Katherine L. McEldoon, Caroline Cochrane-Braswell & Bethany Rittle-Johnson

Outline Current Study Three different problem formats Their effect on problem solving strategy use What is Functional Thinking What makes it challenging What facilitates student understanding 2

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 Out = (In x 2) + 1 Y = 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.41B.51 C. 54D. 77

Functional Thinking Performance – Grade 4 4 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 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 XY

Fostering Functional Thinking Alternate Surface Feature Format In novices, surface features compete with deep features attentionally (Sloutsky & Yarlas, 2000) 6 XY Julia 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 Study Investigate the effect of problem context on problem solving strategy within function table problems Contexts (between subjects) Baseline Alternate Surface Story 7

Contexts & Hypotheses XY XY Cost of Present Cost of Present w Gift Wrapping BaselineAlternate SurfaceStory At 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 functional strategy use, less recursive More functional strategy use, less recursive Hypotheses What is the rule for this function table?

Method Participants 232 students in grades two through six in a middle class community Procedure Three 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. 9

Assessment Three isomorphic versions of the assessment Baseline (or) Alternate Surface (or) Story Three function table problems Additive Y = X + 4 (and) Multiplicative Y = 3X (and) Combo Y = 3X

Strategy Coding The student’s strategy code was based on their function table values and their rule for the table 11 We compared the effect of problem context (baseline, alternate surface, and story problem) on strategy use (functional or recursive) XY Functiona l Recursive Other XY 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 12 FunctionalRecursive

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 Surface BaselineStory

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 15

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