From the introduction: After many years of teaching mathematics, I still fall into the trap of assuming that my students think as I do. Indeed, this failure to recognize my own assumptions and to acknowledge that others may not share them is at the root of most of my teaching problems. However, when I am able to identify the thinking behind my assumptions and recognize that students may see a situation in different ways, I can be proactive. I find myself coming up with new ways of presenting material and structuring my classroom that are more aligned with what I know about teaching, what I know about mathematics, and what I know about my students. A brief anecdote illustrates my point.