On this page we discuss Problem Solving Strategies under three headings.
What Are Problem Solving Strategies?
Strategies are things that Pólya would have us choose in his second stage of problem solving and use in his third stage (What is Problem Solving?). In actual fact he called them heuristics. To Pólya they were things to try that he couldn’t guarantee would solve the problem but, of course, he sincerely hoped they would. So they are some sort of general ideas that might work for a number of problems. And then again they might not.
As speaking in riddles isn’t likely to be of much assistance to you, let’s get down to some examples. There are a number of common strategies that children of primary age can use to help them solve problems. We discuss below several that will be of value for problems on this web-site and in books on problem solving. In this site we have linked the problem solving lessons to the following groupings of problem solving strategies. As the site develops we may add some more but we have tried to keep things simple for now.
Common Problem Solving Strategies
- Guess (this includes guess and check, guess and improve)
- Act It Out (act it out and use equipment)
- Draw (this includes drawing pictures and diagrams)
- Make a List (this includes making a table)
- Think (this includes using skills you know already)
We have provided a copymaster for these strategies so that you can make posters and display them in your classroom. It consists of a page per strategy with space provided to insert the name of any problem that you come across that uses that particular strategy (Act it out, Draw, Guess, Make a List). We have found that this kind of poster provides good revision for children. It also establishes links across curriculum areas. Through these links, children can see that mathematics is not only connected by skills but also by processes.
An In-Depth Look At Strategies
We now look at each of the following strategies and discuss them in some depth. You will see that each strategy we have in our list is really only a summary of two or more others.
This stands for two strategies, guess and check and guess and improve.
Guess and check is one of the simplest strategies. Anyone can guess an answer. If they can also check that the guess fits the conditions of the problem, then they have mastered guess and check. This is a strategy that would certainly work on the Farmyard problem but it could take a lot of time and a lot of computation.
Because it is such a simple strategy to use, you may have difficulty weaning some children away from guess and check. If you are not careful, they may try to use it all the time. As problems get more difficult, other strategies become more important and more effective. However, sometimes when children are completely stuck, guessing and checking will provide a useful way to start and explore a problem. Hopefully that exploration will lead to a more efficient strategy and then to a solution.
Guess and improve is slightly more sophisticated than guess and check. The idea is that you use your first incorrect guess to make an improved next guess. You can see it in action in the Farmyard problem. In relatively straightforward problems like that, it is often fairly easy to see how to improve the last guess. In some problems though, where there are more variables, it may not be clear at first which way to change the guessing.
2 Act It Out
We put two strategies together here because they are closely related. These are Act it Out and Use Equipment.
Young children especially, enjoy using Act it Out. Children themselves take the role of things in the problem. In the Farmyard problem, the children might take the role of the animals though it is unlikely that you would have 87 children in your class! But if there are not enough children you might be able to press gang the odd teddy or two.
There are pros and cons for this strategy. It is an effective strategy for demonstration purposes in front of the whole class. On the other hand, it can also be cumbersome when used by groups, especially if a largish number of students is involved. We have, however, found it a useful strategy when students have had trouble coming to grips with a problem.
The on-looking children may be more interested in acting it out because other children are involved. Sometimes, though, the children acting out the problem may get less out of the exercise than the children watching. This is because the participants are so engrossed in the mechanics of what they are doing that they don’t see through to the underlying mathematics. However, because these children are concentrating on what they are doing, they may in fact get more out of it and remember it longer than the others, so there are pros and cons here.
Use Equipment is a strategy related to Act it Out. Generally speaking, any object that can be used in some way to represent the situation the children are trying to solve, is equipment. This includes children themselves, hence the link between Act it Out and Use Equipment.
One of the difficulties with using equipment is keeping track of the solution. Actually the same thing is true for acting it out. The children need to be encouraged to keep track of their working as they manipulate the equipment.
In our experience, children need to be encouraged and helped to use equipment. Many children seem to prefer to draw. This may be because it gives them a better representation of the problem in hand. Also, if they’re a little older, they may feel that using equipment is only 'for babies'. Since there are problems where using equipment is a better strategy than drawing, you should encourage children’s use of equipment by modelling its use yourself from time to time.
It is fairly clear that a picture has to be used in the strategy Draw a Picture. But the picture need not be too elaborate. It should only contain enough detail to solve the problem. Hence a rough circle with two marks is quite sufficient for chickens and a blob plus four marks will do for pigs. There is no need for elaborate drawings showing beak, feathers, curly tails, etc., in full colour. Some children will need to be encouraged not to over-elaborate their drawings (and so have time to attempt the problem). But all children should be encouraged to use this strategy at some point because it helps children ‘see’ the problem and it can develop into quite a sophisticated strategy later.
It’s hard to know where Drawing a Picture ends and Drawing a Diagram begins. You might think of a diagram as anything that you can draw which isn’t a picture. But where do you draw the line between a picture and a diagram? As you can see with the chickens and pigs, discussed above, regular picture drawing develops into drawing a diagram.
Venn diagrams and tree diagrams are particular types of diagrams that we use so often they have been given names in their own right.
It’s probably worth saying at this point that acting it out, drawing a picture, drawing a diagram, and using equipment, may just be disguises for guessing and checking or even guessing and improving. Just watch children use these strategies and see if this is indeed the case.
4 Make a list
Making Organised Lists and Tables are two aspects of working systematically. Most children start off recording their problem solving efforts in a very haphazard way. Often there is a little calculation or whatever in this corner, and another one over there, and another one just here. It helps children to bring a logical and systematic development to their mathematics if they begin to organise things systematically as they go. This even applies to their explorations.
There are a number of ways of using Make a Table. These range from tables of numbers to help solve problems like the Farmyard, to the sort of tables with ticks and crosses that are often used in logic problems. Tables can also be an efficient way of finding number patterns.
When an Organised List is being used, it should be arranged in such a way that there is some natural order implicit in its construction. For example, shopping lists are generally not organised. They usually grow haphazardly as you think of each item. A little thought might make them organised. Putting all the meat together, all the vegetables together, and all the drinks together, could do this for you. Even more organisation could be forced by putting all the meat items in alphabetical order, and so on. Someone we know lists the items on her list in the order that they appear on her route through the supermarket.
In many ways we are using this strategy category as a catch-all. This is partly because these strategies are not usually used on their own but in combination with other strategies.
The strategies that we want to mention here are Being Systematic, Keeping Track, Looking For Patterns, Use Symmetry and Working Backwards and Use Known Skills.
Being Systematic, Keeping Track, Looking For Patterns and Using Symmetry are different from the strategies we have talked about above in that they are over-arching strategies. In all problem solving, and indeed in all mathematics, you need to keep these strategies in mind.
Being systematic may mean making a table or an organised list but it can also mean keeping your working in some order so that it is easy to follow when you have to go back over it. It means that you should work logically as you go along and make sure you don’t miss any steps in an argument. And it also means following an idea for a while to see where it leads, rather than jumping about all over the place chasing lots of possible ideas.
It is very important to keep track of your work. We have seen several groups of children acting out a problem and having trouble at the end simply because they had not kept track of what they were doing. So keeping track is particularly important with Act it Out and Using Equipment. But it is important in many other situations too. Children have to know where they have been and where they are going or they will get hopelessly muddled. This begins to be more significant as the problems get more difficult and involve more and more steps.
In many ways looking for patterns is what mathematics is all about. We want to know how things are connected and how things work and this is made easier if we can find patterns. Patterns make things easier because they tell us how a group of objects acts in the same way. Once we see a pattern we have much more control over what we are doing.
Using symmetry helps us to reduce the difficulty level of a problem. Playing Noughts and crosses, for instance, you will have realised that there are three and not nine ways to put the first symbol down. This immediately reduces the number of possibilities for the game and makes it easier to analyse. This sort of argument comes up all the time and should be grabbed with glee when you see it.
Finally working backwards is a standard strategy that only seems to have restricted use. However, it’s a powerful tool when it can be used. In the kind of problems we will be using in this web-site, it will be most often of value when we are looking at games. It frequently turns out to be worth looking at what happens at the end of a game and then work backward to the beginning, in order to see what moves are best.
Then we come to use known skills. This isn't usually listed in most lists of problem solving strategies but as we have gone through the problems in this web site, we have found it to be quite common. The trick here is to see which skills that you know can be applied to the problem in hand.
One example of this type is Fertiliser (Measurement, level 4). In this problem, the problem solver has to know the formula for the area of a rectangle to be able to use the data of the problem.
This strategy is related to the first step of problem solving when the problem solver thinks 'have I seen a problem like this before?' Being able to relate a word problem to some previously acquired skill is not easy but it is extremely important.
Uses of Strategies
Different strategies have different uses. We’ll illustrate this by means of a problem.
The Farmyard Problem: In the farmyard there are some pigs and some chickens. In fact there are 87 animals and 266 legs. How many pigs are there in the farmyard?
Some strategies help you to understand a problem. Let’s kick off with one of those. Guess and check. Let’s guess that there are 80 pigs. If there are they will account for 320 legs. Clearly we’ve over-guessed the number of pigs. So maybe there are only 60 pigs. Now 60 pigs would have 240 legs. That would leave us with 16 legs to be found from the chickens. It takes 8 chickens to produce 16 legs. But 60 pigs plus 8 chickens is only 68 animals so we have landed nearly 20 animals short.
Obviously we haven’t solved the problem yet but we have now come to grips with some of the important aspects of the problem. We know that there are 87 animals and so the number of pigs plus the number of chickens must add up to 87. We also know that we have to use the fact that pigs have four legs and chickens two, and that there have to be 266 legs altogether.
Some strategies are methods of solution in themselves. For instance, take Guess and Improve. Supposed we guessed 60 pigs for a total of 240 legs. Now 60 pigs imply 27 chickens, and that gives another 54 legs. Altogether then we’d have 294 legs at this point.
Unfortunately we know that there are only 266 legs. So we’ve guessed too high. As pigs have more legs than hens, we need to reduce the guess of 60 pigs. How about reducing the number of pigs to 50? That means 37 chickens and so 200 + 74 = 274 legs.
We’re still too high. Now 40 pigs and 47 hens gives 160 + 94 = 254 legs. We’ve now got too few legs so we need to guess more pigs.
You should be able to see now how to oscillate backwards and forwards until you hit on the right number of pigs. So guess and improve is a method of solution that you can use on a number of problems.
Some strategies can give you an idea of how you might tackle a problem. Making a Table illustrates this point. We’ll put a few values in and see what happens.
|pigs||chickens||pigs legs||chickens’ legs||total||difference|
From the table we can see that every time we change the number of pigs by one, we change the number of legs by two. This means that in our last guess in the table, we are five pigs away from the right answer. Then there have to be 46 pigs.
Some strategies help us to see general patterns so that we can make conjectures. Some strategies help us to see how to justify conjectures. And some strategies do other jobs. We’ll develop these ideas on the uses of strategies as this web-site grows.
What Strategies Can Be Used At What Levels
In the work we have done over the last few years, it seems that children are able to tackle and use more strategies as they continue with problem solving. They are also able to use them to a deeper level. We have observed the following strategies being used in the stated Levels.
Levels 1 and 2
- Draw a Picture
- Act it Out
- Use Equipment
- Guess and Check
Levels 3 and 4
- Draw a Diagram
- Act it Out
- Use Equipment
- Guess and Improve
- Make a Table
- Make an Organised List
It is important to say here that the research has not been exhaustive. Possibly younger children can effectively use other strategies. However, we feel confident that most children at a given Curriculum Level can use the strategies listed at that Level above. As problem solving becomes more common in primary schools, we would expect some of the more difficult strategies to come into use at lower Levels.
Strategies can develop in at least two ways. First children’s ability to use strategies develops with experience and practice. We mentioned that above. Second, strategies themselves can become more abstract and complex. It’s this development that we want to discuss here with a few examples.
Not all children may follow this development precisely. Some children may skip various stages. Further, when a completely novel problem presents itself, children may revert to an earlier stage of a strategy during the solution of the problem.
Draw: Earlier on we talked about drawing a picture and drawing a diagram. Children often start out by giving a very precise representation of the problem in hand. As they see that it is not necessary to add all the detail or colour, their pictures become more symbolic and only the essential features are retained. Hence we get a blob for a pig’s body and four short lines for its legs. Then children seem to realise that relationships between objects can be demonstrated by line drawings. The objects may be reduced to dots or letters. More precise diagrams may be required in geometrical problems but diagrams are useful in a great many problems with no geometrical content.
The simple "draw a picture" eventually develops into a wide variety of drawings that enable children, and adults, to solve a vast array of problems.
Guess: Moving from guess and check to guess and improve, is an obvious development of a simple strategy. Guess and check may work well in some problems but guess and improve is a simple development of guess and check.
But guess and check can develop into a sophisticated procedure that 5-year-old children couldn’t begin to recognise. At a higher level, but still in the primary school, children are able to guess patterns from data they have been given or they produce themselves. If they are to be sure that their guess is correct, then they have to justify the pattern in some way. This is just another way of checking.
All research mathematicians use guess and check. Their guesses are called "conjectures". Their checks are "proofs". A checked guess becomes a "theorem". Problem solving is very close to mathematical research. The way that research mathematicians work is precisely the Pólya four stage method (What is Problem Solving?). The only difference between problem solving and research is that in school, someone (the teacher) knows the solution to the problem. In research no one knows the solution, so checking solutions becomes more important.
So you see that a very simple strategy like guess and check can develop to a very deep level.