This article reports on a study of Kuwaiti college students' knowledge about the concept of functions. It also shows the connections these students make between different representations of functions. The results indicated that most students (96%) were unable to provide a definition of a function and only a third could provide examples of a…

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…

In this reflection of teaching, we describe a series of activities that introduce the Taylor series through dynamic visual representations with explicit connections to students' prior learning. Over the past several decades, educators have noted that curricular materials tend to present the Taylor series in a way that students often interpret as…

Traditional lecturing is the dominant method of teaching mathematics in undergraduate mathematical courses in many countries, whereas active student-centered approaches such as inquiry-based learning have been shown in some instances to be more effective. Most teaching resources (inquiry-based tasks) available at the tertiary level are related to…

On the one hand, mathematical software is ubiquitous in mathematics education. On the other hand, word problems are an important part of the curriculum, and they often require modelling skills. This is especially true with optimisation and extrema problems proposed to high school and undergraduate students. We propose two activities around extrema…

While proof has been studied from different perspectives in the mathematics education literature for decades, students continue to struggle to build proof comprehension. Complicating this, the manner in which proof comprehension is assessed largely remains to be the definition-theorem-proof format in which students are asked to reproduce proofs or…

The learning of calculus concepts is considered challenging for students. This claim is actual for calculus in general and for specific concepts in particular. In this paper, we focus on the concept of the inflection point. We argue that one of the roots of this problem is the lack of a useful and productive meaning of the concept--the…

Crystalline concept is the main concept used as the reference by students in algebraic verification. This concept divided the way of solving algebraic verification into two types: symbolic and embodied compression. This research aimed to explore the students' mathematical thinking process in solving algebraic verification based on the Crystalline…

This study sought to explore whether access to definitions and general representations influences the construction of general direct arguments. Data was collected in college mathematics courses for prospective elementary school teachers. Participant arguments were analyzed along two variables: the generality of the representations and the…

With this paper, we share an activity for the undergraduate mathematics classroom called "The Harp Project" that leverages the aesthetic nature of both the mathematical and the musical arts. This project was conceived as a STEAM/PBL project with the added feature that it was carried out in pieces by an entire class. Anecdotal evidence…

This article presents the results of our analysis of a sample of 254 Calculus I final exams (collectively containing 4,167 individual items) administered at U.S. colleges and universities. We characterize the specific meanings of foundational concepts the exams assessed, identify features of exam items that assess productive meanings, distinguish…

The Calculus Concept Inventory (CCI), Epstein (2013) aims to test the understanding of calculus ideas, rather than ability to perform calculations. In this paper the CCI is used to measure the effect of the undergraduate calculus cohort over the understanding of calculus in a heterogeneous population including recipients of the program Ser Pilo…

This study aims to address the teaching of integrals in multivariable calculus concerning the role taken by geometry, specifically, geometrical content dealing with boundaries in integrals that appear as curves and surfaces in R[superscript 2] and R[superscript 3]. Adopting the framework of the Anthropological Theory of the Didactic, we approached…

LEGO® stop-motion animations were developed to engage first-year university students in the topics of matrices, vectors, linear geometry and linear transformations. LEGO® provided a versatile medium through which a wide range of concrete and abstract concepts could be physically demonstrated in both two and three dimensions. It was also familiar…

In this article special sequences involving the Butterfly theorem are defined. The Butterfly theorem states that if M is the midpoint of a chord PQ of a circle, then following some definite instructions, it is possible to get two other points X and Y on PQ, such that M is also the midpoint of the segment XY. The convergence investigation of those…