The problem-solving process can be described as a journey from
meeting a problem for the first time to finding a solution,
communicating it and evaluating the route. There are many models of
the problem-solving process but they all have a similar structure.
One model is given below. Although implying a linear process from
comprehension through to evaluation, the model is more of a flow
backward and forward, revisiting and revising on the
This stage is about making sense of the problem by using
strategies such as retelling, identifying relevant information and
creating mental images. This can be helped by encouraging students
to re-read the problem several times and record in some way what
they understand the problem to be about (for example by drawing a
picture or making notes).
This stage is about "homing in" on what the problem is
actually asking solvers to investigate.
- Can they represent the situation mathematically?
- What is it that they are trying to find?
- What do they think the answer might be (conjecturing and
- What might they need to find out before they can get
Central to this stage is identifying what is unknown and what
Planning, analysis and synthesis
Having understood what the problem is about and established what
needs finding, this stage is about planning a pathway to the
solution. It is within this process that you might encourage pupils
to think about whether they have seen something similar before and
what strategies they adopted then. This will help them to identify
appropriate methods and tools. Particular knowledge and skills gaps
that need addressing may become evident at this stage.
Execution and communication
During the execution phase, pupils might identify further
related problems they wish to investigate. They will need to
consider how they will keep track of what they have done and how
they will communicate their findings. This will lead on to
interpreting results and drawing conclusions.
Pupils can learn as much from reflecting on and evaluating what
they have done as they can from the process of solving the problem
itself. During this phase pupils should be expected to reflect on
the effectiveness of their approach as well as other people's
approaches, justify their conclusions and assess their own
learning. Evaluation may also lead to thinking about other
questions that could now be investigated.
Using and Applying Mathematics
Aspects of using and applying reflect skills that can be
developed through problem solving. For example:
In planning and executing a problem, problem solvers may need
- select and use appropriate and efficient techniques and
strategies to solve problems
- identify what further information may be required in order to
pursue a particular line of enquiry and give reasons for following
or rejecting particular approaches
- break down a complex calculation problem into simpler steps
before attempting a solution and justify their choice of
- make mental estimates of the answers to calculations
- present answers to sensible levels of accuracy; understand how
errors are compounded in certain calculations.
During problem solving, solvers need to communicate their
mathematics for example by:
- discussing their work and explaining their reasoning using a
range of mathematical language and notation
- using a variety of strategies and diagrams for establishing
algebraic or graphical representations of a problem and its
- moving from one form of representation to another to get
different perspectives on the problem
- presenting and interpreting solutions in the context of the
- using notation and symbols correctly and consistently within a
- examining critically, improve, then justifying their choice of
- presenting a concise, reasoned argument.
Problem solvers need to reason mathematically including
- exploring, identifying, and using pattern and symmetry in
algebraic contexts, investigating whether a particular case may be
generalised further and understanding the importance of a
counter-example; identifying exceptional cases
- understanding the difference between a practical demonstration
and a proof
- showing step-by-step deduction in solving a problem; deriving
proofs using short chains of deductive reasoning
- recognising the significance of stating constraints and
assumptions when deducing results
- recognising the limitations of any assumptions that are made
and the effect that varying the assumptions may have on the
solution to a problem.
Functional maths requires learners to be able to use mathematics
in ways that make them effective and involved as citizens, able to
operate confidently in life and to work in a wide range of
contexts. The key processes of Functional Skills reflect closely
the problem solving model but within three phases:
- Making sense of situations and representing them
- Processing and using the mathematics
- Interpreting and communicating the results of the analysis