What Problem Solving
Finally, the better students especially, may be able to generalise or extend the problem.Generalising a problem means creating a problem that has the original problem as a special case. The last part of that problem asks how many towers can be built for any particular height.Pólya’s second stage of finding a strategy tends to suggest that it is a fairly simple matter to think of an appropriate strategy.However, there are certainly problems where children may find it necessary to play around with the information before they are able to think of a strategy that might produce a solution.This is useful to show others what they have done and it is also helpful in finding errors should the right answer not be found.At this point many children, especially mathematically able ones, will stop.So a problem about three pigs may be changed into one which has any number of pigs. The answer to this problem will contain the answer to the previous three questions.
Children (and adults too for that matter) will often not be able to absorb all the important information of a problem in one go.When you think about it, the whole aim of education is to equip children to solve problems.In the Mathematics Curriculum therefore, Problem Solving contributes to the generic skill of problem solving in the New Zealand Curriculum Framework.But it is worth getting them into the habit of looking back over what they have done. First of all it is good practice for them to check their working and make sure that they have not made any errors.Second, it is vital to make sure that the answer they obtained is in fact the answer to the problem and not to the problem that they thought was being asked.This is the side of mathematics that enables us to use the skills in a wide variety of situations.Before we get too far into the discussion of Problem Solving, it is worth pointing out that we find it useful to distinguish between the three words "method", "answer" and "solution".Third, in looking back and thinking a little more about the problem, children are often able to see another way of solving the problem.This new solution may be a nicer solution than the original and may give more insight into what is really going on.On the other hand, we use "answer" to mean a number, quantity or some other entity that the problem is asking for. It is frequently the case that children move backwards and forwards between and across the steps.Finally, a "solution" is the whole process of solving a problem, including the method of obtaining an answer and the answer itself.method answer = solution But how do we do Problem Solving? Pólya enunciated these in 1945 but all of them were known and used well before then. The Ancient Greek mathematicians like Euclid and Pythagoras certainly knew how it was done. In fact the diagram below is much more like what happens in practice There is no chance of being able to solve a problem unless you are can first understand it.