Mathematical Sketching: A New Approach for Creating and Exploring Dynamic Illustrations

Diagrams and illustrations are often used to help explain mathematical concepts. They are commonplace in mathematical and physics textbooks providing a form of physical intuition to otherwise abstract principles. Similarly, students often use pencil and paper to create diagrams for math problems as an intuitive aid in visualizing relationships between variables, constants, and functions. However, such static diagrams generally assist only in the initial formulation of mathematical expressions but not in the “debugging” or analysis of those expressions which can be a severe visualization limitation even for simple problems.

To overcome these limitations I developed mathematical sketching, a novel approach to rapidly interacting with and visualizing mathematical concepts through the fluid association of handwritten mathematical notation and free-formed diagrams. Mathematical sketching derives from the familiar pencil-and-paper process of drawing supporting diagrams to facilitate the formulation of math expressions; however, with a math sketch, users can also leverage their physical intuition by watching their hand-drawn diagrams animate in response to continuous or discrete parameter changes in their written formulas. In this talk, I will discuss my current work on mathematical sketching in the context of a prototype application called MathPad².