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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

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April 2007AERA Presentation, Chicago2 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)

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April 2007AERA Presentation, Chicago3 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)

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April 2007AERA Presentation, Chicago4 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)

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April 2007AERA Presentation, Chicago5 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)

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April 2007AERA Presentation, Chicago6 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)

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April 2007AERA Presentation, Chicago7 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)

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April 2007AERA Presentation, Chicago8 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)

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April 2007AERA Presentation, Chicago9 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

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April 2007AERA Presentation, Chicago10 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

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April 2007AERA Presentation, Chicago11 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

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April 2007AERA Presentation, Chicago12 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

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April 2007AERA Presentation, Chicago13 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

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April 2007AERA Presentation, Chicago14 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

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April 2007AERA Presentation, Chicago15 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 For example

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April 2007AERA Presentation, Chicago16 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 For example

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April 2007AERA Presentation, Chicago17 Purpose of study

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April 2007AERA Presentation, Chicago18 Purpose of study Investigate students’ difficulties with estimation Investigate students’ thinking about what makes an estimate good

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April 2007AERA Presentation, Chicago19 Purpose of study Investigate students’ difficulties with estimation Investigate students’ thinking about what makes an estimate good

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April 2007AERA Presentation, Chicago20 Method Part of a larger study 55 6 th graders Private middle school in US South Worked on packets of problems in pairs 2 days of problem solving Partners interactions audio-taped

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April 2007AERA Presentation, Chicago21 Method Part of a larger study 55 6 th graders Private middle school in US South Worked on packets of problems in pairs 2 days of problem solving Partners interactions audio-taped

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April 2007AERA Presentation, Chicago22 Method Part of a larger study 55 6 th graders Private middle school in US South Worked on packets of problems in pairs 2 days of problem solving Partners interactions audio-taped

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April 2007AERA Presentation, Chicago23 Method Part of a larger study 55 6 th graders Private middle school in US South Worked on packets of problems in pairs 2 days of problem solving Partners interactions audio-taped

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April 2007AERA Presentation, Chicago24 Method Part of a larger study 55 6 th graders Private middle school in US South Worked on packets of problems in pairs 2 days of problem solving Partners interactions audio-taped

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April 2007AERA Presentation, Chicago25 Method Part of a larger study 55 6 th graders Private middle school in US South Worked on packets of problems in pairs 2 days of problem solving Partners interactions audio-taped

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April 2007AERA Presentation, Chicago26 Materials Worked examples with questions Independent practice

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April 2007AERA Presentation, Chicago27 Materials Worked examples with questions Independent practice

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April 2007AERA Presentation, Chicago28 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? Allie’s way: 27 * 43 My estimate is 800. I covered up the ones digits and then multiplied the tens digit like this: 2█ * 4█ = 8 Then I added two zeros because I covered up two digits and got 800. Claire’s way: 27 * 43 My estimate is 1200. I rounded both numbers. I rounded 27 up to 30. I rounded 43 down to 40. Then I multiplied 30 * 40 and got 1200.

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April 2007AERA Presentation, Chicago29 Analysis Listened to audio with attention to students’ perceptions of good estimates

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April 2007AERA Presentation, Chicago30 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

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April 2007AERA Presentation, Chicago31 What makes an estimation “Easy”? The first way Compute “in your head” and not on paper

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April 2007AERA Presentation, Chicago32 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.”

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April 2007AERA Presentation, Chicago33 What makes an estimation “Easy”? The second way Compute “in your head” and not on paper Time spent in using a strategy

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April 2007AERA Presentation, Chicago34 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.”

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April 2007AERA Presentation, Chicago35 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

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April 2007AERA Presentation, Chicago36 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

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April 2007AERA Presentation, Chicago37 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)

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April 2007AERA Presentation, Chicago38 Explanation: Leads to closer answer An estimation is easier if methods can lead to estimates that are closer to the exact answer

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April 2007AERA Presentation, Chicago39 What makes an estimate “close”? The first way Closeness between the initial operand and the altered operand

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April 2007AERA Presentation, Chicago40 Explanation: Closeness of rounded numbers To make an estimation is affected by closeness between rounded and initial operands

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April 2007AERA Presentation, Chicago41 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.”

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April 2007AERA Presentation, Chicago42 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

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April 2007AERA Presentation, Chicago43 Explanation: How far away from exact To determine how far from exact is based on how far the operands are altered

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April 2007AERA Presentation, Chicago44 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.”

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April 2007AERA Presentation, Chicago45 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

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April 2007AERA Presentation, Chicago46 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

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April 2007AERA Presentation, Chicago47 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

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April 2007AERA Presentation, Chicago48 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

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Thank You! Jon R. Star, jonstar@msu.edujonstar@msu.edu Kosze Lee, leeko@msu.eduleeko@msu.edu Kuo-Liang Chang, changku3@msu.educhangku3@msu.edu Bethany Rittle-Johnson, b.rittle-johnson@vanderbilt.edub.rittle-johnson@vanderbilt.edu The poster, the associated paper, and other papers from this project can be downloaded from www.msu.edu/~jonstarwww.msu.edu/~jonstar

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