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> It turned out that a simple drawing could capture the entire statement of the theorem, and from that drawing, the proof was trivial to derive.

Excellent point.

1. A similar example: Feynman diagrams.

2. Another: Venn diagrams.

3. Longer example: On Navy nuclear-powered aircraft carriers, the officer of the deck underway (OOD) must have at least a basic understanding of how the engineering plant works. It's second nature for nuclear-trained OODs, of course, but non-nukes could sometimes have trouble. Back in the day, it turned out that an effective way to help non-nukes learn what they needed to know was to have them: (A) memorize a really-simple block diagram of the reactor and steam system, and also (B) memorize a chant, of modest length, that summarized how things worked. During slow periods while standing OOD watch, I'd make a non-nuke OOD trainee draw the diagram from memory; then I'd quiz him with "what if ..." questions (back then it was always "him"). If he got hung up on a question, I'd tell him, "chant the chant." That usually helped him figure out the answer in short order.

(U.S. submarines don't have that problem, AFAIK, because pretty much every officer who will stand OOD watches is nuclear-trained.)

>I suspect lots of people go through the entire education system like this.

+1. it took me a couple years after getting kicked out of college to get my head sorted out to the point where I felt like I could "think" again

I think one of the most harmful things about schooling is the way it imposes a tracked structure on learning. it demarcates knowledge into discrete subjects and sets up a linear progression through them and says you need to master each step on the track before moving onto the next one. this is poisonous and borderline evil, and I've encountered many people who are crippled for life by it. a lot of people never pursue things they're really interested in and could become extremely passionate about because school has convinced them they need to stack up prerequisite knowledge before they're even allowed to touch it

> says you need to master each step on the track before moving onto the next one. this is poisonous and borderline evil

What’s wrong with it? You do need to understand calculus before classical mechanics, classical mechanics before quantum mechanics, quantum mechanics before quantum field theory, and quantum field theory before the Standard Model. I’ve seen tons of people disregard this and the result is always confused word salad. People waste years of their lives this way, going in circles without ever carefully building their understanding from the ground up. The order in school was chosen for a reason.

It didn't work for the GP. That makes it poisonous and evil.

I've seen this sentiment way too much on HN. X didn't work for me, therefore X is a scam, its perpetrators are evil sociopaths, and if it worked for you you're a cog in the machine, man.

I'm a believer in individual learning styles, and hard as it is for me to understand personally, I do recognize that lots of people may learn best by lecture and drill. I certainly don't think bad of them for it. but many more people are not suited for this, and forcing them through the education system as it exists does them great harm, and I empathize strongly with them

Lecture and drill has been out of vogue for decades. The hot thing is active learning and high-impact practices. And frankly it has the opposite problem of dragging the class down to the slowest common denominator, wasting the opportunity to teach more to the high-achievers.

I suspect you and GP are talking about slightly different things. GP is probably more opposed to artificial compartmentalizing of things. As an example:

> You do need to understand calculus before classical mechanics

Yes, but how much calculus? Do you need all of Calc I, II and III before even attempting classical mechanics? And should calculus even be treated independently of classical mechanics?

There are various traditions when it comes to teaching these subjects, and the tradition in the US involves keeping a strict distinction between these things, in addition to a "theory first" approach. Other people have studied things in a different manner. Some of my physics professors from the UK had studied most of the math they knew only as needed when they would get to relevant topics in physics - including differential equations, all of analysis (complex or real), some of Calc III, etc.

Even amongst mathematicians, it was common in parts of Eastern Europe to focus on a problem, and learn whatever theory is needed to solve that problem. They didn't learn theory and apply to problems - they took a problem, and learned whatever theory is needed to solve them. I recall picking up a Kolmogorov textbook on analysis and being surprised by seeing this approach, along with the informality with which everything is discussed.

And just a minor quibble:

> classical mechanics before quantum mechanics,

You don't really need to know much except the basics. I think the classical mechanics we covered in our typical "Engineering Physics" courses was sufficient to dive into proper quantum mechanics. It's nice to have been exposed to Hamiltonians in classical physics prior to taking QM, but really not needed. There's a reason neither schools I attended made the classical mechanics courses as prereqs to QM. In fact, I would argue we should split things up a bit: Have a course to teach the very basics of energy, momentum, etc. Then make it a prerequisite to both classical mechanics and quantum mechanics.

>And should calculus even be treated independently of classical mechanics?

Thank you. The value of calculus didn't really click for me until the next year in physics, at which point I wished I had paid more attention.

It's really unfortunate that you first have to drill high-speed high-precision symbol manipulation, devoid of any meaning, for so long before getting any indication of why you'd want to be able to do it.

I like Lockhart's analogy: mastering musical notation, transposition, etc. before being allowed to play your first note.

https://www.maa.org/external_archive/devlin/LockhartsLament....

Some people like to work their way in reverse. I'd often pick a really complicated subject I'm after like "stellar fusion" and then work my way downwards and learn whatever I need to learn in order to understand it. If I had to start from differential mathematics, without knowing why I need it, I'd probably give up.

Are there any notable teachers who take or have taken this approach?

I can't think of any from my education, everything was bottom up building. But when I self-teach (as it naturally emerges from pursuing projects related to work or hobbies) it's often top down digging.

This is the big problem I had with engineering in school, so much was based on faith that you needed it. And just now as I'm writing this, I realized that's completely counter to my personality.

side note: I remember Einstein needed to learn particular math skills to help prove his ideas and sought out to learn it.

calculus is required to understand classical mechanics, but intuitions about classical mechanics can inform and accelerate how you learn calculus. I actually tutored calc 1 a little bit in college and most of the time the people I was helping could do the calculations asked of them fine, but felt lost and confused because they didn't know what the output actually meant. everyone learns linear algebra before abstract algebra, but even a little background in the latter creates many opportunities to think "ohhh, this is just like X!" that make the former easier to pick up--and when you revist the latter, you'll probably understand it better too because of the connections you made

is it better to learn python or java to build a cursory understanding of programming, and then c and x86 to see what "really" happens "under the hood?" or the other way around, starting from base operations and then layering abstractions on top? I don't think one is strictly better than the other. when I first learned sql, joins were intuitively obvious to me but bewildered the person I was learning with, despite the fact that he'd been in IT for years and I barely knew how to program, because I happened to know set theory. I wonder what it would be like to do the traditional algorithms and data structures stuff before ever learning to program. it might make picking up a lot of details easier! I wonder why we don't teach children logic gates and binary arithmetic before they learn decimal. it's actually simpler!

in general I don't think there's a good reason why you must teach raw technique before practical application, or a concrete special case before a broader abstraction. even an impefect understanding of something "higher" can give you hooks to attach information about something "lower." familiarity and comfort and an understanding of interrelationships between knowledge is so much more valuable than perfectly executing each step on a given track

when I want to get acquainted with a new field, I usually start by reading current research, barely understand any of it, and then work backwards through the author's previous papers and the papers he cites habitually. my cursory grasp on the latest version makes it easier to understand earlier sketchier versions, and at the same time the development history of the older shines light on the newer. sure you can always "start with the greeks" as it were and work your way up, but I don't think this is objectively better than going the opposite direction

really I think of knowledge as more a highly interconnected graph than a series of linear tracks. as long as you can anchor something somewhere it's valuable to pick it up. it can reinforce other things near it and you can strengthen your understanding of it over time. and getting into the practice of slotting arbitrary things together like this is good training for drawing novel inferences between seemingly disparate topics, which I think is the most valuable form of insight there is

While I completely agree with your general comments, I still want to emphasize how hard this ideal is to implement in practice.

Yes, every subject is linked to every other subject. If I'm tutoring introductory physics, I will use examples and analogies from math, computer science, engineering, biology, or more advanced physics, depending on the background and taste of the student, and it works fantastically. But if I'm lecturing to a crowd, this is impossible, because the students will differ. If I draw a link, some people will think it's enlightening, some will think it's totally irrelevant, some will think it's boring, some will think it's so obvious it's not worth saying, and most will just get more confused.

The same thing goes for the top-down "working backwards from cool results" approach; it's supposed to bottom out in something you know, but whenever you teach multiple people at once, everybody knows different things. The bottom-up linear approach is useful because it gives you a guarantee that you can draw a link. If I'm teaching quantum mechanics I expect to be able to lean on intuition from classical mechanics and linear algebra. If I didn't know the students had that, I would draw a lot fewer links, not more.

Similarly, "if people learned X in school, then Y would be easier to understand later" is true for almost any values of X and Y, because of the interconnectedness of knowledge. But if you ask any math teacher, they'll tell you the school curriculum is already bursting at the seams. You can't just add logic and set theory to existing school math without taking something out. In the 70s we tried taking out ordinary arithmetic to make room for that. It was called New Math, and everybody hated it.

Our school is poisonous (can tell for France), if not evil. It become crystal clear after reading Celine Alvarez. Not sure if she got translated yet. In english, but older you also have Alfie Kohn, but I haven't read him.

When reading Celine, one understand that children are natural born learner, and there is no effort needed to make them learn stuff. Our school model is industrial production of objects. Thinking human machines. We are way more than that. Sadly Pink Floyd description of the school still echo to our modern school. Some peoples don't feel that way about school. I don't really know why. Maybe they never imagined how better it could have been, so they found it great.

Check out Célestin Freinet too (also untranslated in English AFAIK)

Reference appreciated - there is one translated book:

Freinet, C.: Education through work: a model for child centered learning; translated by John Sivell. Lewiston: Edwin Mellen Press, 1993. ISBN 0-7734-9303-4

Thank you for the recommendation. Celine Alvarez does have a book translated to English:

The Natural Laws of Children: Why Children Thrive When We Understand How Their Brains Are Wired

> A powerful, neuroscience-based approach to revolutionize early childhood learning through natural creativity, strong human connections, spontaneous free play, and more.

She has a home page, with an English version of some of her articles.

https://www.celinealvarez.org/en/our-approach

seems like a lot of people are imagining a caricature of what school actually is and blaming it for problems they would have had anyway.

>stack up prerequisite knowledge before they're even allowed to touch it

The way you phrased this reminded me of A Mathematician's Lament by Paul Lockhart[0]

[0] https://www.maa.org/external_archive/devlin/LockhartsLament....

OP here -- that is a fantastic anecdote!

Do you have an example of a "drawing" of a theorem, in this context? (I've seen these for fairly trivial theorems but not for more complex ones, so I'm curious.)

Ahh I did a bit of googling but couldn't find anything nice — sorry! Most of the time, the complex stuff is broken down into smaller "lemmas" with their own manageable proofs, and then the proof of the whole theorem will be something like "Follows from Lemma 2.1, Lemma 2.2, and a basic application of Theorem 1.4"

This is the theorem I was talking about: https://i.imgur.com/1xEH51Z.png (taken from https://taimur.me/posts/thinking-at-the-right-level-of-abstr... which touches on a similar topic to your post)

As someone who very much relates to the GP’s anecdote, I might suggest determinants as a good example.

As an undergraduate studying maths, I encountered a standard theorem in one of my first courses, which says that about 947 different conditions are equivalent to a matrix having a determinant of zero. I dutifully memorised these. I also dutifully memorised the algorithm for how to calculate a determinant. I might even have remembered some verbatim proofs of some of the equivalences.

However, I developed absolutely no intuition about what a determinant is. I had book knowledge, but no insight. It was a long time ago now, but I’m fairly sure that when I graduated I still did not truly understand even this very basic (by undergraduate standards) subject. I think it was probably a few years later, when I came across some of the same theory but in a much more practical context at work, that most of the connections in that equivalence theorem first “clicked”.

Meanwhile, here is what a gifted presenter with the right illustrations can do in about ten minutes:

https://www.youtube.com/watch?v=Ip3X9LOh2dk

The 2,000 or so substantially identical comments below that video are very telling.

Given the understanding you’d get with that quality of presentation, the equivalences I mentioned above would have been obvious and constructing the proofs from first principles would have been straightforward.

Awesome example, thank you!

In the process of learning about a family of algorithms in machine learning I also gained some physical intuition of determinants (same diagram as 3Blue1Brown, but applied in a different context of "squashing and stretching" probability mass): https://blog.evjang.com/2018/01/nf1.html

This is awesome, determinants were one of the things that I never really understood during my degree

I think a good example would be Lagrange's theorem[0]. Once you have this[1] picture in mind, it becomes trivial.

[0] https://en.wikipedia.org/wiki/Lagrange%27s_theorem

[1] https://i.stack.imgur.com/w2GfA.png

Was it this theorem of Lagrange's that you had in mind https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_th... ?

Yes this. In grade school I had trouble memorizing the formula for the area of a right angled triangle. But one day I bisected a rectangle diagonally and was blown away how it all of a sudden made sense: The same thing happened with circles and even calculus stuff. Putting it into diagrams suddenly made it make sense.

So now I try to “do” instead of memorize and it makes all the difference in the world.

Even I can’t learn new programming languages unless I make a real project with them. That’s why I don’t believe in things like code katas or reading programming books cover to cover. It’s a bad representative for using a tool in real life to learn its principles, techniques and methods.

And if you got good grades, then the system wasn't testing whether you understood it - Bad news.

But it's good news that you were able to understand once he drew the picture for you. So there is an effective way to teach that theorem - If only the professor knew it.

I felt the same exact way trying to memorize molecular formulas in organic chemistry, being able to visualize the molecules and some outlandish pneumonics really allowed me to force my brain to make memories that I forgot over time however I gained the ability to "see" the molecules in my mind and deriving the formula became trivial.

Creating really unique pneumonics works similar to how you can recall when something happened out of the ordinary or have a "sixth" sense where you subconsciously can sense something is off in an otherwise ordinary situation or encounter. If I had to guess, this might be an evolved human survival mechanism that helped our ancestors detect and avoid dangerous/deadly situation.

I wonder what other evolutionary mechanism we can hack to encode concepts more deeply. I know spaced repetition is the mainstream approach via flashcards/ankii type apps. I think pneumonics blended with spaced repetition is the most practical methodology. Thoughts?

> It turned out that a simple drawing could capture the entire statement of the theorem, and from that drawing, the proof was trivial to derive.

Couldn't agree more. Some of the best CS books I've read have been ones with a tonne of illustrations, like the Head First series.

And _why’s poignant guide! I’m still a rank amateur, but that little PDF helped me truly understand, not just memorize.

Humans are wired to enjoy and remember stories. So much education is done narrativelessly and as such neglects to harness one of our most potent capabilities.

What happened after this revelation? Did anything change? Did you try to intuitively understand the problems you were working on? I think this boils down to how you were taught (or self taught) to approach the subject and the set of tricks you learned along the way that became your toolset.

Did you do a lot of exercises in school? They usually help build the intuitive part, the aha moment, that comes through repetition.

Yup, definitely changed the game for me, and turned on my 'intuition' spidey sense, of whether I actually understood a concept.

We did do a lot of exercises in school, but they mostly just tested whether you can reliably apply a method that you were taught, rather than testing understanding.

Did that realization at the eve of exams bring up a bout of anxiety ? (edit: I would have freaked out)

Haha, yeah. Thankfully, the Maths exams were set up such that you could always get 60% (the boundary for a "2:1" grade in the UK — an acceptable score for most people) by just knowing the 'bookwork', which you could do by memorising stuff without understanding it.

I always remember the exams frustration of remembering the exact page of some graph but being unable to get it back. I hated it!

The university I went to offered open book/note exams for almost all courses. It literally didn't even matter how much you memorized... open book tests didn't make anything easier. you need to understand or fail.

I'm not into showing off ranking or pedigree but I do genuinely believe that the higher the pedigree your school the more likely the exams will be harder and require total understanding and even creativity over rote memorization.

The reason is because memorization is trivial. Students able to get into any top school will likely all easily achieve full score on an ordinary tests. The professors at top schools need to make these tests brutally hard in order to produce a bell curve.

I literally had one new professor at my school actually give a mid term that was what would ordinarily be called fair in any other school or college... but the entire class ended up getting nearly full score.

He realized his mistake and the final was way, way harder.

> Students able to get into any top school will likely all easily achieve full score on an ordinary tests. The professors at top schools need to make these tests brutally hard in order to produce a bell curve.

This is no longer how “high pedigree” schools function in the US. There is too much blowback from helicopter parents and the kids who haven’t ever had to deal with critical feedback before.

Lookup “grade inflation” if you want to be horrified by the quality decline of “pedigree education”.

You got a good point. My school is listed as a school that has grade inflation but anecdotally I feel my engineering department was pretty brutal. Average GPA was below 3 when I was there.

It could be my personal situation was not representative of the norm.

Rote memorization tests are the easiest to make and grade, both in terms of simple instructor effort as well as in terms of fairness and lack of ambiguity.

In Asia it is common for schools to solely grade base on multiple choice exams which promote cramming and memorization. But in the US I would be shocked if a program that only operated that way was accredited. I would argue students who simply memorize things are trying to hack their way to an unearned grade. The goal of higher education is supposed to be to reach higher levels of Bloom's taxonomy, beyond the rote stuff done in primary and secondary school.