When I think about the type of math class environment I want to foster, I can't help but think about the absolute necessity for students to relate with mathematical content on a deeper level than a simply algorithmic one. Valuable math lessons challenge the brain and interest it, instead of forcing it to work. It always seems so easy and obvious as a student when a math (or any subject) lesson is "good," but as a teacher I'll be tasked with actually creating the lesson myself, which seems a lot harder. (Is this a P=NP question? Is it as hard to tell when a lesson is good as it is to create a good lesson? A million dollars to anyone who knows: http://www.extended-cognition.com/2018/11/17/p-np-a-million-dollar-problem/)
What I'm really saying, though, is that I want my class's mathematical discussions to be thorough, conceptual, and personal. I don't want to just tell my students how to use synthetic division, I want to discuss with them what synthetic division is and how it works and can be helpful for exploring the nature of functions. What does dividing a function by another mean, anyway?
It's my goal to teach students how to "do math," of course, but it's also my goal to have my students go home thinking about math each day. And I don't even mean necessarily thinking about our class's discussion each day - in an ideal world, my students would feel so inspired by math that they would look for patterns and numbers and relationships just because they want to. Sure, I want students to think about our class content, but I want them to get to a place where they just find math to be fun to think about. I've always thought that math, once you're comfortable with it, is just as fun as a crossword puzzle: there are questions, sometimes cryptic, and you've got to dig around a bit to solve the problems, but once you do it's satisfying and interesting. There doesn't need to be a "why learn this" if learning it and thinking about it is just fun. From there, someone who enjoys math will want to dig deeper into it individually, and feel empowered and strengthened by their genuine curiosity. I won't be there to hold my students' hands for their whole lives - all I can hope to do is set them off with the right trajectory toward empowering themselves as learners.
