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The young child hears 'one, two, three...'or 'une, deux,trois, quatre...'as the case may be. Gelman and Meck(1986)show that very young children keep the set of count words(one, two, three)seperate from the object labels(cat, dog, spoon) and use each in their appropriate context.
Objects from the same category must have the same basic-level name (eg.,'spoon'). Once an item has a label it can't be given a different label at the same category level but it can be called by a superordinate name. Therefore for labeling each has the same name: 'spoon', 'spoon' 'spoon', 'spoon', but for counting, each must have a different label: 'one', 'two', 'three', 'four'. Toddlers use the domain -specific priciples to demarcate these different functions. This would be impossible if innately specified principles did not guide the infant/toddler in learning which linguistic entities are part of a given domain and which are not.
Children will later have to learn to apply mathematical language to the principles governing mathematical operations. This is crucial to subsequent number development and leads to a richer understanding of the number domains. In every-day language 'multiply' always implies an increment while this is not the case in mathematical language (eg., the multiplication of a fraction). mathematically gifted children understand and use mathematiical languge whereas the less gifted understand mathematics in terms of every-day language.
mathematic notation embodies constraints that differ from those of writing and drawing. Number notation is often an integral part of number development. Extensive studies suggest that external notation helps the child understand the symbolic nature of number. Children were shown boxes filled with different quantities of toys had to count how many were in each box, close the boxes and write down on the box cover how many were in so as to be able to remember later. Children produced either numbers, analogical representations, or drawings. When brought back to the experiment room sone time later, mainly those using number notation, even if wrong, were able to recall the previous quantities. Analogical notations were of less help for reacall of number even when children had made the right number of marks. So understanding the symbolic nature of number notation and the relationship between encoding and decoding takes time developmentally.
...another seemingly universal fact about number has been highlighted by Resnick (1986) who reporrts that almost every society invents or uses additive composition operations...children spontaneously invent addition algorithms prior to schooling. Gelman and Gallistel place particular emphasis on the difference between operations on counted numerositiesand operations on unspecified quantities. The latter involve a more abstract understanding of number. The true conserver has developed the ability to reason about numerical relations in the absence of representations of instantiated numerosities. In some sense then, the conserving child has started to operate on algebraic inputs rather than on merely numerical ones...one-to-one correspondence is an implicit feature of successful counting procedures. The principle embeded in the procedure must then be abstracted, redescribed and represented in a different format independent of the procedural coding. Tollesfrud-Anderson et al, provide data from a reaction-time study they carried out on the conservation task. Their results are directly relevant to the RR model. All their subjects seemed to conserve but a subtle analysis of reaction times showed that there were three very different levels of performance. Some could conserve and provide oral justification of one-to-one correspondance. Others, though unable to provide oral justification, displayed reaction times as fast as true conservers. Finally, the slower group were found to be making post-transformation one-to-one matching checks.
According to Gelman, Cohen and Hartnet (1989) children's initial theory about number is that number is what you get when when you count. Both zero and fractions are therefore rejected as numbers because they are not part of the counting sequence. Their inclusion involves a fundamental theory change in the core concepts of what constitutes a number. The child's theory moves from number as a property of countable entities to something used to perform mathematical operations. Resnick points ot a paradox central to mathematical thinking: on the one hand the algebraic expression a+b takes its meaning from the situations to which it refers. On the other, it derives its mathematical power from divorcing itself from those situations. The movement from conseving number identity via counting real-word objects to conserving equivalence of nonspecified quantities is of a similar type of abstraction...Wellman and Miller (1986) showed that the children's understanding or zero passes through three steps: First comes familiarity with the name and writen notation of zero. Then comes understanding of its unique numerical quantity - none or nothing. Finally children understand that zero is the smallest number in the series of non-negative integers whereas previously they believed one to be the smallest number...developing a broad theory about number involves fundamental theory changes similar to those with respect to language. The child's explicit theory of what a 'word' is moves from thinking that words denote real-world objects and events to to thinking of words in terms of the linguistic system in which they operate.