Using a Conflict-Inducing Task to Investigate Connections and Compartmentalization Between Implications of Integral

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.