Learning about numbers
Much of what has been known about number development came from studying young babies, animals and people who can no longer deal with numbers. Brian Butterworth, a leader in the study of extreme difficulties with number (dyscalculia), describes an example of an Italian hotelier who, after sustaining a stroke, could not do anything with numbers above 4, where previously he had been without difficulty.
It is possible to use brain imaging to identify what areas of the brain are damaged in such cases, or alternatively what areas are used by those successfully manipulating number. Educators have known for many years what brain science is beginning to confirm, that human ideas are grown from early sensory-motor experience. Learners begin with physical manipulation of apparatus, developing their use of mental images and memories of actions. Later, they learn to manipulate symbols which represent those images.
While in the womb, a baby has experiences which lay the foundation for comparing, for example, loudness of noises. The baby further develops its ability to compare as it develops its sight, for example, nearer or further. This ability to compare seems to be inherent in animals as well as babies. However it tends to be presented as ‘lions can count’ when they may be comparing.
When numbers are small, comparing involves ‘knowing how many’. This ability is known as ‘subitising’ and is demonstrated by both babies and animals. Dr Karen Wynn, a child psychologist at the University of Arizona, showed that babies of six months expect that adding one object to another object will produce a set of two objects, not one or three, and that they are surprised when this does not happen. Some scientists think that the human genome includes a code for building a specific neural mechanism for subitising.
As a baby explores its body, it develops sensitivity to the ends of its fingers. The part of the brain that seems to be responsible for representing numbers lies adjacent to the part that is responsible for touch sensation in the fingers. Many children who have difficulties with number have impaired sensation in their fingers, so the two may be connected.
Meanwhile, using a different part of the brain, the developing baby is busy, listening, sorting and noticing patterns in language. The human child develops a mapping between the process of counting and the poem ‘one, two, three, four, ’. The uniquely human part of the counting process appears to be the ability to represent larger quantities exactly, and to develop a concept of number in the abstract. Where most animals use the prefrontal area of brain for better developed sensory-motor abilities, humans have developed high-order and symbolic thinking. This allows us to learn to extend our ability to subitise in order to reason about larger numbers like 100 or a million.
There are many ways in which the process of developing number sense can go wrong. For many people, the negative experience (bordering on panic) of expectations beyond their capability results in a flooding of emotion into the parts of the brain needed for number. So some difficulty experienced in learning mathematics can be attributed to anxiety. Hence some difficulty experienced in learning mathematics can be attributed to anxiety.
There is need for a great many more studies in this area to develop a fuller understanding. In the meantime, teachers are already being encouraged to use a mixture of visual, auditory and kinaesthetic approaches to number and the rest of us need to learn to approach number with calm, using any approach that works for us!
Butterworth, B. (1999) The Mathematical Brain, London: Macmillan.
Dehaene, S. (1997) The Number Sense: How the mind creates mathematics, Oxford: Oxford University Press.
Lakoff G. and Nunez R. E. (2000) Where mathematics comes from: how the embodied mind brings mathematics into being, New York: Basic Books.
Mason J. and Johnston-Wilder S. (2004) Fundamental Constructs in Mathematics Education, London: RoutledgeFalmer.