A challenge for prospective teachers (PTs) is to determine what students know about a topic through asking appropriate questions and being thoughtful about the wording of these questions so as to capture and reframe students' spontaneous mathematical thinking and eventually unriddle the fuzzy boundary of students' complex thinking. This study…

The aim of this paper is on the one hand to discuss from an APOS (Action-Process-Object-Schema) theory perspective the mental constructions involved in the learning of linear algebra, through examples concerning the linear transformation concept and related notions. On the other hand, methodological issues related to the design of research…

Researchers have argued teachers and students are not developing connected meanings for function inverse, thus calling for a closer examination of teachers' and students' inverse function meanings. Responding to this call, we characterize 25 pre-service teachers' inverse function meanings as inferred from our analysis of clinical interviews. After…

The purpose of this study was to gain insight into 30, first year calculus students' understanding of the relationship between the concept of vertex of a quadratic function and the concept of the derivative. APOS (action-process-object-schema) theory was applied as a guiding framework of analysis on student written work, think-aloud and follow up…

In this article, attempts were made to examine students' thinking about the concepts of infinity and their ideas about transiting from finite to infinite states through the concept of limits of sequences. The participants included 78 senior high-school students ranging in age between 17 and 19 years old. The data were collected through a…

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…

Evidence suggests that some students have learned procedures with little or no underpinning understanding while others have a much more connected and conceptual levels of understanding. In this article, the work of four primary students is discussed in terms of their contextual understanding of multiplicative concepts. The difference between…

In this paper, we analyze two episodes from an inquiry-based didactical research; the complete analysis of our research data is still ongoing. By taking into consideration various developments from the history of the geometry of space-time, our general aim is to explore high school students' conceptions about measurement of length and time in…

Learning experiences regarding the line of best fit are typically students' first encounters with the fundamental topic of statistical association. Students bring with them into these learning experiences prior knowledge and experiences about mathematical lines and their properties, namely slope. This study investigated the role students'…

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…

The purpose of this study was to examine the abstraction thinking or the vertical reorganization activity of mathematical concepts of high school students while taking account of the abstraction that was constructed earlier, and the socio-cultural background. This study was qualitative in nature with task-based interviews as the method of…

Relating mathematical concepts to graphical representations is a challenging task for students. In this paper, we introduce two visual strategies to qualitatively interpret the divergence of graphical vector field representations. One strategy is based on the graphical interpretation of partial derivatives, while the other is based on the flux…

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…

Visual and analytic strategies are features of students' schemes for spatial tasks. The strategies used by six students to anticipate the folding of nets were investigated. Evidence suggested that visual and analytic strategies were strongly connected in competent performance.

Mathematicians use example generation to test and verify mathematical ideas; however, the processes through which undergraduates learn to productively generate examples are not well understood. We engaged calculus students in a teaching experiment designed to develop skills in productively generating examples to learn novel concepts. This article…