<p>The purpose of the article is to study the peculiarities of the problem-solving process and to evaluate the effectiveness of using special strategies to improve students’ mathematical competence. The research methodology was based on a comparison of the learning outcomes of students in the control and experimental groups before and after the implementation of the special methodology. The results of pre-and posttesting showed an increase in the average score in both groups, but the progress of the experimental group was more pronounced. The analysis of the rate of learning progress showed an improvement in the skills of the experimental group, particularly in understanding and solving contextual mathematical problems. The study revealed typical problems of students related to the interpretation of mathematical terms, establishing the correct relationships between numbers, and performing arithmetic operations, especially with fractions. Gender analysis revealed slight differences in approaches to solving problems: girls showed greater progress in understanding the terms of problems and correctly drawing up equations, while boys showed greater progress in the accuracy of arithmetic calculations. The qualitative results showed that the use of the 6-step strategy contributed to the development of analytical thinking and critical skills, although the overall changes did not reach statistical significance. It was found that the use of interactive elements, in particular the mathematical dictionary, “Wall of Words,” helped the pupils of the experimental group to better memorize mathematical terms, associating them with the corresponding operations, and reduce the number of typical errors.</p>

This article explores the complex process of translating mathematical terminology from English into Sesotho, with particular emphasis on the challenges and opportunities inherent in developing contextually appropriate Sesotho mathematical terminology. The objective is to analyse the effectiveness of various term formation strategies that were used in a collaborative initiative at the University of the Free State under the auspices of the Academy for Multilingualism to produce Sesotho mathematical terminology for students. The study is framed by the communicative theory of terminology, which conceptualises mathematical terms as linguistic, cognitive and social constructs necessitating precise translation to preserve domain-specific meanings. The methodology comprises a qualitative textual analysis of an extensive Sesotho mathematical terminology list, developed by a multidisciplinary team of mathematicians and Sesotho language experts. The findings demonstrate that translating mathematical terminology into Sesotho is inherently challenging due to the lack of established vocabulary, lexical ambiguities and the absence of localised terminology standardisation guidelines, which can create inconsistencies and confusion for educators and students. Despite these challenges, developing new Sesotho mathematical terms through multidisciplinary collaboration presents a significant opportunity to enhance academic language, improve student comprehension and promote inclusion and equity in mathematics education.

Abstract The demand to know the structure of functionally independent invariants of tensor fields arises in many problems of theoretical and mathematical physics, for instance for the construction of interacting higher-order tensor field actions. In mathematical terms the problem can be formulated as follows. Given a semi-simple finite-dimensional Lie algebra g and a g-module V, one may ask about the structure of the sub-ring of g-invariants inside the ring freely generated by the module. We point out how some information about the ring of invariants may be obtained by studying an extended Lie algebra. Numerous examples are given, with particular focus on the difficult problem of classifying invariants of a self-dual 5-form in 10 dimensions.

Phylogenetic comparative methods are a major tool for evaluating macroevolutionary hypotheses. Methods based on the mean-reverting stochastic Ornstein-Uhlenbeck process allow for modelling adaptation on a phenotypic adaptive landscape that itself evolves and where fitness peaks depend on measured characteristics of the external environment and/or other organismal traits. Here, we give an overview of the conceptual framework for the many implementations of these methods and discuss how we might interpret estimated parameters. We emphasize that the ability to model a changing adaptive landscape sets these methods apart from other approaches and discuss why this aspect captures long-term trait evolution more realistically. Recent multivariate extensions of these methods provide a powerful framework for testing evolutionary hypotheses but are also more complicated to use and interpret. We provide some guidance on their usage and put recent literature on the topic in biological rather than mathematical terms. We further show how these methods provide a starting point for modelling reciprocal selection (i.e., coevolution) between interacting lineages. We then briefly review some critiques of the methodologies. Finally, we provide some ideas for future developments that we think will be useful to evolutionary biologists.

This article discusses the algebraic structure of codon sequences as a representation of DNA nitrogen base sets in mathematical terms. The study aims to prove what algebraic structures are obtained for codon sequences from DNA bases. The methods used include qualitative research methods in the form of literature studies and quantitative research methods in the form of experiments on DNA base sets. In mathematical notation, the nitrogen bases of DNA can be collected in a set and connected into algebraic structures through a bijective mapping on the Galois field of order 4. This results in the set B being viewed as a Galois field of order 4. Additionally, DNA base triplets or codons can be represented in mathematical form. Furthermore, these codons are bijectively mapped onto the Galois field of order 64, so that the resulting algebraic structure is a field. The result of this study show that the codon sequences have an algebraic structure in the form of a one-dimensional vector space over the Galois field on the codon. For further research, the Lie structure in codon can be investigated through the construction of its Lie brackets, where this vector space is a necessary condition for Lie algebras.

This research aims to improve the mathematics learning outcomes of Grade VII B students at SMP Darul Ulum 5 Rebalas through the implementation of Madurese language-based ethnomathematics. The study is motivated by the students' difficulties in understanding mathematical concepts delivered in formal Indonesian, as their daily communication relies on the specific Madurese dialect of Rebalas Village. This study employs the Classroom Action Research (CAR) method using the Kemmis and McTaggart model, conducted in two cycles. Each cycle consists of planning, action, observation, and reflection stages. The subjects of this research were 28 students from class VII B. The research instruments included test instruments in the form of group Student Worksheets (LKPD) and observation sheets for both teacher and student activities. The material focused on the topic of Lines and Angles. The results indicated an improvement in student learning outcomes from Cycle I to Cycle II. In Cycle I, the average group score was 80.2, which increased to 88 in Cycle II. Observations of student activities also showed a shift from passive participation in Cycle I to active and enthusiastic engagement in Cycle II. Furthermore, teacher observations showed significant improvement; initial issues regarding time management and excessive workloads were resolved, resulting in more enthusiastic and punctual teaching delivery. These findings demonstrate that the use of Madurese-based ethnomathematics helps students grasp mathematical terms more easily, increases discussion participation, and creates a more comfortable and interactive classroom atmosphere, thereby leading to improved learning outcomes.

The article discusses the cases of incorrect mathematics in textbooks. For example, instead of defining of a mathematical concept,textbooks explain mathematical symbols and terms. In other cases, false or non-trivial facts are used to justify or prove statements as if they were self-evident. We also provide examples of logical reasoning errors. We explain some cases of incorrect mathematics in textbooks by the problematic nature of mathematics syllabus. Other cases of incorrect mathematics arises from the different treatment of the same topics in different textbooks. In all the cases discussed, we offer alternative correct solutions.

The article deals with the analysis of mathematical terms, i.e. simple nouns. Complex terms are avoided in order to prepare a detailed lexico-grammatical analysis. The research examines the term formation within the word class of nouns. Origins of the English and Slovak terms are analysed, described and compared. Latin and Greek languages are languages of science for the English language. The original languages of terms in both languages. Furthemore, Slovak terminological equivalents are compared with the English ones in term preference the Latin/Greek term or the native term if exists. According to term formation processes, the following suffixes representing bound morphemes are analysed: -er, -or, -ion. The terms formed by affixation using the selected suffixes are compared with the Slovak counterparts in terminology of mathematics. Terms are analysed according to principles for term formation. The following principles should be followed in the formation of terms and appellations, as far as possible and as appropriate to the language: transparency; ⎯ consistency; ⎯ appropriateness, linguistic economy; ⎯ derivability and compoundability; ⎯ linguistic correctness; ⎯ preference for native language. Subject of analysis are terms taken from the Slovak National Corpus 10.0, i.e. English –Slovak Parallel Corpus 4.0 en, Slovak –English Parallel Corpus 4.0 sk and the British National Corpus. In the research, electronic corpora are applied. They use mathematical ad statistical methods to evaluate e.g. occurrence, frequency, collocability of words. The previous mentioned methods are taken into consideration and help to prepare the effective, precise and objective analysis of the planned analysis. The research is based on both languages and comparison of terms in both parts of the corpus. Specific trends and tendencies in the strategies of term formation are analysed. Descriptive method is used and the methods of quantitative and qualitative analysis are applied. Due to analysis of two languages, contrastive and comparative approaches are entailed. Terminology records of terms are examined from lexico-grammatical point of view and on the basis of term-formation tendencies in each analysed language. i.e. Slovak language and English language. Records include entry, identification number, reference to the term, synonyms, subject field, formula, abbreviation, context, reference to the context, definition, reference to the definition.

This paper explores the translation and interpretation of ancient Chinese mathematical texts, with a focus on Liu Hui's commentary on The Nine Chapters on Mathematical Procedures (hereafter The Nine Chapters ). Since the mid-19th century, European missionaries and sinologists have gradually introduced Chinese mathematical texts to Europe through various translations, a process that attracted increasing scholarly attention in the 20th century. The Nine Chapters , along with Liu Hui's commentary, has been one of the most translated and studied works, although in-depth research and translations of Liu Hui's work began to increase gradually only after the 1970s. The paper examines the challenges of translating ancient mathematical terms and concepts, highlighting the importance of maintaining terminological consistency to reflect the original mathematical ideas. Various translation strategies are discussed, each revealing differing scholarly approaches and perspectives. Examples of different translations of Liu Hui's terminology and his explanations illustrate how translators made their choice between historical accuracy and modern understanding. Through detailed analysis of specific mathematical problems and their translations, the paper underscores the richness and complexity of the meanings implied in Chinese mathematical text and the challenges of faithfully translating the subtle ideas. Ultimately, the paper argues that translations that try to carefully preserve the terminological features and textual structure can serve as valuable research tools, deepening our understanding of the algorithms and proofs in ancient Chinese mathematical texts.

Recent advancements in deep learning-based geophysical inversion have drawn considerable attention. Most of these inversions are supervised, which requires the creation of training models that capture as much prior geological information as is available in an area of interest. However, creating such geologically informed training models is challenging because some geological knowledge is difficult to be expressed in mathematical terms. Moreover, geological prior information is not always tied to specific spatial locations. To address these challenges, we propose a novel method based on alpha shapes, a concept from computational geometry, to generate training models that can easily integrate five key types of geological prior information, namely, (1) top boundaries, (2) dip angles, (3) surface outcrop contacts, (4) mineralization zones intersected by drillholes, and (5) measured physical property values on rock samples. We present three distinct scenarios to demonstrate how the proposed method can be employed to systematically generate geologically informed training models. We also show that deep generative models, such as the conditional variational autoencoder, trained on these geologically informed models, can not only output inversion results that align with prior geological knowledge but also quantify the associated uncertainties. To validate our approach, we apply it to a set of magnetic measurements collected in Qinghai Province, China, for the exploration of Cu-Mo critical mineral deposits. The resulting susceptibility models reveal a major dipping structure that is consistent with both surface geology and the magnetic data. The proposed method offers a flexible and unified framework for generating geologically informed training models for deep learning-based geophysical inversions.

Mathematical literacy equips students with the skills to solve real-life problems effectively. However, many students struggle to understand and apply mathematical concepts effectively. This study aims to analyze the types of errors made by 35 students of the X2 class of SMA Negeri 9 Banjarmasin in solving mathematical literacy problems on the material of arithmetic sequences and series based on Newman's procedure. This procedure was chosen because it provides a structured framework that identifies specific stages where learners commonly make mistakes in solving mathematical problems, such as reading, comprehension, transformation, process skills, and encoding. Data were collected through written tests and interviews, then analyzed using a descriptive qualitative method based on the Miles and Huberman technique. The results showed that encoding errors (27.48%) were the most frequent, where students often skipped verification and struggled to draw accurate conclusions. Process skills errors (26.06%) occurred when students were confused about the next step despite knowing the correct formula. Transformation errors (21.81%) stemmed from incorrect application of formulas due to reliance on memorization without contextual understanding. Comprehension errors (21.53%) resulted from misunderstanding the problem, especially in identifying known and unknown information. Reading errors (3.12%), the least frequent, are typically caused by unfamiliarity with mathematical terms or symbols. These findings indicate that students"™ difficulties are both procedural and conceptual. Therefore, instructional strategies should emphasize concept mastery, promote strategic problem-solving, encourage self- verification, and connect mathematics to real-world situations.

Definitions of mathematical terms are often presented as part of formal mathematical discourse, which students are expected to accept even when these definitions conflict with their existing thinking. Considering defining as part of students' developing mathematical discourse, we examined students’ progression from initial discourse to formal discourse as they reinvented a definition of the limit of a convergent sequence with the instructor’s guidance. Initially, students faced a conflict when their definition failed to classify graphs of sequences that they created – intended as examples and nonexamples – into the examples and nonexamples as they had planned. This conflict was eventually resolved through activities in which students engaged with elements of their definitions using routines familiar to them, but in a manner deemed appropriate by the instructor. Our analysis focuses on interdiscursivity, the blending of students’ existing discourse and formal discourse. Our results show how students’ application of familiar routines led to new meanings and uses for these elements, ultimately introducing quantities and their relations to define sequence convergence – characteristics of formal discourse. This approach supports intrinsically motivated learning by allowing students to build mathematical discourse from their own discourse, and our study reports on how the interdiscursivity could promote such learning.

The article examines the problem of using the ideas of set theory and general topology in philosophy on the example of book by Japanese philosopher Tanabe Hajime (1885–1962) The Development of Historicism in Mathematics. It is known that topological models have been successfully applied in physics in quantum field theory. Tanabe Hajime, along with Pavel Florensky and Vladimir Ern, was one of the first to raise the question of the applicability of topology in the social and humanitarian sciences, generating with its help a model of history. Tanabe uses the tools of topology, such as the concepts of a set, its density, neighborhood, con­tinuum, section, etc., to describe and explain the mechanisms underlying social reality not as a spatial phenomenon, but as a historical time continuum. The article focuses on such components of Tanabe’s philosophy as the concept of the conti­nuity of time and history, the features of understanding the present and the relation­ship between time modes, criticism of the unification of time and space, and the con­cept of historicism. The result that the author of the article came to is the clarification of the role and place of mathematical terms and theories in the philosophical system of Tanabe Hajime, who was not limited to the use of mathematical analogies, but also actually raised the question of mathematical modeling of history.

Abstract Mathematical thinking can reveal growth and insights into conceptual understanding and mathematical processes and procedures. It is widely recognised that mathematical thinking is a multi-faceted, multi-modal, and complex process. In communicating mathematical thinking, students can use both specific mathematical terms and language or generic language. The use of generic language can facilitate mathematical thinking. However, there is little known about the processes that primary-aged students use in communicating their mathematical thinking to others. To address this, Edward de Bono’s (1971) practical thinking was used to identify specifically the use of ambivalent or vague terms or what de Bono refers to as porridge words that children use to facilitate the communication of their mathematical thinking. To identify the occurrence and frequency of porridge words, qualitative data in the form of children’s drawings, descriptions of their drawings, and responses were analysed using a retroductive process. Employing discourse analysis uncovered patterns in how students used porridge words to communicate their mathematical thinking. Discourse analysis identified four categories of porridge words: (1) generic porridge words; (2) conceptual use; (3) process or procedural use; and (4) invisible thinking. An aim of applying porridge words to students’ mathematical thinking was to make an ambiguous aspect of mathematical practice transparent and relatable. Encouraging students to clarify vague terms, or porridge words, explain their reasoning, and reflect on their word choices can support deeper understanding and promote more accurate communication of mathematical thinking.

The Aubry transition is a phase transition between two types of incommensurate states, originally described as a transition by "breaking of analyticity." Here, we present Denjoy's (anachronistic) viewpoint, who almost 100 years ago described certain mathematical properties of circle homeomorphisms with irrational rotation numbers. The connection between the two lies in the existence of a change of variables from the incommensurate ground state variables to new simple phase variables that rotate by a constant irrational angle. This confers a cyclic order, an essential property of models with the Aubry transition. Denjoy's description indicates that there are two types of cyclic order, distinguished by the regular or singular nature of the change of variables or, in mathematical terms, by the distinction between topological conjugacy vs semiconjugacy. This allows rephrasing the breaking of analyticity as a breaking of topological conjugacy. We illustrate this description with numerical calculations on the Frenkel-Kontorova model.

Mathematical terminology often presents significant semantic ambiguity in machine translation (MT) systems, especially in language pairs with structural and conceptual asymmetry such as English and Uzbek. This paper discusses the linguistic and technological principles for creating an ontology-based lexical base aimed at enhancing the translation accuracy of mathematical terms. The proposed ontological model integrates linguistic analysis, conceptual structuring, and semantic relations to ensure consistent and contextually appropriate translation of polysemous terms. A bilingual corpus of English–Uzbek mathematical texts was used for term extraction, normalization, and semantic mapping within an ontology framework developed in Protégé using OWL/RDF representation. The research findings demonstrate that the ontology-based approach significantly improves semantic coherence and reduces contextual translation errors compared to conventional neural MT models. The proposed principles contribute to the development of a domain-specific linguistic base for English–Uzbek machine translation and can serve as a foundation for future multilingual ontological systems in scientific and technical fields.

This study explored how 31 upper-elementary school students perceived the learning sequence of 14 key mathematical terms in the ‘Number and Operations’ domain from a macro-metacognitive perspective. Students arranged these terms based on their own criteria and reasoning, thereby revealing their internal conceptual structuring. The analysis revealed that students tended to sequence terms reflecting the inherent hierarchy and interconnections of number concepts at the elementary level, consistently placing ‘natural number’ as foundational and clustering related terms such as those for division and fractions. However, these student-perceived sequences showed notable discrepancies from the textbook's presentation order, particularly concerning the introduction timing of ‘natural number,’ the internal sequencing within specific conceptual clusters, and the placement of ‘decimal’ terms. These findings underscore the dynamic interplay between students' intuitive knowledge structures and the formal curriculum design, providing foundational data for developing learner-centered mathematical vocabulary instruction and curriculum enhancement.

Abstract. We describe environmental gamma spectrometry data for >700 soil samples collected from >35 high-resolution quantitative soil profiles spanning global sites. The data are collected for the purpose of modern soil chronometry based on fallout radionuclides (FRNs) 7Be and 210Pb, using the Linked Radionuclide Accumulation model (LRC). Cumulative gamma counting time for samples in the database exceeds 6.5 years. This is a living database to be augmented as data become available and corrected with improvements in data reduction, or identification of errors. Versions and changes will be indexed. Special attention is paid to measurement uncertainties in the dataset, and to how atmospheric or excess 210Pb is defined in both geochemical and mathematical terms for use in the LRC model. Basic familiarity with gamma spectrometry and radionuclide decay chains is assumed. The data set can be accessed at https://doi.org/10.17632/cfxkpn6hj9.1 (Landis, 2025).

Characterizing the minimum, necessary and sufficient components to generate the dynamics of a biological system has always been a priority to understand its functioning. In this sense, the canonical form of biological systems modeled by Boolean networks accurately defines the components in charge of controlling the dynamics of such systems. However, the calculation of the canonical form might be complicated in mathematical terms. In addition, computing the canonical form does not consider the dynamical properties found when using the synchronous and asynchronous update schemes to solve Boolean networks. Here, we analyze both update schemes and their connection with the canonical form of Boolean networks. We found that the synchronous scheme can be expressed by the Chapman-Kolmogorov equation, being a particular case of Markov chains. We also discovered that the canonical form of any Boolean network can be easily obtained by solving this matrix equation. Finally, we found that, the update order of the asynchronous scheme generates a set of functions that, when composed together, produce characteristic properties of this scheme, such as the conservation of fixed-point attractors or the variability in the basins of attraction. We concluded that the canonical form of Boolean networks can only be obtained for systems that use the synchronous update scheme, which opens up new possibilities for study.

Abstract. Storm surge is one of the most significant marine dynamic disasters affecting coastal areas worldwide. A comprehensive study of its mechanisms is vital for improving forecasting capabilities and developing more prevention strategies. In this study, a two-dimensional (2D) numerical model based on the Advanced Circulation Model (ADCIRC) was employed to examine the characteristics of storm surges and the mechanisms of tide–surge interaction in the Pearl River Estuary (PRE) during Typhoon Nida (2016). Three distinct model runs were conducted to differentiate between variations in water levels attributable to astronomical tides, storm surges, and their combined effect. The results indicated that storm tides are primarily modulated by tides through tide–surge interactions. The nonlinear effect of tide–surge interactions is primarily generated by the nonlinear local acceleration term and convection term from the tide–surge interactions in the study area, as derived from mathematical terms. However, in regions of shallow water, such as the northern part of the island of Qi'ao and Shenzhen Bay, they are predominantly governed by the nonlinear wind stress term and the bottom friction term. Furthermore, the variations in the y components of the nonlinear momentum terms are more significant than those in the x components. To investigate the impact of the tidal phase on the storm surge response to Typhoon Nida, the timing of the landfall was altered in order to introduce variations in PRE characteristics. The results demonstrate that the contribution ratio of each nonlinear term remains relatively constant, while the magnitudes exhibit fluctuations contingent on the timing of the landfall. However, further studies on additional typhoon events, especially with onshore winds, are needed, together with a comprehensive consideration of the meteorological processes and mechanisms of tidal-wave propagation inside and outside the estuary. The model system could still be improved in the future.