People like Timothy Gallwey and Betty Edwards should be studied. There ought to be a system of identifying people like them and trying to find out what makes them so outstandingly good at teaching.
"This was a huge shock, and it turned out that an unusual teacher was the culprit. She was a natural kindergarten and first grade teacher who was also a natural mathematician. She figured out just what to do with 6 year olds and was able to adapt other material as well for them. The results were amazing, and defied all the other generalizations we and others had made about this age group."
What the heck was she doing? How many other people have it figured out but don't share it? Why isn't there an effectivity assessment on curricula or instructional word choice- "mindset" type stuff or whatever it is that actually works.
I've tried the Betty Edwards printed material quite a bit and it is just amazing what you can do when you stop thinking about it in the same way. You can have a non-productive thought pattern getting in the way of understanding.
A good professor explained this to me in 2 minutes after I had struggled with it several times.
Losing a lot by not doing this in person, but what he said, pretty much, was:
Prove that something is true for 1.
Prove that if it's true for 1, it must be true for 1+1. Then it's true for 2. Recycle the proof to prove that it's true for 2+1. Then it's true for 3. Generalize the proof so that it's true for (anything it's true for)+1. Then it's true for any finite number, by induction.
Drawing a number line on the board helps, or a bunch of lines connected by dots.
Practice practice practice. I had a great deal of difficulty as well and only after a SHIT tone of practice did I realize that long term memory of theorems and doing proofs is 90% of it.
