This research contributes to the need to identify and expand learning environments that encourage undergraduate students to develop collaborative work skills and apply their classroom knowledge to solve real-world problems. Using qualitative methods, we examine the effects of the interaction between two teams of students when solving a…

Special tutorials both online and off-line were experimented in order to provide extra support for the senior pre-service mathematics teachers at an Australian regional university to improve their learning experience and achieve the best possible learning outcomes in an advanced mathematics course focusing on solving ordinary differential…

Students need a robust understanding of the derivative for upper-division mathematics and science courses, including thinking about derivatives as ratios of small changes in multivariable and vector contexts. In "Raising Calculus to the Surface" activities, multivariable calculus students collaboratively discover properties of…

The cases of constant and quadratic damping of free oscillations are missing from standard textbooks, even at college and university level. The case most examined is that of linear damping, the reason being that the student can work out a closed form which describes all stages of motion. The case of constant damping is straightforward to be…

Previous studies have suggested that problem-posing activities could be used to improve the teaching, learning, and assessment of mathematics. The purpose of this study is to explore undergraduate engineering students' problem posing in relation to the integral-area relationship. The goal is to help fill a gap in tertiary level research about…

The aim of this paper is to present a teaching proposal for the theoretical part relating to the first- and second-order linear difference equations with constant coefficients suitable for the first-year students at various types of universities. In contradistinction to the methods often applied (memorization of algorithms without a proper…

In this study, we examined one experienced mathematician's class practices, with particular attention to cognitive model described in genetic decomposition. Our findings indicate that students only had limited opportunities to be familiar with the first three steps in genetic decomposition, which may potentially lead students to answer limit tasks…

Why do people shift their strategies for solving problems? Past work has focused on the roles of contextual and individual factors in explaining whether people adopt new strategies when they are exposed to them. In this study, we examined a factor not considered in prior work: people's evaluations of the strategies themselves. We presented…

Combinatorial proof serves both as an important topic in combinatorics and as a type of proof with certain properties and constraints. We report on a teaching experiment in which undergraduate students (who were novice provers) engaged in combinatorial reasoning as they proved binomial identities. We highlight ways of understanding that were…

This keynote address explores the history and role of college math requirements with a focus on ensuring math courses serve to expand students' horizons, rather than serve as gatekeepers. It discusses the advent of general education math courses, which brought more students into math departments, which ultimately contributed to broadening the…

The distant magnetic field of a magnetic dipole is usually derived via the magnetic vector potential and substantial vector calculus. This paper presents an alternate proof that is less mathematically intensive, and that ties together various problem-solving tricks (the principle of virtual work, observation that only instantaneous quantities…

In a recent submission to "The Physics Teacher," we related how trigonometric identities can be used to find the extremes of several functions in order to solve some standard physics problems that would usually be considered to require calculus. In this work, the functions to be examined are polynomials, which suggests the utilization of…

We present our analysis of 254 Calculus I final exams from U.S. colleges and universities to identify features of assessment items that necessitate qualitatively distinct ways of understanding and reasoning. We explore salient features of exemplary tasks from our data set to reveal distinctions between exam items made apparent by our analytical…

This paper extends work in the areas of quantitative reasoning and covariational reasoning at the undergraduate level. Task-based interviews were used to examine third-semester calculus students' reasoning about partial derivatives in five tasks, two of which are situated in a mathematics context. The other three tasks are situated in real-world…

In this dissertation I explore instructors' work with instructional tasks as they plan to introduce various representations of derivatives. Drawing on the notion of "framing" from sociology, I adopt a situative lens to study calculus instructors' planning of instructional tasks. I rely on Herbst and Chazan's (2012) notion of…