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

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)


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)




















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)



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)



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


















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


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


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


Purpose of study


Purpose of study

Investigate students’ difficulties with estimation

Investigate students’ thinking about what makes an estimate good


Purpose of study

Investigate students’ difficulties with estimation

Investigate students’ thinking about what makes an estimate good


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


Method




Method




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


Method




Method




Materials

Worked examples with questions Independent practice


Materials

Worked examples with questions Independent practice


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.


Analysis

Listened to audio with attention to students’ perceptions of good estimates


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


What makes an estimation “Easy”?The first way

Compute “in your head” and not on paper


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.”


What makes an estimation “Easy”?The second way


Time spent in using a strategy


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.”


What makes an estimation “Easy”?The third way


Time spent in using a strategy Using particular strategies


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


What makes an estimation “Easy”?The fourth way


Time spent in using a strategy Using particular strategies Leads to closer answer (proximity)


Explanation: Leads to closer answer

An estimation is easier if methods can lead to estimates that are closer to the exact answer


What makes an estimate “close”?The first way

Closeness between the initial operand and the altered operand


Explanation: Closeness of rounded numbers

To make an estimation is affected by closeness between rounded and initial operands


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.”


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


Explanation: How far away from exact

To determine how far from exact is based on how far the operands are altered


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.”


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


Discussion





Discussion





Discussion




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

Documents

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