1 / 49

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 .

faith
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related