Its all about understanding

In my email signature, I have a quote by William Paul Thurston that says: "Mathematics is not about numbers, equations, computations or algorithms. It is about understanding." I have embraced this feeling as far back in my teaching career as I can remember. I know what exactly what it feels like when I am just giving students the information. When I am just telling them step by step about how to complete a problem. I am sure that this happened in school for you. You were having trouble with a problem and the teacher told you that if you just follow these easy steps, you would arrive at the correct answer. Maybe you asked why you were doing these steps or maybe you just took the teacher's word for it and followed their instructions. I would bet that was not the only time you had to ask for those steps. My students are constantly asking me how to complete a problem or what are the steps. They complete the problem but then the next class they are asking me again. As I go through the steps with them for the umpteenth time, I think that if I could just help them to understand the why of what they are doing they would understand better, it would make more sense to them and they could access the knowledge about this type of problem the next time they approached this type of problem. So for 32 years, my goal has been understanding math and not memorization of algorithms, formulas, or tricks to help them through the class