So often in our work with teachers, we are asked what equipment we would recommend using with children and how we would use it. As a response to these questions we have put together some resources to help teachers develop children’s number sense. This was made possible by funding from the Nuffield Foundation which gave us the opportunity to explore what works and how.
What did we find out?
We asked teachers what they did already and the results were not very surprising – manipulatives are used mainly with younger and lower attaining children. We also found from existing research and our own experience that manipulatives can be much more useful than many people think and help children to understand complex ideas. They also help children to show how they are thinking and give them a way of making sense of mathematics, but only when used in ways that support that process of making sense and constructing understanding.
Our exploration of the field showed that teachers tend to use manipulatives to help children follow written procedures rather than understand numbers. We also identified three key aspects of number: counting, comparison and composition and then set about developing activities that would help children develop them.
From previous research we learnt that deep understanding of small numbers helps to build a secure basis for getting to grips with larger numbers and being able to see their structure in tens, as odd or evens, in ones or as part of other sets of numbers.
So we decided to focus on overlapping ranges of numbers: 0 to 12, 9 to 20, 15 to 50 and 25 to 100 and beyond. Counting, comparison and composition are all important in all of these ranges but the manipulatives that can be used form overlapping sets so we suggest different things for different ranges. The key in all our suggestions is to engage children in tasks they will enjoy whilst helping them make sense of increasingly sophisticated mathematical ideas.
So what have we done to help?
The main output from the project is our book, Making Numbers, which has just been published by Oxford University Press. We are thrilled with it as it captures, in full colour photos, all the different ways in which manipulatives can be used to help children develop number sense. There is an idea on nearly every page that you will be able to try out with children.
We believe that if we can establish the understanding of arithmetic on a sound basis by really delving deep into relationships and structure with young children, then they will be able to go on and achieve great things mathematically as they get older. We have also developed some stop frame animations which show just how children could use various objects to explore the structure of numbers: counting, comparing them to one another and thinking about how they are composed from other numbers. These animations are available to view when you register for free at www.oxfordowl.co.uk.
Here we also explain our rationale and offer some more examples . You can see how we would move the manipulatives to show different aspects of number sense. Take a look now.
We would love to hear what you make of the resources and hope you find them useful and enjoyable to use with children. We certainly found them to be so when we were trialing them.
Professor Rose Griffiths teaches at the University of Leicester on the Primary PGCE and on courses for experienced teachers. Rose has worked in pre-school settings, primary, special and secondary schools, and in adult and family education.
Dr Sue Gifford works in primary mathematics education at the University of Roehampton. She was previously a London primary teacher and a mathematics advisory teacher. She was a member of the Advisory Committee for Mathematics Education (ACME) and works with NRICH at the University of Cambridge to develop early years mathematics activities.
Dr Jenni Back is a freelance consultant in mathematics education with experience in teaching mathematics to children aged 3 to 19 years. She worked for the NCETM (National Centre for Excellence in the Teaching of Mathematics) on research and development projects in continuing professional development.