Search Journal of Online Mathematics and its Applications:
Journal of Online Mathematics and its Applications
Tool Building: Web-based Linear Algebra Modules
Student Responses, Part 1
Each of these comments provides evidence of how students interact with the concept of linear transformations in manners atypical of classroom discussion. Even though some of the statements may reveal misinterpretations and mis-generalizations, the overarching goal of the tool is to help students experience the mathematics and then lead them to examine additional examples that help them recognize the misinterpretation or mis-generalization. For instance, the last comment came from a pair of students who at first examined a particular example and made a particular observation. Then, after changing the values of u and v, they noticed that the previous observation was incorrect and they adjusted their observation. Getting students to interact directly with the linear algebra concepts allows them to formulate conjectures and then test them.
For example, one student wrote in his e-mail dialogue journal:
"I have been working on the problems presented in the transformer worksheet and had a breakthrough. I think I understand how the transformations effect [sic] the vectors. The unit grid portion of the program is very useful. I think I understand a little better how the signs in different areas change the grid and the different values of vectors affect the transformation."
Another student wrote,
"The project you had us do was very helpful. I learned a lot about transformations when you asked us to explain what was going on without knowing what a transformation was. Now if I were to go back and explain what was going on in each matrix, I would use terms like, reflection, contractions and expansions, shears (vertical and horizontal) and projections. When using the unit circle or unit square, you can see what each matrix does to it and therefore you can conclude some sort of transformation."
To get a better feel for how students interacted with the Transformer2D tool and its correspondent cognitively-guided activity, we show on the next page a sample student response to the entire activity.
Next page: 10. Eigenizer Tool and Sample Activity
Next or page: 8. Student Responses, Part 2