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Primary Mathematics, Vol 8, Issue
1 Leicester: Mathematical Association, Spring
A lot of interest is being voiced at the moment in primary schools
about the possibility of 'enriching' the mathematics that we offer
to children whilst delivering the stipulations of the Numeracy
Framework. This interest is frequently linked to the possibility of
developing children's problem solving skills. At NRICH our work has
always focused on problem solving and enrichment, and we have
recently been considering in some depth what we mean by these two
ideas and how they impinge on children's mathematical learning.
What started as a resource to support able young mathematicians who
might be socially isolated and who needed additional stimulation,
has developed into a resource bank of great size that is used by a
far wider audience than pupils who have been involved in
masterclasses. We have found that many teachers have been using our
resources with children from a wide age and attainment range.
We have consciously embraced these developments and changed our own
thinking so that we now regard enrichment as a priority for all
students regardless of attainment. However we do need to clarify
what we mean by enrichment and the purpose of this article is to
illustrate our current thinking in this area. Fundamental to this
clarification is the idea that enrichment is not only an issue of
content but a teaching approach that offers opportunities for
exploration, discovery and communication. We also suggest that
effective mediation offers a key with which to unlock the barriers
to engagement and learning.
A number of problems, such as poor performance on international
tests, meeting the needs of the most able through acceleration
programmes that leave them short changed at a later date, and the
lack of take-up of maths courses at sixth form and beyond are
becoming evident. Although some of these issues appear to be beyond
the remit of the primary school, they do impinge upon it and we
would contend, need to be considered from the beginning of
children's mathematical education. We would suggest that a
programme of enrichment can be used to address these problems and
that enrichment has a place in the curriculum for every child not
only the highest attainers.
In the current literature, 'enrichment' is used almost exclusively
in the context of provision for the mathematically most able.
However, there is strong evidence from the use of the NRICH
website, and our experiences from working with teachers and pupils,
that this fails to address the value of an enrichment approach to
teaching mathematics generally. Problems which offer suitable entry
points can be used with pupils of a wide range of ability and
therefore can be used in the "ordinary" classroom. The labelling of
pupils with titles such as "gifted" and then treating them for this
handicap is replaced by an opportunity for pupils to '"describe"
themselves by what they can and want to do. The teacher or mentor
can use such materials in flexible ways that respond to the needs
and experience of the learner. We see enrichment as an approach to
teaching and learning mathematics that is appropriate for all, not
simply the most able. Good enrichment education is good education
for all. Good mathematics education should incorporate an approach
that is an enriching and stimulating experience for all
It naturally follows that enrichment can be used to support the
most able alongside all the children in a class, often offering
differentiation by outcome. It can also be used to promote
mathematical reasoning and thinking skills, preparing pupils
through breadth and experience to tackle higher level mathematics
with confidence and a sense of pattern and place.
Thus enrichment is dependent on two main components: content
opportunities and teaching approaches. Content opportunities need
to extend the mathematical repertoire of the pupils using them, as
well as to take account of the historical and cultural contexts of
the classrooms in which they are offered. We are aiming to offer
approaches to mathematical learning that encompass more than simply
learning facts and demonstrating skills, and which support pupils'
problem solving. In pursuing these ideas we aim to improve pupil
attitudes to mathematics, enable them to develop their appreciation
of mathematics and develop their conceptual structures. The
teaching approach that we are advocating reflects a constructivist
view of learning and stresses non-assertive mediation, group work,
discussion and communication. It respects the variety of different
approaches and solutions available to any given mathematical
problem and values exploration, flexibility, making mathematical
connections, extending boundaries and celebrating ideas rather than
just answers. It acknowledges that maths is hard but that success
is all the more enjoyable when a hurdle is overcome.
For the March 2004 website we took Codes and Hidden Meanings as our
theme. This is not part of the standard curriculum but offers
plenty of opportunities for exploring challenging mathematical
problems. To illustrate our approach we have selected two problems
from this month. The first was offered as a 'Content level 1,
challenge level ***' which means that the curriculum content would
be approachable by any child who had covered the maths curriculum
at Key Stage 1 but the challenge level is very high for children of
this Key Stage. The problem is about semaphore signalling:
Semaphore is a way to signal the alphabet using two flags, one held
in each hand. To send a message, your left and right hands have to
be in two different positions. You start with both hands pointing
down. Here are the signals for the letters of the alphabet:
What does this message say?
You might want to send a message that contains more than just
letters (exclamation marks, question marks,full stops etc). How
many other symbols could you send using this code?
The second problem is on the same theme but more appropriate to
children who have covered the maths curriculum at Key Stage 2. The
challenge level is once again very high in this problem so it was
presented as Content level 2, challenge level ***:
Someone at the top of the hill sends a message to a friend in the
valley. A person in the valley behind also sees the message being
sent. They get the same message. What is it?
Are there any words that can make sense when viewed from the front
and the back? We have found at least one!
In offering problems linked in this way on the same theme we hope
to give teachers and children the opportunity to develop their
thinking within a particular context and so develop their problem
One key aspect of our work at NRICH has always been the maintenance
of a dialogue with our users, both teachers and children, and the
publishing of children's mathematical work as solutions on our
website. This helps to emphasise the communicative aspects of
mathematics and gives pupils a real audience to which to present
their work. Problems appear without solutions in the month in which
they are first published and we have a number of contributors who
regularly rush to get their solutions to us so that they can see
them posted on the web. Why not encourage your class to have go
next month and see whether they will be successful in providing us
with a solution that we can publish? We are looking for clear
explanations and evidence of sound mathematical thinking. In this
context, children often feel motivated to offer us clear
communication about their mathematics.
Semaphore diagrams taken with permission from http://inter.scoutnet.org/semaphore/semaphore.html