Conclusion And Directions For Future Research

The data reviewed in this chapter suggest that language is not a necessary prerequisite for representing number and reasoning arithmetically. Many animal species represent numerosities independently of continuous variables such as time, density, and area and represent the ordinal relations between numerical values. Similarly, when surface area and density are well controlled, infants too are capable of representing number. A key question is whether human infants, adult humans, and nonhuman animals use the same system to represent number nonverbally. Do all three populations use analog magnitude representations of number? If so, are these representations formed by the same mechanism? Does the process conform to the counting principles defined by Gelman and Gallistel (1978)?

As reviewed above, animals and adult humans exhibit distance and magnitude effects when making numerical comparisons (Brannon and Terrace, 1998, 2000; Moyer and Landaeur, 1967) and exhibit scalar variability in their memory for specific magnitudes (Platt and Johnson, 1971; Whalen et al., 1999). Such data provide strong evidence that both nonhuman animals and adult humans represent number as continuous mental magnitudes. However, as of yet there are no estimates of the variability in infants' numerical representations (Gelman and Cordes, 2001). Methods have not been established that would allow one to assess whether infants show distance and magnitude effects. Such data would help delineate the similarities and differences between animal and human infant numerical representations.

A great deal of research has been done to characterize young children's early understanding of number (for a review, see Gelman and Meck, 1983). While there is abundant evidence that children have rich numerical concepts before they enter school, it is also apparent that the verbal counting system is mastered slowly. A challenge for the field is to determine how young children map the language-specific count words onto nonverbal representations of number. Do young children map count words onto analog magnitude representations of number, as suggested by Gelman and Gallistel (1978; Gallistel and Gelman, 1992), or are the first mappings accomplished by mapping count words onto plural-singular distinctions in language (Bloom and Wynn, 1997) or object-file representations (Carey, 1998; Spelke, 2000)?

It will be important to determine whether there are multiple formats for representing number and the developmental time course of these representations. Other questions that remain unanswered are whether the representations of number held by animals and infants are independent of sensory modality. What is the relative salience of number vs. continuous dimensions such as surface area, density, and time? Does the ability to discriminate and operate upon representations of time or surface area develop before the ability to discriminate number? Is one system used to represent time and number in infancy? Is the enumeration process parallel or serial?

It is ultimately essential to map the neural substrates of numerical processing and determine whether the same substrates underlie numerical abilities in animals, adult humans, and human infants. Are numerical mental magnitudes represented by the parietal cortex in human infants, nonhuman animals, and adult humans? If so, how specific is this neural substrate? Are all analog magnitude comparisons performed by this brain area, or only those that are numerical in nature? Do single cells represent cardinal and ordinal aspects of number, or is there a more distributed representation of number?

In conclusion, while we have outlined some of the many unanswered questions in the study of the nonverbal representation of number, it should be clear that there is an abundance of evidence that the sophisticated and dazzling numerical abilities seen in modern humans have both evolutionary and developmental precursors. Like most of complex cognition, we can map the seeds of mathematical prowess by studying babies and nonhuman animals. It is at the intersection between these two fields that we can begin to understand the nature of thought without language.

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