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The influence of instructional intervention on children's understanding of fractions

指導教授： Chen, Ming-puu

報 告 者： Jheng, Cian-you

報告日期： 2007/01/02

Hajime Yoshida, Yoshinobu Shinmachi. (1999). The influence of instructional intervention on children's understanding of fractions. Japanese Psychological Research, 41(4), 218-228.

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Introduction

Difficulty of learning fractions:

1. The part-whole, the quotient, and the ratio subconstructs are central to understanding fractions as rational numbers.

2. Prior knowledge of whole numbers.• A number of researchers have also found that

partitioning plays a critical role in a student’s understanding of fractions(Behr et al., 1993; Pothier &Sawada, 1983).

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Introduction

• Streefland (1991, 1993) proposed “fair share” as an important schema for teaching fractions.– A fair-share schema means that when N pers

ons share M objects, this should be done equally.

• Implemented this schema, was successful in teaching fractions.

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Introduction

• The present study assumes that the traditional Japanese formal curriculum lacks the basic framework for understanding fractions.– fair-share schema– invariance of the whole (equal-whole)

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hypothesize

1. Experimental classes would perform better on both the ordering and drawing tasks than students given the traditional curriculum.

2. Traditional curriculum would perform better on both the transforming fractions and finding equivalent ones than students in the experimental classes.

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Method

• Fourth graders from three classes in a public elementary school– experimental group ： 59 → E1 、 E2– textbook group ： 27

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Method• same numerators and different denominators

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Pre-test

order three fractions

draw figures illustrating the sizes of two fractions

• same denominators and different numerators

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Retention test

ordering task

drawing task

assessment test

• finding and composing equivalent fractions• convert a given fraction (mixed fraction → improper fraction)• computational skills in addition and subtraction on fractions

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Result- ordering task

• There was no significant difference between the E1 score and E2 score.– pre-test, t(84) = .549

– retention test, t(84) = .601. The data for the E1 and E2 classes

were therefore combined for statistical analysis.

A 2 (E and T) ×2 (pre- and retention tests) ANOVA with repeated measures was conducted on the correct performances on the ordering task.

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Result- ordering task

• There was no difference in performance between the E and T groups in the pre-test, there was significantly better performance on the retention test in the E group than the T group.– main effects of the intervention factor, F(1, 84)= 7.238, p < .01

– main effects of the test factor, F(1, 84)= 21.842, p < .001– significant interaction between the intervention and test, F(1, 84) = 4.072, p <.05– A test of simple effect showed a significant difference between

the E and T groups in the retention test, F(1, 168) = 5.318, p <.025, but no significant difference between the groups in the pre-test.

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Result- drawing task

• There were no significant differences between E1 and E2 .– pre-test, t(84) = .719– retention test, t(84) = .491.

The data for the E1 and E2 classes were again combined for statistical analysis.

A 2 (E and T) ×2 (pre- and retention tests) ANOVA with repeated measures was conducted on the mean number of figures drawn appropriately for each group.

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Result- drawing task

• There was no difference between the two groups in the ability to draw figures illustrating the magnitude of fractions before the instructional intervention.

• The experimental program strongly influenced student understanding of fractions as representing real quantities.

– This indicated a significant main effect of the intervention factor, F(1, 84) = 6.192, p < .01

– The main effect of the test was also significant, F(1, 84) = 27.081, p < .001– The ANOVA indicated a significant interaction between the intervention and the

test, F(1, 84) = 5.550, p <.01– A test of simple effect on the pre-test scores showed that there was no significant

difference between the E and T groups.– the test of simple effect on the retention test scores indicated a significant

difference between the two groups, F(1, 168) = 6.906, p <.01

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Result- assessment test

• The hypothesis 2 was not supported.– The t-tests indicated no difference between the E and

T groups: t(84) = 1.01 for finding equivalent fractions, t(84) = .66, for converting fractions, or for t(84) = .75 for computational skills.

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Discussion

• students in the E group had no opportunities for drawing illustrations of the magnitudes of fractions or for ordering fractions, and yet they performed better on these tasks than the students in the T group.

• It is safe to conclude that the equal-whole schema is indeed critical for understanding fractions.