An alternative route to the Mandelbrot set: connecting idiosyncratic digital representations for undergraduates

Mathematics undergraduates often encounter a variety of digital representations which are more idiosyncratic than the ones they have experienced in school and which often require the use of more sophisticated digital tools. This article analyses a collection of digital representations common to undergraduate dynamical systems courses, considers the significant ways in which the representations are interconnected and examines how they are similar or differ from those students are likely to have experienced at school. A key approach in the analysis is the identification of mathematical objects corresponding to manipulative elements of the representations that are most essential for typical exploratory tasks. As a result of the analysis, augmentations of familiar representations are proposed that address the gap between local and global perspectives, and a case is made for greater use of isoperiodic diagrams. In particular, these diagrams are proposed as a new stimulus for students to generate their own explorations of fundamental properties of the Mandelbrot set. The ideas presented are expected to inform the practice of teachers seeking to develop visually rich exploratory tasks which pre-empt some of the issues of instrumentation that mathematics undergraduates experience when introduced to new digital tools. The overarching aim is to address significant questions concerning visualization and inscriptions in mathematics education.