For as long as I can remember, the dominant philosophy in college level mathematics education has been what I call transmissionism. Teachers know and students don't. Teachers are knowers who transmit knowledge to learners. Then learners know as well. But never quite as well, because, after all, we're the experts and they're the novices. So what we see when we test the effectiveness of our crystal clear lectures is a pale shadow of the brilliance we emitted. And that's their fault, not ours.
As soon as we insisted that students write about their thought processes, we realized that transmissionism is an indefensible philosophy. In fact, what our students know about mathematics, for the most part, was not transmitted to them by any teacher or textbook -- none of us would want to take credit for it. And yet, that knowledge was getting them through courses with passing grades, sometimes with excellent grades. We call this form of mathematical knowledge coping skills.
Where do coping skills come from, and how do they work? Early on, we were discovering a constructive proof of the dominant paradigm in education, which we now know is called constructivism. Learners construct their own knowledge in response to challenges to their current state of knowledge. Given the steady stimulus over many years of having to pass tests on dimly understood material, students continually discover and share ways to get apparently correct answers -- without wasting time on understanding. Once we understood -- even dimly -- how this works, we were able to focus on challenges that demanded understanding as an integral part of finding sensible answers.
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Send comments to the author <das@math.duke.edu>Last modified: May 17, 1997