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Comment by sushisource | Hacker News Reader

I love explainers like this, but it frankly makes me a little angry that the vast majority of the math teachers I had in highschool and college taught in the awful way described in the setup to the piece.

Why is that? Has anyone studied it, or is there even a solid anecdotal explanation? The best one I can imagine is many of these professors simply don't care much for teaching and are more focused on their research, which is still infuriating but at least an explanation.

I ended up with an undergrad in applied math, though I'm a software engineer now. I like math, but I feel like I never got to be all that great at it. I suspect I would've enjoyed it more and achieved more with explanations like these.

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How much of this explainer however seems better to us precisely because of the more comprehensive knowledge and understanding we already have?

For example the author uses the word function liberally in the explanation. However, when studying functions in school in math it was super complicated for me. It was only after I started programming, learning the programming language meaning of function, and then when I was reintroduced to mathematical functions through functional programming that I truly grasped mathematical functions and all the stuff I was taught in school.

I’m not arguing that school level math does not need to be improved. I just think we should be cautious that because an explanation that seems intuitive to us now after having gained a complete introduction to all math concepts as well as programming (esp on HN) may not necessarily be as intuitive when students who are only exposed to a very small subset of mathematical concepts encounter them.

As someone who learned to program before taking any interesting math I was always so confused by the amount of early math classes spent explaining function notation.

It wasn't until I tried teaching people that I realized how odd the notation can seem.

you should be happy someone explained it to you- I feel that notation was overlooked tremendously in my math education- I actually just brought this up in a thread yesterday. It was like "oh, we have this dx now... k."

For sure there is an effect on Reddit and other places where someone will post a question such as, "I'm having trouble with my Calc class, what is a good book?" and people seriously answer Calculus on Manifolds.

Now, CoM is a classic, a real great book, but it is useful only to people who have reached a certain level of mathematical maturity. That, presumably, is not the questioner.

A version of this is that I also see people who write, "I didn't understand this topic when I took the course but now years later, I see it is all actually very easy."

(I call this Second Book Syndrome because I don't know of a common name for it. I understand this to be what Zen people mean by the "Gateless Gate," that after struggling with something at great length a person can come to see that there is no real difficulty. But I've never heard anyone else apply that name to this phenomenon and I'm not Zen trained so I'm not sure that is right either.)

Once you understand monads, you lose the ability to explain monads. Hence the number of monads tutorials grows at an exponential rate as every new understander tries to explain them and fails.

it's a fun problem in teaching

Ironically, I think the original paper that introduced Monads as a useful computational abstraction is the clearest explanation I've seen.

https://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/b...

> Once you understand... you lose the ability to explain

Sorry, this does not make sense to me.

Douglas Crockford on Monads:

In addition to its being good and useful, it’s also cursed and the curse of the monad is that once you get the epiphany, once you understand - "oh that's what it is" - you lose the ability to explain it to anybody else.

https://youtu.be/dkZFtimgAcM

Here is a link that helps me think about it: https://www.alanwatts.org/3-3-10-gateless-gate/ (I teach Math and a common complaint, against all math teachers, is "They obviously know the material but they cannot teach it." When I first heard AW talk about this it was a relevation for me because it so closely linked with my experience. You say to people "A vector space is a place for linear combinations to happen" and they don't get it, of course they don't get it, I wouldn't have gotten it, but that's what a vector space is. So you have to do lots of examples, and work around the edges, and somehow sit with it for a while. Anyway, I find that for me this link conveys the point of that quote about monads.)

Corollary: a good teacher hasn’t understood anything. Well, that’s wrong, I think.

However, this might be true: A good teacher remembers not having understood it.

Actually a rather famous example of your point are the Feynman lectures. They are often hailed as a great, "easy" and intuitive explanation of physics. However, supposedly at the time, the undergrad students Feyman was teaching did not think their were all the rage. It was actually the grad students who retook the material and really enjoyed them. To support your point I think sometimes it is really necessary to have been exposed to the material and been able to grasp some part of it, then you are open to these type of explanations which really lift you over the hurdle of the things that you didn't quite get before.

I agree with your post. I had the same problem where match was harder for me to comprehend. It wasn't until I started programming that I started to understand the concept you cite (Functions.).

Have you ever taught?

I taught Astronomy to the people in the years below me (a school tradition because it was optional), and it was absolutely exhausting trying to plan good lessons that didn't involve getting them to memorise stuff by wrote. I ""derived"" Kepler's laws for a bunch of 14-15 year olds who didn't even know logarithms yet and it was pretty brutal intellectually.

Also, I think teaching mathematics "intuitively" requires a bit of cooperation from the student - not in the sense of intelligence, just that for for every guy (or girl) who watches 3Blue1Brown (and looks deeper into the pure mathematics) there's another who's just along for the ride. I think the frequencies of those personalities are a product of how they were taught, but it's very difficult to convince people in "teaching time" as opposed to naturally (I was in the lowest maths group for years until I picked up a calculus book on a whim, but no teacher could've convinced old me that mathematics can be beautiful)

Quick Edit: To quote Tim Minchin, "Be a teacher". I might come across as moaning in the above but it's really interesting to explain things (in my experience at least - I love writing documentation!), and a good bullshit-test on yourself

I had an eye opening experience and some closure about my own eduction after reading a certain book, a collection of short biographic accounts of famous Germans and their horrific experience in school during the last few centuries. School has always been bad and actually used to be borderline insane, and yet these great characters, politicians, artists and scientists still developed fine.

Rant: If society gave a shit teacher would be the highest paid job with the highest standards, both in knowledge and in teaching skill. I mean literally 500k/year for the most noble calling on earth. The kind of progress humanity would make with a generation taught by the best, and teaching itself revolutionized, would be crazy.

https://www.amazon.de/Deutsche-Schulzeit-Erinnerungen-Erz%C3...

Here are two reasons I think are important, there are more but these stand out. (1) Teaching is hard, and the amount of training you get in teaching higher level maths (rather than the just learning higher level maths itself) is very limited. (2) intuition in maths will lead you on a merry path to very wrong ideas. There are functions that are continuous everywhere without a derivative anywhere, Russell's paradox, classically zenos paradox, etc, etc. A big part of higher level maths training is in being precise with what you mean so that you can learn to do proofs and not trick yourself. I think a synthesis of these approaches is the sweet spot. Be precise but show applications to motivate the material.

I think the problem might be that the kids best at math become teachers. The ones that it ”came to easily”.

This creates the not-so-obvious problem that the easier you learn something the harder it might be to teach it to someone who doesn’t get it.

If you on the other hand struggle hard to find a way to understand a complex concept, you might be good at transfering that knowledge forward.

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