Solving Mathematical Problems
The most important thing to realize when solving difficult mathematical problems is that one never solves such a problem on the first attempt. Rather one needs to build a sequence of problems that lead up to the problem of interest, and solve each of them. At each step experience is gained that’s necessary or useful for the solution of the next problem. Other only loosely related problems may have to be solved, to generate experience and insight.
Students (and scholars too) often neglect to check their answers. I suspect a major reason is that traditional and widely used teaching methods require the solution of many similar problems, each of which becomes a chore to be gotten over with rather than an exciting learning opportunity. In my opinion, each problem should be different and add a new insight and experience. However, it is amazing just how easy it is to make mistakes. So it is imperative that all answers be checked for plausibility. Just how to do that depends of course on the problem.
There is a famous book: G. Polya, “How to Solve It “, 2nd ed., Princeton University Press, 1957, ISBN 0-691-08097-6. It was first published in 1945. This is a serious attempt by a master at transferring problem solving techniques.
Summary taken from G. Polya, “How to Solve It”
- UNDERSTANDING THE PROBLEM
- First. You have to understand the problem.
- What is the unknown? What are the data? What is the condition?
- Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
- Draw a figure. Introduce suitable notation.
- Separate the various parts of the condition. Can you write them down?
- DEVISING A PLAN
- Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually aplan of the solution.
- Have you seen it before? Or have you seen the same problem in a slightly different form?
- Do you know a related problem? Do you know a theorem that could be useful?
- Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.
- Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?
- Could you restate the problem? Could you restate it still differently? Go back to definitions.
- If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other?
- Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?
- CARRYING OUT THE PLAN
- Third. Carry out your plan.
- Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?
- Looking Back
- Fourth. Examine the solution obtained.
- Can you check the result? Can you check the argument?
- Can you derive the solution differently? Can you see it at a glance?
- Can you use the result, or the method, for some other problem?
Acquiring Mathematical Understanding
Since this is directed to undergraduate students a more specific question is how does one acquire mathematical understanding by taking classes? But that does not mean that classes are the only way to learn something. In fact, they often are a bad way! You learn by doing. For example, it’s questionable that we should have programming classes at all, most people learn programming much more quickly and enjoyably by picking a programming problem they are interested in and care about, and solving it. In particular, when you are no longer a student you will have acquired the skills necessary to learn anything you like by reading and communicating with peers and experts. That’s a much more exciting way to learn than taking classes!
Here are some suggestions regarding class work:
- Always strive for understanding as opposed to memorization.
- If this means you have to go back, do it! Don’t postpone clarifying a point you miss because everything new will build on it.
- It may be intimidating to be faced with a 1,000 page book and having to spend a day understanding a single page. But that does not mean that you’ll have to spend a thousand days understanding the whole book. In understanding that one page you’ll gain experience that makes the next page easier, and that process feeds on itself.
- Read the sections covered in class before you come to class. That’s one of the most useful ways in which you can spend your time, because it will dramatically increase the effectiveness of the lecture.
- Do exercises. The teacher may suggest some, put you can pick them on your own from the textbook or make up your own. Select them by the amount of interest they hold for you and the degree of curiosity they stimulate in you. Avoid getting into a mode where you do a large number of exercises that are distinguished only by the numerical values assigned to some parameters.
- Always check your answers for plausibility.
- Whenever you do a problem or follow a new mathematical thread explicitly formulate expectations. Your expectations may be met, which causes a nice warm feeling (and you should probably also look for a new and different problem). But otherwise there are two possibilities: you made a mistake from which you can recover, now that you are aware of it, or there is something genuinely new that you can figure out and which will teach you something. If you don’t formulate and check expectations you may miss these opportunities.
- Find a class mate who will work with you in a team. Have one of you explain the material to the other, on a regular basis, or switch periodically. Explaining math to others is one of the best ways of learning it.
Be open and alert to the use of new technology. (I know you are because you are reading this web page.) You can go from here directly to computing help. But don’t neglect thinking about the problem and understanding it, its solution, and its ramifications. The purposes of technology are not to relieve you of the need to think but:
- To check your answers.
- To take care of routine tasks efficiently.
- To do things that can’t possibly be done by hand (like the visualization of large data sets).
- Keep in mind R.W. Hamming’s famous maxim: The purpose of computing is insight, not numbers.
- Once you are done with a course Keep Your Textbook and refer back to it when you need to. You have spent so much time with that book that you know it intimately and know how to use it and where to find the information you need. The small amount of money you might get by selling it does not come close to offsetting the loss in time and energy you waste being thwarted by a lack of understanding a particular piece of mathematics that you easily refamiliarize yourself with by consulting your old friend, the textbook. Here’s a more passionate elaboration on this theme.