The Importance of Technology-Free Visualization
Activities
Tevian Dray* & Corinne
A. Manogue
Mathematics may be the universal language of science, but at the very least
other scientists speak a different dialect. As part of our work [1] on
bridging the gap between mathematics and physics, we have realized that the
differences can be summed up in a single sentence:
Mathematicians teach algebra; physicists do
geometry.
A key ingredient in bridging the gap is therefore an increased emphasis on
geometric reasoning. We discuss here several activities we use in the
classroom to develop this skill, each of which is designed to address known
student misconceptions. However, none of these activities involve
technology.
As Ken and Pat Heller have noted [2], it is essential to force students to
do the things they find difficult. In each of these examples, we believe
forcing
the students to work through the geometry themselves is essential to
mastering the concepts.
If time permits, we plan to discuss the following activities, which we have
developed for multivariable calculus courses at OSU:
I. Which Way is North? (An introduction to vector bases.)
II. The Hill. (The geometry of the gradient.)
III. The Valley. (What do conservative vector fields look like?)
VI. Linear Transformations. (The geometry of eigenvectors.)
Each of these activities provides a geometric context for known algebraic
manipulations; each produces an "Aha!" reaction from many students.
We also hope to briefly discuss other techniques, such as representations of
functions and diagramming equations.
This work forms part of the Vector Calculus Bridge Project [1], and has been
partially supported by NSF grants DUE-9653250, DUE-0088901, and DUE-021032.
1. The Vector Calculus Bridge Project,