When students look at a matrix of transformation, at best they examine it as the coefficient matrix of a system of linear equations, and at worst they consider it a random array of numbers completely devoid of meaning. In addition, when we as teachers describe a matrix, we often read the elements row-by-row, further separating the potential geometric considerations of column vectors. Consequently, students have a difficult time when faced with geometrically interpreting matrices of transformations other than simple ones involving positive and negative ones and zeros. In response, we began to search for a means to help students interact with matrices of transformations that would lead them to viewing the column vectors as having powerful information.
Our use of Sketchpad for investigation of linear transformations from R2 to R2 allows students to investigate the geometry underlying linear transformations as well as to grapple with the way vector spaces are mapped into other vector spaces. Getting students interested, exploring, and excited can lead them to making deep insights into the content and is one of the primary goals of these web-based tools. As students investigate a large number of examples in a short period of time, they are forced to revise their conjectures over and over again until settling upon insights that are compatible with mathematical definitions of the concepts.
Consequently, we designed the Transformer2D tool to get students to interact with the concept of linear transformation that acts as a "unifying and generalizing concept" permeating the whole of linear algebra (Dorier 1991, 1995; Dorier et al. 2000a, 2000b). A perspective on the geometric meaning of the column vectors in a matrix of transformation can potentially help students overcome some of their difficulties, as identified by Carlson (1993) and Meel (1999b), with related concepts such as null and column spaces.
We developed this tool over an extended period of time. During the Spring 2002 and Fall 2003 terms, we pilot-tested the web-based tools to determine students' reactions to them and to test the transparency of the interface. Based on findings from a beta version written in The Geometer's Sketchpad 3.0 (Jackiw, 1995), elements of the module were not sufficiently transparent. Students were frustrated that particular elements could be moved which should have been held static. This and other observations gave rise to our developing the web-based modules that could lock elements of the sketch. In addition, moving to web-based modules allowed students to access the modules remotely without having to own Geometer's Sketchpad. During pilot testing of Transformer2D we found that students needed a primer on how to interact with the interface and displayed difficulties with interpreting the information they were receiving from the module. For instance, one student stated:
"I found the programs very difficult to understand. I do not learn well by computers, but I do think they are a good way to visualize the problem. They helped me see what was going on. . . . I think in the future it would be helpful to hand out a ditto explaining the programs, so that students can become familiar with all the features the program has to offer."
Another student said:
"The most difficult/frustrating thing of working with the programs was not knowing what I was looking for, and making connections and discovering things on my own. However, this turned into the best thing at the end because it is a good feeling once you make the connection."
As a consequence, we present in this article the revised interface that is accompanied by a more thorough explanation of the role of particular buttons and features.
Transformer2D Tool and Sample Activity