數學學習困難學生與非數學學習困難學生分數數值比較能力之差異與具體－表徵－抽象策略之教學成效

Differences in Comparison Ability on Fraction Magnitude for Students with and without Mathematics Learning Difficulties and Teaching Effectiveness of the Concrete-Representational-Abstract Strategy

本研究的目的是探究數學學習困難學生（數困生）與非數學學習困難學生（非數困生）分數數值比較能力的差異，再以具體－表徵－抽象策略提昇數困生分數數值比較能力。研究一以四年級一般生(4TA)、六年級一般生(6TA)為非數困生，以六年級閱障數困生(6RDMD)與數障生(6MD)為數困生，施測後以二因子變異數分析說明研究結果。研究二以三名數困生為對象，自編介入方案進行教學，以圖示法、目視分析法與統計分析說明研究結果。研究結果如下：(1)數困生與非數困生的分數數值比較能力有顯著差異；(2)不同題目表徵形式對數困生與非數困生的分數數值比較能力有顯著差異；(3)不同學生組別與不同題目表徵形式對分數數值比較能力有顯著的交互作用，即不同學生組別的分數數值比較能力在不同題目表徵形式有顯著差異；(4)研究參與者依循具體操作、表徵圖示逐步建立抽象概念，具體－表徵－抽象策略對增進數困生之分數數值比較能力有立即成效；(5)數困生能以抽象數字或圖示解題，具體－表徵－抽象策略對增進數困生之分數數值比較能力有短期維持效果。

 

Purpose

The mathematics education in junior and elementary schools focuses on integers and fractions (Bailey et al., 2011). According to the Curriculum Guidelines of 12-year Basic Education in Taiwan (Ministry of Education, 2018), the Mathematics curriculum includes fraction in the first learning stage, emphasizing the recognition of equal parts and unit fractions. Geary (2004) categorizes mathematical knowledge into conceptual and procedural types. One core ability of conceptual knowledge is accurate representation of magnitude (Hecht & Vagi, 2012). Literature indicates that students with learning disabilities (LD) experience difficulties in fraction magnitude representation compared to same-grade students without LD. That the intervention improving students’ magnitude knowledge based on the previous notion, Jordan and colleagues (2013) suggested the incorporation of magnitude instruction in elementary mathematics curriculum. A retrospective review of two decades (1995-2017) of domestic research on mathematical difficulties indicates that over 70% of the articles in domestic journals focused on teaching strategies, highlighting the importance of intervention (Wang & Hung, 2019). However, practical researches accounted for less than 30%, indicating a gap between research and practice. The purposes of this study were to explore the differences in the ability to compare fraction magnitude between students with and without mathematics learning difficulties, and to use CRA strategy as intervention for students with mathematics learning difficulties to improve their fraction magnitude comparison ability.

Methods

The researcher recruited fourth and sixth graders with typical achievement (4TA and 6TA) as participants without mathematics learning difficulties, as well as sixth graders with reading disabilities (6RDMD) and with mathematics disabilities (6MD) as participants with mathematics learning difficulties. In the first part of the study, a researcher-made test was administered to measure participants’ fraction magnitude comparison ability, and the results were explained by mixed design two-factor analysis of variance. In the second section, the researcher designed intervention based on CRA strategy to teach three students with mathematics learning difficulties. The results of the study were illustrated by graphic method, visual analysis method and statistical analysis.

Results

The results are as follows. (1) There is a significant difference in the ability to compare fraction magnitude between students with and without mathematics learning difficulties. In all three subtests, the test scores of 6RDMD and 6MD are significantly lower than 6TA; test scores in fraction magnitude comparison tasks with graphics and number representation (subtest 1) and fraction magnitude comparison tasks with number representation (subtest 2) of 6MD are significantly lower than 4TA. (2) Different representations of test questions result in significant differences in the ability to compare fraction magnitude between students with and without mathematics learning difficulties. The scores of 6TA in all three subtests are not significantly different; the test scores in addition comparison task with number representation (subtest 3) of 4TA are significantly lower than in fraction magnitude comparison tasks with graphics and number representation (subtest 1) and fraction magnitude comparison tasks with number representation (subtest 2); the scores in fraction magnitude comparison tasks with number representation (subtest 2) of 6RDMD and 6MD are significantly lower than in addition comparison task with number representation (subtest 3). (3) Different student groups and different representations of test questions have a significant interactive effect on the fraction magnitude comparison ability. Students’ magnitude comparison ability in different groups shows significant differences in test questions with different representations. (4) Students with mathematics learning difficulties learned abstract concepts through physical operations and graphics. For students with mathematics learning difficulties, the CRA strategy has an immediate effectiveness on improving their fraction magnitude comparison ability. (5) Students with mathematics learning difficulties could use abstract numbers or graphics to solve fraction magnitude comparison problems. For students with mathematics learning difficulties, the CRA strategy has a short-term maintenance effect on improving their fraction magnitude comparison ability.

Conclusions

Based on the findings the researcher points out three teaching suggestions: select teaching aids carefully, encourage students’ participation in discussion and monitor individual learning proficiency. For future researches, the researcher suggested that the number and the age range of students being tested could be expanded to understand the differences between students with and without mathematical difficulties at different education level, and the topics could focus on the reasons for the difficulty of numerical magnitude based on students’ mental ability. Additionally, the effectiveness of CRA strategy on different numbers and age range of students and on different mathematics knowledge is also worth exploring.