Limited knowledge of fraction representations differentiates middle school students with mathematics learning disability (dyscalculia) versus low mathematics achievement (original) (raw)
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Journal of learning disabilities, 2016
The goal of the present article is to synthesize findings to date from the Delaware Longitudinal Study of Fraction Learning. The study followed a large cohort of children (N = 536) between Grades 3 and 6. The findings showed that many students, especially those with diagnosed learning disabilities, made minimal growth in fraction knowledge and that some showed only a basic grasp of the meaning of a fraction even after several years of instruction. Children with low growth in fraction knowledge during the intermediate grades were much more likely to fail to meet state standards on a broad mathematics measure at the end of Grade 6. Although a range of general and mathematics-specific competencies predicted fraction outcomes, the ability to estimate numerical magnitudes on a number line was a uniquely important marker of fraction success. Many children with mathematics difficulties have deep-seated problems related to whole number magnitude representations that are complicated by the i...
Developmental Science, 2008
Many middle-school students struggle with decimals and fractions, even if they do not have a mathematical learning disability (MLD). In the present longitudinal study, we examined whether children with MLD have weaker rational number knowledge than children whose difficulty with rational numbers occurs in the absence of MLD. We found that children with MLD failed to accurately name decimals, to correctly rank order decimals and/or fractions, and to identify equivalent ratios (e.g. 0.5 =); they also 'identified' incorrect equivalents (e.g. 0.05 = 0.50). Children with low math achievement but no MLD accurately named decimals and identified equivalent pairs, but failed to correctly rank order decimals and fractions. Thus failure to accurately name decimals was an indicator of MLD; but accurate naming was no guarantee of rational number knowledge-most children who failed to correctly rank order fractions and decimals tests passed the naming task. Most children who failed the ranking tests at 6th grade also failed at 8th grade. Our findings suggest that a simple task involving naming and rank ordering fractions and decimals may be a useful addition to in-class assessments used to determine children's learning of rational numbers.
Early predictors of middle school fraction knowledge
Developmental Science, 2014
Recent findings that earlier fraction knowledge predicts later mathematics achievement raise the question of what predicts later fraction knowledge. Analyses of longitudinal data indicated that whole number magnitude knowledge in first grade predicted knowledge of fraction magnitudes in middle school, controlling for whole number arithmetic proficiency, domain general cognitive abilities, parental income and education, race, and gender. Similarly, knowledge of whole number arithmetic in first grade predicted knowledge of fraction arithmetic in middle school, controlling for whole number magnitude knowledge in first grade and the other control variables. In contrast, neither type of early whole number knowledge uniquely predicted middle school reading achievement. We discuss the implications of these findings for theories of numerical development and for improving mathematics learning.
Fraction Development in Children: Importance of Building Numerical Magnitude Understanding
Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts
This chapter situates fraction learning within the integrated theory of numerical development. We argue that the understanding of numerical magnitudes for whole numbers as well as for fractions is critical to fraction learning in particular and mathematics achievement more generally. Results from the Delaware Longitudinal Study, which examined domain-general and domain-specific predictors of fraction development between third and sixth grade, are highlighted. The findings support an approach to teaching fractions that centers on a number line. Implications for helping struggling learners are discussed.
Learning and Individual Differences, 2017
Fraction magnitude understanding is linked to student achievement in mathematics, but the direction of the relation is not clear. To assess whether fraction magnitude knowledge and mathematics achievement develop in a bidirectional fashion, participants (N = 536) completed a standardized mathematics achievement test and two measures of fraction magnitude understanding-fraction comparisons and fraction number line estimation (FNLE)-twice yearly in 4th-6th grades. Cross-lagged panel models revealed significant autoregressive paths for both achievement and magnitude knowledge, indicating longitudinal stability after accounting for correlational and cross-lagged associations. Mathematics achievement consistently predicted later FNLE and fraction comparison performance. FNLE and fraction comparisons predicted mathematics achievement at all time points, although this relation diminished over time. Findings suggest that fraction magnitude knowledge and broader mathematics achievement mutually support one another. FNLE predicted subsequent mathematics achievement more strongly than did fraction comparisons, possibly because the FNLE task is a more specific measure of fraction magnitude understanding.
Sources of Individual Differences in Children's Understanding of Fractions
Child Development, 2014
The study assessed the relations among acuity of the inherent approximate number system (ANS), performance on measures of symbolic quantitative knowledge, and mathematics achievement for a sample of 138 (64 boys) preschoolers. The Weber fraction (a measure of ANS acuity) and associated task accuracy were significantly correlated with mathematics achievement following one year of preschool, and predicted performance on measures of children's explicit knowledge of Arabic numerals, number words, and cardinal value, controlling for age, sex, parental education, intelligence, executive control, and preliteracy knowledge. The relation between ANS acuity, as measured by the Weber fraction and task accuracy, and mathematics achievement was fully mediated by children's performance on the symbolic quantitative tasks, with knowledge of cardinal value emerging as a particularly important mediator. The overall pattern suggests that ANS acuity facilitates the early learning of symbolic quantitative knowledge and indirectly influences mathematics achievement through this knowledge.
Students with Mathematics Learning Disabilities and Their Ways of Thinking in Fraction Learning
Learning Disabilities [Working Title]
This chapter presents the result of research on ways of thinking of students with mathematics learning disabilities in fraction learning. We conducted a class of fraction learning with Lesh translation model. From the class discussion, interview, and students' work, we then explore the students' ways of thinking when they learn fraction. In the class, students with mathematics learning disabilities perform two mental acts with corresponding ways of thinking and ways of understanding; those are interpreting and problem-solving. We find some interesting findings and they are: (1) students know the common denominator method in the addition of fractions; however, they incorrectly apply the method; (2) students use the common denominator approach (for fraction addition) in the multiplication of fraction; and (3) in the division of fraction, students mistakenly apply the invert multiply algorithm.
Journal of Educational Psychology, 2012
First- to fifth-grade mathematics and word reading achievement were assessed for children with mathematical learning disability (MLD, n = 16), persistent low achievement (LA, n = 29), and typical achievement (n = 132). Intelligence, working memory, processing speed, and in-class attention were assessed in 2 or more grades, and mathematical cognition was assessed with experimental tasks in all grades. The MLD group was characterized by low school-entry mathematics achievement and poor word reading skills. The former was mediated by poor fluency in processing or accessing quantities associated with small sets of objects and corresponding Arabic numerals, whereas the latter was mediated by slow automatized naming of letters and numbers. Both the MLD and LA groups showed slow across-grade growth in mathematics achievement. Group differences in growth were mediated by deficits or delays in fluency of number processing, the ability to retrieve basic facts from long-term memory and to deco...
Journal of Learning Disabilities, 2019
Many children in the United States currently perform below proficient levels in the area of mathematics, with even more bleak outcomes for students with disabilities. For example, on the most recent National Assessment of Educational Progress (NAEP; 2015), only 14% of students with disabilities were proficient and 2% were advanced on the fourthgrade mathematics assessment. On the eighth-grade assessment, that number had decreased to 7% proficient and 1% advanced. The National Mathematics Advisory Panel (NMAP; 2008) cited the critical mathematics foundations of number sense, fractions, geometry, and measurement as key components to master in presecondary course work. Likewise, early proficiency with mathematics skills is vital for later mathematics success (e.g., algebra; Siegler et al., 2012). Scholars consider fraction knowledge to be a critical area of need when considering that many students struggle with these concepts and that future course work demands a certain level of automaticity with fractions (e.g., Siegler et al., 2012). Students with and at risk for mathematics disabilities are broadly defined as students with and at risk for learning disabilities in the area of math (e.g., dyscalcula). This group may include students in general education as well as those receiving special education services (Berch, 2016). Likewise, many students with difficulties in the area of math also have behavioral deficits and excesses that affect their learning and classroom performance (Johns, Crowley, & Guetzloe, 2008; Wagner & Cameto, 2004). According to Geary, Hoard, Byrd-Craven, Nugent, and Numtee (2007), students with mathematics disabilities often have deficits in processing of number sets, estimation tasks with number lines, retrieval of simple and complex arithmetic from long-term memory, accurate use of backup strategies, and working memory and processing speeds. Fraction knowledge is predicated on several, if not all, of these prerequisite mathematics skills. Automaticity in basic number sense (e.g., fact fluency), expanded knowledge of the number line (including all rational numbers), and working memory affect success with fractions, are weaknesses of many students with disabilities, and may prevent a student from properly calculating fractions (Fuchs et al., 2016). This is analogous to reading in that limited foundational skills (e.g., phonemic awareness) hinder later comprehension skills as working memory may be used up on decoding instead of comprehending the 859509L DXXXX10.