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
77
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
5
X Y
2 8
3 12
4
5
6
7
16
20
24
28
20
Fostering Functional Thinking
Alternate Surface Feature Format In novices, surface features compete with
deep features attentionally (Sloutsky & Yarlas, 2000)
6
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
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