Here are suggestions for how to study for comps written by Grant Ingersoll and Conny Morrison, math majors who graduated in 1996 and 2012 respectively. (Grant's comments have been updated to take into account changes in the comps made since he graduated).
Grant Ingersoll, '96: As someone who has taken the math comps twice, I recommend the following study habits:
1. Comps in the math department are not to be taken lightly. My first time I didn't take them as seriously as I should have and I paid for it. I read through the books and did a few problems and just figured it would come back to me when it came time to take the test.
2. Get the old tests and do every single problem*. This is more important than passively reading over the theory in your notes or books. You are bound to get one or two more problems right simply because the department recycles old tests (they may change some of the numbers, but for the most part they are the same). Furthermore, they ask the same types of questions. I can almost guarantee that the multivariable calculus section will include a problem asking you to compute partial derivatives from the definition. Doing the old tests will tell you what the department expects you to know.
3. My basic plan of attack the second time around was to do a review of all of the theory about 3 weeks prior to the exam. This took about one week. I then did problems off of old comps for the remaining two weeks, concentrating mostly in Algebra, which was my chosen elective, but I also did plenty from the core part of the exam.
4. Don't be afraid to ask for help from the department. Go see the professors you have had the most experience with, I am sure they will help.
5. I can't say I recommend huge study groups, but one other person seemed to be a perfect fit for me. Larger groups tend to overwhelm as everyone just seems to shout out problems they are having, completely disregarding the fact that you may not be studying what they are at a given moment and thus disrupting your train of thoughts.
6. Don't stress out completely over the exam. It's hard, but it is nowhere near impossible. However, don't shoot to just pass (as I did the first time around). Aim to high pass or better.
7. Finally, try to high pass your junior year, even if you don't intend on writing a thesis. It will be a load off your mind come senior year and you will be glad you did it. Another possibility is to take the exam for practice as a junior. This will give you a good sense of what the exam is like and how much preparation is needed.
* Note that up until March 2012, the core part of the math comps covered four courses, two of which (Introductory Calculus and Intermediate Calculus) are no longer on the exam. So when looking at the core part of an old comp from March 2012 or earlier, you need only do the problems from Multivariable Calculus and Linear Algebra. We have also made some small changes to the syllabi for these courses as well as the syllabi for Algebra and Analysis. We urge you to read all of the syllabi carefully when preparing for the comps.
Conny Morrison, '12: Here are my additions to Grant's advice:
1. Highlight, star and tab pages from class notes when your Professor says a problem will be on comps. It was really helpful to be able to easily find these problems and refresh myself on how to do them without having to spend a lot of time paging through my notebooks. Your Prof will have taught you exactly how to do them correctly and it's much faster to refresh yourself of their method than it is to have to come up with your own again.
[Note: this advice assumes that you have kept your notes from previous classes. Keep your notes!!]
2. Definitely take the exam for practice (or for real) your junior year. Then follow this up by meeting with a professor to go over what you missed so that you are already reviewing and thinking about the areas in which you are weakest. Try to grade for a class the semester before so you are re-exposing yourself to the concepts.
3. Come back to campus for all of interterm before you take comps. It is crucial to be able to bounce ideas off of other students (like Grant, I recommend one-on-one sessions rather than large groups) and attend office hours. In addition to preparing you to pass comps, comps studying is an amazing way to re-build relationships with professors you had early on in college so take advantage of how accessible most profs make themselves during interterm. Also, their offices get busier as comps nears so the earlier you are there asking questions, the more one-on-one attention you’ll get.
4. Spend more time studying the elective portion than the multivariable/linear alg part. Do all the relevant problems from old exams and make sure you really understand them. It's worth going back to a professor's office to ask them to re-explain the proof you went over 3 days ago so that you really understand how to do it. Also make sure you adequately review theorems that apply to the material listed on the review guide -- you need to have them fresh in your mind in order to be able to recall and apply them during comps. Even though you probably took this class most recently, you will be amazed at how much you need to refresh yourself to actually do well.
5. Although it’s frustrating to receive a Conditional Pass, don’t freak out. If you work out as many of the problems as you can on your own and then review them with a professor that will be there for the oral portion, you'll be fine. If you make sure the first few are done very clearly, this will help you figure out the last few, as the problems are progressively more difficult. Go over them with a professor as many times as it takes you to really fully understand the problems. You will be asked to prove a problem similar to those already given to you, as well as to show some of the proofs of the problems you had time to prepare, so review the proofs on your problem sheet before the oral portion and make sure you can apply these rules to similar proofs. The Conditional Pass is a great opportunity to dedicate concentrated time to absorbing material that you probably breezed over much more quickly in class.