Tool Building: Web-based Linear Algebra Modules

Author(s):

David E. Meel and Thomas A. Hern

The Work of Jack

Our purpose for the Transformer activity is to get students to explore, make conjectures, and adjust their conjectures. To give a sense of how students use Transformer2D, we provide the work of a randomly selected student named Jack.

The worksheet link on the preceding page downloads a one-page MS Word document with the full statement of the project. However, students were working with a multi-page version with space to fill in their answers. We provide that version here as a PDF file.

Here are the parts of Jack's work as scanned images -- each opens in a separate window, and you may want to have more than one open at a time.

It is evident from Jack's work that he has a growing need for language to describe the phenomena he is encountering. Terms such as "shear," "reflected," and "rotated" are being drawn from the text to help explain the changes he observes in the geometric figures. As a consequence of this type of encounter, students come to appreciate the discussion of the terms and their meanings because they have a need for precise definitions of what they are experiencing.

Jack's described his reactions to the transformer project in his weekly journal entry:

"In class this week we started looking at transformation matrices, still in the form Ax = b. Now this is the same equation we have been using, but now we are looking at it in a different way. Going into the computer lab we started a project using the transformer on Dr. Meel's personal website. This was an interesting tool, because it allowed us to see the effects of different matrices on the range of values in R^2. The different kinds of transformations include reflections, contractions, expansions, vertical and horizontal shear, and projection onto the x-axis and y-axis.

"The project at first seemed a little overwhelming to be honest. I really had no clue how to tell what each matrix was doing by looking at them. But as I tried more and more matrices, and experimented to see what changes I made would create, it all started to make sense. After messing around for a while with the transformer and reading the book, I was able to go through the packet and do my best to answer the questions. Some of them I am not entirely sure I approached the correct way, but I am sure that I understand transformations a lot better in spite of that. So even if I am slightly off from the answers, I at least have an understanding of what transformations are and how to create some myself."

We are immersing students in a learning situation in which making sense of the environment is one major component as they grapple with their own limited perspectives and enhance their understandings of one or another concept piece by piece. In such an environment, they keep exploring, keep conjecturing, keep trying to organize thinking in new ways to accommodate the new bits of information being displayed, while at times experiencing considerable frustration because they lack the big picture. The tools and the corresponding projects are designed to eliminate the frustration and also to permit the teacher to assist, probe for understanding, point out significant hurdles, suggest alternative lines of thinking, and help equip students to manage their frustration and continue to pursue knowledge.

Next or page: 10. Eigenizer Tool and Sample Activity