But the real goal of mathematics teaching should be to induce our students to, so far as possible, think like mathematicians.
For example, we should want them to ask, “What’s so special about a ninety-degree triangle? Could there be a “sine” for a 91 degree triangle?” What if the triangle were drawn on the surface of a sphere? Or on a saddle?
Geometry software can help in this, but the spirit of inquiry cannot be induced by software.
That takes a human being, who must embody the spirit of a mathematician herself, as well as having deep empathy with the mental state of a beginner.
People who can do that are rare, sadly. But it is the ideal we must try to emulate.
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