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In actual fact, the best teaching approach is probably some combination of the two.Most of the problems used in problem solving have more than one solution.
(We say more about this in the 'Reporting Back' section of Organising the Teaching of Problem Solving.)Positive attitudes.
Because the children seem to enjoy the problems, and get quite involved with them (we have seen children work through their breaks in order to settle a problem), it helps them to gain a positive attitude towards the subject.
And it’s partly because children enjoy having "ownership" of the problem. By working on a problem, children become involved with it and can get quite deeply involved with the mathematics that is both required to solve it, and that may be required to solve it. In the process of struggling with a problem, children can often obtain a fairly deep understanding of the mathematics surrounding the problem.
This understanding is often enhanced, when, in a whole class setting, teachers draw together the various threads from all of the children in the class.
This also makes them more interesting and relevant to the children.
Then again the questions can be very interesting in themselves.So each of them can be approached in a variety of ways, some of which are sophisticated and some of which are less sophisticated.Hopefully, every child in your class can find one approach that they can use to solve the problems that you present.This proposes that we construct our knowledge through our experience rather than absorbing what we are told.The constructivist views the child as an active learner.Very little encouragement has been given to cooperation in the traditional didactic approach to teaching mathematics.There has not been the emphasis on children working together that there has been in other curriculum areas. But there are other reasons for it to be part of the mathematics curriculum.The following are some reasons that are frequently suggested as to why you should include problem solving in your maths programme. Currently much credence is being given to a theory of learning called constructivism.Even apparently relatively weak children may have ideas that turn out to be fruitful.Sometimes though, it can take a little work on the your part before the consequences of some ideas are seen. It is important to point out at this stage, that though we are concentrating here on mathematical problem solving, many of the strategies and techniques that are used in mathematics are used in any type of problem.