In order to be successful at mathematics children need to understand the basic concept of counting. ACME, 2008. This principle underpins the mathematical curriculum and is the starting block for early number (Hansen, 2011). However, issues in relation to progression through the principles of learning to count can prevent children from constructing number concepts. The issues I will focus on are the lack of coordination when performing the one to one principle, and difficulties with conservation of number.
While very young children have ideas about counting, often as playful activities, they frequently learn to recite numbers with no real concept of their purpose (Baroody,1999). Children's first experiences of counting take place in the home before they have entered formal education, and research shows that many parents believe that their children have an understanding of counting if they can recite numbers to 10, the stable order principle (Munn, 1997). The ability to recite numbers in order is important, yet it is the one to one principle, understanding that each item receives one and only one label, that is required for children to progress through the five principles of counting (Gelman & Gallistel,1978).
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Despite research suggesting that pupils difficulties with early counting could be the result of differences in the teaching approaches of parents, a small scale study of counting in the home revealed that many valuable types of counting were taking place (Griffiths, 2008). However, the reliability of observations made during this research could be questioned because of the sessions being recorded. Nevertheless, the results show that the parents involved showed varying levels of understanding regarding learning to count effectively.
Several of the parents that took part in the research were unsure why counting was important, and referred to the health visitor as having conducted similar activities with their child, so in turn felt that this was a valuable activity. Despite this, the research showed that parents identified the value of teaching counting skills, as is evident in them frequently being observed engaging in meaningful counting activities (Griffiths, 2006). While Munn suggests that parents believe that their children can count if they can repeatedly rote count, it is evident that parents, even if they are unaware of this, are in fact teaching their children the one to one principle. Further research states that in order to minimise this issue, children need to experience counting from an early age, through 'variation in the complexity of everyday counting activities' (Saxe et al, 1987), as witnessed during Griffiths study.
On entering formal schooling some pupils have difficulty in grasping the one to one principle. For example, during a recent teaching placement in Year 1, Child A was asked to count a number of objects. She moved each object as she counted, however, she labelled the first object as '..1,2,3..' and the second object as '...4,5,6...' , resulting in a count that was incorrect. This suggested that Child A had learnt to recite numbers in groups, but had not fully grasped the idea that each object represented one number. The rhythm of the oral counting defined how child A counted each object and this in turn influenced the speed at which she counted (Staves, nd). As child A counted each object she looked directly at me to ascertain, from my reaction, if she was performing the task correctly and it was evident that she lacked confidence in her own ability to perform the one to one principle.
I found this interesting and through research discovered that young children often have difficulty in coordinating the skills of number sequence and one to one labelling (Baroody, 1999 :53) and it was clear that this was Child As difficulty. Research shows, that in time, children learn to coordinate these skills and that the main difficulty is identifying which objects have been counted (Fuson, 1988).
It was clear from the activity that Child A could count to 10, and had some idea that each object was represented by a number, but that she had lost the correspondence between the two actions. In order to improve Child Aââ‚¬â„¢s coordination, counting activities were modelled for her and her sequential touching of objects was controlled, with the TA placing her hand on Child Aââ‚¬â„¢s as she counted and moved each object. This slowing down of the child's actions enabled her to see and make the connection between each object and its number.
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Other activities, suggested by experts to develop co-ordination, include pointing at large objects, controlled clapping and counting and moving items in order to keep track of them (Staves, nd). However Munn (1997) states that the concentration needed to perform these physical counting tasks often detracts from the required outcome, of finding quantity. Once the child has learnt this concept, that the last number said has significance, they have mastered the cardinal principle (Threlfall, 2008.) However, this precedes an additional issue with counting, the conservation of number.
In order for children to add two arrays, they need to be able to conserve number. If shown two arrays, the first array containing three objects and the second containing four objects, the child not only needs to count the first array but also needs to recognise that the final number said is how many objects there are, the cardinal term. The child can then count on from that number, to find the total for both arrays. When performing simple addition, children use many strategies to count, including using their fingers, for example, counting out four fingers and then two more to show which numbers are to be counted (Thompson,1995). However, if the child cannot conserve the first count they will have to count the first group of fingers again.
During a recent placement Child B was asked to use one hand to show the teacher four fingers. The child counted his fingers and showed them to the teacher and was then asked to show the teacher three more fingers and say how many there were altogether. Child B was unable to provide the answer without counting the first four fingers again and then continuing the count. This inability to count on, suggested that the child lacked conservation of number. Despite the teacher confirming that the first number was four, and encouraging the child to say 'four' in their head and then count on, the child once again counted all four fingers.
Following this activity Child B was then asked to count a group of seven objects, which he did easily. He was then asked to split the objects into two groups of three and four objects, and to ascertain how many there were altogether. Again the child counted each of the objects and then informed the teacher that there were seven. This confirmed that Child B did not have conservation of number and that he had not made the necessary connections to be able to move on to the sum
4 + 3 =.
Piagetââ‚¬â„¢s (1965) well known number conservation task showed that children were focussing on the appearance of the groups of objects and, though the children were often aware they were contradicting themselves, they stuck to their final answer. This anomaly, which theorists are yet to fully understand due to the complexities of children's acquisition of number (Damon et al, 2006), can be counter acted by encouraging children to regularly count objects in different arrangements and in different contexts ( Baroody, 1999). Giving children varied experiences of counting groups of objects enables them to see that appearances can be deceptive, and that if a group contains five, whether it is five cubes or five cars, it will still contain five, no matter how it is arranged. However, it has been suggested that by giving children physical objects to count we are inadvertently encouraging the re-count that we are trying to discourage, and so are not developing children's conservation of number ( Maclellan, 1997).
This transition between the one to one principle and conservation of number, enables children to learn, both kinaesthetically and visually, in a context that is relevant to the mathematics classroom. ICT, in particular use of the interactive whiteboard to count and move objects, can be a tool for enabling shared visualisation and for capturing childrenââ‚¬â„¢s interest. Furthermore, developing children's ability to visualise number through visualisation activities and memory games ( McGrath, 2010) will help to promote their conservation of number and in turn their counting skills.
Through this visual representation of number, children can begin to steer away from counting small groups of objects and move forward to seeing the group as a whole, subitising of number. Research suggests that subitising of small numbers shows that young children do know something about conservation of number (Langford, 1987), however when the group contains more than five objects this process fails, with children returning to counting all.
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Piaget (1965) showed that typically, children of 6 years old can solve problems similar to those performed by Child B, but only after counting the objects again (Bjorklund, 2012), and that until they can perform these tasks without relying on physically touching or re-counting, they do not have a real understanding of addition. It is therefore essential for children to master conservation of number as soon as possible, in order for them to progress through the principles of learning to count.
With counting being a key concept in the development of mathematics, it is important that children learn all of the principles. It is clear from research that preschool counting is valuable. Parents can provide one to one counting opportunities that are personalised in a way that would be difficult to replicate in the classroom, as discovered in the previously mentioned research by Griffith's (2006) At the beginning of her research Griffiths suggested making a DVD to inform parents about the principles of counting, conjecturing that it could be helpful in developing children's early counting skills. However she discovered that many counting activities were already being practised over and over again, therefore giving children opportunities to learn and develop their counting strategies. The parents studied were unaware of the principles of counting and yet they were witnessed performing the one to one principle with their children, experiences that can only serve to place the child in a good position when starting formal schooling. The DVD was never made as Griffith's concluded that the formality of recorded instruction could alter parents relaxed and intuitive teaching of number and therefore the good practice already taking place.
Research has shown that the conservation of number and the one to one principle are essential if children are to progress from counting to more complicated applications of mathematics, such as addition and subtraction. These principles underpin the early stages of number and therefore we need to ensure, through various methods, that every child is given regular opportunities to practise. This will enable them to overcome their issues, to make progress and to master each of the counting principles.