Mathematical skills in heritage bilingual children with and without developmental dyscalculia: A comparative study.

In a single glance at a collection of objects, we can appreciate their numerosity. But what are the “objects” over which this number sense operates? Most work in this domain has implicitly assumed that we estimate the number of discrete, bounded individuals actually present in the visual field. However, in many instances we can construe such individuals as potential parts of composite objects that they can create—as when we assemble furniture or complete a jigsaw puzzle. Here, we demonstrate that visual numerosity estimation is sensitive to such part–whole relations, such that the number of items in a display is underestimated when it contains spatially separated but easily combinable objects. Participants saw brief displays containing noncontiguous “puzzle-piece” stimuli, and reported which display had more pieces. Crucially, most of the pieces appeared in pairs that either could or could not efficiently combine into new objects. In four experiments, displays with combinable pieces were judged as less numerous than displays with noncombinable pieces—as if the mind treated two geometrically compatible pieces as being the single whole object they could create. These effects went beyond various low-level factors, and they persisted even when participants were explicitly trained to treat individual pieces as the units that should be counted. Thus, despite the many ways that sets of objects may be construed for the purposes of counting, visual perception automatically takes into account the ways that object parts may combine into wholes when extracting numerosity from visual displays.

We utilize concepts of numeracy including number sense, reading and interpreting graphs, basic probability and statistics, and reasoning to estimate guessing and verify our earlier findings on human self-assessment as replicable. Our field study employed a low-stakes paired measures assessment (11,229 scores from the validated Science Literacy Concept Inventory and postdicted global self-assessment ratings generated upon completion of the Inventory) in conjunction with a five-category taxonomy of self-assessment proficiency. We also simulated 11,229 random guessing responses, to model responses that disengaged, purely random-guessing participants should produce. At least 90% of participants sincerely engaged with the instruments of measure, self-assessed imperfectly but reasonably well, and exhibited equal tendencies to underestimate or overestimate their scores by modest amounts. Results contradict the prevalent claim that most people overestimate their actual abilities, with the least knowledgeable being grossly overconfident (ie, the Dunning-Kruger effect). In this study, disengaged, random guessers could account for nearly all participants who grossly overestimated. Confirming that significant numbers of low-scoring participants are aware of their poor performance removes support from the "dual-burden hypothesis," which states that low-scoring participants lack both the competence and metacognitive competence needed for accurate self-assessment. The amount of guessing in a populace does not attenuate the "effect," as claimed in recent psychology literature, but magnifies it. Studies of paired measures yielded a new understanding of guessing. The "effect" is better explained as an illusion produced by probability than as an accurate portrayal of human self-assessment.

Understanding how non-numerical visual features systematically distort numerosity perception holds promise for unveiling the processes that give rise to our visual number sense. Recent studies show that increasing visual coherence systematically increases perceived numerosity, with this effect strengthening over development (DeWind et al., 2020; Qu, Bonner, et al., 2024; Qu et al., 2022). Here, we investigate the cognitive mechanisms underlying the coherence illusion from a view of perceptual decision processes. Specifically, we applied a drift diffusion model (DDM) to a previously described dataset from participants aged 5-30 tested in an ordinal numerical comparison task with color entropy systematically manipulated (Qu et al., 2022). By jointly modeling choice data and response times, we decomposed numerical discrimination performance into distinct decision components: the speed of numerical evidence accumulation (drift rate), the amount of evidence required for a decision (boundary separation), and the response bias reflecting a prior tendency of selecting one side over the other. We found that color coherence affected only the drift rate but not response bias or boundary separation, demonstrating that color coherence distorts numerical calculation through biased accumulation of evidence of quantity. Moreover, the impact of coherence on the drift rate coefficient increased with age as quantitative information is accumulated more efficiently over development. Our results offer a framework for understanding how numerical illusions arise from perceptual decision-making dynamics.

This study investigated the effective teaching strategies employed by elementary mathematics teachers in addressing numeracy gaps among struggling learners in selected private schools in Laoag City, Ilocos Norte, and Sanchez Mira, Cagayan. Anchored on constructivist and learner-centered learning theories, the research sought to identify the specific numeracy difficulties experienced by learners and the pedagogical practices teachers adopt to address them. Using a descriptive research design, the study involved ten (10) elementary mathematics teachers who participated through an online questionnaire consisting of structured checklists and open-ended questions. Both descriptive statistics and thematic analysis were applied in interpreting the data. Results revealed that learners frequently experience difficulties in fractions, decimals, basic operations, and word problem solving, often arising from weak number sense, poor conceptual understanding, and limited ability to apply mathematical ideas to real-life contexts. Teachers addressed these gaps using visual aids and manipulatives, gamified and technology-based instruction, peer tutoring, remediation, differentiated teaching, and contextualized learning tasks. These adaptive and learner-centered approaches were found to enhance engagement, comprehension, and confidence in mathematics. The study concludes that inclusive, evidence-based, and contextually relevant strategies play a vital role in improving numeracy performance. It recommends sustained professional development for teachers, early diagnostic interventions, strengthened home–school partnerships, and integration of technology-enhanced instruction to promote equitable and lasting numeracy improvement in Philippine education.

Math language matters: math-specific verbal skills, not analog representation of non-symbolic magnitudes, predict symbolic fraction abilities in primary school children.

The development of number sense and its predictive relation to mathematical achievement among young elementary students

Background: In response to persistent challenges in early grade mathematics achievement in South Africa, two intervention projects – Base-Ten Thinking (BTT) and the Mental Starters Assessment Project (MSAP) – were implemented to enhance number sense through mental mathematics in Grades 2 and 3. These projects integrate Freudenthal’s theory of number structuring with context-specific mental strategies, namely Jump and Bridging-through-ten. Aim: To examine how the fidelity of implementation of mental mathematics interventions influences learner performance in early grade classrooms. Setting: The study was conducted in four South African primary schools participating in the BTT and MSAP interventions, each reflecting varying levels of fidelity (high to no fidelity) to the instructional model. Methods: A design research approach was adopted, using 15 video-recorded lessons and pre- and post-test data from 155 learners taught by five teachers. Implementation fidelity was assessed using descriptors derived from components of Lemire et al.’s framework for conceptualising implementation fidelity. Both quantitative and qualitative analyses were employed. Results: Schools with high fidelity demonstrated statistically significant gains, particularly in jump strategy tasks, compared to schools with moderate, low or no fidelity. This contrasts with earlier findings where moderate fidelity, which allows for teacher agency, yielded better outcomes. The study highlights the value of structured instructional coherence. Conclusion: The study underscores the importance of well-structured, evidence-based and well-researched intervention models targeted towards addressing foundational gaps in mathematics in early grade classrooms. Contribution: In addition to the contribution on the importance of well-structured and thoroughly researched instructional models, the study also makes a methodological contribution by providing an analytical framework linking fidelity, professional development and early grade mathematics instruction.

Abstract This article presents a study of 6-year-old Tommy’s first interaction with the educational iPad app Fingu. Fingu is designed to support the development of early number sense, and in particular the understanding of numbers 1–10 as part–whole relations through multi-touch input. Tommy’s mathematical learning was studied in terms of emergent fluency with the mathematical instrument Fingu, with attention to the transformations of his perceptuomotor activity. A fine-grained qualitative analysis of a 13-min video was conducted, following the model of microgenetic learning analysis. The analytical framework adopted allows for capturing many nuances of Tommy’s learning experience, in terms of the development of specific number sense skills, such as finger gnosis and control, representing numbers with fingers, and recognizing part–whole relationships. This research contributes to a broader understanding of how young children’s number sense can develop through the use of multi-touch technologies and how it can be studied in detail.

This paper presents a conceptual framework for teaching basic arithmetic operations – addition, subtraction, and multiplication – at the kindergarten level using the Hundred Chart. Based on developmental learning theories and supported by a comprehensive literature review, the proposed approach bridges the gap between concrete manipulatives and abstract mathematical reasoning. While not reporting empirical findings, this framework lays the groundwork for future experimental research into developmentally appropriate numeracy strategies in early childhood education. We contend that this visual and structured method, when incorporated in the early grade curriculum, will foster counting fluency, number sense, and operational understanding in a way that is accessible and engaging for young learners. Researchers and teacher educators looking for developmentally appropriate methods, as well as classroom practitioners and curriculum developers hoping to make maths education more tangible and interesting for younger students, can all benefit from this framework.

Mathematics plays a vital role in developing students’ logical reasoning and problem-solving abilities, which are essential for both academic success and everyday decision-making. However, many Junior High School learners still struggling with the basic numeracy that leads to low performance in mathematics as a whole. Addressing this issue calls innovative and engaging teaching strategies that will help make mathematical ideas more understandable and appealing. This study investigates the effectiveness of counters-based activities and gamification interventions in improving the basic numeracy skills of struggling learners in Junior High School within Borongan City District IV. The research aims to determine which intervention will enhance learners’ understanding of fundamental mathematical concepts such as number sense, operation, and problem-solving. Using a quasi-experimental design, two groups of low-performing students were exposed to different interventions: one utilizing traditional counters and other employing gamified mathematical activities. Pre-test and post-test scores measured numeracy gains, while learners feedback provided insights on engagement and motivation. The findings aim to guide educators in adopting evidence-based, learner-centered strategies to strengthen numeracy development among struggling students in Junior High School. The pretest results showed that learners in all groups had low numeracy proficiency. The gamification group scored 35.5 (Low Proficient), the counters group 23.3 (Non-Proficient), and the control group 28.6 (Low Proficient), indicating that all the participants struggled with basic operations in Mathematics. After the intervention, post-test results revealed notable improvement. The gamification group achieved 68.5 (Nearly Proficient), the counters group 62.8 (Nearly Proficient), and the control group 58.6 (Nearly Proficient). Statistical analysis confirmed that these improvements were significant for all groups: gamification (t = -8.51, df = 29, p < .001), counters (t = -9.07, df = 29, p < .001), and control (t = -7.48, df = 29, p < .001). the null hypothesis was rejected in all cases, indicating that the interventions had a measurable effect on learners’ numeracy skills. Keywords: Numeracy Skills, Counters, Gamification, Intervention, Struggling Learners, Mathematics Education.

Cognitive biases, which come from the shortcuts our brains use to make decisions, often lead people to misjudge sizes, amounts, or other magnitudes. However, it is still unclear how these biases evolved. Nonhuman animals like primates and corvids share with humans a nonsymbolic number sense. Whether biases in magnitude estimation are evolutionarily widespread in animals remains an open question. We investigated numerical estimation in macaque monkeys and carrion crows-two distantly related but numerically proficient vertebrates-using a delayed match-to-numerosity task with dot arrays. Both species exhibited key biases commonly observed in humans during numerical estimation tasks: scalar variability (increasing response variability with numerosity), regression to the mean (overestimation of small and underestimation of large numerosities), and a sequential effect (estimates biased toward previous trial numerosities). Traditionally, these biases have been attributed either to static influences of overall task history or to dynamic effects from recent trials. Here, we demonstrate that a Bayesian model incorporating dynamic priors based on multiple previously encountered numerosities can account for all observed behavioral patterns. Our findings suggest that numerical judgments in both species are shaped by uncertainty and the integration of recent experience, revealing shared Bayesian-like mechanisms for cognitive biases across distantly related animals.

This study assessed the extent to which Grade 6 completers of Claver Central Elementary School demonstrated essential mathematical learning competencies as defined in the K to 12 Mathematics Curriculum Framework. Using a quantitative descriptive-correlational design, the research evaluated the proficiency of learners in Numbers and Number Sense, Measurement, Geometry, Patterns and Algebra, and Statistics and Probability. Respondents included both pupils and their teachers, with data gathered through validated questionnaires. Statistical tools such as mean, standard deviation, t-test, ANOVA, and Pearson correlation were employed for analysis. Findings revealed that the Grade 6 completers achieved a moderate level of proficiency across the five mathematical domains. Significant differences were observed when competencies were grouped according to age, sex, and general weighted average in Mathematics. Moreover, discrepancies were identified between teachers’ and pupils’ assessments of competencies, while a significant correlation existed between learners’ general weighted average and their self-assessed mathematical proficiency. The study highlights the need for sustained efforts in improving mathematics instruction, including enhanced teacher professional development, curriculum refinement, and increased parental involvement. These findings provide useful implications for school administrators and policy makers in strengthening mathematics education outcomes. Keywords: Mathematics Education; Essential Learning Competencies; Grade 6 Learners; Philippines; Student Performance

This article advances an argument-based approach to presenting validity evidence for a brief assessment of 3 rd –5 th grade students’ number sense. Improving students’ number sense has been a goal of K-12 mathematics education in the US since the late 1980’s, and several popular instructional practices are hypothesized to meet this need. However, there is limited evidence in the literature that elementary students’ number sense is improving. We worked to develop a brief assessment for 3 rd –5 th grade students and then conducted a series of studies to collect evidence that students’ scores reflect their number sense. We present the assessment here along with our intended interpretations and ways to use student scores. We then outline our validity argument, organized by four specific claims addressing the connection between student responses on the assessment and their number sense. Researchers and educators can now use the assessment to efficiently measure elementary students’ number sense.

Purpose: This study investigates the role of number sense performance in the relationship between math anxiety and math achievement among second-grade primary school students. Design/Methodology/Approach: The study sample consisted of 138 second-grade students from a school representing a middle socioeconomic status. Data were collected using a math anxiety scale, a math achievement test, and a number sense test. The data were analyzed through descriptive statistics, Pearson correlation, mediation, and moderation analyses. Findings: The results revealed a strong positive correlation between number sense performance and math achievement (r = .64) and a moderate negative correlation between math anxiety and math achievement (r = -.34). Mediation analysis indicated that number sense performance partially mediated the relationship between math anxiety and math achievement. On the other hand, moderation analysis showed that number sense did not have a significant moderating effect on this relationship. Slope analysis suggested that students with high number sense performance were less affected by math anxiety. Highlights: These findings highlight that number sense is a significant predictor of early mathematical achievement and that math anxiety indirectly influences achievement through this skill. The findings indicate that number sense-based interventions in early education may significantly enhance academic achievement by fostering both cognitive development and affective factors.

Purpose: This study, it was aimed to develop a valid and reliable measurement tool by evaluating the psychometric properties of the Mathematics Learning Disabilities Screening Scale (MLDSS) for elementary and secondary school students. In this context, a measurement tool that is culturally and linguistically appropriate for the Turkish context and overcomes the limitations of translated scales was developed. Design/Methodology/Approach: The study was conducted as a scale development study within a survey research design. The scale's items were developed based on DSM-5 criteria and an extensive literature review. The sample consisted of 644 students, identified by their teachers, from 120 schools across Türkiye's seven geographical regions. The psychometric properties of the scale were evaluated using Exploratory Factor Analysis (EFA) and Confirmatory Factor Analysis (CFA), along with internal consistency (Cronbach's Alpha) and test-retest reliability analyses. Findings: The EFA results revealed a three-factor structure (Number Sense, Calculation, and Mathematical Reasoning) that explained 68.5% of the total variance. The CFA confirmed this structure, with goodness-of-fit indices indicating an excellent model fit (e.g., CFI = .95, IFI = .95, RMSEA = .069, χ²/df = 2.41). The scale demonstrated high internal consistency (Cronbach's Alpha = .93) and strong test-retest reliability (r = .90). Highlights: The mathematics learning disability screening scale is a valid and reliable instrument developed for elementary and secondary school students. Its three-factor structure is consistent with modern theories of mathematics learning disabilities and its strong psychometric properties make it a valuable tool for educators and experts in early identification and intervention planning.

Number sense, the ability to rapidly estimate object quantities in a visual scene without precise counting, is a crucial cognitive capacity found in humans and many other animals. Recent studies have identified artificial neurons tuned to numbers of items in biologically inspired vision models, even before training, and proposed these artificial neural networks as candidate models for the emergence of number sense in the brain. But real-world numerosity perception requires abstraction from the properties of individual objects and their contexts, unlike the simplified dot patterns used in previous studies. Using novel synthetically generated photorealistic stimuli, we show that deep convolutional neural networks optimized for object recognition encode information on approximate numerosity across diverse objects and scene types, which could be linearly read out from distributed activity patterns of later convolutional layers of different network architectures tested. In contrast, untrained networks with random weights failed to represent numerosity with abstractness to other visual properties and instead captured mainly low-level visual features. Our findings emphasize the importance of using complex, naturalistic stimuli to investigate mechanisms of number sense in both biological and artificial systems, and they suggest that the capacity of untrained networks to account for early-life numerical abilities should be reassessed. They further point to a possible, so far underappreciated, contribution of the brain's ventral visual pathway to representing numerosity with abstractness to other high-level visual properties.

Abstract Context: Children with sensorineural hearing loss (HL) are at risk for difficulties in language, cognition, and academic learning, particularly in mathematics. Early diagnosis and rehabilitation have improved auditory and verbal outcomes, but less is known about their effects on mathematical skills. Aims: This study aims to assess the mathematical skills in children with and without HL through the standardized BDE-2 test and to explore the potential correlations between audiological parameters and cognitive-mathematical performance. Settings and Design: A case–control study was performed on 37 children aged between 9 and 15 years, enrolled at the Audiology and Phoniatrics Unit of the University of Naples Federico II. Subjects and Methods: The sample included 25 children with bilateral sensorineural HL (study group) and 12 normal-hearing peers (control group). Each participant underwent a full audiological assessment – comprising unaided and aided pure-tone and speech audiometry – and completed the BDE-2 test. The two groups were compared in terms of mathematical skills, and correlations were analyzed between the degree of HL and BDE-2 assessment. Statistical Analysis Used: Chi-square, Fisher’s exact, t -test, Mann–Whitney, Wilcoxon, and Spearman correlation coefficients were applied. Significance was set at P < 0.05. Results: Children with HL scored significantly lower in calculation (75.4 ± 22.9 vs. 95.0 ± 12.7; P = 0.0021) and in total BDE-2 (82.2 ± 21.9 vs. 95.2 ± 6.9; P = 0.0113) indices than controls, while differences in number recognition and sense were not statistically significant. No correlations emerged between HL severity and BDE-2 outcomes. All children with HL showed a statistically significant improvement in pure-tone thresholds and speech recognition when tested with hearing aids or cochlear implants. Conclusions: Despite modern rehabilitative technologies, children with HL exhibit persistent weaknesses in complex calculation and global mathematical performance. Early screening and interventions targeting linguistic, mnemonic, and symbolic reasoning abilities should be integrated into standard follow-up programs to enhance academic outcomes.

This research explored the connection between students’ computational abilities and their performance in solving word problems in Mathematics. A quantitative approach was utilized, using researcher-developed assessments to evaluate arithmetic proficiency, number sense, mathematical fluency, and word-problem solving skills to 240 students, selected through simple random sampling. The results indicated that a significant number of students faced difficulties in computation, especially in performing arithmetic operations and demonstrating number sense. Moreover, students demonstrated generally low performance in solving word problems. The correlation analysis revealed a strong positive relation between computational skills and problem-solving ability, suggesting that students who are more proficient in computations tend to do better in word problem tasks. Furthermore, regression analysis indicated that the strongest success predictor in problem-solving are arithmetic skills. These results highlight the importance of developing computational proficiency to enhance mathematical problem-solving abilities. Educators may implement targeted instructional strategies to improve students’ arithmetic skills and number sense. Future research may explore interventions that strengthen computational skills at an early stage to improve overall Mathematics achievement.

This study aims to systematically analyze the early number concepts developed among lower elementary school students and to draw implications for the development of teaching and learning strategies. To achieve this, we assessed the early number concepts of conservation, group recognition, and quantity comparison in first and second graders, compared and analyzed correct answer rates and strategies based on mathematics achievement level by grade, and identified specific error patterns. The results indicate that lower-grade students with low mathematics achievement may fail to sufficiently develop number sense as they progress through grades due to a lack of understanding of early number concepts. Therefore, it is necessary to improve approaches to number and support the formation of early number concepts. Providing opportunities to count groups using various arrangements and configurations of objects, conducting comparisons focused on the essence of quantity (more and less), and implementing hands-on activities utilizing one-to-one correspondence strategies can help students develop a deeper understanding of number concepts. The results of this study will provide foundational data for developing effective early number concepts instruction and offer implications for textbook development.