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Numerical competence in animals: Definitional issues, current evidence, and a new research agenda

Published online by Cambridge University Press:  04 February 2010

Hank Davis  and
Rachelle Pérusse
Hank Davis
Affiliation:
Department of Psychology, University of Guelph, Guelph, Ontario, CanadaNIG 2WI
Rachelle Pérusse
Affiliation:
Department of Psychology, University of Guelph, Guelph, Ontario, CanadaNIG 2WI
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Abstract

Numerical competence is one of the many aspects of animal cognition that have enjoyed a resurgence of interest during the past decade. Evidence for numerical abilities in animals has followed a tortuous path to respectability, however, from Clever Hans, the counting horse, to modern experimental studies. Recent surveys of the literaturereveal theoretical as well as definitional confusion arising from inconsistent terminology for numerical processes and procedures. The term “counting” has been applied to situations having little to do with its meaning in the human literature. We propose a consistent vocabulary and theoretical framework for evaluating numerical competence. Relative numerousness judgments, subitizing, counting, and estimation may be the essential processes by which animals perform numerical discriminations. Ordinality, cardinality, and transitivity also play an important role in these processes. Our schema is applied to a variety of recent experimental situations. Some evidence of transfer is essential in demonstrating higher-order ability such as counting or “sense of number.” Those instances of numerical competence in which all viable alternatives to counting (e.g., subitizing) have been precluded, but no evidence of transfer has been demonstrated might be described as “protocounting.” To show that animals are capable of “true” counting future research will have to demonstrate generality across situations.

Keywords

animal cognitioncardinalitycountingnumbernumerical competencenumerousness judgmentordinalityrhythmsubitizingsymbol usetemporal patternstransitivity
Type
Target Article
Information
Behavioral and Brain Sciences , Volume 11 , Issue 4 , December 1988 , pp. 561 - 579
Copyright
Copyright © Cambridge University Press 1988

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References

Allport, D. A. (1975) The state of cognitive psychology. Quarterly Journal of Experimental Psychology 27:141–52.CrossRefGoogle Scholar
Antell, S. E. & Keating, D. P. (1983) Perception of number invariance in neonates. Child Development 54:695–701.CrossRefGoogle ScholarPubMed
Arndt, W. (1939) Abschliessende Versuche zur Frage der “Zähl” Vermögens der Haustaube. Zietschrift für Tierpsychologie 3:88–142.CrossRefGoogle Scholar
Ascher, M. & Ascher, R. (1971) Numbers and relations from ancient Andean quipus. Archive for History of Exact Sciences Vol. 8. Springer-Verlag.Google Scholar
Bartley, S. H. (1980) Introduction to perception. Harper & Row.Google Scholar
Beach, F. A. (1960) Experimental investigations of species-specific behavior. American Psychologist 15:1–18.Google Scholar
Beckwith, M. & Restle, F. (1966) Process of enumeration. Psychological Review 73:437–44.CrossRefGoogle ScholarPubMed
Bennett, T. L. (1978) The sensory world. Brooks/Cole.Google Scholar
Bever, T. G. (1984) The road from behaviorism to rationalism. In: Animal cognition, ed. Roitbalt, H. L., Bever, T. G. & Terrace, H. S.. Erlbaum.Google Scholar
Bloomfield, T. M. (1967) Behavioral contrast and relative reinforcement frequency in two multiple schedules. Journal of the Experimental Analysis of Behavior 10:151–58.CrossRefGoogle ScholarPubMed
Boring, E. G. (1942) Sensation and perception in the history of experimental psychology. Appleton.Google Scholar
Boysen, S. T. (1987) Scrutinizing subitizing: Further studies of numerical skills in the chimpanzee. In: Numerical skills in animals, Boysen, S. (Chair). Symposium conducted at the annual meetings of the Midwestern Psychological Association, Chicago, 111.Google Scholar
Boysen, S. T. (1988) One banana, two banana, three banana, four: Counting in a chimpanzee who knows less from more. Paper presented at the Southern Society for Philosophy and Psychology, Miami, Fla.Google Scholar
Boysen, S. T. & Bernston, G. B. (1986) Clever Kermit: Counting capabilities in the chimpanzee (Pan troglodytes). Paper presented at the meetings of the Midwestern Psychological Association, Chicago, III.Google Scholar
(in press) The emergence of numerical competence in the chimpanzee (Pan troglodytes). In: Language and intelligence in animab: Developmental perspectives, ed. Parker, S. T. & Cibson, K. R.. Cambridge University Press.Google Scholar
Brackbill, Y. & Fitzgerald, H. E. (1972) Stereotype temporal conditioning in infants. Psychophysiology 9:569–77.CrossRefGoogle ScholarPubMed
Brainerd, C. J. (1979) The origins of the number concept. Praeger.Google Scholar
Braun, H. (1952) Über das Vermogen der Papageien unbenante Anzahlen zu unterscheiden. Zietschrift für Tierpsychologie 9:80–91.Google Scholar
Breslow, L. (1981) Re-evaluation of the literature on the development of transitive inference. Psychological Bulletin 89:325–51.Google Scholar
Brooks, R. A. (1986) Achieving artificial intelligence through building robots. MIT Al Laboratory Memo 895:1–11.Google Scholar
Brouwer, L. E. J. (1913) Intuitionism and formalism. Bulletin of the American Mathematical Society 20:81–96.Google Scholar
Brown, D. P. F., Lenneberg, E. H. & Ettlinger, G. (1978) Ability of chimpanzees to respond to symbols of quantity in comparison with that of children and monkeys. Journal of Comparative and Physiological Psychology 92:815–20.CrossRefGoogle ScholarPubMed
Bryant, P. E. & Trabasso, T. R. (1971) Transitive inferences and memory in young children. Nature 232:456–58.CrossRefGoogle ScholarPubMed
Burns, R. A. & Gordon, W. U. (in press) Some further observations on serial enumeration and categorical flexibility. Animal Learning and Behavior.Google Scholar
Burns, R. A. & Sanders, R. E. (1987) Concurrent counting of two and three events in a serial anticipation paradigm. Bulletin of the Psychonomic Society 25:479–81.Google Scholar
Burns, R. A., Sanders, R. E. & Hulbert, L. G. (submitted) Conditional serial enumeration of three, four, and five motivational events.Google Scholar
Capaldi, E. J. (1964) Effect of N-length, number of different N-lengths, and number of reinforcements on resistance to extinction, journal of Experimental Psychology 68:230–39.CrossRefGoogle ScholarPubMed
Capaldi, E. J. (1966) Partial reinforcement: A hypothesis of sequential effects. Psychological Review 73:459–77.Google Scholar
Capaldi, E. J. (1967) A sequential hypothesis of instrumental learning. In: Psychology of learning and motivation, vol. 1, ed. Spence, K. W. & Spence, J.. Academic Press.Google Scholar
Capaldi, E. J. & Miller, D. J. (1988) Counting in rats: Its functional significance and the independent cognitive processes that constitute it. Journal of Experimental Psychology: Animal Behavior Processes 14:3–17.Google Scholar
Capaldi, E. J. & Verry, D. R. (1981) Serial anticipation learning in rats: Memory for multiple hedonic events and their order. Animal Learning and Behavior 9:441–53.CrossRefGoogle Scholar
Caramuel, J. (1977) Meditatio prooemialis, mathesis biceps. In: L'aritmetica binaria e le altre aritmetiche di Ciocanni Caramuel, Vescovo di Vigevano, ed. Parea, A., Soriano, P. & Terzi, P.; trans. Glasersfeld, E. von. Vigevano, Italy: Academia Tibernia. (Original work published 1670.)Google Scholar
Carlson, A. D. & Copeland, J. (1985) Flash communication in fireflies. Quarterly Review of Biology 60:415–36.CrossRefGoogle Scholar
Carston, R. (1988) Review of D. Premack: Gavagai or the future history of the animal language controversy. Mind & Language 2:333–49.CrossRefGoogle Scholar
Chi, M. T. H. & Klahr, D. (1975) Span and rate of apprehension in children and adults. Journal of Experimental Child Psychology 19:434–39.Google Scholar
Church, R. M. & Meek, W. H. (1974) The numerical attribute of stimuli. In: Animal cognition, ed. Roitbalt, H. L., Bever, T. G. & Terrace, H. S.. Erlbaum.Google Scholar
Conant, L. L. (1896) The number concept. Macmillan.Google Scholar
Cooper, R. G. (1984) Early number development: Discovering number space with addition and subtraction. In: Origins of cognitive skills, ed. Sophian, C.. Erlbaum.Google Scholar
Cooper, R. G., Campbell, R. or Blevins, B. (1983) Numerical representation from infancy to middle childhood: What develops? In: The acquisition of symbolic skills, ed. Rogers, D. R. & Sloboda, J. A.. Plenum Press.Google Scholar
Cooper, W. S. (1987) Decision theory as a branch of evolutionary theory; A biological derivation of the Savage Axioms. Psychological Review 94(4):395–411.Google Scholar
Coren, S., Porac, C. & Ward, L. M. (1979) Sensation and perception. Academic Press.Google Scholar
Czemy, P. & Thomas, R. K. (1975) Sameness-difference judgments in Saimiri sciureus based on volumetric cues. Animal Learning and Behavior 3:375–79.Google Scholar
Dantzig, T. (1930) Number: The language of science. Macmillan.Google Scholar
Davis, H. (1984) Discrimination of the number three by a raccoon (Procyonlotor). Animal Learning and Behavior 12:409–13.Google Scholar
Davis, H. (1986) Self-imposed feeding restraint in the rat: A numerical discrimination. Paper presented at the 27th annual meeting of the Psychonomic Society, New Orleans, La.Google Scholar
Davis, H.(in press) A theoretical note on the moral development of rats. (Rattus norvegicus). Journal of Comparative Psychology.Google Scholar
Davis, H. & Albert, M. (1986) Numerical discrimination by rats using sequential auditory stimuli. Animal Learning and Behavior 14:57–59.Google Scholar
(1987) Failure to transfer or train a numerical discrimination using sequential visual stimuli in rats. Bulletin of the Psychonomic Society 25:472–74.CrossRefGoogle Scholar
Davis, H., Albert, M. & Barren, R. W. (1985) Detection of number or numerousness by human infants. Science 228:1222.CrossRefGoogle ScholarPubMed
Davis, H. & Bradford, S. A. (1986) Counting behavior by rats in a simulated natural environment. Ethology 73:265–80.Google Scholar
(1987) Simultaneous numerical discrimination by rats. Bulletin of the Psychonomic Society 25:113–16.CrossRefGoogle Scholar
Davis, H., MacKenzie, K. & Morrison, S. (in press) Numerical discrimination using body and vibrissal touch in the rat. Journal of Comparative Psychology.Google Scholar
Davis, H. & Memmott, J. (1982) Counting behavior in animals: A critical evaluation. Psychological Bulletin 92:547–71.Google Scholar
Davis, H. & Memmott, J. (1983) Autocontingencies: Rats count to three to predict safety from shock. Animal Learning and Behavior 11:95–100.Google Scholar
Davis, H., Memmott, J. & Hurwitz, H. M. B. (1975) Autocontingencies: A model for subtle behavioral control. Journal of Experimental Psychology: General 104:169–88.Google Scholar
Davis, H. & Péusse, R. (1988) Human-based social interaction can reward a rat's behavior. Animal Learning and Behavior 16:89–92.CrossRefGoogle Scholar
Dawson, B. V. (1961) Counting in jackdaws. Behaviour 18:229–38.Google Scholar
Dewsbury, D. A. (1973) Introduction. In: Comparative psychology: A modern survey, ed. Dewsbury, D. A. & Shafer, D. A. Rethling. McGraw-Hill.Google Scholar
Dooley, G. B. &. Cill, T. (1977) Mathematical capabilities in Lana chimpanzee. In: Progress in age research, ed. Bourne, G. H.. Academic Press.Google Scholar
Douglas, J. W. B. or Whitty, W. M. (1941) An investigation of number appreciation in some sub-human primates. Journal of Comparative Psychology 31:129–43.CrossRefGoogle Scholar
Douglass, H. R. (1925) The development of number concept in children of pre-school and kindergarten ages. Journal of Experimental Psychology 8:443–70.CrossRefGoogle Scholar
Dowling, W. J. or Harwood, D. L. (1986) Music cognition. Academic Press.Google Scholar
Egremont, P. & Rothschild, M. (1979) The calculating cormorants. Biological Journal of the Linnaean Society 12:181–86.CrossRefGoogle Scholar
Fernandes, D. M. & Church, R. (1982) Discrimination of the number of sequential events by rats. Animal Learning and Behavior 10:171–76.Google Scholar
Ferster, C. B. (1958) Intermittent reinforcement of a complex response in a chimpanzee. Journal of the Experimental Analysis of Behavior 1:163–65.CrossRefGoogle Scholar
Ferster, C. B. (1964) Arithmetic behavior in chimpanzees. Scientific American 210:98–106.Google Scholar
Fouts, R. (1978) Sign language in chimpanzees: Implications of the visual mode and the comparative approach. In: Sign language and language acquisition in man and ape, ed. Peng, F. C. C.. Westview Press.Google Scholar
Fraisse, P. (1982) Rhythm and tempo. In: The psychology of music, ed. Deutsch, D.. Academic Press.Google Scholar
Fuson, K. C. (1988) Children's counting and concepts of number. Springer-Verlag.CrossRefGoogle Scholar
Fuson, K. C. & Hall, J. W. (1982) The aquisition of early number word meanings: A conceptual analysis and review. In: Children's mathematical thinking, ed. Ginsburg, H. P.. Academic Press.Google Scholar
Gabrielsson, A. (1981) Music psychology: A survey of problems and current research activities. Royal Swedish Academy of Music Press (Whole No. 32).Google Scholar
Gallistel, C. R. (1988) The formal characterization of counting processes and mechanisms. Paper presented at the annual meeting of the Southern Society for Philosophy and Psychology, Miami, Fla.Google Scholar
Gallistel, C. R. (1988) (in press)Animal cognition: The representation of space, time and number. Annual Review of Psychology 40:Google Scholar
Gardner, R. & Gardner, B. (1969) Teaching sign language to a chimpanzee. Science 165:664–72.Google Scholar
Gast, H. (1957) Der Umgang mit Zahlen und Zahlgebidden in der Fruhen Kindheit. Zeitschrift fur Psychologie 161:1–90.Google Scholar
Geldard, F. A. (1953) The human senses. Wiley.Google Scholar
Gellatly, A. R. H. (1987) The acquisition of a concept of logical necessity. Human Development 30:32–47.CrossRefGoogle Scholar
Gellatly, A. R. H.(in press a) Human inference. In: Human and machine problem solving, ed. Gilhooly, K. K..Google Scholar
Gellatly, A. R. H.(in press b) The myth of cognitive diagnosis. In: Cognition and social worlds, ed. Gellatly, A. R. H., Rogers, D. R. & Sloboda, J. A.. Oxford University Press.Google Scholar
Gelman, R. (1982) Basic number abilities. In: Advances in the psychology of human intelligence, vol. 1, ed. Steinberg, R. J.. Erlbaum.Google Scholar
Gelman, R. & Callistel, C. R. (1978) The child's understanding of number. Harvard University Press.Google Scholar
Gibbon, J. (1981) On the form and location of the psychometric bisection function for time. Journal of Mathematical Psychology 24:58–87.Google Scholar
Gibson, E. J. & Spelke, E. S. (1983) The development of perception. In: Handbook of child psychology, vol. 3, 4th ed., ed. Mussen, H.. Wiley.Google Scholar
Gillan, D. J. (1981) Reasoning in the Chimpanzee: II. Transitive inference. Journal of Experimental Psychology: Animal Behavior Processes 7:150–64.Google Scholar
Ginsburg, H. (1977) Children's arithmetic. Van Nostrand.Google Scholar
Gleitman, H. (1974) Getting animals to understand the experimenter's instructions. Animal Learning and Behavior 2:1–5.CrossRefGoogle Scholar
Guttinger, H. R. (1979) The integration of learnt and genetically programmed behaviour: A study of hierarchical organization in songs of canaries, greenfinches and their hybrids. Zeitschrift für Tierpsychologie 49:285–303.Google Scholar
Hall, V. G. & Kaye, D. B. (1977) The necessity of logical necessity in Piaget's theory. In: Alternative to Piaget, ed. Siegel, S. & Brainerd, C. J.. Academic Press.Google Scholar
Harter, M. R. & White, C. T. (1967) Perceived number and evoked cortical potentials. Science 156:406–8.Google Scholar
Hartline, H. K. (1934) Intensity and duration in the excitation of single photoreceptor units. Journal of Cellular & Comparative Physiology 5:229–47.CrossRefGoogle Scholar
Hasher, L. & Zacks, R. T. (1979) Automatic and effortful processes in memory. Journal of Experimental Psychology: General 108:356–88.CrossRefGoogle Scholar
Haugeland, J. (1978) The nature and plausibility of cognitivisim. Behavioral and Brain Sciences 2:215–60.Google Scholar
Herman, L. M., Richards, D. G. & Woltz, J. P. (1984) Comprehension of sentences by bottlenosed dolphins. Cognition 16:129–219.CrossRefGoogle ScholarPubMed
Herrnstein, R. J. (1970) On the law of effect. Journal of the Experimental Analysis of Behavior 13:243–66.Google Scholar
Herrnstein, R. J. & Hineline, P. M. (1966) Negative reinforcement as shock frequency reduction. Journal of the Experimental Analysis of Behavior 9:421–30.Google Scholar
Heyting, A. (1956) Intuitionism: An introduction. North-Holland.Google Scholar
Hicks, L. H. (1956) An analysis of number-concept formation in the rhesus monkey. Journal of Comparative and Physiological Psychology 49:212–18.Google Scholar
Hilbert, D. (1899) Crundlagen der Ceometrie. Teubner.Google Scholar
Hoban, E. (1986) The promise of animal language research. Ph.D dissertation. University of Hawaii, Manoa.Google Scholar
Hobson, S. L. & Newman, F. (1981) Fixed-ratio-counting schedules. In: Quantitative analyses of behavior: Discriminative properties of reinforcement schedules, vol. 1, ed. Commons, M. L. & Nevin, J. A.. Ballinger.Google Scholar
Honig, W. K. (1978) Studies of working memory in the pigeon. In: Cognitive processes in animal behavior, ed. Hulse, S., Fowler, H. & Honig, W. K.. Erlbaum.Google Scholar
Honigman, H. (1942) The number conception in animal psychology. Biological Review 17:315–37.Google Scholar
Huber, F. & Thorson, J. (1985) Cricket auditory communication. Scientific American 253(6):60–68.Google Scholar
Hughes, M. (1987) Children and number. Basil Blackwell.Google Scholar
Hulse, S. H., Fowler, H. & Honig, W. K. (1978) Cognitive processes in animal behavior. Erlbaum.Google Scholar
Hulse, S. H. & O'Leary, D. K. (1982) Serial position learning: Teaching an alphabet to rats. Journal of Experimental Psychology: Animal Behavior Processes 8:260–73.Google Scholar
Hunter, W. S. & Sigler, M. (1940) The span of visual discrimination as a function of time and intensity of stimulation. Journal of Experimental Psychology 26:160–80.CrossRefGoogle Scholar
Horwitz, W. A., Kestenbaum, C., Person, E., & Jarwick, L. (1965) Identical twin “idiot-savants” Calendar Calculator, The American Journal of Psychiatry 126:1075–77.Google Scholar
Hurwitz, H. M. B. (1963) Facilitation of counting-like behavior. Animal Behavior 11:449–54.Google Scholar
Jenkins, H. M. & Harrison, R. H. (1960) Effect of discrimination training on auditory generalization. Journal of Experimental Psychology 59: 246–53.Google Scholar
Jevons, W. S. (1871) The power of numerical discrimination. Nature 3:281–82.Google Scholar
Johnson-Laird, P. N. (1982) Thinking as a skill. Quarterly Journal of Experimental Psychology 34A:l–30.Google Scholar
Johnson-Laird, P. N. (1983) Mental models. Cambridge University Press.Google Scholar
Johnson-Laird, P. N. (1986) Reasoning without logic. In: Reasoning and discourse processes, ed. Myers, T., Brown, K. & McGonigle, B.. Academic Press.Google Scholar
Johnson-Laird, P. N. & Wason, P. C. (1977) A theoretical analysis of insight into reasoning task. In: Thinking: Readings in cognitive science, ed. Johnson-Laird, P. N. & Wason, P. N.. Cambridge University Press.Google Scholar
Kacelnik, A. & Houston, A. I. (1984) Some effects of energy costs on foraging strategies. Animal Behaviour 32:609–14.CrossRefGoogle Scholar
Kalmus, H. (1964) Animals as mathematicians. Nature 202:1156–60.Google Scholar
Karmiloff-Smith, A. (1979) Micro- and macro-developmental changes in language and other representational systems. Cognitive Science 3:91–118.Google Scholar
Karmiloff-Smith, A.(1983) A new abstract code or the new possibility of multiple codes? Behavioral and Brain Sciences 6:149.Google Scholar
Kaufman, E. L., Lord, M. W., Reese, T. W. & Volkmann, J. (1949) The discrimination of visual number. American Journal of Psychology 62:498–525.Google Scholar
Klahr, D. (1973) Quantification processes. In: Visual information processing, ed. Chase, W. G.. Academic Press.Google Scholar
Klahr, D. (1984) Transition processes in quantitative development. In: Mechanisms of cognitive development, ed. Stemberg, R. J.. Freeman, W. H..Google Scholar
Klahr, D. & Wallace, J. G. (1973) The role of quantification operators in the development of conservation. Cognitive Psychology 4:301–27.Google Scholar
Klein, F. (1927) Elementary mathematics from an advanced standpoint. Dover.Google Scholar
Klein, A. & Starkey, P. (1987) The origins and development of numerical cognition: A comparative analysis. In: Cognitive processes in mathematics, ed. Sloboda, J. A. & Rogers, D.. Clarendon Press.Google Scholar
Knopp, K. (1952) Elements of the theory of functions. Dover.Google Scholar
Koehler, O. (1937) Konnen Tauben zāhlen? Zietschrift für Tierpsychologie 1:39–48.Google Scholar
Koehler, O. (1940a) Vom Erlemen unbenannter Anxahlen bei Tauben. Veröffentlichungen der Reichsstelle für den Unterrichtsfilm Nr.B 521:1–14.Google Scholar
Koehler, O. (1940b) Wellensittiche erlemen unbennte Anzahlen. Veroffentlichungen der Reichsstelle für den Unterrichtsftlm Nr.B 523:1–14.Google Scholar
Koehler, O. (1950) The ability of birds to “count.” Bulletin of Animal Behaviour 9:41–45.Google Scholar
Koehler, O. (1924/1959) The mentality of apes. (Winter, E., translator.) Vintage Books. (Original work published 1924.)Google Scholar
Koehler, O., Muller, O. & Wachholtz, R. (1935) Kann die Taube Anzahlen erfassen? Verhandlungen Deutscher Zoologischen Cesellschafi, 39–59.Google Scholar
Krebs, J. R. (1978) Optimal foraging: Decision rules for predictors. In: Behavioral ecology, ed. Krebs, R. & Davies, N. B.. Sinauer.Google Scholar
Krebs, J. R., Kacelnik, A. & Taylor, P. (1978) Test of optimal sampling by foraging great tits. Nature 275:27–31.Google Scholar
Krebs, J. R., Ryan, J. C. & Charnov, E. L. (1974) Hunting by expectation or optimal foraging? A study of patch use by chickadees. Animal Behaviour 22:953–64.CrossRefGoogle Scholar
Lack, D. (1954) The natural regulation of animal numbers. Clarendon Press.Google Scholar
Lechelt, E. C. (1971) Spatial numerosity discrimination as contingent upon sensory and extrinsic factors. Perception & Psychophysics 8:76–80.Google Scholar
Lechelt, E. C. (1975) Temporal numerosity discrimination: Intermodal comparisons revisited. British Journal of Psychology 66:101–8.CrossRefGoogle ScholarPubMed
Lee, D. N. & Reddish, P. E. (1981) Plummeting gannets: A paradigm of ecological optics. Nature 293:293–94.CrossRefGoogle Scholar
Le Gare, M. (1987) The use of general systems as a metatheory for developing and evaluating theories in the neurosciences. Behavioral Sciences 32:106–20.Google Scholar
Lenneberg, E. H. (1971) Of language knowledge, apes, and brains. Journal of Psycholinguists Research 1:1–29.Google Scholar
Lögler, P. (1959) Versuche zur Frage des Zähl-Vermogens an einem Graupapegei und Vergleichsversuche an Menschen. Zeitschrift für Psychologie 16:179–217.Google Scholar
Luchins, A. S. & Luchins, E. H. (1965) Logical foundations of mathematics for behavioral scientists. Holt, Rinehart and Winston.Google Scholar
Lunzer, E. A. (1960) Some points of Piagetian theory in the light of experimental criticism. Journal of Child Psychology and Psychiatry 1:91–202.Google Scholar
Luria, A. R. (1980) Higher cortical functions in man (2nd ed.). Basic Books.Google Scholar
Mandler, G. & Shebo, B. J. (1982) Subitizing: An analysis of its component processes. Journal of Experimental Psychology: General 11:1–22.Google Scholar
Marler, P. & Tamura, M. (1962) Song “dialects” in three populations of white-crowned sparrow. Condor 64:368–77.CrossRefGoogle Scholar
Marold, E. (1939) Versuche an Wellensittichen zur Frage des “Zähl”-vermögens. Zietschrift für Tierpsychologie 3:170–223.Google Scholar
Masserman, J. H. & Rubinfine, D. L. (1944) “Counting” behavior in cats. Journal of General Psychology 30:87–88.Google Scholar
Matlin, M. W. (1988) Sensation and perception. Allyn & Bacon.Google Scholar
Matsuzawa, T. (1985) Use of numbers by a chimpanzee. Nature 315:57–59.Google Scholar
McCloskey, M., Sokol, S. M. & Goodman, R. A. (1986) Cognitive processes in verbal-number production: Inferences from the performance of braindamaged subjects. Journal of Experimental Psychology: General 115:307–30.CrossRefGoogle ScholarPubMed
McGonigle, B. (1985) Can apes learn to count? Nature 315:16–17.Google Scholar
McConigle, B. O. & Chalmers, M. (1977) Are monkeys logical? Nature 267:694–96.CrossRefGoogle Scholar
McNamara, J. M. (1982) Optimal patch use in a stochastic environment. Theoretical Population Biology 21:269–88.Google Scholar
Mechner, F. M. (1958) Probability relations within response sequences under ratio-reinforcement. Journal of the Experimental Analysis of Behavior 1:109–21.Google Scholar
Mechner, F. M. & Guevrekian, L. (1962) Effects of deprivation upon counting and timing in rats. Journal of the Experimental Analysis of Behavior 5:463–66.Google Scholar
Meek, W. H. & Church, R. M. (1983) A mode control model of counting and timing processes. Journal of Experimental Psychology: Animal Behavior Processes 9:320–34.Google Scholar
Menninger, K. (1969) Number words and number symbols: A cultural history of numbers. MIT Press.Google Scholar
Migler, B. (1964) Effects of averaging data during stimulus generalization. Journal of the Experitnental Analysis of Behavior 7:303–7.Google Scholar
Millenson, J. R. (1962) Acquired counting behavior in mice maintained under two reinforced procedures. Animal Behavior 10:171–73.Google Scholar
Miller, G. A. (1956) The magical number seven plus or minus two: Some limits on our capacity for processing information. Psychological Review 63:81–97.CrossRefGoogle ScholarPubMed
Mintz, D. E. (1962) Force of response during ratio reinforcement. Science 138:516–17.CrossRefGoogle ScholarPubMed
Mitchell, R. W., Yao, P., Sherman, P. & O'Regan, M. (1985) Discriminative responding of a dolphin (Tursiops truncatus) to differentially rewarded stimuli. Journal of Comparative Psychology 99:218–25.Google Scholar
Munn, N. F. (1950) Handbook of psychological research on the rat. Oxford University Press.Google Scholar
Nelson, T. M. & Bartley, S. H. (1961) Numerosity, number, arithmetization, and psychology. Philosophy of Science 28:178–203.Google Scholar
Nelson, T. M. & Lechelt, E. C. (1970) Socioeconomic status, value, and response to number. Perception & Psychophysics 8:76–80.Google Scholar
Nevin, J. A. (1968) Differential reinforcement and stimulus control of not responding. Journal of the Experimental Analysis of Behavior 11:715–26.Google Scholar
Nottebohm, F. (1969) The song of the chingolo, Zonotrichia capensis, in Argentina: Description and evaluation of a system of dialects. Condor 71:299–315.Google Scholar
Otte, D. (1970) A comparative study of communicative behavior in grasshoppers. Miscellaneous Publications, Museum of Zoology, University of Michigan, No. 141.Google Scholar
Pastore, N. (1961) Number sense and “counting” ability in the canary. Zietschrift für Tierpsychologie 18:561–73.Google Scholar
Payne, R. B. (1978) Local dialects in the wingflaps of flappet larks Mirafra rufodnnamomea. Ibis 120:204–7.Google Scholar
Peano, G. (1894) Notations de logique mathematique. Cuadagnini.Google Scholar
Pepperberg, I. M. (1983) Cognition in the African Grey Parrot: Preliminary evidence for auditory vocal comprehension of the class concept. Animal Learning and Behavior 11:179–85.Google Scholar
Pepperberg, I. M. (1987) Evidence for conceptual quantitative abilities in the African Grey Parrot: Labelling of cardinal sets. Ethology 75:37–61.Google Scholar
Pepperberg, I. M. (1988) The importance of social interaction and observation in the acquisition of communicative competence: Possible parallels between avian and human learning. In: Social learning: Psychological and biological perspectives, ed. Zentall, T. R. & Galef, B. G. Jr, Erlbaum.Google Scholar
Piaget, J. (1937) La construction du reel chez l' enfant'. Neuchatel: Delachaux et Niestle.Google Scholar
Piaget, J. (1952) The child's conception of number. Routledge & Kegan Paul.Google Scholar
Piaget, J. (1960) Introduction. In: Problemes de la construction du nombre, ed. Greco, P., Grize, J. B., Papert, S. & Piaget, J.. Etudes d'epistemologie genetique, vol. 11. Presses Univer. France.Google Scholar
Platt, J. R. & Johnson, D. M. (1971) Localization of position within a homogeneous behavior chain: Effects of error contingencies. Learning and Motivation 2:386–414.Google Scholar
Popov, A. V. & Shuvalov, V. F. (1977) Phonotactic behavior of crickets. Journal of Comparative Physiology 119:111–26.Google Scholar
Posner, M. I. & Snyder, C. R. (1975) Attention and cognitive control. In: Information processing and cognition: The Loyola symposium, ed. Solso, L.. Erlbaum.Google Scholar
Premack, D. (1976) Intelligence in ape and man. Erlbaum.Google Scholar
Premack, D. (1983) The codes of man and beasts. Behavioral and Brain Sciences 6:125–67.CrossRefGoogle Scholar
Premack, D. (1986) Gavagail or the future history of the animal language controversy. MIT Press.Google Scholar
Razran, G. (1965) Empirical codifications and specific theoretical implications of compound-stimulus conditioning: Perception. In: Classical conditioning, ed. Prokasy, W. F.. Appleton-Century-Crofts.Google Scholar
Reese, E. P., Robinson, H. B., Stevenson, J. G. & Volkman, J. (1960) Relative effectiveness of presenting information to selected sense modalities. Technical Report: NAVTRA DEVCEN 512–1, U.S. Naval Training Device Center, Port Washington, New York.Google Scholar
Reynolds, G. S. (1961a) Behavioral contrast. Journal of the Experimental Analysis of Behavior 4:57–71.Google Scholar
Reynolds, G. S. (1961b) Relativity of response rate and reinforcement frequency in a multiple schedule. Journal of the Experimental Analysis of Behavior 4:179–84.Google Scholar
Rilling, M. E. (1967) Number of responses as a stimuls in fixed interval and fixed ratio schedules. Journal of Comparative and Physiological Psychology 63:60–65.Google Scholar
Rohles, F. H. & Devine, J. H. (1966) Chimpanzee performance on a problem involving the concept of middleness. Animal Behaviour 14:159–62.Google Scholar
Rohles, F. H. & Devine, J. H. (1967) Further studies of the middleness concept with the chimpanzee. Animal Behaviour 15:107–12.Google Scholar
Roitblat, H. L. (1982) The meaning of representation in animal memory. Behavioral and Brain Sciences 5:353–406.Google Scholar
Roitblat, H. L. (1987) Introduction to comparative cognition. W. H. Freeman.Google Scholar
Roitblat, H. L., Bever, T. G. & Terrace, H. S. (eds.) (1984) Animal cognition. Erlbaum.Google Scholar
Rozin, P. (1976) The evolution of intelligence and access to the cognitive unconscious. In: Progress in psychobiology and physiological psychology, ed. Sprague, J. M. & Epstein, A. N.. Academic Press.Google Scholar
Rozin, P. & Kalat, J. W. (1972) Learning as a situation-specific adaptation. In: Biological boundaries of learning, ed. Seligman, M. E. P. & Hager, J. L.. Appleton Century Crofts.Google Scholar
Rumbaugh, D. M. (1985) Comparative psychology: Patterns in adaptation. In: The G. Stanley Hall Lecture Series, vol. 5, ed. Rogers, A. M. & Scheirer, C. J.. Washington, D.C. American Psychological Association.Google Scholar
Rumbaugh, D. M., Savage-Rumbaugh, E. S. & Hegel, M. T. (1987) Summation in the chimpanzee. Journal of Experimental Psychology: Animal Behavior Processes 13:107–15.Google Scholar
Rumbaugh, D. M., Savage-Rumbaugh, E. S. & Pate, J. L. (1988) Addendum to summation in the chimpanzee (Pan troglodytes). Journal of Experimental Psychology: Animal Behavior Processes 14:118–20.Google Scholar
Salman, D. H. (1943) Note on the number conception in animal psychology. British Journal of Psychology 33:209–19.Google Scholar
Sauter, U. (1952) Versuche zur Frage des “Zähl” Vermögens bei Elsten. Zietschrift für Tierpsychologie 9:252–89.Google Scholar
Savage-Rumbaugh, E. S. (1986) Ape language: From conditioned response to symbol. Columbia University Press.Google Scholar
Saxe, G. B. (1982) Culture and the development of numerical cognition: Studies among the Oskapmin of Papua New Guinea. In: Children's logical and mathematical cognition, ed. Brainerd, C. J.. Springer-Verlag.Google Scholar
Schiemann, K. (1939) Vom Erlennen unbenanntes Anzahlen bei Dohlen. Zietschrift fir Tierpsychologie 3:292–347.Google Scholar
Schiffman, H. R. (1982) Sensation and perception. John Wiley.Google Scholar
Schiffman, H., Lore, R., Passafiume, J. & Neeb, R. (1970) Role of vibrissae for depth perception in the rat (Rattus norvegicus). Animal Behaviour 18:290–92.Google Scholar
Schusterman, R. J. & Gisiner, R. (1988) Artificial language comprehension in dolphins and sea lions: The essential cognitive skills. Psychological Record 38:311–48.Google Scholar
Schusterman, R. J. & Krieger, K. (1984) California sea lions are capable of semantic comprehension. Psychological Record 34:3–23.CrossRefGoogle Scholar
Secada, W. G. (1984) Counting in sign: The number string, accuracy and use. Unpublished doctoral dissertation, Northwestern University.Google Scholar
Seibt, U. (1975) Instrumentaldialekte der Klapperlerche Mirafra rufocinnamomea (Salvadori). Journal für Ornithologie 116:103–7.Google Scholar
Seibt, U. (1982) Zahlbegriff und Zahlverhalten bei Tieren. Neue Versuche und Deutungen. Zielschrifl für Tierpsychologie 60:325–41.Google Scholar
Seidenburg, A. (1962) The ritual origin of counting. Archive for History of Exact Sciences 2:1–40.Google Scholar
Seidenberg, M. S. & Petitto, L. A. (1979) Signing behavior in apes: a critical review. Cognition 7:177–215.Google Scholar
Seligman, M. E. P. (1970) On the generality of the laws of learning. Psychological Review 77:406–18.Google Scholar
Seyfarth, R. M., Cheney, D. L. & Marler, P. (1980) Vervet monkey responses to three different alarm calls. Evidence of predator classification and sematic communication. Science 210:801–3.Google Scholar
Sherrick, C. E. (1964) Effects of double simultaneous stimulation of the skin. American Journal of Psychology 77:42–53.Google Scholar
Sherry, D. F. & Shacter, D. L. (1987) The evolution of multiple memory systems. Psychological Review 94:439–54.Google Scholar
Sidman, M., Ftauzin, R., Lazar, R., Cunningham, S., Tailby, W. & Carrigan, P. (1982) A search for symmetry in the conditioned discriminations of rhesus monkeys, baboons, and children. Journal of the Experimental Analysis of Behavior 37:23–44.Google Scholar
Sidman, M. & Taitby, W. (1982) Conditional discrimination vs. matching to sample: An expansion of the testing paradigm. Journal of the Experimental Analysis of Behavior 37:5–22.Google Scholar
Simons, D. & Langheinrich, D. (1982) What is magic about the magical number four? Psychological Research 44:283–94.Google Scholar
Smith, J. N. M. & Dawkins, R. (1971) The hunting behaviour of individual great tits in relation to spatial variations in their food density. Animal Behaviour 19:695–706.Google Scholar
Sperber, D. (1985) Anthropology and psychology: Towards an epidemiology of representations. Man 20:73–89.Google Scholar
Starkey, P., Spelke, E. & Celman, R. (1983) Detection of intermodal numerical correspondences by human infants. Science 222:179.Google Scholar
Steffe, L. P. (1988) Lexical and syntactical meanings: Tyrone, Scenetra, and Jason. In: Construction of arithmetical meanings and strategies, ed. Steffe, P., Cobb, P. & von Clasersfeld, E.. Springer-Verlag.Google Scholar
Steffe, L. P., Cobb, P. & von Clasersfeld, E. (1988) The construction of arithmetical meanings and strategies. Springer-Verlag.Google Scholar
Steffe, L. P., von Clasersfeld, E., Richards, J. & Cobb, P. (1983) Children's counting types: Philosophy, theory, and application. Praeger Scientific.Google Scholar
Steirn, J. (1985) Concept formation in Japanese Quail (Coturnix coturnix japonica). Unpublished doctoral dissertation. University of Georgia.Google Scholar
Stevens, S. S. (1939) On the problem of scales for the measurement of psychological magnitudes. Journal of Unified Science 9:94–99.Google Scholar
Stevens, S. S. (1951) Mathematics, measurement, and psychophysics. In: Handbook of experimental psychology, ed. Stevens, S. S.. Wiley.Google Scholar
Stevens, S. S. (1961) To honor Fechner and repeal his law. Science 133:80–86.Google Scholar
Strauss, M. S. & Curtis, L. E. (1984) Development of numerical concepts in infancy. In: Origins of cognitive skills, ed. Sophian, C.. Erlbaum.Google Scholar
Suppes, P, Cohen, M., Laddaga, R., Anliker, , & Floyd, (1983) A procedural theory of eye movements in doing arithmetic. Journal of Mathematical Psychology 27:341–69.Google Scholar
Taborsky, M. (1985) On optimal parental care. Zeitschrift für Tierpsychologie 70:331–36.Google Scholar
Taves, E. H. (1941) Two mechanisms for the perception of visual numerousness. Archives of Psychology 37: No. 265.Google Scholar
Terrace, H. S. (1985) In the beginning was the “name.” American Psychologist 40:1011–28.Google Scholar
Terrace, H. (1987) Chunking by a pigeon in a serial task. Nature 325:149–51.Google Scholar
Terrell, D. & Thomas, R. K. (1983) Conceptual discrimination of polygons by squirrel monkeys. Paper presented at Southern Society for Philosophy and Psychology, Atlanta, Ca.Google Scholar
Thomas, R. K. (1988) Methodological missteps make concepts of percepts. Paper presented at the Southern Society for Philosophy and Psychology, Miami, Fla.Google Scholar
Thomas, R. K. & Chase, L. (1980) Relative numerousness judgments by squirrel monkeys. Bulletin of the Psychonomic Society 16:79–82.Google Scholar
Thomas, R. K., Fowlkes, D. & Vickery, J. D. (1980) Conceptual numerousness judgments by squirrel monkeys. American Journal of Psychology 93:247–57.Google Scholar
Thomas, R. K. & Ingram, D. K. (1979) Conceptual volume judgment by squirrel monkeys. American Journal of Psychology 92:33–43.Google Scholar
Thomas, R. K. & Noble, L. (1988) Visual and olfactory oddity learning in rats: What evidence is necessary to show conceptual behavior? Animal Learning and Behavior 16:157–63.Google Scholar
Thomas, R. K. & Walden, E. L. (1985) The assessment of cognitive development in human and nonhuman primates. In: Nonhuman primate models for human growth and development, ed. Watts, E. S.. R, Alan. Liss.Google Scholar
Thompson, R. F., Mayers, K. S., Robertson, R. T. & Patterson, C. J. (1970) Number coding in association cortex of the cat. Science 168:271–73.Google Scholar
Thorpe, W. H. (1956) Learning and instinct in animals. Harvard University Press.Google Scholar
Thorpe, W. H. (1958) The learning of song patterns by birds with special reference to the song of the chaffinch (Fringilla coelebs). Ibis 100:535–70.Google Scholar
Thorson, J., Weber, T. & Huber, F. (1982) Auditory behavior of the cricket, II. Journal of Comparative Physiology 146:361–78.Google Scholar
Todd, R. R., Barber, P. J. & Jones, D. (1987) The internal representation of number: Analogue or digital? In: Cognitive processes in mathematics, ed. Sloboda, J. A. & Rogers, D. R.. Oxford University Press.Google Scholar
Trabasso, T. R. (1975) Representation, memory and reasoning: How do we make transitive inferences? In: Minnesota symposium on child psychology, vol. 9, ed. Pick, A. D.. University of Minnesota Press.Google Scholar
Vincent, S. (1912) Function of the vibrissae in the behavior of the white rat. Behavior Monographs 1:No.5.Google Scholar
von Frisch, K. (1967) The dance language and orientation of bees. Translated by Chadwick, L.. Harvard University Press.Google Scholar
von Glasersfeld, E. (1981) An attention model for the conceptual construction of units and number. Journal for Research in Mathematics Education 12(2):33–96.Google Scholar
von Glasersfeld, E. (1982) Subitizing: The role of figural patterns in the development of numerical concepts. Archives de Psychologie 50:191–218.Google Scholar
von Glasersfeld, E. (1982a) The abstraction of counting units. In: Proceedings of the fourth annual meeting of the North American Croup for Psychology in Mathematics Education, ed. Wagner, S.. Athens, Ca.Google Scholar
von Glasersfeld, E. & Richards, J. (1983) The creation of units as a prerequisite for number: A philosophical review. In: Children's counting types: Philosophy, theory, and applications, ed. Steffe, L. P., Clasersfeld, E. von, Richards, J. & Cobb, P.. Praeger Scientific.Google Scholar
von Clasersfeld, E., Steffe, L. P. & Richards, J. (1983) An analysis of counting and what is counted. In: Children's counting types: Philosophy, theory, and applications. Praeger Scientific.Google Scholar
Vygotsky, L. S. (1978) Mind in Society. Harvard University Press.Google Scholar
Wagman, A. M. I. (1968) Effects of frontal lobe lesions upon behavior requiring use of response-produced cues. Journal of Comparative & Physiological Psychology 66:69–76.Google Scholar
Watson, J. B. (1914) Behavior: An introduction to comparative psychology. Holt, Rinehart & Winston.Google Scholar
Wertheimer, M. (1912/1938) Numbers and numerical concepts in primitive peoples. Abridged in: A source book ofCestalt psychology, ed. and trans. Ellis, W. D.. Harcourt, Brace. (Original work published 1912.)Google Scholar
Wesley, F. (1961) The number concept: A phylogenetic review. Psychological Bulletin 58:420–28.CrossRefGoogle ScholarPubMed
White, C. T. (1963) Temporal numerosity and the psychological unit of duration. Psychological Monographs 77:1–37.Google Scholar
Whitehead, A. N. & Russell, B. (1927) Prindpia mathematicia, Vol. 1. Cambridge University Press.Google Scholar
Wickler, W. & Seibt, U. (1983) Optimal maternal care. Zeitschrift für Tierpsychologie 63:201–5.Google Scholar
Wilkie, D. M., Webster, J. B. & Leader, L. G. (1979) Unconfounding time and number discrimination in a Mechner counting schedule. Bulletin of the Psychonomic Society 13:390–92.Google Scholar
Williams, B. A. (1983) Another look at contrast in multiple schedules. Journal of the Experimental Analysis of Behavior 39:345–84.Google Scholar
Williams, B. A. & Wixted, J. T. (1986) An equation for behavioral contrast. Journal of the Experimental Analysis of Behavior 45:47–62.Google Scholar
Wolters, C., van Kempen, H. & Wijlhuizen, G. (1987) Quantification of small numbers of dots: Subitizing or pattern recognition? American Journal of Psychology 100:225–37.Google Scholar
Woodruff, G. & Premack, D. (1981) Primitive mathematical concepts in the chimpanzee: Proportionality and numerosity. Nature 293:568–70.Google Scholar
Woodruff, G., Premack, D. & Kennel, K. (1978) Conservation of liquid and solid quantity by the chimpanzee. Science 202:991–94.Google Scholar
Wundt, W. C. (1907) Outlines of psychology. Wilhelm Engelmann.Google Scholar
Zaslavsky, C. (1973) Africa counts: Number and pattern in African culture. Prindle Weber & Schmidt.Google Scholar
Zorina, Z. A. (1982) Reasoning ability and adaptivity of behaviour in birds. In: Evolution and environment, ed. Novak, V. J. A. & Mlikovsky, J.. Praha: CSAV.Google Scholar