Teaching mathematical modeling is a demanding task. Thus, fostering teachers' competencies in this regard is an essential component of teacher education. Recent conceptualizations of teachers' competencies include situation-specific skills based on the concept of noticing, which is of particular interest for the spontaneous reactions needed when…

Previous research in mathematics education has explored teachers' conceptions of mathematics and its teaching and learning, and how their instructional tendencies (e.g., "traditional", "technological", "spontaneous" and "investigative") relate to these conceptions. However, empirical evidence on this topic…

Coming from the commognitive standpoint, we consider proof-based mathematics as a distinct discourse, the transition to which requires special rules for endorsement and rejection of mathematical statements. In this study, we investigate newcomers' learning of these rules when being taught them explicitly. Our data come from academically motivated…

In mathematics education research, proofs are often conceptualized as sequences of mathematical assertions. We argue that this ignores proofs that contain instructions to perform mathematical actions, often in the form of imperatives, which are common both in mathematical practice and in undergraduate mathematics textbooks. We consider in detail a…

In this study, we investigated undergraduate mathematics students' (N = 267) attitudes towards proving. The students were taking an introduction-to-proof type course that was situated at the beginning of the mathematics curriculum and lasted for one term. Four attitude variables were measured at the beginning and at the end of the course with a…

In this paper, we identify various forms of geometric work carried out by student teachers who were asked to perform a geometric task for the estimation of a land area. The theory of Mathematical Working Spaces is used to analyze and characterize the work produced. This study provides evidence that students developed forms of geometric work that…

This article presents a research apparatus for investigating and making sense of stories that emerge from feedback that mathematicians provide on students' proofs. Using the commognitive framework, the notion of "didactical discourse on proof" is developed as a lens for conceptualizing mathematicians' practice of feedback provision. The…

This paper deals with writing a proof text as the final step of the proving process at university level, particularly when it results in a disorganized, unclear draft. The reported study concerns third year university students when dealing with proof tasks for which the proving process has to be built up, as opposed to tasks that students may…

This paper explores how in-service teachers enrolled in a graduate proof course interpret, understand, and use generic examples as part of their proving and justification activities. Generic examples, which are capable of proving and justifying with strong explanatory power, are particularly important for teachers considering teaching proof in…

This manuscript focuses on how students make sense of proofs. Participants were students who engaged in peer-review conferences of each other's attempted proofs in a graduate-level real analysis course for mathematics teachers. Building on the concept of distance from conversational analysis, we distinguish how three types of distance (epistemic,…

This paper presents the findings from a survey used to investigate how mathematicians perceive the genre of mathematical proof writing at the undergraduate level. Mathematicians were asked whether various proof excerpts highlighted in four partial proofs were unconventional in each one of three pedagogical contexts: undergraduate mathematics…

The notion of flow of a proof encapsulates mathematical, didactical, and contextual aspects of proof presentation. A proof may have different flows, depending on the lecturer's choices regarding its presentation. Adopting Perelman's New Rhetoric (PNR) as a theoretical framework, we designed methods to assess aspects of the flow of a proof. We…

This paper deals with the competence of Cultural (epistemological, historical and anthropological) Analysis of the Content (CAC), which is important for teachers' awareness and autonomy when dealing with educational choices in a changing cultural and institutional context. We report on an 18-hour intervention in a teacher education course at the…

The research is a study of the Husserlian approach to intuition, informed by Merleau-Ponty's theory of perception, in the case of a prospective teacher of mathematics. It explores the two major stages-categories of intuition, the essential relations between them, and their vital role in the emergence of empirical and abstract mathematical…

Researchers have increasingly focused on how gestures in mathematics aid in thinking and communication. This paper builds on Arzarello's (2006) idea of a "semiotic bundle" and several frameworks for describing individual gestures and applies these ideas to a case study of an instructor's gestures in an undergraduate abstract algebra…