EIGHT PROBLEM SOLVING STRATEGIES.
1. WORK BACKWARDS
To solve some problems, you may need to undo the key actions in the problem. This strategy is called Work Backward.
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.
2. GUESS AND CHECK
Some problems cannot be solved directly. You need to use a strategy called Guess and Check.
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.
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.
3. LOOK FOR A PATTERN
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.
4. DRAW A PICTURE
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. 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.
5. MAKE A TABLE
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.
6. 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 organize things systematically as they go. This even applies to their explorations.
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.
7. ACT IT OUT
Meaning that two strategies combine together because they are closely related.
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 are 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.
8. USING SYMMETRY
It helps us to reduce the difficulty level of a problem. Playing Noughts and crosses, for instance, you will have realized 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 analyze. This sort of argument comes up all the time and should be grabbed with glee when you see it.