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Investigating Student Thinking about Estimation: What Makes a Good Estimate?
Jon R. StarKosze Lee, Kuo-Liang ChangTharanga WijetungeMichigan State University
Bethany Rittle-Johnson Vanderbilt University
April 2007 AERA Presentation, Chicago 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)
April 2007 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)
April 2007 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)
April 2007 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)
April 2007 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)
April 2007 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)
April 2007 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)
April 2007 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
April 2007 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
April 2007 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
April 2007 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
April 2007 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
April 2007 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
April 2007 AERA Presentation, Chicago 15
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 2007 AERA Presentation, Chicago 16
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 2007 AERA Presentation, Chicago 18
Purpose of study
Investigate students’ difficulties with estimation
Investigate students’ thinking about what makes an estimate good
April 2007 AERA Presentation, Chicago 19
Purpose of study
Investigate students’ difficulties with estimation
Investigate students’ thinking about what makes an estimate good
April 2007 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
April 2007 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
April 2007 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
April 2007 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
April 2007 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
April 2007 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
April 2007 AERA Presentation, Chicago 26
Materials
Worked examples with questions Independent practice
April 2007 AERA Presentation, Chicago 27
Materials
Worked examples with questions Independent practice
April 2007 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?
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.
April 2007 AERA Presentation, Chicago 29
Analysis
Listened to audio with attention to students’ perceptions of good estimates
April 2007 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
April 2007 AERA Presentation, Chicago 31
What makes an estimation “Easy”?The first way
Compute “in your head” and not on paper
April 2007 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.”
April 2007 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
April 2007 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.”
April 2007 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
April 2007 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
April 2007 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)
April 2007 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
April 2007 AERA Presentation, Chicago 39
What makes an estimate “close”?The first way
Closeness between the initial operand and the altered operand
April 2007 AERA Presentation, Chicago 40
Explanation: Closeness of rounded numbers
To make an estimation is affected by closeness between rounded and initial operands
April 2007 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.”
April 2007 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
April 2007 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
April 2007 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.”
April 2007 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
April 2007 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
April 2007 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
April 2007 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
Thank You!
Jon R. Star, [email protected] Kosze Lee, [email protected] Kuo-Liang Chang, [email protected] Bethany Rittle-Johnson, [email protected]
The poster, the associated paper, and other papers from this project can be downloaded from www.msu.edu/~jonstar