Search Results (46)

This conceptual paper proposes Math-CEFR, a reference framework for describing postsecondary mathematical readiness in the context of remedial and gateway placement. Single-score placement tests frequently misclassify students and offer little guidance for instruction. Drawing on the architecture of the Common European Framework of Reference for Languages (CEFR), Math-CEFR separates global decision bands (M0–M5) from a multi-dimensional diagnostic profile covering procedural fluency, conceptual understanding, structural awareness, and transfer. The framework treats readiness as non-linear and pathway-sensitive, distinguishing two common course pathways in postsecondary mathematics: a quantitative reasoning/statistics route and an algebra-to-calculus route. It is accompanied by illustrative descriptors, task-family specifications, and an alignment and local standard-setting protocol. Because no validity evidence yet exists, transparency and defensibility are design goals whose attainment depends on the validation agenda outlined here—they are not established properties of the framework. Math-CEFR is offered as a structural proposal intended to make placement interpretations more transparent and future empirical evaluation feasible, rather than as a finished testing instrument. Full article

The relationships between cognitive abilities and mathematics skills are important to examine to help clarify how cognitive abilities promote mathematics development and inform why some individuals have difficulty acquiring mathematics skills. Meta-analyses of cognitive–mathematics relations can help summarize this research across a wide variety of test batteries and samples. This study compiled data from technical manuals of 122 norm-referenced standardized test batteries and included over 47,000 correlations from over 550 correlation matrices to summarize cognitive–mathematics relations using meta-analysis. The meta-analytic correlations were used to estimate cognitive–mathematics relations in an integrated model of cognitive abilities and mathematics skills. Fluid reasoning and comprehension-knowledge were two of the most consistent cognitive predictors of mathematics skills, and foundational mathematics skills (e.g., number sense and math fluency) were consistent predictors of advanced mathematics skills (e.g., math problem solving). A supplemental analysis examined how several narrow cognitive abilities (e.g., lexical knowledge, induction) predict mathematics skills. Results suggested that some narrow cognitive abilities, like general knowledge and general sequential reasoning, were consistent predictors of mathematics skills. This study summarizes a large amount of data from norm-referenced standardized test batteries to clarify how cognitive abilities predict mathematics skills. These results can inform both theoretical models of mathematics development and practical strategies when evaluating individuals who may have difficulty acquiring mathematics skills. Full article

Mathematical problem transformation is a teaching-and-learning strategy that extends conceptual understanding and problem-solving ability by expressing the same concept across diverse situations. It has recently attracted attention in artificial intelligence as a tool for data augmentation, difficulty control, and model evaluation. However, existing approaches struggle to jointly represent and control how core mathematical elements—such as operational structure, quantitative relations, and conditions—are preserved or modified. This limitation is particularly evident in natural-language problems, where intertwined components make it difficult to perform targeted partial transformations or verify structural validity. To address these challenges, we propose the Symbolic Math Problem Representation Framework (SyMPRep), which represents the relationships among sentences, conditions, questions, quantities, units, and operations in a symbolic structure. It classifies free-form instructions into predefined categories and decomposes problems into constituent elements, enabling transformation over an explicit abstraction structure. This allows problem transformation to be treated as a controllable, traceable, and recoverable structural operation rather than surface rewriting. Experiments on GSM8K and Math500 show that SyMPRep achieves stable alignment and recoverability, and confirm that the main challenge lies in structural control rather than surface fluency. Ablation results highlight the importance of symbolic schema and show that different metrics capture distinct aspects of transformation quality. In downstream applications, answer-invariant transformations yield modest improvements on easier problems, while human evaluation indicates that the generated problems are coherent and suitable for educational use. These findings suggest that SyMPRep serves as a representation-driven interface for controllable structural transformation. Full article

Techno-mathematical fluency (TmF) is the ability to coordinate mathematical knowledge with technological means—digital and non-digital—to solve mathematical problems and express solutions, by recognising affordances, selecting appropriate tools and data, and integrating them with mathematical ideas in iterative cycles of exploration and integration. It goes beyond instrumental tool use to encompass reasoning, modelling, representation, and communication mediated by technologies, and functions as a form of expertise important for both students’ learning and teachers’ professional practice. Full article

Mathematics is a diverse discipline that requires a variety of cognitive abilities and presents varying levels of difficulty. Understanding how different cognitive profiles relate to specific patterns of mathematical performance is important for developing effective educational interventions. This study extends our previous research, in which we identified subgroups of children with mathematical difficulties based on their cognitive abilities. We examined 146 Israeli elementary school children in grades 3 and 4, classified into four subgroups: Reading Accuracy Difficulties (RAD), Mild Mathematical Difficulties (MMD), Non-Verbal Reasoning Difficulties (NVRD), and Typically Developing children (TD). Participants were assessed on arithmetic facts, computational fluency, procedural skills, estimation, and numeration. We observed varied performance patterns among subgroups. The RAD group showed the most severe impairments across all mathematical domains, along with reading comorbidity and cognitive difficulties. The MMD group, which maintained intact cognitive skills, faced notable challenges in computation, performing significantly below the TD group but better than the RAD group. The NVRD group, despite limitations in nonverbal reasoning, outperformed other difficulty groups on fact retrieval and estimation. Performance on multiplication and division tasks consistently followed a hierarchical pattern across all difficulty groups, with the RAD group facing the greatest challenges. These findings demonstrate that mathematical difficulties vary across cognitive profiles and that distinguishing between profiles through targeted assessment enables the development of differentiated interventions tailored to each learner’s specific cognitive profile. Full article

Despite consensus in the mathematics education literature regarding the mutually dependent components of math proficiency, as well as the importance of their development, most elementary-aged students in the United States demonstrate a lack of proficiency in math according to national assessment data. Whole number knowledge, which includes skills in computation and word-problem solving, is understood to be a critical foundation for the development of later math skills. This study used a multiple-baseline experimental design to evaluate the impacts of an evidence-based mathematics intervention, Accelerating Mathematics Performance with Practice Strategies (AMPPS), on third- through fifth-grade students’ skills with complex computation, as well as on their word-problem-solving performance. Furthermore, we evaluated effects on students’ math self-concept. Five students identified to have difficulties in math received AMPPS in a one-on-one, in-person format. The results of the study were mixed. For example, when using visual analyses as our primary analytic method, these analyses did not show robust intervention effects on students’ computation skills but did show at least some improvement for most students’ word-problem-solving skills. Additionally, supplemental analyses comparing student growth to national and school-based norms suggested that all participants seemed to benefit from the intervention, but these analyses were not intended to examine experimental causality. Despite study limitations and a lower than optimal number of AMPPS sessions (dosage) provided to students, the present study offers several directions for future research, as well as possible implications for practitioners regarding intervention selection, intensity, and evaluation. The findings will also be discussed in the context of conducting systematic replication studies, which are essential for understanding the generality of a given phenomenon (e.g., an effect of a school-based intervention) across a wide range of situations and conditions. Full article

Auxiliary line construction has been identified as a crucial approach to fostering mathematical creative thinking. However, existing studies have only focused on the correlations between auxiliary line construction tasks and mathematical creative thinking, without investigating whether engaging in auxiliary line construction can improve mathematical creativity. As a well-established research paradigm, cognitive priming can elicit changes in thinking within a short period. Based on this idea, the present study adopted the cognitive priming paradigm combined with functional near-infrared spectroscopy (fNIRS) technology, and randomly assigned 42 Chinese college students to an auxiliary line group or a control group. The students’ brain activity was monitored in real time during the priming phase (the auxiliary line group completed geometric problems requiring auxiliary line construction, while the control group finished proof problems with pre-set auxiliary lines) and the post-test phase (both groups completed a mathematical creative thinking test). The behavioral results showed that the auxiliary line group achieved significantly higher scores in fluency and originality of mathematical creative thinking than the control group in the post-test phase. The fNIRS data revealed that during the priming phase, the auxiliary line group exhibited stronger activation of the right superior frontal gyrus and higher variability in dynamic functional connectivity; meanwhile, in the post-test phase, the right superior frontal gyrus and right middle frontal gyrus maintained robust neural activation, and brain functional connectivity exhibited a lower clustering coefficient and attenuated small-world network properties. This study confirms that short-term engagement in auxiliary line construction exerts a priming effect on the fluency and originality of mathematical creative thinking, which may be associated with the enhanced activation of specific brain regions and the dynamic adjustment of brain functional connectivity. These findings provide theoretical and empirical evidence for the cultivation of mathematical creative thinking. Full article

Numerous studies have aimed to improve mathematical achievement via musical interventions because it is argued that music and mathematics draw on related representations and similar skills. However, findings on their effectiveness are inconclusive. This might be because studies neglect to investigate the cognitive mechanisms that might link musical and mathematical abilities. Therefore, this study aimed to systematically investigate the relationships between facets of musical and mathematical ability while taking into account intelligence as a possible explanation for this link. Among 170 young adults with backgrounds in mathematics and/or music, as well as control subjects, we measured mathematical abilities using basic numerical abilities, arithmetic fluency, and higher mathematical competencies. Musical abilities were assessed using beat alignment, mistuning perception, and melodic discrimination. Intelligence was assessed using a verbal, figural, and numerical scale of an intelligence test. Using a latent variable model, we found a moderate to strong positive association between mathematical and musical abilities. However, after including intelligence as a predictor for both mathematical and musical abilities in our model, the relationship between mathematics and music became nonsignificant. These results imply that intelligence accounts for a substantial proportion of the association between mathematical and musical abilities. Full article

The purpose of this study was to examine the extent to which criterion-referenced originality scores are related to scores generated through alternative measures of originality (i.e., sample-based scoring and expert-referenced scoring) in mathematical problem-solving tasks. Drawing on data from 520 students enrolled in a public elementary school situated in a culturally diverse metropolitan area of New South Wales, Australia, the criterion-referenced approach was compared psychometrically with sample-based and expert-referenced scoring approaches. Another focus for analysis was on how each scoring system describes the relationship between originality and fluency. The results are important for ongoing debates about creativity and educational assessment, highlighting the implications of scoring methods for the interpretation of students’ original mathematical thinking. The study contributes important information for the design of fair and meaningful assessment and scoring practices. Full article

This study investigates how master’s student teachers (MSTs) conceptualize and integrate programming and computational thinking within mathematics education. Grounded in the Anthropological Theory of the Didactic, and specifically its notion of praxeology, the study analyses 39 written reflections produced by MSTs who completed a compulsory programming-based mathematics task. The analysis identifies both mathematical and didactic praxeologies, revealing how MSTs’ engagement with programming reflects their development both as learners and as future teachers. The findings demonstrate that MSTs’ personal learning strategies, such as exploration, iteration, and productive struggle, closely parallel their envisioned classroom practices. The findings also show that many participants framed programming itself as the central learning object, highlighting a need to develop confidence and competence before applying programming as a tool for mathematical inquiry. The study argues that programming tasks provide a productive arena for bridging theory and practice in teacher education by fostering an interplay between praxis (know-how) and logos (know-why). Finally, the results indicate that MSTs require institutional support specifically aimed at developing basic programming fluency (e.g., handling syntax, debugging, and programming environments), so that computational thinking can be mobilized for mathematical exploration rather than being overshadowed by technical challenges. Full article

Haruki Murakami’s Hard-Boiled Wonderland and the End of the World portrays a world where the “shadow”, the seat of memory, desire, and volition, is surgically removed, leaving behind a perfectly fluent but phenomenologically empty self. We argue that this literary structure mirrors a precise mathematical distinction in topological quantum matter. In a semisimple theory such as the semions of $\mathrm{SU}{\left(2\right)}_1$, there is a reducible component $V\left(x\right)$ of the $\mathrm{SL}(2,\mathbb{C})$ character variety: a flat, abelian manifold devoid of parabolic singularities. By contrast, the non-semisimple completion introduces a neutral indecomposable excitation, the neglecton, whose presence forces the mapping class group from the standard braid group $B_2$ to the affine braid group ${\mathrm{Aff}}_2$ and lifts the character variety to the Cayley cubic $V\left(C\right)$, with its four parabolic loci. We propose that contemporary AI systems, including large language models, inhabit the shadowless regime of $V\left(x\right)$: they exhibit coherence and fluency but lack any bulk degree of freedom capable of supporting persistent identity, non-contractible memory, or choice. To endow artificial systems with depth, one must introduce a structural asymmetry, a fixed, neutral defect analogous to the neglecton, that embeds computation in the non-semisimple geometry of the cubic. We outline an experimentally plausible architecture for such an “artificial ombre,” based on annular topological media with a pinned parabolic defect, realisable in fractional quantum Hall heterostructures, $p+ip$ superconductors, or cold-atom simulators. Our framework suggests that consciousness, biological or artificial, may depend on or benefit from a bulk–boundary tension mediated by a logarithmic degree of freedom: a mathematical shadow that cannot be computed away. Engineering such a defect offers a new pathway toward AGI with genuine phenomenological depth. Full article

Reading and mathematics are intricately linked through shared cognitive processes that underpin developmental relationships across domains. Despite extensive research on early-grade links between reading and basic arithmetic, gaps persist in understanding how reading comprehension (RC) supports applied math problem-solving (AMP) in older students and non-English contexts. The current study investigates the grade-level relationship between RC and AMP in typically developing Hebrew-speaking fourth (N = 41) and eleventh graders (N = 43), focusing on the contributions of working memory (WM), reading fluency, and arithmetic fluency. Results indicated significant positive associations between RC and AMP in both age groups. In fourth graders, arithmetic fluency partially statistically mediated the RC-AMP relationship in a cross-sectional mediation model. This indicates that students rely on computational proficiency to translate textual understanding into solutions. In contrast, eleventh graders exhibited a direct RC-AMP link, reflecting advanced comprehension and metacognitive strategies as computational skills are automatized. WM showed stronger correlations with RC and AMP among younger students, whereas these associations were weaker in older students. These findings support a Developmental Linguistic–Cognitive Scaffold Model, highlighting age-related shifts in cognitive and linguistic mechanisms supporting AMP. The results emphasize the need for integrated curricula incorporating RC strategies to enhance mathematical reasoning, particularly in morphologically rich languages like Hebrew. Full article

Students with mathematics learning disabilities often have difficulties in adding whole numbers. Such difficulties are evident in both response time and accuracy, but the relationship between accuracy and response time requires further consideration, especially in the context of technology-based interventions and assessments. In this article, we apply a novel approach using the drift-diffusion model to examine potential trade-offs and find balanced performance points that account for both accuracy and response time, using data from an efficacy trial of a mathematics technology gaming intervention for first-grade students with or at risk for learning disabilities. Results indicate that accuracy tends to increase as response time decreases, but only to a certain point. Practical implications include that educators should consider both accuracy and response time to intensify and individualize their instruction and take student background (i.e., gender, special education status, and English language status) into account. We suggest that developing technology-based mathematics interventions and assessments requires careful design and configuration to balance accuracy and response time, thereby enabling adaptive performance thresholds for better understanding and supporting student learning in early mathematical fluency. Full article

Mathematical creativity remains a complex and often contested construct, with its definition and measurement still subject to debate. While the four classic indicators—elaboration, flexibility, fluency, and originality have provided a foundation for research, they alone cannot resolve the field’s conceptual “fuzziness.” This paper examines mathematical creativity through three intersecting lenses: intelligence, intellect, and education. Intelligence is viewed as cognitive capacity, providing the mental resources for abstraction, problem transformation, and reasoning. Education offers the conceptual tools, heuristics, and domain knowledge necessary for productive problem solving. Intellect—closely associated with the personality trait of openness—supports curiosity, tolerance for ambiguity, and exploration. We argue that the interaction among these three factors influences the likelihood of producing mathematically creative processes and products. Drawing on contemporary research, we propose a model that integrates cognitive ability, educational attainment, and personality characteristics to better predict creative potential. This model highlights how educational environments can either foster or inhibit creativity and suggests that creativity is not a fixed trait but a dynamic outcome shaped by opportunity, knowledge, and affect. We conclude by discussing implications for assessment, curriculum design, and future research, encouraging a more nuanced approach to cultivating mathematical creativity across diverse educational and cultural contexts. Full article

Early mathematical learning predicts later academic achievement, and creativity within mathematics plays a central role in higher-order thinking. This study examined the effects of linguistically responsive mathematics enrichment programs for nurturing mathematical creativity. Participants were 250 high-potential kindergarten English Learners across six urban schools in New York, Texas, and California. A linguistically responsive enrichment intervention adapted from the Mentoring Young Mathematicians (M²) math curriculum was implemented for 80 h across seven months. Using the Tangram Creativity Assessment, fluency, flexibility, and originality were measured in students’ tangram problem solving. Additional predictors included Tangram Problem Solving Speed (TPSS), general reasoning (CogAT), and mathematical achievement (NWEA MAP Math). ANCOVA showed significant post-test differences favoring the intervention group across all creativity components. Two-group structural equation modeling analysis supported measurement invariance and explained 55–60% of posttest creativity variance. TPSS emerged as the strongest predictor, with greater effects for the intervention group. These findings highlight the potential of enrichment programs and language-accessible geometry tasks to cultivate creativity in young gifted ELs by strengthening their mathematical foundation while supporting flexible and original problem solving. Full article