, students cannot enter the matrix of transformation directly -- rather they must move vectors to obtain the necessary values. Many students do not recognize the elements of a transformation as a linear combination of column vectors, and this particular exercise of having to manipulate the column vectors reinforces the idea that the column vectors in a matrix of transformation have meaning. In particular, some of our students made the following explicit observations about the impact of changing the column vectors and the influences those changes made:
- "Scaling a vector v while leaving u alone expands the unit grid in the direction of v, makes the unit square a rectangle, and alters the unit circles diameter to make it an oval with the long dimention [sic] parallel with v."
- "The shape of the transformation of unit circle is dependent on u & v and not on x."
- "Making the vectors u and v scalars of one another makes a scalar of T(x) and changes the length and direction of by moving the domain of the transformation."
- "If the vectors are the same length but opposite directions -- T(x) is at the center."
- "Rotating u, rotates the quad about the v vector."
- "As long as the vectors are not colinear [sic], we can map to any place in R2."
- "The direction of x does not directly effect [sic] the direction of T(x). We now observe that after changing u and v, x does effect [sic] the direction of T(x)."
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.
Eigenizer Tool and Sample Activity
Next or page: 8. Student Responses, Part 2