I love a fire. Because I know how to build a fire, tend a fire, maintain a fire, I will never be cold. When I built my stone house I built a massive fireplace. In the winter I would sleep next to the fire and I was content. Where I live now we heat with gas and electric. I know how to adjust the thermostat and where the emergency shutoffs are but if something doesn’t work I need to call in a specialist with his rules of thumb and repair manuals. I am warmed by gas and electric heat but I also know I am subject to things I can’t control and don’t understand.
I am teaching a unit on angles and circles. I know all circles are similar geometric objects. I know how to use this fact to convert angles from degrees to radius measure and back and to find the lengths and areas of sectors. There are formulas for these calculations but I don’t need them because I know where they come from – geometric similarity and proportional reasoning.
Before I started teaching if I needed the quadratic formula I derived it and didn’t bother to memorize it. When I was building my house and needed to solve a non-right triangle I worked from first principles essentially proving the law of sines or cosines. When we started making payments on our land, I spent a pleasant hour deriving the formula using geometric series.
Students say there are too many formulas to memorize. There aren’t that many anyway but I agree. Know where the formula comes from. Use it only for reasons of efficiency. The real work of learning mathematics is the effort to understand the concepts behind formulas and algorithms.