Tommy Dreyfus

We investigate the meanings three mathematics education graduate students have for integrals, following participation in a calculus course centered around rate of change and accumulation. We conducted interviews on a task designed to elicit both antiderivative and accumulation considerations with possibly conflicting outcomes. We investigate this using a new approach to compartmentalization based on Thompson et al.’s recent theory of meaning (2024) and an associated novel methodology. Our findings suggest that for two of the three students, antiderivative and accumulation were only partially cognitively connected. This led to behavior indicating compartmentalization as well as cognitive conflict. We identify a more complex relation between compartmentalization and cognitive conflict than the literature identifies. Our results suggest that area, constant of integration, and integration bounds are potential ‘bridges’ between accumulation and antiderivative that could enable the formation of connections between them.

The concept of function traverses the mathematics studied at school and university and is an important transitional link between the two. We examine how an interval perspective on function may help with constructing graphical antiderivative functions. In doing so a number of important constructs needed to build such a graphical understanding are considered, along with how the students in the study linked them and built with them. In addition, some of the difficulties they faced and possible reasons for them are explained. The evidence presented shows that an interval perspective on function was important in being able to construct antiderivative functions graphically. Hence, we propose that this interval perspective on function may prove useful in helping students in transition to construct a local perspective on function. In turn we suggest what a potential path for thinking about graphical antiderivatives could look like and the kind of activities that could assist transition students along it.

Integral calculus presents persistent challenges for students in general and for those transitioning from secondary to tertiary education in particular. This study examines the development of "Accumulative Thinking" as a foundation for understanding integration. In a purposely designed learning activity, pairs of Grade 11 students explored accumulation concepts using the context of water flowing into a pool. Using the Abstraction in Context framework, we analysed students’ processes of constructing knowledge during this activity. Our findings indicate that most students constructed elements of Accumulative Thinking, preparing them for future studies; the findings also demonstrate how real-world contexts can facilitate the development of Accumulative Thinking.

Calculus concepts such as derivatives and integrals play an important role in some areas of chemistry, in particular, topics in physical chemistry such as chemical kinetics and thermodynamics. The ways chemists use mathematics suggests how calculus can support students in making meaning of phenomena and processes in chemistry; however, we posit that chemistry can also be used to support students in making meaning of calculus concepts. In this work, we begin by describing the discipline of chemistry to identify an abridged list of topics that use calculus. We then discuss the nature of thinking with rates of change in kinetics and the behavior of gases in thermodynamics, focusing on the insight they provide in building a conception of what derivatives and integrals may represent. In each of these examples, we show how the nature of thinking in chemistry and calculus can be mutually beneficial.

The International Journal of Research in Undergraduate Mathematics Education (IJRUME) celebrates a decade of academic excellence. Over the past ten years, IJRUME has established itself as a leading forum for disseminating innovative research, fostering collaboration, and shaping the discourse in undergraduate mathematics education. This paper reflects on the journal’s growth over its first 10 years and explores themes, evolutions, and trends in its publications, highlighting IJRUME’s contributions to the field.