Problem Posing and Generalization

An important part of mathematical thinking involves considering limiting situations, going beyond the initially imposed constraints, and generalizing to broader domains. Dynamic manipulation lends itself, in very seductive ways, to fostering these attitudes towards mathematics. As you drag things around, you often stumble onto unexpected treasures. Another example will help.

Example: Conic Conundrum

The illustration above shows a Geometer's Sketchpad construction of the set of perpendicular bisectors to a given segment as one end of that segment moves around a circle. The envelope of these lines is a hyperbola.

As you drag the right end of the line segment, the envelope changes. Inevitably you drag the point inside the circle, at which point the envelope appears to be an ellipse. When your dragged point reaches the center, the ellipse becomes a circle. Hyperbola, ellipse, circle--what's missing? "Aha," you think, "dragging the point onto the circle will give me a parabola!" Wrong. Now you have to explain how these three envelopes are related and figure out how to get the missing conic.

Proposed Standard: Students should be given ample opportunity to make mathematical discoveries, to propose generalizations, to ask "what if" questions, and to engage in open-ended investigations. Dynamic manipulation software, as it provides an environment in which serendipity and the unexpected abound, should be used throughout a student's mathematical career to encourage such behavior.

Summary

Dynamic manipulation of mathematical objects provides a way of learning and understanding mathematics that has already proven itself in the classroom. In thinking about the next round of mathematics standards, we recommend that there be explicit mention of use of dynamic manipulation technologies, and we have proposed example standards that do so.

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