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Advice on solving problems
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Many students try to solve problems by staring at the statement of the problem until they can see the solution. This kind of strategy works tolerably well with problems for which a standard method of solution exists. If you spot the relevant method then all you have to do is apply it. And if most of the problems you have met in the past simply involved using a standard method then it might seem reasonable to try to apply this strategy to all the problems that you face in future.
As your level of study gets more advanced you will find that many problems cannot be solved using standard methods of solution. You will increasingly be required to solve problems which at first you have no idea how to solve.
The structured approach that we will follow here is based on work by the mathematician George Polya. He divided problem-solving into four phases:
Once you have looked through each of these sections, try your hand at the exercises .
The first stage in our structured approach is to make sure that you actually understand the problem. It is surprising how many students try to solve problems without understanding the ideas involved. If you are dealing with a standard problem then you might be able to just mindlessly apply a stock procedure. But complex problems demand a different approach.
A particularly effective way of identifying what you need to understand is to answer the following two questions:
What do I know?
What do I want?
Answering these two questions helps you to focus your attention on the information given in the problem and on the goal of the problem.
In order to fully understand the answers to these two questions you will need to explicitly employ some of the skills that are second nature to mathematicians, one of which is the ability to use examples of ideas. List all of the ideas that are involved and provide a varied set of examples for each idea.
The next stage in our structured approach is to engage in planning.
Once you have gained a genuine understanding of what you know and of what you want to know, the next stage is to begin to plan how to bridge the gap between them. This is of course the goal of all problem-solving.
There are two strategies we will consider for bridging this gap between what is known and what is wanted. The first of these is to identify and draw on resources. The second is to ensure that you take control of your work.
- Identify and draw on resources
It is often possible to link your problem to other relevant information. So you need to ask what additional resources you can draw upon to bridge the gap between what is known and what is wanted. Clearly to do this it is necessary to have gained a thorough grasp of the area in which you are working. Look through your notes or a textbook to see if any procedure or theorem is relevant. Does any mathematical result link what you know to what you want to know?
- Take control of your work
Once you have identified some of the relevant resources that you can draw upon you are now in a position to take control of your work by devising a plan. The aim is to propose a way of using the resources you have identified to bridge the gap between what you know and what you want to know.
Now in any given problem there will be a vast range of ways to try and bridge this gap. Some options will reach a solution quickly, others will reach a solution after plenty of work and yet others will never yield the solution. It therefore makes little sense to spend most of your time trying to carry out the first option that comes into your head as it could well fail to lead to a solution.
Before you rush in to a particular course of action, brainstorm as to which strategies might work-generate at least two different options.
It is worth noting that your options need only concern how to start work on solving the problem. There is no need to be convinced that any option will lead to a solution. The gap between what is known and what is wanted is often too wide to bridge straight away, even using additional resources.
The next stage in our structured approach is to carry out the plan.
Once you have devised at least a couple of plans, you need to carry out the option that you think is most likely to be effective. And make sure that you write down your work, as this often triggers new ideas about how to proceed. It is also important to monitor how well the solution is proceeding.
You should feel tension between a certain measure of confidence that your plan might well lead to the solution, and a recognition that you could be wasting your time.
Is it worth continuing with this line of attack? Is the solution getting too complicated? As a general rule, the longer the solution is taking without any hopeful signs, the more likely you are to be heading towards a dead-end.
Now, in general, either your work will lead to a solution or your monitoring will suggest you that you are heading towards a dead-end. Congratulations are in order if you manage to solve the problem with your first plan (although it will be worth checking your solution). But if then first plan fails then we simply move onto the next stage in the structured approach, to review the situation.
If a solution does not appear in sight then there are several options open to you. Note that giving up straight away is not on the list of options!
- Review your work and check for mistakes. Making mistakes is an occupational hazard of being a mathematician. But leaving them uncorrected is another matter.
- Try your second plan of attack on the problem.
- Return to the original statement of the problem and repeat the stages of the structured approach. Make sure that you fully understand the problem. Are there any other resources you can draw upon to bridge the gap between what is known and what is wanted? Are there any other options for solving the problem?
- Analyse how you have been tackling the problem. Perhaps the easiest way to do this is to write down answers to the following questions at regular intervals. What am I doing? Why am I doing it? How is it going to help me to solve the problem?
- Leave the problem until the next day. It is surprising how much difference a fresh mind can make.
- Consult a friend or tutor. Even if your friend is not studying mathematics or one of its applications, simply explaining your problem to them may help you to gain the insight that you need.
One of the strategies in our list above, or a combination of them, will often lead to the solution of the problem. Or alternatively the time allotted to spend on the problem will elapse. In both cases, however, it can be very useful to provide a written summary as to how you tackled the problem. Summarise what you did, why you did it and how it related to the solution of the problem.
The most effective way of developing control of your problem-solving is not by an expert telling you what to do, but for you to become aware of how you solve problems yourself.
Try out the exercises to see how aware you are of how you solve problems.
1. One plan of attack to solve the problem immediately below would be to measure the required angle. Why is this plan inappropriate and what other ways might there be to tackle this problem?
Let A,B,C be points on the circumference of a circle, with centre O. Further, let the angle ABC be the angle subtended by the chord AB to the point C and let the angle subtended at the centre of the circle by the arc AB be equal to 40 degrees. Find the angle ABC.
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2. Consider the following problem. What resources are relevant to bridging the gap between what is known and what is wanted? Further, outline at least two ways of trying to solve the problem.
Find the constant force that needs to be applied to a mass of 5 kilograms to enable it to travel a distance of 20 metres in 10 seconds.
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3. Carry out each of the plans that you devised in Exercise 2 above. Describe how you monitored the effectiveness of your work.
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This content has been written by Peter Kahn, author of Studying Mathematics and its Applications.
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