# Early Stages in Mathematical Development

## Understanding Counting

The verb ‘to count’ has origins from the old French word ‘conter’ which means to add up. However it has many meanings; to determine the total number of items in a set; to calculate or compute a total or to list or name the numerals. Nowadays the most widely used numeral system used for counting is the decimal or base-10 system which when used in combination with Arabic numerals enables us to represent number using just ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).

There are many numeral systems but these tend to be found in small ethnic groups. For example, in Wales (part of the UK), the traditional numeral system is vigesimal or base-20 but more recently they have moved towards adopting a decimal system.

The understanding of counting is closely linked to the concept of number. Indeed a child often has many concrete experiences of both as a young child. For example, counting stairs as you climb them or counting out food as you share it amongst family or peers. Similar to understanding number, for counting a learner has to make connections between key components.

### Pre-counting experiences

Counting begins before formal teaching. One common home activity is to sort objects that are the same. The sameness property can be colour, size, etc., in order to separate members of a set which can be counted. Furthermore young children are usually exposed to language which involves counting. They may be asked if they want ‘another one’ which essentially sets up the experience of counting. In addition to these external experiences, children have an innate ability to perceive and distinguish between small sets of objects without actually counting them. This ability is called subitizing. Thus if a child is presented with a visual display of 3 items they can state that there are ‘three’ more quickly and accurately than if they had taken the time to count the items.

Subitizing only occurs for small sets of objects; usually up to 4 items. However the number of items in a set can be extended if the items are placed in a recognisable pattern. For example, four items in the shape of a square or five in the shape of house. There is some evidence that some children that go on to show difficulties in learning arithmetic and mathematics have problems with subitizing, even from a very young age. However the evidence for this is not clear and more research is needed to explore the link between subitizing, counting and moving on to arithmetic.

### Formal counting experiences

When children first learn to count they memorize a pattern, a bit like learning a nursery rhyme, the actual words do not have much meaning but reflect sequential learning. However over time, children gradually begin to learn the principles of counting. There are five principles:

1. One-to-one principle: the correspondence between number names and counted items, where each item only has one number name.
2. The stable order principle: number names must always come in the same order or sequence.
3. The cardinal principle: the last number named has a special meaning because it represents the number of counted items. An additional element of cardinality is that the arrangement of the objects in a set does not make any difference to the counting process. In other words, it doesn’t matter which one is ‘one’, ‘two; or ‘three’, there are still three things.
4. The abstraction principle: the arrangement of the objects in a set is irrelevant; if you have seven items in a set, it doesn’t matter if they are arranged in a long line or together in a circle.
5. The order-irrelevance principle: the objects in a set can be counted in any order. It does not matter if you count from the left to the right or right to left as long as each item is matched with one number name.

These principles are not thought to be acquired together, but are learned with experience. Errors learning these principles are often the source of many errors in children’s counting.

### Finger Counting

It may seem surprising that using our fingers could be an integral part of understanding counting and hence mathematics but at least a transient phase of finger counting and finger calculation almost invariably precedes competent mathematical cognition, and the use of fingers to represent number is ubiquitous across ages and cultures. Children use finger counting as an initial strategy to understand and keep track of counting and calculate, and even amputees and children with congenital agenesia of hands and fingers will use phantom fingers.

Furthermore performance in tests of finger gnosis (i.e. being able to recognize and name the fingers) before formal schooling selectively predict mathematical outcomes at a later age. It has also been reported that early finger training may improve numerical abilities at a later stage. In contrast finger counting strategies tend to be used by older children and adults with MD, to make up for deficient mental number representations. The use of fingers to represent numerical knowledge may mean that number can be linked with sensory and motor features that are engaged during learning. However, another possibility is that the crosstalk between numerical and body representations is not integral to numerical representations but provide a means to offload and free some memory resources while processing numerical information in a task-dependent way.

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