Seeing numbers at a glance
When numbers are written on paper or shown on a calculator screen, they are represented by a set of abstract squiggles. It takes children a while to learn which squiggle to associate with which number - 1 with one, 2 with two, and so on. They recite the numbers in turn as they learn to count a group of objects.
So, as the key objectives for Reception Year in the National Numeracy Strategy explain, they learn to: lSay and use the number names in order in familiar contexts.
lCount reliably up to 10 everyday objects.
lRecognise numerals 1 to 9.
But the result of all this counting and sequential recitation is to build an understanding of each number as a collection of ones, not as a concept in its own right. Five, for example, has meaning and existence primarily as the number that comes after four. There is no overall understanding of the fiveness of five. Rather, it is seen as the result of one and one and one and one and one.
But this emphasis on the sequential nature of numbers is not the only one possible. It has perhaps been forced on us by our reliance on print, particularly in the school curriculum. But we, as teachers, can use other approaches and these can give a different emphasis. For some children - especially those for whom pictures come more easily than words - a more holistic approach to numbers will make much more sense.
Most people can learn to scan up to four objects and see how many there are without having to count them. We can see a collection of four dots, or fingers or goldfish, or whatever and know that there are four, with no need to go through the sequential process of counting one, two, three, four.
Some arrangements are easier to see than others but we can learn to recognise the number of objects in any group of up to four.
So children can learn to understand four not as one add one add one add one, but as an image of four objects. Similarly for one, two or three, they can see the whole, not just the separate parts.
So much for numbers up to four. But how can we go beyond ?
Our first and most readily available resource is literally to hand.
Children can learn to see (see, not count) up to four fingers on one hand.
But they can also see the whole hand and learn that this is a representation of five - although other, random representations of five are much harder to just "see". Most people have to count them.
So now we have the numbers one to five, each able to be represented by the digits on one hand.
But we have two hands. So, just as children can learn to see (not count) that this pattern of fingers is three: so, in the same way, they can learn to see that this pattern is eight: So the numbers one to 10 can be represented as patterns of fingers on a pair of hands. This approach helps to establish each number as a whole, rather than as a part of a sequence. It also provides a concrete, rather than a symbolic image of number. And finally, the representation of each number involves physical movement. So children develop an understanding of number that is based on aural, visual and kinetic images. How much more powerful than any merely symbolic representation!
But how can we go beyond 10 without symbols? This is where some types of abacus comes in.
The majority of readily available abacuses in the UK lead naturally to a "counting" approach to number. There are typically five or 10 rows of 10 beads, with each row painted a different colour - 10 red beads, 10 blue, 10 yellow and so on. Nothing about the row of ten red beads gives any help in just seeing (not counting) numbers up to 10. Given a row of seven identical beads to look at, for example, children have no choice but to count them to discover how many there are.
But the Slavonic abacus is one of a type which supports the seeing rather than the counting approach to number. Slavonic abacuses are much more common on the Continent than here or the US, but they are well worth finding or making. They have the usual 10 rows of 10 beads, but these are coloured with only two, or at the most four, colours in such a way that each row is made up of five beads of one colour and five of another.
One row of beads on the Slavonic abacus allows us to represent numbers up to 10 in the same way as on a pair of hands. So work on the first row of beads follows naturally from simple finger-pattern arithmetic carried out by seeing whole numbers of fingers, not counting them one by one.
Children can see (not count) the numbers one to four in the usual way.
But now they can also see (not count) five beads because they are distinguished by their colour from the rest of the row.
And they can learn to see (not count) eight as a pattern of five of one colour, and three of the other colour.
This approach to numbers up to 10 also has the advantage that every number is seen with its complement to 10, so each number becomes deeply associated with its complement. The two that goes with eight, for example, becomes an inbuilt aspect of eight.
And the way in which the beads on the Slavonic abacus are coloured allows us to go further and see numbers up to 100, with their complements to 100.
For example, children can see that this abacus shows 72 beads and that 28 are left.
So the Slavonic abacus offers children a way to understand each number up to 100, not as a point in a recited sequence, but rather as a whole that is seen, not counted.
This is not the approach to number advocated in the national curriculum.
Nowhere in the numeracy strategy is there an emphasis on teaching children to "subitise" - see a number of objects at a glance. Throughout their Reception Year and beyond they are expected to count. But for children whose "spatial intelligence" outstrips their "numerical intelligence", the sequencing activity of counting may be at best tedious and at worst virtually meaningless. The holistic approach of the Slavonic abacus offers a worthwhile alternative.
If this type of abacus is unavailable then an overhead projector slide with a coloured transparent plastic sheet cut to mask dots to the right and below the number being shown, or a block of Multilink cubes, may be used instead to convey the same ideas.
A master for the OHP slide is on the Association of Teachers of Mathematics website, www.atm.org.uk The site also gives a supplier of Slavonic abacuses and an electronic version for downloading. Further information on Slavonic abacuses can be found in the following publications:Elementary Mathematics and Language Difficulties by Eva Grauberg (Whurr Publishers pound;22.50) Spatial Ability: a Handbook for Teachers edited by Tandi Clausen-May and Pauline Smith (National Foundation for Educational Research pound;8).
Tandi Clausen-May is principal research officer, Department of Assessment and Measurement, National Foundation for Educational Research