
Ask HN: How to Study Mathematics? - hypersurface
For context, I am a university student just finished my first year, studying mathematics and a mostly unrelated humanities subject. That said, general advice in the topic is probably welcome for other readers. I was drawn to mathematics from a young age; however, in recent years my attention has been taken by my other subject. That said, I don&#x27;t regret taking the course at all and I enjoy it, although some topics I prefer to others.<p>My institution gives a very significant incentive to those who excel academically in their early years. I would beat myself up if I did not use the unique opportunity of vast free time this summer to advantage myself in this regard.<p>My question is essentially for those who have experience studying the subject at university level, or self-studying to an approximately equivalent degree, with emphasis on the pure and problem-solving end over the applied. What strategies, resources, and tips would you advise for learners?<p>Also, I understand that this type of question is asked fairly regularly. Due to the nature of this forum, much has been said about approaching applications of mathematics for those in other disciplines or for the purpose of self-study. But I think that most would agree that from the outset, pure mathematics and abstract problem skills appear arcane from university-level on.
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nimonian
For my own part, it was all about finding the right book. I enjoyed my degree
in pure maths, and after finishing I really missed studying so I tried out
dozens of books. Eventually, I hit upon Calculus by Spivak. This book
fundamentally changed me and I solved every problem in it fastidiously - it
was just so well made, funny in parts, it had cliffhangers, and the problems
were just amazing, linking together over chapters like characters in a novel.
For the first time, I was solving problems just for fun and I was improving
much faster than I did during my degree. I wrote all my solutions in LaTeX so
I had an organised body of work to look back on and watch grow, adding to the
satisfaction (I lose paper; pdf is better for me).

So my advice: find a great book, make notes and expand on what the author is
saying with diagrams or additional lines of working, focus on solving
problems, do it out of love.

~~~
harperlee
Spivak was great, I remember it fondly from my university years.

Is there any kind of crowdsourced "canonical" list of great books? Ideally
something that does not bring 5 answers per question to the table. For
example, what is the Spivak of discrete mathematics? Of statistics? What is
Spivak for multivariate calculus?

~~~
nimonian
I would love to know the answer to your question. On the one hand, it's too
subjective to answer, one student's Spivak is another's Apostol; on the other
hand, certain books just seem special in an almost objective way and rise
above the scores of other books on the subject.

There is this: [https://mathblog.com/mathematics-
books/](https://mathblog.com/mathematics-books/) which looks like a pretty
well curated list

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chadcmulligan
Maybe some history of mathematics, to give you some perspective on it all?
Something I wish I'd known when I did my degree, Norman Wildberger has a great
course:

[https://www.youtube.com/watch?v=dW8Cy6WrO94&list=PL55C7C8378...](https://www.youtube.com/watch?v=dW8Cy6WrO94&list=PL55C7C83781CF4316)

Visual Group Theory, by Nathan Carter is a book you can get something from no
matter your level of maths, imho.

If you know some linear algebra, Gilbert Strang's course on YouTube may be
interesting - [https://ocw.mit.edu/courses/mathematics/18-06-linear-
algebra...](https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-
spring-2010/)

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jstrieb
Currently finishing my junior year of undergrad in math. I'd give different
advice to a casual enthusiast than I would a full-time math student, but here
are some tips that have helped me:

\- Reread everything. The first time, only read definitions and theorems;
proceed steadily and focus on high-level ideas. The second time, try to recall
definitions and focus on details of theorems. The third time, try to recall
definitions and recall proofs of key theorems; work on memorization and
fluency. The fourth time, try to modify the conditions of theorems and see
what changes. If you are struggling with reading, get as far as you can and
then go around again - it gets easier every time. Take as much time as you can
afford before reading the same section again to let the material sink in.

\- Do lots of problems - start with trivial examples and slowly scale up the
difficulty. Lock down the basics before moving on to harder problems. Ask for
recommendations for books that have particularly good problems.

\- Practice proving theorems and claims. If you have a question while reading,
make a conjecture and try to prove it formally. It took me too long to realize
that understanding abstract math is not the same as being able to do proofs
and formalize intuition. Following along while reading doesn't mean you can
quickly or easily prove facts about the material, and the proof is what counts
in the end. Knowing why something is true is not helpful if you cannot
formalize it.

\- Hone intuition. Proving things is just performing a (directed) graph search
on a set of facts with a well-defined start and end goal. Having good internal
heuristics will help the search proceed quickly, in the right direction, while
avoiding dead-ends; the only way to develop this sense is by practice.

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Atiscant
The single most important thing IMHO is to get the basics right

For context, I have a PhD in theoretical computer science and a master in
mathematics. I've been teaching mathematics for several years ranging from
high school to university.

The biggest problem for students, especially at the university level, is that
they don't have the basics down. If you are not fluid in fractions and
manipulating equations, you can't focus on the important part of learning
induction proofs. This is just one example, but the return on investment on
getting the basics down is huge.

For each new subject, you need to take the prerequisites serious. If you can't
read a new definition and focus only on the new part, you should go back. The
is of course very time consuming if you are missing a lot of prerequisites but
if you want to get good, this is the way.

A well planned program will do this for you to some degree but the reason most
people struggle is that they don't take prerequisites seriously or simple
haven't learned them probably. You should be able to solve exercises in each
prerequisite subject to a fairly good degree before tackling the next.

For anybody starting or doing university level mathematics I suggest your
start refreshing pre-uni. Make sure you have fractions, linear and quadratic
equations, exponentiation and logorithms and basic properties of functions are
really down. If you have time, it can be very beneficial for pure mathematics
to start out with some formal logic and natural deduction to really understand
the way mathematicians reason in proofs and definitions.

Finally, ask all the stupid questions. Nobody, except yourself maybe, care if
you misunderstood a basic point and got something wrong because of it. If "you
just don't get" something, there is a high chance you missed something. If you
don't understand the answer, same thing applies. Mathematics should be, if
done properly, straight forward.

~~~
justaj
What online coursework do you recommend for getting the basics / fundamentals
down? Is Khan Academy sufficient for this?

~~~
Atiscant
Khan Academy is not bad if you like their style of presentation, but as
another comment noted, the number of exercises is too few. Unfortunately, I
have yet to find a good recourse with focus on exercises, so I tend to create
them myself for courses I teach. If you (or any other for that matter) are
interested, I can upload a work sheet somewhere.

There properly are tons of sites providing work sheets, but the focus then
tends to be on K-12 education and they are often to simple IMO.

For university the importing thing is often manipulating say fractions rather
then "calculating" with fractions.

Tangentially, would there be any interest in a service providing this? Pay say
2$ or something and get 50 exercises in the basics with an eye towards
university with sample solution? That space is properly swamped but maybe
there is a niche? I often think that there is room for textbooks of this short
but then again, that market is also swamped with big publishers.

~~~
yonatano
I am interested in this. Do you have a website / would you mind posting your
contact info?

I think there is absolutely a niche: students (or graduates) who have some
experience with undergraduate-level math courses but that need to improve
their fundamentals. Most US universities give credit for "Advanced Placement"
courses taken in high school, even though these courses are rarely taught to
university standards. I skipped the intro calculus sequence at my school, and
since the proof-based math courses are usually fairly self-contained, I never
got a chance to address the gaps in my fundamentals. I've now found myself in
this somewhat awkward position where I can e.g. explain to you what sequential
compactness means but can't solve for the roots of a third-degree polynomial.

It doesn't make sense to read a high-school level textbook, and books on
competition math are either too hard or the focus is more on exposing you to
hard problems rather than teaching fundamentals. There needs to be something
which reviews fundamentals in a rigorous way.

~~~
Atiscant
I've added an e-mail to my profile.

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bjourne
1\. Find pens and paper. 2\. Find a quiet place. 3\. Find a book with lots and
lots of exercises suitable for your level. 1/3 easy ones, 1/3 medium ones and
1/3 hard ones could be a good mix. The more motivated you are, the more harder
ones you want. 4\. Solve exercises. 5\. Repeat step 4 forever. Also don't
bring any electronic devices with you because they are distracting. If you
don't, you'll be much more efficient. And 3-4 hours/day is more than enough to
learn whatever you want if you study efficiently.

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lurker137
The standard advice is to get the assigned books (or highly recommended books)
for each of the courses in your coming semester and work through them early.
You can support them with other resources but you’ll save yourself a lot of
time by using them as your main guide - especially for higher level math. The
hard choice is how to study from the books. The only general tip I can give
you is to not get too caught up in the early chapters. It’s better to do a
fast first pass then come back to it later.

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currymj
if it's something you're self-studying, don't be afraid to put something down
and pick it up later. There are topics and materials that I completely failed
to understand the first time I encountered them, but months or years later
they were easy to understand.

Of course, if your goal is to get good grades in specific courses which have
exams on specific dates, then this isn't very useful advice.

Some other advice:

\- don't underestimate the value of memorization when it comes to definitions
and important theorems.

\- do lots of practice problems which are like those on the exams (obviously)

\- as much as possible, find peers to study with and talk with. it's easy to
fool yourself into thinking you understand something, but when you try to talk
about it with a real person, the holes in your knowledge are revealed pretty
quickly.

\- if you can't do that, try to explain the concepts and ideas you're learning
to an imaginary friend.

\- you might be able to find lecture notes or slides online from the actual
versions of classes taught at your institution.

I agree you should ask the professors for materials to look at, or at least
try to focus on the materials and courses actually taught at your institution.
This is the path of least resistance and will probably make it easier to stay
motivated too.

~~~
hypersurface
The end goal is of course, a grade; however I will have to take supplementary
exams. Although past papers are available, there is a degree of ‘aptitude
testing’ involved which is strictly mathematical but still difficult to
prepare for directly. Nevertheless practice problems are good for that and I
will be doing a lot.

You are right, the advice to skip/quickly pass the first chapters [thank you
lurker137] is good. I think this is something I have known subconsciously but
I will try to keep aware. Same for those who have suggested seeking further
material.

> as much as possible, find peers to study with and talk with

This is good advice. Thank you.

~~~
thomashobohm
In terms of aptitude testing, going through some books like "How to Prove It"
and working on rigorous proofs-based material (perhaps an introductory
analysis book with lots of diagrams) will be helpful. As will brushing up on
geometry-lots of people memorize a lot of geometry in middle school/high
school, do well in their class, and forget it. Geometric concepts continue to
appear throughout math and having a deep grasp of them is super helpful for
your first "rigorous" math classes, where the secret to finishing a proof
often lies in visualizing or drawing it correctly.

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avmich
Few high-level advices, in arbitrary order.

1\. Don't panic. If you don't understand everything immediately, that's
normal. Try second or third time - and even fourth and fifth. Not necessarily
immediately - take some time.

2\. Try different books - for different people different explanations "click"
differently.

3\. Math is often understood better a few months after you studied the
material. In many courses you don't have time to internalize new concepts in
real time - but after some pause you feel better adjusted to them.

4\. When you see a math expression, read it aloud, for better understanding.
The goal is to force you to actually pay attention to every feature of the
math notation used. Even reading this once helps. After reading a few times,
you may start to see patterns and understandable pieces of the expression.

5\. Scientists sometimes need quite a bit of time to understand what their
peers - even in close areas - write in their papers. Reading one paper, even
not too long, for the whole week to get a good understanding could be normal.

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nsainsbury
I wrote about my journey learning mathematics for the last ~6 years quite
recently actually! Might be of interest to you as well:
[https://www.neilwithdata.com/mathematics-self-
learner](https://www.neilwithdata.com/mathematics-self-learner)

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Durgasoft
Obtain Terrence Tao's Analysis I book and start from scratch, there's an
ongoing workshop for this
[https://learnaifromscratch.github.io/math.html](https://learnaifromscratch.github.io/math.html)

However you should probably ask professors at your school what you should be
doing in your free time, since they are the one's who will be determining said
significant incentive so probably have a better idea what you should be doing
than any of us.

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battery_cowboy
Do what the professors tell you: buy (or download for free) the books, do all
the practice exercises and proofs, go to office hours, listen to the professor
in class and don't write everything they write, just the key stuff, and get
together with other students to go over the harder problems.

That'll get you through the courses, and your level of effort and capabilities
will determine how much you learn. Forget about grades, if you just do your
best you'll get the best grade you can.

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cesarosum
Some practices and advice that I gained from my time doing a PhD in
mathematics:

1\. Find a time of day to read and work on problems that works for you and
make it interruption-free. For me it was up at 6AM and working until 9 or 10AM
with no phone or laptop. If you can do longer or fit in another session, go
for it.

2\. I found that I process new material in the background, so either going to
the gym or taking a long walk gave my mind the break and mental space to
process what it had just ingested.

3\. In mathematics, it's common to present the most terse, stripped-down
version of a result which can rob it of the background and context. I found it
was often helpful to find the author's previous work, PhD thesis or talks in
order to understand how they got to their result.

4\. Don't be afraid to get a little side-tracked or obsessed with something
that's not directly related to what you're currently working on. Part of
training to be a mathematician is learning the ability to hold seemingly
unrelated concepts in your head for long periods of time until the connection
becomes apparent. Learn to love difficult things!

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credit_guy
Feedback.

You can try to do math on your own, but you'll have a high chance of failing.
You need to get feedback.

Just like you can't learn a foreign language without interacting with the
speakers of that language, it's very hard to learn math without speaking with
people who know math.

So, by all means, find good books, good youtube videos (you can't go wrong
with 3blue1brown), Khan academy, all those will help. But if you don't get
feedback from people, it's easy to think you make solid progress when you
actually don't. It's easy to not know what direction to pursue among hundreds
of options.

How do you get feedback? If you can afford it, just pay for it. Nowadays there
should be various online services that offer math tutoring. Get in touch with
them, and tell them you'd like to go through Calculus 1, or Ordinary
Differential Equations, for example. With a competent tutor, you can learn
calculus in, let's say, 3 weeks, on your own, you could spend 3 months and
still have serious gaps.

Edit: Because of feedback, it's actually easier to get good at programming
than at math. In programming, you write programs, if they run and product the
result you expect, that's great, if they don't, you realize right away you
made a mistake.

So, if you can find a little coding project that forces you to learn math, you
can get valuable math feedback by seeing that the program runs as expected.
Ray tracing comes to mind. Maybe building an n-body simulator could be a fun
exercise too. The phenomenal book "Turtle Geometry" teaches you math in this
way, it asks you to write and run (and play with) various programs to produce
quite some pretty pictures. You could go through it, and write all those
programs in Python rather than Logo (Python has a turtle package embedded).
With that book you can really learn a ton of math, including some Riemannian
geometry.

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9wzYQbTYsAIc
A few things that helped me during my studies as a math major:

\- get a small-ish whiteboard; one that you can fit in a larger bag. Do
problems on that whiteboard first, then work them out on paper for submission
once you have either solved the problem or know you are on the right track.
It’s way easier to wipe the whiteboard clean to start over than it is to erase
your pencil work.

\- invest in a quality mechanical pencil and a separate eraser.

-work at least 3 extra problems beyond what was assigned for each category of problem

\- don’t memorize anything that you can derive quickly - memorize the
derivation. Use your scratch paper to derive whatever you need, when you need
it, instead of fretting over memorizing the right formulas, etc.

\- read up on the topic of lecture, before lecture, so that you can listen to
the meta-information and ask questions during lecture instead of scrambling to
follow the topic.

\- make a notebook of techniques and tricks of the trade

\- definitions are everything.

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pknerd
I recently bought Alex in numberlansld. Quite an interesting book. Also try
What is mathematics? The book helped Einstein to grasp math concepts.

~~~
jonny789
Could you tell me details about 'What is mathematics' book ? Which book
specifically & Who's the author ?

~~~
liendolucas
The authors of "What is Mathematics" are Richard Courant and Herbert Robbins.
This one is good as well: "Mathematics: Its Content, Methods and Meaning" by
Aleksandrov, Kolmogorov and Lavrent'ev (Dover publications). It's a big book
covering a wide amount of topics.

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soberr
Learn from the countless of people who went before you. Read the book, A Mind
For Numbers by Dr. Barbara Oakley.

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thomashobohm
This depends super heavily on the math courses you’ve already taken.

