Mathematical Discourse Research Articles

Embodied mathematical pedagogy to liberate racialized and multilingual bodies.

When language is defined narrowly in mathematics classrooms, racially and linguistically minoritized students in classrooms could be systematically positioned as “learners of deficiency.” Recent scholarship calls for expanding the notion of language to emphasize embodied expression of mathematical ideas. Taking a critical perspective to understand racialized experiences of using languages in disciplinary learning spaces, this article proposes the reconceptualization of embodiment as a language for racialized multilingual learners. This study was conducted in a Grade 1 classroom in a linguistically and racially diverse school in Canada. Through a series of professional development sessions, we worked with an experienced teacher to redesign the normalized and institutionalized pedagogy toward greater mobility of racialized multilingual learners’ bodies, which was intertwined with their intellectual liberation. Focusing on the spatiality of pedagogy, the previously restrictive areas in the school were transformed into a place that augments embodied expression of mathematical ideas and agentive participation of minoritized learners. The analysis focused on the embodied discourse that participating racialized multilingual students used to actively engage in mathematical discussion. Our findings show that the designed pedagogy, characterized by the spatial and temporal expansion of the learning environment, offered more spaces for uncertainty and spontaneity with the decreased control of the teacher as an explicator. Our article furthers anti-colonial approaches to understand the intersection of racialized bodies and language in mathematics education.

Ante rem structuralism and the semantics of instantial terms

Ante rem structures were posited as the subject matter of mathematics in order to resolve a problem of referential indeterminacy within mathematical discourse. Nevertheless, ante rem structuralists are inevitably committed to the existence of indiscernible entities, and this commitment produces an exactly analogous problem. If it cannot be sorted out, then the postulation of ante rem structures is futile. In a recent paper, Stewart Shapiro argued that the problem may be solved by analysing some of the singular terms of mathematics not as genuinely referring expressions, but as instantial terms. In this paper, I discuss several competing accounts of the semantics of terms of this kind, and argue that they are all untenable for the ante rem structuralist. Shapiro, then, still owes us an account of the semantics of instantial terms that suits the ante rem structuralist project. Without it, the ante rem structuralist is still unable to determine the reference of the singular terms of mathematics.

Truth pluralism without domains

Truth pluralists say that truth-bearers in different “discourses”, “domains”, “domains of discourse”, or “domains of inquiry” are apt to be true in different ways – for instance, that mathematical discourse or ethical discourse is apt to be true in a different way to ordinary descriptive or scientific discourse. Moreover, the notion of a “domain” is often explicitly employed in formulating pluralist theories of truth. Consequently, the notion of a “domain” is attracting increasing attention, both critical and constructive. I argue that this is a red herring. First, I identify the theoretical role for which pluralists appeal to domains, which is to answer what I call the “Individuation Problem”: saying what determines the way in which a particular truth-bearer is apt to be true. Second, I argue that pluralists need not appeal to domains for this purpose. I thus conclude that, despite the usual way of glossing the view, there is no role for the notion of a “domain” to play in the pluralist’s theory of truth. I argue that this defuses the “Problem of Mixed Atomics” and allows the pluralist to sidestep potentially intractable disputes about the nature of domains.

The influences of social, cognitive, and teaching presence on pre-service teachers’ online engagement in productive mathematical discourse

The increase in online education has implications for mathematics teacher educators who are tasked with teaching mathematics content and pedagogy to pre-service teachers (PSTs) who do not attend on-campus. Concerns about PSTs’ content knowledge continues to be an ongoing issue, and ways to address this, especially in an online environment, require further investigation. While online courses offer accessibility and flexibility, it can be challenging to translate synchronous on campus teaching and experiences into the online environment. The importance of productive mathematical discourse has been identified as an appropriate pedagogy, yet the opportunity to engage in such discourse can be limited in the online environment. This paper provides the results from a case study of an online forum, which highlights how pre-service teachers can be engaged in productive online mathematical discussions, particularly when facilitated by the instructor's and other learners’ interactions and presence. The findings indicated that social, cognitive, and teaching presence influenced the nature and frequency of online mathematical discourse. The study contributes to existing research through conceptualizing how productive mathematical conversations can occur outside of the classroom setting, with implications for mathematics teacher educators who are tasked with teaching content, pedagogy and practices to PSTs.

Mathematical discourses on propositional equivalence: An exploration through the commognitive lens

This paper describes the mathematical learning of purposively sampled second-year Bachelor of Secondary Education mathematics majors from a state university in Bulacan province on propositional equivalence concept within the Logic and Set Theory course via a commognitive lens. This small-scale study employed exploratory qualitative research with one class recording, one focus group, and select activity outputs. Four participants in the focus group were sampled based on commognitive conflict occurrences. The teacher-researcher operated as a co-participant in the mathematical discourse. The dean's approval and participants' informed consent were observed, explaining the research objectives and confidentiality scope. The findings present accounts and descriptions of participants' mathematical discourses through the commognitive lens: word use, visual mediators, endorsed narratives, and routine practices that describe Logic and Set Theory discourses on the propositional equivalence concept from a participationist's learning standpoint.

Mathematical discourses on propositional equivalence: An exploration through the commognitive lens

This paper describes the mathematical learning of purposively sampled second-year Bachelor of Secondary Education mathematics majors from a state university in Bulacan province on propositional equivalence concept within the Logic and Set Theory course via a commognitive lens. This small-scale study employed exploratory qualitative research with one class recording, one focus group, and select activity outputs. Four participants in the focus group were sampled based on commognitive conflict occurrences. The teacher-researcher operated as a co-participant in the mathematical discourse. The dean's approval and participants' informed consent were observed, explaining the research objectives and confidentiality scope. The findings present accounts and descriptions of participants' mathematical discourses through the commognitive lens: word use, visual mediators, endorsed narratives, and routine practices that describe Logic and Set Theory discourses on the propositional equivalence concept from a participationist's learning standpoint.

Teacher Educators’ Word Use when Teaching Student Teachers how to Teach Place Value to Early Years Learners

This article discusses the findings of a case study that explored the use of explanatory talk in mathematics teacher education lessons on how to teach the concept of place value. The Mathematics Discourse in Instruction framework guided the study, in particular the element of naming in explanatory talk. Three teacher educators’ lessons on the teaching of place value of whole numbers were observed and video recorded. Findings indicate that across all the lessons, 66% of the naming was mathematical and was seen to provide opportunities for student teachers to learn to use mathematical terms correctly, while 34% of the naming was non-mathematical and might have constrained student teachers’ understanding of some concepts related to place value. Furthermore, two-thirds of the non-mathematical naming was use of ambiguous pronouns, which could have been avoided. The findings suggest that teacher educators need to pay more attention to correct mathematical naming of terms related to place value. How mathematical concepts are named in teacher education is important because correct naming is likely to help student teachers learn to regard mathematical talk as an important part of their work of teaching.

An analysis of teacher knowledge for teaching functional thinking to elementary school students

Given the significance of functional thinking at the elementary school level, there has been growing interest in teachers who play a major role in enhancing students’ functional thinking. This study surveyed 119 elementary school teachers in Korea to investigate their knowledge for teaching functional thinking. A written assessment was developed for this study regarding the knowledge of mathematical tasks, instructional strategies, and mathematical discourse to teach functional thinking. The results of this study showed that many teachers could design mathematical tasks corresponding to simple relationships of two quantities but some of them had difficulties in constructing the task for [Formula: see text]. Teachers could explain the affordances of key instructional strategies to foster functional thinking and identify students’ typical errors in recognizing or representing a functional relationship but some explanations included a superficial understanding of the core ideas of functional thinking. Based on these results this article closes with a discussion of several implications regarding the aspects needed for elementary school teachers to further promote students’ functional thinking.

Arguing with Numbers: The Intersection of Rhetoric and Mathematics

Arguing with Numbers: The Intersection of Rhetoric and Mathematics

Entanglement Hamiltonians: From Field Theory to Lattice Models and Experiments

AbstractResults about entanglement (or modular) Hamiltonians of quantum many‐body systems in field theory and statistical mechanics models, and recent applications in the context of quantum information and quantum simulation, are reviewed. In the first part of the review, what is known about entanglement Hamiltonians of ground states (vacua) in quantum field theory is summarized, based on the Bisognano–Wichmann theorem and its extension to conformal field theory. This is complemented with a more rigorous mathematical discussion of the Bisognano–Wichmann theorem, within the framework of Tomita–Takesaki theorem of modular groups. The second part of the review is devoted to lattice models. There, exactly soluble cases are first considered and then the discussion is extended to non‐integrable models, whose entanglement Hamiltonian is often well captured by the lattice version of the Bisognano–Wichmann theorem. In the last part of the review, recently developed applications in quantum information processing that rely upon the specific properties of entanglement Hamiltonians in many‐body systems are summarized. These include protocols to measure entanglement spectra, and schemes to perform state tomography.

The technique to solve equations of the first degree. A case study

In Mexico we can find upper secondary teachers who, sometimes, are forced to teach mathematics even though their profile is different. This document is aimed to identify the technique –mathematical methods– that a teacher from the Mexican Telebachillerato subsystem (high school) uses to help students to solve the type of tasks related to first-degree equations. This proposal is part of a research project where thirteen teachers participated, and who allowed to be observed twice teaching mathematical contents according to their study program. The purpose of observation works was to make them a discussion object with the participants. This is a study case research. The results show that the teacher creates a technique focused on obtaining determinants to solve types of task about system of equations, but the theory which proves its functionality is reduced to a more didactic than mathematical discourse. In the classroom, techniques to solve problems are taught, but these remain unprotected from mathematical theories (mathematical argument) that can validate their functionality, the student is the one who has to infer o build his own theory from the teacher’s technique.

Using Multiple Representations to Foster Multiplicative Reasoning in Students With Mathematics Learning Disabilities

Developing and supporting understanding of the meaning of multiplication and multiplicative relationships in students with mathematics learning disabilities requires carefully designed instruction that emphasizes strategic representation use. This article discusses three ways in which teachers can incorporate multiple representations within multiplication instruction to develop a deep understanding of underlying mathematical structures. First, it is important to select concrete and semi-concrete representations (e.g., discrete objects and those that illustrate continuous quantities as length or area) that will help students most accurately model the multiplication or division concept or procedure. Next, provide students with multiple representations that are presented concurrently within the same lesson so that they can make connections among the different types of representations. Last, teachers can support students to use and connect representations through mathematical discourse that elicits student thinking by not only structuring tasks around three processes, reversibility, flexibility, and generalization, but also by posing questions to elicit different ways that students can communicate their ideas using accurate and precise vocabulary.

Mathematical Discourses on Propositional Equivalence: An Exploration through the Commognitive Lens

This paper describes the mathematical learning of purposively sampled second-year Bachelor of Secondary Education mathematics majors from a state university in Bulacan province on propositional equivalence concept within the Logic and Set Theory course via a commognitive lens. This small-scale study employed exploratory qualitative research with one class recording, one focus group, and select activity outputs. Four participants in the focus group were sampled based on commognitive conflict occurrences. The teacher-researcher operated as a co-participant in the mathematical discourse. The dean's approval and participants' informed consent were observed, explaining the research objectives and confidentiality scope. The findings present accounts and descriptions of participants' mathematical discourses through the commognitive lens: word use, visual mediators, endorsed narratives, and routine practices that describe Logic and Set Theory discourses on the propositional equivalence concept from a participationist's learning standpoint.

Exploring the relationship between commognition and the Van Hiele theory for studying problem-solving discourse in Euclidean geometry education

This article is an advanced theoretical study as a result of a chapter from the first author’s PhD study. The aim of the article is to discuss the relationship between commognition and the Van Hiele theory for studying discourse during Euclidean geometry problem-solving. Commognition is a theoretical framework that can be used in mathematics education to explain mathematical thinking through one’s discourse during problem-solving. Commognition uses four elements that characterise mathematical discourse and the difference between ritualistic and explorative discourses to explain how one displays mastery of mathematical problem-solving. On the other hand, the Van Hiele theory characterises five levels of geometrical thinking during one’s geometry learning and development. These five levels are fixed and mastery of one level leads to the next, and there is no success in the next level without mastering the previous level. However, for the purpose of the Curriculum and Assessment Policy Statement (CAPS) we only focused on the first four Van Hiele levels. Findings from this theoretical review revealed that progress in the Van Hiele levels of geometrical thinking depends mainly on the discourse participation of the preservice teachers when solving geometry problems. In particular, an explorative discourse is required for the development in these four levels of geometrical thinking as compared to a ritualistic discourse participation.

Knowledge demands placed on mathematics teachers by textbooks: Case of decimal fractions

Textbooks remain an important and sometimes the only resource for teachers in developing nations. The participative relation between the teachers and the textbooks can be studied through an analysis of how the object of learning is mediated through a careful selection and sequencing of examples, tasks, and representations. In this paper, we use the Mathematical Discourse in Instruction framework to abstract the knowledge demands placed on the teachers by textbooks. We compare the textbooks of two countries, India and South Africa, for the topic of introduction to decimal operations. Findings indicate that these textbooks promote reliance on whole number thinking while deemphasising the use of fractional notation in generalizing the algorithm for decimal addition using a variety of examples, tasks and multiple representations. The demands on teacher knowledge include examining the affordances of using mediational tools while making connections between learners’ prior and emerging knowledge. The study offers an approach for unpacking the relational transparency of textbooks and can therefore be usefully extended to analyze and inform the design of educative materials for teachers.

Enablers and obstacles in teaching and learning of mathematics

This paper presents results of a systematic review of papers published at the LUMAT journal on the current issues positively and negatively affecting teaching and learning in mathematics, in concurrence with the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guidelines. The analysis also offers insight into the most studied topics in mathematics education research, including key demographic and methodological characteristics such as year of publication, participants, education level, research methodologies, and research focus. Data was gathered from the studies published in the LUMAT: International Journal on Math, Science and Technology Education, starting from its first volume in 2013. So far, 225 articles were published in this journal, with 133 studies written in English and 51 studies related to mathematics. Although earlier studies support the notion that mathematics education is mostly traditional, this review suggests current research has thorough and positive outcomes, such that mathematics educators are likely to implement non-traditional approaches, encouraging student engagement, peer collaboration, and mathematical discourse. Certainly, in such learning environments, students tend to feel more motivated and less anxious about learning mathematics. They may also be more active and responsible in their learning, collaborate with peers, and get into mathematical discussions. Yet, there are also a number of difficulties and obstacles highlighted both in teaching and learning of mathematics. The findings might inspire several instructional implications for mathematics educators, curriculum developers, and researchers. Recommendations are given to add into what the existing literature claims and offer greater empirical evidence to support the verdicts.

Using Prior Knowledge to Advance Students’ Mathematical Understanding: Geometric Translation as a Function

Matematik eğitimi araştırmacıları ve öğretim programı geliştiricileri, matematik öğretiminin öğrencilerin ön bilgileri dikkate alınarak tasarlanması gerektiği konusunda hemfikirdir. Bu araştırmanın amacı, öğrencilerin herhangi bir matematiksel konuyla ilgili matematiksel anlamalarını geliştirmek için, ön bilgilerin öğretim dizaynında nasıl kullanılabileceğine dair bir yaklaşım sunmaktır. Konu olarak, okul matematiği ve matematik öğretmen adaylarının eğitimi için önemli konularından birisi olan öteleme seçilmiş ve Pirie-Kieren teorisinde yer alan geriye katlamalar ile öğretim müdahaleleri temel alınarak bir eylem araştırması dizayn edilmiştir. Katılımcılar, bir devlet üniversitesinin Matematik Öğretmenliği programının dördüncü sınıfında öğrenim gören 28 matematik öğretmen adayıdır. Araştırmanın verileri, öteleme dersleri başlamadan önce uygulanan ön teste katılımcıların verdiği yazılı cevaplar, dersler boyunca gerçekleşen matematiksel tartışmalara yönelik kayıtlar, ders gözlem notları ile dersi yürüten eğitimcilerin ders öncesi veya sonrası görüşmelerinde alınan notlardan oluşturmaktadır. Dersler boyunca, öğretmen adaylarının ön bilmeleri üzerinde çalışmalarını sağlayan geriye katlamaları teşvik eden öğretim müdahalelerin, adayların ötelemeyi bir fonksiyon olarak anlamlandırmalarına yardımcı olduğu belirlenmiştir. Araştırmanın, öğrencilerin matematiksel anlamalarını desteklemek amacıyla ön bilmeleri referans alarak öğretim tasarlamak isteyen matematik eğitimcilerine fikir vereceği düşünülmektedir.

Math in Arabic and other languages: exploring the possibilities of translanguaging practices in the classroom

ABSTRACT This article presents the findings from a study on lower secondary math teaching (year 7 and 9) in a Danish school. The study is part of a larger action research project aiming at developing new ways of teaching and learning in culturally and linguistically diverse classrooms. The math teachers hypothesised that their students would be able to benefit more from math instruction if they were able to draw on their full linguistic repertoire in class. As a response to this, a series of teaching activities addressing mathematical discourse in Danish, Arabic, and the other languages present in the classrooms were developed and delivered to the two classes. A group of Arabic-speaking students worked with math assignments and mathematical discourse using learning materials in both Danish and Arabic, but other languages present in the linguistically heterogenetic classes were also involved in the teaching and learning activities, integrating translanguaging practices in math teaching activities. Findings indicate that translanguaging practices in the math classroom have the potentials to elicit linguistic awareness about the nature of mathematical discourse in students, but also raise awareness about the role of languages in the classroom as such.

Teachers’ noticing to promote students’ mathematical dialogue in group work

How can teachers refine their strategies for purposefully engaging students in mathematical discussions when students are working in groups and the teacher enters an ongoing group conversation? In three educational design research cycles, four teachers collaborated with a researcher for one year to analyse, design and evaluate strategies for engaging students in small-group mathematical discussions. The idea of noticing (Mason in Researching your own practice: the discipline of noticing, RoutledgeFalmer, London, 2002; Sherin et al. in Mathematics teacher noticing: seeing through teachers’ eyes, Taylor & Francis, New York, 2011) was used to organize the findings—by paying attention to aspects in the mathematical discussions and interpreting the interactions, teachers could together refine their own actions/responses to better support students’ work. The Inquiry Co-operation Model of Alrø and Skovsmose (Dialogue and learning in mathematics education: intention, reflection, critique, Kluwer Academic Publishers, Dordrecht, 2004) was used as a theoretical base for understanding qualities in mathematical discussions. Ehrenfeld and Horn’s (Educ Stud Math 103(7):251–272, 2020) model of initiation-entry-focus-exit and participation was for interpreting and organizing the findings on teachers’ actions. The results show that teachers became more aware of the importance of explicit instructions and their own role as facilitators of mathematical questions to students, by directing specific mathematical questions to all students within the groups. In this article, by going back and forth between what happened in the teachers’ professional development group and in the classrooms, it was possible to simultaneously follow the teachers’ development processes and what changed in students’ mathematical discussions.

Cu-kerosene and Al2O3-kerosene boundary layer nanofluid flow past a stretching/shrinking surface

A mathematical discussion of two different classes of nanofluids, such as Copper (Cu)-Kerosene and Aluminum oxide (Al2O3)-Kerosene, over a stretching/shrinking surface, has been discussed in this manuscript. Here Kerosene based nanofluids carry Copper and Aluminum oxide as nanoparticles. The ODEs are obtained from basic equations by introducing the similarity approach. The respective coupled nonlinear ODEs are solved with the help of a suitable numerical technique named as Runge-Kutta fourth-order method. It is found that Al2O3-Kerosene possesses a slightly greater velocity than Cu-Kerosene, but a reverse effect is found in the case of temperature and nanofluids. The presence of volume concentration is important due to the presence of nanoparticles as nanofluids property depends on the physical properties of nanoparticles.