Old hands at the numbers game
A new Millennium demands a new idea and the Government's pride in the Numeracy Strategy suggests that it fits the bill. However, a closer look reveals that it is putting forward mathematical ideas which go back centuries.
As children in infant classes build up their knowledge and understanding of the number system, it can be pointed out to them that they are using methods known to ancient civilisations. Similarly children in junior classes are likely to be carrying out mathematics borrowed from the societies they may study in history and geography.
The Reception section of the framework suggests using fingers to aid counting or calculating. The existence of the base 10 system is a major clue to the fact that early number systems were probably based on counting fingers. Not all societies used base 10, base 20 was a popular option. It is tempting to think of base 20 as invented by bare-footed peoples in sunny climes who counted on their toes as well. Unfortunately this doesn't quite hold true as, for example, the Inuit developed a base 20 system, presumably without baring any more of their bodies than necessary.
There are also linguistic clues that counting started with fingers and toes, as many number words betray their origins. In the Banda language from central Africa the word for 15 can be translated as "three fists", and 20 as "take one person". The Malinke word for 40 means mattress, so we can assume they had double beds.
The framework suggests that children may use objects as well as fingers. It seems likely that past civilisations used stones, seeds or the like as mathematical aids. The word calculate comes from the Latin "calculus" meaning pebble, proof presumably that the Romans either calculated using pebbles, or threw stones in maths lessons. For less harmful objects we can look to the Greeks and Plato's description of distributing apples in schools. Apparently they weren't all gifts for the teacher, but were to be used to model division.
Of course the Numeracy Strategy does not advocate overuse of either fingers or objects, and children are encouraged to develop mental strategies. In the publicity leading up to the strategy there seemed to be a particular emphasis on doubling. Not a newidea: the Ancient Egyptians were great advocates of doubling and also saw the importance of being able to multiply and divide by 10. There is evidence of this and other Egyptian mathematics in the British Museum's Rhind Papyrus. This suitably yellowed and mysterious looking document has a surprising amount in common with the framework. The papyrus includes not just simple calculations, but fractions, proportion and calculation of area. There are even word problems about sharing loaves and a few mathematical puzzles. I like to fantasise that the papyrus was written on the orders of a Pharaoh worried about falling standards. Perhaps someone had told him that the Mayans were building the best pyramids.
Another mathematical device known to the Egyptians was the Nilometer, a way of measuring water levels in times of potential flooding. It is likely that marked pieces of stick or bone have been used for centuries for measuring, keeping count, or calculating. One famous example is the Ishango bone, discovered in Zaire and probably dating back to between 9000 BC and 6500 BC. Given the existence of such objects I couldn't help smiling when the idea of a counting stick appeared in numeracy strategy publicity.
When I tried to buy one, I was told it was such a new idea that they weren't yet being made. If manufacturers have been slow in producing counting sticks, they have made up for it with the speed with which they have produced items such as number cards and hundred squares.
A popular number card activity is the formation of magic squares. One way to do this is to get number cards from 1 to 9 and arrange them in a three-by-three square so that the total of each row, column or diagonal is the same. Like so many good ideas, it is hard to say who had this one first. One claim is that the first magic square appeared on the back of a divine turtle in a Chinese river more than 4,000 years ago.
The framework may be produced with the Government's blessing, possibly even some of its money, but the ideas have far-off origins. It is positive that in world mathematics year primary children are offered a curriculum owing much to other societies. It shows children how different cultures have enriched mathematics - possibly a useful reminder to us all.
Jenny Houssart is research fellow at the Open University's Centre for Mathematics Education