About the Test
The GRE Mathematics Subject Test is generally taken by students who are looking to enter graduate school. It is meant to cover the most important topics learned during undergraduate mathematics.
The test is 2 hours and 50 minutes long and consists of 66 multiple-choice questions, each with 5 answer choices labeled A-E. Although you (hopefully) don’t need me to tell you this, this gives you an average of about 2½ minutes per question. This doesn’t mean you should be aiming to spend 2½ minutes on every question — some can be solved in less than a minute, while others will require careful consideration and insight that may take longer.
- Calculus 50% 50%
- Algebra 25% 25%
- Additional Topics 25% 25%
A full half of the material on the test is Calculus. There’s plenty of single-variable calculus, so know know your limits, derivatives, integrals, and series by heart if you want to survive! (If you’ve taken AP Calculus, you should recognize the style of these questions.) The test also contains multivariable calculus and vector calculus (partial derivatives, gradients, multiple integrals, line integrals, and all that good stuff), as well as differential equations and their solution methods. You’ll also see questions that use calculus techniques to solve questions in other branches of math, such as geometry and trigonometry.
One-fourth of the test is dedicated to Algebra. Of course you need excellent command of all the basic algebra you learned in high school, but this involves much more. Linear algebra makes up a large portion of this material, so you need to be comfortable with matrices and transformations, vector spaces, bases and dimensions, eigenvalues and eigenvectors, and so on. There’s also plenty of abstract algebra, so you’ll see a number of questions on groups, rings, and fields, and all the concepts related to them (think ideals and such). You’ll even see some questions on number theory, as the study of abstract algebra is closely related to the properties of prime numbers and modular arithmetic.
The last fourth of the test covers Additional Topics from the wide expanse of undergraduate mathematics. You’ll find some questions from topics in discrete mathematics — combinatorics and graph theory make a showing, as do logic and set theory, and even some analysis of algorithms (written in pseudocode). This is also where you’ll see questions from topology and analysis, both real and complex, so you need to be comfortable with concepts like continuity and convergence. There are problems from probability and statistics, as well as some numerical analysis. You’ll even find some problems that require no more than basic geometry … though don’t expect the problems to be easy!
The way the test is scored has changed as of 2017. In previous years, each question answered correctly would be worth one point, each question left blank would earn zero points, and each question answered incorrectly would subtract 1/4 of a point. This is no longer the case! Now, for every question you leave blank or get incorrect, you earn zero points — there are no more penalties for incorrect answers. Therefore, it is in your best interest to put an answer for every single question, even if that answer is a guess.
Much like the SAT, the score you receive is not just the number of points you earn (your raw score), but a scaled score between 200 and 990. The actual possible range of scores varies each time the test is given, and the raw scores that correspond to each scaled score change, even between different forms given on the same day. This is partially to account for slight differences in the difficulties of the questions. For example, on a harder version, a student with a raw score of 50 might yield a scaled score of 800, while a student with an easier version given the same day might have to get a raw score of 52 to obtain that same scaled score of 800. What really matters, though, is your percentile score — if you’re in the 80th percentile, for example, then you scored better than 80 percent of other students.
Here’s my top tips for doing well on the GRE Math Subject Test:
1. Have your basics down.
While the GRE covers a considerable breadth of topics, the vast majority of the test just requires you to be comfortable with the big ideas, important theorems, and common formulas in undergraduate mathematics. Know the basics of calculus like the back of your hand (since it makes up half the test!), study the essential theorems and definitions from linear and abstract algebra, and review the most important results in analysis and topology, and you’ll be fine. That doesn’t mean that the more minor topics won’t make an appearance, of course — you never know what’ll show up on test day — but even those tend not to stray too far from the main body of knowledge.
2. Know lots of examples and counterexamples.
Plenty of questions on the GRE require that you know when certain theorems do and don’t apply and when certain properties are and aren’t always true. Because of this, it’s useful to have an arsenal of instructive examples and counterexamples that you can pull out when evaluating the truth of a statement, especially in the I/II/III types of questions. Don’t be afraid to try so-called “trivial” examples either — for instance, the identity matrix and the zero matrix can be quite useful for linear algebra questions!
3. Use process of elimination.
This one seems like a no-brainer, since you’ve likely been using it since grade school, but the tried-and-true strategy of process of elimination is one of the most useful techniques for doing well on the GRE. This ties nicely in to the previous tip as well — the right example or counterexample can sometimes knock out four out of the five answer choices! Remember, you don’t have to prove anything on the GRE — you just need to be able to get the right answer.
4. Look for shortcuts.
While few questions on the GRE flat-out require a shortcut to be able to solve, some questions can be answered more quickly with a bit of cleverness. One common shortcut is to substitute simple values into an expression to eliminate some answers — don’t forget that variables can stand for numbers! Or, if you can understand a key fact about what type of answer you should be getting — say, a real number — then you can often eliminate choices that don’t fit that form. If you find yourself doing a long and arduous computation, 9 out of 10 times there’s a simpler way to do the problem.
5. Visualize it.
It’s easy to get caught up in the symbolic nature of mathematics, but geometric intuition is one of the strongest tools you have at your disposal on the Math Subject Test. Sometimes an integral that would be annoying to calculate via antiderivatives can be made much easier by simply finding thea area of a region. Even on the toughest, most abstract questions, a quick sketch can go a long way in eliminating incorrect answer choices and may even lead you to the insight you need to pick between two equally attractive options.
6. Skip around.
Despite the sequential numbering of the questions, you’re under no obligation to actually do them in that order! I’ve found that the best strategy for me is to start off answering the questions in order, but if I find that a particular calculation is taking too long or isn’t coming out quite right, I mark it with a star in the test booklet and go on to the next problem. Getting the questions you know how to do out of the way first means that you can spend your precious time on the tougher questions later, and sometimes a break from a problem is what you need to be able to freshly tackle it after more time has passed. And sometimes, the last question can even be downright easy!
7. If nothing else works, guess.
This would have been terrible advice until very recently, but now that the ETS has removed the guessing penalty, there is no reason you should leave even a single answer blank at the end of the test. That being said, you should at least attempt to narrow down the choices for every question using process of elimination so that you can maximize your chances of getting the correct answer.