Mathematical Texts Research Articles

Infinite analytical procedures for the computation of logarithms in works by Benito Bails (1731–1797)

During the Spanish eighteenth century, a process of modernization took place in scientific knowledge, partly driven by the circulation and appropriation of new scientific ideas. In this context, the Spanish mathematician Benito Bails (1731–1797) published his course Elementos de Matemática (Elements of Mathematics) consisting of ten volumes (1779–1799), in which, among other subjects, he presented one of the most complete mathematical developments of logarithmic calculation methods of his time, by using the infinity through infinite series. The aim of our article is to demonstrate how algebraic analytical reasoning enabled Bails to obtain new and more efficient infinite algorithms that converge more quickly in the computation of logarithms in any system. We show how Euler's number ‘e’ is calculated, probably for the first time in Spanish teaching, in an eighteenth century mathematical text. Our analysis concludes that Bails’ course constituted an innovation and provides evidence of its creativity, originality and ingenuity.

The Dynamic Vectors-Based Attention Model for Chinese Mathematical Term Extraction

Term extraction aims to extract domain-specific terminology from documents. Secondary school mathematics terminology plays a guiding role in the learning and teaching of secondary school mathematics. We found that it needs to be more corpus in this field and that there are two main issues in extracting the terms from the middle school mathematics text: polysemy of one character problem and nested terms problem. Accordingly, we have constructed a secondary school mathematics corpus and proposed a model to resolve these problems, termed BERT-LLA-CRF. Using pre-training to obtain dynamic word vectors to improve the model’s understanding of text semantics and solve the problem of multiple meanings of one word, the proposed model integrates attention mechanisms and Chinese character information to solve the nested terminology problem. The experiment results show that the proposed model effectively solves those problems and achieves good middle-school mathematics term extraction results.

George Kiraz and Sabine Schmidtke, eds. Scribal Habits in Near Eastern Manuscript Traditions

George Kiraz and Sabine Schmidtke, eds. <i>Scribal Habits in Near Eastern Manuscript Traditions</i>

Homotopy cartesian squares in extriangulated categories

Abstract Let ( C , E , s ) \left({\mathcal{C}},{\mathbb{E}},{\mathfrak{s}}) be an extriangulated category. Given a composition of two commutative squares in C {\mathcal{C}} , if two commutative squares are homotopy cartesian, then their composition is also a homotopy cartesian square. This covers the result by Mac Lane [Categories for the Working Mathematician, Second edition, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York, 1998] for abelian categories and by Christensen and Frankland [On good morphisms of exact triangles, J. Pure Appl. Algebra 226 (2022), no. 3, 106846] for triangulated categories.

Mathematics in the Spanish press: a case study of the 18th century journal Semanario de Salamanca

Old mathematics books and textbooks have always been sources of relevant information for researchers working on the history of mathematics and mathematics education. However, helpful information about this field can also be found in journals, ministerial decrees, or notes. Consequently, this article analyzes the contents related to mathematics or mathematics education in an 18th-century weekly journal published in the Spanish city of Salamanca, Semanario de Salamanca. To this end, we conducted a qualitative and descriptive investigation using a content analysis technique, which is widely used in research in this related area. Our results show that despite the fact that the weekly publication under investigation was specialized neither in mathematics nor in science, it included mathematical problems and solutions, mathematical texts, job advertisements for mathematics teachers, mathematics book reviews, and opinions about mathematics. Therefore, the present work expands knowledge of the presence and dissemination of mathematics and mathematics education in 18th-century Spanish society.

Мовностилістичні вимоги до кваліфікаційних робіт майбутніх учителів математики

The article reveals the general requirements for the qualification papers of future Mathematics teachers in the context of observing linguistic and stylistic norms of the scientific style, forming the language culture of a modern teacher. The social and communicative importance of the influence on the development of the teachers’ mathematical and scientific language has been analyzed. The peculiarities of the scientific language and the main flaws of its written presentation – the scientific style text – in the qualification papers of Mathematics teachers, who obtain the second educational level, have been revealed. Considerable attention has been paid to the formation of mathematical texts, in particular to issues of modern mathematical terminology. Typical linguistic and stylistic mistakes of authors, which they make when writing, editing and translating mathematical texts, have been analyzed. These mistakes to the specifics of scientific technical terminology, lexical and stylistic features of the style of scientific language have been related. Examples of the most frequently used lexical constructions and stable inflections in mathematical texts have been given. Recommendations on avoiding mistakes when constructing a text in a scientific style have been given. Selecting appropriate Ukrainian equivalents of commonly used lexemes of the Russian language, using established word combinations, and observing the grammatical norms of the literary language have been provided.

Формирование у студентов педагогических направлений подготовки коммуникативной компетенции в области математического образования

We present a theoretical substantiation and practical implementation of the educational environment for students to study a mathematics course focused on the formation of communicative competence. The purpose of the study is to theoretically substantiate, practically determine and experimentally create an educational environment to develop communicative competence in future primary school teachers in the study of mathematics, as communicative competence is a necessary condition for the further successful pedagogical activity of a graduate. Research methods: a) theoretical: analysis, evaluation, interpretation of scientific literature on the research issue; b) empirical: pedagogical experiment. Conclusions and recommendations: in conducted theoretical study we substantively characterize the main approaches to the definition of the category of “communicative competence”, show the importance of communicative competence of future elementary school teachers, taking into account the specifics of the language of meaningful and formal mathematics. The importance of forming the linguistic and pragmatic components of communicative competence in students in the course of studying the mathematical content at the university is theoretically substantiated. The effectiveness of the use of dialogue situations during the work of students in small groups with a mathematical text in order to develop the communicative competence of future teachers has been experimentally proven. The practical significance of the study lies in the development of an educational environment for studying the course of mathematics in the pedagogical areas of the university, focused on the formation of students' communicative competence in mathematical education. The practical results of the study can be used in the practice of the work of mathematics teachers at universities to form the communicative competence of future teachers in the study of the discipline “Mathematics”.

CALCULATOR PREVALENCE IN MATHEMATICS CLASSROOMS: CASE OF EKUDIBENG CIRCUIT IN SOUTH AFRICA

This study influenced by the Technology adoption theory, explored the prevalence of calculators in Ekudibeng circuit mathematics classrooms in South Africa. Pragmatism guided assessment of technology adoption by allowing application of both quantitative and qualitative data analysis to portray reality. A sequential design of document analysis, a survey, lesson observations, focus group discussions and in-depth interviews was used. Empirical data was collected from a purposive sample of 154 participants, composed of 41 teachers of mathematics and 113 learners from grade 8 to 12 in Ekurhuleni South district secondary schools. A sample of 26 teachers’ schemes and mathematics school syllabuses was analysed. The study found (71%) prevalence for calculator use in both public and private schools mathematics classrooms. A hypothesis test at 5% level of significance confirmed an association between gender and calculator use preference. Although all teachers reported willing to use the calculator, 54 (35%) reported, not possessing a personal calculator. Study found five calculator models in use. The majority used the SHARP model. National examinations disallowed the graphic FX-CG50 because of its perceived complexity. Calculators were used for computation of ( r2) regression coefficient correlation and evaluation for logarithms and trigonometry ratios for transformation functions such f(x) = 2Sin θ. Factors retarding calculator adoption include; lack of financial resources for high density public school learners, limited teacher commitment for calculator use and low teacher technical knowhow of using calculator for instruction.. National examinations not emphasising use of calculators, mathematics text books do not show how calculators can be used for instruction. Study recommends use of a single calculator model as a district policy. Authoring of a teachers’ handbook for the use of calculators for instruction is called for. Mathematics teachers’ staff development workshop on the application of calculators is long overdue.

Students’ Perception and Academic Performance in Mathematics in Secondary Schools in Handeni District – Tanga, Tanzania

This paper presents part of a wider study on Student’s perception and performance in Mathematics in secondary schools of Hendeni District, Tanzania. Using data collected from 100 respondents, the paper reports that there is a statistical significant relationship between students’ perception and their performance in Mathematics. In other words, the students who perceive Mathematics to be a hard subject are more likely to perform poorly than their counterparts who perceive it to be a simple subject. Though other factors, as per the oral informants (such as teacher-student ratio, poor school facilities, inadequate Mathematics text books and the methods of teaching Mathematics) also accounted for students’ poor performance in Mathematics, their contribution was minimal. The study recommended the need for school managers and classroom teachers to build the self-esteem of students to develop a feeling that they have the capacity to perform all subjects excellently.

Why Mathematics at the High School Level Evolved to Include Linguistics

This article foregrounds the fundamental role that language can play in communicating mathematical ideas and contexts. The findings from this article contribute to an understanding of different ways textbook series with particular orientations make opportunities available for students to develop forms of agency and autonomy during classroom learning. The article also contributes methodology for analyzing mathematical texts of different genres.

Understanding mathematical texts: a hermeneutical approach

The work done so far on the understanding of mathematical (proof) texts focuses mostly on logical and heuristical aspects; a proof text is considered to be understood when the reader is able to justify inferential steps occurring in it, to defend it against objections, to give an account of the “main ideas”, to transfer the proof idea to other contexts etc. (see, e.g., Avigad in The philosophy of mathematical practice, Oxford University Press, Oxford, 2008). In contrast, there is a rich philosophical tradition dealing with the concept of understanding and interpreting texts, namely philosophical hermeneutics, represented, e.g., by Schleiermacher, Dilthey, Heidegger or Gadamer. In this tradition, “understanding” generally refers to the integration in a comprehensive (historical, existential, life-worldly, ...) context. In this article, we take some first steps towards exploring the question how the ideas from philosophical hermeneutics presented in Gadamer’s “Truth and Method” apply to mathematical texts and what (if anything) can be learned from these for the didactics and presentation of mathematics.

Texts, Practice and Practitioners: Computational Cultures at Work in Early Modern South India.

This essay will discuss the hegemonic role that texts have come to play in the historiography of subcontinental mathematical traditions. It will argue that texts need to be studied as records of practices of people's working lives, grounded in social hierarchies. We will take particular mathematical texts to show how different occupational registers have come to shape practices that defy the binaries of concrete and abstract, high and low mathematics or the pure and applied conundrum. Measuring, counting and accounting practices as part of the routine work of practitioners performing their caste occupations then provide us with a spectrum of the computational activities that controlled and regulated the lives of people in the past. In the process the act of computing itself gained certain political values such as cunning and manipulation, identified with professions of village accountant and merchant, for example. Drawn from my earlier work on these records, I discuss the occupational role of the accountant as a political functionary who assessed and authenticated the measurements of land and produce in the village, making values of the labor performed by others, and creating avenues for his own proficiency as a mathematical practitioner.

Comprehension of argumentation in mathematical text: what is the role of interest?

This article describes the research design and findings from a use-inspired study of online text-based mathematics resources. We sought to understand whether and how existing mathematics interest, together with the learner characteristics of prior coursework in mathematics and proof scheme, influenced comprehension of mathematical argumentation and triggered interest in two types of mathematics text: (1) text featuring concrete, real-world applications (public domain) and (2) text with abstract and generalized modes of expression and content (abstract domain). Using an online assessment and person-centered analyses, we studied 64 (32 M, 32 F) undergraduate students who were and were not pursuing advanced mathematics coursework. Cluster analysis revealed two participant groups. Less mathematically immersed (LMI) participants improved comprehension of mathematical argumentation when working with public domain text, performing comparably to the more mathematically immersed (MMI) cluster in this domain; those in the MMI cluster performed comparably across text domains. In addition, LMI participants were more likely to identify public domain text as more interesting than abstract text, and they also were more likely than those in the MMI group to explain this by citing public rather than abstract domain reasons. Taken together, study findings suggest that interest in coordination with other learner characteristics scaffolds comprehension of mathematical argumentation. This study makes contributions to interest theory, understanding the role of interest in comprehension of mathematical argumentation, and ways in which practitioners might leverage student interest to promote comprehension.

The Akkadian Expression Lā Watar

Abstract The Akkadian expression lā watar (lā matar) appears in two dozen Old Babylonian letters, in Standard Babylonian lexical lists and, arguably, in an Old Babylonian mathematical text and in an Old Assyrian letter. Though the expression has been known since the early twentieth century, it has never been an object of a special study. Commentaries in the editions and the dictionaries posit for lā watar four meanings: ‘that's enough’, ‘no longer’, ‘really’ and ‘finally’. These quite different meanings can hardly be expressed by one and the same word. The question is therefore: Which of the four are correct, if any? This study attempts to answer this question by analysing all available examples of lā watar and using cross-linguistic evidence. The inquiry shows that lā watar means ‘now, but not before’. Contextual English translations are ‘finally’, ‘at last’, ‘only now’, ‘(only) just’ and ‘from now on’.

Scaffolding Close Reading of Mathematical Text in Pre-service Primary Teacher Education at the Tertiary Level: Design and Evaluation

Even in the digital age, learning mathematics at an academic level still requires much reading of mathematical text. Research has shown that reading mathematical text requires readers to engage with all the structures of the book and with its pedagogical voice, making connections, and plausible reasoning. Specific practices and strategies that support the close reading of mathematical text have been suggested; however, descriptions and empirical evaluations of materials designed to support these activities are rare. We present the design and first evaluation cycle of materials developed in a design research project that aims to scaffold close reading of mathematical text. The materials were designed and evaluated in a German university course on elementary geometry for first-year teacher education students who study mathematics to become primary teachers. The reading strategies were explained and modeled for students in reading-strategy videos. Additionally, close reading of mathematical text was scaffolded by close-reading tasks and homework tasks and problems that build on the reading strategies and were specifically designed to foster understanding of the mathematical text. Survey data were collected from 296 students to evaluate their use of and attitude toward the different materials. The quantitative results indicate that students used the materials and were generally able to learn the course content by themselves. From all provided materials, they found the close-reading tasks most helpful. A qualitative analysis of answers to open questions revealed issues with different materials, particularly with the script, and requests for additional materials. The issues with the script were categorized inductively. The categories are presented as a qualitative result of the study and discussed.

Первоклассник и математика: особенности обучения по обновлённому стандарту

Teaching mathematics to younger schoolchildren in accordance with the Federal State Educational Standard for Primary General Education (FSES PGE) has a number of features. Leading among them is the development of spatial thinking, logical and algorithmic thinking, mathematical speech, functional mathematical literacy in accordance with the age capabilities of children. The article discusses the organization of teaching first-graders, focused on the implementation of the requirements of the updated version of the Federal State Educational Standard of the PGE for mathematical preparation: listening and reading a mathematical text, discussing and modeling a mathematical situation, searching for different solutions, drawing up an action plan. Appropriate skills and actions of first-graders can be successfully formed on the basis of traditional subject content.

Anachronisms in the History of Mathematics: Essays on the Historical Interpretation of Mathematical Texts , by Niccolò Guicciardini, ed.

Anachronisms in the History of Mathematics: Essays on the Historical Interpretation of Mathematical Texts , by Niccolò Guicciardini, ed.

What Can We Learn from the Different Understandings of Mathematical Literacy?

Mathematical literacy, quantitative literacy, numeracy, matheracy, disciplinary literacy, and content-area literacy are among a plethora of terms used to link mathematics and literacy. In addition to this abundance of terms, the content of the terms is not precise, and in some cases the terms are used interchangeably. I delve into this landscape and dissect the meaning of each of them. From such an analysis of mathematical literacy, numeracy, and quantitative literacy, we learn about the importance of quantitative practices and elementary mathematics in society, including both how mathematics is necessary for everyday life but also how it can be used to change the world. By discussing content-area literacy in mathematics, we learn about text in mathematics and the shortcomings of general text-competence approaches are revealed. By analyzing disciplinary literacy in mathematics, we learn to appreciate mathematics as a subject with its own language, tools, techniques, and ways of thinking that emphasize inquiry and experimentation following the heritage from Euclid, Archimedes, Newton and Euler. Having so many competing terms and approaches has a negative effect on the academic discourse but also on the communication with the world outside of academia.

Kant on mathematical axioms

This article is intended to explain the notion of “mathematical axioms” presented in Kant’s Critique of Pure Reason.This notion is developed mainly within the framework of a justification of the thesis of the methodological dualism of the rational sciences (mathematics and metaphysics). We argue that there are significant differences between the critical notion of mathematical axioms, the pre-critical developments and the Wolffian definitions. The notion of “axiom” that Kant intends to take from mathematical procedures is inscribed in his peculiar way of thinking this science. This paper studies the considerations of (i) Wolff’s mathematical texts, (ii) the pre-critical texts and (iii) the Critique of Pure Reason, and mentions the differences between them in the conclusion.

Comparing lexical bundles across three advanced mathematical text types: a corpus-based genre-focused investigation

Although much research has been conducted on academic multi-word units across a range of academic disciplines, little attention has been given to the distribution of such units in mathematics. Drawing on three parallel corpora of doctoral dissertations, textbook chapters, and research journal articles, this study compares the use of recurrent bundles across three mathematical text types using a combination of corpus methodology and linguistic analysis. The corpus treatment of bundles is then supplemented with an analysis of their structural forms and discourse functions. A total of 291 four-word bundles that recur at least 25 times per million words, and appear in 15% or more of the texts were retrieved and their structural and functional attributes examined. Results indicate that student-produced texts tend to use fewer and less varied lexical bundles than research articles and educational texts. The three text types also show considerable variation in the use of structural forms of recurrent bundles, with the student writing incorporating more phrasal patterns, while expert texts employing clausal constructions. Functionally, the three text types exhibit striking similarities, making greater use of research- and text-oriented bundles but producing a much narrower range of participant-based bundles.