Solving Problems in Mathematics

About Solving Problems

Now we've seen a problem and worked out a solution, however rough, let's look at the whole business of problem solving. There is no way that at the first reading i can expect you to grasp all the infinite subtleties of the following discussion. So read it a couple of times and move on. But do come back to it from time to time. Hopefully you'll make more sense of it all as time goes on.

Holton's Analysis of solving problems

a. First take one problem. Problem solving differs in only one or two respects to mathematical research. The difference is simply that most problems are precisely stated and there is a definite answer (which is known to someone else at the outset). All the steps in between problem and solution are common to both problem solving and research. The extra skill of a research mathematician is learning to pose problems precisely. Of course he/she has more mathematical techniques to hand too.

b. Read and understand. It is often necessary to read a problem through several times. You will probably initially need to read it through two or three times just to get feel for what's needed. Almost certainly you will need to remind yourself of some details in mid solution. You will definitely need to read it again at the end to make sure you have answered the problem that was actually posed and not something similar that you invented along the way because you could solve the something similar.

c. Important words. What are the key words in a problem? This is often a difficult question to answer, especially on the first reading. However, here is one useful tip. Change a word or phrase in the problem. If this change the problem then the word or phrase is important. Usually numbers are important. In the problem of the last section, "jug" is only partially important. Clearly if "jug" was changed to "vase" everywhere, the problem is essentially not changed. However "3" can't be changed to "7" without affecting the problem.

Now you've come this far restate the problem in your own words

d. Panic!. At this stage it's often totally unclear as to what to do next. So, doodle, try some examples, think "have I seen a problem like this before?". Don't be afraid to drink "i'll never solve this (expletives deleted) problem". Hopefully you'll get inspiration somewhere. Try another problem. Keep coming back to the one you're stuck on and keep giving it another go. If, after a week, you're still without inspiration, then talk to a friend. Even mother (who may know nothing about the problem) are marvellous sounding boards. Often the mere act of explaining your difficulties produces an idea or two. However, if you've hit a real toughie, then get in touch with your teacher -- that's way they exist. Even then don't ask a solution. Explain your difficulty and ask for a hint.

e. System. At the doodling stage and later, it's important to bring some system into your work. Tables, charts, graphs, diagrams are all valuable tools. Never throw any of this initial material away. Just as soon as you get rid of it you're bound to want to use it.

Oh, and if you're using a diagram make sure it's a big one. Pokey little diagrams are often worse than no diagram. And also make sure your diagram covers all possibilities. Sometimes a diagram can lead you to consider only part of a problem.

f. Patterns. Among your doodles, tables and so forth look like for patterns. The exploitation of pattern is fundamental to mathematics and is one of its basic powers.

g. Guess. Yes, guess! Don't be afraid to guess at an answer. You'll have to check your guess against the data of the problem or examples you've generated yourself but guesses are the lifeblood of mathematics. OK so mathematicians call their guesses "conjectures". It may sound more sophisticated but it comes down to the same thing in the long run. Mathematical research stumbles from one conjecture (which may or may not be true) to the next.

h. Mathematical technique. As you get deeper into the problem you'll know that you want to use algebraic, trigonometric or whatever techniques. Use what methods you have to. Don't be surprised though, if someone else solves the same problem using some quite different area of mathematics.

i. Explanations. Now you've solved the problem write out your solution. This very act often exposes some case you hadn't considered or even a fundamental flaw. When you're happy with your written solution, test it out on a friend. Does your solution cover all their objections? If so, try it on your teacher. If not, rewrite it.

j. Generalisation. So you may have solved the original problem but now and then you may only have exposed the tip of the iceberg. There may be a much bigger problem lurking around waiting to be solved. Solving big problems is more satisfying than solving little ones. It's also potentially more useful. Have a crack at some generalisations. 

In conclusion though, problem solving is like football or chess or almost anything worthwhile. Most of us start off with more or less talent but to be really good you have to practice, practice, and practice.