David Womack suggests that children's intuitive understanding of numbers can be a hindrance in mastering formal mathematics.
Tara was a bright-as-a-button five-year-old from a tiny village school in the heart of the Lake District. I had asked her to tell me the number of letters in her name. She told me that Tara had three letters. I sat beside her and counted out loud the letter T-one, A-two, R-three, A... But as I was pronouncing the final A, she stopped me and said "You've said that one!" After some reflection, I realised she was right. There are only three letters in "Tara". The incident gave me much food for thought; I began to reconsider the logic on which I based my teaching of number. It seems the problem lies deeper than just counting; it involves what we mean by a letter and a number.
To give another example. Heather, an equally bright and eager-to-please child who lives next door, was proudly showing me how she was learning to write at school. She began carefully to form a series of letters and "words" on a piece of paper and eventually handed me the result. I pored over it for a minute or two but was unable to make out exactly what was written. Finally, I gave up.
"I don't know. I'm afraid, you'll have to tell me," I confessed, with some disappointment. "But I don't know," she replied innocently "I can't read!" Our adult familiarity with symbols sometimes breeds contempt for others who fail to understand the meanings that we attach to them. But can we really say that children understand if they understand in a different way to us? My experience of teaching suggests that children are very capable of understanding the use of symbols - intuitively for example, in drawings - provided that the symbols are theirs and not ours.
Mathematics is particularly prone to use non-intuitive symbols in its teaching. For example when we teach children addition through "sets", we teach that the answer to 9+2 is a combination of two sets - but this contrasts sharply with the intuitive view of children. They believe that 2 added to 9 is not necessarily the same thing as 9 added to 2. Sets is therefore not an intuitive model for addition, although it is the mathematicians' model.
Again, how often do children in the classroom of their own accord take up counters to solve number problems? Isn't this just our way of doing things rather than theirs? John Holt in How Children Fail (1969) noted that children do not spontaneously transfer their mathematical knowledge of wooden rods to other situations. Placing rods along a track and reading off the number at the end is neither an intuitive method of adding nor a mental exercise.
Some Ministry advisers might agree with the discarding of manipulation and advocate instead, the completion of pages of "sums". But what exactly are children practising when they do sums?
Some of my teacher-training students suggest that the purpose of sums is to familiarise children with the signs for addition and subtraction. How much familiarisation is needed to learn the meaning of two or three signs - and what do the signs mean anyway? My opinion is that operation signs are ungrammatical, inconsistent, ambiguous and in places simply bizarre. In short, sets, counters and sums are established methods for teaching but are far from intuitive learning practices.
And yet learning intuitively is the most powerful way to learn because it is based on an accumulation of one's own experiences rather than the experiences of others. In fact, a common thread in my own teaching of mathematics to slower-learning children and, also, with science to rural African students, (not to mention English to sophisticated European adults), is that we learn best if we learn it in our own way. Even the traditional teaching of algebra conflicts with intuitive learning, causing one pupil to lament, "How can I multiply by x when I don't know what x is?" Might it not be the case, that some children's inability to grasp what is going on in maths is not due to dullness, but a reluctance to exchange one way of understanding numbers for an untried and potentially problematic "mathematical" system?
Paradoxically, it may be that the more confident we are in using our own powers of reasoning, the more difficult it may be to understand the formal mathematics taught in school.
David Womack is a researcher at the University of Manchester