The retention of fundamental mathematical skills is imperative to provide a foundation on which new skills are developed. Educators often lament about student retention. Cognitive scientists and educators have explored teaching methods that produce learning which endures over time. We wanted to know if using spaced recall quizzes would prevent our…

This study examined college calculus instructors' preferences in solving two calculus tasks to examine college calculus instructors' use of important cognitive roots in understanding derivatives of function. Our results showed that only one instructor consistently uses cognitive roots while other instructors either focus on algebraic methods or…

The concept of derivative is used in many areas including applied problems and requiring mathematical modelling in different disciplines. One of the most important approaches for teaching the derivative is to support students in visualizing the concept. Also, it is necessary to shift researchers and teachers' focuses to students' dynamic mental…

Over the last several centuries, mathematicians have developed sophisticated symbol systems to represent ideas often imperceptible to their five senses. Although conventional definitions exist for these notations, individuals attribute their personalized meanings to these symbols during their mathematical activities. In some instances, students…

This dissertation reports on three studies about students' conceptions and learning of the idea of instantaneous rate of change. The first study investigated 25 students' conceptions of the idea of instantaneous rate of change. The second study proposes a hypothetical learning trajectory, based on the literature and results from the first study,…

Virtual reality (VR) research in mathematics education has centred largely on geometry content. This paper contributes by describing VR modules developed for another area that heavily involves three-dimensional objects: multivariable calculus. This paper also contributes by describing an empirical study on students' experiences inside the VR…

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…

Undergraduate students study the topic of binomial series expansion as part of their Calculus course. The purpose of this study was to explore the mental constructions of binomial series expansion of a class of 159 students. Data were collected through a written assessment task by each member of the class. A convenient sample of eleven students…

Conceptual understanding of calculus is crucial in the fields of applied sciences, business, and engineering and technology subjects. However, the current status indicates that students possess only procedural knowledge developed from rote learning of procedures in calculus without insight of core ideas. Hence, this paper aims to assess students'…

This study investigates one Calculus student's meanings for quantifiers in Calculus statements involving multiple quantifiers. The student was asked in a two-hour long clinical interview to evaluate and interpret the Intermediate Value Theorem (IVT) and three other statements whose logical structure was similar to the IVT except for the order of…

Knowing when students have the prerequisite knowledge to be able to read and understand a mathematical text is a perennial concern for instructors. Using text describing Newton's method and Vinner's notion of concept image, we exemplify how prerequisite knowledge influences understanding. Through clinical interviews with first-semester calculus…

This study investigated the cognitive and social processes through which high school students acquire the differential concepts through communication in a dynamic geometry environment through some cases. Additionally, we observed how a dynamic geometry environment affects these processes. To achieve this objective, eight students were recruited by…

In this article, we report the results of research that explores the intra-mathematical connections that high school students make when they solve Calculus tasks, in particular those involving the derivative and the integral. We consider mathematical connections as a cognitive process through which a person relates or associates two or more ideas,…

When establishing connections among representations of associated mathematical concepts, students encounter different difficulties and successes along the way. The purpose of this study was to uncover information about and gain greater insight into how student processes connections. Pre-calculus students were observed and interviewed while…

Cognitive Load Theory's Four Component Instructional Design (4C/ID) Model has been used in mathematics education but not confirmed as an instructional theory. Using the Factors Influencing College Success in Mathematics (FICSMath) project and confirmatory factor equation modeling, we empirically validated the model and created the 4C/IDMath Model.…