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How to Study and Prepare for a Math Test

As a math teacher, one of the questions I get asked the most often is, "What do I need to do to study for this test?" Or, unfortunately, often after someone doesn't do so well on one test, they will ask me, "What can I do to do better on the next test? I really studied for this test and it didn't seem to do much good."

Well, I have some bad news and some good news. The bad news is that there is no "magic bullet" that guarantees that you will do well on any test. There is no recipe that I can give you that guarantees you will ace your next math (or English or history or...) test. However, there are some things you can do to prepare for a test and some tricks you can use when you are taking a test that can greatly improve how well you do on it.

First things first

I don't imagine it will come as a big surprise to anyone that the most important step to doing well on a test is to know the material covered on the test really, really well. This is not something you can take care of the last week (or day or hour) before the test. This is something you have to do all along, throughout the semester. So, for a math course, I suggest you do the following things throughout the semester:

• Do your homework regularly - This is probably the biggest reason people don't do well on tests: they don't keep up with their homework regularly. I know, I know, you are taking 37 credit hours and working 86 hours a week and you have a family (with 2.3 kids, half of whom have been sick all week) and none of your teachers seem to understand that you have other things to do in your life besides their class. This may all be true. Unfortunately, it just doesn't matter. Teachers don't assign homework to make you miserable (well, most of us don't); we assign you homework because we know that the only way to really learn math is to actually do it and practice it and spend time on it. That's just the way it is. If you are just too overloaded to possibly keep up, then you really should consider cutting back somewhere (i.e., drop a class or see if you can reduce your hours at work, for example).
• Keep up with the schedule - If you are in a classroom course, it is essential that you keep up with the schedule so that the lectures will actually do you some good (once you get more than a class or two behind, the lectures really aren't likely to make lots of sense to you anymore). In any kind of course, however, you have to keep caught up with the class schedule because otherwise you won't be ready for the tests in time. Again, you may have the best of reasons for being behind, but when it comes time for the test, they just don't matter. I'm not trying to sound mean or "hard-nosed" about this; it's just that math courses are designed so that when you finish them, you have a certain level of knowledge (enough to take the next course), so things have to be done at the right time in order to get everything in.
• Keep your head when you do get behind - Well, as we all know, despite the best of intentions, all of us get behind at times. The important thing is how you deal with this. Some people (even some very dilligent, hard working people) take the approach that they will just keep working through the homework until they get caught up. If you don't get too much behind and you have enough time for this, then this is a great approach. However, if you get more than a little behind or your time is very limited, you should probably resort to what I call the "slash and burn" method for homework:
  
  The slash and burn method for homework - What I mean by this is that if you get more than a little behind or you know you won't have time to do all the homework assigned, it is always better to be caught up than to be complete. So, if you run into this situation, you need to be sure that you do some of every type of problem in the homework; if they have different instructions, you should probably do a few from each. Don't pick all the easy ones (or all the hard ones), but a few easy ones, a few medium ones, and a few of the harder ones. The exact number you need to choose depends on how far behind you are and how much time you have. You can always go back and pick up the others later. This way, though, you are sure you get some of everything, so you don't get to the test and discover some problems on it that you never got around to covering beforehand. Are you going to lose points on your homework grade? YES, but not anywhere near as much as you would miss on the test if you hadn't covered something. Is this just as good as working all the homework? Nope, but it beats getting further and further behind and having to drop the course. Can you use this method for a long period of time (like an entire class)? Probably not. (I call this the "slash and burn" method because it is sort of like slash and burn agriculture, not a sustainable method of raising crops...)
• Get help as soon as you start running into problems - If you start having problems with the work or you are having problems understanding things, go get help right away. Go see the tutors, get together in study groups with your classmates, go to your teacher's office hours, look for help on the internet, but do something right away. If you do this right away, you probably won't get behind as a result of it. If you just skip it and go on, it could keep you from understanding later material (which means you will get behind anyway). When you do go for help, go prepared. Be sure you have actually tried to understand and work the problem beforehand; if you don't show that you have made an effort to figure things out first, don't expect the tutors (or your teacher or other students) to be very enthusiastic about helping you. Especially since most of these methods to get help don't cost anything, you should take advantage of them, but respect the time of these people.
• Don't abuse the solutions manual (or the tutors) - No, I'm not talking about violence here. (Well, okay, I don't recommend violence either...) One of the biggest problems I find many students (even good, hard working students) have is that they think that, because they can understand how the solutions manual worked a problem or how a tutor explained a problem that this means that they really understand how to do a problem. This just isn't true. You don't really understand how to work a problem until you can work it on your own without looking back to see how the solutions manual did it or without looking back at your last problem to see what you did or without getting help from the tutors. If you get help on one problem (which is perfectly normal and nothing to be ashamed of), you need to turn around and work several more problems like it without that help. In fact, if you really want to be sure you know it, you should try to work a similar problem the next day without any help. If you can do that, then you probably do understand how to work it.
• Study groups are a Good Thing - It is useful to connect up with a study group right before a test, but it helps even more if you can get tied in with a group of people who get together to study and help each other with problems. Even people who already know the material can benefit from study groups; you really learn things even better when you have to explain them to others.
• A little review can go a long way if you spread it out - I realize that the main time most of us think about reveiwing the material we have already covered is right before a test. However, if you can manage to do a few review problems (even 2 or 3) each day whenever you do your new homework, you will be surprised how much better you retain things (and how much easier your official "test review" will be).

The test starts to loom on the horizon

Well, okay, so you have been a Good Person and done the things I suggest above. However, a test is now coming up shortly. What additional things should/could you do to prepare for it?

• Don't wait until the last minute to prepare - Personally, I suggest you start seriously preparing for the test about a week in advance (longer if it is a midterm or a final exam). This gives you time to prepare and, if necessary, to get some extra help on things you still don't quite have a grip on. If you wait until a few days before, you probably won't have time to review everything and you certainly won't have time to learn things you didn't get the first time through.
• Make yourself a list of the types of problems you need to review - This helps you to organize your thoughts and make yourself a checklist to make sure you have reviewed everything. You can usually do this by skimming through your textbook, your homework, and your class notes (for classroom courses). Try to decide what were the most important types of problems you covered. (Hint: this doesn't mean the ones you had the most trouble with, particularly. Look for what types of problems were stressed by your teacher and in your homework.) Also, make a list of those you had trouble with. Pay particular attention to items that are high on both lists. Often, you teacher will mention particularly important types of problems you should pay attention to (or particularly common mistakes you should avoid).
• Review examples of problems from each item in your list - Some teachers will give you a review sheet for the test, but even if they don't you can make your own review sheet by going through the book and picking out example problems to work, being sure to get examples of everything on your list. If you have trouble working any of these problesm, go back and review how to work them (get help from the tutors, your classmates, or the teacher as needed).
• Test yourself beforehand - The best way to see how well you are likely to do on a test is to make yourself take a "practice test" beforehand. Now, this doesn't mean that you just work through the review sheet (you should have done that already). It means that you make yourself a list of problems based on your list of what is on the test (about the same length as a real test). Then, you need to sit down and work through them like it really was a test, so no notes or looking things up in the book or looking back at old homework or anything else like that. Really treat it like a test. After you have finished it completely, go back through and "grade" yourself; check your answers and see what kind of grade that would give you. If you don't do so well, go back and review your mistakes and then make yourself another practice test and do it again (preferably the next day, if you have the time). If you can do a good job on a practice test like this (without yielding to the temptation to cheat "just a little"...), then you stand a very good chance of doing well on the real test.
• Study groups are a Good Thing - As I mentioned earlier, studying together with other students in your class can be very helpful. This can be a very efficient way to review for a test (assuming everyone is focused on what you are doing) and you can even make up practice tests for each other. (Try not to be too rough on each other...)
• Really try to keep a positive attitude - A really negative attitude towards a test ("Math sucks" or "I just can't do math") can really hurt your preparation for the test. I'm not suggesting you tell yourself things you don't really believe, but obsessing about negative thoughts is just going to make things worse. Try to break out of negative emotional spirals ("I hate math. I'm no good at math. I suck at math. I suck. I really suck..." You get the idea). I think you will find, if you really seriously try the things I am suggesting here that you can do much better in math than you think you can. You might be surprised, you could even like it (I won't tell...).
• Try to know more than you actually need to on the test - Try to give yourself a safety margin on your preparation. If you know the material 10% better than you really need to, you will find that you remember more important things on the actual test (when most of us tend to forget things we knew right beforehand). This just gives you a margin to compensate for "test anxiety" on the actual test day.

The test arrives

So, the day of the test finally arrives. What sorts of things can you do going into the test and during the test to help you be successful on it?

• Get a good night's sleep beforehand - Taking a test on very little sleep is a Very Bad Idea. Trust me on this, your mental performance drops very dramatically if you are operating on too little sleep (especially if it has been going on for a while). The extra adrenaline rush you get when you go into the test may help a little, but it won't make up for a rested mind (and if the test takes very long, that adrenaline wears off after a while). If you have been studying properly for the test (i.e., you didn't wait until the last minute), the extra few hours of studying you might get by skipping sleep will not be worth it.
• Eat reasonably before you go in to the test - Protein is your friend. Be sure you don't just eat a bunch of carbohydrates right beforehand; they will give you a lot of quick energy, but your body (and mind) will probably crash pretty quickly after that. To do well on a test, you need to have an energy source that will last throughout the test; this means protein. I'm no health food, granola eating fanatic, but I have found this simple rule very useful.
• Save your moaning and self-criticism until after the test - I know many of us get very uptight and start beating ourselves up when under this kind of stress ("I'm never going to pass. I always have problems on math tests..."). Let me give you some of the best (and hardest to follow) advice I can give you here: SAVE IT. Just put it off until after the test and you can make yourself feel as miserable as you want. There is nothing that will drain your ability on a test faster than this sort of "self-talk". Even if you truly believe this deep down (something to work on for later), it won't help you to do better on the test by dwelling on it now. Just tell yourself that you will think about that later. Concentrate on the math on the test, not how you feel about the math. This is very hard to do, but it is really key to being successful in math.
• Write down important things you need to remember right away - If you teacher doesn't allow you notes on the test (I don't), then one of the first things you should do after you sit down with the test is to flip over to a blank page and write down any formulas or important points you need to remember for the test. That way, you won't have to try to remember it again and again later, when your mind won't be as fresh. (Do be careful to do it right, however. You wouldn't want to write down the wrong formula and then keep using it throughout the test.)
• Don't try to work through the test from start to finish - The problem with this approach is that, if you hit a problem that you don't know, you could spend a really long time on that one problem and then, by the time you move on, you could be so frustrated or tired that you can't give your best work on the rest of the test. Also, if the test is timed, you might run out of time. I suggest a "3 pass" approach:
  1. The first time through, work all the easy problems and the ones you know you can do without much effort. This makes sure you get these done (and hopefully it clears out a lot of problems right away). If you don't see how to do a problem pretty quickly, skip it and move on. Don't even worry about how you would work it (you'll come back to think about it later).
  2. The second time through, work the ones you are pretty sure you can figure out given a little time and effort. If something seems really hard or you can't remember how to do it, skip it and move one.
  3. The third time through, go back and work on all the hard ones and the ones you were having trouble remembering. This way, if you run out of gas (or time), you have gotten as many problems done as possible. Also, it helps you to put off the serious freaking out about the math until the end. These are the problems that give you fits and you have saved them for last; that way you got all the ones you really knew out of the way before you started getting really upset.

  Of course, if you use this method, you need to clearly keep track of which problems you have already worked and which ones you still need to work. (Try putting a check mark by each problem as you finish it.)

• If you make a mistake, just put an X through it - Don't erase (especially big mistakes) and don't scratch it out so it isn't readable. You would be amazed how many times I have seen students erase the correct answer. If you just cross it out once, the work is still there so the instructor can read it, so that if you cross out something correct, your teacher might have some pity on you and give you some partial credit.
• If you work a problem more than once, make it really clear to your teacher which one you want graded - Just put an X through the one you don't want counted. Otherwise, if you don't, your teacher may think you meant the wrong work you didn't cross out.
• Always clearly (and legibly) show your work - If you show your work, your teacher can see what you were trying to do. That way, if you make a mistake (like you add 2 and 3 to get 6), your teacher might give you partial credit for the problem.
• Before you turn your test in, go through and make sure you didn't forget any problems - It would be very frustrating to lose points because you just forgot a problem. This happens more often than you might think.