Yeah, I must say I prefer the math teaching approach of:
1. Explain some problem that is difficult to solve using other techniques. Ideally the problem should be interesting, even if potentially somewhat abstract.
2. Propose a technique to solve it, without requiring full rigor, but that allows for intuitively feeling that the technique is probably valid.
3. Show that the technique also works on other different problems.
4. Show show some flaws (or limits of generality) of the technique. Propose fixes for the flaw, and/or better outline where the technique viable.
5. Now either prove the technique's validity more rigorously (need not be a perfect proof, especially if a full proof requires far more complicated mathematics) or extend this technique to be able to solve more (but eventually coming back to proving the validity).
This sort of approach is has been used in educational videos like some of threeblueonebrown's on youtube, and I've also seen this used as a faux historical development of algebra and high school calculus in "Algebra: the easy way", and "Calculus: the easy way", which did a great job of motivating the development of most of the ideas in those courses. I'm not familiar with similar texts for higher level mathematics (which would have a different tone, since they would be targeting adults, not teens/children), but surely some must exist, right?
This is infinitely better than the all too common higher level math textbook approach of:
1. Here is some unfamiliar theorem, that you might not even really understand, and certainly have no clue of the relevance.
2. Now here is how to prove the theorem. (Which you might be able to follow, but you probably don't really care about right now).
3. Now we finally explain what the theorem is, and hint at (but may fail to show why) it might be useful.
Unfortunately what we typically end up with when learning math is:
1. Explain some problem that is difficult to solve using other techniques. The problem is painfully abstract and completely absurd.
2. Propose a technique to solve it. Never mind how we got to this technique or why it makes sense, just apply this equation and you'll get the answer. Memorize the numbers and symbols, that's all that matters.
3. Show that the technique also works on different (but really the same) problems.
4. Exam time! Here's a problem that looks nothing like the ones you've seen so far, but use the magic technique you memorized! You did memorize it right? Even though you didn't understand it at all?
5. This class is over so you will never use that technique again--even though it's generally useful, but since you never learned how to actually apply it or why it's useful, you'll forget it the day the exam is finished.
1. Explain some problem that is difficult to solve using other techniques. Ideally the problem should be interesting, even if potentially somewhat abstract.
2. Propose a technique to solve it, without requiring full rigor, but that allows for intuitively feeling that the technique is probably valid.
3. Show that the technique also works on other different problems.
4. Show show some flaws (or limits of generality) of the technique. Propose fixes for the flaw, and/or better outline where the technique viable.
5. Now either prove the technique's validity more rigorously (need not be a perfect proof, especially if a full proof requires far more complicated mathematics) or extend this technique to be able to solve more (but eventually coming back to proving the validity).
This sort of approach is has been used in educational videos like some of threeblueonebrown's on youtube, and I've also seen this used as a faux historical development of algebra and high school calculus in "Algebra: the easy way", and "Calculus: the easy way", which did a great job of motivating the development of most of the ideas in those courses. I'm not familiar with similar texts for higher level mathematics (which would have a different tone, since they would be targeting adults, not teens/children), but surely some must exist, right?
This is infinitely better than the all too common higher level math textbook approach of:
1. Here is some unfamiliar theorem, that you might not even really understand, and certainly have no clue of the relevance.
2. Now here is how to prove the theorem. (Which you might be able to follow, but you probably don't really care about right now).
3. Now we finally explain what the theorem is, and hint at (but may fail to show why) it might be useful.