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..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.