Separating the sound flow into separate words, and establishing the order of those words comes as a later phase of elaboration. A further part of the elaboration phase is when children begin to use the words as number names and the words become "objects of thought" that symbolise quantity, and can be used for counting things.Cognitive field interactionist learning theory when this knowledge is absorbed the child will begin to recognise that there is a fixed order on which they can progress up and down, they will become able to say number names onwards or backwards from a given point.Cognitive field interactionist learning theory

To complicate matters different aspects of number sequences − teen structure 13 to 20 and the decade structure 20 to 90 may be being learned at the same time but be in different phases of acquisition.Cognitive field interactionist learning theory E.G. When one to ten has moved into the phase of elaboration, the teens may just be starting to be acquired as a sound sequence. The whole process takes a number of years and the rate at which children develop the skills is varied.Cognitive field interactionist learning theory

For young children touching is an important part of this process of itemisation, and provides physical prompt to help with timing the saying of the naming word.Cognitive field interactionist learning theory martin hughes (1986) notes how children still resort to pointing and tapping to assist their counting even when objects are out of sight. He also reminds us of the powerful use made of finger counting all over the world as a means of assisting both itemisation and partitioning.Cognitive field interactionist learning theory

In the early stages of learning to count children may be vague or imprecise about their pointing, they wave their fingers in the general direction, but let the rhythm of the verbal counting sequence dominate the speed at which they count, and consequently lose correspondence.Cognitive field interactionist learning theory later they become more aware of the importance of co-ordinating the itemising and tagging, and they develop strategies for keeping track, and noticing if they have double counted or missed items.Cognitive field interactionist learning theory gelman, R. Meck,E. (1983).

This presents the child with the problem of remembering a long list. Bearing in mind the general limitations of short-term memory (miller 1956), the human mind only being able to keep track on around seven items at once, we will recognise the valuable role of intonation and rhythm.Cognitive field interactionist learning theory they offer prompts, and connections that make memorable "chunks" and so help learning the number sequence.

When the child understands this principle they recognise that earlier numbers were temporary steps towards the last number tag, which is special, because it is the cardinal number and represents "how many" items have been counted.Cognitive field interactionist learning theory appreciating the importance of cardinality is an important milestone in a child's mathematical development, it is a keynote in understanding that the process of counting has a meaningful and useful purpose.Cognitive field interactionist learning theory

It is necessary to grasp this principle of abstraction in order to be able to generalise the use of counting as a tool. It helps us confirm the consistency of the quantity of a set, and it is confidence in that consistency that enables us to be sure about making comparisons.Cognitive field interactionist learning theory such confidence helps us to override the messages of perception that may confuse us when spatial changes make things appear bigger, and it may therefore underlie our ability to recognise the conservation of number.Cognitive field interactionist learning theory

This early ability enables children to make out and compare small groups and may support the refinement of concepts of quantity. It only works with small groups and this may account for why quite young children can be accurate with processes involving small numbers, but lose track when the groups involved are larger than the perceptual range of subitising.Cognitive field interactionist learning theory

Symbolic representation becomes a vital part of the language of mathematics, and as with reading working with numerals often presents difficulties for SLD pupils.Cognitive field interactionist learning theory whilst a discussion of symbolic development would take more space than is available a brief overview of strategies children use to keep track during counting, and aid the making of comparisons, may illustrate how some forms of symbolism may help our special pupils.Cognitive field interactionist learning theory

Perhaps the first symbols which children use are their fingers, they keep tally with them during a count using them as symbols to represent objects.Cognitive field interactionist learning theory fingers are very useful in that they can be both objects and symbols (hughes 1986). They can be used to keep tally but are also useful to provide a visual aid representing the parts of simple arithmetic processes.Cognitive field interactionist learning theory using fingers may be seen as a development beyond itemising objects. Their use extends the pointing and touching we discussed earlier, and as such may be a link between sensorimotor learning and symbolic representation.Cognitive field interactionist learning theory

It is self evident that making marks to represent things is important because it is a means of recording and remembering. What may be less evident to us an yet vitally important is the role that mark making plays in helping children move towards abstract thinking, drawings become symbols, and children come to understand that representations and symbols can "stand for things" we can even represent things that are not here right at this moment!Cognitive field interactionist learning theory

The use that children make of their fingers is apparent to any observer and the development of keeping tally by making iconic marks can be seen as a step extending finger counting.Cognitive field interactionist learning theory both provide a mode of symbolisation which helps the child retain focus firstly on accounting for if an object is there or not, and secondly as a cumulative record which helps memory and can be worked on later.Cognitive field interactionist learning theory

Pictorial and iconic recording help children keep a track on quantities and provide concrete visual records for them. It may be useful to encourage children to use such systems of representing quantities and keeping tally to support counting activities.Cognitive field interactionist learning theory it seems likely that they need to be fully grasped before the relationships between quantity and more symbolic numerals can be realised.

Looking back over the skills and processes I have described it is easy to regard the course of learning as a linear sequence where skills develop in a specific order and we should teach one skill at once.Cognitive field interactionist learning theory however such a view is too simple, children have surprising abilities at ages much earlier than a rigid view of the sequence would lead us to expect for example:

cognitive field interactionist learning theory

These examples might be taken to indicate that children have both innate understandings of quantity, and a disposition towards using mathematical thinking, even before they can count and express numerical comparisons.Cognitive field interactionist learning theory exposure to experiences enables them to use their early concepts and abilities in parallel, each contributing their part to the refinement and consolidation of the way that they are used together.Cognitive field interactionist learning theory

This process is reflected in the development of counting. For example children who have not yet mastered number names and their order still practice itemising and enumerating, using their fingers and their own versions of number names.Cognitive field interactionist learning theory as they develop their knowledge of number names their perceptions of quantity help them to understand order and comparative value etc. Many people assume that children do not learn to add or subtract until they can count, but counting itself is a process of addition, and counting backwards is subtraction.Cognitive field interactionist learning theory

Children can apply mathematical concepts to practical tasks long before they can express their understanding. Martin hughes (1986) describes how pre school children were able to add and subtract small groups when the words used in the question related to real items.Cognitive field interactionist learning theory in contrast they were unable to answer when the reference to the objects was left out. He suggests that relating the problem to real objects allowed the children to visualise the items and use their practical knowledge to give the answer.Cognitive field interactionist learning theory without reference to real objects the child was puzzled by the question, and could not translate the mathematical language into real terms. E.G.Cognitive field interactionist learning theory

Throughout the article we have discussed the complexity of what is usually regarded as a simple operation. We have seen that the parts of counting, - includes physical co ordination, mental and verbal skills, and learning rules.Cognitive field interactionist learning theory we have also seen that in order to understanding mathematical language children need to relate it to real objects, or mental images of them.Cognitive field interactionist learning theory these aspects of learning have practical and presentational implications that we need to bear in mind when we arrange how to present mathematical experiences to help our children learn, including as we teach them to count.Cognitive field interactionist learning theory

The framework of our teaching needs to be reality, we need to exploit everyday activities, create events where counting takes place, and games that give opportunities to notice, practise and integrate the constituent skills.Cognitive field interactionist learning theory the analysis of constituent skills made in this article can provide a framework through which we can add detail that will enhance the national curriculum programs of study.Cognitive field interactionist learning theory the knowledge that has been outlined about children's inclination towards using their concrete experience, and how they use visualisation, should provide guidance to us.Cognitive field interactionist learning theory it illustrates that we need a style of delivery that refers to real events and illustrates the language that is used when referring to changes in quantity.Cognitive field interactionist learning theory

The information that has been highlighted in the article will help teachers see targets that are aspects within the skill of counting. Armed with this level of consciousness about detail of counting processes the teacher will be able to maximise many opportunities within everyday situations.Cognitive field interactionist learning theory develop events or games when there is opportunity to tally; record; match; visualise; compare; name etc.