Planar cell polarity represents a fundamental mechanism by which cells within epithelial sheets align their orientation, enabling coordinated tissue morphogenesis and function. Disruption of PCP leads to developmental defects and disease, highlighting the importance of understanding its establishment and maintenance. While experimental studies have identified key protein molecules that drive PCP, mathematical and computational modeling have become indispensable in connecting molecular interactions to tissue-level outcomes. Over the last couple of decades, diverse approaches, such as agent-based models, Cellular Potts frameworks, Petri nets, continuum theories, and phenomenological models, have been developed to capture distinct aspects of PCP dynamics. These frameworks allow systematic exploration of nonlinear feedback, intracellular and intercellular signaling, and the influence of geometry and mechanics, and noise on polarization. This review summarizes these mathematical and computational developments in PCP modeling, emphasizing methodological assumptions, insights gained, and open challenges. By bridging experiment and theory, PCP modeling advances both mechanistic understanding and predictive capacity for tissue-scale organization.

ABSTRACT While school mathematics is valuable, it often limits authentic problem-solving, adaptable thinking, and curiosity-driven exploration. In contrast, home environments offer opportunities to engage children in informal, everyday mathematics that connects to their cultural and lived experiences. The purpose of this study was to examine how seven families engaged in humanistic approaches to mathematics within home-based engineering tasks. We utilized video observation with interaction analysis across 12 sessions (~21 h) involving children in grades 2–6. Findings indicate that engineering design tasks in the home environment allow children to engage in humanistic and authentic approaches to mathematical concepts, with parents prompting co-reasoning and household materials functioning as improvised tools that surface “hidden mathematics.” We highlight this through geometrical reasoning, informal measurement, and proportional and covariational reasoning. This study highlights the potential of out-of-school settings, particularly the home environment, to support early mathematical development, serving as a foundation for later formal learning.

Purpose: This study aims to examine the effects of the RADEC (Read, Answer, Discuss, Explain, and Create) learning model and local cultural integration on early childhood mathematical development. The research specifically analyzes the independent and interaction effects of instructional model and cultural context on children’s mathematical performance. Method: A quantitative approach was employed using a 2×2 factorial experimental design. The participants consisted of 52 kindergarten children divided into four groups: RADEC with cultural integration, RADEC without cultural integration, conventional learning with cultural integration, and conventional learning without cultural integration. The cultural component incorporated elements of Minangkabau traditions into mathematics learning activities. Children’s mathematical abilities were assessed using a validated instrument covering numbers, classification, measurement, sequencing, geometry, and patterns. The data were analyzed using Two-Way Analysis of Variance (ANOVA) at a 0.05 level of significance. Findings: The results revealed significant main effects of both the RADEC learning model and local cultural integration on children’s mathematical development (p < .05). Learners exposed to RADEC-based instruction achieved higher mathematical performance compared to those receiving conventional instruction. Similarly, integrating cultural contexts significantly enhanced children’s conceptual understanding. However, no statistically significant interaction effect was found between the learning model and cultural integration. Significance: The findings provide empirical support for the role of structured constructivist pedagogy and culturally responsive instruction in early childhood mathematics education, offering practical implications for designing meaningful and context-based learning experiences.

The fundamental characteristics of math ability in individuals with autism spectrum disorder (ASD), specifically proficiency and variability, remain inadequately understood. Here, in this systematic review and meta-analysis, we addressed this gap by synthesizing evidence on math ability in autistic individuals relative to the non-autistic population. Searches in multiple databases yielded 66 studies. Risk of bias was assessed using an adapted Joanna Briggs Institute checklist. Random-effects meta-analyses used Hedges' g and natural logarithm of variability ratio (lnVR) as effect sizes. Publication bias was adjusted for using the precision-effect test and precision-effect estimate with standard errors, as well as a three-parameter selection model. Results show that, compared with the non-autistic population, as represented by standardized norms (mean 100, s.d. 15; 3,051 participants) and typically developing (TD) control groups (2,351 participants), individuals with ASD exhibit significantly lower math scores (ASD versus norms: Hedges' g = -0.360, 95% confidence interval (CI) -0.605 to -0.114; ASD versus TD: Hedges' g = -0.696, 95% CI -0.947 to -0.445) and greater variability (ASD versus norms: lnVR 0.159, 95% CI 0.102 to 0.216; ASD versus TD: lnVR 0.298, 95% CI 0.199 to 0.396). Group discrepancies were moderated by intelligence, age or their interactions. The math-intelligence relationship in ASD provides a theoretical framework for understanding their mathematical development. In addition, the ASD-TD discrepancy has widened over the past four decades. These findings underscore the need for sustained, individualized mathematical education for ASD and investigation of the developmental trajectories of mathematical skills in ASD. Methodological challenges in the field included potential publication bias and insufficient rigour in sample matching.

ABSTRACT This article explores the estimation of unknown parameters and reliability characteristics under the assumption that the lifetimes of the testing units follow an Inverted Exponentiated Pareto (IEP) distribution. Here, both point and interval estimates are calculated by employing the classical maximum likelihood method, a pivotal estimation method and a hierarchical Bayesian estimation method. Also, existence and uniqueness of the maximum likelihood estimates are verified. Further, asymptotic confidence intervals are derived by using the asymptotic normality property of the maximum likelihood estimator. Moreover, generalized confidence intervals are obtained by utilizing the pivotal quantities. Also, the 95 % highest posterior density (HPD) intervals are constructed based on a Markov chain Monte Carlo (MCMC) algorithm within the Bayesian estimation context. Additionally, some mathematical developments of the IEP distribution are discussed based on the concept of order statistics. Furthermore, all the estimations are performed on the basis of the block censoring procedure, where an adaptive progressive Type-II censoring is employed to every block. In this regard, the performances of the three estimation methods, namely, maximum likelihood estimation, pivotal estimation and the hierarchical Bayesian estimation are evaluated and compared through a simulation study. Finally, a real data is illustrated to demonstrate the flexibility of the proposed IEP model.

Domain 1 of the Philippine Professional Standards for Teachers (Content Knowledge and Pedagogy) underscores the need for pre-service to possess strong subject matter expertise to ensure quality instruction. This study investigated the mathematical content knowledge (MCK) of sixty-three elementary pre-service teachers to provide a basis for proposing action plans to enhance their proficiency. Using a descriptive research design, a researcher-developed fifty-item proficiency test was administered across five distinct domains: numbers and number sense, measurement, geometry, patterns and algebra, and statistics and probability. Results show an over-all low mean score of 17.2 out of 50, with participants showing a nearing mastery level in geometry and low mastery level on the other domains—reflecting an alarming gap in mathematical understanding among elementary pre-service teachers. These findings align with international evidence, such as the Teacher Education and Development Study in Mathematics (TEDS-M), which identified similar deficiencies among pre-service educators worldwide, particularly in algebraic reasoning and numerical operations. To address these gaps, three action plans were developed using the Plan-Do-Check-Act (PDCA) cycle: (1) integration of key mathematical concepts into related bachelor of elementary education (BEEd) courses, (2) development of Core Mathematics: A Power-Up Guide for Elementary Teachers as a learning resource, and (3) implementation of Project Foundations Forward: Math for Elementary Educators. It is strongly recommended that teacher education institutions review and enhance their course syllabi to include core mathematical topics aligned with the revised K to 12 Curriculum. Future research should explore interventions that strengthen the content knowledge to ensure pre-service teachers’ readiness for classroom instruction. Strengthening MCK among pre-service teachers is vital for improving mathematics instruction and providing better learning outcomes for students.

The Concrete-Representational-Abstract (CRA) instructional method has arisen as a systematic and evidence-based pedagogy meant to solve mathematics education difficulty. This study explores primary school teachers’ perceptions of the effectiveness of the Concrete–Representational–Abstract (CRA) instructional approach, supported by Ganitha Kalika Andolana (GKA) Maths Kits, in enhancing mathematical understanding among elementary learners. The CRA approach follows a structured progression from hands-on concrete experiences to pictorial representations and finally to abstract symbolic reasoning, thereby supporting conceptual development in mathematics. GKA kits, developed by the Akshara Foundation, provide activity-based learning resources aligned with this sequence and aim to make mathematics learning more engaging and meaningful in primary classrooms.A descriptive survey design was adopted, and data were collected from 30 primary school teachers selected through stratified random sampling from 15 upper primary schools in Subarnapur District, Odisha. A structured Likert-scale questionnaire was used to examine teachers’ perceptions regarding the knowledge, student-centeredness, usability, and practical implementation of the CRA approach, as well as the usefulness of GKA kits in classroom instruction. The findings indicate that teachers generally perceive the CRA approach as effective in promoting students’ conceptual clarity, active participation, and engagement in mathematics learning. Teachers also reported that GKA kits contribute to joyful and interactive learning experiences, supporting creativity and classroom involvement. However, respondents also highlighted challenges related to time requirements, lesson planning, and the need for sustained professional training for effective implementation.

This study examines the effects of focused instructional strategies on young children’s conceptual growth in mathematical reasoning and explores how early numeracy abilities influence later mathematical achievement and broader cognitive development. Drawing on empirical research from diverse educational settings, it compares traditional instruction and play-based learning (PBL), both of which are shown to support the development of early arithmetic skills, with particular attention to student engagement, cognitive growth, and numeracy competence. The review analyzes how early mathematical skills, such as number sense, spatial reasoning, and problem-solving—shape academic performance into adolescence and discusses the roles of play-based learning, technology-enhanced approaches, and structured instruction in fostering these foundational concepts. Employing a qualitative approach, the study synthesizes relevant literature on cognitive readiness, early childhood mathematics development, instructional and play-based interventions, and conceptual growth in early mathematics, as well as the longitudinal impact of early childhood mathematics on later academic outcomes. Overall, the findings underscore early intervention as a critical foundation for early childhood mathematics learning and highlight the need for carefully designed pedagogical strategies that integrate play, explicit instruction, and rich learning environments to optimize young children’s mathematical and cognitive development.

Russia was devastated by the aftermath of the First World War and the Bolshevik Revolution that started in 1917. As a result, many people, particularly members of the intelligentsia, were forced to flee their country, prompting waves of migration. They were reasonably safe in some European countries and the USA, and the newly established Kingdom of Serbs, Croats, and Slovenes was one such destination. Here I discuss the case of two Russian mathematicians, Anton Bilimovich (1879–1970) and Nikolay Saltykov (1872–1961), who were established figures within the Russian scientific community, but upon arrival in Belgrade their status shifted immediately to that of refugees. Nevertheless, they promptly continued their scientific careers as integrated members of Yugoslav society, ultimately reaching the ranks of academicians of the Serbian Academy of Sciences and Arts, the highest academic distinction in Serbia to this day. Their lives in Yugoslavia were characterized by new beginnings and continuing with mathematical practices. I also argue that their arrival in Belgrade was mutually beneficial for them and for the state, since their contributions to the development of Yugoslav mathematics were substantial at a time when the country was struggling to establish its national identity.

Indonesian students’ mathematics performance remains below the OECD average, particularly on items requiring reasoning, interpretation of data, and problem-solving in real-world contexts. Although PISA outcomes are influenced by multiple factors, including curriculum alignment, learning opportunities, and socioeconomic conditions, prior studies indicate that students’ difficulties with non-routine, reasoning-based tasks reflect limitations in higher-order mathematical thinking. In this context, strengthening students’ mathematical critical thinking remains an urgent instructional challenge. This study aimed to determine whether students’ mathematical critical thinking skills improved more significantly after using a Google Classroom-based Comprehensive Mathematics Instruction (CMI) model than after regular learning. Additionally, this research examined changes in students’ mathematics self-efficacy before and after the intervention and explored the relationship between mathematical critical thinking skills and mathematics self-efficacy. A quasi-experimental nonequivalent control group design was employed. The population consisted of all tenth-grade students at SMAN 1 Jalancagak, with a purposive sample of 70 students drawn from two classes (X-10 and X-9). Data were collected using a mathematical critical thinking test and a mathematics self-efficacy questionnaire. The results show that students using the Google Classroom-based CMI model achieved significantly greater improvements in mathematical critical thinking skills than those in standard learning. Moreover, students’ mathematics self-efficacy increased significantly following implementation of the model. The analysis also found a statistically significant, though modest, relationship between mathematical critical thinking skills and mathematics self-efficacy. These results indicate that combining structured cognitive instruction with digital learning environments can enhance students’ mathematical critical thinking and mathematics self-efficacy, while recognising that broader contextual factors influence their mathematics development achievement. Keywords: CMI model, google classroom-based CMI model, google classroom, mathematical critical thinking skills, mathematics self-efficacy.

Abstract Several studies have demonstrated that understanding repeating patterns is associated with early mathematical development in young children. In these studies, the structure of repeating patterns typically follows a linear sequence, requiring children to visually process a one-dimensional line. However, the repeating patterning ability of Deaf and Hard-of-Hearing (DHH) children remains largely unexplored. This raises the question of whether DHH children can effectively engage with the standard linear task format or if their enhanced peripheral visual perception, developed through visual-spatial sign language competence, might lead to better performance with an alternative, two-dimensional pattern format. The present study investigated pattern recognition and structuring abilities in linear patterns (LP) and circular patterns (CP) among 41 six-year-old DHH and typically hearing children. Analysis of the activities copy (reconstructing a pattern sequence from memory), translate (reproducing a given pattern sequence with different elements), and repair (completing a missing part in a pattern sequence) revealed high correlations both within and between LP and CP. Typically hearing children consistently performed better in LP, whereas DHH children’s performance in LP and CP varied depending on the activity. Additionally, early access to sign language appeared to influence patterning ability in the two-dimensional circular pattern. The circular pattern may thus serve as a complementary format in studies of pattern recognition and structuring abilities, particularly in inclusive early mathematics education for children with diverse sensory and linguistic backgrounds.

ABSTRACT Exploring teacher education needs is essential to improve educational quality, especially in specific areas such as teaching mathematics in early childhood education. Different studies and highly recognized organizations endorse the importance of promoting the development of early mathematics and the need to have properly prepared teachers. Therefore, the objective of this exploratory study is to identify the training needs of early childhood education (0–3 years) teachers with regard to the teaching of mathematics. To do so, a mixed methodology has been used that combines quantitative and qualitative analysis. Data was collected through questionnaires, observation and discussion groups, with a sample of 28 teachers from the network of Municipal Nursery Schools of Vic (Barcelona). The results show a lack of teacher education in both mathematical and didactic knowledge to teach mathematics, highlighting the need to design teacher education programmes that are more in line with the realities of the classroom. This study concludes by highlighting the importance of improving the teacher education of teachers in nursery schools to cover the identified shortcomings, both in the discipline of study and in its teaching.

Mathematical modelling and development methods for high-speed flow control over spiked blunt bodies: a review

Abstract While peer-to-peer conversations can be beneficial for children’s linguistic and mathematical development, the specific conditions needed to support optimal conversations remain elusive. As part of a larger project to infuse peer-to-peer interactions into mathematics instruction for multilingual students, 8- to 11-year-old children in the U.S. were videotaped by their teachers interviewing one another about their solution strategies to equal sharing problems. Partner Interviews were analyzed to determine the quality of the interactions between pairs using Barwell’s (2023) definition of negotiating meaning. This study examines the relationships among the quality of student negotiations, accuracy of strategies, similarities between strategies, and grade levels. Findings indicate that the degree to which students negotiated each other’s ideas varied, and the accuracy of students’ solutions was related to the quality of their negotiations. We provide a framework for assessing the quality of the peer-to-peer negotiations as well as concrete examples of the structures and scaffolds used to elicit these conversations.

Liu Hui is a seminal figure in the history of Chinese mathematics. In 263 CE, he commented on The Nine Chapters on Mathematical Procedures ( Jiuzhang Suanshu 九章算术). Prior research has highlighted his contributions in formulating rigorous arguments for the correctness of mathematical procedures, employing logical reasoning through nearly infinite segmentation and systematizing Chinese mathematics. This article examines three other aspects of his work that have a profound impact on mathematics but have rarely been discussed before. First, it explores his discussion on the early development of mathematics, which profoundly influenced the cultural foundations of Chinese mathematics. Second, it analyzes his contributions to what is known as ‘geometrical algebra’, a precursor to the Confucian mathematical methods that emerged during the Southern and Northern dynasties (420–589 CE) and later evolved into the so-called Chinese algebra developed between the 11th and 13th centuries. Third, it examines his innovative concept and application of counting rods, which can be regarded as an early precursor to the textualization and symbolization of Chinese mathematics during the 13th century. Finally, the article argues that Liu Hui's achievements should also be understood in the context of his interactions with political power and his engagement with contemporary philosophical ideas.

The development of Android-based mathematics learning media using Unity 3D at SD Negeri 4 Padangkerta was conducted to address the limited use of interactive learning media in mathematics instruction. This community service activity aimed to enhance teachers’ competence in developing Android-based mathematics learning media through participatory training and hands-on practice. The program was implemented through stages of socialization, technical training on Unity 3D, prototype development assistance, and evaluation of training outcomes. The results indicate that 80% of participating teachers successfully produced Android-based learning media prototypes, accompanied by a 75% increase in teachers’ understanding of learning technology based on pretest and posttest evaluations. In addition, teachers demonstrated increased motivation and creativity in integrating digital technology into classroom learning. Overall, this activity contributes to strengthening teachers’ digital competence and improving the quality of mathematics learning using interactive and contextually relevant digital media.

Abstract This study explores the role of a competence model specifically designed for an open digital testing and training tool. It aims at serving bridge courses during the transition phase from school to STEM studies at universities. Planned as structural guide and flexible access point for tutoring purposes, it is merged from curricula of five Austrian upper secondary school types. Adopting a design research approach, this article examines the alignment between the identified phases of mathematical concept development and item difficulty, operationalized through solution frequencies in national final examinations. Rather than testing a predictive hypothesis, the study investigates where concept development phases correspond to empirical difficulty patterns and where refinements to the framework may be needed. Analysis of 197 examination items from four consecutive years reveals no overall alignment between concept development phases and solution frequencies; however, topic-specific patterns emerge. Functions, Calculus and Statistics & Probability show tendencies toward increasing difficulty with higher concept development phases, while Geometry displays an inverse pattern. These divergent findings are interpreted with reference to contextual factors, including technology use during examinations and item-writing conventions for large-scale assessment. The results provide a foundation for iterative refinement of the competence model and identify specific areas requiring further investigation as the adaptive testing platform develops.

This article offers a philosophical reflection based on an analysis of Inca Garcilaso de la Vega’s discourse regarding the Andean nodal notation system known as kipu. To this end, it first reviews selected passages from the Comentarios Reales in which the characteristics of this archival method are described, highlighting its importance for Inca administration. The main focus of the analysis, however, lies in the attempt to uncover the internal logic behind Garcilaso’s explicit discursive contradictions and his ongoing comparison between phonological writing and kipus. Subsequently, still following the reading of Comentarios Reales, the study examines the inseparable link between the nodal system and the mathematical, architectural, and astronomical development of ancient Andean societies. It emphasizes that this native science did not lead to a mechanistic conception of the cosmos but rather emerged within an ontological reflection in which humans could ritually connect to the sacred web of life. In this way, the article demonstrates that Amerindian peoples were not illiterate societies; they had their own technical systems for creating archives, which allowed them to perform advanced calculations without breaking from a broad cosmogonic worldview in which all existence possessed life and intelligence and formed part of a single existential fabric.

Development of metallurgy provides for further increase and improvement of steel production volumes through the introduction of various advanced resource- and energy-saving technologies. The main and most universal control actions that affect the course and technical-economic indicators of the process are inextricably linked to the optimization of technology parameters which is focused on achieving the best results in the field of productivity, product quality and reduction of resource costs. This is achieved through the regular monitoring and analysis of key indicators, as well as making necessary adjustments to process management. A successful combination of these factors contributes to maximizing the production efficiency and increasing the competitiveness of products on the market. To calculate the process static parameters, it is advisable to use the resources of mathematical modeling and development of an instrumental system. When creating a static calculation model, the electric steelmaking process was considered as a complex thermodynamic system into which condensed and gaseous input media enter, and the final products are metal, slag and gas. Calculation of the static modes of the electric steelmaking process is carried out on the basis of calculations of material and thermal balances based on the laws of mass and energy conservation relative to the components of a heterogeneous system. The solution of the optimization problem based on formal methods involves selection of various criteria and setting a system of restrictions (requirements for metal composition; ranges of change in the cost of components of charge materials and system state parameters; compliance with the law of mass conservation at the level of fluxes, substances and elements; compliance with the law of energy conservation). A feature of the developed method of mathematical modeling and optimization of the electric steelmaking process is the systematic solution of a set of interrelated optimization problems to determine the optimal conditions for the processes in the metallurgical system and the optimal solutions for implementation of electric smelting technology.

A bstract A growing body of evidence suggests that the complexity of Feynman integrals is best understood through geometry. Recent mathematical developments [arXiv:2402.07343] have illuminated the role of exponential integrals as periods of twisted de Rham cocycles over Betti cycles, providing a structured approach to tackle this problem in many situations. In this paper, we apply these concepts to show how families of physically relevant integrals, ranging from exponentials to logarithmic multivalued functions, can be recast as twisted periods of differential forms over homology cycles. In the case of holomorphic exponents, we provide explicit decompositions as thimble expansions and reveal a geometric wall-crossing structure behind the analytic continuation in parameters. We then show that the generalization to multivalued functions provides the right framework to describe Feynman integrals in the Baikov representation, where the multivaluedness is governed by the logarithm of the Baikov polynomial. In this context, the thimble decomposition aligns with the decomposition into Master Integrals. We highlight how the wall-crossing structure allows for a sharp count of independent Master Integrals (or periods), circumventing complications arising from Stokes phenomena. Additionally, we study the large-parameter expansions of these integrals, whose coefficients correspond to periods of standard (co-)homology associated with families of algebraic varieties, and which reveal the dominant basis elements in different sectors of the wall crossing structure. This unifies perturbative expansions and geometric representation theory under a single cohomological framework.