Mathematical Discourse Research Articles

Investigating the Mathematical Discourses of Secondary School 8th Grade Students: Case of Linear Equations

Cebir öğretiminde denklem sistemlerinin, kavramsal olarak zorluğu ve karmaşık yapısının dikkate alınması gerekmektedir. Diğer taraftan, öğrencileri kuralları ezberlemeye teşvik eden, çoğunlukla uygulamaya dönük, gündelik hayatla ilişkili bulunmayan, yalnızca sembol ve işlemi kapsayan uygulamaları içeren cebir öğretimi, bu yapının açığa çıkmasında engel oluşturmaktadır. Doğrusal denklemler kapsamında öğrencilerin gelişimsel öğrenmelerini ve öğretmenlerin doğrusal denklemler konusunun öğretimini sosyo-kültürel bakış açısından ele alan öğrenme ile öğretimi birlikte inceleyen çalışmaların gerekliliği öngörülmektedir. Bu doğrultuda araştırmada, ortaokul öğrencilerinin doğrusal denklemlerle ilgili söylemlerinin değişimini ve öğretimle olan bağını ortaya çıkarmak hedeflenmiştir. Bu hedef doğrultusunda yedi öğrenciden oluşan katılımcılarla yarı yapılandırılmış klinik görüşme ve sınıf gözlemleri gerçekleştirilmiştir. Bulgular değerlendirildiğinde öğrencilerin doğrusal denklemlerle ilgili matematiksel söylemleri oluştururken güçlük çektikleri belirlenmiştir.

Using Interactive Presentations to Promote Mathematical Discourse

<p style="text-align: justify;">The current study investigated whether: (1) using an interactive presentation (IP) platform could affect the amount of usage of the practices of making orchestrating mathematical discourse- sequencing and connecting students' responses. (2) using an interactive presentation (IP) platform could affect the amount of narratives constructed by students. Fifty seventh-grade students participated in the study; those students were divided into control and experimental groups. Qualitative and quantitative analyses were performed based on voice recordings and field notes. The results revealed that the teacher using (IP) asked nearly three times more questions that connected students’ responses (i.e., questions that involved valuing students' ideas, exploring students' answers, incorporating students’ background knowledge, and encouraging student-to-student communication). We also saw that the students participated in the learning processes. The students in the experimental group presented three times as many narratives as those in the control group. We present several excerpts from the transcripts of the classroom discussions to illustrate our findings. Discussion of the implications and limitations of these results and make recommendations based on those results.</p>

Interactions and tensions between mathematical discourses and schoolwork discourses when solving dynamic geometry tasks: what is internally persuasive for students?

ABSTRACT I explore students’ discourses in small groups working on mathematical problems using GeoGebra, focusing on the Cartesian connection between algebra and geometry. Specifically, the interest lies in what is internally persuasive for students in upper-secondary school (11th grade) with histories of low attainment. Three problem-solving episodes are presented in detail to illuminate different intertwining and interacting discourses that students produce. In all cases, students engaged in visual explorations and expressed joy in the work. Students used discourses of visual appearance and technical symbolism to talk about screen objects and problem solutions. Two cases illustrate how a student makes a conjecture internally before verifying it for themselves and then convincing their peers of the validity of their solution. In a contrasting case, students used visual trial and error and asked an authority to confirm their solution. Discourses concerning the institutional demands of school interfered with students’ mathematical dialogue.

Development and analysis of moments preserving finite volume schemes for multi-variate nonlinear breakage model

Modelling and simulation of collisional particle breakage mechanisms are crucial in several physical phenomena (asteroid belts, molecular clouds, raindrop distribution etc.) and process industries (chemical, pharmaceutical, material etc.). This paper deals with the development and analysis of schemes to numerically solve the multi-dimensional nonlinear collisional fragmentation model. Two numerical techniques are presented based on the finite volume discretization method. It is shown that the proposed schemes are consistent with the hypervolume conservation property. Moreover, the number preservation property law also holds for one of them. Detailed mathematical discussions are presented to establish the convergence analysis and consistency of the multi-dimensional schemes under predefined restrictions on the kernel and initial data. The proposed schemes are shown to be second-order convergent. Finally, several numerical computations (one-, two- and three-dimensional fragmentation) are performed to validate the numerical schemes.

Sharon Berry.A Logical Foundation for Potentialist Set Theory

This book offers a foundation for mathematics grounded in a collection of axioms for logical possibility in a first-order language. The offered foundation is argued to have various epistemological benefits, in particular as regards our justification for believing certain axioms of set theory, and more generally as regards the ‘access’ problem for mathematical objects. The foundation for mathematics is provided in stages. First, Berry argues for a ‘potentialist’ interpretation of standard ZF set theory in terms of some axioms for logical possibility that are argued to ‘seem clearly true’. Then, she argues for a Carnap-inspired account of the semantics of ordinary mathematical language in support of the idea that all mathematical discourse as usually pursued can be interpreted faithfully in terms of the foundation provided by her potentialist set theory. The end result, it is argued, is something approximating a logicist foundation for mathematics: standard mathematical truths are interpreted in terms of a pure logic of logical possibility.

Promoting Equitable PST Participation in Mathematical Discourse: Rough Drafts on an Asynchronous Discussion Board

Issues of equity in mathematics classrooms existed prior to COVID-19. For many students, however, meaningful participation in mathematical discussions became nearly impossible in online settings during the pandemic. In this study, we note the diversity in and nature of participation in mathematical discourse in an online course for preservice teachers (PSTs). We investigate the influence of implementing two support strategies for discussion: (a) establishing a “rough-draft/revision” orientation to mathematical tasks; and (b) providing time and structure (tasks and prompts) in an online discussion board for PSTs to post their initial thoughts, react to peers’ solutions, and collectively revise their ideas. In this article, we highlight several benefits of these support strategies to equitable PST participation in a unit on number theory. For example, as compared with oral discussions where only a few PSTs offered their ideas, the written discussion format encouraged every PST to post their ideas. Using a rough-draft/revision stance in the prompts fostered sharing and revealed diverse mathematical approaches, perspectives, and ideas. We argue that giving students opportunities to interact with one another and the mathematics in a variety of ways promotes equitable participation.

Rethinking Wait Time: What Can 3 Seconds Do?

A high-leverage strategy first discussed more than 50 years ago, wait time has many benefits for both teachers and students yet is not used to its full potential. See how it can enhance your students’ mathematical discourse.

Enriching Capacity and Transmission of Hybrid WDM-FSO Link for 5G Mobility

A dramatic increase in user and capacity demands has been noted after the COVID-19 pandemic. These challenges have damaged the 5G communication system mobility. Therefore, developing mobility and enhancing capacity transmission of 5G advanced services are the focused research gaps in the current era. In this paper, the free space optics (FSO) link is modeled with wavelength division multiplexing (WDM) technology based optical fiber system, purposing to enhance the 5G capabilities in multi-channel, high distance, and bidirectional transmissions. In addition, the presented hybrid FSO-WDM supported optical fiber network is analyzed for 4, 8, and 16 × 10 Gbps downlink and uplink transmission. The paper also includes the mathematical discussion of merged fiber length (SMF = 30 km) and FSO (600 m) with improved mobility management. In another contribution, the tolerance against Rayleigh backscattering (RB) noises is developed through various wavelengths of downlink and uplink channels. Finally, we perform the simulation analysis and reliability of the proposed structure for the 5G advanced communication system.

Analysing the Mathematical Discourse of Biology Assignments: The Case of a Graduate Fisheries Management Course

Analysing the Mathematical Discourse of Biology Assignments: The Case of a Graduate Fisheries Management Course

Teaching routines and student-centered mathematics instruction: The essential role of conferring to understand student thinking and reasoning

We compare two lessons with respect to how a teacher centers student mathematical thinking to move instruction forward through enactment of five mathematically productive teaching routines: Conferring To Understand Student Thinking and Reasoning, Structuring Mathematical Student Talk, Working With Selected and Sequenced Student Math Ideas, Working with Public Records of Students’ Mathematical Thinking, and Orchestrating Mathematical Discussion. Findings show that the lessons differ in the enactment of teaching routines, especially Conferring to Understand Student Thinking and Reasoning which resulted in a difference in student-centeredness of the instruction. This difference highlights whose mathematics was being centralized in the classroom and whether the focus was on correct answers and procedures or on students’ mathematical thinking and justifying.

Teachers' Classroom Discourse and Academic Achievement Prediction in Mathematics among Secondary School Students in Kakamega County, Kenya

Teachers' Classroom Discourse and Academic Achievement Prediction in Mathematics among Secondary School Students in Kakamega County, Kenya

의사 인지 이론 관점에서 본 2015 개정 고등학교 수학에서 대학 수학으로의 정적분 담론의 이행

According to the 2015 national curriculum revision, the high school mathematics 2 textbook introduces the definite integral as the difference of the anti-derivatives at the endpoints. However, because most calculus and elementary analysis textbooks at the university level define definite integral as the limit of Riemann sums, students who learned mathematics according to 2015 curriculum revision in high school are forced to use the word “definite integral” differently in university. Sfard views mathematics as a discourse, and she explains the historical development of mathematics and mathematics learning as the change of discourse in the participationist’s perspective. This article looks at the transition problem of the definite integral from school and university in the commognitive framework of Sfard. The comparative analysis of high school and university mathematics discourse about definite integral demonstrates that the two discourses are incommensurable not only in the different word use of definite integral but also the different meta-rule of using definitions and proving. Furthermore, the historical analysis shows that this meta-level change in the transition can be traced to the mathematical revolution from 18th result-oriented calculus to 19th rigorous calculus based on precise definitions and proof. This analysis helps instructors to understand the nature and meaning of the challenges of students in the transition of definite integral between school and university mathematics, to be aware of the commognitive conflict in the communication with students about definite integral, and to design a lesson that helps student change their discourse. The commognitive perspective can contribute to instructors seeing the didactical possibilities in the transition and improve their teaching approach.

Лексические и синтаксические параметры академического текста: дискриминантный анализ

This article presents three mathematical models to differentiate academic texts from three subject discourses written in Russian (i.e., Philological, Mathematical, and Natural Sciences) which further enable design and automated profiling of corresponding typologies. Our models include 5 indices, one at surface level (i.e., sentence length) and 4 syntax features (i.e., mean verbs per sentence, mean adjectives per sentence, local noun overlap, and global argument overlap). We identified and validated the five statistically significant features out of 45 linguistic features extracted from our research corpus consisting of 91.185 tokens. The shortest sentence length is found in Russian language textbooks while the longest sentences are identified in Natural Science texts. The mean number of verbs, nouns, and adjectives per sentence is higher in Natural Science textbooks, whereas Mathematics discourse is characterized by the shortest word length, highest local noun overlap, and highest global argument overlap. We assign the metric differences between the three discourses to their functions: Natural Science texts are characterized by descriptions and narrative passages in contrast to Philology that is associated with opinions. Mathematical discourse operates with precise definitions, explanations and justifications thus exercising numerous overlaps. The discriminant analysis built on top of the features supports the development of text profilers targeting parametric analyses. The automation of these features and the provided formulas for classification enable the design and development of text profilers required for textbook writing and editing. Our findings are useful for professional linguists, technologists, and academic writers to select and modify texts for their target audience.

Identifying the Characteristics of Non-Digital Mathematical Games Most Valued by Educators

Non-digital games are frequently used to support primary mathematics instruction. Moreover, we know from the literature that to increase the likelihood that a chosen mathematical game is educationally rich it should reflect specific principles, such as offering a balance between skill and luck and ensuring that a key mathematical focus is central to gameplay. However, there is limited research informing us, from a teacher’s perspective, of the specific characteristics of mathematical games that are most indicative of a game’s value for supporting learning, and the likelihood that teachers will use the game with students in the future. To help address this gap, the current study invited 122 educators to complete an on-line questionnaire, including 20 Likert-scale items designed to assess the characteristics of educationally-rich mathematical games (CERMaGs) that aligned with six ‘good practice’ principles previously identified in the literature, in relation to a specific mathematical game of their choosing. In total, educators chose a broad range of mathematical games to be evaluated (n = 53). On average, they reported that their chosen game was highly valuable for supporting mathematics learning and that they were very likely to use this game with students if given the opportunity. Our results revealed that the extent to which educators perceived a game to be suitably challenging, engaging, enjoyable, modifiable to support different learners, and transformable into an investigation or broader mathematical inquiry, were particularly important characteristics associated with perceptions of a game’s educational value. Similarly, perceived levels of student enjoyment, engagement and a game’s potential to lead to a rich mathematical investigation were important characteristics for evaluating the likelihood that an educator would use a particular game in the future with students if given the opportunity, as was the capacity of a game to support mathematical discussion. The implications of these findings for supporting classroom practice and teacher professional learning are discussed.

Pre-Service Teachers' Discourse Moves During Whole Class Mathematical Discussions: An Analysis and Proposed Framework

Learning to facilitate whole-class mathematical discussions is a complex task. Teachers must become adept at using discourse moves to elicit and extend student thinking and move discussions towards a mathematical point. Practice-based mathematics methods courses give pre-service teachers opportunities to rehearse these moves in a controlled setting. Providing opportunities for practice, often called rehearsals or approximations of practice, have been shown to be effective at developing pre-service teachers' abilities to use discourse moves, but we still do not have a clear understanding of the types of discourse moves pre-service teachers use during discussions or how these moves change over the course of a discussion. In this article, we analyze. whole-class discussions facilitated by pre-service teachers to determine the types of discourse moves used as the discussion progresses. Based on this data and relevant literature, we propose a framework for the structure of whole-class mathematical discussions.

Mathematical Discussions: Revealing Biases

Do you use mathematical discussions to increase engagement in your classroom? In this Front and Center article, authors provide a discourse tool that can be used to reveal potential biases found in the implementation of the Five Practices.

ICT 환경에서 수학 교사의 수업 조직과 권위 분배에 대한 연구

Extant literature on mathematics instruction in the ICT environment has reported its potential and the role of teachers. However, there is a paucity of literature regarding how teachers specifically organize the use of ICT instruments in instruction and distribute authority with and for students during mathematical discussion. This study analyzes the instructional organization and practices in Korea that promote students’ inquiry-based learning in an ICT environment by using the framework of instrumental orchestration. This framework considers how teachers utilize various artefacts and adjust teaching to promote students’ learning. In addition, the study examines how teachers distribute authority in the classroom to promote students’ inquiry activities. The study reports different types of instrumental orchestration and compares the distribution of authority in four teachers’ classes. The study also discusses implications for inquiry-based learning in ICT environments.

Degrees of Objectivity? Mathemata and Social Objects

A down-to-earth admission of abstract objects can be based on detailed explanation of where the objectivity of mathematics comes from, and how a ‘thin’ notion of object emerges from objective mathematical discourse or practices. We offer a sketch of arguments concerning both points, as a basis for critical scrutiny of the idea that mathematical and social objects are essentially of the same kind—which is criticized. Some authors have proposed that mathematical entities are indeed institutional objects, a product of our collective imposition of function onto reality (the phrase comes from Searle) and of surrogation or hypostasis. Yet there are significant disanalogies between the typical social objects and mathemata, on which basis I argue that one should make a clear distinction between both. The comparison of mathematical with social objects helps understanding how non-physical objects can figure prominently in our explanations of reality. Yet mathematical objects have a different kind of cognitive grounding, and the more elementary of them emerge under relatively very simple sociocultural conditions. The differences are also reflected in the wide scope of use of mathematical concepts, and the much higher degree of variation found among social objects. On the basis of all of these features, I defend the thesis that one can significantly distinguish degrees of objectivity, and I use the distinction to articulate a graded ontology where one can locate the different kinds of mathematical and social objects.

Examining a Coaching Routine to Support Teacher Learning

ABSTRACT Mathematics specialists tasked with the responsibility of supporting teacher learning face both the opportunity and the challenge of transforming the organization of the school workplace to support educators’ collective, ongoing learning, which is not the norm in most school settings. In this study, we examine a coaching routine called Teacher Time Out (TTO), which was organically developed by a school-based mathematics coach and the teachers with whom she worked. Through the routine, coaches and teachers work through complex, in-the-moment pedagogical decision making while collectively facilitating mathematics discussions among students. The routine thus opens opportunities for educators to learn about ambitious teaching alongside their colleagues. We report findings from an analysis of 360 TTOs that occurred over three years of one coach’s work supporting a school-wide, multi-year instructional reform effort in mathematics teaching and learning. We found that the coaching routine fostered teachers’ collective inquiry into practice, as they engaged with the unpredictability of teaching during real-time instruction with students. We discuss the potential of this routine to support coaching as a lever for organizational reform, reshaping mathematics teaching across many classrooms.

A teacher’s positioning of a multilingual learner in the first month of the academic year

ABSTRACT Positioning theory [van Langenhove and Harré (1999). Introducing positioning theory. In R. Harre, & L. van Langenhove (Eds.), Positioning theory: Moral contexts of intentional action (pp. 14–31). Blackwell Publishers] provides a useful analytical lens to examine how teachers’ positioning in moment-to-moment interactions can expand or restrict opportunities for multilingual learners to participate in mathematical discussions and develop robust mathematical identities. In this study, I examined the discursive practices of one white, third-grade monolingual teacher, Keri, and the ways she interactively positioned Alexia, a Latina multilingual learner, in the first full month of the academic year during whole class interactions. Findings demonstrate Keri used a range of positioning acts to recursively position Alexia as a communicator of mathematics and mathematical thinker that simultaneously situated Alexia within the broader classroom storyline that successful mathematics students communicate their ideas to others. The findings confirm that monolingual teachers in elementary classroom settings appear to use similar positioning acts as bilingual teachers/teacher-researchers with individual multilingual students.