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Teachers' questioning plays an essential role in shaping collective argumentative discourse. This paper demonstrated that rationality dimensions in teacher questions can be assessed by adapting Habermas' three components of rationality. By coordinating Habermas' construct with Toulmin's model for argumentation, this paper investigated how two secondary mathematics teachers used rational questioning to support student participation in collective argumentation. This paper identified various ways in which two participating teachers used rational questioning to support student participation in argumentation via contributions of argument components. The results establish a theoretical connection between the use of rational questions and students' contributions of components of arguments. The results indicated that not all rational questions were associated with a component of argument, and rational questions may additionally support argumentation in general for the development of a culture of rationality. The study has implications in terms of theory and professional development of teachers.

Teacher education programs have a critical role in supporting prospective teachers' connections between theory and practice. In this study, we examined three prospective secondary mathematics teachers' discourses regarding collective argumentation during and after a unit of instruction addressing collective argumentation and ways they recontextualized their on-campus coursework (theory) into their student teaching (practice) as demonstrated by their support for students' mathematical arguments during student teaching. Through a recursive process of coding data from interviews, reflections, and classroom discussions, we constructed descriptions of participants' discourses about argumentation based in their coursework and identified three themes about collective argumentation in the prospective teachers' discourses: the purposes of argumentation, the role of the teacher in argumentation, and characteristics of effective mathematical arguments. Analysis using extended Toulmin diagrams of classroom data from participants' student teaching showed that these three interrelated themes were visible in the participants' enactment of moves supporting collective argumentation. Our study opens spaces for future investigations into how teachers recontextualize their learning from coursework (theory) into their practice.

Supporting students in making mathematical arguments is an important part of discourse practices in mathematics classrooms. Differences in teachers' support for collective argumentation have been observed and documented, and the importance of the teacher's role in supporting collective argumentation is well established. This article seeks to explain differences in teachers' support for argumentation by examining two student teachers' beliefs about mathematics, teaching, and proof to see which beliefs are visible in their support for argumentation. Assisted by a framework for argumentation and a commitment to teachers' beliefs and actions as sensible systems, we found that teachers' beliefs about the role of the teacher, particularly with respect to giving explanations, were more visible in their support for collective argumentation than other beliefs about mathematics or proof.

Cooney, Shealy, and Arvold (1998) wrote a widely cited article describing the belief structures of prospective teachers and argued that the structures can aid in describing how beliefs change and the influence of authority on the individual. We investigate the impact of this manuscript on the field. To do this, we conducted a literature review (n = 59) of journal articles and proceedings published since 1998 covering the same population and goals of Cooney, Shealy, and Arvold (changing prospective teachers' beliefs) and then conducted an analysis of 101 journal articles citing Cooney, Shealy, and Arvold (1998) to see why the authors cited the piece. We conclude that the impact of Cooney, Shealy, and Arvold's article differs from that of their results and suggest that belief structures should be more carefully investigated by the field.

"Collective argumentation" occurs when a group works together to arrive at a conclusion (supporting it with evidence). Simplistically, this occurs when students give answers to questions and tell how they arrived at the answer, perhaps prompted by a teacher. But collective argumentation can be much richer, with a focus on the process of solving a problem and why that process is appropriate. In this article, the authors describe a study in which they videotaped and analyzed units of instruction (for at least seven class days) by two teachers who exhibited a diverse range of teaching practices. They used 277 diagrams to represent the arguments that took place during our observations. The study sought to answer the following questions: (1) What parts of arguments were provided by the teacher and by the students? (2) What kinds of questions from the teacher prompted students to contribute parts of arguments? and (3) What kinds of warrants were provided by the teacher and by the students? Herein, they provide an example from the classroom in the study to exemplify the most crucial aspects of the teacher's role in collective argumentation.

We combine Peirce's rule, case, and result with Toulmin's data, claim, and warrant to differentiate between deductive, inductive, abductive, and analogical reasoning within collective argumentation. In this theoretical article, we illustrate these kinds of reasoning in episodes of collective argumentation using examples from one teacher's practice. Examining different kinds of reasoning in collective argumentation can inform how students engage in generating and examining hypotheses using inductive and abductive reasoning and move toward the deductive reasoning required for proof. Mathematics educators can build on their understanding of these kinds of reasoning to support students in reasoning in productive ways.

In this article, we report on a study of beliefs about mathematics, teaching, and proof conducted with six prospective secondary mathematics teachers as they completed a two-semester sequence of a content course and a methods course. The initial beliefs of the participants were identified using interview and survey data, and potential shifts in beliefs were examined through further interview and survey data combined with classroom observations and written work. While their beliefs about mathematics and proof appeared to be relatively stable, their beliefs about teaching shifted from a more teacher-centered view to beliefs that foreground the activities and understandings of the students. These shifts are analyzed using the construct of belief structures, and activities and events from the courses that may have facilitated the shifts are identified. The results are consistent with the literature in some respects, such as the stability of the participants' beliefs about mathematics. On the other hand, our results present new information about how prospective secondary mathematics teachers' beliefs about teaching may be impacted. (Contains 2 figures and 4 footnotes.)

Developing deep conceptual understanding of what Ma (1999) calls fundamental mathematics is a well-accepted goal of teacher education. This paper presents a microanalysis of an intriguing episode within a course designed to encourage such understanding. An adaptation of Krummheuer's (1995) elaboration of Toulmin's (1958/2003) diagrams is used to examine video recordings and transcripts of a group of graduate students in secondary mathematics education grappling with the idea of a three-dimensional line having negative slope. The graduate students' understandings of slope are examined using an expansion of Stump's (1999, 2001b) categories of conceptions of slope. The episode ends in an interesting impasse, in which the graduate students agree to pursue the idea no further, purposely ignoring the question of negative slope, despite the clear intention of the task. The analysis explores the argumentation, factors of the learning environment, and conceptions of slope that may have contributed to this impasse.

Prior research on students' uses of technology in the context of Euclidean geometry has suggested that it can be used to support students' development of formal justifications and proofs. This study examined the ways in which students used a dynamic geometry tool, NonEuclid, as they constructed arguments about geometric objects and relationships in hyperbolic geometry. Eight students enrolled in a college geometry course participated in a task-based interview that was focused on examining properties of quadrilaterals in the Poincare disk model. Toulmin's argumentation model was used to analyze the nature of the arguments students provided when they had access to technology while solving the problems. Three themes related to the structure of students' arguments were identified. These involved the explicitness of warrants provided, uses of technology, and types of tasks. (Contains 22 figures, 2 tables and 3 footnotes.)

Views of mathematical modeling in empirical, expository, and curricular references typically capture a relationship between real-world phenomena and mathematical ideas from the perspective that competence in mathematical modeling is a clear goal of the mathematics curriculum. However, we work within a curricular context in which mathematical modeling is treated more as a venue for learning other mathematics than as an instructional goal in its own right. From this perspective, we are compelled to ask "how learning of mathematics beyond modeling may occur" as students generate and validate mathematical models. We consider a diagrammatic model of mathematical modeling as a process that allows us to identify how mathematical understandings may develop or surface while learners engage in modeling tasks. Through examples from prospective teachers' mathematical modeling work, we illustrate how our diagrammatic model serves as a tool to unpack the intricacies of students' mathematical experience while engaging in modeling tasks.