Getting students to interact with linear algebra content can be a daunting task, especially when you want them to recognize the algebraic and geometric meanings behind many of the concepts. We present a glimpse of the possible -- here is a student voice to introduce what we mean by "the possible":
"I learned alot [sic] through the use of Eigenizer & Gridmaster & Transformer. Although they were difficult to use at first, because I did not know what I was looking for, and could not see the connections, by the end of the year, eigenizer taught me basically all of the connections with eigenvalues, eigenvectors, character poly & so forth. At first, I did not understand and had trouble following class discussions about eigenvalues & so forth, but after doing the project, everything came together because I could visually see the connection. I know this is a specific example, but I think all of the programs were helpful to see the linear algebra concepts in the program."
David E. Meel and Thomas A. Hern are in the Department of Mathematics and Statistics at Bowling Green State University.
In no way do we claim to provide a final word on the subject, but we provide
several tools that are helpful for students when investigating difficult linear algebra concepts,
student reaction to these tools from classroom use, and
some technical details of how the tools were built.
Some of the tools produced to study and visualize linear algebra concepts are described by
We have noticed in these sources a lack of stand-alone tools. For instance, the tools generated by the ATLAST project were designed as MATLAB applications, and others have been based on Mathematica, Maple, and other software systems. This usually requires the user to own or have access to a particular software product, which may not be available for all platforms. In addition, some basic skill in using the software is usually required.
We present in this paper several interactive web-based tools for visually exploring various linear algebra concepts, tools that are not constrained by the requirement of owning particular software. By building these tools with JavaSketchpad, the only requirement for a user is a Java-enabled browser. In particular, we present three tools, GridMaster, Transformer2D, and Eigenizer, and a couple of bonus items for advanced linear algebra.
Each figure below shows a small part of the JavaSketchpad window -- click on each image or its title to see a picture of the full window.
Each of these tools permits students to interact with both algebraic and geometric representations of concepts such as change of bases, coordinate systems, linear transformations, eigenvalues, and eigenvectors. We include student reactions to enhance the discussion and identify the effects of using such tools as part of linear algebra explorations.
Published May, 2005
© 2005 by David E. Meel and Thomas A. Hern