Monday, January 5, 2015

Math Test-taking Strategy

Math testing strategy varies a bit from the strategies that would normally be useful for classes requiring a lot of rote memorization. This is mostly because, in addition requiring a student to interpret and regurgitate information beforehand, math tests also involve a performance element that tests short-term problem solving ability. You may recognize this as an important skill to have! Before we list some specific ways to prepare, keep in mind that:
  • Mathematical ability can be improved. Some people have a tendency to give up on math if they aren't good at it immediately. However, if you spend more time on math than your classmates; whether it is going to math camp, working with a private tutor, thinking about outside problems, or figuring out how things work; you will get better faster. Nobody comes out of the womb being able to solve every type of math problem. It's like playing a musical instrument or learning to dance: the more practice, the better.
  • Getting an A in most classes requires a level of understanding that is not taught in the class. In American schools, an A grade is meant to signify that a student is going above and beyond what is required, even if it looks like all of the testing material is being taught in the class. There is often far more to the material! For example, a lot of students understand the general material being taught, but get slammed on small mistakes such as minus sign errors. Catching and being aware of these errors is something that the student must develop on their own, and it is hard to explicitly teach. Small things like this are often the difference between an A and a C.
Got that? Good! If you're studying for a test and aren't quite sure what to start with, there are several 'levels' of understanding the material, which, for your purposes, we'll express in three separate categories:
  • Basic understanding. You're at this level if you can read and understand everything in the textbook chapter, and know how to do the problems that aren't word problems.
  • Familiarity. You're at this level if you understand what the answers should 'look' like, why the answers 'look' that way, and how to fix something if you've made a mistake. To get to this level, you have to be observant and look for patterns in the work you're doing. Knowing where mistakes can arise in a certain type of problem is very powerful!
  • Creative application. If basic understanding is like knowing how to get home, and familiarity is knowing how to get home even if you've taken a wrong turn, creative application is like getting home by parkour. You're at this level if you understand the technique so well that even when it's not mentioned, you know when it has to be used. This type of understanding is the most important for word problems.
Let's use (scalar) multiplication as an example. Someone would have basic understanding if they could multiply 45x25 on paper (or in their head! Can you?). They're familiar with how multiplication works if they can explain why the answer cannot be 725, or 329670, or 2621. Lastly, they will have mastered multiplication as a concept if they can use it to solve problems such as 'how much do 45 vending-machine gumballs cost'? or use multiplicative identities to prove the exponent rules.

So, in a nutshell, if you understand the book and can do all the problems, you're still not guaranteed to do well on tests. Nooo! How can this be?

The primary problem is that very few books go into detail on how things work and how to recognize mistakes. Yet recognizing where mistakes happen and how to fix them is vital---not to mention that knowing how methods work is the only way to understand word problems! Here are a few helpful things you can do to improve your math testing ability:
  • Know exactly where your abilities are for each type of problem. Can you do mental multiplication? Can you do basic algebra problems? Can you prove that L^2 is complete? Being honest about where your weaknesses are makes it much easier to conquer them. Needing more practice on a specific type of problem isn't a bad thing, and you'll save time if you focus on only the hardest problems.
  • Develop 'sanity checks'. 'Sanity checks' are what I call pieces of information you remember to check whether you've made a mistake in the problem. Using the multiplication example above, a good example of a sanity check would be 'an odd number times an odd number cannot be an even number.' This helps build your familiarity with the material and could save you from losing tons of points.
  • Build the test. Even if you don't know which specific problems will be on the test, you can generally work out how many of each type of problem will be on the test. This will help you direct your attention towards whatever will win you the most points. For example, you might be okay with everything on the test except one very, very hard type of problem: would it be better to focus on figuring out the hard problems, or making sure you don't mess up on the moderate problems? This depends on how many of the hard problems will be on the test.
  • Talk with friends. We don't mean about videogames. There aren't a lot of ways to develop creative thinking for mathematics other than 'think about math a lot and try to come up with and solve math problems outside of school', but this is one of the more fun ones. Your friends may have some insight about math that hasn't occurred to you yet, or you may be able to solve a hard problem by working together. In any case, talking directly to people who know more than you will teach you a lot.
  • Look for patterns. If you don't want to talk to your friends about math, or have some pride about developing things on your own, remember that math is all about finding and exploiting patterns. Once you've found a pattern, try to figure out where it comes from. This line of thinking often leads to developing newer, faster ways of solving problems. See our mental math post for some simple examples of pattern exploitation.
  • Hire a private tutor. We said earlier that talking to people who know more than you will improve your skills very quickly: well, private tutors know a lot of math and they can teach you a lot about math. There's no shame in needing one! If someone is very good at the guitar and wants to become even better, they hire a private guitar teacher. The same is true for math. This is how you should see a private math tutor: someone who can help you improve very, very quickly and understand mathematics far beyond what you are taught in class.
These are all things you would do before the test happens (we've left out the obvious "read the textbook and do the problems" advice.) While the test is happening, try to:
  • Read the test beforehand and do the easiest problems first.
  • Temporarily improve performance by stretching, chewing gum, drinking caffeine, or listening to energetic music.
Now you should be focused and in the moment!

Lastly: even if progress seems slow sometimes, or you're having trouble catching up, don't feel hopeless. If you spend time thinking about how the methods work, you will improve.

Good luck, and happy testing!

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