> Most of that foundational information is lost when it's not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject.
From "A Mathematician's Apology", G. H. Hardy:
> Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain.
From another angle: when developing new theories or models, your thoughts are all over the place and frankly it's boring to go over your own crappy notes afterwards and try to reconstruct them in a way that others can understand. And much of the time you forget exactly what happened along the way as well, so any story you reconstruct is going to have some hindsight bias, which defeats the purpose of trying to "teach the story".
Really, from the first answer:
> Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through "guided struggle".
This is the only way to "properly" learn mathematics or science. Anything else is only making you think you've learnt something.
>> Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through "guided struggle".
> This is the only way to "properly" learn mathematics or science. Anything else is only making you think you've learnt something.
My point is not just that most textbooks make it too hard/inefficient to learn this stuff. My point is that most of it is never learned. The Legendre transform connects the Lagrangian and Hamiltonian mechanics, the two fundamental formulations of both quantum and classical physics, and yet most physicist cannot tell you why the transform is defined as it is. The reason is they don't take seriously the possibility that we'll find non-Lagrangian phenomena, and so they have not been forced to consider what observational and theoretical evidence led to it's identification in the first place.
I learned math through guided struggle. My high school had a kind of macho attitude about it, we were all about math competitions and pushing through university level stuff etc. Today I feel that was a good way to study, but much of the struggle could've been avoided. You must solve problems to progress, but they don't have to be hard problems. They just need to be formulated at level n but require solutions at level n+1. Devising such problems is hard and many teachers don't bother, instead they give you definitions at level n+1 right away and make you solve problems about those. That's the root of the problem IMO.
> mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain.
Yes, Hardy was a great mathematician and he did say this --- but most mathematicians have tremendous respect for peers who strive for clear exposition in their lectures, their papers, and (if they write them) their books.
I am a professional mathematician, and Hardy's attitude is one I have never heard expressed by any of my peers.
"A Mathematician's Apology" is a fascinating read, but his description of mathematicians' attitudes is certainly not accurate today.
It indeed can be boring to reconstruct your thoughts in a way so that others can understand -- but many of us make the effort anyway, and doing so often leads to new insights.
The full paragraph (in fact, the very first paragraph of the essay) reads:
> It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.
I interpreted "men who explain" not as mathematicians that can explain their work well, but as people who try to explain mathematics in a "lay" way to cater for a large audience, whose explanation can very often become inaccurate, non-mathematical or just downright false, yet still get public credit for seeming to know the field very well, despite these inaccuracies, and even though they are not directly pushing the advancement of the field itself.
It's of course a good thing to try to reconstruct your own thoughts, but I wouldn't say it's unreasonable for a mathematician to omit doing that. Could you go into more detail on the examples you mention, where doing so led to new insights?
> Could you go into more detail on the examples you mention, where doing so led to new insights?
Good question. It's a bit hard to do so (especially without going into mind-numbing technical detail) -- in math you never quite know where insights really come from. "Fortune prepares the prepared mind."
But generally speaking, I would say that good exposition gets you thinking about: Why does the technique work? What is the key insight? What are its limitations? And if you think about such questions, you naturally get a better sense of for which other questions your techniques are also likely to work.
How's about a half way step here: I agree that guided struggle is probably the most likely path to learning, but giving the student a concrete context they can apply does wonders. Back when I was first learning Calc, I had a professor notice I was also writing game software, at the time I was making a "Missile Command" clone. He pointed out that I could use calc to create "guided missiles" and more calc for various other things I had solvers. The light dawned in my head, and suddenly just the realization of the concrete application caused several calc concepts I'd been struggling with evaporate. Additionally, that created a reason, a justification for interest and context, and math almost instantly became peer to my interest in software, practically erasing any conceptual difference they had in my head. I cite that memory as the moment I became a mathematician.
>> Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through "guided struggle".
>This is the only way to "properly" learn mathematics or science. Anything else is only making you think you've learnt something.
I agree with this statement but I think it misses the point entirely. Guided struggle is indeed necessary, but learning some theorem without seeing the impetus for its discovery is like learning how to play an instrument without understanding that the intent is to make music. Yeah, with enough time and struggle, you might be able to go through the motions and play some scales, but most people don't learn to play instruments that way, they learn to play a simple song or two, then go back and start with the scales and building musical theory.
Math textbooks sometimes try to do the same thing, but it seems like they always come up with the most inane and pointless exercises.
All of this is just to say, learning they why of math can help someone learn the how.
No, definitely not, and I agree with the other sibling replies. I was more specifically responding to the part that was dismissive of the "so you should too" point.
Certainly, we can and do develop newer and simpler ways of understanding previous theories. And teaching the historical sequence of events can help with understanding; I myself experienced that with [1] for modern analysis. However, these understanding-aids don't teach you how to do mathematics, and only marginally improve your ability to apply those models and theories to existing real-world problems. To improve your ability to do mathematics, active exercises are necessary. Really, it's the same with many other fields, you don't get to be a good musician merely by reading about music and music theory.
I assumed they used the term "struggle" poetically, it certainly doesn't have to be unpleasant. But you have to put in some active mental exploratory effort. I found this post [2] a good summary of the skill set. But it's very abstract and likely won't make much sense unless you've been through the experience yourself.
These understanding-aids are also sometimes unnecessary. If you've done enough of the right kinds of exercises, they are of themselves an aid to understanding. For example, I could understand category theory better, not by learning about how this theory was developed historically, but by writing lots and lots of similar programs, and having a natural tendency to syntactically (and without much thought) refactor my code to be less repetitive, eventually leading me to various "category theory aimed at programmers" blog posts and papers. This one [3] of course deserves a mention, but there are many more.
To further emphasise this point, very brilliant mathematicians can just "pick up" models and concepts and work creatively and productively on them, without needing these aids.
My other point was that, the understanding-aids are very rarely what actually happened in the head of the people that developed a theory. Even historical narratives have distortions, and they are rarely detailed or precise enough to describe the rejected options, nor why they were options in the first place. (This fact, is also why they are not useful for teaching how to do mathematics.) There are exceptions, but reconstructing them is a boring process with little reward, especially since new developments 10 years later might explain it in even simpler terms.
That said, I would disagree with this part (from the top answer to the OP):
> a) The goal is to learn how to do mathematics, not to "know" it.
Modern mathematics has so much damn material these days that it's impossible to learn everything you need in order to solve modern-level problems, merely by teaching yourself all models and all theories "the hard way". Understanding-aids are certainly needed, and I use them very often myself, and I certainly prefer resources that teach using good analogies, proper context, descriptions of the motivations behind a theory, step-by-step "n/n+1" exercises, and everything else that other people mentioned here.
From "A Mathematician's Apology", G. H. Hardy:
> Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain.
From another angle: when developing new theories or models, your thoughts are all over the place and frankly it's boring to go over your own crappy notes afterwards and try to reconstruct them in a way that others can understand. And much of the time you forget exactly what happened along the way as well, so any story you reconstruct is going to have some hindsight bias, which defeats the purpose of trying to "teach the story".
Really, from the first answer:
> Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through "guided struggle".
This is the only way to "properly" learn mathematics or science. Anything else is only making you think you've learnt something.