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Mathematical Representations and Technology
Imagine a Geometry classroom with no blackboard, whiteboard, or any other place to construct figures or make drawings. Imagine teaching Algebra without graphs or tables of values. There is no doubt that mathematics education is unique in its utter reliance on mathematical representations. The dependency is so complete that most of us forget that the objects of mathematics are purely in our minds (or "between" the minds of those engaged in mathematical discourse). Instead, we mistake the representations for the objects rather than the representatives of those objects. Technology blurs the line even further by endowing representations with properties of the underlying objects that the previous generations of paper and pencil representations did not have.
Suppose I wish to introduce students to the topic of solving trigonometric equations. I start with the equation sin(x)=sqrt(3)/2, where "sqrt" is used to denote the square root function. I wish my students to understand first that there are an infinite number of solutions, second that these solutions fall into two branches, and third that there is a conventional way of representing these two branches. After that, I can concentrate on the solution steps required to obtain the conventional representations.
Using the HP Prime graphing calculator, I enter sin(X)=sqrt(3)/2 in Symbolic view (first figure below) and then press Plot to see the graph (second figure below).
Figure 1
The graphical representation is a set of vertical lines, arranged in groups of two. The vertical lines suggest solutions of the form X=C, where C is a real nummber. The lines appear in pairs, suggesting two sets of solutions. I can zoom out to see that the pairs appear to extend indefinitely, suggesting that the number of solutions is infinite. In just a few keystrokes, I have created a representation that has a high degree of mathematical fidelity with respect to my first two objectives. In the figures below, I use a pinch gesture to zoom out and then back in to focus on just 2 or 3 of the pairs.
Figure 3
In Figure 4, the cursor is on X=1.04719755 or X=pi/3, the first line to the right of the yaxis. Tapping on the second line, I see X=2.0943951 (not shown here). It is not difficult to establish that this second value is twice the first, or X=2*pi/3. I have a start on my two solutions. Now I need to see how far apart these line pairs are. The Numeric view shown below displays the first 4 pairs of solutions numerically.
If I examine every other value, starting with the first, I see 1.047..., 7.330..., 13.6
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