Mathematical Visualization

Inspection of any high school math textbook reveals that visualization is an essential part of the learning process. These books are filled with graphs and pictures used to represent the abstract ideas of mathematics in a pictorial form. A graphing calculator is an extension of these textbook visualizations: these allow a student to create his or her pictures to gain deeper understanding of general mathematical principals. Our goal is to use recent advances in programming language design to go beyond the graphing calculator and exploit the full power of modern computers.

Computers allow students to visualize abstract concepts in new and creative ways, providing learning experiences that mix traditional problem solving with interaction and creativity. We give students the ability to build concrete objects that embody abstract mathematical ideas and also give measurable educational insight into the subject material. Our goal is to construct software according to these principals:

• Generality: a tool usable in wide areas of the curriculum, not just a few selected lessons.
• Simplicity: a student should not have to deal with issues outside the intended learning domain; mastery of advanced computer programming techniques must not be required.
• Expressiveness: students must be able to explore concepts in many different manners, adapting to the individual needs and interests.
• Integration: software should be tightly incorporated into the standard curriculum, with lessons and activities targeted at well defined subject material.

A formal notion of language is an essential aspect of our approach. Language provides the expressiveness and, more importantly, basis for understanding that a student needs to master abstract concepts. We augment the standard language of mathematics with just enough computational devices to support interactive visualization. A good language is the software equivalent of a science laboratory. A student of chemistry can, for example, learn much from freely experimenting with a well-stocked chemical lab. A language of mathematical visualization similarly supplies the basic elements of mathematics and allows a student to experiment freely with their combinations.

Go to the Pan# page to find out more about our language or download software. Many of the examples in the sofware distribution are part of presentations to high school math students. Slides associated with these presentations are also available.