Sara Cordes

Variability signatures distinguish verbal from nonverbal counting for both large and small numbers

Humans appear to share with animals a nonverbal counting process. In a nonverbal counting condition, subjects pressed a key a numeral-specified number of times, while saying “the” at every press. The mean number of presses increased as a power function of the target number, with a constant coefficient of variation (c.v.), both within and beyond the proposed subitizing range (1–4 or 5), suggesting small numbers are represented on the same continuum as larger numbers and subject to the same noise process (scalar variability). By contrast, when subjects counted their presses out loud as fast as they could, the c.v. decreased as the inverse square root of the target value (binomial variability instead of scalar variability). The unexpected power-law relation between target value and mean number of presses in nonverbal counting suggests a new hypothesis about the development of the function relating number symbols to mental magnitudes.

The Difficulties of Representing Continuous Extent in Infancy : Using Number Is Just Easier

This study investigates the ability of 6-month-old infants to attend to the continuous properties of a set of discrete entities. Infants were habituated to dot arrays that were constant in cumulative surface area yet varied in number for small (< 4) or large (> 3) sets. Results revealed that infants detected a 4-fold (but not 3-fold) change in area, regardless of set size. These results are in marked contrast to demonstrations that infants of the same age successfully discriminate a 2- or 3-fold change in number, providing strong counterevidence to the claim that infants use solely nonnumerical, continuous extent variables when discriminating sets. These findings also shed light on the processes involved in tracking continuous variables in infants.

Crossing the Divide: Infants Discriminate Small From Large Numerosities

Although young infants have repeatedly demonstrated successful numerosity discrimination across large sets when the number of items in the sets changes twofold (E. M. Brannon, S. Abbott, & D. J. Lutz, 2004; J. N. Wood & E. S. Spelke, 2005; F. Xu & E. S. Spelke, 2000), they consistently fail to discriminate a twofold change in number when one set is large and the other is small (<4 items; F. Feigenson, S. Carey, & M. Hauser, 2002; F. Xu, 2003). It has been theorized that this failure reflects an incompatibility in representational systems for small and large sets. The authors investigated the ability of 7-month-old infants to compare small and large sets over a variety of conditions. Results reveal that infants can successfully discriminate small from large sets when given a fourfold change, but not a twofold change, in number. The implications of these results are discussed in light of current theories of number representation.

Quantitative competencies in infancy

We review recently published papers that have contributed to our understanding of how the preverbal infant represents number, area and time. We review evidence that infants rely on two distinct systems to represent number nonverbally and highlight the similarities in the ratio-dependent discrimination of number, time and area. Contrary to earlier assertions that continuous dimensions are more salient (and thus more discriminable) to the infant than numerosity, we argue that the opposite conclusion is better supported by the data. The preverbal infant may be better able to extract numerosity than continuous variables from arrays of discrete items.

The relative salience of discrete and continuous quantity in young infants

Whether human infants spontaneously represent number remains contentious. Clearfield & Mix (1999) and Feigenson, Carey & Spelke (2002) put forth evidence that when presented with small sets of 1-3 items infants may preferentially attend to continuous properties of stimuli rather than to number, and these results have been interpreted as evidence that infants may not have numerical competence. Here we present three experiments that test the hypothesis that infants prefer to represent continuous variables over number. In Experiment 1, we attempt to replicate the Clearfield & Mix study with a larger sample of infants. Although we replicated their finding that infants attend to changes in contour length, infants in our study attended to number and perimeter/area simultaneously. In Experiments 2 and 3, we pit number against continuous extent for exclusively large sets (Experiment 2) and for small and large sets combined (Experiment 3). In all three experiments, infants noticed the change in number, suggesting that representing discrete quantity is not a last resort for human infants. These results should temper the conclusion that infants find continuous properties more salient than number and instead suggest that number is spontaneously represented by young infants, even when other cues are available.

Nonverbal arithmetic in humans: Light from noise

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.

Comment on “Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures”

Dehaene et al. (Reports, 30 May 2008, p. 1217) argued that native speakers of Mundurucu, a language without a linguistic numerical system, inherently represent numerical values as a logarithmically spaced spatial continuum. However, their data do not rule out the alternative conclusion that Mundurucu speakers encode numbers linearly with scalar variability and psychologically construct space-number mappings by analogy.

Common Representations of Abstract Quantities

Representations of abstract quantities such as time and number are essential for survival. A number of studies have revealed that both humans and nonhuman animals are able to nonverbally estimate time and number; striking similarities in the behavioral data suggest a common magnitude-representation system shared across species. It is unclear, however, whether these representations provide animals with a true concept of time and number, as posited by Gallistel and Gelman (2000). In this article, we review the prominent cognitive and neurobiological models of timing and counting and explore the current evidence suggesting that nonhuman animals represent these quantities in a modality-independent (i.e., abstract) and ordered manner. Avenues for future research in the area of temporal and mathematical cognition are also discussed.

Attending to One of Many: When Infants are Surprisingly Poor at Discriminating an Item's Size

Despite a prevailing assumption in the developmental literature that changes in continuous quantities (i.e., surface area, duration) are easier to detect than changes in number, very little research has focused on the verity of this assumption. The few studies that have directly examined infants’ discriminations of continuous extent have revealed that infants discriminate the duration of a single event and the area of a single item with similar levels of precision (Brannon et al., 2006; vanMarle and Wynn, 2006). But what about when items are presented in arrays? Infants appear to be much worse at representing the cumulative surface area compared to the numerosity of an array (Cordes and Brannon, 2008a), however this may be due to a noisy accumulation process and not a general finding pertaining to representations of the extent within an array. The current study investigates how well infants detect changes in the size of individual elements when they are presented within an array. Our results indicate that infants are less sensitive to continuous properties of items when they are presented within a set than when presented in isolation. Specifically we demonstrate that infants required a fourfold change in item size to detect a change when items were presented within a set of homogeneous elements. Rather than providing redundant cues that aided discrimination, presenting a set of identical elements appeared to hamper an infants ability to detect changes in a single elements size. In addition to providing some of the first evidence to suggest that the presence of multiple items may hinder extent representations, these results provide converging lines of evidence to support the claim that, contrary to popular belief, infants are better at tracking number than continuous properties of a set.

Intact interval timing in circadian CLOCK mutants.

While progress has been made in determining the molecular basis for the circadian clock, the mechanism by which mammalian brains time intervals measured in seconds to minutes remains a mystery. An obvious question is whether the interval-timing mechanism shares molecular machinery with the circadian timing mechanism. In the current study, we trained circadian CLOCK +/- and -/- mutant male mice in a peak-interval procedure with 10 and 20-s criteria. The mutant mice were more active than their wild-type littermates, but there were no reliable deficits in the accuracy or precision of their timing as compared with wild-type littermates. This suggests that expression of the CLOCK protein is not necessary for normal interval timing.