Official Practice Tests
When it comes to preparing for the Math Subject Test, you can’t get better than the practice tests put out by the ETS themselves! These are actual tests given in previous years, and are the best reflection of the way that the questions tend to be asked.
Not all practice tests are created equal, though: In October 2001, the ETS noticed that a large proportion of test takers were receiving perfect scores, so they amped up the difficulty by adding more conceptual questions and diving further into the more advanced topics such as abstract algebra and real analysis. As a result, the tests from before this time are not quite representative of the level of difficulty you can expect from the test nowadays. That being said, they’re still excellent practice for getting your basic skills down pat, so that you have enough time to think through the harder questions.
The newest test is hosted on ETS’s website https://www.ets.org/. A big thanks to Ian Coley at Rutgers for hosting the other practice tests!
One more source of official practice material is the “Descriptive Booklet (97-99)” released by the ETS, containing a 35-question miniature practice test.
(I’ve decided to label this practice set as “GRDB67”.)
The scaled scores and percentile scores found at the back of these tests are somewhat outdated and generally do not reflect your actual performance. Therefore, I have compiled an unofficial approximate conversion chart that you can use to convert your raw scores on each of these tests to an approximate scaled score and percentile, based on recent data (as of 2019) and my own estimations.
Click here for my hints to all of the above materials!
(Why hints? Click to find out.)
Solutions to the practice tests can be found at the following pages:
Ian Coley‘s solutions
GR8767 | GR9367 | GR9768 | GR0568 | GR1768
Brian Sittinger‘s solutions
GR8767 | GR9367 | GR9768 | GR0568 | GR1768
Charles Rambo‘s solutions
Practice Test (NEW!!)
That’s right — a brand-new practice test! I’ve put a good bit of effort into making sure that this test is as representative as possible of the actual difficulty level of recent subject tests. There are topics here that you won’t find on any of the official practice tests, but that are more than fair game for the real thing!
Scratch paper version | No scratch paper version | Hints (coming soon!) | Solutions (free!)
Even with the topics listed on the official test guide, it can be hard to know what to study! I’ve put together an unofficial list of all the topics I think could conceivably show up on test (without giving away any actual questions of course). It’s a bit overkill, but I think that’s a good thing!
The scaled scores and percentile scores found at the back of these tests are somewhat outdated and generally do not reflect your actual performance. Therefore, I have compiled an unofficial approximate conversion that you can use to convert your raw scores on each of these tests to an approximate scaled score and percentile, based on recent data (as of 2019) and my own estimations.
More resources are in development for the future, so stay tuned!
Test Prep Books
There are a few books available to prepare for the GRE Math Subject Test. Here is my ranking of these books from most to least helpful.
Cracking the GRE Mathematics Subject Test (4th Edition)
Author: Steven LeDuc
If you’re going to spend your money on one resource, this one should be it. The Princeton Review book gives the most comprehensive review available of the majority of topics covered by the GRE test. Each chapter gives an in-depth summary of the various topics, formulas, and theorems, with plenty of examples and exercises sprinkled throughout. While these exercises are not GRE-style multiple choice problems, they are nevertheless worth working through as a refresher. At the end of each chapter is a set of true GRE-style review questions, for which detailed solutions can be found at the back of the book. Finally, there’s a full practice test at the end of the book, as well as solutions. Throughout the book, the GRE-style questions tend to stay true to the “spirit” of the test, containing both relatively routine problems and tougher abstract questions.
The only slight gripe I have with this book is that there are a few topics that are underrepresented in the book. Most notably, there is absolutely nothing on ideals of rings or uniform continuity/convergence, which I have seen show up on a number of Math Subject Tests, and there are a few minor topics are left out of other sections such as multivariable calculus and linear algebra. That being said, as long as you thoroughly review ideals and uniform continuity/convergence and refresh on the other topics mentioned on your own, you should hit all of the main concepts you’ll need to know for the test.
You can buy this book on Amazon, as well as find it at many retailers. Definitely worth it!
Practice for the GRE Math Subject Test: One Practice Test and Solutions (1st Edition)
Author: Charles Rambo
Charles Rambo has created a full-length practice test of excellent quality. You can actually get the practice test and answers for free on his website, but purchasing the book (on Amazon) will get you full solutions to each problem. The questions are very much in line with the topics, difficulty, and overall “feel” of the GRE — it was clearly inspired by the GR0568 and GR1768 practice tests.
There are a couple of instances where I feel like the solution presented would be too time-consuming during the GRE, and one instance of an early question (#14) that I think turned out to be much harder than he realized (and the solution given doesn’t really do much to demystify it). Even with these minor flaws, though, it’s great to have an extra practice test that so closely matches the real thing.
SubjectMath.com Practice Tests (2017 Edition)
Author: Gilad Pagi
SubjectMath.com is a site that offers a complete course in GRE Math Subject Test Preparation. The owner of the site, Gilad Pagi, has also created two practice tests that are for sale on Amazon. Each practice test is formatted to be very similar to the actual test, including the amount of area usually reserved for scratch work, and is then followed by detailed solutions to each question. While these books certainly aren’t the worst I’ve seen (that honor goes to REA!), they do unfortunately suffer from their share of flaws.
The main issue with these materials is in the level of difficulty. Generally, on the actual GRE, the questions start off relatively easy and on average get progressively harder as the test goes on. The SubjectMath.com tests, on the other hand, just go from hard to harder. Most of the questions seem deliberately designed to be intensely laborious to calculate unless one notices a clever “trick” (which the solution guide often mentions is covered in the online courses). While it’s certainly useful to have some tricks up your sleeve, in reality you don’t need one for every problem. My biggest issue in particular is how often questions have “(E) None of the above” as an answer choice. The actual GRE has almost never used this as an answer choice since at least form GR9768 (except sometimes on the I/II/III questions), and by including this as an answer choice, it becomes much harder to use process of elimination — arguably one of the most vital test strategies! — to arrive at a correct answer. One last minor issue is that the questions can sometimes be needlessly verbose, and often there’s gratuitous notation and obfuscated expressions thrown in that just make the questions artificially harder.
SubjectMath’s practice tests do have their value though. They may not be the best books for the average test-taker, but if you’re looking to do a sort of “strength training” to raise your already-good score a few percentile points by doing more challenging problems, these practice tests may be useful to you.
GRE Mathematics (GRE Test Preparation) (5th Edition)
Authors: O. P. Agrawal, T. Elsner, J. Robertson, J. W. Samples, J. T. Wilson
This book seems at first glance it would be a Math Subject Test review powerhouse — after all, it boasts six whole practice tests. But don’t be fooled: this book is not worth the money. Like the Princeton Review book, the first half of the REA book contains a review of topics, formulas, and theorems. However, the depth of presentation is all over the place, with some topics barely touched on at all and others going entirely too far. The Numerical Analysis section in particular goes into an excruciating amount of depth — 30 pages worth of Lagrange interpolation, Bernstein polynomials, and splines! Imagine my own reaction when I dutifully reviewed all of these topics, just to find that absolutely none of these topics were on a single GRE I ever took.
Then you get to the six practice tests. From the get-go, you encounter topics that not only are never seen on the Math Subject Test, but weren’t even covered in the preliminary review. I can say with 99.9% confidence that you’ll never see fringe topics like the height of a polynomial, the minimax solution to a matrix game, or the Laplace transform of a function show up on an actual Math Subject Test. There are still plenty of questions that do stick to the GRE topics, but these also vary greatly in quality — some are right on point, whereas others are either entirely too easy or way too hard. Add to that the numerous errors and atrocious mathematical typesetting throughout the book, and you’ll find that you’d spend more time trying to decipher this book’s contents than preparing for the test.
In sum: don’t waste your money.
To get an even more thorough review of specific GRE Math Subject Test topics, it’s often helpful to go back to the textbooks themselves for loads of practice problems. While everyone has their own favorite textbooks*, what follows are my suggestions for which textbooks give the best review for the GRE.
I have not included any particular “editions” for these — any version you can get your hands on will do. (If you can borrow one from a friend, even better — textbooks are expensive!)
Calculus (Early Transcendentals) by James Stewart
Not only does Stewart’s powerhouse of a calculus book cover both single-variable and multivariable/vector calculus, but it has appendices that provide just enough precalculus review and even a chapter on second-order differential equations. The last number of exercises for each section also provide some meaty problems that can develop the skills and intuition needed to solve the toughest calculus problems on the GRE. Considering that calculus makes up a whopping halfof the test, if you want only one textbook from which to pull lots of exercise, this is the one you want.
Differential Equations and Linear Algebra by Stephen W. Goode and Scott A. Annin
Why get separate textbooks for Differential Equations and Linear Algebra when you can have two in one? Goode and Annin beautifully link the two subjects (which too often are studied separately) together, while providing plentiful practice problems in each subject individually. One thing I particularly like is the set of True/False questions at the end of each section — essential practice for the I/II/III questions on the GRE.
Contemporary Abstract Algebra by Joseph A. Gallian
Gallian’s text has plentiful examples and diagrams throughout the text to help build intuition for the various algebraic structures and their properties. These examples are well organized, often being summarized in tables for quick reference. The exercises are just as numerous, but most strikingly, there are answers at the back of the book for the odd exercises, even including sketches and hints for the proofs — somewhat of a rare occurrence for a proof-based book, but undoubtedly useful!
Introduction to Topology: Pure and Applied by Colin Adams and Robert Franzosa
Not only do Adams and Franzosa give clear presentations and explanations of the definitions in point-set topology, but they provide plenty of illustrated geometric examples from both familiar and unfamiliar spaces, more than I’ve seen in many other books on the subject. The first half of the book (through Chapter 7) is really all you need to be ready for the Math Subject Test, and while the exercises are more on the proof-heavy side, they’re certainly doable.
Elementary Analysis, The Theory of Calculus by Kenneth A. Ross
Finding a good real analysis textbook for GRE review was one of the toughest parts of writing this list. There are a few things that make this book stand out to me. First of all, its exposition is clear and even conversational compared to some of the more terse traditional analysis textbooks. Second of all, it has an abundance of examples, both in the sections themselves and the exercises — exactly the sorts of examples you want to have on quick recall for the GRE. But the best part is that in the exercises, rather than every single question being a proof, there are quite a few computations, where all you need to do is take your definitions and see how they apply to actual functions you can sink your teeth into, so that you can really get a feel for them.
A First Course in Complex Analysis with Applications by Dennis G. Zill and Patrick D. Shanahan
Zill and Shanahan’s introduction to complex analysis was born from a much larger engineering textbook on which Zill was also an author. As a result, their text feels less like an analysis book and more like a calculus book, just where the complex numbers are now the object of study. There is no shortage of problems, from routine exercises (much like the ones you’d see as actual GRE problems) to deeper explorations to really make sure you understand the concepts. Answers to most odd problems are provided at the end.
Discrete Mathematics and Its Applications by Kenneth H. Rosen
This book is another powerhouse. In just one book, Rosen covers such topics as logic, set theory, algorithms, number theory, combinatorics, discrete probability, and graph theory — all of which are likely to show up throughout the Math Subject Test. Numerous examples, tables, and exercises round out each section. There’s even an appendix dedicated to understanding pseudocode algorithms. Definitely one of the best textbooks for preparation!
Probability and Statistical Inference by Robert V. Hogg, Elliot A. Tanis, and Dale L. Zimmerman
This book has everything you could ask for to prepare for probability & statistics: thorough explanations, plenty of examples, and a ton of exercises. Like the Topology book above, you don’t really need everything from this book, so take things like moment generating functions with a grain of salt. A good rule of thumb to use when studying might be that the longer a formula or derivation is, the less likely it’s something you’d need to be able to reproduce on the test. Focus on the more basic formulas and the concepts, and this book will be your best friend.
These books are excellent sources of problems in many of the GRE topics (with worked solutions!), containing both routine computations and proofs of important concepts, including some that may be left out of the above books. The ones I have found are most helpful are Advanced Calculus, Complex Variables, Discrete Mathematics, General Topology, and Linear Algebra.
* If you’re used to textbooks of a more traditional and theoretical nature, you may find some of my choices of textbooks surprising. Where’s Spivak? Apostol? Dummit & Foote? Rudin? While these authors’ books may be good for a rigorous, proof-based grounding in mathematics, the GRE is not about proofs. It’s about calculation, examples, and good intuition — you either know it or you don’t. The books I’ve selected above may not always be the “proofiest”, but they have clear explanations, abundant examples and visualizations, and loads of relevant practice.
Here are some other websites that may prove helpful in reviewing for the GRE Math Subject Test.
This forum is an excellent source of advice about not only preparing for the GRE but also navigating the treacherous waters of applying to graduate school in mathematics. If you have questions about preparing your application, getting letters of recommendation, deciding between a master’s or a PhD, or anything in between, this is the place to ask.
This page contains lecture notes by Dr. Padraic Bartlett at the University of California, Santa Barbara, for a Math 94: GRE Seminar class. The page contains notes on many of the GRE topics, as well as a slightly shorter practice test with original practice problems.
Charles Rambo is a math tutor in North San Diego County. On his GRE page, you can find his solutions to GRE Math Subject Tests, as well as a full-length practice test with original practice problems. His site also hosts the UCLA GRE Bootcamp Problems that used to be on Dr. Charlie Marshak’s page (which is now unfortunately defunct).