Use Of Mathematical Language Research Articles

A geometric interpretation of the Peano Axioms as a didactic strategy for teaching and learning natural numbers

Peano’s axioms are a set of propositions that allow a formal definition of the set of natural numbers. In these axioms, Peano rescues the essence of counting objects and materializes it by assuming the existence of a set, of a function in this set (which he calls successor) and an axiom of induction. In this work a geometrical representation and interpretation of each of these axioms is made, which allows understanding the independence and necessity of each one of them, which facilitates giving the conceptual rigor of them by making use of mathematical language. Finally, using Euclidean geometry, a geometric construction of a system of natural numbers on a straight line is made.

An Examination of Mathematics Teachers’ Mathematical Language Usage

The aim of this research is to examine the mathematical language use of mathematics teachers in detail within the framework of the theoretical approach of “Three Worlds of Mathematics” put forward by Tall (2007). The research was carried out in the case study pattern, which is one of the qualitative research methods. The study group of the research consists of 27 math teachers. In the research, “Worksheet for the Use of Mathematical Language”, which includes four mathematical situations, was used as a data collection tool. In the analysis of the data collected within the scope of the research, the interpretation of the “Three Worlds of Mathematics” theoretical approach towards the use of mathematical language was adopted. As a result of the research, it was seen that mathematics teachers used mathematical language mostly in the context of the “conceptual-embodied world”, and least in the context of the “axiomatic-formal world”. At the same time, while the mathematical world in which the mathematical situation is presented affects the mathematical language used by the teachers for that situation, the teachers could not fully reflect the mathematical language to different worlds whilst working on mathematical situations. Finally, based on the conclusion that teachers use mathematical language in a more qualified way while performing high-level skills, it is suggested to include activities that allow the development of high-level skills in mathematics lessons and that serve to understand and develop mathematics

GeoGebra Software on the Mathematical Language Developments and Self-Efficacy Perceptions of Students

The aim of the study is to investigate the effects of using GeoGebra in teaching functions on mathematical language development and self-efficacy perceptions of tenth grade students. The study, which used the action research method, changes in participants’ language structures were examined with the worksheets, mathematical language questions, researcher’s logs and participant’s logs; participants' self-efficacy perceptions were also examined with the self-efficacy perception scale. The ability of the participants to switch between the sub-dimensions of mathematical language was observed. The research showed that GeoGebra-Assisted Education improved the participants’ perceptions of mathematical self-efficacy and positively affected their mathematical language skills. Since the effective use of mathematical language is an important component of mathematics lessons, the results present important findings.

The feelings of knowing – fundamental interoceptive patterns (FoK-FIP) system: connecting consciousness to physics

ABSTRACT The feelings of knowing – fundamental interoceptive patterns (FoK-FIP) theory is both a theory of the mind and a unification theory. It includes cosmological and cellular frameworks. The cellular frameworks occur through the cosmological frameworks. This framework within a framework approach allows the connection between physics and consciousness to be envisioned in new ways, expanding current understanding and definitions. The cosmological frameworks refer to the astrophysics and theoretical physics constructs (e.g., string theory) that, without the use of mathematical language, conceptually expand the theory. In contrast, the cellular frameworks are the constructs represented by living organism models with DNA. In this way, the FoK-FIP theory represents an efficient framework for understanding consciousness and its phenomena. The transdisciplinary modeling of the FoK-FIP theory creates contextual bridging between classical theory and quantum theory as well as a broad range of empirical research so that biology and information connect, creating new avenues for disease diagnosis, intervention, and prevention. This article intends to render the FoK-FIP theory more robust and accessible for practical application by introducing the FoK-FIP system, which includes figures to promote clarity. Further, the theory aims to stimulate reasoning that challenges current notions about ‘life’ and the concept of ‘self.’ Through this process, the theory might contribute to the transdisciplinary collaboration needed to address some of the world’s complex issues. It is suggested that a significant contributor to the current complex matters in the world is the lack of understanding of how things are and how they appear.

An Investigation into the Effects of Secondary Mathematics Trainee Teachers' Use of Mathematical Language in Reducing Mathematical Errors: A Jamaican Context

An Investigation into the Effects of Secondary Mathematics Trainee Teachers' Use of Mathematical Language in Reducing Mathematical Errors: A Jamaican Context

Developing students' problem posing skills with dynamic geometry software and active learning framework

In the present study, the effects of using dynamic geometry software in active learning framework on students’ problem posing skills and their views about problem posing were examined. The participants consisted of 16 eighth-grade students. Data were collected by problem posing tests, open-ended questions, student diaries, and dynamic geometry software supported tasks. The study, designed with the embedded mixed method, lasted 13 weeks. The dependent t-test was used in the analysis of quantitative data, and descriptive analysis was performed in qualitative data. Problem posing skills of students examined according to use of mathematical language, grammar and expression, suitability to acquisitions, quantity and quality of data, solvability, originality, solution of the problem criteria. In the present study, it was determined that the use of dynamic geometry software in active learning framework developed students’ problem posing skills. The problems that students posed in the dynamic geometry software during the implementation process indicated improvement in terms of problem posing skills as the weeks passed. It was found that students had positive views about the problem posing process, while they experienced some difficulties in this process.

EVALUATION OF MOBILE APPLICATIONS IN THE TEACHING OF GEOMETRY

Today’s world implies more and more frequent use of smartphones and their applications in every place and at every moment. In this paper, we will first talk about mobile educational applications in general, and then we will present various research related to the evaluation of mobile educational applications in the teaching of geometry (Bos, Dick, Larkin). Analyzing the relevant literature, the conclusion is drawn that the evaluation of mobile educa- tional applications is important for learning if geometrical content can be successfully carried out according to three aspects: pedagogical, mathematical and cognitive. The pedagogical as- pect implies the effectiveness of the application to assist in learning; the mathematical aspect of mobile educational applications is very often not fully satisfied because the incorrect use of mathematical language is noticeable, as well as the incorrect classification of shapes and objects; the cognitive aspect determines to what extent the application affects the development of students’ thought processes. According to previous research, mobile educational applications such as Co-ordinate Geometry, Transformations and Attribute Blocks were rated highly in all mentioned aspects of the evaluation.

“Shh!”: Shifting discourse patterns and techniques of surveillance in middle-school mathematics classrooms

The growing body of empirical studies on discourse in mathematics education have drawn on, broadly, sociocultural or situated views of learning but do not make issues of power central and/or explicit. For this reason, then, we situate the present analysis within the intersection of two bodies of literature: professional development focused on mathematics classroom discourse and Foucauldian analyses of power relations. In our poststructural analysis of a mathematics classroom, we interrogate the use of surveillance and its relation to meaningful mathematical discourse. Using Foucault’s techniques of surveillance and modalities of power, we analyze videos from one teacher-researcher’s (TR) classroom at two snapshots in time to answer the questions: What happens to a mathematics TR’s use of teacher discourse moves (TDMs) and surveillance practices across three years of doing action research on their classroom discourse? and How might we know whether these TDMs are doing ‘good’ things in the classrooms? Our findings show the complex ways that a TR’s work on classroom discourse is intertwined with modes of power and techniques of surveillance. First, we found that shushing and silencing, normalized as a part of teaching, decreases when a teacher positively values the use of (mathematical) language by students. This decrease in shushing and silencing details what we understand to be ‘the good’ and constitutes the primary finding of this study as we seek to expand what is understood by enhancing student language: an overall increase in language use by students and an overall increase in their use of mathematical language.

Investigation of Students' Cognitive Learning in Mathematics Lessons Supported with Writing Activities

The aim of the present research was to examine the contribution of expository and journal writing activities to the cognitive development of 7th-grade students with varying levels of academic achievement in mathematics education. The qualitative research approach has been adopted in this research. The lessons were conducted in one 7th grade class of 28 students using a total of 26 expository writing activities and 6 journal writings designed to address the topics of lines, angles and measuring angles, operations with integers, rational numbers, algebraic expressions, equations, ratio and proportion.‖ During the 14 weeks of lessons, three themes were created and various codes were established under these themes by means of content analysis of the written explanations obtained from students' expository writing activities and diaries. In addition, the expository writing activities were analyzed using an analytical scoring rubric (ASR). The findings were coded under three themes: (1) features of the students' explanations (2) use of mathematical language and (3) mathematical algorithms and calculations. According to the data obtained, it has been determined that writing activities contribute to the cognitive development of particularly those students with a moderate level of academic achievement. In addition, it was revealed that the responses provided in writing activities by students with different levels of academic achievement varied. Based on the results of the study, it is recommended that writing activities be implemented over a long and continuous period of time.

A spiral pattern investigation: making mathematical connections

Nowadays there is considerable agreement among educators that learning mathematics fundamentally involves making mathematics [1]. Students learn mathematics while working on tasks that they consider meaningful and worthwhile, and their interest is aroused when they can see the point of what they are being asked to do. Given that learning mathematics involves a process of meaning-making - the use of mathematical language, symbols and representations as learners negotiate ideas – activities should provide students with a variety of challenging experiences through which they can actively construct mathematical meanings for themselves.

Enhancing Mathematical Communication in the Classroom: A Case Study

Mathematical communication competence was mentioned the first time in the Mathematics curriculum of Vietnam, which was published in 2018. It includes different skills such as listening, reading, presenting, expressing mathematical ideas, and knowing how to use mathematical language to communicate, discuss interaction with other people. The problem is how to organize teaching to promote mathematical communication. In 2004, Radford and Demers based on the theory of Zones of Proximal Development [12] to develop a teaching strategy called the fourth strategy [9]. This teaching strategy has seven steps; it provides groups of the student with different activities to discuss, exchange, review their learning products. We have adopted this teaching strategy in a teaching situation about the concept of function in junior high school in Vietnam. Twelve students who participated in our study case worked in pairs. This teaching process focused on each group of two students reviewing products of another and basing on comments from other groups to improve their products. Furthermore, our teaching situation puts students in front of the various representations of function and thus enhances the use of mathematical language and mathematical communication for students. The results of our study showed that the teaching strategy of Radford and Demers increased students' mathematical communication. Lessons drawn from the study are that this strategy is really an effective teaching method if teachers assign open problems that stimulate the curiosities of students; in the classroom, they feel free and confident in discussion; if these conditions are met, students who learn in this approach will acquire and reinforce the knowledge as a result of a mathematical communication process.

Development of Competent Mathematical Speech of Students at Technical University

The development of literate speech of students, including mathematics, as one of the areas of communicative component of learning outcomes, is a requirement of higher education standards. The authors have analysed the educational standards for the content of the requirements to the general education of a graduate related to the development of literate speech, as well as the works of domestic and foreign teachers-researchers on the development of mathematical literacy. The literature review and the authors’ own experience of teaching mathematics to engineering students at technical University showed that the ability of students to use logically correct, reasoned and clear oral and written speech is developed insufficiently to apply the mathematical apparatus in their educational and professional activities. The authors have identified the criteria of competent speech and theoretical and methodological conditions of the educational process which enable the ensure the organizational and methodological support of educational process aimed at the development of competent mathematical speech of students. The main applied methods of teaching are advanced self-directed work of students, lecture-discussion, mutual dictation, interchange of tasks, interchange of themes, study of the text fragments, repetition training-game, etc. The effectiveness of educational activities is achieved through the use of active and interactive forms of teaching. The prepared organizational and methodological support of teaching, the systemic educational work allow students to improve their speech skills and skills of active use of mathematical language as a universal language of science, to develop logical, algorithmic and mathematical thinking, the ability to apply methods of mathematical analysis and modeling, theoretical and experimental research in solving professional problems.

Investigating Seventh Grade Students’ Mathematical Communication Skills: Student Journals

This study aims to investigate students’ mathematical communication skills through investigating their defining, using mathematical concepts and mathematical language skills. Additionally, this study aims to investigate the relationship between students’ mathematical communication skills with their academic achievements. A seventh grade classroom at a public middle school in the Eastern Anatolia region and 34 students and their mathematics teacher participated in the study. Students were asked to keep journals during seven weeks for sixteen Geometry standards after each standard. Student journals were analyzed in four hierarchical levels—Level 0 (Avoidance), Level 1 (Incorrect Use), Level 2 (Incomplete Use) and Level 3(Correct and Complete Use) — by two researchers individually. Later, the researchers compared their categorizations and discussed till solving the disagreements. According to the findings of the study, participating students did not comprehend the concept definitions. It was seen that students attempted to use mathematical language in their journals as a way of communication. However, it should be noted that some incorrect use of mathematical language was apparent in student journals. This study also showed that it was hard to conclude that student mathematical communication skills differ based on their academic achievement levels.

Preservice Secondary Mathematics Teachers' Conceptualizations of Equity: Access and Power as Seen Through Vignette Responses

Attention to equity in the mathematics education field has been growing in recent years. We have evidence that many novice secondary mathematics teachers do not feel prepared to teach in regards to diverse populations. We need to know more about how secondary preservice mathematics teachers (PSMTs) conceptualize equitable environments. This study investigates 30 secondary PSMTs' proposed responses to two hypothetical vignettes from mathematics department conversations regarding calculator usage and mathematical discourse, respectively, utilizing two of Gutiérrez's four dimensions of equity: Access and Power. Results suggest these PSMTs considered equity, equality, and creating a classroom that invites participation among other factors when thinking of an equitable approach with respect to calculator usage. When considering mathematical discourse, PSMTs cited the need to “model” proper use of mathematical language as well as to allow students to themselves verbalize it. Implications mathematics education and teacher education more broadly are to integrate equity and equality discussions in methods courses and to include strategies to facilitate productive discourse.

Investigating Alignment Between Elementary Mathematics Teacher Education and Graduates' Teaching of Mathematics for Conceptual Understanding

In this article, Amanda Jansen, Dawn Berk, and Erin Meikle investigate the impact of mathematics teacher education on teaching practices. In their study they interviewed six first-year teachers who graduated from the same elementary teacher education program and who were oriented toward teaching mathematics conceptually. They observed each teacher teaching two lessons: one on a mathematics topic that was developed in their teacher education program (target topic) and one on a mathematics topic that was not addressed in their program (control topic). Based on their observations, the authors identified four instructional practices for teaching mathematics conceptually that the participants used in their classroom practice and found that these teachers were more likely to enact two of these instructional practices when teaching target topics: use of mathematical language to support students' sense making and use of visual representations. They also found that the teachers enacted two other instructional practices—use of story problems and pressing students for mathematical explanations—in both target and control topic lessons but did so with limitations in control topic lessons. For teacher education to influence teaching, the authors assert, it is important to develop content knowledge for teaching and pedagogical knowledge in tandem with developing beliefs.

Conocimiento y uso del lenguaje matemático en la formación inicial de docentes en matemáticas

Este artículo analiza los niveles de competencias en el conocimiento y uso del lenguaje matemático que poseen y emplean los estudiantes durante su formación inicial en la carrera de docencia en matemáticas, en una universidad pública colombiana. La investigación se propone establecer el nivel de conocimiento y uso del lenguaje matemático entre estos estudiantes, qué nivel de habilidad matemática emplean y su competencia para comunicar por escrito sus conocimientos. El diseño de la investigación es descriptivo y de campo, para lo cual se empleó como muestra un grupo de 92 estudiantes de un programa de formación inicial de docentes que cursan las asignaturas del eje disciplinar de matemáticas. Sobre la base de la resolución escrita de problemas se determinó el nivel de conocimiento y uso del lenguaje matemático empleado por los estudiantes. Los resultados muestran desconocimiento y mal uso del lenguaje y la simbología matemática, de las reglas de la lógica y escasa capacidad de razonamiento, lo cual genera dificultades para comprender y resolver problemas matemáticos. Se determinó asimismo que los estudiantes presentan mayores dificultades en la traducción del lenguaje cotidiano al formal, en la identificación de objetos matemáticos y en el trabajo con conceptos que ameritan demostraciones lógicas. Los estudiantes poseen un nivel insuficiente de conocimiento y uso del lenguaje matemático, habilidad matemática también baja y mayor competencia para descifrar el lenguaje matemático hacia el verbal que en cifrar lenguaje verbal en lenguaje matemático.

The Evaluation of the Problem Solving in Mathematics Course According to Student Views

This study was conducted to determine the problem solving skills of the third grade students studying at the department of elementary school mathematics teaching. The study was conducted in the second semester of the academic year of 2015-2016. The study group consists of 47 third year student who study at Ondokuz Mayis University, Faculty of Education Elementary School Mathematics Teaching ad take the selective course of Problem Solving in Mathematics. Within the scope of this course, the researchers explained subjects related to problem and problem solving, problem solving skills and solved problems during the first 4 weeks of the course. For the rest of the weeks, the students were divided into groups. They have solved two non-routine problems each week for 8 weeks. The method of study is the interview method, which is one of the qualitative research methods. In light of the retrieved findings, the answers given by the students have been thematized as the stages of problem solving, understanding the problem, implementing the problem, evaluation of the problem, reasons for taking the courses, association problems, ways of finding different solutions, development of procedural skills, creating formulas, mathematical thinking, use of mathematical language, suitability of the course, views on problem solving, and the contribution of the course.

Making a case for exact language as an aspect of rigour in initial teacher education mathematics programmes

Pre-service secondary mathematics teachers have a poor command of the exact language of mathematics as evidenced in assignments, micro-lessons and practicums. The unrelenting notorious annual South African National Senior Certificate outcomes in mathematics and the recognition by the Department of Basic Education (DBE) that the correct use of mathematical language in classrooms is problematic is reported in the National Senior Certificate Diagnostic Reports on learner performance (DBE, 2008-2013). The reports further recognise that learners do not engage successfully with mathematical problems that require conceptual understanding. This paper therefore highlights a need for teachers to be taught and master an exact mathematical language that for example, calls an 'expression' an 'expression' and not an 'equation'. It must support the call of the DBE to use correct mathematical language that will support and improve conceptual understanding rather than perpetuate rote procedural skills, which are often devoid of thought and reason. The authentic language of mathematics can initiate and promote meaningful mathematical dialogue. Initial teacher education programmes, as in the subject methodologies, affords lecturers this opportunity. The language notions of Vygotskian thought and language, Freirian emancipatory critical consciousness and Habermasian ethical and moral communicative action frame the paper theoretically. Using a grounded approach, after examining examples of student language in a practice based research intervention, the design and development of a repertoire of language categories, literal, algebraic, graphical (Cartesian) and procedural (algorithmic) emerged from three one-year cycles of an action research methodology. The development of these repertoires of language was to assist teachers in communicating about mathematical objects through providing a structured framework within which to think and teach. A course model encompassing small group discussions, an oral examination and a self-study action research project, that helped sustain the teaching of an exact mathematical language, is presented. This is supported by student reflections on the usefulness of implementing them.

Negotiation Engineering: A Quantitative Problem-Solving Approach to Negotiation

Negotiations are complex problems whose solutions require special, practical methods. We introduce a problem-solving approach to negotiation inspired by the established solution-oriented discipline of engineering, which we call Negotiation Engineering. The key elements of this approach are the reduction of problems to their basic structures and the application of heuristic methods to problem solving. We argue that the use of mathematical language in negotiations enables an increased logical accuracy in negotiation analysis and allows the use of a variety of existing helpful mathematical tools to achieve a negotiation agreement. We demonstrate the practicability and usefulness of this approach in four case studies in the area of international diplomacy in which Negotiation Engineering was applied to achieve negotiation solutions.

The Use of the Language of Mathematics as an Inspiration for Contemporary Architectural Design

The purpose of the article is to present the evolution of the use of mathematical language as an inspiration for creating spatial, three-dimensional forms in art and architecture. The article focuses on the possibilities for art and architectural design ideas gained by contemporary mathematics, algorithms and computational parametric approach. The analysis of various examples represents the relationships between the composition of spatial forms and the rules of mathematics. It is evident in different time frames, different styles and different approaches to thinking about and creating art and architecture. The starting point for this analysis is the symbolic Vitruvian golden ratio and its impact on the principles of spatial forms composition. Next, using the study of the geometric art of folding and cutting paper to create three-dimensional spaces, elements of applied art or even furniture or clothing, the article reaches to the mathematical issue of fractals, as the most accurate illustration of the aims of the parameterization in contemporary architecture. Parametric architecture, as a way of thinking about building as a set of numerically coded aspects, demonstrates the possibilities of using mathematical resources to improve functioning of the building by using optimization algorithms. The article shows possibilities of such uses of mathematics in generating spatial forms. Further, this analysis asks about the possible risks and disadvantages of such an approach, wondering if the correct definition of architecture is possible to achieve simply by using language of mathematics without all the other immeasurable aspects.