1

Designing Knowledge Scaffolds to Support Mathematical Problem Solving

Rittle-Johnson, B., Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem solving. Cognition and Instruction, 23(3), 313-349.

指導教授： Chen, Ming-puu

報 告 者： Jheng, Cian-you

報告日期： 2007/03/03

2

Introduction• DFA(difficulty factors assessment) can be used to identify

what problem features (i.e., factors) facilitate problem solving.

• three types of knowledge for problem solving:

contextual, conceptual, and procedural knowledge.

• Contextual → candy bar

Conceptual → fraction bars

Procedural → common denominator

• To evaluate whether each scaffold facilitated addition and subtraction of fractions, we used DFA.

3

Method• 223 sixth-grade students ： urban(137) 、 suburban(86)

• Procedure：pretest (incorporated DFA)

→implement an intervention

→posttest (identical to the pretest)

• Pretest & Posttest：– same denominators– unlike denominators– adding three fractions – subtracting mixed numbers– identifying a verbal description of the conventional procedure

2

1

5

1

10

9

8

13

2

113

4

• correct

• combine-both error– combine both

numerator and denominator

• fail-to-convert error– fail to convert

numerators after finding a common denominator

• other error

5

fading of scaffolding

找公分母

6

Result-pretest• Average accuracy：

– All ─ 45%– Same denominator ─ 80%– unlike denominators ─ 40%– other three items ─ 37% to 42%

• suburban schools had higher accuracy scores than students at the urban schools at pretest (Ms = 62% vs. 35% correct), F(1, 221) = 51.9, p <.0001.

7

the least the most

52%

33%

8

Result-pretestSummary

• Each of the scaffolds reduced combine-both errors, but only the conceptual scaffold consistently reduced fail-to-convert errors.

9

Result-posttest

51% 66%

43% 53%

22% 13%

10

Result-posttestSummary

• children were more accurate across a range of problems

• made many fewer common errors such as adding the numerator and denominator

• had less need for the scaffolds

• seemed more likely to correctly use the conventional procedure.

11

Discussion Why the conceptual and contextual knowledge scaffolds

may have facilitated accurate problem solving?

• The rationale for that approach is that students need to understand the key ideas in order to have something to connect with procedural rules.

Three general design suggestions emerged from integrating these findings with past research:

• story contexts may be useful scaffolds for introducing new tasks or problem types

• visual representations may facilitate problem solving

• scaffolding intermediate procedural steps and then fading the scaffolding may support learning and problem solving.