He explains that it was a "triumph of Greek mathematics" to prove that multiplying the top and bottom of a fraction is how you multiply two fractions. That is, we accept as true that you can multiply fractions this way, but we kind of just got lucky that it does work that way. Proving that it always works is non-trivial, but that is totally glossed over in math classes, doing a disservice to students.

This is in context of dividing two fractions where kids are taught an arbitrary rule too early in their studies and it turns them off to math overall because it's become a class of memorizing arbitrary procedures instead of learning concepts.

(That's all my summary of that segment with a little reading between the lines, not my personal view of it. Or it might be later, but have to consider it more.)

The problem with teaching math is you must give them the tools that were created last, so they have the tools to keep up with math. They then must use those tools to understand and prove what they first had to accept as true, lacking proof.

Tend to think about constructive mathematics for this particular insight.

Dividing fractions is also easy. Did you mean summing fractions? Because, yeah, summing fractions is significantly harder than multiplying fractions, simply due to pure misfortune.

This is easy to derive if you have a little algebra, but it is shoved at kids before they can derive it, so they don't understand it (which is anti-math). The dull ones go along with it, but the brighter ones often think they are stupid. I (and independently others) have worked out visual proofs of both fraction multiplication and division for younger kids.

But I mostly think that "fractions as ratios" can be left until the child is older and has more chops for derivations.

The Greeks thought about these as different ways to measure intervals, and they worked out the idea of "common measure" as a good solution to dealing with these problems.

I may be wrong, but I think the issue is: what does it mean to divide x by 1/2? It means to multiply it by 2. We can easily understand that dividing by 2 means to split something into 2 pieces. What does it mean to split something into 1/2 pieces? Why does dividing by 1/2 mean to multiply it by 2 and does it always work that way? Can you prove it?

Summation of fractions is much easier to visualise, even if it is harder to calculate.

Informally speaking, that can be visualised by asking "how many half pieces can you get if you have x pieces?" If I have two pieces I can break them down into 4 half pieces, which nets the same result as multiplying by two.

I'm pretty sure he means "it is an easy equation to memorise, but it's not trivial". People can remember the formula, but have no idea how it actually works.

Memorizing equations is "religion" not "math". "Understanding" is "math".

I wish more math teachers would actually understand this.

The amount of crap I got for using the "wrong way" to calculate something in math class, was probably one of the major reason I never really got into it.

I remember math classes in the beginning actually being fun, I loved figuring out more convenient/to me more logical ways to calculate something, only to be constantly shut down by the teachers for using them, as he/she demanded I use the solution given by the book.

I do understand the value of getting to the same solution in different ways, I even did back then, but teachers didn't frame it like that; their insistence on the book solution just felt like an accusation against me of not understanding, telling me my way of calculating is wrong (even tho they usually proof calculated right) when to me it looked like it was them not understanding, by insisting on that one formalized, and often awkward, solution, and only that.

So as a somewhat rebellious character, it didn't take long for me to start actually despising math classes.

If I could go back I'd probably try to do it differently, I'm in my mid-30's now and consider myself "math illiterate". Sure I can do most basic stuff and even the occasional Pythagorean theorem (which fascinates me to this day, take two knowns to figure out the third unknown!), but I still feel like I missed out on something I could probably really have enjoyed.

Easiest way of visualising the pythagorean theorem is to think about it the way it was initially written: think about areas formed by tree squares.

This image[0] from wikipedia shows it all: the area of square a plus the area of square b equals the area of square c. It's just really convenient that we can use that to calculate triangle sides without resorting to trigonometry.

[0] https://upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Py...

I find that that's a terrible way of visualising the theorum, as it offers no indication of why the theorem is true. I much prefer the below picture: https://www.mathalino.com/sites/default/files/images/01-pyth...

It could be improved by putting -(ab/2) inside the triangles, but I'm on mobile, so making something is out of the question.

I consider this a tragic story. It sounds like you have the type of genuine curiosity and will to understand (not just get the answer) that would make you a really good math student, but your dopey teachers--who probably don't have your natural disposition towards math--beat it out of you.

I happen to like a little metaphor of mine; Math is like cooking: either you blindly follow a recipe (rote "learning") or you truly learn how ingredients combine together (from discovering, to understanding, to — ultimately — grokking).