When I was in college I studied Electrical Engineering. I got an MS. I spent 5 years in the field before getting laid off. Then I switched to teaching for reasons I usually make up during interviews to sound good.
I only introduce my past like that because when I was studying electrical engineering I was told a story that I would only later come to realize how it not only defined engineering – but also defined math education. And if I was really being bold – education.
It came from Professor Pedrotti at UC Santa Cruz, who was telling us about a time in his life when he was asked to fix a toaster. His grandmothers toaster had stopped working, and Prof Pedrotti was home for the holidays while getting his Ph.d. So his grandmother asked him – “Ken, maybe you could fix my toaster? Have you taken a class on toasters?”
There is no class on toasters.
When learning electrical engineering, you learn basic fundamentals about how electricity works, and how different circuit components behave. You learn about when assumptions about behavior are valid, and when they are not valid. You learn how things work individually, and how they should work when you put them together. And you are expected to apply these fundamentals to a wide variety of applications. For example – here is how such and such behaves. Here is how it is used to blah blah. Now use such and such to design a circuit that’s going to muck muck. Or something like that. You get the point.
That’s conceptual understanding. And that’s mathematics. And that’s education. What mathematical reasoning allows this process to work? What mathematics does this new application remind me of? And if there is some component missing – what might that component need to do?
The story’s conclusion fits in nicely with the idea of a thinking classroom. But on a more micro level probably – in our profession it’s more like “there is no lesson on toasters”. We can’t teach you how to do every specific thing. We can teach you to apply fundamentals so you have the ability to do whatever it is you have been asked to do.
“I must have been absent when we covered this?” Nope. You’re here now.