The Nature of Representations of Number in Early Childhood: Numerical Comparison as a Case Study
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What is the nature of non-verbal representations of number? Broadly speaking, non-verbal representations of number can be divided into two categories: representations of particular numerosities and representations of unspecified numerosities. However, studies of representations of number do not only call for investigations into the representations of numerosities. As philosophers of mathematics (i.e., structuralists) have pointed out, the essence of number lies in numerical relations. Using numerical comparison as a case study, this dissertation asks questions about the nature of the non-verbal representations of particular numerosities and unspecified numerosities. Research in the last few decades has found evidence for a non-verbal representation of particular numerosities – the approximate number system (ANS). The ANS encodes number as approximate numerical magnitudes, and is a dedicated system for representing number. While there is much evidence that the ANS can be used to represent numerical relations, little is known about whether a separate system for representing small sets of individual objects – parallel individuation – can also be used to represent number. In Chapter 2, I ask whether the parallel individuation system can support numerical comparison. In two experiments, children between the ages of 2 ½ and 4 ½ years old were asked to compare either exclusively small sets (< 4) or exclusively large sets (> 6) on the basis of number. The results of these studies suggest that parallel individuation supports numerical comparison prior to the acquisition of numerical language. In addition to representations of particular numerosities, humans are also capable of representing unspecified number. For example, we understand that the numerical statement ‘x + 1 > x’ is true without representing the particular value of x. But when does this representation develop? In Chapter 3, I ask at what age children begin to show the capacity to reason about unspecified numerosities. In three experiments, children between the ages of 3 and 6 years were asked to reason about the effects of numerical and non-numerical transformations on the numerosity of a set. These sets were large enough to be outside the range of parallel individuation and involved comparisons that are not computable by the ANS (Experiment 3), or were hidden so that the specific numerosity was unavailable (Experiments 4 and 5). After the transformation, children were asked whether there were more objects in the set. The results of these studies suggest that the ability to represent unspecified numerosities emerges at around age 4, and is fully in place by ages 5 to 6. Together, these studies provide evidence that 1) in addition to the ANS, parallel individuation is one of the developmental roots of the representation of numerical relations and 2) the numerical reasoning principles that operate over representations of unspecified numerosities may develop later than computations that operate over representations of particular numerosities.