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. KEYWORDS—analog magnitudes; interval timing; number representation; abstract quantities Interactions with the environment—from balancing a checkbook to coordinating motor movements—consistently rely upon an understanding of time and number. These quantities are everywhere and, by virtue of their abstract nature, are not tied to any particular domain or modality. Five Shih Tzus, five bell chimes, five taps on your shoulder, five scents, and five ideas are all instances of the number five, regardless of whether they were seen, heard, felt, smelled, or thought. In both animals and humans, skills as diverse as language acquisition, visual perception, and foraging are dependent upon representations of quantities (Gallistel, 1990; Gallistel, Gelman, & Cordes, 2005), but whether this means animals have a true concept of time and number is unclear. In addition, quantity representations are selectively impaired in persons affected by Parkinson’s disease, schizophrenia, and dyscalculia (developmental disability in mathematics comprehension; Buhusi & Meck, 2005). An increased understanding of both the behavioral and neurobiological mechanisms responsible for time and number representations will not only further our understanding of learning in general but may also lead to improvements in the diagnosis and treatment of these debilitating disorders. The ability to represent abstract quantities is not dependent upon language; demonstrations in nonhuman animals confirm that this ability is shared across species. Meck and Church (1983) trained rats to discriminate two signals in which time and number were confounded. Lever presses on the left side were rewarded following a 2-second signal consisting of two noise bursts, and presses on the right were rewarded following eight bursts lasting a total of 8 seconds (0.5 seconds on, 0.5 seconds off; Fig. 1a). Following training, unrewarded probe trials were presented. On number-relevant probe trials, rats experienced anywhere from two to eight bursts, with total stimulus duration set at 4 seconds. On duration-relevant trials, the rats heard four bursts, but total stimulus duration varied from 2 to 8 seconds. Data revealed that the probability of a right lever response was clearly a function of the relevant stimulus value (Fig. 1b), suggesting the rats attended to both temporal and numerical stimulus cues. That is, on number-relevant trials, they responded based on the number of bursts they heard, and on duration-relevant trials, they timed the bursts and responded accordingly. Representations of abstract quantities served as the basis for responding in this task, as they have in many other studies involving a variety of species (Gallistel, 1990). The results obtained from such counting and timing tasks ubiquitously obey Weber’s law: The speed and accuracy with which two values are discriminated is dependent upon their ratio (2 vs. 8 is just as easy as 20 vs. 80, despite a 10-fold discrepancy in the absolute difference between magnitudes). In the behavioral data, this is evidenced by scalar variability— response variability (the standard deviation in the distribution of responses, or more simply, the magnitude of estimation errors) increases in direct proportion to the magnitude of the quantity (Cordes, Gelman, Gallistel, & Whalen, 2001; Gallistel, 1990). Address correspondence to Sara Cordes or Warren H. Meck, Department of Psychology and Neuroscience, 572 Research Dr., GSRB2 Box 91050, Duke University, Durham, NC 27708-91050; e-mail: [email protected], [email protected]. CURRENT DIRECTIONS IN PSYCHOLOGICAL SCIENCE 156 Volume 16—Number 3 Copyright r 2007 Association for Psychological Science Obviously, humans also represent abstract quantities. By looking at a clock, we can tell exactly what time it is. By counting students in a class, we know precisely how many copies of an exam to make. What may be less obvious is that humans share the same primitive representational capacities as nonhuman animals, independent of language. We know this, in part, because human timing also reveals scalar variability (Buhusi and Meck, 2005). Essentially, this means that when you head into the kitchen for a snack during a 3-minute TV commercial break, you may misestimate the amount of time you have to return by, say, 30 seconds, but when judging how much longer before the 30-minute show is over, you may be off by as much as 5 minutes. 100