On using a core idea to foster the transition to advanced mathematics–transferring the idea of average to complex path integrals

Abstract

The transition from Riemann integrals to other real integrals has been discussed previously in mathematics education literature. In this transition, the interpretation of integrals as averages of the integrated functions generalises smoothly to measure and probability theory. In this paper, I address the transition to complex path integrals. Based on the premise that using core ideas throughout mathematics curricula may facilitate the transition from earlier to more advanced courses, I analyse epistemologically to what extent the idea of averaging may be transferred to complex path integrals. In this case, the transition poses special epistemological challenges. Using an example from an expert interview, in which the expert aims to apply the interpretation of integrals as averages to complex path integrals, I illustrate the aforementioned challenges empirically.