..or... life is never as straightforward as you think. Jenny Piggott and Jenni Back ask what are problem solving and mathematical thinking, and how do they relate to what we do in the classroom?

Primary Mathematics,Autumn 04, Mathematical Association, Leicester

Problem solving skills are generic skills that help us tackle the unknown. A problem is something we do not immediately know the answer to and often do not immediately know how to go about solving:

"A problem is only a problem when you cannot do it"

Problem solving involves a structuring of our thought and activities in order to help us move from looking at the problem towards a solution. There are many models of problem solving, including, possibly most famously, that offered by Polya (1957). All these models have a similar structure which comprises (very briefly):

- Getting to know the problem
- Thinking about how you might solve the problem
- Solving the problem
- Reflecting on the problem.

It is worth saying here that we do not often move neatly
through these four stages of problem solving but sometimes jump
stages or revisit stages as part of a process of clarification and
restating the problem.

These skills are valuable not just to support mathematics but
also as a generic aid to problem solving across the curriculum (and
beyond). Having a problem solving weapon in your armoury seems to
be a skill of value from the cradle to the grave. But these skills
are not easily learnt or taught, not least because many of us have
not had personal experience of real problem solving in the
classroom. Problems and problem solving are key features of
mathematics and can pervade mathematics experiences of pupils from
the early years. Putting pupils in challenging situations where the
route to a solution is not immediately obvious and giving them
experience of that slightly uncomfortable feeling of "not knowing"
is valuable in helping to develop their thinking skills. It also
helps to develop their understanding of maths as requiring more
than rote learning, and being about reasoning and experimenting. It
is this aspect of maths that contributes to its having a label of
being "hard" but it does seem to be an essential part of the
subject. At NRICH we believe that by giving pupils experiences of
problem solving we are inducting them into what mathematics is
really about and giving them the skills to tackle its difficulties
with greater confidence as they move through the Key Stages.

One thing we have been describing in the last paragraph is teaching "for" problem solving. That is, regarding it as a key mathematical activity and worthy of being taught -just like we teach percentages because it is both a fundamental part of mathematics but also has generic benefits. However to do this we also need to teach problem solving skills and one might assume that the obvious way to teach such skills is through problem solving itself. This is sometimes termed "teaching about" problem solving. Finally we can also teach mathematical concepts through problem solving. The idea is to teach new concepts by introducing them through problem solving. So here are at least three ways that we might think of problem solving:

1. Teaching for - Teaching for problem solving involves
children in experiencing mathematics as a problem solving arena and
gives them the chance to "be a mathematician" in some sense. Real
mathematics involves solving real problems not just running through
procedures that others have devised and problems that have already
been solved. Offering children the opportunity to have a go at a
problem that may seem totally new and different for them can give
them a sense of what doing real mathematics is like.

2. Teaching about - Teaching about problem solving involves
developing problem solving skills. In this role, we aim to help
children to adopt a model for problem solving (such as the four
stage model above). But, if only life were that easy! Mathematical
problem solvers also need to have at hand some particular
mathematical thinking skills such as , such as generalising and
being systematic which they can employ as part of their problem
solving and these also need to be learnt.

3. Teaching through - Teaching through problem solving
involves using a specific problem to teach a mathematical concept
or result.

NRICH uses problem solving in all three of these ways. Let's look at examples of each of these:

Teaching for :

Here is a problem from the October website that poses a challenge and involves thinking 'outside the box':

You might decide to use this problem to encourage pupils to simply engage in problem solving. We think the problem also provides opportunities to discuss, share ideas and visualise and then convince an observer of a strategy. This process has value as a mathematical activity in its own right.

Teaching through:

Here is a problem that offers a way of introducing the Fibonacci sequence:

In this case you might use the problem to introduce the Fibonacci sequence and illustrates the fact that there are real situations that generate what otherwise seems like a purely mathematical invention.

One could of course be teaching about problems solving every time pupils tackle problems but perhaps we should be making these aspects of problem solving far more explicit as we teach. Perhaps we should include them in our learning objectives. Of course you can add a lot more detail to all of this. In particular, you might notice that the "stages" of problem solving are not particularly mathematical, but skills which would be useful to have in any problem situation, in Science, Art, Music and so on. These skills then have value and are worth learning and becoming efficient in because they are very useful skills to have. They also form a backbone of continuing mathematics across phases -a connection throughout a pupil's lifetime of mathematics learning. Learning to be confident about not knowing will also prove an invaluable piece of armoury as pupils move from Primary into Secondary phases.

At NRICH we recognise that within mathematics there are particular problem solving skills such as "being systematic" or "generalising" that need to be developed. In other words we can recognise some particular components of mathematical thinking that support pupils with their problem solving in mathematics. There are many more of these mathematical thinking threads and a list of some of them can be found on the website (www.nrich.maths.org.uk/content/research/a-framework-for-enrichment.doc). These are not things we tend to have sufficient familiarity with to know "how to teach" in the same way we might be confident about teaching about areas of rectangles and compound shapes. At NRICH we are putting together packages of material that develop these ideas in an "ordered" and we think "developmental" way. We are calling them mathematical thinking "trails". One example of a trail we have recently worked on is "being systematic". We often ask pupils to be systematic but do not often reference it to pupils when we model it in the classroom. Nor do we identify specific curriculum opportunities that can be used to illustrate this skill in different contexts, and that will give pupils the chance to revisit and consolidate the thinking skill over a period of time. This particular group of activities is designed to take pupils through KS2 into KS3. We will be publishing an electronic version of the trail online with teacher notes and guidance but to give you a flavour of the sort of problems that are included here is an investigation about numbers that would be accessible to many children at Key Stage 1 but that will really stretch older children as the investigation develops and a proof is created.

Teaching about:

This problem may be an ideal opportunity to use as a vehicle to identify some of the strategies we adopt as problem solvers -using the techniques of being systematic and generalising within the four stage model of problem solving. Start by getting to know the problem by experimenting with some numbers and seeing what happens.

So the message is that there is a great deal of danger in talking about problem solving or mathematical thinking without trying to make sense of these very complex and difficult ideas and skills -life is never as straight-forward as you think but NRICH is doing its best to help make sense of it for you!

Reference: Polya, G. (1957). How to Solve it, Princeton University Press.