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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.

The importance of fraction knowledge to later mathematics achievement, along with U.S. students’ poor knowledge of fraction concepts and procedures, has prompted research on the development of fraction learning. In the present study, participants’ (N = 536) development of fraction magnitude understanding and fraction arithmetic skills was assessed over 4 time points between 4th and 6th grades. Latent state-trait modeling was used to examine codevelopment of these 2 areas of fraction knowledge. Fraction arithmetic skill predicted later fraction magnitude understanding, and conversely, fraction magnitude understanding predicted later fraction arithmetic skill. The results are consistent with a bidirectional model of the development of fraction concepts and procedures, in which knowledge of one type facilitates learning of the other type. However, transfer in both directions between fraction arithmetic skill and fraction magnitude understanding was more likely to occur later in the development of fraction knowledge, after fraction arithmetic with unlike denominators had been taught in school (during 5th grade in the current sample). Furthermore, the effects of previous knowledge of the other type were small and not nearly as substantial as the effects of previous knowledge on later knowledge of the same type. Findings suggest a need for instruction to link fraction magnitude understanding to fraction arithmetic skill and vice versa.

This study investigated the accuracy of three fraction measures (i.e., fraction number line estimation accuracy, general fraction concepts, and fraction arithmetic) for screening fourth graders who might be at risk for mathematics difficulties. Receiver operating characteristic (ROC) curve analyses assessed diagnostic accuracy of the fraction measures for predicting which students would not meet state standards on the state mathematics test in fourth grade ( n = 411), fifth grade ( n = 362), and sixth grade ( n = 304). A combined measure consisting primarily of fraction number line estimation items and general fraction concept items was the most accurate screener of risk status in fourth, fifth, and sixth grades (area under the curve [AUC] = .84, .81, and .85, respectively). To maximize efficiency for classroom use, the length of the combined screener was reduced using best subset automatic linear modeling. The study highlights the importance of fraction knowledge for predicting mathematics achievement more generally and validates an effective and practical screening tool for the intermediate grades.

The current study investigated the mediating role of number-related skills in the developmental relationship between early cognitive competencies and later fraction knowledge using structural equation modeling. Fifth-grade numerical skills (i.e., whole number line estimation, non-symbolic proportional reasoning, multiplication, and long division skills) mapped onto two distinct factors: magnitude reasoning and calculation. Controlling for participants’ (N = 536) demographic characteristics, these two factors fully mediated relationships between third-grade general cognitive competencies (attentive behavior, verbal and nonverbal intellectual abilities, and working memory) and sixth-grade fraction knowledge (concepts and procedures combined). However, specific developmental pathways differed by type of fraction knowledge. Magnitude reasoning ability fully mediated paths from all four cognitive competencies to knowledge of fraction concepts, whereas calculation ability fully mediated paths from attentive behavior and verbal ability to knowledge of fraction procedures (all with medium to large effect sizes). These findings suggest that there are partly overlapping, yet distinct, developmental pathways from cognitive competencies to general fraction knowledge, fraction concepts, and fraction procedures.

The present study examined predictors of student's knowledge of fraction concepts and procedures in sixth grade (N = 334). Predictors included both math-specific and more general competencies, which were assessed in fifth grade. Multiple regression analyses showed that whole number line estimation, non-symbolic proportional reasoning, long division, working memory, and attentive behavior contributed uniquely to a general measure of students’ fraction concepts; on a measure of fraction procedures, whole number line estimation, multiplication fact fluency, division, and attention made unique contributions. The combined predictability of the measures was lower for fraction procedures than for fraction concepts. Although the unique predictors and the amount of explained variance differed according to the fraction outcome, the ability to locate whole numbers on the number line was a major contributor to prediction in each model. Non-symbolic proportional reasoning was particularly predictive of children's conceptual understanding of fractions.

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.

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 introduction of fractions into the curriculum. Implications for helping students with mathematics difficulties are discussed.

Developmental predictors of children’s fraction concepts and procedures at the end of fourth grade were investigated in a 2-year longitudinal study. Participants were 357 children who started the study in third grade. Attentive behavior, language, nonverbal reasoning, number line estimation, calculation fluency, and reading fluency each contributed uniquely to later conceptual understanding of fractions. Number line estimation, attentive behavior, calculation fluency, and working memory made unique contributions to acquisition of fraction arithmetic procedures. Notably, number line estimation made the largest independent contribution in both models. The results suggest that although there is considerable shared variance among the predictors, both general and number-related competencies are uniquely important for explaining why some children struggle with fractions.