This study examines how collective activity related to multiplication evolved over several class sessions in an elementary mathematics content course that was designed to foster prospective elementary teachers' number-sense development. We document how the class drew on as-if-shared ideas to make sense of multidigit multiplication in terms of…

This study expands contemporary theorising about students' conceptions of equality. A nationally representative sample of New Zealand students' were asked to provide a spoken numerical response and an explanation as they solved an arithmetic additive missing number problem. Students' responses were conceptualised as acts of communication and…

Becoming a skillful prover is critical for success in advanced undergraduate and graduate mathematics courses. In this dissertation, I report my investigations of proof and the proving process in three separate studies. In the first study, I examined the amount of logic used in student-constructed proofs to help in the design of…

In this article, nine mathematicians were interviewed about their why and how they presented proofs in their advanced mathematics courses. Key findings include that: (1) the participants in this study presented proofs not to convince students that theorems were true but for reasons such as conveying understanding and illustrating methods, (2)…

We propose the use of finite topological spaces as examples in a point-set topology class especially suited to help students transition into abstract mathematics. We describe how carefully chosen examples involving finite spaces may be used to reinforce concepts, highlight pathologies, and develop students' non-Euclidean intuition. We end with a…

The purpose of this study was to investigate three second-year graduate students' awareness and understanding of the relationships among the "big ideas" that underlie the concept of derivative through modeling tasks and Skemp's distinction between relational and instrumental understanding. The modeling tasks consisting of warm-up,…

This project classifies groups of small order using a group's center as the key feature. Groups of a given order "n" are typed based on the order of each group's center. Students are led through a sequence of exercises that combine proof-writing, independent research, and an analysis of specific classes of finite groups…

This dataset contains the results of a quasi-experiment, testing Karl Pearson's "drunkard's walk" analogy for an abstract random walk. Inspired by the alternate hypothesis that drunkards stumble to the side of their dominant hand, it includes data on intoxicated test subjects walking a 10' line. Variables include: the…

As computing becomes ubiquitous, software correctness has a fundamental role in ensuring the safety and security of the systems we build. To design and develop software correctly according to their formal contracts, CS students, the future software practitioners, need to learn a critical set of skills that are necessary and sufficient for…

Validating proofs and counterexamples across content domains is considered vital practices for undergraduate students to advance their mathematical reasoning and knowledge. To date, not enough is known about the ways mathematics majors determine the validity of arguments in the domains of algebra, analysis, geometry, and number theory--the domains…

In this paper, we report a pilot study on engaging a group of undergraduate students to explore the limits of sin(x)/x and tan(x)/x as x approaches to 0, with the use of non-graphic scientific calculators. By comparing the results in the pretest and the post-test, we found that the students had improvements in the tested items, which involved the…

A well known result due to Laplace states the equivalence between two different ways of defining the determinant of a square matrix. We give here a short proof of this result, in a form that can be presented, in our opinion, at any level of undergraduate studies.

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…

A comparison of the proof validation behavior of beginning undergraduate students and research-active mathematicians is explored. Participants' eye movements were recorded as they validated purported proofs. The main findings are that (a) contrary to previous suggestions, mathematicians sometimes appear to disagree about the validity of even short…

This note explains how Emil Artin's proof that row rank equals column rank for a matrix with entries in a field leads naturally to the formula for the nullity of a matrix and also to an algorithm for solving any system of linear equations in any number of variables. This material could be used in any course on matrix theory or linear algebra.