Tips on Studying for Math Comps

Here are suggestions for how to study for comps. First is some advice from the faculty. Then come suggestions written by Grant Ingersoll and Conny Morrison, math majors who graduated in 1996 and 2012 respectively. (Grant's and Conny’s comments have been updated to take into account changes in the comps made since they graduated).

General advice from the Math Faculty: 

  1. Many weeks before the exam, assemble your study materials:
    1. (From the Comprehensives links on this website): Download the Syllabi, Study Guides, a bunch of old Comps exams, and some solutions to old Comps exams, so you have them on your computer, ready to go.
    2. Find a friend or two or three who are also taking the Comps, and form a study group with them.
    3. Dust off your notebooks from the courses in question: Multivariable Calculus and Linear Algebra. (Or if you are taking the Junior Honors Qual, then also your choice of Algebra or Analysis.) If you still have the textbooks, get those out, too.
  2. Two or three months before the exam, go over the Comps syllabi for those three courses. You do not need to study everything in all three courses; the Comps syllabi cover only portions of those courses. Plan a calendar of when you will study all the various topics. Coordinate with your study group and share the work if it helps.
  3. Focus your studying on two things: the discussions of all the relevant topics in the Study Guides, and doing lots of practice problems from old Comps exams. (Also use your old notebooks and textbooks for supplemental support and reminders, but use the Study Guides to, well, guide your study.) In particular:
    1. Read the Study Guides with pencil or pen in hand, taking your own notes on the definitions and theorems, as well as working through the sample problems found there. Also flag any points you don’t completely understand, and ask Math Faculty or QCenter staff about them.
    2. Set aside two full old Comps exams for practicing later under timed conditions, but then attempt as many old Comps problems as you can over the course of the weeks you study. (And check the solutions after you finish, of course.) You will notice a lot of repeated themes, and practice will make you faster, more accurate, and more fluid in solving the kinds of problems that show up on the Comps.
  4. Ideally, start studying halfway through the semester before the exam (i.e., late October for most students), spending an hour a week reviewing Multi and Linear, preferably with your study group. (And for Junior Honors Quals, also a bit of Algebra or Analysis.) In the three weeks before the exam (i.e., January for most students), study all the topics more intensely, starting from an hour a day and ramping up to at least two hours a day.
    Of course, adjust this schedule based on your other commitments. For example, if you have a job or coursework or other obligations in January, spread those three weeks of more focused studying over six or eight weeks. In other words, plan a realistic study schedule over the weeks and months leading up to Comps based on your own personal situation.
  5. As you study, besides working with your study group (or any other fellow Comps-takers), get help whenever you have questions. Math faculty are happy to answer questions ranging from general theory to specific questions about individual problems. So are the QCenter staff, who provide lots of support for Comps studying in January, including study and practice exam sessions.
  6. In the final week leading up to the exam, take an old Comps exam under test conditions: two hours, closed book; or for Junior Honors Quals, three hours, closed book. (Remember the recommendation above, to set aside a couple old Comps exams for later.) If time permits, do this a second time with another old Comps exam.
  7. Get a good night’s sleep before the exam. During the exam itself, remember that all of the Math faculty are cheering for you. Comps are challenging, but if you’ve taken the time to prepare as described above, you will be well positioned to succeed.

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. 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.
  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.

Good Luck.

* The comps syllabi have been reduced over the years, and there are some problems on old exams (especially those from 2012 or before) that are no longer on the syllabus. 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. Some other types of problems even from those subjects are no longer covered, so 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. 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 concepts, techniques, and 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.
  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.