This paper demonstrates how questions of "provability" can help students engaged in reinvention of mathematical theory to understand the axiomatic game. While proof demonstrates how conclusions follow from assumptions, "provability" characterizes the dual relation that assumptions are "justified" when they afford…

Many of the central ideas in an introductory undergraduate linear algebra course are closely tied to a set of interpretations of the matrix equation Ax = b (A is a matrix, x and b are vectors): linear combination interpretations, systems interpretations, and transformation interpretations. We consider graphic and symbolic representations for each,…

Starting from an interest in the teachers' use of diagrams and gestures during a traditional front lesson at tertiary level, this research takes a narratologic perspective to see a mathematical lesson as a story, and hence the students' notes as re-tellings of a mathematical story. The first minutes of a traditional mathematics lecture…

In his "Proofs and Refutations," Lakatos identifies the "Primitive Conjecture" as the first stage in the pattern of mathematical discovery. In this article, I am interested in ways of reaching the "Primitive Conjecture" stage in an undergraduate classroom. I adapted Realistic Mathematics Education methods in an…

Weber and Alcock's (2004, 2009) syntactic/semantic framework provides a useful means of delineating two basic categories of proof-oriented activity. They define their dichotomy using Goldin's (1998) theory of representation systems. In this paper, I intend to clarify the framework by providing criteria for classifying student reasoning into…

Many mathematics educators have noted that mathematicians do not only read proofs to gain conviction but also to obtain insight. The goal of this article is to discuss what this insight is from mathematicians' perspective. Based on interviews with nine research-active mathematicians, two sources of insight are discussed. The first is reading a…

Mathematical knowledge used in teaching has attracted the interest of many researchers, but was mainly explored considering teaching at the elementary school level. This paper attends to mathematical knowledge used in teaching at the University level. We present a story about a student suggesting reconsideration of Cantor's diagonal method and the…

Interviews students majoring in mathematics who had passed a required introductory course on algebraic structures on students' difficulties with basic concepts in group theory as part of a research project. Reports data concerning cyclic groups. (ASK)

Presents an exploratory study with expert university students to determine how students could reseal the rupture and restore a sense of unity between the figural and conceptual components of conic sections. Suggests certain tools of semiotic mediation which could be introduced to enable students to achieve the conceptual oversight that is possibly…

Reflects on some aspects of learning mathematics that emerged along the way in creating a turtle-based exploratory tool for non-Euclidean geometry. Contains 14 references. (ASK)

Describes a college geometry course centered around challenging, open-ended problems that invite students to draw upon their experiences and share their understandings. Includes discussion of sample problems; writing assignments; classroom dialog; student comments and insights about geometry, mathematics, and themselves and mathematics; and…

Discusses insufficiency of the linear method and some informal practices (or heuristics) used by expositors in trying to alleviate it. Uses the Cantor-Bernstein theorem to illustrate the linear proof, structuring, and the structure proof. Argues that the informal practices considered be consistently applied to the presentation of pivots and…

Discusses students' writing in mathematics, focusing on (1) creating stories; (2) writing mathematical essays; (3) expressing feelings and beliefs about mathematics through diaries and anecdotes; and (4) the use of dialogs. The strategies considered have been used at the high school and remedial college levels. (JN)

This study concerned conceptual difficulties of college students learning programming and ways the teacher can simplify the process. An analysis of general strategies used in solving problems involving loops is given, with four types of hierarchical strategy categorized. Students had difficulty transforming their algebraic descriptions into…

Some concrete examples of the use of historical materials in developing certain topics from precalculus and calculus are presented. Ideas which can be introduced with a reformulated curriculum are discussed in five areas: algorithms, combinatorics, logarithms, trigonometry, and mathematical models. (MNS)