Jade is three years old. She's sitting, head bent in concentration, over a table scattered with small, flexible plastic cars, boats and trains, muttering to herself as she arranges the toys and starts to point at them and tap them with her index finger. "Got all the boats there. Cars, you're going over there, beside the trains. There. One, two, three boats. One, four, three cars. How many trains? Three, five, six."
Many adults, parents and early years practitioners alike, might, on hearing this running commentary, gently encourage Jade to try counting again in order to help her to "get it right" - that is, to make sure she arrives at the correct number of cars, boats and trains, and uses number names in the right order. They might not realise that Jade is demonstrating the very clever, important and valuable steps that she has already made along the road that leads to being able to count. She is also showing the breadth of her emerging mathematical thinking and, by talking herself through what she is doing, she is actively helping herself learn.
Jade knows that numbers have names. She knows that one number name belongs to each object (the one-to-one principle). She also knows that the last number in a sequence has special importance, because she stresses it in her intonation. This will in time help her grasp the cardinal principle - namely that the last number name she recites identifies the size of the group of items being counted. Jade knows that fingers make handy tools for helping us to count, reflecting the fact that moving the fingers and thinking mathematically are areas of the brain that are closely linked.
Jade knows lots of numbers by name, but is not yet secure in her knowledge of which order they are used in. Once she learns that we use number names in the same order every time, she will have grasped the stable order principle and be able to count accurately. At the moment, accuracy doesn't matter. Jade is enjoying herself, is motivated, is having a go, and is not being told she's got it wrong - and that's what matters, since these are the factors which will help her to develop into someone who explores mathematical concepts with confidence and enthusiasm.
It is always delightful to hear small children uninhibitedly chatting their way through the games they devise. In this instance, Jade is not only demonstrating an awareness of position, but also her knowledge of the words we use to describe it (over there, next to). She can see the difference between the cars, boats and trains, and is eagerly imposing a sense of order. By playing with small items, Jade is developing her manipulative skills, which are useful for when she's a little older and may want to start recording her thoughts on paper.
Many adults, parents and practitioners alike, lack confidence when it comes to helping children with mathematical development, perhaps because of their own experiences in the classroom. Workshops and training sessions are often poorly attended - after all, no one wants to risk being shown up by their lack of knowledge. It is worth remembering that mathematics is taught very differently now, with an emphasis on a secure grasp of the basic concepts.
One of the key ways in which you can help three to five-year-olds in this area is to encourage them to use their whole bodies to explore the environment and gain first-hand experience of concepts such as heavy and light, big and small, fast and slow. The body develops from the head down and from the inside out, which explains why a child can move his arms with precision long before his fingers will co-operate fully with detailed tasks. When children experience crouching, stretching, running, jumping, climbing and rolling, pathways are laid down in the brain and these contribute greatly to learning about position, size, shape and space.
If, like many practitioners, you find you use your indoor space more readily than outside, and have not thought of your garden or outdoor area in relation to mathematical development, take a look at Learning Through Landscapes (www.ltl.org.uk), a website which offers help for schools and nurseries to make the most of outdoor spaces.
Young children may not appear to be learning much about maths while scrambling under, over and through an assault course that you have carefully planned in the garden, but they are, and by encouraging, guiding and praising them, you're giving them the sort of help that really counts.
To encourage whole-body mathematical development, hold a Jumping for Joy Week. Plan activities that encourage jumping with a mathematical focus embedded in a cross-curricular approach.
* Play at being rabbits, kangaroos or frogs (perhaps with the option to dress up).
* Hold jumping competitions, inside or out, and focus on high, highest, up, down, zig-zags, forwards and backwards, over and on to.
* Talk about things that make you jump. This is a good opportunity to explore anxieties and encourage communication against a backdrop of a common reaction - after all, everybody jumps sometimes.
* Read the story of Little Lumpty by Miko Imai (Walker Books, pound;4.99) to the children. It's all about one of Humpty Dumpty's descendants who is irresistibly drawn to high walls; one day he has to summon all his courage and jump!
* Build in an awareness of safety to encourage spatial awareness and social development. Think about the feeling of physical well-being from keeping the body fit.
* Add a number dimension by seeing if everyone can jump once, twice, three times - and don't be afraid to join in. The children will literally jump for joy to see you and your colleagues leaping about and the more fun they have, the more they'll learn.