The notion of definition is a central building block of mathematics. In addition to specific definitions, it is desirable that students learn about meta-mathematical aspects of definitions such as their role of classification, and about characteristics of definitions such as arbitrariness. We investigated how such meta-mathematical aspects emerge…

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…

The emergence of a proof image is often an important stage in a learner's construction of a proof. In this paper, we introduce, characterize, and exemplify the notion of proof image. We also investigate how proof images emerge. Our approach starts from the learner's efforts to construct a justification without (or before) attempting any…

We present a view of knowledge construction processes, focusing on partially correct constructs. Motivated by unexpected and seemingly inconsistent quantitative data based on the written reports of students working on an elementary probability task, we analyze in detail the knowledge construction processes of a representative student. We show how…

This case study deals with a solitary learner's process of mathematical justification during her investigation of bifurcation points in dynamic systems. Her motivation to justify the bifurcation points drove the learning process. Methodologically, our analysis used the nested epistemic actions model for abstraction in context. In previous work, we…

Identified and classified solution strategies of (n=49) 10th-grade students who were presented with linear programming problems in a predominantly visual setting in the form of a computerized game. Visual strategies were developed more frequently than either algebraic or mixed strategies. Appendix includes questionnaires. (Contains 11 references.)…

One sentence answer to the question in the title is that the ability to prove depends on forms of knowledge to which most student are rarely, if ever, exposed. Presents more detailed analysis, drawing on research in mathematics education and classroom experiences. (Contains 44 references.) (Author/ASK)