Katherine L. McEldoon, Caroline Cochrane-Braswell & Bethany

Rittle-Johnson

OutlineCurrent Study

Three different problem formatsTheir effect on problem solving strategy use

What is Functional Thinking

What makes it challenging

What facilitates student understanding

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Functional Thinking

• A type of mathematical thinking which focuses on the relationship between two (or more) varying quantities, specifically the kinds of thinking that lead from specific relationships to generalizations of that relationship across instances (Smith, 2008).

• Encapsulates important core components of early algebraic reasoning, such as generalization and covariation. 3

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Out = (In x 2) + 1Y = 2X + 1

The table shows how the “In” numbers are related to the “Out” numbers. When a 38 goes in, what number comes out?

A.41 B.51 C. 54 D. 77

Functional Thinking Performance – Grade 4

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The table shows how the “In” numbers are related to the “Out” numbers. When a 38 goes in, what number comes out?

A.41B.51C.54D.77

National Assessment of Educational Progress (NAEP), National Performance results in Mathematics at Grade 4; 2007

Functional Thinking in Elementary School

Two Possible Relationships

Recursive Relationship Y1-Y2

Functional Relationship X1-Y1

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X Y

2 8

3 12

4

5

6

7

16

20

24

28

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Fostering Functional Thinking

Alternate Surface Feature Format In novices, surface features compete with

deep features attentionally (Sloutsky & Yarlas, 2000)

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X Y

3 8

5 16

6

9

25Julia is selling boxes of cookies as a fundraiser for her school. Three boxes cost $9 and five boxes cost $15. Her aunt wants to buy 6 boxes, and her neighbor wants to buy 8. How much do they have to pay? Julia sells 25 boxes in all. How much did she earn?

• Story Problem Format

• Depends on linguistic proficency (Kintsch & Greeno, 1985; Koedinger, Alibali & Nathan, 2008)

• When readers can handle it, they help support informal solution methods (Carraher et al., 2003)

Current StudyInvestigate the effect of problem

context on problem solving strategy within function table problems

Contexts (between subjects)

Baseline

Alternate Surface

Story

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Contexts & Hypotheses

X Y

2 6

3 7

4 8

5 9

6

14

25

41

8

X Y

2 6

4 8

5 9

7 11

8

14

25

41

Cost of Present

Cost of Present w

Gift Wrapping

2 6

3 7

4 8

5 9

6

14

25

41

Baseline Alternate Surface StoryAt a gift shop, you can pay extra to have your present gift-wrapped, as shown in the table above. What is the total cost of the present with gift-wrapping if the cost of the present is $6? $14? What about $41? If the total cost of a present with gift-wrapping is $25, what was the cost of the present itself?

Most recursive strategy use, less functional

More functionalstrategy use, less recursive

More functionalstrategy use, less recursive

Hypotheses

What is the rule for this function table?

MethodParticipants

232 students in grades two through six in a middle class community

ProcedureThree different forms of the

assessment (baseline, alternate surface, and story) were randomly distributed to students during their normal math class as a whole class activity.

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AssessmentThree isomorphic versions of the assessment

Baseline (or)

Alternate Surface (or)

Story

Three function table problemsAdditive Y = X + 4 (and)

Multiplicative Y = 3X (and)

Combo Y = 3X + 2

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Strategy CodingThe student’s strategy code was based on

their function table values and their rule for the table

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We compared the effect of problem context (baseline, alternate surface, and story problem) on strategy use (functional or recursive)

X Y

2 6

3 7

4 8

5 9

6 1014 18

21 25

41 45

Functional

Recursive

OtherX Y

2 6

3 7

4 8

5 9

6 1014 11

15 25

41 26

Rule: Add 4 to the X

number to get the Y number

Rule: Add 1 to the

number before it to get the next

one

42% 11%

47%

Strategy Use by Grade & Condition

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Functional Recursive

Results- Functional Strategy

Overall, the alternate surface context was the most conducive to the functional strategy significantly more than the story contex marginally more than the baseline

There was no difference in strategy use in the baseline and story contexts. 13

Alt Surface

Results- Recursive Strategy

The baseline context elicited the most recursive strategy use. Significantly more than alternate surface No sig. difference compared to story

There was no difference in the alternate surface and story contexts. 14

Alt SurfaceBaseline Story

Conclusions

The baseline context resulted in the most use of the naïve recursive strategy

The alternate surface context elicited the most functional strategy use

Story context did not increase recursive strategy use, but it also did not encourage functional strategy use

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Thank you