This paper is a case study of the teaching of an undergraduate abstract algebra course with a particular focus on the manner in which the students presented proofs and the class engaged in a subsequent discussion of those proofs that included validating the work. This study describes norms for classroom work that include a set of norms that the…

This article investigates the numbers [image omitted], originally studied by Catalan. We re-confirm that they are indeed integers. Using the close relationship between them and the Catalan numbers C[subscript n], we develop some divisibility properties for C[subscript n]. In particular, we establish that [image omitted], where f[subscript k]…

A full-cycle assessment of our efforts to improve quantitative reasoning in an introductory math course is described. Our initial iteration substituted more open-ended performance tasks for the active learning projects than had been used. Using a quasi-experimental design, we compared multiple sections of the same course and found non-significant…

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…

We present an analysis of students' formal constructions in mathematics regarding to syntactic, semantic and pragmatic aspects. The analyzed tasks correspond to students of the Course of Mathematics for the admission to the university. Our study was qualitative, consisted in the identification, analysis and interpretation, focused in logic…

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…

A typical first course on linear algebra is usually restricted to vector spaces over the real numbers and the usual positive-definite inner product. Hence, the proof that dim(S)+ dim(S[perpendicular]) = dim("V") is not presented in a way that is generalizable to non-positive?definite inner products or to vector spaces over other fields. In this…

Mathematical elegance is illustrated by strikingly parallel versions of the product and quotient rules of basic calculus, with some applications. Corresponding rules for second derivatives are given: the product rule is familiar, but the quotient rule is less so.

Using a simple trigonometric limit, we provide an intuitive geometric proof of the Singular Value Decomposition of an arbitrary matrix.

Maximal ideals in the ring of continuous functions on the closed interval [0, 1] are not finitely generated. This is well-known. What is not as well-known, but perhaps should be, is the fact that these ideals are not countably generated although the proof is not harder! We prove this here and use the result to produce some non-prime ideals in the…

Proofs that the area of a circle is nr[superscript 2] can be found in mathematical literature dating as far back as the time of the Greeks. The early proofs, e.g. Archimedes, involved dividing the circle into wedges and then fitting the wedges together in a way to approximate a rectangle. Later more sophisticated proofs relied on arguments…

This paper describes how the author's students (in-service and pre-service secondary mathematics teachers) enrolled in college geometry courses use the Geometers' Sketchpad (GSP) to gain insight to formulate, confirm, test, and refine conjectures to solve the classical airport problem for triangles. The students are then provided with strategic…

Examining the actions taken during a teaching experiment can provide insight into practices applicable to the use of mathematics technology to assist adult learners. A case study in the form of a teaching experiment was conducted with a small number of subjects to allow for detailed examination of the influence of technology on student thinking.…

Examining the actions taken during a teaching experiment can provide insight into practices applicable to the use of mathematics technology to assist adult learners. A case study in the form of a teaching experiment was conducted with a small number of subjects to allow for detailed examination of the influence of technology on student thinking.…

This case study examines the salient features of two individuals' reasoning when confronted with a task concerning the cardinality and associated cardinal number of equinumerous infinite sets. The APOS Theory was used as a framework to interpret their efforts to resolve the "infinite balls paradox" and one of its variants. These cases…