S-Space College of Education (사범대학) Program in Special Education (협동과정-특수교육전공) Theses (Ph.D. / Sc.D._협동과정-특수교육전공)
Exploring Growth Patterns in Mathematical Ability of Students with Mathematics Difficulties
수학 학습장애 위험 아동의 수학 능력 발달 특성에 관한 연구: 수학 문장제 해결 능력과 인지 변인 간의 관계를 중심으로
- 사범대학 협동과정 특수교육전공
- Issue Date
- 서울대학교 대학원
- 학위논문 (박사)-- 서울대학교 대학원 : 협동과정 특수교육전공, 2013. 2. 김동일.
- Although many students show proficiency on assessments of general mathematics skills, many of them experience difficulties in solving word problems in mathematics. This may be attributed to a lack of the required skills necessary for the solution of these word problems. In order for students to be able to solve these problems, they first have to be able to read and understand the problem. Next, they must have the ability to translate the texts of the problem described in natural language into arithmetic operations expressed in mathematical language. Finally, they should be able to compute the equations without any error and to check that their calculations and computations are correct. In other words, word problem solving in mathematics requires the integration of computation and application knowledge with basic reading skills in terms of language comprehension.
Students with certain learning disabilities are especially less successful in word problem solving in mathematics compared to other students without learning disabilities. The former group has difficulty understanding word problems because of the fact that they lack the basic reading and computation skills, coupled usually with the condition of diminished working memory. Specifically, they are not capable of representing the relationship in the problem to some comprehensible form which can lead them to solve the problem mathematically. Moreover, students with learning disabilities use even more inefficient strategies for problem solving than typically achieving students.
The notion of learning disabilities has traditionally indicated unexpected underachievement due to a disorder in one or more of the basic psychological process or processes. This notion has been ratified within an intrinsic processing deficits model and has been used as an indicator for identifying learning disabilities. This model attempts to evaluate psychological process or capacity weakness directly, because these two variables form the basis on which learning problems are determined. Although the intrinsic processing deficits model provides the primary criterion for identifying learning disabilities, and although it is a direct approach to characterize and measure learning disabilities as opposed to the indirect methods using exclusionary clauses, it has not been a mandatory requirement for identification of learning disabilities in South Korea. That is not only because theoretical and empirical support for the notion of psychological process deficits and its influence on learning is still vague, but also because there is no agreement as to how to accurately measure cognitive ability This renders results acquired through this test invalid at best.
This research attempts to explore in a defined heterogeneous population whether students difficulties in learning mathematics is based on their growth patterns and/or cognitive abilities. The research attempts this by, first of all, exploring the possibility that students' growth trajectories might be a determinant of the mathematical learning problem as it pertains to solving word problems. . Then, the cognitive predictors of group membership are identified and the similarities and differences in the cognitive characteristics by subtypes among students with mathematical difficulties are determined. For these purposes, the following research questions were established: 1) Are there any identifiable groups within the given population whose mathematical word problem solving ability show correlation with those students growth patterns (intercept and slope) 2) Given that there are multiple growth patterns, what are the effects of students' cognitive abilities on their growth patterns in terms of word problem solving in mathematics? 3) Do cognitive abilities differ among students with mathematical difficulties identified by their growth patterns, depending on whether there is an accompanying problem in computation, in reading, in both computation and in reading, or in neither computation or reading? The implications in conjunction with the interventions, as well as some of the limitations of the study, are discussed at the end of this study.
Research Question 1 was used to examine identifiable subgroups based on growth patterns of word problem solving in mathematics. In order to explore the heterogeneity in growth trajectories of students repeatedly measured data, latent class growth analysis (LCGA) was conducted. As a result of LCGA, four distinct classes emerged based on characteristics of growth patterns (i.e. performance levels and growth rates). Class 1 (15.2%) was characterized as high intercept and slow progress. Class 2 (25.6%) was characterized as average intercept and fast progress. Class 3 (43.1%) was characterized as low intercept and slow progress. Class 4 (16.1%) was characterized as lowest intercept and little progress. The four groups classified by exploratory methods were labeled as high achieving students (HAS), average and fast growing students (AFG), low but steadily growing students (LSG), and students with mathematics difficulties (SDD) respectively.
Research Question 2 was used to examine the relationship between growth patterns of word problem solving and cognitive abilities in mathematics. To investigate this relationship between students' learning progress in word problem solving and their cognitive abilities, a growth mixture modeling approach was used. In the case of setting up a HAS group as a reference, the lower the students' working memory ability is, the higher the possibility for students to be categorized into an AFG group (odds ratio=0.570) and a LSG group (0.582). Also, when the values of a processing speed and a language decrease (odds ratio=0.344 and 0.477, respectively), the probability of being in a SDD group increases. If the AFG group is set up as a reference, the lower the students' processing speed (odds ratio=0.664), the higher the possibility for students to be classified into a LSG group. The estimates of the multinomial logistic regression show that the probability of being in a SDD group increases as the values of a processing speed and a language decreasing (odds ratio=0.340 and 0.540, respectively). When a LSG group is the reference group, having lower processing speed (odds ratio=0.513) and language (odds ratio=0.640) increased the estimated odds of being a SDD group student compared to other groups. When a SDD group is set up as a reference, when the values of a processing speed (odds ratio=1.950) and a language (odds ratio=1.562) increase, the probability that students belong to a LSG group increases. Attention and nonverbal reasoning are not related to group contrasts.
In research Question 3, the different growth patterns and cognitive abilities among subtypes of students with difficulties in word problem solving were explored. Based on the hypothesis that students would have different cognitive characteristics depending on whether they have difficulties in computation, in reading, in both computation and reading, or in neither computation nor reading, multivariate analysis of variance (MANOVA) was used to explore their differences on cognitive abilities. For group formation, 25th percentile and 40th percentile were selected as cutoffs and four subgroups were identified by difficulty status – word problem solving difficulty (PD), computational difficulty with word problem solving difficulty (CPD), reading difficulty with word problem solving difficulty (RPD), computational and reading difficulty with word problem solving difficulty (CRPD). Observed mean trajectories revealed CPD, RPD, and CRPD showed significantly lower growth levels compared to PD. CRPD showed the lowest growth levels among the four groups. Significant differences between PD and RPD were found in working memory, processing speed, language, and nonverbal reasoning. Working memory, processing speed, and language between PD and CRPD, and working memory and language between CPD and RPD/CRPD were significantly different. No difference in attention was found in any contrast. In neither PD versus CPD, nor RPD versus CRPD, were cognitive profile differences found.