As I begin this blogging adventure, I thought it would make sense to start by outlining my teaching philosophy. There are four basic parts to the "why" of how I teach, outlined below. You must do mathematics to learn mathematics. Seems obvious, right? Yet, our default way of teaching mathematics is to talk mathematics at students, or lecture. Don't get me wrong, I've fallen into this trap. By virtue of being the teacher, we know what we are talking about, so talking about it is the easiest way to present content. However, it is also probably the most inefficient way to teach, for teaching cannot happen if learning does not happen. Lecturing is an instructor-focused activity. Learning, however, is an internal process. Students need to be working their own brains in order to learn, not passively listening to someone else. In my classrooms, I put student thinking and exploration as the focus of class time. That is not to say that I don't ever lecture. In fact, I...