Equations Mean Things
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A small correction: the probability of "avoided archdevil & died" should be 18%, not 8%. This isn't used in the subsequent calculation, but if the question had been "Looks like he's not coming back. What's the chance an archdevil got him?" it would. (28% = 7/(7+18).)
I think you could've done better with integration by parts.
In physics, integration by parts is usually applied for a definite integral in which you can neglect the uv term. Thus, integration by parts reads: "The integral of udv = integral of -vdu, that is, you can trade what you differentiate in a product, as long as the functions in question have a small integral over the boundary".
Common examples are when you integrate over some big volume, as most physical quantities are very small far away from the stuff.
I also think the intuition behind Bayes rule as usually interpreted here on LW, that is, it provides the updating rule posterior odds = prior odds*likelihood ratio and thereby also provides a formalization of how good evidence is. As for the derivation from P(A|B) defined as equal to P(A and B)/P(B), I think this is best described by saying that P(A|B) is the probability of A once you know B, so you take the mass associated to the worlds where A is true once B is true and compare to your total mass, which is the mass associated to the worlds where B is true. The former is really just "mass of A and B", so you are done.
Now, P(A and B) = P(B)P(A|B), which I think of as "First, take probability B is true, then given that we are in this set of worlds, take the probability that A is true". Essentially translating from locating sets to probabilities.
From here, Bayes theorem is the simple fact that A and B = B and A. So P(B)P(A|B) = P(A and B) = P(A)P(B|A). If you draw a square with 4 rectangles where the first row is P(A), where the second row is P(-A), where the first column is P(B), and where the second is P(-B), and each rectangle represents a possibility like P(A and -B), then this equation just splits the rectangle P(A and B) into (rectangle compared to row) * row = (rectangle compared to column) * column. Divide by P(B) (that is, the row) to get Bayes law.
For the sine rule, I think it also helps to show that the fraction a/sin(a) is the diameter of the circumcircle. Wikipedia has good pictures.
For an extra math fact that totally doesn't need to be in the post, it is interesting that for spherical triangles, the law of sines just needs to be modified so that you take the sine of the lengths as well. In fact you can do similar in hyperbolic space (by using sinh), and there's a taylor series form involving the curvature for a version of sine that makes the law of sines still true in any constant curvature space. (you can find this on the same wiki page).
which I worry your teachers didn’t
Oh it can be so much worse than that - actively pushing students away from that kind of understanding. I've had math teachers mark answers wrong because I (correctly) derived a rule I'd forgotten instead of phrasing it the way they taught it, or because they couldn't follow the derivation. Before college, I can think of maybe two of my teachers who actually seemed to understand high school math in any deeper way.
Wanted to add:
I think this post is great for here on LW, but if someone wanted to actually start teaching students to understand math more deeply, calling it common sense probably comes off as condescending, because it doesn't feel that way until you get comfortable with it. There's a lot to unlearn and for a lot of people it is very intimidating.
Personally I wish we treated math class at least some of the time as a form of play. We make sure to teach kids about jokes and wordplay and do fun science-y demonstrations, but math is all dry and technical. We assign kids books to read like A Wrinkle in Time and The Phantom Tollbooth. But, I don't think my elementary school teachers had any clue what a tesseract was, or what the Mathemagician and Dodecahedron are all about, and so that whole aspect of these books was just a lost opportunity for all but maybe 3 kids in my grade.
More specifically, the correctness of the proof (at least in the triangles case) is common sense, coming up with the proof is not.
The integrals idea gets sketchy. Try it with e^(1/x). It's just a composition of functions so reverse the chain rule then deal with any extra terms that come up. Of course, it's not integrable. There's not really any utility in overextending common sense to include things that might or might not work. And you're very close to implying "it's common sense" is a proof for things that sound obvious but aren't.
1Sure. And I'm of the opinion that it is only common sense after you've done quite a lot of the work of developing a level of intuition for mathematical objects that most people, including a significant proportion of high school math teachers, never got.
Overall, I think this is a much better way to teach math - in some sense it's similar to removing date memorization from history classes, which I also agree with. I do have an issue with the phrase "a triangle has the same area as itself." A more user-friendly intuition for me is "if you describe the same thing two ways, it's still the same thing." This seems more generalizable and also gets more directly at the point that sin(A)/a is a complete description of a triangle's proportions.
Agreed with respect to better way to teach maths. However, noting that teaching like this requires students who want to learn like this which is almost always going to be the minority. For folks interested/enjoy this perspective, I encourage you to read A Mathematician's Lament by Paul Lockhart.
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