The development of the mathematical concept of the infinite, through the reflections that arise from personal notions and perceptions and the analysis of some ideas of Galileo and Cantor, invites us to investigate the relationship between intuition and formalization for the understanding of the said concept. This paper aims to observe and describe…

This study focuses on students' understanding of multiple representations of functions. It examines student responses to a task in which calculus students are asked to evaluate the derivative at a point of the cubing function when represented piecewise. Results suggest that attending to the graph of the piecewise function does not improve…

In this paper, we address the variety of ways in which students conceive of binary operations and the metaphors they might leverage when working with binary operations in group theory. We use open-ended surveys paired with interviews to qualitatively explore student's conceptions of binary operation. Through this analysis, we identified three…

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.…

Trigonometry is one of the fundamental topics taught in high school and university curricula, but it is considered as one of the most challenging subjects for teaching and learning. In the current study Mason's theory of attention has been used to examine undergraduate student's perception of the transformation of sinusoidal functions. Two types…

In this study I investigate Saldanha and Thompson's (1998) claim that conceptualizing a coordinate pair in the Cartesian coordinate system as a multiplicative object, a way to unite two quantities' values, supports students in conceptualizing graphs as emergent representations of how two quantities' values change together. I presented three…

In this research we discuss relating the description of certain harmonic phenomena, with models that represent them, with the goal of promoting the understanding of the period, the frequency, and the amplitude of the trigonometric functions, particularly referring to sinusoidal functions; in this way the frontier between sinusoidal functions and…

Student understanding of integration has become a topic of recent interest in calculus research. Studies have shown that certain interpretations of the definite integral, such as the area under a curve or the values of an anti-derivative, are less productive in making sense of contextualized integrals, while on the other hand understanding the…

In this paper we explore how students construe what it means for an informal argument to be the basis of a formal proof and what students pay attention to when assessing whether a proof is based on an informal argument. The data point to some undergraduate mathematics students having underdeveloped conceptions of what it means for a proof to be…

The purpose of this study was to investigate how a set of physical and digital instructional activities can serve as an example space to help further develop a concept image that is aligned with the formal concept definition for the limit of a sequence. In addition, the unique affordances and constraints allowed by using either a physical or…

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…

Formative assessment is the process by which teachers and students gather evidence of learning and then use this to adapt the way they teach and learn. I describe a design research project in which we integrated formative assessment strategies into lesson materials that focus on developing students' conceptual understanding and their capacity to…

This study uses APOS Theory to study students' construction of the sine and cosine functions and their inverses. The conjecture of the constructions students would need to do in order to attain a process conception of these functions is tested by using semi-structured interviews with a group of 11 undergraduate students. Analysis of the interviews…

This study expands contemporary theorising about students' conceptions of equality. A nationally representative sample of New Zealand students' were asked to provide a spoken numerical response and an explanation as they solved an arithmetic additive missing number problem. Students' responses were conceptualised as acts of communication and…

We explore ways that university students handle proving statements that have the overall structure of a conditional implies a conditional, i.e., (p [right arrow] q) [implies] (r [right arrow] s). We structure our analysis using the theory of conceptual blending. We find conceptual blending useful for describing the creation of powerful new ideas…