Most people think that mathematics is a body of knowledge and/or procedures that must be passed from teacher to student. For many students, this is both irrelevant and unexciting. My belief is that mathematics is a way of thinking and sense making. It is creative, beautiful, individual, and dynamic. It is being curious, asking questions, figuring out why things work, breaking problems apart, seeking regularity, making predictions, and creating logical arguments. These things we all engage in on a daily basis and have relevance in our lives outside of a mathematics classroom. My goal is to create an educational environment in which mathematical thinking is not only the core of what we do, but runs through ALL of what we do.Bryan is inspiring because he does things many teachers wouldn't think of doing, but that strike me upon reflection as a very good thing to do. For instance, in one post, he describes his class working on and discussing a probability problem, at the end of which some students thought one answer was right, while others felt that another answer was right. But he never jumped in to tell them which one was the right answer, even at the end.
This is unnerving to read at first because, of course, the instinct is to say --- wait, shouldn't the teacher announce the correct answer so no one is left with the wrong impression? He chose differently. He writes:
I'm curious what you think and what you would do in this same situation. I let it go. I felt I did my job by helping students test their ways of thinking, not by telling them what to think.Bryan's point is that in the long term, showing his authority by presenting the right answer could potentially do harm to their ability and confidence to test out their own ideas. As he writes in a different article,
I am inclined to think that my distaste is not actually for right answers but rather for the students' lack of authority in deciding that answer. As it stands now, students' ways of thinking are always subject to some greater authority (teacher, textbook, video, etc.). As Schoenfeld puts it, students:
"...have little idea, much less confidence, that they can serve as arbiters of mathematical correctness, either individually or collectively. Indeed, for most students, arguments are merely proposed by themselves. Those arguments are then judged by experts, who determine their correctness. Authority and the means of implementing it are external to the students."I am very excited about discovering Bryan's blog, because his thinking and most admirably, his doing, is very much along the lines of what I myself noted as the ideal in discussing the dangers of the classroom:
The learning process should leave the learner with an innate, ever-improving 'filter' of quality, to be able to evaluate for oneself, when encountering some piece of information, whether it is reliable, new, significant, important, or useful... [however] what is taught is often only what is already well understood, processed and neatly packaged. This makes it harder for students to think outside the box or understand the current limits of our knowledge, let alone how they can be extended... The loss of autonomy and the presentation of processed information deprives students of the confidence to develop their own filter...
Bryan Meyer also has a nice set of recommended readings I plan to go through. I know of a few of them, but am unfamiliar with most.