This study investigated the effects of a fraction concept understanding intervention with six fifth-grade students with special educational needs in mathematics. Explicit instruction and CRA were cornerstones in the intervention program, with a specific focus of fraction as measurement. A single case study with a multiple baseline across student design was used. The results revealed a functional relation between intervention and fraction concept knowledge in four cases. However, all six students maintained their scores on the fraction magnitude understanding measure at a 6-week follow-up. Additionally, all participating students improved their scores on a generalization measure of fraction concept knowledge, assessed both before and after the intervention. The findings reveal that a series of nine lessons fraction intervention supports fraction learning for students with diverse characteristics, but with similar mathematical struggles and special educational needs in mathematics.

Numerical cognition constitutes a set of hierarchically related skills and abilities that develop-and may subsequently begin to decline-over developmental time. An innate "number sense" has long been argued to provide a foundation for the development of increasingly complex and applied numerical cognition, such as symbolic numerical reference, arithmetic, and financial literacy. However, evidence for a direct link between basic perceptual mechanisms that allow us to determine numerical magnitude (e.g., "how many" objects are in front of us and whether some of these are of a "greater" or "lesser" quantity), and later symbolic applications for counting and mathematics, has recently been challenged. Understanding how one develops an increasingly precise sense of number and which neurocognitive mechanisms support arithmetic development and achievement is crucial for developing successful mathematics curricula, supporting individual financial literacy and decision-making, and designing appropriate intervention and remediation programs for mathematical learning disabilities as well as mathematics anxiety. The purpose of this review is to provide a broad overview of the cognitive, neural, and affective underpinnings of numerical cognition-spanning the earliest hours of infancy to senior adulthood-and highlight gaps in our knowledge that remain to be addressed. (c) 2025 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

The core assumptions of the integrated theory of numerical development were tested using a cross-sectional sample of 379 fifth-grade students, who were assessed on their understanding of whole numbers and fractions, including arithmetic skills. An additional factor of part-whole knowledge was also assessed and analyzed in four structural equation models. All models showed non-significant chi-square values and good fit indices according to common modelling benchmarks. As hypothesized, whole number magnitude knowledge predicts fraction magnitude knowledge and arithmetic proficiency. Additionally, part-whole knowledge mediates numerical magnitude understanding but does not predict arithmetic outcomes. These findings are in line with the integrated theory of numerical development, suggesting that numerical magnitude knowledge is crucial for both numerical development and arithmetic learning. Our findings also reveal that part-whole knowledge may serve as a bridge to more advanced magnitude understanding. Theoretical and practical implications are discussed.

The intention of the study was to examine the effects of a fraction intervention in a whole-class environment. The intervention aimed to enhance students conceptual fraction knowledge, with a major focus on fraction magnitude understanding. This study included 120 fifth-grade students in standard classroom settings. Utilizing a cluster randomized controlled trial design, students were divided into either an intervention group (n = 64) or a control group (n = 56). Students in the intervention condition received a series of seven 35-minute lessons. Students in the control condition received "treatment as usual". Both post-test and delayed post-test results revealed that students in the intervention group performed significantly better than those in the control group on fraction concepts, with a stronger effect in measurement aspects compared to part-whole aspects. The intervention group also outperformed the control group on fraction arithmetic on both post-tests, while no significant difference was observed on fraction word problems.

Numerical cognition can take place in multiple representational formats, such as Arabic digits (e.g., 1), verbal number words (e.g., “two”), and nonsymbolic (e.g., •••) numerical magnitude. Basic numerical discrimination abilities are key factors underlying the development of arithmetic abilities, acting as an important developmental precursor of adult-level numeracy. While prior research has begun to detail the neural correlates associated with basic numerical discrimination skills in different representational formats, the interactions between functional neural circuits are less understood. A growing body of evidence suggests that the functional networks recruited by number discrimination tasks differ between children and adults, which may provide valuable insights into the development of numerical cognition. To this end, we posed two questions: how do the interactions between functional circuits associated with number processing differ in children and adults? Are differences in functional network connectivity modulated by numerical representational codes? A theoretically motivated 22 ROI analysis indicated significant functional connectivity differences between children and adults across all three codes. Adults demonstrated sparser and more consistent connectivity patterns across codes, indicative of developmental domain-specialization for number processing. Although neural activity in children and adults is similar, the functional connectivity supporting number processing appears subject to substantial developmental maturation effects.

A modified pathways to mathematics model was used to examine the cognitive mechanisms underlying arithmetic skills in third graders. A total of 269 children were assessed on tasks tapping the four pathways and arithmetic skills. A path analysis showed that symbolic number processing was directly supported by the linguistic and approximate quantitative pathways. The direct contribution from the four pathways to arithmetic proficiency varied; the linguistic pathway supported single-digit arithmetic and word problem solving, whereas the approximate quantitative pathway supported only multi-digit calculation. The spatial processing and verbal working memory pathways supported only arithmetic word problem solving. The notion of hierarchical levels of arithmetic was supported by the results, and the different levels were supported by different constellations of pathways. However, the strongest support to the hierarchical levels of arithmetic were provided by the proximal arithmetic skills.

Context: The aim was to further understand the heterogeneity of  developmental dyscalculia (DD). Utilizing four children (8-9 year-old) performance was contrasted against predominant hypotheses of DD.

Case report: Despite showing similar mathematical deficits, these children showed remarkable interindividual variability regarding cognitive profile and deficits. Two cases were consistent with the approximate number system deficit account, and the general magnitude-processing deficit account. One case had an access deficit in combination with a general cognitive deficit. One cases suffered from general cognitive deficits only.

Conclusions: The results showed that DD cannot be attributed to a single explanatory factor. These findings support a multiple deficits account of DD and suggest that some cases have multiple deficits, whereas other cases have a single deficit. We discuss a previously proposed distinction between primary DD and secondary DD, and suggest hypotheses of dysfunctional neurocognitive correlates responsible for the displayed deficits.

Background

Developing sufficient mathematical skills is a prerequisite to function adequately in society today. Given this, an important task is to increase our understanding regarding the cognitive mechanisms underlying young people's acquisition of early number skills and formal mathematical knowledge.

Aims

The purpose was to examine whether the pathways to mathematics model provides a valid account of the cognitive mechanisms underlying symbolic-number processing and mathematics in adolescents. The pathways model states that the three pathways should provide independent support to symbolic-number skill. Each pathway's unique contribution to formal mathematics varies depending on the complexity and demand of the tasks.

Sample

The study used a sample of 114 adolescents (71 girls). Their mean age was 14.60 years (SD = 1.00).

Methods

The adolescents were assessed on tests tapping the three pathways and general cognitive abilities (e.g., working memory). A structural equation path analysis was computed.

Results

Symbolic-number comparison was predicted by the linguistic pathway, the quantitative pathway, and processing speed. The linguistic pathway, quantitative pathways, and symbolic-number comparison predicted arithmetic fact retrieval. The linguistic pathway, working memory, visual analogies, and symbolic-number comparison predicted percentage calculation.

Conclusions

There are both similarities and differences in the cognitive mechanisms underlying arithmetic fact retrieval and percentage calculation in adolescents. Adolescents’ symbolic-number processing, arithmetic fact retrieval, and percentage calculation continue to rely on the linguistic pathways, whereas the reliance upon the spatial pathway has ceased. The reliance upon the quantitative pathway varies depending on the task.