This study used task-based interviews to examine students' reasoning about multivariable optimization problems in a volume maximization context. There are four major findings from this study. First, formulating the objective function (i.e. the function whose maximum or minimum value(s) is to be found) in each task came easily for 15 students who…

The ability to distinguish between exact and inexact differentials is an important part of solving first-order differential equations of the form Adx + Bdy = 0, where A(x,y) [not equal to] 0 and B(x,y) [not equal to] 0 are functions of x and y However, although most undergraduate textbooks motivate the necessary condition for exactness, i.e. the…

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…

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…

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…

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…

We offer an analysis of calculus assessment items that highlights ways to evaluate students' application of important meanings and support their engagement in generative ways of reasoning. Our central aim is to identify characteristics of items that require students to apply their understanding of key ideas. We coordinate this analysis of…

In calculus, related rates problems are some of the most difficult for students to master. This is due, in part, to the nature of the problems, which require constructing a nuanced mental model and a solid understanding of the function. Many textbooks present a procedure for their solution that is unlike how experts approach the problem and elide…

Adolescent and children's concepts of multiplication and fractions have been linked to differences in the number of levels of units they coordinate. In this paper, we discuss relationships between adult students' conceptual structures for coordinating units and their pre-calculus understandings. We conducted interviews and calculus readiness…

This paper sets forth a construct that describes how many undergraduate students understand mathematical terms to refer to mathematical objects, namely that they only refer to those objects that satisfy the term. I call this students' pronominal sense of reference (PSR) because it means they treat terms as pronouns that point to objects, like…

In introductory level math classes, writing prompts can be used as part of weekly homework assignments to encourage students to think more deeply about the subject at hand. These writing prompts present scenarios related to recently learned material in a new context and require students to submit a short written response online. Writing prompts…

We explore student performance on True-False assessments with statements in the conditional form "If P then Q" in order to better understand how students process conditional logic and to see whether logical misconceptions impede students' ability to demonstrate mathematical knowledge. We administered an online assessment to a population…

The complexity in understanding the [epsilon-delta] definition has motivated research into the teaching and learning of the topic. In this paper I share my design of an instructional analogy called the Pancake Story and four different questions to explore the logical relationship between [epsilon] and [delta] that structures the definition. I…

Previous studies have shown that students who have completed differential and integral calculus often accept and employ empirical arguments as proofs, but this is not the case for students who have had at least one upper-level proof course; these students tend toward the use of deductive proofs. This paper finds that a majority of the students…