poster:how to do,from:
how to do
About a third to a half of any math course deals with technique---the process of making theorems work for you in specific situations rather than in the general setting in which they are usually stated. Sometimes this is fairly easy: many proofs give explicit constructions which you follow for the special case. In these situations the only problems are with algebraic and trigonometric manipulations and keeping track of where you are in the process. In other situations (technique of integration is a good example) there are lots of approaches which might apply to a given problem and several tricks which might be used to make the problem more tractable. For these you need to develop judgment.
Read through the theorems and examples.
Some students make the whole process of learning how to do problems more difficult by acting like it had no connection with the other material in the course. Often problems follow a pattern which is given explicitly in the proof of the major theorem they follow. Knowing the general pattern in advance is easier than trying to find it by trial and error.
Work enough problems
At this stage you should work enough problems so that the single technique which the problems illustrate is firmly in your mind. Since you have ultimate responsibility for your education, you should take the initiative to work enough problems for your own practice needs. This may well be more problems than are assigned to be turned in.
Work a few problems in as many different ways as possible
Too often the practice obtained in step 2 leads the student to think that there is only one approach to each problem. Sometimes one approach is easy and another is complicated, but often several different attacks will work equally well. Complicated approaches give the student practice in solving problems which take more than one step and more than one technique.
Make yourself a set of randomly chosen problems
One difficulty with learning many techniques to solve a particular kind of problem is that you have to figure out which technique to use before you can get to work on a solution. This is exacerbated by the tendency for problems to be grouped so that the appropriate technique to use is the one which immediately preceded the problem set. Putting two or three problems from each of the problem sets in a chapter on technique on 3 by 5 cards and then shuffling the cards will give you a set of problems on which to practice deciding which technique to use.
