Title

Mathematical Sketching: An Approach To Making Dynamic Illustrations

Abstract

Diagrams and illustrations are often used to help explain mathematical concepts. They are commonplace in math and physics textbooks and provide a form of physical intuition about abstract principles. Similarly, students often draw pencil-andpaper diagrams for mathematics problems to help in visualizing relationships among variables, constants, and functions, and use the drawing as a guide to writing the appropriate mathematics for the problem. Unfortunately, static diagrams generally assist only in the initial formulation of a mathematical problem, not in its "debugging", analysis or complete visualization. Consider the diagrams in Fig. 4.1. In both cases, a student has a particular problem to solve and draws a quick diagram with pencil and paper to get some intuition about how to set it up. In the diagram on the left of Fig. 4.1, the student wants to explore the difference between the motion of two vehicles, one with constant velocity and one with constant acceleration. In the diagram on the right, the student wants to understand how far an object pushed off a table will fall before it hits the ground and how long it will take to do so. The student can use these diagrams to help formulate the required mathematics to answer various possible questions about these physical concepts. However, once the solutions have been found, the diagrams become relatively useless. The student cannot use them to check her answers or see if they make visual sense; she cannot see any time-varying information associated with the diagram and cannot infer how parameter changes affect her solutions. The student could use one of many educational or mathematical software packages available today to create a dynamic illustration of her problem, but this would take her away from the pencil and paper she is comfortable with and create a barrier between the mathematics she had written and the visualization created on the computer. Because of these drawbacks, statically drawn diagrams have a lack of expressive power that can be a severe limitation, even in simple problems with natural mappings to the temporal dimension or in problems with complex spatial relationships. © 2011 Springer-Verlag London Limited.

Publication Date

12-1-2011

Publication Title

Sketch-based Interfaces and Modeling

Number of Pages

81-118

Document Type

Article; Book Chapter

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/978-1-84882-812-4_4

Socpus ID

84884601750 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84884601750

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