The most valuable part of a
problem is the solution, not the answer. The solution to a problem is just like
a proof to a theorem; it contains the ideas, the techniques, and the properties
of mathematics that work to solve the problem. These ideas, techniques, and
properties can be used again to solve other similar problems and sometimes even
problems which are quite different. The answer has no use or meaning beyond the
problem to which it applies.
You can be successful in mathematics if you begin to
collect solutions and observe their critical steps and key techniques. Where
can you collect good solutions? There are several places. College textbooks are
written today with many examples of problems whose solutions are completely
worked out. Such solutions usually are very similar to the kinds of problems
you will be asked to do in the exercises at the end of the section. Another
good source is your instructor.
As you collect solutions to problems make it a point
to look for the problem solving techniques in the solutions; look for the steps
that seem like they can be used elsewhere to solve other problems, and make a
point of trying to remember them. In time and with practice, you will soon find
that your mathematical memory is improving. Your memory is not really improving, rather your ability to study mathematics is what
is really improving.
A final word of advice: don’t get too caught up in
the importance of the answers in the back of the book. It is alright to look up
the answer to a solution after you have solved a problem. But if your answer
agrees with the book’s answer, then don’t run off to the next problem too
quickly. Pause after every solution, whether it is yours or someone else’s, to
examine it for the essential ideas and techniques. The best time of all to
learn some mathematics for the future is to review, while it is still fresh,
your own correct work. You are close to it; you are familiar with your own thinking
and the steps that you took in your solution. So take advantage of this great
moment to learn mathematics.