Cardinality and Child Development

About Cardinality

Cardinality is defined by Sophian (1996) as numbers that are used to quantify a set. If a child understands cardinality they understand that the final number in the sequence refers to the total objects in the set, a necessary skill in everyday life to transfer information to another person. It is not yet known when children understand cardinality.

Sophian (1996) cites Vygotsky’s (1978) socio-cultural theory that cognitive number development is achieved through social interactions. The child is supervised externally and takes more responsibility when they gain experience until they can do the task alone. This he called internalization. Wood (1998) cites that Vygotsky refers to gaps in assisted and unassisted ability as the zone of proximal development. Wood (1998) uses the term scaffolding to represent the correct amount of help a caregiver needs to give. Too much help given by a caregiver means the child will not learn, and not enough help means they will fail. So the amount of scaffolding needs to be judged. These are social theories that need to be tested.

Maternal help with counting was studied by Fluck (1995) on children with a mean age of 2 years and 10 months. He found mothers offered high support assume their child understands that counting’s goal is to identify the total number of a set and often used how many as a prompt. In a counting task he noted that when children were asked how many objects there are, they rarely give a single number, but count a sequence. He found no indication that the children understood a connection between cardinality and counting.

Linnel and Fluck (2001) investigated maternal helping during a give x task. Children at 32 months scored poorly on the give x task but their performance had significantly improved by 44 months when unassisted. When assisted they found the mothers help to be either constant or non existent, and when the mother was assisting results improved significantly at both ages, even more so when the mother supplied the last word. They also conducted a sequential counting task at the two ages and found the mother helped more in the give x task. Therefore this study will focus on the give x task.

A Study…

This repeated measures study observes a child at the age of 32 months and 44 months completing a give x task both assisted and unassisted. Give x tasks test whether the child understands that the last number counted in the sequence refers to the total number of objects, and therefore tests their understanding of cardinality. As the verbal assistance of the mother has been investigated in previous studies such as last number emphasis, this study looks at whether the mother uses physical prompts on the child. Based upon previous research the hypotheses are that the child will perform better in the give x task at 44 months than 32 months unassisted, and that the mother will use more verbal than physical assistance with the child performing better when assisted.

The objectives of the study were to test whether a child’s understanding of cardinality improves between the ages of 32 months and 44 months, and to see whether the mother’s assistance is mostly verbal or physical. This was investigated through an observation of archived footage form a previous repeated measures s study of one child completing a give x task at these ages. At each age there was an assisted and unassisted condition.


The results showed that the child performed better assisted than unassisted, and there was an improvement both assisted and unassisted between 32 and 44 months. The mother used more verbal assistance than physical. The conclusion drawn was that although the child performed better at 44 months her unassisted results suggest she did not understand the concept of cardinality, and that her assisted success was mainly due to the verbal scaffolding from her mother.

The results from the give x task show a slight improvement in performance at 44 months in both the assisted and unassisted conditions. Assisted at 32 months showed two poor attempts where the child was wrong by at least five, two where the child was only wrong by one and two correct performances. At 44 months this had improved to four correct performances and two where the child was wrong by one. Unassisted at 32 months the child was out by at least ten every time. This slightly improved at 44 months to the child being out by ten or more twice, out by five or more three times and out by two or more once. The hypothesis that the performance would improve at between 32 and 44 months and would perform better when assisted is therefore accepted. Fluck (1995) noted that the children in his study made a sweeping gesture to wards the objects as they counted the, this was evident in this study. He believed this was because counting had become a routine. The give x task requires this routine to be broken at the appropriate point, which the child in this study could not achieve.

The unassisted condition may show that the principle of cardinality was beginning to develop as she was closer giving the correct number of objects. However, even though the child had improved according to the coding scheme in that she was not wrong by so much, the fact remains that she was still wrong every time. This may be a flaw in the coding choice because as long as she was wrong at all it shows she did not understand cardinality. The fact that the child was so much better when assisted suggests that she was heavily dependent on her mother’s scaffolding and not her own understanding.

The findings support Fluck (1995) who found that children with a mean age of 2 years and 10 months did not understand cardinality. This is strong support as the children in this study still didn’t understand, and at 44 months are older than the children in his study. The improvement in the unassisted condition supports Linnel and Fluck (2001). Due to the much improved results when assisted, Vygotsky’s (1978) zone of proximal development idea is supported as the mothers’ expertise was vital in the success.

At 32 months the mother used verbal assistance every time, and did also five times at 44 months. On task five at 44 months she used a physical assistance. Therefore the hypothesis that the mother would show more verbal than physical assistance is accepted.

During this experiment assistance this came mostly in the form of counting for the child. There was also evidence of emphasising the last word which hinted to the child that she had completed the task, sometimes even after the mother and child had been counting different numbers, this supports Linnel and Fluck (2001). At times the mother actually told the girl to stop, this was the only reason the child got the correct answer. The only physical assistance came when the child would not stop even though the mother told her to. This caused the child to give the correct amount. This suggests that physical help is either unnecessary, only used when the child does particularly badly, or the mother becomes frustrated as she expects the child to perform better. The increased performance in the assisted condition was due to the mother intervening.

Unfortunately however, studying one child is insufficient to base a study on. This decreases external validity as it is not representative of the population. This child may have developed faster or slower than an average child and standard deviation cannot be calculated. The results could also be confounded by the sequential counting task conducted before the give x task these results were based on. The child may have become fatigued and their was evidence of boredom.

Further studies could use more children and gain a general overview from their combined results. A longitudinal study on a group of children would be able to establish a reasonable window for which cardinality develops. The studies should be conducted in the child’s natural environment. To gain a further insight into the child’s thoughts, it might be they should be asked why if they believe counting to be important to them, if they do not understand its purpose it may correlate with poor results. In any case, the concept of cardinality and what it can tell us about child development is a fascinating one and worthy of further study.

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Adam Sinicki

Adam Sinicki is a full time writer who spends most of his time in the coffee shops of London. Adam has a BSc in psychology and is an amateur bodybuilder with a couple of competition wins to his name. His other interests are self improvement, general health, transhumanism and brain training. As well as writing for websites and magazines, he also runs his own sites and has published several books and apps on these topics.

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