Abstract
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.
Article PDF
References
Balakrishnan, J. D., &Ashby, F. G. (1992). Subitizing: Magical numbers or mere superstition.Psychological Research,54, 80–90.
Brannon, E. M., &Terrace, H. S. (2000). Representation of the numerosities 1–9 by rhesus macaques (Macaca mulatta).Journal of Experimental Psychology: Animal Behavior Processes,26, 31–49.
Brannon, E. M., Wusthoff, C. J., Gallistel, C. R., &Gibbon, J. (2001). Numerical subtraction in the pigeon: Evidence for a linear subjective number scale.Psychological Science,12, 238–243.
Butterworth, B. (1999).The mathematical brain. London: MacMillan.
Carey, S. (1998). Knowledge of number: Its evolution and ontogeny.Science,282, 641–642.
Carey, S. (2001). Cognitive foundations of arithmetic: Evolution and ontogenesis.Mind & Language,16, 37–55.
Dehaene, S., Dehaene-Lambertz, G., &Cohen, L. (1998). Abstract representations of numbers in the animal and human brain.Trends in Neuroscience,21, 355–361.
Dehaene, S., Dupoux, E., &Mehler, J. (1990). Is numerical comparison digital? Analogical and symbolic effects in two-digit number comparison.Journal of Experimental Psychology: Human Perception & Performance,16, 626–641.
Fetterman, J. G. (1993). Numerosity discrimination: Both time and number matter.Journal of Experimental Psychology: Animal Behavior Processes,19, 149–164.
Gallistel, C. R., &Gelman, R. (1992). Preverbal and verbal counting and computation.Cognition,44, 43–74.
Gallistel, C. R., &Gelman, R. (2000). Non-verbal numerical cognition: From reals to integers.Trends in Cognitive Sciences,4, 59–65.
Gallistel, C. R., Gelman, R., & Cordes, S. (in press). The cultural and evolutionary history of the real numbers. In S. Levinson & P. Jaisson (Eds.),Culture and evolution. Oxford: Oxford University Press.
Gelman, R. (1993). A rational-constructivist account of early learning about numbers and objects. In D. Medin (Ed.),Learning and motivation (Vol. 30, pp. 61–96). Academic Press: New York.
Gelman, R., &Cordes, S. (2001). Counting in animals and humans. In E. Dupoux (Ed.),Language, brain and cognitive development: Essays in honor of Jacques Mehler (pp. 279–303). Cambridge, MA: MIT Press.
Gelman, R., &Meck, E. (1992). Early principles aid early but not later conceptions of number. In J. Bideaud, C. Meljac, & J. P. Fischer (Eds.),Pathways to number ( pp. 171–189). Hillsdale, NJ.: Erlbaum.
Gibbon, J., &Church, R. M. (1981). Time left: Linear versus logarithmic subjective time.Journal of Experimental Psychology: Animal Behavior Processes,7, 87–107.
Gibbon, J., Church, R. M., &Meck, W. H. (1984). Scalar timing in memory. In J. Gibbon & L. Allan (Eds.),Timing and time perception (New York Academy of Sciences, Vol. 423, pp. 52–77). New York: New York Academy of Sciences.
Hauser, M., &Carey, S. (1998). Building a cognitive creature from a set of primitives: Evolutionary and developmental insights. In E.D. D. Cummins, E. C. Allen, et al. (Eds.),The evolution of mind (pp. 51–106). Oxford: Oxford University Press.
Joram, E., Subrahmanyam, K., &Gelman, R. (1998). Measurement estimation: Learning to map the route from number to quantity and back.Review of Educational Research,68, 413–449.
Leslie, A. M., Xu, F., Tremoulet, P. D., &Scholl, B. (1998). Indexing and the object concept: Developing What and Where systems.Trends in Cognitive Sciences,2, 10–18.
Logie, R. H., &Baddeley, A. D. (1987). Cognitive processes in counting.Journal of Experimental Psychology: Learning, Memory, & Cognition,13, 310–326.
Mandler, G., &Shebo, B. J. (1982). Subitizing: An analysis of its component processes.Journal of Experimental Psychology: General,11, 1–22.
Meck, W. H., &Church, R. M. (1983). A mode control model of counting and timing processes.Journal of Experimental Psychology: Animal Behavior Processes,9, 320–334.
Moyer, R. S., &Landauer, T. K. (1967). Time required for judgments of numerical inequality.Nature,215, 1519–1520.
Moyer, R. S., &Landauer, T. K. (1973). Determinants of reaction time for digit inequality judgments.Bulletin of the Psychonomic Society,1, 167–168.
Platt, J. R., &Johnson, D. M. (1971). Localization of position within a homogeneous behavior chain: Effects of error contingencies.Learning & Motivation,2, 386–414.
Reynvoet, B., &Brysbaert, M. (1999). Single-digit and two-digit Arabic numerals address the same semantic number line.Cognition,72, 191–201.
Rumbaugh, D. M., Savage-Rumbaugh, S., &Hegel, M. T. (1987). Summation in the chimpanzee (Pan troglodytes).Journal of Experimental Psychology: Animal Behavior Processes,13, 107–115.
Simon, T. J. (1999). The foundations of numerical thinking in a brain without numbers.Trends in Cognitive Sciences,3, 363–364.
Spelke, E. S. (2000). Core knowledge.American Psychologist,55, 1233–1243.
Starkey, P., &Cooper, R. G. (1995). The development of subitizing in young children.British Journal of Developmental Psychology,13, 399–420.
Stevens, S. (1956). The direct estimation of sensory magnitudes— Loudness.American Journal of Psychology,69, 1–25.
Stevens, S., &Harris, J. R. (1962). The scaling of subjective roughness and smoothness.Journal of Experimental Psychology,64, 489–494.
Trick, L. M., &Pylyshyn, Z. W. (1994). Why are small and large numbers enumerated differently?A limited-capacity preattentive stage in vision. Psychological Review,101, 80–102.
Washburn, D. A. (1994). Stroop-like effects for monkeys and humans: Processing speed or strength of association?Psychological Science,5, 375–379.
Whalen, J., Gallistel, C. R., &Gelman, R. (1999). Nonverbal counting in humans: The psychophysics of number representation.Psychological Science,10, 130–137.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research and preparation of the manuscript was supported by NSF Grants SRB-97209741 to R. Gelman and C. R. Gallistel and DFS-9209741 to R. Gelman.
Rights and permissions
About this article
Cite this article
Cordes, S., Gelman, R., Gallistel, C.R. et al. Variability signatures distinguish verbal from nonverbal counting for both large and small numbers. Psychonomic Bulletin & Review 8, 698–707 (2001). https://doi.org/10.3758/BF03196206
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.3758/BF03196206