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. They are a collection of general approaches that might work for a number of problems. There are a
number of common strategies that students of primary age can use to help them solve problems. We discuss below several that will be of value for problems on this website and in books on problem solving. Common Problem Solving Strategies 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 strategywith 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). This kind of poster provides good revision
for students. 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 includes two or more subcategories. 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.What Are Problem Solving Strategies?
An In-Depth Look At Strategies
This stands for two strategies, guess and check and guess and improve.
Because it is so simple, you may have difficulty weaning some students away from guess and check. As problems get more difficult, other strategies become more important and more effective. However, sometimes when
students are completely stuck, guessing and checking will provide a useful way to start to explore a problem. Hopefully that exploration will lead to a more efficient strategy and then to a solution.
We put two strategies together here because they are closely related. These are Act it Out and Use Equipment.
This 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.
Sometimes the students acting out the problem may get less out of the exercise than
the students watching. This is because the participants are so engrossed in the mechanics of what they are doing that they don’t see the underlying mathematics.
Some students need to be encouraged and helped to use equipment. Many students seem to prefer to draw. This may be because it gives them a better representation of the problem in hand. Since there are problems where using equipment is a better strategy than drawing, you should encourage students' use of equipment by modelling its use yourself from time to time.
Here we discuss
both Drawing a Picture and 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.
Making Organised Lists and Tables are two aspects of working systematically. Most students 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 students 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.
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.
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
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 |
60 | 27 | 240 | 54 | 294 | 28 |
50 | 37 | 200 | 74 | 274 | 8 |
40 | 47 | 160 | 94 | 254 | -12 |
41 | 46 | 164 | 92 | 256 | -10 |
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 students 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 students can effectively use other strategies. However, we feel confident that most students 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 students' 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 students may follow this development precisely. Some students may skip various stages. Further, when a completely novel problem presents itself, students 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. Students 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 students 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 students, 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 students couldn’t begin to recognise. At a higher level, but still in the primary school, students 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.