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In this study we investigated qualities of the mathematical discourse in four kindergarten classes in which kindergarten teachers and 5-year-old children engaged in mathematical learning activities. We analysed differences in the mathematical discourses in two experimental kindergarten classes and two control kindergarten classes, in a research and development project. The overarching research question guiding our study was as follows: what characterises the mathematical discourse unfolding in kindergarten classes? In our study we drew on the theoretical framework Mathematical Discourse in Instruction coined by Adler and Ronda, as we quantified the collected qualitative data. Our analyses identified significant characteristics of mathematical discourse with respect to the children’s opportunities to contribute with ideas and arguments. The discourse in the kindergartens differed both with respect to the extent and nature of verbal utterances among the participants, as well as the mathematical engagement nurtured amongst the children. Moreover, the mathematical discourse within the experimental kindergarten classes, to a greater extent than that in the control kindergarten classes, initiated opportunities for the participating children’s mathematical learning.

The discussion on the development of mathematical discourse plays a key role in the determination of the in-classroom interactions in mathematics learning and instruction. The present study aims to present a theoretical framework for the nature of mathematical discourse that addresses the teacher and student interaction in the classroom. Previous studies attempted to discuss the theoretical structure in mathematical discourse with the embedded theory approach. The findings revealed the core of mathematical discourse that reflected the structure of mathematical discourse based on open, axial and selective codes determined based on the embedded theory approach. The external structure of this core reflects the types of in-classroom interaction, while the internal structure reflects the development of the mathematical discourse. The external structure included four types of interaction: teacher, teacher-class, teacher-student, and student-student. The internal structure includes mathematical discourse movements associated with three stages: motivation, explanation of mathematical ideas, and achievement of mathematical ideas. The external structure of mathematical discourse core revealed the general state of in-classroom interaction core, and the internal structure revealed the specific mathematical discourse based on the mathematical content. It could be suggested that the discourse movements in the mathematical discourse core determined in the present study would provide guidelines for mathematical communications. The study also includes recommendations for future studies on the employment of this general and specific theoretical mathematical discourse framework.

This paper focuses on students’ mathematical discourse emerging from interactions in the digital environment GeoGebra, in which one can construct virtual objects that realize mathematical signifiers and then interact with them. These virtual object realizations can become dynamic interactive mediators (DIMs) that influence the development of the learners’ mathematical discourse. In this case study, I analyze in fine detail the discourse developed by two dyads of students in response to an unfamiliar interview question. One dyad came from a class in which GeoGebra was not part of classroom practice and included students who, according to the teacher’s evaluation, were standard-to-high achieving. The other dyad was from a generally demotivated and low-achieving class in which GeoGebra had become part of classroom practice. The analyses, focused especially on the low-achieving dyad, are guided by the question of how DIMs shaped these students’ discourse. According to the analysis, these students ended up succeeding where standard-to-high-achieving peers did not. Moreover, the detailed analysis of the ways in which the DIMs supported this dyad’s learning showed mechanisms that may be general rather than specific to this one case. This suggests that appropriate integration of DIMs into the teaching and learning of high school algebra can be beneficial for low-achieving students.

The teaching and learning of mathematics in sub-Saharan African countries is dominated by teacher-centred pedagogies rather than student-centred ones. Observations of mathematics teachers at two private schools in South Sudan confirmed such practices. This inspired the researchers to design an intervention to help six primary mathematics teachers shift their practices through problem-solving and mathematical discourse. Design-based research methods were implemented, and data were gathered using observations supported by video and audio recordings and field notes. The participants were selected using convenience sampling, and the data were analysed using Stephan’s checklist of student-centred teaching as a framework. The findings revealed that initially, teachers were using tasks from textbooks, and the teachers themselves were engaged in solving the tasks while their students reproduced their actions. Additionally, all the teachers dominated classroom discussions. After the intervention, the teachers began to select tasks that could enhance learning through problem-solving and mathematical discourse among the students, shifting the teacher’s role to facilitation as the students engaged in solving the tasks on their own. However, the shift in practice was highly dependent on the intervention, as the teachers’ tasks were adopted from the workshop. Contribution: This study contributes to the literature on student-centred mathematics teaching in sub-Saharan Africa by providing insights into the factors that enable or hinder teachers’ adoption of student-centred approaches, and by suggesting ways to support teachers’ professional development and learning within their own contexts.

The discussion on the development of mathematical discourse plays a key role in the determination of the in-classroom interactions in mathematics learning and instruction. The present study aims to present a theoretical framework for the nature of mathematical discourse that addresses the teacher and student interaction in the classroom. Previous studies attempted to discuss the theoretical structure in mathematical discourse with the embedded theory approach. The findings revealed the core of mathematical discourse that reflected the structure of mathematical discourse based on open, axial and selective codes determined based on the embedded theory approach. The external structure of this core reflects the types of in-classroom interaction, while the internal structure reflects the development of the mathematical discourse. The external structure included four types of interaction: teacher, teacher-class, teacher-student, and student-student. The internal structure includes mathematical discourse movements associated with three stages: motivation, explanation of mathematical ideas, and achievement of mathematical ideas. The external structure of mathematical discourse core revealed the general state of in-classroom interaction core, and the internal structure revealed the specific mathematical discourse based on the mathematical content. It could be suggested that the discourse movements in the mathematical discourse core determined in the present study would provide guidelines for mathematical communications. The study also includes recommendations for future studies on the employment of this general and specific theoretical mathematical discourse framework.

The importance of the role of language/discourse in the learning and teaching of mathematics is noted in many mathematics curricula and standards documents. In the research literature, this role has been widely theorised from a Vygotskian perspective. This perspective is limited by some of its underlying assumptions, including an instrumental and systemic view of language as tool and its basis in dialectic. In this paper, I propose a Bakhtinian, dialogic perspective as an alternative. I focus my discussion on the long-standing issue of the relationship between formal and informal mathematical language in the learning and teaching of mathematics. I illustrate this discussion with an examination of interaction in an elementary school mathematics classroom in Québec, Canada. Based on Bakhtin's ideas, I argue that mathematical meaning emerges through locally produced, situated dialogic relations between multiple discourses, voices and languages in mathematics classroom interaction. From this perspective, students do not follow a linear path from informal to formal mathematical discourse; rather, they work with the teacher to expand the repertoire of possible ways to make meaning in mathematics.

Although complex analysis is part of the study programs of many mathematics undergraduates, little research has been done on how individuals interpret basic concepts from complex analysis. To address this gap, this paper investigates how experts individually think about complex path integrals. For this purpose, the commognitive framework is used to conceptualize experts’ interpretations of mathematical concepts discursively, namely in terms of so-called intuitive mathematical discourses. A total of nine interpretations of complex path integrals, so-called discursive images, as well as eight sets of rules governing their construction, so-called discursive frames, are derived from expert interviews. These interpretations range from a rejection of intrinsic meaning to connections with real and vector analysis, mean values, and individual formulations of theorems. The paper also raises questions for the inclusion of the results into teaching and addresses further research.

This study aims to reveal the structure of learning as routinization by providing empirical examples. The participation of a high school student in discourse in task situations involving systems of equations and inequalities was examined in terms of objectification of the discourse, flexibility, bondedness, substantiability, performer’s agentivity, and applicability. The study was designed as a case study, one of the qualitative research methods. Elif, the study participant, is an 11th-grade student studying courses focused on mathematics and science-oriented fields in a public high school. The data were collected through semi-structured individual and focus group interviews. Findings showed that Elif exhibited ritualistic participation by imitating others in the individual interview. In contrast, she exhibited more explorative participation in the focus group interviews as the participants became aware of the structure of the concepts. It was found that students’ participation in discourses was more explorative when they reflected on the routines of those they considered more experienced. It was also determined that for learning to occur as routinization, individuals should participate in discourse and make an effort to develop their routines from process-oriented to result-oriented.

This study explores the relationship between students’ perceptions and teachers’ discourse practices in mathematics classrooms. It reframes the sequence of Initiate-Response-Follow-up (IRF) with a renewed discourse structure that focuses on teachers’ follow-up actions including listening, thoughtful questioning, and effective talk moves. Specifically, the study analyzes how these follow-up actions were related to positive student perceptions about their teachers’ discourse practices around sustaining productive discussions in mathematics classrooms. Participants were secondary mathematics teachers (n = 57) and their students (n = 875) in U.S. schools. The study first considered the students’ perceptions of their teachers’ discourse practices, identifying which teachers were perceived by students to implement mathematics discussions. Next, the study identified and examined patterns of teacher practices in discussions—the teachers’ talk moves, duration, and frequency in asking follow-up questions. Findings indicate that the teachers identified by students as promoting mathematics discussion tended to ask follow-up questions that increased and sustained students’ participation in mathematics discussions. What this finding implies is that in asking follow-up questions, the teacher listened and responded to students’ ideas, and students felt heard. The study asserts that there is much potential for enhancing mathematics instruction by learning more about how teachers listen to and build on students’ responses.