Differences between experts’ and students’ conceptual images of the mathematical structure of Taylor series convergence

Abstract

Taylor series convergence is a complicated mathematical structure which incorporates multiple concepts. Therefore, it can be very difficult for students to initially comprehend. How might students make sense of this structure? How might experts make sense of this structure? To answer these questions, an exploratory study was conducted using experts and students who responded to a variety of interview tasks related to Taylor series convergence. An initial analysis revealed that many patterns of their reasoning were based upon certain elements and actions performed on elements from the underlying mathematical structure of Taylor series. A corresponding framework was created to better identify these elements and how they were being used. Some of the elements included using particular values for the independent variable, working with terms, partial sums, sequences, and remainders. Experts and students both focused on particular elements of Taylor series, but the experts demonstrated the efficiency and effectiveness of their reasoning by evoking more conceptual images and more readily moving between images of different elements to best respond to the current task. Instead of moving between images as dictated by tasks, students might fixate on “surface level” features of Taylor series and fail to focus on more relevant features that would allow them to more appropriately engage the task. Furthermore, how experts used their images, supports the idea that they were guided by formal theory, whereas students were still attempting to construct their understanding.