1. 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),

2. 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.

3. Introduction
找出相同分割數

4. Introduction
找出等值分數

5. 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.

7. 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?

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

9. 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

10. Methods Learning and teaching activities

12. 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)

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

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

15. 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.

16. 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

17. 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.

18. 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.

19. 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.

20. 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)

21. 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.