The aim of this study is to characterize ways of reasoning and arguing that first year university mathematics students exhibit in problem-solving activities from a course that emphasizes the importance of formulating conjectures and the search for different ways to support or validate them. The use of a Dynamic Geometry System in the…

Assessment is a topic of concern to all stakeholders in our educational system. Pattern Based Questions are an assessment tool which is an alternative to the standardized assessment tool, and they are based on generative learning pedagogy, which shows promise in engaging all learners and usefulness in teaching and learning but validity has not yet…

This study investigates the associations of spontaneous "dynamic gesture" and "transformational speech" with the production of "deductive proofs" in participants' reasoning about geometric conjectures (N=77). Although statistical analysis showed no significant association, the result suggests that purposefully…

The purpose of this study is to examine the characteristics of students' thinking about graphs while evaluating statements from Calculus. We conducted clinical interviews in which undergraduate students evaluated mathematical statements using graphs to explain their reasoning. We report our classification of students' thinking about aspects of…

In this paper I explore eleven undergraduate students' comprehension of a proof taken from an undergraduate abstract algebra course. My interpretation of what it means to understand a proof is based on a proof comprehension model developed by Mejia-Ramos, et al. (2012). This study in particular examines the extent to which undergraduate students…

This study investigates one Calculus student's meanings for quantifiers in Calculus statements involving multiple quantifiers. The student was asked in a two-hour long clinical interview to evaluate and interpret the Intermediate Value Theorem (IVT) and three other statements whose logical structure was similar to the IVT except for the order of…

This paper sets forth a concept (Simon, 2017) of contrapositive equivalence and explores some related phenomena of learning through a case study of Hugo's learning in a teaching experiment guiding the reinvention of mathematical logic. Our proposed concept of contrapositive equivalence rests upon set-based meanings for mathematical categories and…

Feedback on assessed work is invaluable to student learning, but there is a limit to the amount of feedback an instructor may provide. Peer feedback increases the volume of feedback possible, but potentially reduces the quality of the feedback. This research proposes a model of collaborative peer feedback designed to increase quality of peer…

In this paper we explore how students construe what it means for an informal argument to be the basis of a formal proof and what students pay attention to when assessing whether a proof is based on an informal argument. The data point to some undergraduate mathematics students having underdeveloped conceptions of what it means for a proof to be…

We studied the linguistic norms of mathematical proof writing at the undergraduate level by asking two mathematicians and five mathematics undergraduate students to read seven partial proofs based on student-generated work and to identify and discuss uses of mathematical language that were out of the ordinary with respect to standard mathematical…

This case study investigates the effectiveness of a lecture in advanced mathematics. We video recorded a lecture delivered by an experienced professor. Using video recall, we then interviewed the professor to determine the content he intended to convey and we analyzed his lecture to see if and how this content was conveyed. We also interviewed six…

We explore ways that university students handle proving statements that have the overall structure of a conditional implies a conditional, i.e., (p [right arrow] q) [implies] (r [right arrow] s). We structure our analysis using the theory of conceptual blending. We find conceptual blending useful for describing the creation of powerful new ideas…

To support university students' understanding of mathematical proofs, pictures accompanying text are frequently used in textbooks as well as in lectures. However, it is unclear if such pictures influence the individual's reading behaviour. By recording the eye movements of eight mathematicians, we investigated whether and how adults with high…

Today's college ACCUPLACER System's Computerized Placement Test (CPT) plays a major role for determining whether post-secondary students are ready for placement into Arithmetic, Elementary Algebra, Intermediate Algebra, or College Mathematics courses. It is widely used in most college systems in the U.S. This paper discusses some of the issues to…

In this paper the notion of "procept" (in the sense of Gray & Tall, 1994) is extended to advanced mathematics by considering mathematical proof as "formal procept". The statement of a theorem as a symbol may theoretically evoke the proof deduction as a process that may contain sequential procedures and require the synthesis…