Over the years I became increasingly concerned about the students (mainly female) who never spoke a word; the students who sorely needed individual attention yet never used my office hours; the students who were heading for certain failure but were lulled into thinking they might pass the course because, as they said, "It seems so easy when you do it on the board." I began to consider on my own experiences as a mathematics undergraduate at Oxford. It shocked me to realize how well I was reproducing the same structures of power and domination which had so effectively silenced and disempowered me then.
Reflecting on the differences between what I was doing and what I believed needed to be done, I realized that the approach I sought is one that demystifies the "doing" of mathematics; that calls attention to its cultural construction and engages students in purposeful, meaningful activity. In other words, I needed to make visible the means by which mathematical ideas come into being, and the process by which they are polished for public consumption. At the same time, to deny the institutional authority and power I possess would be profoundly dishonest. In sum, I seek to open the gates to the mathematical community by teaching students the skills they need to join the club and to operate within it.
At the beginning of a course I discuss my goals with the students and make them available in written form. My main goals fall into three categories: content-specific, process, and social. Content-specific goals include helping students develop the skills necessary for constructing their own proofs of mathematical statements and writing them clearly and precisely in correct form. Process goals involve the development of independent working skills (so that they might eventually become free of the need for a teacher). Such skills include reading a mathematics text with understanding, finding, analyzing and correcting their own mistakes, and asking questions. Since mathematics is a cultural activity, social goals are also stressed. These include developing the ability to work with others by communicating ideas clearly and with confidence both orally and in writing, active listening, offering constructive criticism, and by asking and responding to questions.
I try to create a climate of safety and trust in my classroom so that risk-taking occurs; so that students are enabled to test their ideas and thoughts and so that fluency is developed in mathematical language. In other words, I build a community of mathematicians.
Following is a brief description of some of the teaching techniques I use most frequently.
Full class dialogue