Enrichment at NRICH
Primary Mathematics, Vol 8, Issue 1 Leicester: Mathematical Association, Spring 2004.
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 pupils.
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 Signals
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 ***:
Symmetrical Semaphore
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 solving skills.
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 scoutnet