Problem Solving and Me . .
Robinson, James, Mathematics Teaching
In the Cockcroft report of 1982 the following list of types of activities appeared, with a very similar list appearing in the KS3 strategy :
'Direct teaching and good interaction are as important in group-work and paired work as they are in whole-class work. Organising pupils as a whole class . . . helps to maximise their contact with you . . .'
Key Stage 3 strategy
The KS3 strategy encourages us to use an open investigative approach with provision for work on consolidation etc., and it also shows that the Cockcroft report and the KS3 strategy are not opposites, but complementary in terms of advice.
One conflict I feel here is that the sample medium term plans contain far too much for us to cover in detail. Was it ever said that we need perfect detail? The KS3 strategy is built upon a spiral, and thus if students do not gain an understanding this time, they have at least one more chance this year, with more in years 8, 9 and beyond!
Another conflict is the concept of sharing the objective with the pupils before you begin to teach the topic. Inevitably, using a problem solving strategy they may not learn topic A at all, but B or C instead. Thus the 'objective' cannot usually be revealed until the end. However, it is possible to say that you wish students to engage with a certain topic after an initial period of discussion.
I have come to the decision that a problem solving strategy is the best for pupil learning. However, I have heard this phrase many times without any discussion of what it means. Investigations are often experienced by pupils as a means unto themselves - they 'do' investigations as a separate part of mathematics - rather than as a way of delivering curriculum aims.
My definition of a problem solving strategy
'We have assembled some starting points and have tried to indicate a few of the many roads that can be taken. There are no finishing posts.'
Banwell, Saunders and Tahta; 1972 
A problem solving strategy is one in which students are encouraged to engage with an open task in order to come to some understanding of a topic themselves, preferably before any formal teacher input. A plenary session can then occur before further tasks can be used to consolidate the learning. Assessment can take place in the form of a write-up, formal written test or presentation.
To plan in this way requires identification of key mathematical ideas, in pedagogic terms, and thought about how a student might come to understand them.
Mike Ollerton & Anne Watson; 2001 
The basis is to decide what single method or idea you wish the pupils to learn and find a problem that will lead to understanding or consolidation of this. Extensions are then used to help students understand related objectives from this or other units of work.
I suggest the following models for delivery using a problem-solving approach:
One example of where a formal method may be required is factorisation of quadratics. This does not preclude the use of problem solving to consolidate the learning of the methods. One suggestion here is to get students to split a page vertically and on one side multiply out pairs of brackets writing the answer on the other. The page can then be cut in half and another student given the 'questions'.
Methods of introduction include: the whole class working on a single problem, a linked set of work-cards on a single topic, whole class discussion leading to a problem.
Why problem solving?
'Being able to do something is close to wanting to do something'
I forget where I heard this quotation but the next one is surely one we have all heard:
'Why are we doing this, sir?'
I have never been asked this when working on a problem-solving task.
Given a suitably simple starting point anyone can begin to engage with mathematics. And, if you accept the principle of the first quotation, this causes students to engage. …