Action-Process-Object-Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help…

Our investigation is concerned with new teachers developing their ability to understand student thinking. We conducted individual interviews with graduate students teaching calculus for the first time, interviewing a representative sample of graduate students before and after their first teaching assignment. The interviews were transcribed and…

In this article, we elaborate methodologies to study the argumentation speech of a teacher involved in argumentative activities. The standard tool of analysis of teachers' argumentation concerning pedagogical matters is Toulmin's model. The theory of argumentation schemes offers an alternative perspective on the analysis of arguments. We propose…

We propose a new course structure to address the needs of college students with previous calculus study but no course validations as an alternative to repeating the first year of calculus. Students are introduced directly to topics from Calculus III unpreceded by a formal review of topics from Calculus I or II, but with additional syllabus time…

Interest in understanding mathematics teaching and learning phenomena, and to develop new and effective methodologies to teach Differential Calculus and Linear Algebra led me to look for ways to dialogue between APOS Theory and other mathematics education theories. This enterprise has facilitated a better understanding of them. Bridges between…

We present a case study of how five instructors used gesture when introducing the second derivative test in a first semester calculus class. The second derivative test and optimization naturally evoke hand motions while teaching, making this a fertile ground for studying gesture use in the classroom. Each of the five instructors used a classic…

Novice students have difficulty with the topic of limits in calculus. We believe this is in part because of the multiple perspectives and shifting metaphors available to solve items correctly. We investigated college calculus instructors' personal concepts of limits. Based upon previous research investigating introductory calculus student…

Our research focuses on the learning of series as a consequence of institutional choices for their teaching. Our analyses of textbooks and teaching practices led us to conjecture the existence of some implicit contract rules in the teaching of series: in particular, the teaching of series is made almost exclusively in the algebraic setting, with…

Understanding Maxwell's equations in differential form is of great importance when studying the electrodynamic phenomena discussed in advanced electromagnetism courses. It is therefore necessary that students master the use of vector calculus in physical situations. In this light we investigated the difficulties second year students at KU Leuven…

By describing the motion of a charged particle in the well-known nonuniform field of a current-carrying long straight wire, a variety of teaching/learning opportunities are described: 1) Brief review of a standard problem; 2) Vector analysis; 3) Dimensionless variables; 4) Coupled differential equations; 5) Numerical solutions.

Recent studies of the transition from school to university mathematics have identified a number of epistemological gaps, including the need to change from an emphasis on equality to that of inequality. Another crucial epistemological change during this transition involves the movement from the pointwise and global perspectives of functions usually…

Two philosophical ideas motivate this paper. The first is an answer to the question of what is an appropriate activity for a physicist. My answer is that an appropriate activity is anything where the tools of a physicist enable him or her to make a contribution to the solution of a significant problem. This may be obvious in areas that overlap…

This article empirically tests a previously developed theoretical framework for mathematics identity based on students' beliefs. The study employs data from more than 9,000 college calculus students across the United States to build a robust structural equation model. While it is generally thought that students' beliefs about their own competence…

Derivatives and integrals are two important concepts of calculus which are precondition topics for most of mathematics courses and other courses in different fields of studies. A majority of students at the undergraduate level have to master derivatives and integrals if they want to be successful in their studies However, students encounter…

We investigate the nature of Calculus I homework at five PhD-granting universities identified as having a relatively successful Calculus I program and compare features of homework at these universities to comparable universities that were not selected as having a successful program. Mixed method analyses point to three aspects of homework that…