Test Material

According to the official website from the ETS, the subject test content is divided as follows:
  • 50% Calculus,
  • 25% Algebra,
  • 25% Additional Topics.
I took the test twice, and have seen copies of four additional tests. (I linked these four tests in the web resources section.) Based upon those six tests, I would say that the test material has the following breakdown:
  • Basic Function Theory: 15-20 Questions. This is essentially high school material. Do you remember the quadratic formula? How about those annoying trig identities? Can you solve or approximate the solutions to some basic functional equations (e.g. can you ballpark the two real solutions to e^x = x^2 + x + 1?).
         Aside from the trig identities (which I still don't remember), you probably will not need to review any of this              material.
  • Computational One Variable Calculus: 15-20 Questions. This is mostly AB/BC Calc material. Do you remember the derivatives and anti-derivatives of rational functions, trig functions and their inverses, exponetials and logs? Do you remember those falling ladder (related rates) problems? Do you remember the error bound for Taylor series?

         For me, the biggest surprise here was the emphasis upon approximation. Remembering all of the differential              methods for ballparking figures (e.g. estimate 5^(2.001) - 25) will be important. I'd recommend going through           some review books for AB/BC calculus. Warning: When reviewing this material, computational speed is key. I          made the mistake of reading through things, thinking "Yeah, yeah. I remember how to do that," and never                  bothering to actually perform the relevant computations. But as I mentioned in the general section, speed is              everything. Sure, given 2 minutes you could figure out the derivative of sin^{-1}(x). But you will need to know          the answer immediately; 2 minutes is far too long to spend on such a thing.
  • Linear Algebra: 10-15 Questions. Can you solve a system of linear equations? More importantly, can you do quickly? (Time yourself: How long does it take you to solve 3x + 4y + 5z = 1, 2x + z = 2, 6x + 7y + 3z = 8. If it takes more than 2 minutes and 35 seconds, then you are falling behind: You will need to do another question in less time in order to complete the exam.) Can you quickly compute determinants? Can you quickly compute characteristic polynomials? Can you quickly compute ranks? Given a set of five vectors in R^3, can you quickly determine which subsets are linearly dependent?

         I think that, much like the test as a whole, the ability to quickly compute is key for this portion of the exam.              I personally found the linear algebra questions to be the most computationally intensive: On both of the tests I          took, there were questions that required you to take the determinant of a 4x4 matrix as an intermediate step.             (Granted, the matrices involved had entries in [0,10); but they were not particularly sparse.) I recommend                performing many computations while practicing for this part of the exam. Be able to quickly multiply matrices,          take their determinants, perform Gaussian elimination, do Gram-Schmidt, etc.
  • Multi-Variable Calculus: 3-5 Questions. The test is very predictable in this part. There will almost certainly be exactly one Lagrange optimization problem. There will be one or two basic problems involving easy integrals or partial derivatives. There will almost certainly be exactly one line integral problem; it will almost certainly be an application of the multivariable fundamental theorem. There will almost certainly be exactly one surface integral problem; it will almost certainly be an application of Smith's theorem.

         I would not spend much time studying for this portion of the exam, since the material is predictable and                      constitutes a very small portion of the test. Make sure that you remember how to evaluate basic integrals and            apply the results mentioned above; then you should be fine.
  • Counting and Probability: 3-5 Questions. There are usually a couple of questions to check whether you know some definitions from probability theory (I didn't). Can you define variance? Standard deviation? Can you compute these things in a reasonable amount of time?

         There are also a few counting/discrete probability questions. For example, you might have to find a closed                   formula for some awful sum involving powers of binomial coefficients. It is likely that you will also have to                  explicitly calculate (or estimate) some percentages in discrete problems. (E.g. I flip 100 coins. Approximately            what is the probability that more than 70 of them turned up heads?)
  • Complex Analysis: 1-2 Questions. Almost without fail, there is an integral or two that you should solve by using contours. If you remember Cauchy's theorem and the residue calculus, then you should be fine.
  • Other Material: 4-6 Questions. There is almost always a question from general topology (e.g. Are there nontrivial clopen sets in a Hausdorff space?). Usually, though, the relevant definition is provided and the question boils down to some logical pencil pushing. There is often a question or two about real analysis in a single variable (e.g. here are some random facts about a differentiable function. Must it be bounded?) There are usually two or three questions about basic number theory / group theory. How many Abelian groups of order 156 are there? What is the greatest common divisor of 1237 and 5329? Sometimes you have to trace your way through some pseudocode for an inane algorithm. You may have to be able figure out which cut-and-paste diagram yields a torus.

        Again, it is probably not worth expending much time preparing for this part of the exam. The material is not             very predictable, and there few points to be gained. If you know the answer, great; but if you don't, there is no             need to worry about it. Move on to a different one.
Overall, let me reiterate my main advice: The subject test is a "compute first, think later" kind of exam. Virtually no pure math is tested. You can forget about homological algebra, you don't need any algebraic topology, you can set algebraic geometry aside, you can ignore Galois, you can eschew Lebesgue, you can avoid commutative algebra, you can sleep in your representation theory class, you can stop studying Hilbert spaces, etc. Even when it comes to the tested material, theory takes second place. You understand the general (differential forms) version of Stokes' theorem; great, but you won't need it, nor will you need to prove the mean value theorem, nor will you need to understand why the rank-nullity theorem holds.

But you will need to be able to compute, and compute quickly. This is why it is crucial that when studying for this exam, you actually force yourself to go through the computational hell necessary to solve elementary problems. Don't just trust that you could compute the inverse of that matrix; do it, and time yourself so that you know how quickly you can perform the computation. Above all else, it is important to be able to quickly and accurately perform the calculations that things like Wolfram Alpha, MatLab, Mathematica, etc. normally save you from doing.