As you go further into advanced math or advanced applied math they do a good job of making you think that you can apply what they teach to all manner of things.
However, there is a tendency for all the examples known to the subject to be put in the book as examples or problems for the student.
In physics, econ, optimization, and finance, quadratic problems are often solvable in multiple different ways. But as soon as you go beyond quadratic, the solvable cases are few. But they don’t tell you that.
A lot of math is just finding the solutions of the few problems that have been solved and learning them. The same applies to good definitions and proofs.
Expecting you can find a really novel solution for your problem is usually being overconfident. This is particularly the case if you expect to find a closed formed formula to a math problem that is not yet solved with one.
Learning to imitate is perfectly good for most cases. If you are going to replace the ice maker in your refrigerator, you look up a youtube video that teaches it to you step by step. There are some of these. You just imitate it. You don’t have to learn how to do explanatory youtube videos yourself. You just want to replace your ice maker. Remember to unplug the refrigerator first.
Steve shows you how it is done.
Notice very carefully that Steve shows you the tools you need. A video camera is not one of the tools you need to replace your ice maker.
You don’t have to learn to be Steve. You just need to replace your icemaker. Math is like this.
We get good models and learn them. It is like learning a foreign language, a musical instrument, or in Obama’s case, basketball.
The human brain is designed to learn by imitation. Sit around the camp fire with the elders and learn to imitate them. Generation after generation. If math ed teachers believe in evolution, then they should think about learning by imitation being an easily heritable skill. Inventing the quadratic formula for the first time, or the first guy to make a stone axe is not so heritable a task since it has a big luck component.
Learning to use language is heritable. This also is imitation. So push what is heritable, imitation.