Teaching and Learning through Problem Solving
Ollerton, Mike, Mathematics Teaching
Mike Ollerton relates some problem solving work with primary schools to DfES support.
The following piece of writing emerged from working in four primary schools in the West Midlands. The focus was teaching mathematics through problem solving. The schools had already begun to work through materials published on the DfES 'standards' site (www.standards.dfes.gov.uk), so I looked at these materials as part of my planning.
I noted the way problem solving was broken down into specific sections by DfES:
* finding all the possibilities
* logic problems
* finding rules and describing patterns
* diagram problems and visual puzzles
* word problems.
Each section contains suggested problems to engage pupils with each of these aspects of problem solving. Whilst I wholeheartedly celebrate the 'official' recognition of the role of problem solving to support the learning of mathematics, I am concerned about the prescribed way the materials are to be used, both in lessons and at staff meetings. However, as a way of bringing problem solving to the fore, this initiative is worth celebrating. In this article I hope to offer constructive ways forward and, at the same time, describe an interesting surprise event that occurred in a KS1 classroom.
Problem solving does embrace 'finding all the possibilities', 'logic', etc. I wonder, however, about breaking it down into specific steps or ways of working. Problem solving is rarely a clean, clear set of procedures, otherwise where would the 'problem' reside? Problem solving, by definition, is likely at times to be messy and ambiguous. When used effectively, problem solving becomes the vehicle for developing pupils' mathematics and simultaneously helps develop autonomous decisionmaking and independence of thought and action. Problem solving also provides learners with a purpose or a context for learning mathematics. Finding all the possibilities and solving logic problems are only subsets of problem solving. I am far more interested in finding problem solving approaches that enable pupils to process, and therefore develop, their mathematical content knowledge.
For example, a problem such as 'Find all the whole number pairs that add together to make 10' can lead to pupils working on a variety of processes and concepts. Some of these are:
* ordering information or working systematically (processes)
* pattern spotting and generality (processes)
* determining that all possible answers have been found; and proof (processes)
* recognising commutativity, eg 6 + 4 = 4 + 6 (concept)
* practising basic addition (concept)
If these pairs of values are then turned into coordinate pairs and then graphed, a range of other concepts can emerge, such as:
* working with the co-ordinate system
* drawing a graph
* the non-commutativity of the system of coordinates (except when y = x)
* negative numbers (if one of the pair of numbers is > 10)
* decimals (if non-integer values are allowed)
* finding the equation of the graph
* recognising that if the initial value of 10 is changed to 12 a parallel graph is produced.
The starting question can be made quite simple and a variety of extension tasks are possible; thus, issues of access and depth have to be considered. These depend upon a teacher's working knowledge of their class.
I suggest there are five key criteria involved in setting up problem solving situations:
1 Any problem must be aimed at developing pupils' content knowledge
2 Problems need to be accessible, whilst at the same time puzzling
3 Use can be made of more 'open' type questions
4 To support differentiated learning, problems must be extendable
5 Independent learning is fostered.
1 Developing pupils' content knowledge
Mathematical processes help pupils make sense of mathematics. …