Studying for the Exam
When preparing for an exam, a common question is what steps to take. In my opinion, the following is a good progression that will maximize understanding. You may not have time for everything on this list, but try to do what you can.
- Re-read the notes and work through the proofs yourself. Understand each step and maybe write a note as to wha each step of the proof means and why it's there.
- Look over the discussion and homework solutions. This will give you a reminder as to what applications and extensions you learned about.
- Go over the morals and your notes to improve your intuitive understanding of the material.
- Practice exam problems. This is definitely the most important element in the sequence of preparation. The more ideas you get from solving problems, the more ideas you will have on the exam.
- Once you've practiced a healthy variety of problems, think to yourself how changing small details will change the solution. This is how you'll truly understand the point of the exam problems. You'll know the material requisite to solve it, but the little changes will require the type of creativity that will go a long way on the real thing.
General Problem-Solving Tips
This is just a collection of tips and tricks I have compiled after looking at a variety of practice exams. I hope it will serve you well.
- Write down what you know. If you're stuck and can't progress, writing down ideas will either give you inspiration or get you partial credit (sometimes both).
- Exam problems will sometimes have intractable numbers. If not computationally difficult, the problems will be tough to imagine. In this case, try smaller examples and look for patterns.
- Similar to the previous point, try testing out hypotheses and look for counterexamples in order to understand a claim. You want to convince yourself of a fact before you prove it.
- Write out scratch work when working on a proof and once you have everything, work backwards to convey your thoughts cleanly.
- Don't spend too much time staring at a problem - get warmed up with other questions you can solve first and when your gears are moving, come back to the problem where you got stuck.
- Keep track of the clock. This is very important. You want to make sure you're spending your time wisely and not focusing too much on a small subset of the exam.
- When solving a problem about a concept, in addition to looking up relevant facts in your cheat sheet, remind yourself of the intuition behind the problem and similar problems you've solved before. This will give you the best idea of how to approach the problem.
- If you're unsure about what a problem is saying, ask a TA to clarify. You can't ask TAs to give you hints or any content-based reminders, however.
Before and During the Exam
Your mental preparation for the exam is just as important as material preparation (okay maybe not as important, but it's pretty close). If possible, try to end your preparation early on the day before the exam. I don't think I need to say this, but math can be tiring. You want to approach the exam with a fresh mind. Get a good night's rest (if only you had a dollar for every time you heard that before an exam). This is especially important before an exam for a class like this. If you wake up early as well then you will feel less tired during the test.
As the exam time approaches, the important thing is to remain calm. Not many people are expecting this exam to be easy, and that's alright. When in doubt, try to remember a similar problem that you've worked on before. If you are just completely stuck on a question, come back to it later, don't spend time worrying about it. The last thing you want to worry about while working on this exam is your performance. I understand that the academic climate pressures you to get the best score you can, but, at least in my experience, this is counterproductive when you're doing math. Just do the best you can and remain calm. At the end of the day, you are much more than a test score and I am already very proud of y'all. Good luck :)
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