Academic journal article Memory & Cognition

The Grammar of Approximating Number Pairs

Academic journal article Memory & Cognition

The Grammar of Approximating Number Pairs

Article excerpt

In the present article, we studied approximating pairs of numbers (a, b) that were used to estimate quantity in a single phrase ("two, three years ago"). Pollmann and Jansen (1996) found that only a few of the many possible pairs are actually used, suggesting an interaction between the ways in which people estimate quantity and their use of quantitative phrases in colloquial speech. They proposed a set of rules that describe which approximating pairs are used in Dutch phrases. We revisited this issue in an analysis of Swedish and American language corpora and in a series of three experiments in which Swedish and American adults rated the acceptability of various approximating pairs and created approximating pairs of their own in response to various estimation tasks. We found evidence for Pollmann and Jansen's rules in both Swedish and English phrases, but we also identified additional rules and substantial individual and cross-language variation. We will discuss implications for the origin of this loose "grammar" of approximating pairs.

It is likely that humans have some form of inherent system for representing quantity-specifically, for representing exact quantities up to three and for making reasonable estimates for larger quantities (see, e.g., Dehaene, 1997; Gallistel & Gelman, 1992). Number words (e.g., "five") and other quantitative words and phrases (e.g., "more than") are mapped onto these inherent representations in ways that allow people to communicate about quantity. Much is known about children's mapping of number words onto cognitive representations of specific quantities (Fuson, 1988) and about how children estimate on mathematical tasks, such as a number line (Siegler & Opfer, 2003). Much less is known about how quantitative phrases are used in colloquial speech and whether there are discernible patterns in these phrases that reflect constraints on how people inherently represent quantities, whether they are based on the structure of the school-taught base- 10 number system and other cultural systems (e.g., time), or whether they are some combination of constraint and cultural conventions.

Dehaene and Mehler's (1992) cross-linguistic study of people's number use uncovered an inverse relationship between the magnitude of numbers and the frequency of usage, with local peaks at numbers such as 10, 12, 15, 20, 50, and 100. The same basic pattern was found for all seven languages assessed in this study. Dehaene and Mehler concluded that the preference for small numbers may have been due to the more precise underlying quantitative representations, whereas the prevalence of "reference numerals," such as 10 and 50, reflects use of our less precise approximations for larger quantities. Reference numerals may provide an anchor for mapping numerals and number words onto the underlying approximate representational system and for guiding a variety of daily activities in modern cultures. When searching for a street address at 1615, an understanding of the quantitative sequence of 100, 200, . . . 1000 allows people to scan for the correct block and to focus attention on individual addresses once the 1600 block is reached. At the same time, studies of traditional cultures without formal education indicate a much more limited range of number words that likely map onto the precise representations for small quantities (Chagnon, 1997; Pica, Lemer, Izard, & Dehaene, 2004), although number words for quantities beyond this range are sometimes found (Saxe, 1981).

The cross-cultural pattern suggests that the development of representations for quantity-number words, Arabic numerals-results from a combination of a social use for these words and numerals and an ability to map these onto precise or approximate representations of quantity. Dehaene and Mehler's (1992) fascinating finding that the same reference numerals (10, 12, 15, 20, 50, and 100) appear in many languages and modern cultures suggests a regularity in either the culturally derived number systems that map onto the approximate range or in how people intuitively use the approximate system for developing number systems. …

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