Professors in proof-based mathematics courses often intend that the feedback they provide on students' flawed proofs will promote proof comprehension. In this theoretical article, we investigate how such feedback can be formulated. Drawing on Lakatos's process of proof and refutation, we propose the notion of "heuristic refutation…

Mathematics content courses for prospective elementary teachers (PTs) should provide them with opportunities to deepen their understanding of and develop productive dispositions toward mathematics. Supporting mathematics teacher educators (MTEs) to learn how to develop practices that create rich learning opportunities for novice teachers comprises…

In this paper, we draw on our recent research to inspect again some of the theoretical perspectives we have been using to analyze data and to characterize teaching-learning in university settings. We focus particularly, within a sociocultural perspective on Activity Theory and the construct the "Teaching Triad," seeking to embed the…

Although mathematical modeling plays an important role in many curricula worldwide, significant discrepancies persist in the importance of mathematical modeling in ordinary mathematics classrooms and teacher education. This paper compares pre-service mathematics teachers' professional mathematical modeling competencies in three different…

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…

Medical students generally learn to calculate drug doses using paper problems containing numerical information. In contrast, once qualified, they have to extract the relevant information from treatment guidelines and use equipment, such as syringes, to dose patients. We compared students' success in calculating drug doses using those different…

This paper reports on gender and participation in one inquiry-oriented undergraduate mathematics course. While there is evidence that inquiry can better support all learners, especially women, less is known about the distribution of participation in inquiry classrooms. Here, we focus on how men and women participated in both public spaces (in…

Non-mathematics specialists' competence and confidence in mathematics in their disciplines have been highlighted as in need of improvement. We report from a collaborative, developmental research project which explores the conjecture that greater integration of mathematics and biology in biology study programs, for example through engaging students…

This paper argues that rationality and belief are mutually formative dimensions of school mathematics, where each term is more politically embedded than often depicted in the field of mathematics education research. School mathematics then presents not so much rational mathematical thought distorted by irrational beliefs but rather a particular…

This study reports the results of a study examining how easily students are able to transfer frequency solutions to conditional probability problems to novel situations. University students studied either a problem solved using the traditional Bayes formula format or using a natural frequency (tree diagram) format. In addition, the example problem…

Although students' invented strategies typically prove to be meaningful and effective in improving the students' mathematical understanding, much remains unexplored in the current literature. This study examined, through a teaching-scenario task, the nature of 80 preservice teachers' reasoning and responses to students' informal and formal…

A common theme in accounts of choosing mathematics is that of persistence in the face of troubles or difficulties which are often associated with the structuring effects of gender, class, culture and ethnicity. Centring on an analysis of one woman's account of becoming a mathematician, we build on our understanding of multiple and developing…

Changes in school mathematics curricula, pedagogy and assessment, and as a consequence the pressure for changes in initial teacher preparation, are consequences both of government policy initiatives and pressure from unofficial agents such as the research community, teachers and others, though with great variation across the world. In the case of…

This paper builds on previous work (Black et al., "Educational Studies in Mathematics" 73(1):55-72, 2010) which developed the notion of a "leading identity" (derived from Leont'ev's concept of "leading activity") which, we argued, defined students' motive for studying during late adolescence. We…

We report on our analysis of data from a dataset of 26 videotapes of university students working in groups of 2 and 3 on different proving problems. Our aim is to understand the role of example generation in the proving process, focusing on deliberate changes in representation and symbol manipulation. We suggest and illustrate four aspects of…