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

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)

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)

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)

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)

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)

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 x 30 = 300 (LeFevre, GreenHam & Waheed, 1993; Reys & Bestgen, 1981)

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)

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

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

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

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

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

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

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

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

April 2007AERA Presentation, Chicago17 Purpose of study

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

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

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

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

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

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

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

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

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

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

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 I rounded both numbers. I rounded 27 up to 30. I rounded 43 down to 40. Then I multiplied 30 * 40 and got 1200.

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

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

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

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.”

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

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.”

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

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

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)

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

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

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

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.”

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

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

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.”

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

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

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

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

Thank You! Jon R. Star, Kosze Lee, Kuo-Liang Chang, Bethany Rittle-Johnson, The poster, the associated paper, and other papers from this project can be downloaded from