Academic journal article Perception and Psychophysics

Nonverbal Arithmetic in Humans: Light from Noise

Academic journal article Perception and Psychophysics

Nonverbal Arithmetic in Humans: Light from Noise

Article excerpt

Animal and human data suggest the existence of a cross-species system of analog number representation (e.g., Cordes, Gelman, Gallistel, & Whalen, 2001; Meck & Church, 1983), which may mediate the computation of statistical regularities in the environment (Gallistel, Gelman, & Cordes, 2006). However, evidence of arithmetic manipulation of these nonverbal magnitude representations is sparse and lacking in depth. This study uses the analysis of variability as a tool for understanding properties of these combinatorial processes. Human subjects participated in tasks requiring responses dependent upon the addition, subtraction, or reproduction of nonverbal counts. Variance analyses revealed that the magnitude of both inputs and answer contributed to the variability in the arithmetic responses, with operand variability dominating. Other contributing factors to the observed variability and implications for logarithmic versus scalar models of magnitude representation are discussed in light of these results.

Humans and other animals appear to compute descriptive statistics in a variety of domains-from language (e.g., Aslin, Saffran, & Newport, 1999), to foraging (Gallistel, 1990), to vision (e.g., Ariely, 2001) and motor skills (Trommershauser, Maloney, & Landy, 2003). These statistics may derive from mental magnitudes representing elementary abstractions like number, duration, and distance (Gallistel, Gelman, & Cordes, 2006). These magnitude representations are approximate (i.e., fuzzy estimates) and nonverbal-animals, preverbal human infants, and adult humans engaged in conflicting verbal tasks use them, suggesting a common cross-species mechanism for representing quantity (e.g., Cordes, Gelman, Gallistel, & Whalen, 2001; Meck & Church, 1983; Xu & Spelke, 2000).

There is a growing psychophysical literature on the nonverbal representation of these basic quantities (Brannon & Roitman, 2003; Dehaene, 1997; Gallistel & Gelman, 200S). A well-established finding is that performance obeys Weber's law; that is, the ease with which two subjective quantities (e.g., numbers) can be ordered is proportional to their ratio (Brannon & Terrace, 2000, 2002; Buckley & Gilman, 1974; Dehaene, Dupoux, & Mehler, 1990; Moyer & Landauer, 1973; Parkman, 1971). This Weber-characteristic is also evidenced when subjects must repeatedly state the number of rapid arrhythmic flashes they have seen or rapidly press a button a given number of times without verbal counting; the standard deviation of the resulting distribution is proportional to its mean (Cordes et al., 2001; Whalen, Gallistel, & Gelman, 1999). This scalar characteristic of the variability has led to the widely accepted hypothesis that discrete quantity (number) is represented nonverbally by continuous, noisy symbols (or signals) called mental magnitudes.

It has been shown that both animal and human subjects can do arithmetic with these nonverbal numerical symbols (Barth et al., 2006; Barth, La Mont, Lipton, & Spelke, 2005; Brannon, Wusthoff, Gallistel, & Gibbon, 2001; Cordes, King, & Gallistel, 2007; Gallistel & Gelman, 2005; Gibbon & Church, 1981; McCrink & Wynn, 2004; Pica, Lemer, Izard, & Dehaene, 2004). For example, one particular line of work with bilingual adult humans comparing "exact" (language-dependent) versus "approximate" (thought to employ nonverbal magnitudes) arithmetic finds that when subjects must choose the exact answer to arithmetic problems (e.g., 16 + 18 = 41 or 34), the reaction time is longer when the problem is posed in the subject's second (normative) language (Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999; Spelke & Tsivkin, 2001). By contrast, when subjects must only approximate the answer (give the response closest to the correct answer-e.g., 16 + 18 = 41 or 36), they choose equally rapidly regardless of the language of presentation. These studies and others suggest that while exact arithmetic engages arithmetic skills taught in school and dependent on language, approximate arithmetic taps into primitive nonverbal calculation abilities. …

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