What can we learn from infant studies of the development of numerical knowledge?
Infant studies often tell us about the development of skills and knowledge that human beings possess (by innate means or through learning). Piaget and Szeminska (1952) believe that, even though children may be able to count at certain ages, this does not necessarily mean that they understand counting.
Therefore, this indicates that numerical knowledge is very different from other basic properties such as orientation, distance or size. The reason being that though perceptual appearances may change with things like the angle of viewing, number usually remains unchanged (Samuel and Bryant (1974).
To study numerical knowledge, Paiget (1952) and other contenders have widely used the ‘conservation task’ to understand children’s development of numerical and arithmetical understanding. Conservation refers to the development of adult-like understanding that quantity (whether continuous or discontinuous) remains the sane (i.e. is conserved), unless the same portion is removed or added to the original quantity; and in addition, it refers to the understanding of the situation under which one is consider a portion to have been removed or added. In basic terms the Conservation Paradigm is a test of whether young children can understand the principles of ‘invariance’ i.e. that counted-entities (such as buttons) or mass entities (such as liquid) only changes in quantity when something is added to them or taken away from them. In other words, researchers ask Do children realise that certain properties of objects, such as volume or mass, remain unchanged when the objects appearances are disrupted in some superficial way.
The most well documented variant on the classical Paigetian conservation task is that for number. This task focuses on two rows of counters. The infant is first asked whether there are the same numbers of counters in the two rows, or whether one row has more counters than the other. This is termed the pre-transformation question. After this question is asked, one of the rows is compressed or spread out in full view (i.e. the transformation), with the child’s attention deliberately drawn to this fact. The child is then asked the question again (this is termed the post-transformation question). Children tend only to get this task correct at around the age of seven or beyond. Paiget believes that if a child gives the conservation response and can justify it, then they understands conservation and if the child gives the non-conservation response, then they do not understand the process of conservation. The conservation tasks is said to test the notion that children acquire capabilities that approximate close to development of ‘logical rules’. In regards to numerical and arithmetic understanding, studies that have tested the Paigetian conservation paradigm reported that counting can be achieved without the need for true understanding of the concept of number (Cowan, 1991). This implies that infant studies involving number tasks will sometimes be inappropriate for properly testing numerical knowledge and understanding (McShane, 1991).
For example, Light (1986) suggests that it is possible for the child to give the conservation response, when they do not understand why, or the child may understand the process of conservation but response by giving the incorrect answer. Light (1986) argues that with the traditional conservation task, you may get a ‘false positive’ response if the researcher accidentally or deliberately misleads the child, or if instead, following the instructions given by the experimenter, the child follows the context of the experimenter’s behaviour. But, as well as leading to false positives, a child may change their response even though they have fully understood the process of conservation. Therefore, what does this experiment show that we can learn from infant studies of the development of numerical knowledge?
In general, results from the original design of the conservation task are extremely reliable, it that the results show repetitively that children give either a conservation response or a non-conservation response. However, the task has seemingly weak validity. For example, it is questionable to whether the task is actually measuring the level of conservation knowledge present within the child. Therefore, it becomes questionable as to what this research can inform us about the development of numerical knowledge. To combat this factor, researchers have adapted the original conservation task and employed greater controls upon things such as the interpretation of the questioning, linguistic limitations of the child, context and experimenter bias.
One attempt at a more valid paradigm came in the form of the naughty teddy experiments (McGarrigle & Donaldson, 1975). Their design aimed to prevent the child from focusing on the experimenter’s behaviour. Two conditions were employed, differing only in how (or why) the materials were transformed. In the standard condition, the experimenter overtly moved the counters in the rows, after the child’s first judgement (this is a judgement of equality). In the modified condition, the naughty teddy would escape from a nearby cage (manipulated by the experimenter), and manages to mess up the rows a little, before being recaptured and returned to his cage. The rearrangement of materials was therefore ‘incidental’ to the task in hand, and so, does not mislead the child. Under this design 4 and 5 year olds conserved at above chance in the incidental condition, a finding that challenged the validity of the classical paradigm. Another challenge to the validity of the standard task came from Rose and Blank (1974).
Hughes (1993) argued that when the experimenter lengthens one row, the child assumes that the post-transformation question is to do with the length, rather than the quantity. Hughes (1993) argues that this could explain why children fail to the task, although they can conserve. Rose and Blank (1974) avoided this issue by asking only the transformation question. They displayed counters one to one correspondence, and asked children to attend to the transformation itself. Under these conditions, most 6 year olds responded correctly, indicating that they could indeed conserve. However, the results of this research have proved difficult to replicate. It has been suggested that variability could be due to memory factors. For example, not asking a pre-transformation question may lead to the child’s inability to remember the original array. Therefore, are infant studies of the development of numerical knowledge, finding out more about the development or use of memory?
Moore and Fry (1986) found evidence that for numbers up to 7; children tend to be distracted by the teddy. This leads them to judge the post-transformation array according to its appearance, or to strategies of such as counting, subsidizing and length. One way of avoiding the potential misleading effects of the teddy in McGarrigle and Donaldson’s (1975) experiment is to make the transformation ‘incidental’ rather than ‘accidental’. This was the basic idea behind the experiment conducted by Light, Buckingham and Robbins (1979). They constructed a conservation task around a game to be played with pasta shells by pairs of children.
A highly believable reason was given, for the need to transfer the pasta shells from one container to the other, (i.e. that the containers was damaged). The results were very similar to the McGarrigle and Donaldson’s experiment. This implies that the accidental transformation had not infact mislead the children into not considering the pre-transformation array. One way for accounting for the differences between the two researchers results is to argue that both the accidental and incidental conditions biased children away from considering the pre-transformation array. Therefore, is this implying that incidental and accidental transformations artificially lead the children towards conserving responses? Therefore, if this is true, then it is questionable as to whether the conservation task adequately informs us about the development of numerical knowledge.
