Advice
Graduate
 Bjorn Poonen on math talks.
 Kristen DeVleming on COVID job advice
 Dan Margalit’s advice on choosing an advisor
 Sara Billey’s advice collection
Undergraduate
On Learning and Being a Student

Stay organized: Find and settle on a good way to track all of your due dates, and in particular have a good idea of when each upcoming major exam is for all of your classes. I personally use things like Google Calendar and Todoist, and map out my entire semester within the first week or two, but there are many other great apps. A physically planner is also an excellent choice.
Fitting in a lot of work at the lastminute isn’t usually an optimal strategy, so actually planning when you’ll work on things can be helpful too. Be sure to slot in enough time to avoid lastminute stress, since this tends to hurt how much you can absorb/retain the material.

Start everything early: Starting early gives you time to mull over questions and potentially visit professors/TAs for help.

Use a wide variety of resources: Books, lectures, notes, PDFs – use them all! The search modifiers
inurl:edu filetype:pdf
on Google are incredibly useful. 
Extensively use good resources: There is also some value in working a single, particularly wellregarded book, front to back.

Spaced repetition: Read up on it (there is some pretty good science on its effectiveness in learning) and work it into your study habits. Cramming can have diminishing returns in proofbased classes, so it’s worth visiting and revisiting material often enough to be familiar with it. Apps like Anki are great for this.

Sleep like it is your hobby. Two hours of extra sleep is almost always a more efficient use of your time than two hours of tired studying.
On Learning Mathematics Specifically

Learn the vocabulary. Mathematics is (partially) a humanities discipline, so it can help to (partially) study it like one. I.e., the techniques you use to study in an English, Art History, or language class can also be put to good use here.
One useful trick is keeping an ongoing list of new words you don’t know and their definitions, and continually reviewing and adding to it.

Spaced Repetition: When you run into things you don’t know, memorizing can be a first step (although this will also come naturally with practice). I usually recommend keeping a list of words/terms you’re less comfortable with or don’t know, or longer equations or formulas, and then continually reviewing all of them over the semester.

Practice a lot! The wonderful violinist Jascha Heifetz had this to say about practicing music:
If I don’t practice one day, I know it; two days, the critics know it; three days, the public knows it.
In some ways, Maths is very similar.
Moreover, doing tiny bits of math every day really does add up: a nice mathematical analogy is that $(1.00)^{365} = 1.00$, but $(1.01)^{365} \approx 37.0$. I.e., a tiny bit of extra work/studying/review every day (in this equation, an extra 1% over the course of a year) multiplies your skill level many times over.

Explain/teach to others: A really great way to learn (anything) is to try teaching it to someone else. This is especially true in Math, since communicating your solutions and process is a huge component. If you stumble a little in your explanation of some particular concept, that is a great way to diagnose and pick out areas you can work on (e.g. restudying the section, working more problems, etc).
If none of your friends jump at the opportunity to hear about wonderful Maths all of the time, it can also be useful to try recording yourself explaining/summarizing a concept!

Use multiple books. For most undergraduate topics, there are many “standard” references. Use them all! Crossreference liberally as well.

Get the definitions down cold. Record every single definition you come across and just do whatever you have to do to memorize them! Some people like flash cards, I personally make a “dictionary” document for each subject. Meaning and understanding is often a gradual process, but knowing the full statement of a definition should just be automatic.
I like to think of definitions as the analog “multiplication tables” within higher Mathematics – is it essential to memorize? Well no, but you also wouldn’t want to waste time stopping to multiply out $12^2$ when you’re trying to solve an integral.

Synthesize often. This includes things like revising class notes, making your own “cheat sheets”, writing up explanations of particular theorems or motivations for certain concepts, etc. Even better, find ways to give talks or teach other people. Present things as if you were trying to teach yourself the topic 612 months in the past.

Google for notes. For just about any topic at any level of specificity, someone has probably written up notes or expository articles and posted them online. So it can be extremely beneficial to look through the first page or two of Google results using
filetype:pdf
for any concept or proof you happen to be studying and see how other mathematicians think about and present them. 
Work and rework many problems and proofs. Do more than you are assigned, but be judicious with your time. If you stare at a problem for more than an hour or two without making progress, pivot. Sometimes the best way to make progress on a problem is to take a break, work on something different, or sleep on it.

Work the exams. If possible, do them in a timed setting.

Maths is cumulative: Previous classes tend to be broken up: you’ll use a concept from a unit, then move on to something else. At the university level, we start bringing together all kinds of tools from previous sections/courses and using them to build more complex things. The more tools you have in your tool belt (that you remember how to use!), the easier things will be.
On the other hand, something complicated can be much more complicated if you’re rusty with a concept from an earlier section. So continually “honing your old tools” with practice becomes very important (reviewing notes, looking back over old homeworks, reworking old problems, etc)
For Incoming Math Majors

Start thinking about whether or not grad school as an option. You don’t have to decide within your first year, but you’ll want to pick courses accordingly.
 Get an idea of what Math is all about  there are great things beyond Calculus!
 To get an idea of what you’re in for as a Math major, take a look at the GRE Mathematics Subject Exam. It covers a broad array of pretty standardized topics.
 Also take a look Garrity’s book (linked in the resources section) for a short, condensed survey of some of the major topics and theorems you’ll encounter.
 One thing that was helpful to me was to make graphs and diagrams of different areas of Math I encountered. Take a look, knowing the landscape you’re traversing is valuable.

Learn Latex and Mathjax early, and then make efforts to regularly typeset your written notes.

It may be intimidating, but try to find ways to talk to your professors outside of class. Don’t hesitate to ask for advice related to Math or your academic career. In my experience, they are usually happy to help.
 For those aiming for grad school, particularly in pure Math: here are what I would consider some of the most essential, core classes to take:
 Calculus (of course)
 Linear Algebra
 Ordinary Differential Equations
 Abstract Algebra
 Real Analysis
 PointSet Topology
Here are some “nicetohaves” – still important, still beautiful, and great electives, but missing out on them in undergrad isn’t the end of the world:
 Probability
 Combinatorics
 Complex Analysis
 Number Theory
 Discrete Math (e.g. Graph Theory)
 More Linear Algebra (e.g. Numerical Analysis)
Of course, I recommend taking as many Math courses as possible – explore the subject and discover what you like! But the ideas and concepts from these particular courses are relevant to just about any area you might go into, so having some exposure to them makes life much easier (regardless of what you pursue).
The Math Subject GRE
If you are thinking about applying to graduate school in Mathematics (pure or applied), I would recommend that you start looking at the material for this exam within your first few years, and take it as soon as your Junior year.
Note that the Math Subject GRE is vastly different than the Math portion of the General GRE. The latter covers primarily highschool level mathematics and requires no courses beyond Calculus (although the questions can still be tricky). The former is over 50% Calculus, and often includes a wide variety of upperdivision topics as well. Many sources confuse these two exams!
Topics
 Single Variable Calculus
 Multivariable Calculus
 Ordinary Differential Equations
 Linear Algebra
 Complex Analysis
 Abstract Algebra
 Primarily groups and rings, virtually no module or Galois theory.
 Probability/Statistics
 Real Analysis
 Convergence of sums/sequences, topological properties of $\RR$.
 Combinatorics
 Graph theory, counting problems
 PointSet Topology
 Numerical Analysis
 Set Theory and Logic
Resources
 :faexternallinkalt: You can find five sample exams here.
 :fabook: Princeton Review, Cracking the Math Subject GRE
Advice
 Most problems don’t require extensive computations; there is usually a trick that solves it very quickly.
 Time is the most difficult factor, be sure to take timed practice tests. You have right around two minutes per problem.
 Sign up early, as these exams are often only held a few times per year.
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