One of my math professors always insisted on showing us around thirty minutes of why a concept works and how to get the formula that we use in the final product. Once he showed us the short cut (formula) the work went from ten minutes of calculating to ten seconds. Back then everyone groaned about the “waste of time” but now that we are free of his tyranny, we are coming into contact with the “formula people”. They are the folk who panic every time they have to use an old concept in a new way. Why are they so unsure? It is because they were told to take the math on faith. They were taught that “It just works” and that was okay for them. My math mentor would routinely go through some fake absurd process on the board and watch the majority of the room follow his false trail. He awarded bonus points to anyone who rightfully challenged him by understanding the concept and working it out for yourself. The sad thing is that for my entire education until now the teachers taught instead of mentored.
It wasn’t until I was 29 yrs. old that I finally experienced why any number raised to the zero exponent is one. See my short video explanation, Powers of 0 and 1. I had been told that it was one and sent on my way, just relying on my memory to retain that knowledge. After a long gap in school between high school and college, I had forgotten even that simple rule. I have often heard that math is not like riding a bike and I have said it myself. You have to keep it up or the knowledge melts away. Could we be wrong? When I think back to my trigonometric identities, I can’t remember them! BUT, I remember one concept that was experienced and I can reliably derive the rest of the identities from that one branch. Think about bikes for a minute. Did you get tested on the methods of balance? Were you drilled on the names of a bicycles parts and their purposes? Was everyone made to ride on standardized bikes and had to all learn at the same pace? Do you recall the physics class you had to pass in order to take your training wheels off? How is it that just about anyone who ever learned the difficult task of riding a bike can go for years without practice and still retain the skill? It is because we all experienced the process of learning instead of memorizing the concept. I bet a shiny nickel that anyone who had never learned how to ride a bike, but had studied it in a book, would not remember the necessary information needed to describe the process to someone else. Years ago I had memorized the algorithm for solving a Rubiks Cube. Give me one today and I can’t do it! I failed to develop the fundamental experience necessary to achieve mastery of that skill. If you want to learn mathematics instead of memorize formulas, you need to know the why that sadly is too often left out in s foolhardy quest for time and effort savings. So the next time a “teacher” asks you to memorize a formula, be brave, and ask “why?” and turn them into a mentor.