Mathematical Language Research Articles

Understanding mathematics preservice teachers’ beliefs about English learners using the Mathematics Education for English Learners Scale (MEELS)

ABSTRACT This article outlines the development and validation of the Mathematics Education of English Learners Scale (MEELS) to understand mathematics preservice teachers’ beliefs about teaching mathematics to English learners (ELs). Nine hundred and twenty-five PSTs from 17 states across the U.S. were divided randomly into two samples for the analysis. A Principal Component Analysis was conducted with one sample of 463 respondents, which suggested a four-factor model and a five-factor model. A Confirmatory Factor Analysis with the other sample of 462 respondents suggested that the four-factor model was a better fit. The 19 item four-factor model consisted of the following factors – Classroom Strategies, Native Language, Language in Math, and Comparison with non ELs. The factors show that preservice teachers’ beliefs about teaching ELs cluster around teaching moves in the mathematics classroom with ELs, the use of students’ native language in the class, the role of language in mathematics, and balancing the teaching and learning of both ELs and non ELs in the mathematics class. The MEELS can be used in teacher preparation to evaluate preservice teachers’ beliefs in the program. The factors of MEELS can highlight areas where preservice teachers need more focus.

Unity as an Indicator of Theory Choice

While choosing scientific theories scientists often show natural biases, such as regarding unity, identity, simplicity, and novelty as indicators of theory choice. Unity is determined by the heterogeneous conceptual and empirical scope directly involved in theory through a logical intermediary. In physics, the pursuit of unity is one of the unremitting motives for scientific innovation and progress. Maxwell’s Electromagnetic theory unifies electricity, magnetism, and light. Einstein unified not only time and space but also matter and energy. Einstein failed to complete the unified field theory in his lifetime, but his pursuit of unification was not wrong. Quantum mechanics is a theory of describing nuclear forces in mathematical language, which successfully unifies three forces except for gravity. In today’s view, string theory is not generally acceptable to most people, but it has some strong local advantages, so it can be pursued

The influence of semantic alignment on the performance of addition and division operation: age-related differences.

Semantic alignment refers to the cognitive tendency that people always seek for the matching of semantic relations between mathematical cognition and objects in specific situations. In developing individuals, language ability and mathematical ability are in the development stage, and the development of semantic consistency is more helpful to reveal the relationship between the two. From the perspective of semantic alignment, this study aims to explore the age-related difference in the effects of semantic relationship on arithmetic operations. Sixty-two children and 62 adults were recruited to perform an arithmetical verification task. The results showed that: (1) Compared with children, adults had shorter response times and higher accuracy in mathematical operations. (2) Both adults and children showed longer response times in division condition than that in addition condition. (3) Children are more affected by semantic alignment than adults in addition. (4) For addition operation, participants performed better under semantic consistent condition than under semantic inconsistent condition, but the opposite result was found for division operation. It indicated that semantic consistently promoted the addition operation and semantic inconsistently promoted the division operation. This suggests that teachers should pay attention to integrating mathematical conceptual knowledge and language context in future educational practice.

Survey of Ideal Lecturer Interested by Students IKIP Budi Utomo Malang

Education is one of the means to answer various challenges related to the development of information, globalization, free markets, and even the problem of national and state harmony. In college, the lecturer is a figure who has an important role in the college life of a student. The most common factors used by students as parameters for the assessment of their lecturers are seen from three factors, namely the way of teaching, giving lectures, and giving grades. The research approach used is descriptive quantitative, with survey method. This research was conducted at IKIP Budi Utomo Malang during July 2022. The population in this study were students of IKIP Budi Utomo Malang while the research sample consisted of 7 study programs, namely physical education, health and recreation, mathematics education, biology education, language education and education. Indonesian literature, English language education, sociology history education, and economic education with 50 students. The results of the study indicate that lecturers have a fairly dominant influence on the quality of learning because it is the lecturer who is responsible for the learning process in the classroom even as an education provider in the campus environment.

Mathematical Model Assisted Six-Sigma Approach for Reducing the Logistics Costs of a Pipe Manufacturing Company: A Novel Experimental Approach

This research addresses and analyzes the results of a six-sigma approach used to optimize the logistics costs of a pipe manufacturing company. Two mathematical models are developed for containers to control the company’s logistics. The Mathematical Programming Language (MPL) software is used to generate and solve these models. The results verify that the proposed mathematical models result in the company’s logistics improvement, especially in the DMAIC (define, measure, analyze, improve, and control) cycle by providing flexibility in choosing the most appropriate containers for logistics.

Mathematics and Physics; Brain and Universe machine languages

Revelations of the nature of existence, through humans’ cognitive systems, have been evolving since the development of language. Part of such revelations, appearing early in the history of civilization, were geometrical forms and numbers, whose beauty and order, both wondrous and mysterious, conveyed a sense of unreality beyond the physical reality normally discerned; and to Plato, these were indications of the “other reality” of which only a few glimpses had occurred. Access to all the attributes of this reality required breaking out of the shackles of (mental) captivity; beautifully portrayed in Plato’s cave allegory. The present era’s sense of wonder about the effectiveness of mathematics in formulating the discoveries of the fundamental and natural laws of existence, on the one hand, and advances in computation, especially in artificial intelligence (AI), on the other have revived the Platonic idea, in one form or other. This revival is supported by some ardent adherents among the very minds who have contributed to furthering our understanding of the physical universe. The work presented here is an attempt to prove that a “physical world” precludes the prospects for “mathematics” to be its “other reality.” This end is achieved by bringing to light the processes involved in the perceptions and discernment of the world by humans and, for that matter, by any creature with a physical central nervous system. The process involved makes it clear that the phenomena of the “physical universe” are played out (one way or other) in our “physical brains” as electrochemical operations, which are (fundamentally) physics-based computations. Therefore, the historically configured “symbolic language of mathematics” is the physical brain’s “machine language” for the specific outputted expressions (formalisms) of the underlying physics of some of the (physically) sensed phenomena behind such computations. This rapidly developing language, much of it emerging in “appearances” foreign to the physical reality, has been taken by many great minds, including Plato, Galileo, Kant, and Wigner, as an indication of an incontrovertible mysterious mathematical world, or its language. However, it is inadvertently the “machine language” whose progress is owed to the successful efforts of the “computational brains” that are blazing the trail for discoveries of more novelties in the physics of the physical universe. The overall claim in this work is based on the ideas that (1) brains only sense the world physically and operate physically, and (2) mathematics has been known on occasion to be predictive of physics, and the latter of new frontiers in the former.

За чистоту математического языка

Математический язык считается одним из самых точных и строгих. Его особенность в том, что каждый символ и знак имеют свой уникальный смысл и значение, что позволяет точно и ясно выражать математические факты и формулировать выводы. Засорение языка математики может препятствовать правильному пониманию математических концепций и увеличивать вероятность ошибок и недоразумений в различных сферах человеческой деятельности. В этой связи при использовании математического языка, необходимо следить за его чистотой, не допуская ни каких вольностей. В статье рассмотрены проблемы засорения языка математики, которые как снежный ком появляются «благодаря» некоторым средствам массовой информации, производителям рекламы, чиновникам, работникам культуры, учителям и преподавателям (в том числе учителям и преподавателям математики), школьникам, студентам и др. Проанализированы наиболее распространенные ошибки, встречающиеся в математической речи не только в быту, но и на интернет-сайтах, в учебниках и учебных пособиях по математике: неправильное название десятичных дробей; неправильное название цифр и чисел; неправильное название букв, используемых в математике; неправильное название математических терминов; неправильное оформление математических текстов; неправильное название математических понятий и разделов математики и др. В статье проводятся параллели с аналогичными проблемами современного русского языка. Mathematical language is considered one of the most accurate and rigorous. Its peculiarity is that each symbol and sign has its own unique meaning and meaning, which allows you to accurately and clearly express mathematical facts and formulate conclusions. Clogging the language of mathematics can hinder the correct understanding of mathematical concepts and increase the likelihood of errors and misunderstandings in various spheres of human activity. In this regard, when using mathematical language, it is necessary to monitor its purity, not allowing any liberties. The article deals with the problems of clogging the language of mathematics, which appear like a snowball "thanks" to some mass media, advertising producers, officials, cultural workers, teachers and teachers (including teachers and teachers of mathematics), schoolchildren, students, etc. The most common errors encountered in mathematical speech not only in everyday life, but also on Internet sites, in textbooks and teaching aids in mathematics are analyzed: the wrong name of decimals; the wrong name of digits and numbers; incorrect name of letters used in mathematics; incorrect name of mathematical terms; incorrect design of mathematical texts; incorrect name of mathematical concepts and sections of mathematics, etc. The article draws parallels with similar problems of the modern Russian language.

MATHEMATICS IN NATURAL SCIENCES AND EDUCATION: THEORETICAL ASPECT

The need to modernize modern education in order to increase the level of students' interest in studying subjects, mathematical and natural sciences, was noted. The role of mathematics and natural sciences for the versatile development of the personality in general, their necessity from the first years of education is defined and integration as an important condition for unification and mutual use in the educational process of mathematical and natural knowledge. The relevance of the problem of using mathematics both in the learning process and for various areas of scientific knowledge is substantiated. The article describes general historical information about mathematics as a science and gives examples of the interpretation of its content by scientists of both the past and the present. The factors determining the importance of the role of mathematics are determined. Its general aspects are characterized from the point of view of mathematical language, its elements, namely, sign, symbol, model, mathematical modeling. The important role of the language of mathematics both in the cognitive activity of a person and in the research of natural sciences at various stages of their development is substantiated.
 As a result of the analysis of scientific and pedagogical literature and generalizations of the use of mathematical methods and tools, examples of the interpretation of the concept of "mathematization of scientific knowledge" are given. The main aspects of the mathematization of sciences and, in particular, its necessity in the formation and development of natural sciences are theoretically substantiated. The necessary conditions for the effectiveness of the application of the concepts and methods of mathematics and the strengthening of the mathematization of knowledge have been identified. Examples of the application of mathematical methods for such sciences as astronomy and chemistry are given. The need for mathematization in natural science is also mentioned in the context of biological sciences, and the stages of this process are characterized.
 The degree of reality of mathematical concepts and structures in natural science has been clarified; of mutual dependence, bilateral connection of mathematics and natural sciences are clarified. That is, that natural science is necessary for modern mathematics, just as it is necessary for it. The significance of mathematization in the integration of natural knowledge in today's conditions is indicated. The author also drew attention to the issue of mathematization of natural sciences in the context of the educational process, i.e., that for the subjects of the study, it involves the penetration of mathematics into natural science; on the problem of conditioning the integration of mathematics with science subjects. It was emphasized that science and mathematics education is gaining importance today and the need for active implementation of STEM education in the New Ukrainian School and, in particular, in the primary level.

DimSum: A Decentralized Approach to Multi-language Semantics and Verification

Prior work on multi-language program verification has achieved impressive results, including the compositional verification of complex compilers. But the existing approaches to this problem impose a variety of restrictions on the overall structure of multi-language programs (e.g. fixing the source language, fixing the set of involved languages, fixing the memory model, or fixing the semantics of interoperation). In this paper, we explore the problem of how to avoid such global restrictions. Concretely, we present DimSum : a new, decentralized approach to multi-language semantics and verification, which we have implemented in the Coq proof assistant. Decentralization means that we can define and reason about languages independently from each other (as independent modules communicating via events), but also combine and translate between them when necessary (via a library of combinators). We apply DimSum to a high-level imperative language Rec (with an abstract memory model and function calls), a low-level assembly language Asm (with a concrete memory model, arbitrary jumps, and syscalls), and a mathematical specification language Spec. We evaluate DimSum on two case studies: an Asm library extending Rec with support for pointer comparison, and a coroutine library for Rec written in Asm. In both cases, we show how DimSum allows the Asm libraries to be abstracted to Rec-level specifications, despite the behavior of the Asm libraries not being syntactically expressible in Rec itself. We also verify an optimizing multi-pass compiler from Rec to Asm, showing that it is compatible with these Asm libraries.

PATHS TAKEN FOR MATHEMATICAL ARGUMENTATION IN THE CONSTRUCTION AND USE OF MATHEMATICAL POEMS

Literary writing is the democratic form of inclusion and development of student aptitude, in which everyone can participate, it is a challenging and stimulating pedagogical activity, in addition to enhancing the development of skills and competencies for meaningful and reflective learning with Mathematics . The school is a space for this “habit-forming force, as it provides those who are directly or indirectly subjected to its influence, not so much particular and particularized thought schemes, but a general disposition that generates particular schemes capable of being applied in different contexts”. different fields of thought and action which can be called cultivated habutus”. (Bourdieu, 2005, p.211). The objective of the article is to present reports of experience of encouraging literary writing with Mathematics with students of the 9th year of Elementary School, stimulating in them the taste for reading and writing with the culture of reading more and writing better, turning to the Mathematics area. The present research also reflects on the importance of the creative didactic transposition of Mathematics contents in the school environment and emphasizes the need for teachers to be equipped to build this knowledge in an innovative and integrated way to the students’ daily lives. in a private teaching unit in Nossa Senhora da Glória, Sergipe, specifically with 9th grade students. The applied methodology is Brainstorming. This technique consists of showing that the student can create a list of ideas on a topic, pointing out everything they can think of on a given topic or subject, transporting it to the types of literary genres: poetry, short stories, chronicles, associating these categories for the universe of Mathematics such as: numbers, geometry, magnitudes and measurements, algebra and statistics. Using mathematical concepts and language for the formulation and writing of texts. Thus contributing for the student to think, do differently, lead to the fantastic world of mathematical knowledge.

Research on Strategies of Improving Mathematics Reading Ability in Secondary Vocational School

Reading is the premise of learning. In the information age, the cultivation of secondary vocational school students' mathematics reading ability is an important way to cultivate students' comprehensive learning ability and improve the core quality of mathematics. Therefore, more and more teachers attach importance to the cultivation of students' mathematics reading habits and the improvement of reading ability, so as to promote students' mathematical language expression ability, arouse students' thinking about mathematical problems, and promote students' mathematical language transformation ability. This paper discusses the relationship between middle school students' reading habits and mathematics achievement, and gives some suggestions on how to cultivate secondary vocational school students' good reading habits and improve their mathematics reading level.

Could Galileo Discover the Law of Universal Gravitation in 1611, Was There Newton’s Apple and What Is “Modern Physics”?.

The central problem of the article is the paradox in the history of Newton’s mechanics: prominent researchers of the genesis of the Principia did not believe Newton’s words about the origin of the idea of universal gravity. They did not believe that he could have come up with this idea as early as 1666, considering circular orbits, and believed that Newton invented the story of the falling apple. The article proposes a “subjunctive” scenario leading to the law of universal gravity and feasible at the level of Galileo’s knowledge and skills in 1611. The basis for such a scenario is the description of a thought experiment in Newton’s manuscript “The System of the World”, preceding the creation of Principia. The proposed reconstruction helps to consider and clarify the concept of “modern physics”, the birth of which was the main event of the Scientific Revolution of the XVI–XVII centuries. The traditional understanding reduces the essence of modern physics to a reliance on experience and on the language of mathematics. Such a definition, however, is not sufficient. The geometry of Euclid and the physics of Archimedes were mathematically perfect, and their axioms were based on objective experience. Despite the importance of the tools of mathematics and experiment, the key innovation of modern physics has become the belief in the hidden fundamental laws of the Universe and in the right of the researcher to invent invisible, “illogical”, “absurd” concepts and postulates, experimentally verifiable only together with the theory based on them. This postulate of fundamental cognitive optimism combines bold ingenuity with a humble need for empirical verification.

An algorithm for estimating the number of objects in a selected area of space based on ordinary photos

There are many areas of activity where the problem of counting objects in three-dimensional space appears. It is often the case that the primary source of information in such a situation are photos or digital videos. In mathematical language, a two-dimensional photo can somewhat be treated as a projection of a selected space area. The significant issue in such a situation is that the counted objects in the image may overlap many times, which may cause some objects to obscure others. This approach addresses this issue by using appropriate statistical estimations of the amount of obscured objects based on computer image analysis. The proposed algorithm has already been initially used to count the number of birds in digital photos and films.

A Study on the Implementation of Teaching Mathematical Modeling Classes Based on Mathematical Learning Objectives

Mathematical modeling literacy is one of the six core literacies in high school mathematics and occupies an important place in the objectives of the high school mathematics curriculum. “The core element of mathematical modeling literacy is to abstract mathematically from real problems, express them in mathematical language, and construct models to solve them with mathematical methods. The STEAM education concept has received much attention in the education field because of its effective integration of “science, technology, engineering, art and mathematics” in the practice of education and teaching, which not only makes up for the shortage of traditional teaching in knowledge inquiry, but also helps to improve the problems of traditional classrooms. This paper explores the development of mathematical modeling literacy in high school by drawing on the STEAM education concept, and examines the effectiveness of the integration of the two through specific teaching cases.

ОБ ОДНОЙ ИЗ ФОРМ ПРОВЕДЕНИЯ ЭКЗАМЕНА ПО МАТЕМАТИКЕ В ВУЗЕ

The materials of the article consider the issues of organizing and conducting intermediate certification of students of specialties related to information security on the discipline «Mathematics». An approach is proposed, in the implementation of which a multi-level system of assessing students' knowledge is used, the range of opportunities for unlocking the potential of students is expanded, communication skills in a mathematical language are formed. A distinctive feature of the described methodology is the possibility of using both standard sources and digital information environment in the preparation process.

Lexis and Structure in Mathematics Language

Lexis and Structure in Mathematics Language

Habilidade de comunicação matemática da aluna no processo de comprovação: uma revisão baseada na ansiedade matemática

Abstract This qualitative research aimed to explore the mathematical communication profile of prospective mathematics teachers in-depth, while also examining their levels of math anxiety. Mathematical communication involves delivering mathematical ideas and understandings through written and verbal means, using mathematical language in the form of symbols, signs, terms, formulas, images, and words. The study’s subjects were two female fifth-semester students from mathematics education study programs, one with High Math Anxiety (HMA) and one with Low Math Anxiety (LMA). Data were collected through various methods, including tests, assignments, questionnaires, and interviews. The findings of the study indicated that the HMA student accurately conveyed ideas in writing, but was less systematic and logical, whereas the LMA student expressed ideas correctly, systematically, and logically, but sometimes made accidental technical errors. Additionally, the HMA student did not pay much attention to the regularity and accuracy of her thoughts during oral communication. In contrast, the LMA student demonstrated a strong interest in learning, and was able to convey her ideas in a systematic, logical, detailed, complete, and correct manner.

Faze konstrukcije i teškoće u izradi testova znanja u obrazovno-vaspitnoj praksi

The paper presents the results of an empirical research which determined the stages of construction of knowledge tests in educational practice, the difficulties faced by teachers in the process of test development, as well as the ways of overcoming these difficulties. The descriptive-analytical re-search method, content analysis and survey techniques were used in the research. The research sample included 362 subject teachers employed in primary schools in Serbia. All respondents included in the research indicated the difficulties they face in the process of making tests, i.e., the ways of solving these difficulties. A sub-sample of a total of 80 teachers consisted of Serbian language and mathematics teachers who reported on the phases of the knowledge test construction in their practice. The results of the research indicate that teachers perceive only a small number of theoretically defined stages in the construction of knowledge tests. Certain differences were identified in explaining the phases among the teachers of Serbian language and teachers of mathematics. At the same time, two-thirds of the surveyed teachers face difficulties when they prepare knowledge tests independently. The existing difficulties are usually solved by consulting the more experienced colleagues or by searching for a solution independently. The author's recommendation is to enable teachers by means of an additional teacher training courses, to acquire the appropriate knowledge and skills that would help them create the knowledge tests with satisfactory methodological characteristics, and at the same time adequate for the needs of their teaching practice.

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”.

Affine geometric description of thermodynamics

Thermodynamics provides a unified perspective of the thermodynamic properties of various substances. To formulate thermodynamics in the language of sophisticated mathematics, thermodynamics is described by a variety of differential geometries, including contact and symplectic geometries. Meanwhile, affine geometry is a branch of differential geometry and is compatible with information geometry, where information geometry is known to be compatible with thermodynamics. By combining above, it is expected that thermodynamics is compatible with affine geometry and is expected that several affine geometric tools can be introduced in the analysis of thermodynamic systems. In this paper, affine geometric descriptions of equilibrium and nonequilibrium thermodynamics are proposed. For equilibrium systems, it is shown that several thermodynamic quantities can be identified with geometric objects in affine geometry and that several geometric objects can be introduced in thermodynamics. Examples of these include the following: specific heat is identified with the affine fundamental form and a flat connection is introduced in thermodynamic phase space. For nonequilibrium systems, two classes of relaxation processes are shown to be described in the language of an extension of affine geometry. Finally, this affine geometric description of thermodynamics for equilibrium and nonequilibrium systems is compared with a contact geometric description.