Ad Majorem Dei Gloriam
Trying to solve a problem requires energy and time, both are precious. If you don't know that the work you are doing will get to the solution you risk losing that energy and time. Being willing to take that risk is necessary to even try. Courage is needed.
Euclid's exposition of geometry is one of the landmark achievements in intellectual history. And it has set the pattern for the teaching of mathematics. Start with some common notions, undefined but clear, and axioms and derive, step by step the towering edifice. This is done for rigorous calculus as well. Starting with the most basic material, results are proved, and the subject builds up. Result upon result, and the soundness is assured.
It takes a while to get to the interesting stuff. Suppose instead, we start with the interesting stuff. Not in it's full generality, simplified. Play with the ideas, see where the results come from, and argue for the correctness. Look for counterexamples. Like learning to play the piano. I can learn the notes, but it takes practice to be able play well.
The world is a complex place; theories are approximations. We use our imperfect understanding to make decisions. If our understanding is good then the decisions will likely turn out well.
A decision can turn out well even with poor understanding. Sometimes we get lucky.
Because our understanding is imperfect, it is not reality. Mistaking our model for the Truth is a mistake.
Sounds wise.
Watch little kids, after learning to take a few steps they start running.
Careful mathematics requires stating the precise premises which are required for a theorem.
Greater generality is a benefit, but it makes the underlying idea harder to suss out.
Learning to solve problems will let you solve problems you know how to solve is a useful skill. Having the creativity to solve problems you haven't seen before is still more useful.
This started with an idea that it would be an alternative approach to teaching math. I've thought better of that, and will use these notes as my own exploration to deepen my own understand. In the event, unlikely as that is, that these notes might be useful to another, they are posted.
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