Mathematical Texts Research Articles

Higher Arithmetic

This paper investigates the historical context, internal content, and educational significance of Higher Arithmetic, a 1919 mathematics text by David Eugene Smith and George Wentworth, which has been insufficiently studied. Through historical and content analysis, this study reveals the authors’ intention to provide a review and extension of knowledge in a modern context of post-World War I industrial United States, intended for both students and pre-service teachers who need thorough teacher preparedness. Although Higher Arithmetic may not be well- recognized today, its progressive intentions for teacher training and the modernization of material remain goals that are as important today as in Smith and Wentworth’s time.

Evolution and Significance of Sine Function in Ancient Indian Mathematics

This research article delves into the historical and mathematical intricacies of the construction of sine functions, particularly focusing on ancient Indian mathematical concepts and their evolution. It begins by highlighting the significance of sine (jya) as a fundamental trigonometric function in Indian mathematics, alongside its synonyms in Sanskrit. The historical journey of the term “sine” is traced back to ancient Indian mathematical texts, which were later translated into Arabic and then into Latin, ultimately leading to the adoption of the term “sine” in modern mathematics. The article elaborates on the definitions and interpretations of sine (ज्या), cosine (कोटिज्या), versed sine (उत्क्रमज्या), and co-versed sine (कोट्युत्क्रमज्या) in Indian mathematical traditions. It explains how these functions were conceptualized and utilized in astronomical calculations, emphasizing their geometric and trigonometric significance. Furthermore, the article discusses the division of a circle into quadrants (Vṛttapādas) and the concept of Trijyā (radius), highlighting the mathematical sophistication and terminology used by ancient Indian mathematicians. Several mathematical formulas related to the construction and calculation of sine functions are presented, including formulas for finding sine values at specific angles such as 45 degrees, 18 degrees and 72 degrees. These formulas showcase the mathematical ingenuity and problem-solving skills of ancient Indian mathematicians. Overall, this research article provides a comprehensive overview of the historical development and mathematical foundations of sine functions in ancient Indian mathematics, shedding light on their enduring legacy and contributions to modern trigonometry.

The manuscript diagrams of Theodosios’ Spherics

This paper is a study of the visual characteristics of the medieval manuscript diagrams found in Theodosios’ Spherics. It provides a new taxonomy of eight types of diagrams found in the manuscript sources, and examines the way in which geometric information is encoded in the manuscript diagrams using a number of visual encodings. It then argues that these diagrams can be used to address ongoing research debates about whether the striking visual characteristics of medieval diagrams in mathematical texts are the result of deliberate choices by the premodern authors, the result of accidents of transmission, or both. Along the way, the paper identifies a further type of transmission error that is prevalent in the medieval transmission of mathematical diagrams.

Deep transformer-based architecture for the recognition of mathematical equations from real-world math problems

Identifying mathematical equations from real-world math problems presents a unique and challenging task within the field of Natural Language Processing (NLP). It has a wide range of applications in various areas, such as academics, digital content design, and the development of automatic or interactive learning systems. However, the accurate understanding of these equations still needs to be enhanced due to the intrinsic complexity of mathematical symbols and various structural formats. Additionally, the unique syntax, diverse symbols, and complex structure of mathematical equations present significant obstacles that traditional NLP methods and Optical Character Recognition (OCR) systems need help to overcome. In this research, we utilize deep transformer architecture to recognize mathematical equations, and we have utilized our novel dataset with 3433 distinct observations. This dataset, which we have collected to include a diverse range of mathematical equations, is used to predict six (z=x+y, z=x−y, z=xy, z=x/y, y=x! and y=x) basic mathematical equations. We applied different transformer-based architectures, such as BERT, ELECTRA, XLNet, RoBERTa, and DistilBERT, and BERT performs best with 99.80% accuracy. To the best of our knowledge, this is the first NLP work in any language where we recognize the equation from the mathematical text.

Диагностика сформированности смыслового чтения при работе учеников средней школы с учебными математическими текстами

Learning is working with a variety of educational texts based on reading, understanding, evaluating and interpreting information. To identify the result of reading an educational mathematical text and the quality of understanding the meaning of the text space created by educational materials, it is necessary to diagnose reading, namely, the level of semantic reading. In the modern study of reading, the PISA international monitoring, the concept of reading literacy, its components and levels are defined. The success of understanding more complex educational mathematical texts will depend on the level of reader literacy. If the levels of reading literacy have evaluation criteria and diagnostic apparatus, then how to assess the level of semantic reading? The article examines the relationship between the levels of reading literacy and text comprehension, models for assessing the semantic complexity of the text and semantic reading skills, various level structures for diagnosing the understanding of a mathematical educational text. Based on the studied scientific material, the complexity of the educational mathematical text "Triangle" was assessed according to criteria such as the number of words, simple words, scientific terms and their complexity, the number of mentions of scientific terms in each paragraph. Diagnostic material has also been created to check the understanding of this text, each task of which is aimed at determining the understanding of each information block and assessing the degree of development of semantic reading.

Triggering A Renaissance in Historical Textual Studies in Turcology: Text Encoding Initiative (TEI) and the Initial Standardised Ottoman Mathematics Text

This article seeks to answer a critical question about the impact of digital transformation, which global technological and digital developments have made a necessity, on historical textual studies in Turcology. The article draws attention to the need for digital transformation in historical textual studies and the lack of literature on the use of TEI in Turcology. The research considers historical textual studies as an object of digital cultural heritage, digital humanities, and digital scholarship. It is evaluated through the Text Encoding Initiative (TEI) standard. The aim is to establish a symbiotic relationship between historical textual studies and the Text Encoding Initiative (TEI) and offer a new perspective on the latest digital environment for studying Turkish manuscripts and the future of this research. It also explains how manuscripts (especially in the Ottoman field) should be encoded at TEI to enable interdisciplinary cooperation in digital editions. A case study of an 18th-century Ottoman mathematical text was encoded in TEI, and the process is explained in detail for those working in Turcology and those who want to do text editions with this method. The article's results are shared with the reader by presenting a Github link. These results constitute a new source for the international literature on digital encoding of historical texts and set standards for digital text editions to be used in historical text studies in Turcology. In this way, it aims to trigger a renaissance promoting the use of TEI in historical textual studies and to take the first step towards its eventual transformation into a renewed field of research.

Web-Based Math Braille Translator Using Indonesian Braille Symbols

According to estimates from the Indonesian Ministry of Health in 2017, at least 3,750,000 people in Indonesia have visual impairments. This significant number highlights the need for special attention to their educational needs, especially in mathematics. Educational media is essential to facilitate learning for visually impaired students and educators, as they often struggle with understanding math due to limited access to visual content. One solution is to use Mathematical Braille symbols, yet technology that translates these symbols into mathematical text is still limited. To address this, the author has developed a web-based Math Braille Translator application aimed at assisting visually impaired individuals and educators. This website translates mathematical Braille into notation by inputting Braille using the keys s, d, f, j, k, and l on the keyboard. The application is developed using the waterfall methodology and is based on the Indonesian Mathematical Braille Symbol Reference NOMOR 056/U/2000.

Logical Models of Mathematical Texts: The Case of Conventions for Division by Zero

Arithmetical texts involving division are governed by conventions that avoid the risk of problems to do with division by zero (DbZ). A model for elementary arithmetic texts is given, and with the help of many examples and counter examples a partial description of what may be called traditional conventions on DbZ is explored. We introduce the informal notions of legal and illegal texts to analyse these conventions. First, we show that the legality of a text is algorithmically undecidable. As a consequence, we know that there is no simple sound and complete set of guidelines to determine unambiguously how DbZ is to be avoided. We argue that these observations call for further explorations of mathematical conventions. We propose a method using logics to progress the analysis of legality versus illegality: arithmetical texts in a model can be transformed into logical formulae over special total algebras that are able to approximate partiality but in a total world. The algebras we use are called common meadows. Our dive into informal mathematical practice using formal methods opens up questions about DbZ which we address in conclusion.

Effects of AR mathematical picture books on primary school students' geometric thinking, cognitive load and flow experience

As an important branch of mathematics, geometry plays a very significant role in developing students’ thinking, but many students need to improve their geometric thinking abilities, especially in understanding graphics and forms. Therefore, it is worthwhile to explore techniques and methods for developing students’ geometric thinking, and Augmented Reality (AR) technology can visualize geometric objects to enhance mathematic learning. This study designed an interactive mathematical picture book by utilizing AR technology, and used a quasi-experimental setting to compare the effects of three different reading materials, including AR mathematical picture books, mathematical picture books and mathematical texts. Eighty-three fourth-grade students at an elementary school in China were divided into three groups to participate in the experiment. The effects on primary school students were measured in terms of geometric thinking level, cognitive load and flow experience. The quantitative findings indicated that after reading the AR mathematical picture book, students’ geometric thinking levels 0–3 were significantly improved (p < 0.05). Moreover, compared with the other two groups, the students who finished reading the AR mathematical picture book scored significantly higher in geometric thinking and flow experience and significantly lower in cognitive load. Analysis of the semi-structured interviews showed that students enjoyed using AR mathematical picture books and felt that it resulted in a better flow experience and lower cognitive load. As a result, this research indicates that AR mathematical picture book has a significant role in increasing students’ geometrical thinking, and provide a successful teaching case for the application of AR/VR in the classroom.

The Schur–Potapov Algorithm in the General Matrix Case and Its Application to the Matricial Schur Problem

This paper is a generalization of the topic handled in Bogner et al. (Oper Theory 1(1):55–95, 2007a, Oper Theory 1(2):235–278, 2007b) where the Schur–Potapov algorithm (SP-algorithm) was handled in the context of non-degenerate p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur sequences and non-degenerate p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur functions. In particular, the interplay between both types of algorithms was intensively studied there. This was itself a generalization of the classical Schur algorithm (Schur in J Reine Angew Math 148:122–145, 1918) to the non-degenerate matrix case. In treating the matrix case a result due to Potapov (Potapov in Trudy Moskov Mat Obšč 4:125–236, 1955) concerning particular linear fractional transformations of contractive p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} matrices was used. For this reason, the notation SP-algorithm was already chosen in Dubovoj et al. (Matricial version of the classical Schur problem, volume 129 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1992). We are going to introduce both types of SP-algorithms as well for arbitrary p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur sequences as for arbitrary p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur functions. Again we will intensively discuss the interplay between both types of algorithms. Applying the SP-algorithm, a complete treatment of the matricial Schur problem in the most general case is established. A one-step extension problem for finite p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur sequences is considered. Central p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur sequences are studied under the view of SP-parameters.

Students’ reception of two alternative arrangements of mathematical symbols and words: Differences in focus and text navigation

To engage with specialized subject content, students must develop adequate reading skills. In mathematics, this includes to integrate information from different semiotic resources. This study elucidates how differences in the structural connections between mathematical symbols and written language in mathematics texts can affect the reading process. With the help of eye-tracking techniques, we investigated differences in focus and navigation when 15-year-olds read task texts in two distinct designs: a traditional design with written language presented in lines and all connections based on semantics; and a design including a graphic emphasizing links between symbols and explanations. While the graphic design was found to facilitate fast interpretation of the symbol–language connections, the traditional design seemed to encourage global reading, involving more text parts. When designing texts for mathematics learning, structural connections may be chosen to adapt texts to various student groups and purposes.

Третьеклассник и математика: освоение учебного предмета в рамках требований действующего стандарта

The article discusses the organization of education taking into account the requirements of the updated educational state standard, federal educational and work (subject) programs for the mathematical preparation of third-graders. Some features of the updated content of the third year of study and methodological approaches to strengthening the work of the teacher in developing in younger schoolchildren the ability to construct and verify the truth of statements, justify their point of view, understand practical problems, and apply educational and subject-related actions when working with different course contents are presented. Methodological approaches are characterized that allow schoolchildren to ensure better mastery of updated subject content: the ability to work with mathematical text, conduct independent testing and evaluate the results of the solution. Thus, the article substantiates the possibility of taking into account the standard requirements for student independence when organizing the learning process.

How Ecologists Develop the Logistic Model

ABSTRACT The logistic differential equation is ubiquitous in calculus and differential equations textbooks. If the model is developed from first principles in these courses, it is usually done in an abstract mathematical way, rather than in one based in ecology. In this short note, we look at examples of how the model is derived in mathematical texts and compare them to that in an ecology text.

Extraction of the author’s terminology and definitions&#x0D; from mathematical texts

Extraction of the author’s terminology and definitions&#x0D; from mathematical texts

Integration Strategies of Information Technology in Mathematics Education and Student Motivation Enhancement

Abstract Integrating information technology with the curriculum has been closely watched by the educational community, and how well the two are integrated determines the effectiveness of mathematics education. This paper proposes an implementation scheme for mapping mathematical knowledge. Firstly, the Word2vec word vector model is used to acquire data from the original mathematical text, and then the Conditional Random Field model is used to extract the knowledge entities, and the Markov Logic Nets are used to extract the relationships between the knowledge entities. Then, the Path Ranking algorithm is utilized on the basis of entity alignment to realize the knowledge inference of the subject knowledge graph and mine the learning paths in the knowledge graph. Through the experimental and questionnaire methods, it was found that the math scores of the experimental class before the comparison experiment increased by 9.4337 points, which was significantly ahead of the control class. The average number of students who thought knowledge mapping teaching greatly improved their interest in math learning was 46.81%, while 31.29% believed that the new method had a relatively positive impact on their motivation to learn. It shows that knowledge mapping can be an effective method of teaching information technology.

Tikaram and Chandrakala Dhananjaya: A collaborative couple in mathematics from Nepal

Within the history of mathematics and mathematics education in Nepal, Tikaram and Chandrakala Dhananjaya are relatively well-known figures for their two books Śiśubodha Taraṅgiṇī and Līlāvatī. This is despite there being almost no archival or manuscript materials offering a window into their lives: we have no letters, notebooks, diaries, or school records. Rather than focusing on either individual in isolation, in this article we present an argument for considering the Dhananjayas as an analytically indivisible collaborative couple in mathematics. Of the two aforementioned books, one is attributed to Chandrakala and the other to Tikaram; but in fact, both are translations of the same Sanskrit source text, Līlāvatī, into Nepali. By comparing the mathematical contents of these two works, which were published within a few years of each other, we explore what it means to be an author or translator of a mathematical text and propose different models of spousal collaboration which could plausibly have been adopted by the Dhananjayas. In the absence of documentary evidence, the impossibility of delineating each individual’s contributions removes the temptation to focus exclusively on apportioning credit. Instead, we offer the alternative perspective of considering what labour must have been undertaken to bring their books to publication.

Students' Inquisitive Questions Predict Their Understanding of Mathematics Texts

Students' Inquisitive Questions Predict Their Understanding of Mathematics Texts

Applying a new framework of connections between mathematical symbols and natural language

A reader of mathematical text must often switch between reading mathematical symbols and reading words. In this study, five different categories of structural connections between symbols and language, which invite such switches, are presented in a framework. The framework was applied in a study of Swedish mathematics textbooks, where 180 randomly selected pages from different educational stages were analyzed. The results showed a significant change in communication patterns as students progress through school. From a predomination of connections based on proximity found in year two, there is a gradual change to a predomination of symbols interwoven in sentences in year eight. Furthermore, more qualitative investigations of the different connections complemented the quantification, both through further explanations of the quantitative results, and through more examples of differences in communication patterns. The implications for readers of mathematics texts are discussed.

Rethinking Proficiency in the Language of Teaching and Learning (LoTL) as a Pillar in the Learning of School Mathematics

Although English has become the preferred language of instruction in most classrooms, including those of mathematics across the world today, it will still remain a second language (L2) to many students and their teachers for a long time to come. In mathematics language research, the attention so far given to the role of English as the Language of Teaching and Learning (LoTL) has been with the regard to the impact of levels of student proficiency in the language. This perhaps explains why those who learn in English as their first language (L1) are perceived as proficient in mathematical language, while L2 learners have to attain a level of proficiency in English first. This is in spite of the current absence of clear benchmarks for satisfactory proficiency in English for successful general learning in school mathematics. The poignant question remains: must L2 learners learn mathematics in English given the double jeopardy they face? This paper presents a critical analysis of findings of students’ interpretation of lexical vocabulary common in mathematics texts and in the classroom language typical of mathematics students in L2 contexts, to highlight the general difficulty of this language (English) to the mathematics learners. The main study from which the paper is extracted employed multiple-case study design to examine Mathematical Language (ML) usage and learners’ conceptual understanding of mathematics in secondary schools in Kenya. Data were collected by questionnaires, classroom observations and interviews. The findings of the paper indicate that a majority of L2 students have low proficiency in LoTL; socio-economic background is a factor of proficiency in the LoTL, and mathematics teachers were not aware of the importance of language in learning mathematical concepts. The main conclusion of the paper is that interpretation of mathematical concepts is language dependent which is a challenge to learners not versed in the LoTL. The paper reco

Perceived Influence of Test Anxiety on the Academic Performance of Secondary School Students in Physics

Purpose: This study aimed to investigate the perceived influence of test anxiety on the academic performance of secondary school students in physics in Benue State, Nigeria. Methodology: The research utilized a descriptive survey design, with a target population of 15,060 students from government-approved secondary schools in the Obi Local Government Area of Benue State. A random sampling technique was employed to select 250 students from five secondary schools. The data collection instruments employed were the Mathematics Text Anxiety Questionnaire (MTAQ) and Mathematics Performance Text (MPT). The collected data were analyzed using measures such as means and standard deviation to address the research questions, while the analysis of variance was conducted to test the hypothesis at a significance level of 0.05. Findings: The findings indicated a significant difference in the influence of test anxiety levels exhibited by students on their average performance in physics within the secondary schools of the Obi Local Government Area in Benue State. Moreover, the results revealed a significant difference in the influence of test anxiety levels exhibited by male students on their average performance in physics in secondary schools. Significance: Similarly, a significant difference was observed in the influence of test anxiety levels exhibited by female students on their average performance in physics within secondary schools. Based on these findings, recommendations for school management are proposed.