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