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Numerical competence: From backwater to mainstream of comparative psychology
... Numerosity alone cannot explain why multiple pieces of food is more rewarding than a single piece. When a collection of pieces is laid out as a stimulus, it also differs visually from a single piece on other factors that have been shown to affect perception-layout (Beran, 2006;Davis & Pérusse, 1988;Ginsburg, 1976Ginsburg, , 1980Xu & Spelke, 2000), density (Davis & Pérusse, 1988;DeWind, Adams, Platt, & Brannon, 2015;Hollingsworth, Simmons, Coates, & Cross, 1991;Stevens et al., 2007;vanMarle & Wynn, 2011;Xu & Spelke, 2000), surface area (Davis & Pérusse, 1988;Feigenson, Carey, & Spelke, 2002), and contour length (Clearfield & Mix, 1999Xu & Spelke, 2000). When the number of items is kept consistent, numerosity estimates in monkeys and humans are biased when comparing them in a regular versus random pattern (Beran, 2006;Ginsburg, 1976Ginsburg, , 1980, in one large cluster versus multiple small clusters (Frith & Frith, 1972), and in regular versus nested patterns (Beran & Parrish, 2013;Chesney & Gelman, 2012). ...
... Numerosity alone cannot explain why multiple pieces of food is more rewarding than a single piece. When a collection of pieces is laid out as a stimulus, it also differs visually from a single piece on other factors that have been shown to affect perception-layout (Beran, 2006;Davis & Pérusse, 1988;Ginsburg, 1976Ginsburg, , 1980Xu & Spelke, 2000), density (Davis & Pérusse, 1988;DeWind, Adams, Platt, & Brannon, 2015;Hollingsworth, Simmons, Coates, & Cross, 1991;Stevens et al., 2007;vanMarle & Wynn, 2011;Xu & Spelke, 2000), surface area (Davis & Pérusse, 1988;Feigenson, Carey, & Spelke, 2002), and contour length (Clearfield & Mix, 1999Xu & Spelke, 2000). When the number of items is kept consistent, numerosity estimates in monkeys and humans are biased when comparing them in a regular versus random pattern (Beran, 2006;Ginsburg, 1976Ginsburg, , 1980, in one large cluster versus multiple small clusters (Frith & Frith, 1972), and in regular versus nested patterns (Beran & Parrish, 2013;Chesney & Gelman, 2012). ...
... Numerosity alone cannot explain why multiple pieces of food is more rewarding than a single piece. When a collection of pieces is laid out as a stimulus, it also differs visually from a single piece on other factors that have been shown to affect perception-layout (Beran, 2006;Davis & Pérusse, 1988;Ginsburg, 1976Ginsburg, , 1980Xu & Spelke, 2000), density (Davis & Pérusse, 1988;DeWind, Adams, Platt, & Brannon, 2015;Hollingsworth, Simmons, Coates, & Cross, 1991;Stevens et al., 2007;vanMarle & Wynn, 2011;Xu & Spelke, 2000), surface area (Davis & Pérusse, 1988;Feigenson, Carey, & Spelke, 2002), and contour length (Clearfield & Mix, 1999Xu & Spelke, 2000). When the number of items is kept consistent, numerosity estimates in monkeys and humans are biased when comparing them in a regular versus random pattern (Beran, 2006;Ginsburg, 1976Ginsburg, , 1980, in one large cluster versus multiple small clusters (Frith & Frith, 1972), and in regular versus nested patterns (Beran & Parrish, 2013;Chesney & Gelman, 2012). ...
Visual cues have an important role in food preference for both rats and humans. Here, we aim to isolate the effects of numerosity, density, and surface area on food preference and running speed in rats, by using a forced-choice maze paradigm. In Experiment 1, rats preferred and ran faster for a group of multiple smaller pellets rather than a single large pellet, corroborating previous research (Capaldi, Miller, & Alptekin Journal of Experimental Psychology: Animal Behavior Processes, 15(1), 75–80, 1989). Further experiments tested the prevailing hypothesis that multiple food pieces are more reinforcing because they occupy a larger surface area. Experiment 2 controlled for numerosity by utilizing a continuous food: mashed potatoes flattened to cover a larger surface area or rounded into a ball. The rats preferred and ran faster for the flattened potatoes, suggesting surface area plays a role in quantity estimations. Finally, in Experiment 3, rats displayed no preference or difference in running speed between a group of scattered and clustered pellets when number of pellets were kept constant. Taken together, these results suggest that density has an important role in food perception—that is, the rewarding effect of higher numerosity or larger surface area is removed when the food does not fill out the entire space. Alternative explanations and implications for human diet are discussed.
... For instance, four apples occupy a larger area than two apples, but if animals choose the four-apple set it is not possible to establish which cues they used (number or the cumulative surface area) as the behavioral output is the same. With respect to this issue, some authors argue that animals preferentially exploit non-numerical cues in quantity judgments and use number only as a "last-resort", when no other cues on which they could base their choices are available ( Davis and Perusse, 1988;Seron and Pesenti, 2001). Support for the "last-resort" hypothesis comes from studies reporting a spontaneous use of continuous quantities. ...
... The use of neutral stimuli (i.e., two-and three-dimensional objects) in training procedures is known to permit a rigorous control for continuous quantities that correlate with number. However, some authors have argued that extensive training creates a numerical competence that is not naturally present in animals ( Davis and Perusse, 1988; Seron and Pesenti, 2001;Hauser and Spelke, 2004). Conversely, spontaneous choice tests are considered ecologically highly valid for studying numerical competence in animals since they simulate problems that animals may encounter in their habitat and reflect numerical capacities displayed in their natural repertoire ( Abramson et al., 2011;Panteleeva et al., 2013;Rodríguez et al., 2015). ...
... Results of incongruent and controlled trials provide evidence in support to the idea that dogs used total amount as a primary cue to make their choices. In the former condition subjects selected the set containing the larger mass of edible food, irrespective to the number of items, whereas in the latter condition, dogs indiscriminately selected the smaller or the larger set when the total volume was equalized, thus excluding the hypothesis that, in foraging contexts, they rely on numerical information as a last strategy when no alternative solution is available ( Davis and Perusse, 1988;Seron and Pesenti, 2001). However, one may argue that in both conditions the largest piece of food was always presented in the less numerous set and a possible explanation for our results might be that dogs had a strong preference for the largest item, as previously reported in chimpanzees ( Boysen et al., 2001;Beran et al., 2008). ...
Numerous studies have reported that animals reliably discriminate quantities of more or less food. However, little attention has been given to the relative salience of numerosity compared to the total amount of food when animals are making their choices. Here we investigated this issue in dogs. Dogs were given choices between two quantities of food items in three different conditions. In the Congruent condition, the total amount of food co-varied with the number of food items; in the Incongruent condition the total amount was pitted against the numerosity; and in the Controlled condition the total amount between the sets was equal. Results show that dogs based their choice on the total amount of edible food rather than on the number of food items, suggesting that, in food choice tasks, amount counts more than number. The presence of the largest individual item in a set did not bias dogs' choices. A control test excluded the possibility that dogs based their choices on olfactory cues alone.
... Thus, when dot arrays are presented in fixed Gestalten (like those on the faces of a die), then the reaction time for judging their numerosity is indeed what one would expect if twoness and threeness were directly perceptible attributes like cowness and treeness, but when the arrays do not have a fixed pattern, then the reaction-time function is what you would expect from a counting process, (p. 586) Davis and Perusse (1988b) opposed Gallistel's view: ...
... (p. 604) Support for Davis and Perusse's (1988b) argument can befound in research by Rosch (1975). Rosch had subjects rank "goodness of example" (p. ...
In Experiment 1, with the number of sides or angles of irregular polygons as cues, programmed training, and a 90% correct criterion (36 of 40), 2 squirrel monkeys’ (Saimiri sciureus sciureus and S. boliviensus boliviensus) best performances were to discriminate heptagons from octagons, a 3rd’s best was hexagons from heptagons, and a 4th’s best was pentagons from heptagons. In Experiment 2, on most trials 2 polygons on one or both discriminanda had to be summed to determine which discriminandum had the total fewer sides. Only 1 monkey met criterion (27 of 30) on the 2 tasks, 6 vs. 8 and 7 vs. 8 sides, but the other 3 performed better than chance on the 6 vs. 8 task. We conclude that previous studies of animals’ discrimination of polygons in terms of complexity were minimally relevant to this work, and counting and subitizing were rejected in favor of a prototype-matching process to explain our monkey’s performances.
... Some canine research, however, requires precise control over the number of treats dispensed. For instance, studies on numerical cognition or how well animals can understand quantities of objects (Davis & Perusse 1988;Brannon 2006) may require that an individual receive a precise number of treats based on their choice. If they are meant to receive three treats and they accidentally receive four, that can be problematic for the study. ...
Some forms of canine cognition research require a dispenser that can accurately dispense precise quantities of treats. When using off-the-shelf or retrofitted dispensers, there is no guarantee that a precise number of treats will be dispensed. Often, they will over-dispense treats, which may not be acceptable for some tasks. Here we describe a 3D-printed precise treat dispenser with a 59-treat capacity driven by a stepper motor drive and controlled by an integrated Raspberry Pi. The dispenser can be built for less than 200 USD and is fully 3D printable. While off-the-shelf dispensers can result in an error rate of 20-30%, the precision dispenser produces a 4% error rate. This lower error rate and the integrated Raspberry Pi allows for new possibilities for using treat dispensers across a range of canine research questions.
... A possible explanation for this difference comes from foraging theory, which predicts that organisms should maximize caloric intake while reducing energy expenditure (Stephens & Krebs, 1986). In some cases, this could occur if an animal attends to non-numerical quantitative variables such as volume and surface area to discriminate the largest food patch available (Davis & Pérusse, 1988). Experimental evidence suggests that a diverse set of species display denseness-sensitivity in foraging scenarios, with a preference for more densely baited food sites among primates (e.g., orangutans (Pongo pygmaeus abelii): MacDonald & Agnes, 1999) and several species of birds (e.g., pigeons (Columba livia): Mitchell, Calton, Threlkeld, & Schachtman, 1996; turnstones (Arenaria interpres): Vahl, Lok, van der Meer, Piersma, & Weissing, 2005). ...
The density bias, documented within the foraging domain for some monkey species and for human infants, emerges when perceived numerosity is affected by interstimulus distance such that densely arranged food items appear more numerous relative to the same amount of food sparsely arranged. In this study, capuchin monkeys and rhesus monkeys were presented with a computerized relative discrimination task that allowed for the control of stimulus size, interelemental distance, and overall array pattern. The main objective was to determine whether the density bias was a more widespread and general perceptual phenomenon that extends beyond the foraging domain, similar to other numerosity illusions and biases. Furthermore, we compared the current results to these same monkeys’ data from a previous study on the Solitaire numerosity illusion to investigate a potential link between a density bias and this related numerical illusion. Capuchin monkeys showed a density bias in their perceptual discrimination of dense versus sparse stimuli; however, rhesus monkeys perceived this bias to a lesser degree. Individual differences were evident, as with the Solitaire illusion. However, there was not a relation between susceptibility to a density bias and susceptibility to the Solitaire illusion within these same monkeys.
... Our data suggest that crows do not use number as a 'last resort' to discriminate quantity, as has been suggested several times (e.g. [56]). Also, previous work indicates that the number of items, and not just continuous quantity, is a salient parameter animals use to discriminate stimuli [3,8,[57][58][59]. ...
The ability to estimate number is widespread throughout the animal kingdom. Based on the relative close phylogenetic relationship (and thus equivalent brain structures), non-verbal numerical representations in human and non-human primates show almost identical behavioural signatures that obey the Weber-Fechner law. However, whether numerosity discriminations of vertebrates with a very different endbrain organization show the same behavioural signatures remains unknown. Therefore, we tested the numerical discrimination performance of two carrion crows (Corvus corone) to a broad range of numerosities from 1 to 30 in a delayed match-to-sample task similar to the one used previously with primates. The crows’ discrimination was based on an analogue number system and showed the Weber-fraction signature (i.e. the ‘just noticeable difference’ between numerosity pairs increased in proportion to the numerical magnitudes). The detailed analysis of the performance indicates that numerosity representations in crows are scaled on a logarithmically compressed ‘number line’. Because the same psychophysical characteristics are found in primates, these findings suggest fundamentally similar number representations between primates and birds. This study helps to resolve a classical debate in psychophysics: the mental number line seems to be logarithmic rather than linear, and not just in primates, but across vertebrates. © 2016 The Author(s) Published by the Royal Society. All rights reserved.
... Some authors have suggested that animals are not naturally attuned to number, as numerical information would be less salient in natural contexts than physical attributes; as a consequence, animals would be expected to use number only as a last-resort strategy, when no other cues are available to discriminate between quantities 41,42 . This hypothesis is supported by evidence showing that mosquitofish, Gambusia holbrooki, could easily select the larger group of social companions, but their performance dropped to chance level when the cumulative surface area between shoals was controlled for 43 . ...
Mammals and birds can process ordinal numerical information which can be used, for instance, for recognising an object on the basis of its position in a sequence of similar objects. Recent studies have shown that teleost fish possess numerical abilities comparable to those of other vertebrates, but it is unknown if they can also learn ordinal numerical relations. Guppies (Poecilia reticulata) learned to recognise the 3rd feeder in a row of 8 identical ones even when inter-feeder distance and feeder positions were varied among trials to prevent the use of any spatial information. To assess whether guppies spontaneously use ordinal or spatial information when both are simultaneously available, fish were then trained with constant feeder positions and inter-feeder distance. In probe trials where these two sources of information were contrasted, the subjects selected the correct ordinal position significantly more often than the original spatial position, indicating that the former was preferentially encoded during training. Finally, a comparison between subjects trained on the 3rd and the 5th position revealed that guppies can also learn the latter discrimination, but the larger error rate observed in this case suggests that 5 is close to the upper limit of discrimination in guppies.
Scientists from many disciplines have been intrigued by the topic of how the mind represents number because of the question’s relevance to controversial topics such as thought without language, the evolution of cognition, modularity of mind, and nature vs. nurture. Number is an abstract and emergent property of sets of discrete entities; two people and two airplanes look nothing alike, and yet the numerosity of the set is the same. Some researchers believe that the abstract nature of numerical representation makes it an unlikely candidate for a cognitive capacity held by nonhuman animals and human infants. However, a growing body of data suggests that both nonverbal animals and preverbal human infants represent number and even perform operations on these representations. In fact, a new synthesis of the data on numerical abilities in animals and infants suggests that there is an evolutionarily and developmentally primitive system for representing number as mental magnitudes with scalar variability (Gallistel and Gelman, 1992, 2000; see also Dehaene, 1997; Wynn, 1995). Furthermore, there is abundant evidence that adult humans also represent number nonverbally as analog magnitudes (Cordes et al., 2001; Dehaene, 1997; Dehaene et al., 1998; Moyer and Landauer, 1967; Whalen et al., 1999). For these reasons, numerical cognition has become an exciting area of research and an exemplary model of a crossdisciplinary field where comparative and developmental studies have influenced current conceptions of adult human numerical processing (e.g., Dehaene, 1997).
Mathematics is a powerful tool for describing and developing our knowledge of the physical world. It informs our understanding of subjects as diverse as music, games, science, economics, communications protocols, and visual arts. Mathematical thinking has its roots in the adaptive behavior of living creatures: animals must employ judgments about quantities and magnitudes in the assessment of both threats (how many foes) and opportunities (how much food) in order to make effective decisions, and use geometric information in the environment for recognizing landmarks and navigating environments. Correspondingly, cognitive systems that are dedicated to the processing of distinctly mathematical information have developed. In particular, there is evidence that certain core systems for understanding different aspects of arithmetic as well as geometry are employed by humans and many other animals. They become active early in life and, particularly in the case of humans, develop through maturation. Although these core systems individually appear to be quite limited in application, in combination they allow for the recognition of mathematical properties and the formation of appropriate inferences based upon those properties. In this overview, the core systems, their roles, their limitations, and their interaction with external representations are discussed, as well as possibilities for how they can be employed together to allow us to reason about more complex mathematical domains. WIREs Cogn Sci 2015, 6:355-369. doi: 10.1002/wcs.1351 For further resources related to this article, please visit the WIREs website.
The authors have declared no conflicts of interest for this article.
© 2015 John Wiley & Sons, Ltd.
A theoretical model is proposed that explicates the generation of conceptual structures from unitary sensory objects to abstract constructs that satisfy the criteria generally stipulated for concepts of “number”: independence from sensory properties, unity of composites consisting of units, and potential numerosity. The model is based on the assumption that attention operates not as a steady state but as a pulselike phenomenon that can, but need not, be focused on sensory signals in the central nervous system. Such a view of attention is compatible with recent findings in the neurophysiology of perception and provides, in conjunction with Piaget's postulate of empirical and reflective abstraction, a novel approach to the analysis of concepts that seem indispensable for the development of numerical operations.
First, we review recent efforts to demonstrate language competence in apes and dolphins. Then, with techniques originally devised by Herman in his artificial language studies with dolphins (Tursiops truncatus), we show that smaller brained sea lions (Zalophus californianus), like larger brained dolphins, are capable of comprehending signals about a relationship between two designated objects. The language we used consisted of signs designating properties of size, brightness, or location (modifiers), types of objects, and actions. The results of our experiments with two sea lions support Premack’s notion that Herman’s dolphins (as well as our sea lions) relied principally on two phrase structure rules to comprehend thousands of novel, unique messages that could be transmitted gesturally by a “blind” human signaler. One rule instructs the animal to perform an action directly on an object designated by an object signal and one or two optional modifiers. The instruction takes the form (Modifier) + Object + Action. The second rule instructs the animal to perform an action relative to two designated objects. The object to be transported and the goal object were assigned their particular roles by their position in the sign sequence. The relational instruction takes the form (Modifier) + Object A + (Modifier) + Object B + Action. Reversing the order of the two object signals in the string reversed the meaning of the message. For both sea mammals we found support for Herman’s notion that the critical constraint on the application of the second rule is memory for Object A (the goal item). The variables influencing memory for the goal item were: (a) the number of objects available, (b) bias for certain goal items, (c) whether the goal item was fixed in space, that is whether it was a transportable or nontransportable object, and (d) whether the goal item (Object A) and the transported item were reversed on successive trials.
