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In January of 2002, we discovered a paper entitled "Multigrid Graph Paper" (Bevis, 2002) that claimed multigrid paper (see Figure 1) would be helpful for students studying a variety of linear algebra concepts such as linear transformations, change of bases, and coordinate systems.
Figure 1. Multigrid paper 
In Figure 1 we can see that vector P has coordinates [1,3] with respect to the natural basis {e_{1}, e_{2}}, but with respect to {u_{1}, u_{2}}, the vector P has coordinates [1,1].
We saw the multigrid paper as a means of streamlining the Lay (2003) presentation of coordinate systems using two different graph papers. In particular, Lay (2003) described the use of two separate grids similar to those shown in Figures 2 and 3 when negotiating between two coordinate systems. For instance, the vector x has coordinates 1 and 6, typically expressed as with respect to the natural basis {e_{1}, e_{2}} (see Figure 2). In contrast, Figure 3 displays the vectors b_{1}, b_{2} and x, but in reference to a grid defined by b_{1} and b_{2}. The position of x has not moved, but .


Figure 2. Standard graph paper 
Figure 3. Bgraph paper 
After examining Bevis' article, which suggested overlaying multiple grids on a single sheet, we felt that static multigrid graph paper could be improved upon by using The Geometer's Sketchpad. The resultant webbased module, entitled GridMaster, permits students to model a portion of R^{2} and define vectors that coordinatize this vector space. Specifically, this first module was designed to help students recognize the meanings of different coordinate systems and change of bases, two topics identified as difficult for students to understand (Carlson, 1993).
In addition, we sought to address an observation of Hillel (2000) that some of the representations used in linear algebra can serve as obstacles to the development of student understanding. Hillel specifically identified that students have difficulty generalizing the notion of an ntuple as no longer representing a single vector but a potential representation of any other vector. This is of significant importance when students have to work across different bases. This observation motivated us to look for different ways, other than the purely computational MATLAB interactions we had used before, to cause students to interact with multiple representations simultaneously.
Next or or page: 4. GridMaster Tool and Sample Activity
David E. Meel and Thomas A. Hern, "Tool Building: Webbased Linear Algebra Modules  Discussion of GridMaster," Convergence (May 2005)
Journal of Online Mathematics and its Applications