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In the mathematics education literature, there is currently a debate about the mechanisms by which group discussion can contribute to mathematical learning and under what conditions this learning is likely to occur. In this paper, we contribute to this debate by illustrating three learning opportunities that group discussions can create. In analyzing a videotaped episode of eight middle school students discussing a statistical problem, we observed that these students frequently challenged the arguments that their colleagues presented. These challenges invited students to be explicit about what mathematical principles, or warrants, they were implicitly using as a basis for their mathematical claims, in some cases recognize the modes of reasoning they were using were invalid and reject these modes of reasoning, and in other cases, attempt to provide deductive support to justify why their modes of reasoning were appropriate. We then describe what social and environmental conditions allowed the discussion analyzed in this paper to occur. [ABSTRACT FROM AUTHOR]

Not only are the problems teachers pose throughout their teaching of great importance but also the ways in which they use those problems make this a critical component of teaching. A problem-posing episode includes the problem setup, the statement of the problem, and the follow-up questions. Analysis of problem-posing episodes of precalculus instructors suggests that their mindset influences how they pose problems. In this paper, we describe the reflexive relationship between an instructor's mindset and problem-posing episodes. [ABSTRACT FROM AUTHOR]

This paper, written within a discursive perspective, explores the co-shaping of public and private discourse, and some of the circumstances under which one occasions the other, in the evolution of mathematical thinking by pairs of 13-year-olds. The discourse of six pairs of students, engaged in interpreting and graphing problem situations involving rational functions, was analyzed by means of recently developed methodological tools. The nature of the mathematics that emerged for each pair was found to be related to several factors that included the characteristics of the interpersonal object-level utterances both before and after the solution path had been generated, the degree of activity of the personal channels of the interlocutors, and the extent to which the thoughts of participants were made explicit in the public discourse. The analysis of the discursive interactions provided evidence that adolescents within novel problem situations can experience some difficulty in making their emergent thinking available to their partners in such a way that the interaction be highly mathematically productive for both of them. [ABSTRACT FROM AUTHOR]

The paper we present is a case study which has two objectives: the identification of the interactions between pairs of 16 and 17-year-old students related to problem solving and the influence of such interactions in their cognitive development. To achieve these aims, we consider problems associated with a definite conceptual structure – problems comparing flat surface areas –, and we focus the analysis of the interactions on the point of view of the thematic and interlocutive dimensions of the discourse. Such analysis has allowed us to identify a wide typology of exchanges, and, from this, to outline four models of interaction - alternative, guided, relaunching, and co-operative. In the cases analysed, these interactions significantly influence the individual development of cognitive and heuristic abilities in the problem solving process. [ABSTRACT FROM AUTHOR]

In this report, we offer a framework for analyzing the ways in which collaboration influences learners' building of mathematical arguments and thus promotes mathematical understanding. Building on a previous model used to analyze discursive practices of students engaged in mathematical problem solving, we introduce three types of collaboration and discuss their influence on the building of mathematical arguments and student agency. The framework is exemplified using data from a study of the development of mathematical reasoning in an urban sixth-grade informal after-school program. [ABSTRACT FROM AUTHOR]

We discuss contemporary theories in mathematics education in order to do research on research. Our strategy consists of analysing discursively and ideologically recent key publications addressing the role of theory in mathematics education research. We examine how the field fabricates its object of research by deploying Foucault's notion of bio-politics-mainly to address the object 'learning'-and Žižek's ideology critique-to address the object 'mathematics'. These theories, which have already been used in the field to research teaching and learning, have a great potential to contribute to a reflexivity of research on its discourses and effects. Furthermore, they enable us to present a clear distinction between what has been called the sociopolitical turn in mathematics education research and what we call a positioning of mathematics education (research) practices in the Political. [ABSTRACT FROM AUTHOR]

In the United States, undergraduates-regardless of their field of study-generally must complete a mathematics course to meet breadth-of-study requirements. This report is aimed at providing a research foundation for practical efforts to improve teaching and learning in such college mathematics service courses (e.g., college algebra, liberal arts mathematics, business calculus). The case participant, Professor Kale, was a PhD mathematician with 12 years of college teaching experience, 6 years as a graduate student and 6 years after the doctorate. He and students in both of his classes agreed to the daily video recording of their meetings for an entire semester. Through constant-comparative analysis of videos and extensive interviews with Professor Kale, as well as brief interviews with his students and other members of the department, we derived a description of discursive patterns in Kale's classes. We conclude with possible implications for future work in college mathematics service course research and teaching. [ABSTRACT FROM AUTHOR]

The question of how to enhance the learning of low-achieving students in mathematics presents an important challenge to researchers and teachers alike. We investigated whether and how the use of a contextual story presented in a video clip facilitated low-achieving students' understanding of the meaning of fraction expansion. To this end, we (a) videotaped one group of three such students during a guided interaction session, (b) interviewed students and teachers about their first impressions of the use of the video clip, and (c) conducted pre-post-tests to examine the discourse students choose to employ to discuss expansion. Despite the interviewees' impression that the use of the video clips makes it easier to remember the story, the analysis of the guided interaction session revealed that the students did not use it spontaneously when asked to explain why a fraction and its expanded form are equivalent. Rather, their explanations revolved around the expansion procedure. It was the tutor's careful interventions in the discourse, building on the students' recall of the story, which led to a synergy effect that facilitated the students' understanding and articulation of the meaning of fraction expansion. This combination proved to be a potentially successful strategy in effectively promoting low-achieving students' understanding in mathematics, as demonstrated in the students' discourse and post-test performance. At the same time, our results highlight the delicate scaffolding required to achieve a beneficial effect. [ABSTRACT FROM AUTHOR]

This article presents an analysis of a phenomenon that was observed within the dynamic processes of teaching and learning to read and elaborate Cartesian graphs for functions at high-school level. Two questions were considered during this investigation: What types of metaphors does the teacher use to explain the graphic representation of functions at high-school level? Is the teacher aware of the use he/she has made of metaphors in his/her speech, and to what extent does he/she monitor them? The theoretical framework was based on embodied cognition theory. Our findings include teachers' expressions that suggest, among other ideas: (1) orientation metaphors, such as 'the abscissa axis is horizontal'; (2) fictive motion, such as 'the graph of a function can be considered as the trace of a point that moves over the graph'; (3) ontological metaphors; and (4) interaction of metaphors. We also show that teachers were not aware of using metaphors. [ABSTRACT FROM AUTHOR]

Prospective elementary teachers’ ( N = 148) motivation to participate in whole-class discussions during mathematics content courses for teachers, as expressed in their own words on an open-ended questionnaire, were studied. Results indicated that prospective teachers were motivated by positive utility values for participating (to achieve a short-term goal of learning mathematics or a long-term goal of becoming a teacher), to demonstrate competence (to achieve performance-approach goals), or to help others (to achieve social goals). Negative utility values for participating were expressed by those who preferred to learn through actively listening. Five motivational profiles, as composed of interactions among motivational values, beliefs, goals and self-reported participation practices, were prevalent in this sample. Self-reported variations among participants’ utility values and participation practices suggested that prospective teachers engaged differentially in opportunities to learn to communicate mathematically. Results provide pedagogical learner knowledge for mathematics teacher educators. [ABSTRACT FROM AUTHOR]

Spontaneous gesture produced in conjunction with speech is considered as both a source of data about mathematical thinking, and as an integral modality in communication and cognition. The analysis draws on a corpus of more than 200 gestures collected during 3 h of interviews with prospective elementary school teachers on the topic of fractions. The analysis examines how gestures express meaning, utilizing the framework of cognitive linguistics to argue that gestures are both composed of, and provide inputs to, conceptual blends for mathematical ideas, and a standard typology drawn from gesture studies is extended to address the function of gestures within mathematics more appropriately. [ABSTRACT FROM AUTHOR]

We report a study of repairs in communication between workers and visiting outsiders (students, researchers or teachers). We show how cultural models such as metaphors and mathematical models facilitated explanations and repair work in inquiry and pedagogical dialogues. We extend previous theorisations of metaphor by Black; Lakoff and Johnson; Lakoff & Nunes; and Schon, to formulate a perspective on mathematical models and modelling and show how dialogues can manifest (i) application of ‘dead’ models to new contexts, and (ii) generative modelling. In particular, we draw in some depth on one case study of the use of a double number line model of the ‘gas day’ and its mediation of communication within two dialogues, characterised by inquiry and pedagogical discourse genres respectively. In addition to spatial and gestural affordances due to its blend of grounding metaphors, the model translates between workplace objects on the one side and spreadsheet-mathematical symbols on the other. The model is found to afford generative constructions that mediate the emergence of new understandings in the dialogues. Finally we discuss the significance of this metaphorical perspective on modelling for mathematics education. [ABSTRACT FROM AUTHOR]

Our approach to emotion in school mathematics draws on social semiotics, pedagogic discourse theory and psychoanalysis. Emotions are considered as socially organised and shaped by power relations; we portray emotion as a charge (of energy) attached to ideas or signifiers. We analyse transcripts from a small group solving problems in mathematics class, and from an individual student. The structural phase of analysis identifies positions available to subjects in the specific setting, using Bernstein's sociological approach to pedagogic discourse. The textual phase examines the use of language and other signs in interaction and describes the positionings taken up by particular pupils. We then focus on indicators of emotion, and find indications of excitement and anxiety, linked to participants' positionings. Finally we consider implications of our approach. [ABSTRACT FROM AUTHOR]

Four groups (three engineering students in each group) were videotaped while constructing concept maps in Linear Algebra. There are two aims of this study. The first is to characterize the discourse in the groups by addressing the following research questions: Do the students communicate in an effective way? Do the students' communications contain the elements typical for a mathematically productive discourse? The analysis indicates that the communication among the students is effective and contains the elements that are characteristic for a mathematically productive interaction. The two types of methods used to analyze the data were focal and preoccupational analysis. The mathematical content and the coherence of the conversations were examined through focal analysis. The participants' engagement in the discourse was examined by preoccupational analysis, carried out by means of interactive flowchart. The second aim of this study is to evaluate the newly developed methodological framework used to characterize the discourses. The study shows that several aspects of the methodological framework need to be developed. [ABSTRACT FROM AUTHOR]

The purpose of this exploratory study was to investigate the impact of writing during mathematical problem solving. The study involved an analysis of ninth grade algebra students' written and verbal descriptions of their mathematical problem solving processes. Through this comparison, a better understanding of the connection between problem solving and writing is realized. The written and verbal data show a relationship between the number of problem solving strategies tried by students and their success. The majority of problem solving behaviors involve execution actions such as carrying out goals and performing calculations. Students who construct global plans are more successful problem solvers. Students engage in verification behaviors at various stages of problem solving though the majority of students do not verify their final answers. While both oral and written descriptions serve as a tool for understanding students' thinking processes, a comparison of the two modes of reporting, using a metacognitive framework as the lens of analysis, reveals some important variations. Students who wrote descriptions of their thinking were significantly more successful in the problem solving tasks (p< 0.05) than students who verbalized their thinking. Differences in metacognitive behaviors also support the premise that writing can be an effective tool in supporting metacognitive behaviors. [ABSTRACT FROM AUTHOR]

A review of literature shows that during the history of mathematics education at school the answer of what counts as `real mathematics' varies. An argument will be given here that defines as `real mathematics' any activity of participating in a mathematical practice. The acknowledgement of the discursive nature of school practices requires an in-depth analysis of the notion of classroom discourse. For a further analysis of this problem Bakhtin's notion of speech genre is used. The genre particularly functions as a means for the interlocutors for evaluating utterances as a legitimate part of an ongoing mathematical discourse. The notion of speech genre brings a cultural historical dimension in the discourse that is supposed to be acted out by the teacher who demonstrates the tools, rules, and norms that are passed on by a mathematical community. This has several consequences for the role of the teacher. His or her mathematical attitude acts out tendencies emerging from the history of the mathematical community (like systemacy, non-contradiction etc.) that subsequently can be imitated and appropriated by pupils in a discourse. Mathematical attitude is the link between the cultural historical dimension of mathematical practices and individual mathematical thinking. [ABSTRACT FROM AUTHOR]