In his recently published book "Mathematica: A Secret World of Intuition and Curiosity", David Bessis argues that the intuition is the "secret" of understanding maths at all levels.
Not sure what conclusion to draw from here, but my (rather dated) experience with university maths tells me that the intuition is a powerful tool in developing the understanding of the subject.
At university level the objects become more abstract, so the intuition we use in normal daily life may no longer apply. New kinds of intuition may develop but it takes work, including lots of time spent with the formal processes and calculations along with reflection on that time, and the active creation of new metaphors to drive the intuition. For example, I still remember a professor using "Ice-9" (from _Cat's Cradle_) as a metaphor for how proving some local property of a holomorphic function on the complex plane made that property true for its global behavior
To me, what it says is "intuition can be honed and it is powerful, but hard to pass along to others". Just that.
Bessis actually mentions examples of how intuition and technique complement each other nicely.
When a high school student looks at a high school math problem they’re drawing on all of their experience in K-12 math to get intuition for how to solve the problem. When they leave high school to study math in undergrad they struggle because their experience is no longer sufficient. They’re faced with a lot more abstract problems and the demands for rigour are much higher. The problems also tend to operate at higher levels on Bloom’s taxonomy [1] than high school math, something with which the average high school student would have little or no experience. It is this unfamiliar territory where intuition is hard to come by.
After gaining more experience (later undergrad and into grad school and beyond) the intuition starts to come back. But it’s fundamentally a different kind of intuition. In high school math it was often a visual/geometric intuition that teachers were trying to build. In higher math it’s an intuition for abstractions and for the tools you need to attack problems. This is really no different from a programmer looking at a problem and saying “I need a hash map and then this problem is trivial.”
As Tao puts it, the value of intuition becomes much higher in the post-rigorous stage once you have sufficiently developed your technical skills.
https://terrytao.wordpress.com/career-advice/theres-more-to-...
The reason part being essentially a one or two line natural language summary of ‘why the proof works’ — something that is almost always possible and is enlightening and conducive to efficient memorisation, but that for some reason is very rarely written down explicitly.
Along the lines of your own argument: even better would be
reason0-definition-reason1-theorem-reason2-proof
The "Reason" as result is true is that it follows from the previously established axioms via logical reasoning.
Theorem: Every subspace Y of a second-countable topological space X is second-countable.
Reason: Intersecting each set in a basis for X with Y yields a basis for Y.
Proof: [formal symbolic stuff involving open sets and unions, and mentioning cardinality, etc.]
(I’m not claiming ‘reason’ is the best word for this — it probably isn’t. But it’s not the same thing as motivation.)
> The "Reason" as result is true is that it follows from the previously established axioms via logical reasoning.
One could argue this is not the reason a result is true; it’s the reason we know it’s true. The fact that true statements follow from established truths by logical reasoning is more a property of the formal system (which hopefully is sound and consistent) than it is to do with the notion of truth itself.
In graduate school that was expanded: take every chapter of the textbook and rewrite it, filling in all the intermediate steps of every proof, those where the author writes "it follows that ..." or "from which it's obvious that ..."
1) Drill/spaced repetition basic definitions. The Cornell note taking method is convenient to do this while taking notes.
2) Keep a diary of thoughts, things you couldn't solve or did solve. Especially identifying problems, what works or doesn't, why something went wrong. Metacognitive thinking was really useful for transferring problems to solving new ones.
3) A study group involving a lot of us explaining to each other.
That strategy in my opinion is not optimal for humans.
What we need to do is develop math resources that can help students learn things analytically & conceptually.
Like how they learn biology.
Maths content is the ne plus ultra of conceptual.
It's totally possible to slog through a chapter of a maths text and feel like we got it. But it turns out our 'understanding' was a facade. We can't apply the concepts in a new situation. Exposing ourselves to feedback via problem sets reveals this.
How do they learn biology?
It really depends on the textbook, isn't it? I find it impossible to solve all the problems in CLRS, for example. Our professor assigned one of the problems about universal hashing, and it took me hours to get the key insights to find the correct proof. I can't imagine how one can solve all the problems given so many competing priorities, except for a few truly talented.
> given so many competing priorities,
Undoubtedly, the best time to do this was when you were young. The second best time is now. Pick a book and work on a problem or 2 every day. It will likely take 6 months or so but you will learn the material. This is an incredible way to level up in a technical area.
Imagine how much knowledge is in CLRS.
I recently-ish had a read through some of my old fundamental/pure maths notes from Uni, including plenty of proofs. The damned things are littered with steps which were "obvious" to my smug self-satisfied 20 year old self but impenetrable to me reading without much context 20 years later.
Git.
Some books can take many many months to finish off like this, and most courses only cover a small percentage of the book.
Yes, you’ll learn more. Also, TAs will recognize you put in the effort so if you’re arguing for partial points or you’re really close to a cutoff grade they will be more likely to bump you up versus someone they’ve not seen or noticed all semester.
On the other hand, I second the suggestion to engage more deeply with the subject material itself: Modify assumptions and see what happens. Can certain proofs be simplified? Try to reconstruct proofs by only memorising certain key details. Try to draw a mental map of a subject and how the different theorems and definitions relate together. Try to implement some proofs in an automated theorem prover, if that's your thing.
Sadly I didn't do that. I graduated and do okay, but I encourage everyone to do better than me. As I get close to retirement I need a few people who are still working to build things (and medical treatments) that makes my life better (and in turn take some of that money I saved up over the years for your own life)
I also don't think that people who develop new medical treatments necessarily did all the exercises in a pure maths textbook. Being able to prove that continuous functions on a compact set are uniformly continuous probably won't help you fight cancer.
How to Study Mathematics (2017) - https://news.ycombinator.com/item?id=26524876 - March 2021 (73 comments)
How to Study Mathematics (2017) - https://news.ycombinator.com/item?id=16392698 - Feb 2018 (148 comments)
Bonuses:
Ask HN: How to Study Mathematics? - https://news.ycombinator.com/item?id=23074249 - May 2020 (31 comments)
Ask HN: How to self-study mathematics from the undergrad through graduate level? - https://news.ycombinator.com/item?id=18939913 - Jan 2019 (227 comments)
Ask HN: How to self-learn math? - https://news.ycombinator.com/item?id=16562173 - March 2018 (211 comments)
Others?
Ask HN: How to self-learn math? - https://news.ycombinator.com/item?id=16562173 - March 2018 (211 comments)
Huh. Any mathematicians who want share their own opinions and experiences about this?
This pretty much goes completely against my experience with other grad school level neuroscience/ML
You don't want to be so familiar with stuff as to make it second nature but NOT from memorization. That, at least an other areas, leads to surface level recognition
Does the author mean internalize and not memorize?
Counterintuitively, mathematicians like being "brain-off" as much as possible -- you want to be able to read a phrase like "closed convex subset of a Hilbert space" and effortlessly think to yourself "oh! there's a unique norm minimizer" -- if you have to piece that together from scratch every time, you're going to have a hard time -- reading papers and learning new fields becomes a dreadful slog, similar to how math in general becomes a slog for kids who don't memorize their times tables.
To be clear, this does not mean memorizing all the theorems. Getting to know the theorems (and solving problems) is what helps you internalize the subject. Math is the art of what's certain, and knowing exactly what the objects of the subject are is necessary for that. Theorems are derived from the definitions, but definitions can't be derived.
In my experience with a math (undergrad and PhD), I realized I had to know definitions to feel competent at all. In my teaching, it's hard to convince students to actually memorize any definitions — so many times students carry around misconceptions (like that "linearly independent" just means that no vector is a scale multiple of any other vector), but if they just had it memorized, they might realize that the misconception doesn't hold up. Math is weird in that the definitions are actually the exact truth (by definition! tautologically so), so it does take some time to get used to the fact that they're essential.
It’s easy to forget that non-math people find this — the idea that the definition is its own ‘model’ rather than an approximation of something more ‘real’ — somewhat hard to stomach. Outside of pure mathematics the idea is that mathematics is a tool for (usually lossy) modelling of reality, not a collection of already perfectly well-motivated objects to be studied in their own right.
When you are studying science and technology, and the math theorem doesn't match experiment, the theory is probably wrong (or incomplete, missing factors), so you can discard it or try to improve it.
When you are studying math, and the intuition doesn't match the theorem, the intuition is probably wrong (or incomplete).
But yeah, while studying math, I think it's similar to learning programming — don't blame the compiler for your mistakes, it's a well-tested piece of software.
But, in my last comment I was just trying to temper my previous comment's claim about how important definitions are. At some point you get so used to a definition that even if you don't know a particular formulation word for word, you could still write a textbook on the subject because you know how the theory is supposed to go.
I don’t have any musical training, but I related it back to the practice and warm up sessions we had before we’d play an actual game in the sports I played as a kid.
Perhaps some explanation like that will get it to click with someone.
I also learned of the existence of soft question tags on Math Overflow and Math Stack Exchange that contained an incredible amount of guidance that I think was never possible in lectures. Sharing links to those websites in the syllabus may be helpful for the odd student that actually looks at the syllabus.
As someone who's gone through the mathematical ringer, the analogy doesn't ring true to me, but it does sound pedagogically useful still (my students will be CS majors, so the math will be for training rather than an end). Even at the highest levels the definitions are of prime importance, though I suppose once you get to "stage 3" in Terry Tao's classification (see elsewhere in the thread) definitions can start to feel inevitable, since you know what the theory is about, and the definitions need to be what they are to support the theory.
Personal aside: In my own math research, something that's really slowed me down was feeling like I needed everything to feel inevitable. It always bugged me reading papers that gave definitions where I'm wondering "why this definition, why not something else", but the paper never really answers it. Now I'm wondering if my standards have just been too high, and incremental progress means being OK with unsatisfactory definitions... After all, it's what the authors managed to discover.
This is okay! They're students on the way to gain that experience. At some point you can and will go over to internalizing, instead. But as advice to students just starting out, memorizing is the way to go.
You have to internalize the meaning also, but you must know the definition precisely.
That hit home. I'm afraid I was one of those lazy math undergrads who struggled with a few of the first year topics, didn't get help or put the hours in and never really recovered. I will maintain I think the teaching was very poor in places (lots of "just trust me" handwashing and "this is obvious so I'll leave it to you to complete" which for an 18 year old frankly sucks). A system that lets you get 30% in "analysis 1" and then just marches you straight into "analysis 2" next semester and expects you to just pull your socks up isn't much of a system to me. Honestly I'm afraid my time at university doing maths was miserable. I should have done something more applied like engineering or CS probably.
Someone once told me "If you like biology at school, do psychology at university. If you like chemistry, do biology. If you like physics, do chemistry. If you like maths, do physics and if you like philosophy, do maths". I should have listened.
This is good advice if the objective is to do well regarding grade results. If you want to get down to the bottom of things, to understand everything and to solve fundamental questions of science, you might well want to invert the advice:
If you liked biology, study chemistry, for the processes of life are (electro-)chemical processes.
If you liked chemistry, study physics, for the processes of molecules and atoms, their formation and reactions, are physical processes.
...
Anyway, if you were interested in it, you can always revisit it, and even try taking a class again if you ever want to. There are lots of great books out there for self-learners, and lots of communities of folks learning together
Seems like I made the right decision! (I did maths.)
A nice thing I realised is that once I did that, almost all of the exercises that were complex before for me, turned out to be straightforward. It was like a cheat code where I almost did not need to do any exercises.
I used to teach at the uni at several levels, and every year I would ask if anyone tried to recall the proofs of the theorems at home. and no one did. They were always shocked when I told then they should do it.
I try to do this as well. If you combine this with understanding the definitions of the various units you more or less will have the textbook in your head, some assembly required.
Creative introspection into how one learns begins to really pay off partway through college.
One's relationship to convention becomes as important as one's relationship to technique. Understanding the "whole" of something involves understanding the biases that shape the presentation you're seeing. You'll probably want to shed them.
This applies whether one wants to change math or just learn it. A passive stance, trying to do what others want, is a recipe for frustration.
At the end of high school, I could do everything. I finished my IB exams with huge amount of time to spare, the only thing holding me back was being able to write fast enough. It had been months since I saw a regular curriculum question that I didn't know how to do. Any marks I lost were just trivial errors.
When I got to university, there would be question sheets where I would look at the questions and wonder what it had to do with the lectures I had just been in. As in "I went to this lecture, and I'm supposed to use the information to answer these questions, but I don't even know what the questions mean".
The learning happens when you are doing this frustrating head-bashing.
You read, you read more, you fill a notebook with useless derivations, and eventually you things start to take shape. This could take the entire week's worth of time, just sitting there fumbling about.
The difference is that in uni, the amount of material is so vast you cannot explain it to someone in the time that you have. The students have to pick up some key ideas, and then fill in all the details themselves by pouring hours into it on their own.
Very well said, at the university where I studied, there was a pre-semester math repetition course. It was a week long and started with addition of natural numbers. After two days, everything I learned in 13 years of school had been repeated. That was a brutal resetting of expectations. But it made everybody clear, that this is not school anymore. That a different kind of work and focus would be necessary.
Not everyone will enjoy mathematics at first sight. But I think at least 50% of that can be explained by the lack of obvious paths to enjoy mathematics. Obviously, most mathematics taught in high-school is not taught as it should be: a cool artistic logical pursuit that has all kinds of fun in it.
So my advice is to really find a mentor who already has found that path and let them show you how to enjoy math.
Believe me, I've tutored a lot of people, many of which initially disliked math and found it difficult. But after a few tutoring sessions, I could see a little sparkle in their eye that said, "hey, this might be cool".
So before you apply logic, studying, and other tedious "productivity" measures to your math learning, make sure you find a way to enjoy it first.
I am not a PhD, but have done a fair bit of tutoring young people in maths. I feel similarly to you there, and am always on the lookout for new ways to foster that feeling of mathematics being fun and wonderful.
It can be hard. The feeling a lot of young people pick up - of maths being roughly akin to pointless abject suffering - is so strongly rooted in young people sometimes, and can be strongly connected to feelings of inadequacy and shame and so on.
I'd filed it away in my head as something to sink more time into when I've the emotional space to do so, but essentially I find it quite courageous to go against the grain in a world where it's perhaps harder and harder to do so. Everyone has a hot take, of course, but the hot takes are usually very much within the bounds of acceptable discourse.
Anyway, more power to you, in your endeavours.
1. Coloring maps (four color theorem) 2. Drawing curves and then looking for pathological ones (quadratic equations ...but more of an exploratory rather than methodological approach) 3. Infinite series (but just discuss some paradoxes people thought of centuries ago...) 4. Triangles on spheres
...if you don't make some of it extremely rigorous, there's a lot of basic facets of these things like the drawing aspect that can be quite fun. And TBH it makes sense to start there because mathematicians started out exploring these topics just with random calculations and doodles rather than rigorous proofs.
Exactly the same for me. Honestly, the satisfaction of seeing that "sparkle" in the eye of an initially unmotivated or discouraged student is probably among the most fulfilling moments of my professional career.
A lot of people confuse enjoying [x] with enjoying being good at [x]. This is why so many students switch subjects later on in life; when a field suddenly doesn't come naturally to them, they seek to play to their strengths elsewhere. Problems occur when they quit too early, and building confidence early on is important for stopping this.
In my experience, when you think you're bad at something, it's almost impossible to enjoy doing it, which makes preliminary mastery actually the first step to enjoyment and therefore downstream success.
On a side note, recently the government of Manitoba in Canada removed requirement for maths teachers to take university maths courses. This is being pushed strongly by the education departments of university, which shows how much these maths teachers hate maths.
That is messed up. Harsh. What is also messed up is that to become a math teacher, you have to go to teacher's college. That's one reason why I never became a teacher. I think I would do better than most at teaching (at least based on the comments I got in my student teacher reviews) but spending another two years at school is rather humiliating and costly.
Taken from Gian-Carlo Rota in The Lost Cafe, a quote I found here http://www.romanpress.com/Rota/Rota.php
Now in my 50’s, I wanted to relearn high school maths from 35 years ago and I scooted through their Foundations series (now half way through Foundations 3, rapidly accruing like 9000 xp in 9 weeks, 2 hours a day). Planning in 2025 to do 1-3 university level courses with them at a slower pace.
It’s suited my way of learning as an autodidact: enjoyable; measurable; adaptable level of hardness; no hitting of a “wall” or “unmet dependencies”; thorough explanation of problems I didn’t solve.
Perhaps my biggest realisation was that one can learn without needing to document many notes to revise/memorise, because experience and spaced repetition suffices. I’m taking a Xmas hiatus which will be the real test of baked learning.
Congrats on your pace, 9k XP in 9 weeks is impressive!
- John Von Neumann
I find the parallels between proofs and programs to be fascinating. We could write an analogous thing for programming:
"A good way of seeing how a sort of program works is to examine one of the popular programs/libraries and see what functions were used in it. Then look inside of those functions and see what functions are used inside of those. Eventually you will work your way back to the lower level primitives."