A cognitive tool for teaching the addition/subtraction of common fractions: a model of affordances 指導教授：Chen, Ming-puu 報 告 者：Jheng, Cian-you 報告日期：2007/04/21 Kong, S. C., & Kwok, L. F. (2005). A cognitive tool for teaching the addition/subtraction of common fractions: A model of affordances. Computers & Education, 45(2), 245-265.

Introduction cognitive tools (CT) are both mental and computational devices that can support, guide and mediate the cognitive processes of learners. (Derry &LaJoie, 1993; Kommers, Jonassen, & Mayes, 1992) The Graphical Partitioning Model (GPM) was designed as the mechanism for providing the learning support. (rectangular bar) GPM Benefit : links the concrete manipulations to finding a common denominator.

Introduction 找出相同分割數

Introduction 找出等值分數

Introduction Key finding 50% of the learners did not know the inverse relationship between the number of parts and the size of a part of a unit. 58% of the learners showed no intention to represent fractions for comparing their equivalence. For example, many of them did not represent fractions with the same unit for comparison. 67% of the learners did not show an ability to use equivalent fractions for adding/subtracting fractions with unlike denominators. 8% of the learners had the concept of fraction equivalence but were unable to relate the concept to ways of finding equivalent fractions.

Research questions What are the learning outcomes of learners after working with the CT? What is the relationship between the knowledge of fraction equivalence and the capability of learners to generate procedural knowledge for adding/subtracting fractions with unlike denominators by using this CT? Do different groups with different mathematics ability differ in their learning outcomes? Do different groups with different mathematics ability differ in their modes of interaction with the model of affordances? Is there any evidence on tool-affordances?

Methods Subject, sampling and profiles fourth graders (9–10 years)

Methods Learning and teaching activities CT support ○→× Individually / groups B Freely exploring the CT C Peer tutoring Use CT as communication Group discussion Methods Learning and teaching activities

Methods Learning and teaching activities

Methods Instruments Niemi’s instruments on assessing conceptual understanding of common fractions (Niemi, 1996a). knowledge of computing equivalent fractions the concept of fraction equivalence pre and post tests (eight items were added to the post-test, procedural knowledge of adding/subtracting fractions with unlike denominators)

Results and discussion (1)Knowledge generated from working with the CT Same starting point

Results and discussion (1)Knowledge generated from working with the CT ANCOVA PreFEConcept score as a covariate Group X Time interaction, F(1, 45) = 22.690, p < 0.001. significant experimental group significant higher than control group (m=2.506, p < 0.001) PreASCompute score as a covariate Group X Time interaction, F(1, 45) = 12.659, p < 0.01. significant experimental group significant higher than control group (m=2.555, p < 0.01)

Results and discussion (1)Knowledge generated from working with the CT Summary : The model of affordances enabled learners to develop the concept of fraction equivalence. With the mediation of the CT, learners were able to generate procedural knowledge on adding/subtracting fractions with unlike denominators.

Results and discussion (2) Knowledge of fraction equivalence and procedural knowledge generation critical role of the knowledge of fraction equivalence in generating the procedural knowledge. Significant variables

Results and discussion (2) Knowledge of fraction equivalence and procedural knowledge generation in our learning model, the concept of fraction equivalence and the knowledge of computing equivalent fractions were almost equally important for learners in generating procedural knowledge for adding/subtracting fractions with unlike denominators. Summary : We had validated the construct of knowledge of fraction equivalence. It had two parts: concept of fraction equivalence and knowledge of computing equivalent fractions. Procedural knowledge on adding/subtracting fractions with unlike denominators would be likely generated by learners working with our CT if knowledge of fraction equivalence were developed from a conceptual understanding of its meaning.

Results and discussion (3) Group differences on learning outcomes and tool-affordances The CT enabled learners with high and medium mathematics. The CT enabled only a handful of learners with low mathematics ability.

Results and discussion (3) Group differences on learning outcomes and tool-affordances three mathematics ability groups showed no significant statistical differences for all tool-interaction features. The mode of interaction might be different if learners work in individualized setting.

Results and discussion (3) Group differences on learning outcomes and tool-affordances second feature : pre-test results indicated that learners already had some knowledge of computing equivalent fractions. make a hypothesis and test its effect was a cognitively demanding task. third feature : most frequently used (table 10)

Future research manipulate the sequence of introducing the concept and computing knowledge of fraction equivalence the effectiveness of teaching the subject domain with different sequences.