AERA Presentation, Chicago

  36. Example: Particular strategies • Students think: Rounding both operands is easier than rounding only one operand • One student said: “It is easier just to round both numbers” • Another student said: “It would be less confusing to round both numbers.” • To illustrate: to estimate 21x39, 20x40 is easier than 21x40 or 20x39. • Students think: rounding two numbers is easier because they are familiar with it AERA Presentation, Chicago

  37. What makes an estimation “Easy”?The fourth way • Compute “in your head” and not on paper • Time spent in using a strategy • Using particular strategies • Leads to closer answer (proximity) AERA Presentation, Chicago

  38. Explanation: Leads to closer answer • An estimation is easier if methods can lead to estimates that are closer to the exact answer AERA Presentation, Chicago

  39. What makes an estimate “close”?The first way • Closeness between the initial operand and the altered operand AERA Presentation, Chicago

  40. Explanation: Closeness of rounded numbers • To make an estimation is affected by closeness between rounded and initial operands AERA Presentation, Chicago

  41. Example:Closeness of rounded numbers • To estimate 11 * 78 • Alter one number v.s. alter two numbers 10 * 78 is closer than 10 * 80 “numbers are close[r] to the [original] numbers used in the problem.” AERA Presentation, Chicago

  42. What makes an estimate “close”?The second way • Closeness between the initial operand and the altered operand • How far the estimate is away from the exact value AERA Presentation, Chicago

  43. Explanation:How far away from exact • To determine how far from exact is based on how far the operands are altered AERA Presentation, Chicago

  44. Example: How far away from exact • Two hypothetical students in a given problem • 11 x 18 - “Anne” estimates 10 x 18 • 11 x 68 - “Yolanda” estimates 10 x 68 Anne’s estimate would be closer “because 10 times 18 is 180, and then 11 is 18 more, [whereas] if Yolanda goes up [one] it is gonna be 68 more.” AERA Presentation, Chicago

  45. Discussion • Students’ thinking about simplicity and proximity is diverse • Should not assume uniformity in students’ evaluation • Perception may be different from experts’ • Informative for effective teaching strategies and for assisting student learning AERA Presentation, Chicago

  46. Discussion • Students’ thinking about simplicity and proximity is diverse • Should not assume uniformity in students’ evaluation • Perception may be different from experts’ • Informative for effective teaching strategies and for assisting student learning AERA Presentation, Chicago

  47. Discussion • Students’ thinking about simplicity and proximity is diverse • Should not assume uniformity in students’ evaluation • Perception may be different from experts’ • Informative for effective teaching strategies and for assisting student learning AERA Presentation, Chicago

  48. Discussion • Students’ thinking about simplicity and proximity is diverse • Should not assume uniformity in students’ evaluation • Perception may be different from experts’ • Informative for effective teaching strategies and for assisting student learning AERA Presentation, Chicago

  49. Thank You! Jon R. Star, jonstar@msu.edu Kosze Lee, leeko@msu.edu Kuo-Liang Chang, changku3@msu.edu Bethany Rittle-Johnson, b.rittle-johnson@vanderbilt.edu The poster, the associated paper, and other papers from this project can be downloaded from www.msu.edu/~jonstar

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