Published November 2009,December 2009,February 2011.
This means we have resources that fit together like this:
An important point to bear in mind is that whilst a problem
normally requires all aspects of a learner's problem solving skills
to be brought to bear, it will probably only cover a limited area
of content knowledge (possibly one key concept or several concepts
which come together and enable connections to be made). In any
single context it is often possible to place different emphases on
different aspects of process skills, such as representing or
communicating, for making particular teaching or assessment points.
However, it is almost impossible
and generally undesirable to try to isolate process skills when
With this model in mind we now want to look at what progression
in process looks like and therefore what we should look for when
assessing learners' process skills.
There is progression in mathematical process/problem solving.
This progression lies within the experiences of the processes
themselves. Processes are not a simple set of six (or so) things to
do in a particular order and once you have done them you can "do"
processes. Process skills are complex beasts, in terms of breadth,
depth and diversity:
So some time needs to be spent on a wide range of problems to
ensure learners are given the opportunity to experience, and
practise, the depth, breadth and diversity of process skills and in
doing so they will become more competent problem solvers.
information on problem solving and mathematical thinking skills can
be found in the
research section of the NRICH
website - articles including "
Developing a framework for mathematical thinking" and "
Mathematics Enrichment: what is it and who is it
You might think of all I have said so far as sitting within the
mathematics. However, much of what makes an effective mathematical
problem solver goes beyond mathematical practice and into
developing more general attitudes and ways of thinking that
encourage confidence. These more generic skills will have value
well beyond mathematics but are crucial in the development of
This is the area of progression that builds upwards from the
content and process elements of the curriculum. In other words, we
have a third dimension:
Effective learners develop through encouraging greater
independence, reflection, resilience, participation and creativity,
whilst offering contexts which are set in more complex and less
familiar settings. As learners become more effective
mathematicians, they also become more confident mathematicians and
therefore better problem solvers. The strands of independence,
reflection, resilience, participation, creativity, complexity and
familiarity are ways of describing aspects of a more effective and
confident mathematician. I cannot say whether confidence results in
development along these strands or the development results in
greater confidence. What is important is that confidence is seen as
underpinning (or overarching) these ideas and is an important
characteristic of an effective learner.
Explanations of the terms I have
used can be found in the
So, how does this link to NRICH problems?
Well, NRICH is full of resources which offer opportunities to
practise and develop problem-solving (process) skills. The
activities give scope for learners to work at a range of levels
within those skills whilst encouraging them to move on in their
thinking and confidence (the third dimension).
Using the problem
Tricky Track I have tried to describe a progression in learner
behaviour that might indicate development in the "third-dimension".
In reality learners might vary considerably in each of the threads
of development. For example, they may be very creative but not very
resilient, or work independently - but only in very familiar
settings or certain areas of mathematics. It is also the case that
learners may score very highly in this dimension but be working at
a low content-knowledge level. In other words, very young
mathematicians will be very mathematical whilst working within
content knowledge in line with their age.
The following table gives examples of what might distinguish
learners working at a low level in the "third-dimension" and those
working at higher levels on the problem
Tricky Track. This can inform us of learners' progression, the
level at which they are working and their potential for moving on.
The table also indicates some of the implications for teacher
action that can influence learner behaviour.
Assessing Pupil Progress is about supporting learning. Its aim
is to help teachers to find out more about their pupils' strengths
and weaknesses, and help them plan for learning that takes into
account learners' individual needs. Whilst there is a need to keep
track of what a pupil can do and where they might go next, APP is
not about ticking hundreds of boxes, levelling every piece of work,
or doing additional assessment activities in addition to those you
already undertake at school. It is about knowing what might come
next on a learner's journey and offering opportunities for that
learner to progress.