Undergraduates' exploration of contour integration: What is Accumulated?

In this work we explored undergraduate students' geometric and visual interpretations of the inscription for contour integrals, integral(C)f(z)dz, without them having any prior knowledge of integration of complex-valued functions. Our research participants drew from various sources of geometric and visual interpretations to productively investigate the components of contour integrals, which they conveyed with diagrams and gesture. Although this enabled significant progress, they were overwhelmed coordinating the multiple quantitative relationships and reverted to simplified interpretations such as summing values of z, f(z), or delta z. In other words, they were unable to maintain focus on what was accumulated. Our participants also engaged in the thinking real, doing complex phenomenon which sometimes provided productive feedback to assess their interpretations. We offer potential reasons for students' struggles including various interpretations for integration of real-valued integration and the layering of inscriptions. We also provide potential instructional strategies based on the participants' interpretations.