Spontaneous versus trained numerical abilities. A comparison between the two main tools to study numerical competence in non-human animals.

G Model

ARTICLE IN PRESS

NSM-6892; No. of Pages 10

Journal of Neuroscience Methods xxx (2014) xxx–xxx

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Journal of Neuroscience Methods journal homepage: www.elsevier.com/locate/jneumeth

Basic Neuroscience

Spontaneous versus trained numerical abilities. A comparison between the two main tools to study numerical competence in non-human animals Christian Agrillo ∗ , Angelo Bisazza Department of General Psychology, University of Padova, Italy

h i g h l i g h t s • • • •

Numerical abilities have been reported in a wide range of non-human animals. Two main approaches have been adopted: spontaneous choice tests and training procedures. The two methodologies seem to lead to different results. We review pros and cons of studying spontaneous and trained numerical abilities.

a r t i c l e

i n f o

Article history: Received 26 November 2013 Received in revised form 23 April 2014 Accepted 24 April 2014 Keywords: Animal cognition Numerical competence Continuous variables Mammals Birds Fish

a b s t r a c t A large body of experimental evidence shows that animals as diverse as mammals, birds, and fish are capable of processing numerical information. Considerable differences have been reported in some cases among species and a wide debate currently surrounds the issue of whether all vertebrates share the same numerical systems or not. Part of the problem is due to the fact that these studies often use different methods, a circumstance that potentially introduces confounding factors in a comparative analysis. In most studies, two main methodological approaches have been used: spontaneous choice tests and training procedures. The former approach consists of presenting to the subjects two groups of biologicallyrelevant stimuli (e.g., food items or social companions) differing in numerosity with the assumption that if they are able to discriminate between the two quantities, they are expected to spontaneously select the larger/smaller quantity. In the latter approach, subjects undergo extensive training in which some neutral stimuli (e.g., a quantity of dots) are associated with a reward and the capacity to learn a numerical rule is taken as evidence of numerical abilities. We review the literature on this topic, highlighting the relevance, and potential weaknesses in controlling confounding factors obtained with either approach. © 2014 Elsevier B.V. All rights reserved.

Contents 1. 2. 3. 4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spontaneous numerical abilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trained numerical abilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A comparison of the two methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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∗ Corresponding author at: Department of General Psychology, Via Venezia 8, 35131 Padova, Italy. Tel.: +39 0498276931; fax: +39 0498276600. E-mail address: [email protected] (C. Agrillo). http://dx.doi.org/10.1016/j.jneumeth.2014.04.027 0165-0270/© 2014 Elsevier B.V. All rights reserved.

Please cite this article in press as: Agrillo C, Bisazza A. Spontaneous versus trained numerical abilities. A comparison between the two main tools to study numerical competence in non-human animals. J Neurosci Methods (2014), http://dx.doi.org/10.1016/j.jneumeth.2014.04.027

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ARTICLE IN PRESS C. Agrillo, A. Bisazza / Journal of Neuroscience Methods xxx (2014) xxx–xxx

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1. Introduction In the last decade the study of numerical abilities has become one of the main issues in cognitive neuroscience. Numbers are an essential feature of everyday life, allowing us to conduct functions as diverse as estimating the shortest queue at the supermarket, checking if our change is correct, or designing a bridge. Behavioral and neuroimaging studies converge to indicate that humans display multiple cognitive systems for number processing. Symbolic numerical abilities are strictly related to culture and language and permit us to learn the wide range of symbols and syntax required in school mathematics. Neuroimaging studies have shown that these abilities rely on a large number of brain regions, such as the intraparietal sulcus, prefrontal cortex, cingulate gyri, the insula, and the cerebellum (reviewed in Arsalidou and Taylor, 2011). However, cross-cultural, developmental and experimental evidences indicate that humans possess other numerical abilities that are not related to language and culture. Mundurukù, an Amazonian population that lacks vocabulary for numbers beyond five, can nevertheless discriminate between much larger quantities (Pica et al., 2004). Infants can discriminate between 6 and 12 objects at six months, well before the emergence of language (Xu and Spelke, 2000). These abilities, often called non-symbolic numerical abilities (Gilmore et al., 2010; Price et al., 2012; Zebian and Ansari, 2012), permit us to discriminate 9 from 10 items in no more than 150 ms (Halberda et al., 2008). Neuroimaging studies suggest that symbolic and non-symbolic numerical abilities recruit partially different neural circuits (Holloway et al., 2010). For instance, while both symbolic and non-symbolic representations activate the right intraparietal sulcus, symbolic numerical abilities are primarily processed in the left hemisphere while non-symbolic numerical abilities recruit the right hemisphere (Chassy and Grodd, 2012). Several lines of research indicate that our symbolic numerical abilities are based on non-symbolic numerical systems (Agrillo et al., 2013; Halberda et al., 2008; Park and Brannon, 2013). Also, deficits in the study of mathematics, such as dyscalculia, seem to be associated with low performance in non-symbolic numerical tasks (Piazza et al., 2010; Furman and Rubinsten, 2012). In this sense, the study of non-symbolic numerical abilities becomes crucial to understanding the foundation of our mathematical abilities. Until a few decades ago, it was assumed that non-symbolic numerical abilities could only be studied in our species. Today we know that other vertebrates share the capacity to discriminate between quantities and make simple calculations (i.e., mammals: Vonk, 2014; Perdue et al., 2012; birds: Emmerton and Renner, 2006; Farnsworth and Smolinski, 2006; fish: Agrillo et al., 2014; GómezLaplaza and Gerlai, 2013). There are many ecological situations in which numerical abilities can be useful. Hyenas, for instance, are more willing to enter social contests when their group outnumbers that of opponents (Benson-Amram et al., 2011). Numerical capacities can be important in guiding foraging decisions such as selecting the larger amount of food (Normand et al., 2009) or the optimal quantity of preys (Panteleeva et al., 2012). The ability to compare numerosities can enable animals to select the group with the more advantageous sex ratio (Flay et al., 2009) or to dilute predation risks by getting protection within the largest group of social companions (Hager and Helfman, 1991). As we have all experienced in everyday life, for example when searching for an uncrowded train carriage or selecting the best fruit basket, we can discriminate between different quantities without necessarily counting the number of objects in each group. Numerosity normally covaries with several other physical attributes, and animals can use the relative magnitude of nonnumerical cues, such as the total area of the stimuli or the overall space occupied by the sets or their density. Human and other animals can estimate which group is larger/smaller by using these

non-numerical cues (hereafter “continuous quantities”, Beran et al., 2008a; Cantlon and Brannon, 2007b; Gebuis and Reynvoet, 2012; Gómez-Laplaza and Gerlai, 2012, 2013). For example, cats were able to learn to discriminate between two and three dots to get a food reward. However, as soon as the cumulative surface area was controlled for, their performance dropped to chance level, suggesting that cats primarily based their choice on continuous quantities instead of numbers (Pisa and Agrillo, 2009). Also, salamanders were able to discriminate between 8 and 16 crickets but careful controls showed that they used the overall quantity of movement of these potential preys instead of their numerosity (Krusche et al., 2010). In this sense, before assuming that a species possesses a specific numerical ability, it is necessary to strictly control for continuous quantities, a challenge that represents one of the most critical issues in this research field (see Sections 2 and 3 for more details). Indeed, as we will see later, several non-human animals proved able to discriminate between quantities even when prevented from using continuous quantities, and numerical abilities are often very similar in distantly related species. These findings prompted a debate as to whether all animal species share the same numerical systems and if these are homologous to our non-symbolic numerical systems. This issue becomes even more relevant regarding the possibility of developing animal models to study neural circuits of number processing and the biological basis of learning disabilities in the acquisition of mathematical abilities. Inter-specific comparisons have led to mixed results. Some studies reported similar performances in distantly related species. For instance, one study showed that New Zealand robins can discriminate between 1 vs. 2, 2 vs. 3, and 3 vs. 4, while their performance significantly decreases in 4 vs. 5 (Hunt et al., 2008) – a similar numerical acuity exhibited by distantly related species such as guppies (Agrillo et al., 2012a), and mosquitofish (Agrillo et al., 2008a). The accuracy in relative numerosity judgments of bears (Vonk and Beran, 2012), dogs (Ward and Smuts, 2007), parrots (Al Aïn et al., 2009), and angelfish (Gómez-Laplaza and Gerlai, 2011a) is affected by the numerical ratio between the matched numerosity, as commonly reported in humans (Revkin et al., 2008) and non-human primates (Beran, 2004; Cantlon and Brannon, 2007a). Even mosquitofish and college students show surprising similarities when tested with the same numerical contrasts (Agrillo et al., 2010). Other studies, however, highlighted differences in performance among different vertebrates. For example, horses, domestic chicks, salamanders, and angelfish discriminate between groups differing by one unit and up to 2 vs. 3 (Gómez-Laplaza and Gerlai, 2011b; Rugani et al., 2008; Uller and Lewis, 2009; Uller et al., 2003), while robins, guppies, and mosquitofish discriminate between 3 vs. 4 (Agrillo et al., 2008a, 2012c; Hunt et al., 2008). Trained pigeons can discriminate up to 6 vs. 7, a numerical acuity not observed in untrained birds (Al Aïn et al., 2009; Rugani et al., 2009). Difference in performances have been reported even between closely related species: the accuracy of African elephants is affected by the numerical ratio (Perdue et al., 2012) while the accuracy of Asian elephants appears to be insensitive to the numerical ratio (Irie-Sugimoto et al., 2009; Irie and Hasegawa, 2012). In sum, while some studies highlighted similar performance among vertebrates, others remarked upon the inter-specific differences. Part of the inconsistencies reported in the literature might be ascribed to the different methodology adopted (Agrillo and Miletto Petrazzini, 2012), such as different paradigms (e.g., spontaneous behavior vs. trained behavior), different stimuli (e.g., food, social companions, dots), and sensory modality (e.g., visual vs. auditory stimuli). In some cases, there is evidence that different methods of measuring numerical abilities can lead to different results in the same species. For instance, goldbelly topminnows could discriminate up to 2 vs. 3 companions with one experimental procedure




while they were unable to solve the same task using a different procedure (Agrillo and Dadda, 2007). Two main approaches have been adopted in the literature to study numerical abilities: spontaneous choices tests and training procedure. Several authors believe that these two methodologies might recruit partially different neural circuits and accordingly lead to different results (Barnard et al., 2013; Hauser and Spelke, 2004). Due to the primary relevance of these methodological questions, here we critically examine and compare studies that investigated numerical cognition in non-human animals. In the first part of this review, we will summarize representative examples of studies using these two methodologies. In the second part, we will compare the two methods, highlighting the pros and cons of using each approach.

2. Spontaneous numerical abilities In spontaneous choice tests, subjects are typically presented with two groups of biologically-relevant stimuli differing in numerosity. In most cases, food items are used as stimuli. For instance, the experimenter might show two plates containing three and four slices of apple pie and the subject is free to reach for the plates. The first plate selected by the subject is recorded as the dependent variable. The assumption is that if subjects are able to discriminate between the two quantities, they are expected to spontaneously select the larger quantity in order to maximize food intake. The simplest procedure to assess whether animals display the capacity to discriminate between quantities consists in simultaneously presenting two groups of items that remain visible until the time of choice. Several non-human primates showed the ability to select the larger group of food items. For instance, using this procedure, Call (2000) tested the spontaneous ability of orangutans to choose the larger quantity of food items (cereals) presented in two separate dishes. In this case, subjects were required to manually point in order to request one of the two groups. The experimenter presented all quantity combinations between one and six, finding that their performance was affected by numerical ratio, with accuracy increasing when the numerical ratio between the smaller and the larger number decreased. For instance, one orangutan was 84% accurate in 1 vs. 2 (0.50 ratio) and 58% accurate in 5 vs. 6 (0.83). Similarly, researchers presented baboons with a different number of peanuts, finding that their accuracy varied as a function of numerical ratio (Barnard et al., 2013). A similar methodology was also adopted by Baker et al. (2011, 2012) to compare the numerical abilities of dogs and coyotes. When required to simultaneously select between two groups of visible food items in the range 1–6, the two species showed the capacity to select the larger one in most numerical contrasts. For instance, both species discriminated 0.50 ratio at approx. 70% accuracy, while their accuracy did not differ from chance levels in 0.75 ratio. The performance of dogs adhered with that reported in coyotes, suggesting the existence of similar quantificational systems in these two related species. Similar abilities have been reported in birds, too. Parrots were presented with all possible combinations in the range 1–5 using seeds as stimuli. Researchers found that these birds were able to select the larger group in all contrasts, even though the overall performance decreased in 3 vs. 4 and 4 vs. 5 (Al Aïn et al., 2009). Northern mockingbirds were shown two potential feeders containing a different number of bamboo sticks at each of the two ends. In order to reach a mealworm (potential prey), the birds were required to remove all sticks from either end of the two feeders. The optimal choice was to select the feeder presenting fewer sticks. Results showed the existence of spontaneous quantity discrimination in this species as subjects selected the smaller group of sticks in 1 vs.

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6 and 2 vs. 5. No discrimination was found in 3 vs. 4 (Farnsworth and Smolinski, 2006). Spontaneous choice tests have often been used to investigate the numerical abilities in basal vertebrates, such as amphibians and fish. When given a choice between tubes containing fruit flies differing in numerosity, salamanders selected the larger group in 1 vs. 2 and 2 vs. 3 but not 3 vs. 4 (Uller et al., 2003). Salamanders were also able to discriminate between larger groups, provided that the numerical ratio was equal to 0.50 (8 vs. 16 but not 8 vs. 12, Krusche et al., 2010). In fish, social companions instead of food are commonly used in spontaneous choice tests. There is substantial evidence that individual fish that happen to be in an unknown environment tend to join other conspecifics and, if choosing between two shoals, they exhibit a preference for the larger one (Bisazza et al., 2010; Buckingham et al., 2007; Hager and Helfman, 1991). This is thought to be an anti-predatory strategy aimed at reducing the chance of being captured by predators (Hamilton, 1971). Several studies took advantage of the spontaneous tendency to go for the larger shoal to assess the limits of quantity discrimination (Buckingham et al., 2007; Gómez-Laplaza and Gerlai, 2011a, b). Subjects are typically inserted into the middle of an unfamiliar tank where two groups of social companions differing in number are visible. As a measure of accuracy, the researchers usually record the proportion of time spent near the larger shoal. When tested with this procedure, angelfish can discriminate between 1 vs. 2 and 2 vs. 3 but not 3 vs. 4. A 0.56 numerical ratio is needed to discriminate between larger numbers, such as 5 vs. 9 (Gómez-Laplaza and Gerlai, 2011a, b). Similarly, it has been shown that guppies can discriminate between 1 vs. 2, 2 vs. 3, 3 vs. 4, and 4 vs. 8 (Agrillo et al., 2012c). These abilities in guppies seem to be partially displayed at birth, with newborns being able to discriminate between 1 vs. 2, 2 vs. 3, and 3 vs. 4, just as it happens in adults. The capacity to discriminate between larger numerosities, however, only emerges later – at 20–40 days – as a result of both maturation and experience (Bisazza et al., 2010). The procedure used in shoal choice tests has also been applied to test quantity abilities in the situation of mate choice. Male mate decisions are often constrained by their cognitive ability to assess the magnitude of sperm competition (Shifferman, 2012). Tested in a binary choice, male mosquitofish were found to select the larger group of females in a comparison between 1 vs. 3 and 2 vs. 4. When the same number of females were available, one with males and the other without, male mosquitofish preferred the latter. Nonetheless, when a different number of males were presented in the two stimulus shoals, subjects were unable to select the group with a more favorable sex ratio (Agrillo et al., 2008b). As illustrated by many of the above examples, a common finding in mammals, birds, and fish is that individuals are increasingly accurate as the numerical ratio between the smaller and the larger group to compare decreases. Several authors hypothesize that, like in humans, numerical acuity in non-human animals is set by Weber’s law, which states that the smallest detectable change is a constant proportion of stimulus magnitude. In short, we can commonly discriminate between 2 and 3 objects (absolute difference = 1 dot; ratio = 0.67) while the same absolute difference cannot be detected with higher numerosities (e.g., 12 vs. 13). On the other hand, the same ratio is discriminated also with larger numerosities (e.g., 12 vs. 18, ratio = 0.67). As stated in Introduction, numerosity normally co-varies with continuous quantities and the fact that non-human animals can discriminate between quantities does not necessarily imply that they can process numerical information. Several experimental procedures have been proposed to avoid the possibility that subjects use alternative routes to solve quantitative problems. A slightly different version of the previous procedure consists of presenting the two groups simultaneously but making the two


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groups visually unavailable at the time of choice. In this case, subjects have to remember the quantity of items included in the two groups. Using this procedure, it was shown that apes (Hanus and Call, 2007) and dogs (Ward and Smuts, 2007) could select the larger group of food items in the range of 1–5, with apparently the same effort exhibited when groups were visually available at the time of choice. However, this procedure does not entirely exclude the possibility that animals use continuous quantities, as subjects can potentially compare the different areas occupied by stimuli when they are visible and then remember the position occupied by the larger amount prior its disappearance. Another experimental strategy to reduce the possibility that an animal uses continuous quantities consists in presenting groups of items controlled for the overall area of the stimuli or other continuous variables. For example, in a 2 vs. 3 discrimination of food items, two larger pieces of food can be compared to three smaller ones so that the overall amount of food is the same in the two groups (Beran et al., 2008a; Feigenson et al., 2002). Nonetheless, it is likely that natural selection has shaped quantificational systems in order to maximize the amount of food (i.e., calories) retrieved rather than the number of pieces. Even though subjects can use numerical information to discriminate between two and three, the possibility exists that they do not exhibit such discrimination when the overall volume of food is the same within the two groups. When social companions are used as stimuli, smaller individuals can be presented in the larger group so that the sum of areas/volumes is similar between the two groups (Agrillo et al., 2008a). This might introduce another confounding factor. It is known that when social companions differ in body length, several fish species prefer to shoal with similar-sized fish in order to reduce the risk of predation by minimizing phenotypic oddity (Landeau and Terborgh, 1986; Wong and Rosenthal, 2005). A better experimental strategy to control for continuous quantities is the “item-by-item presentation.” In its original form, used with infants and non-human primates, the researcher inserts a different number of stimuli sequentially into two opaque boxes; in this way, subjects can never have a global view of the entire contents of the groups and are required to enumerate the number of items presented in the two boxes in order to optimize their choice (Beran, 2004). For example, macaques observed experimenters that placed pieces of apple, one at a time, into each of two opaque containers. Monkeys showed a precise discrimination of up to four items (1 vs. 2, 2 vs. 3, and 3 vs. 4, but not 4 vs. 5), providing one of the first evidence of spontaneous number representation in non-human primates (Hauser et al., 2000). Several confounding factors can potentially affect the performance with this procedure. Subjects can select the group where the last item was visible. To avoid this possibility, the position of the last item must be counterbalanced across trials, so that in half of trials the last item is inserted into the box containing the smaller group, while in the other half, the last item is included in the box containing the larger group. Temporal cues must be controlled as well. If the researcher inserts three pieces of food in one box and two pieces of food in another box, the possibility remains that subjects’ choice is determined by the total time spent by the experimenter in proximity of the two boxes. To prevent this possibility, two different procedures are available: first, the experimenter may spend the same amount of time in proximity of the two boxes, for instance 3 s to insert three items in the left box (one item per second), and 3 s to insert two items in the right box (one item per 1.5 s). Alternatively, the researcher may insert stimuli using the same rate (one item per second) but can also include a neutral stimulus in the box containing the smaller number of items (e.g., three food items in the left box and two food items + one stone in the right box). In both cases, the researcher will spend the same amount of time near the two boxes, thus preventing the possibility that temporal

information might cue subjects’ choice. For instance, Sulkowski and Hauser (2001) sequentially inserted a different number of plums into two opaque containers. To control for temporal cues, a plumsized metal object was inserted in the smaller group in the 1 vs. 2 discrimination. Results showed that monkeys were unaffected by the presence of the metal object, being able to select the larger number of fruits in all comparisons presented. However, even after all the above-mentioned precautions are taken, other confounding factors may affect the results. For instance, in the macaques’ studies, it is difficult to exclude that subjects might have solved the task by using other sensory modalities. Monkeys might have compared different magnitudes of olfactory cues or made their choice on the basis of the auditory feedback provided by the stimuli being dropped into the boxes. Since in spontaneous choice tasks the stimulus to discriminate coincides with the reward, such a problem is intrinsically related to this type of procedure. An interesting variation of the item-by-item procedure involves the use of a condition in which both visible and invisible items are presented in correspondence of the two boxes. Beran et al. (2008b) tested summation ability in capuchin monkeys by inserting two different quantities of food into two cups: both cups were then covered and the experimenter placed additional food items on top of the covered cups. To select the most advantageous cups, monkeys were required to mentally add two different numerosities for each cup (number of items hidden in the cup plus number of items presented on the top of the covered cup, for instance 5 hidden items vs. 1 hidden plus 2 visible items). Capuchin monkeys selected the larger group even when additional food was visible only above the smaller group, thus showing the capacity to inhibit responses to approach the cup where visible food items were placed on the top if those items were part of an overall smaller quantity of food. Item-by-item procedure has also been used with non-primate species, such as elephants (Perdue et al., 2012), horses (Uller and Lewis, 2009), and robins (Garland et al., 2012), showing spontaneous numerical abilities in these species. On the contrary, dogs tested with this procedure did not exhibit the same numerical abilities (MacPherson and Roberts, 2013). A variant of this procedure has been recently used to control for continuous quantities in shoal choice tests. Bisazza et al. (2010) studied whether guppies are able to select the larger number of conspecifics by using a modified version of the shoal choice apparatus (Gómez-Laplaza and Gerlai, 2011a). Each stimulus fish was confined in an adjacent and separate compartment at the two ends of the subject compartment. Opaque screens were also inserted in the experimental tank to prevent the possibility of seeing more than one stimulus fish at a time. To join the larger shoal, guppies were required to add the number of companions on each side of the tank. Results showed that juvenile guppies could discriminate both numerical contrasts presented (2 vs. 3 and 4 vs. 8), suggesting a spontaneous use of numerical information in sub-adult individuals of this species. In sum, spontaneous numerical abilities have been reported in a wide range of species. Several methodological attempts have been done to prevent the use of continuous quantities; however, as biologically relevant stimuli are presented, the possibility to control for all non-numerical cues remains difficult in spontaneous choice tests. For this reason, other studies used training procedures and presented inanimate objects as stimuli.

3. Trained numerical abilities In training procedures, subjects are required to learn a numerical rule in order to receive a reward. Neutral stimuli are commonly associated with a food reward. For instance, a monkey could be trained to select the larger group of dots presented on the monitor




in order to obtain a pellet. This procedure necessarily requires several steps before the subject begins the task. The subject is usually trained to use the apparatus and undergoes a shaping procedure in order to learn the association of reward with the correct response. The subject is then presented with a series of numerical discriminations until it reaches the learning criterion – normally around 70–80% correct choices in two consecutive sessions (Bogale et al., 2011; Emmerton and Renner, 2009; Jordan et al., 2008; Vonk and Beran, 2012). Due to the nature of the stimuli (generally inanimate bidimensional objects), there are several tools that can be used to control for confounding factors. First, a large number of different stimuli can be presented within the same numerical contrast, preventing the possibility that pattern recognition of the arrays can be used to reach the learning criterion (Ashkenazi et al., 2013). Continuous quantities can also be finely controlled for. For example, in a numerical discrimination, cumulative surface area can be matched by placing the smaller objects in the more numerous set. However, as a by-product of this type of control, smaller-than-average objects might be more frequent in the larger group and subjects can use this non-numerical cue to discriminate between the two groups. For this reason, other types of controls are required – for example by doing a series of control probe trials using objects of the same size for all stimuli (Agrillo et al., 2009; Halberda et al., 2008). Other studies have used a different type of control for cumulative surface area by presenting three different sets of stimuli: one in which there is congruence between area and numerosity (the larger group also occupies the larger area), one with a neutral condition (the larger group has the same area of the smaller group), and one in which there is incongruence between area and numerosity (the larger group occupies the smaller area). In this way, the cumulative surface area of the stimuli represents an unreliable cue and cannot permit animals to learn discrimination based on this cue. For instance, in a study comparing monkeys and humans, cumulative surface area was equated in 10% of the trials; in 45% of the trials the larger numerosity had the larger cumulative surface and in 45% of the trials the larger numerosity had the smaller cumulative surface (Cantlon and Brannon, 2006). A similar problem occurs when researchers try to control for the overall space occupied by the two sets (the space delimited by outermost items). The by-product of pairing the overall space occupied by the two sets is that items are usually more spaced out in the smaller group and subjects can use the density of the objects to discriminate between the two groups. As a consequence, half of the stimulus pairs should be controlled for overall space occupied and half for their density (Agrillo et al., 2010). Chimpanzees can be easily trained to discriminate between two quantities of dots in order to get a food reward (Tomonaga and Matsuzawa, 2002). Sometimes trained apes also exhibit remarkable performance. An accuracy of 80% in discriminating a 0.80 ratio was reported by Tomonaga (2008); other studies revealed that chimpanzees can learn to associate Arabic numbers to the correct quantity of objects (Biro and Matsuzawa, 2001; Boysen and Berntson, 1989). Recently a trained orangutan proved able to discriminate between groups in the range of 7–10 with 75% accuracy, even when two groups differed by one unit (Vonk, 2014). Impressive numerical abilities have been similarly reported in monkeys. Trained macaques are able to match stimuli up to numerosities of 12 (Jordan and Brannon, 2006) and exhibit an accuracy of 90% to discriminate even a 0.90 ratio (Beran, 2008), a performance that is nearly the same observed in our species when humans are prevented from using verbal language (Halberda et al., 2008). Cantlon and Brannon (2007a) compared the performance of humans and rhesus monkeys in a task requiring the addition of numerical values from two sets of dots and choosing a stimulus from two options that corresponded to the sum of the two sets. Despite

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obvious quantitative differences between the two species (with humans showing an overall better performance), the performance of monkeys adhered to that of humans, reinforcing the idea of similar non-symbolic numerical systems among primates. In support of this view, neurophysiological studies found the existence of similar neural networks for number processing in humans and rhesus monkeys. Monkeys were involved in a delayed match-tonumerosity task requiring memorizing the numerosity in a 1 s delay period (sample) and then matching it to a subsequent test stimulus. If the second array contained the same number of objects presented in the sample, subjects had to release a lever (Nieder and Miller, 2003). Behavioral data adhered with Weber’s law, with accuracy decreasing while decreasing the numerical distance between the sample and test stimulus. In addition, recording the neural activity from the pre-frontal cortex, the researchers found neurons with peak activities specific for a given numerosity. Monkeys’ neural activity acts as a numerosity filter, with activity declining progressively with increasing numerical distance from their preferred number (see also Nieder et al., 2002; Moskaleva and Nieder, 2014). The pre-frontal cortex is known to be involved in several numerical tasks also in humans (reviewed in Arsalidou and Taylor, 2011), a fact that further reinforces the idea of similar neuro-cognitive systems for number processing between human and non-human primates. Training procedures have been used with several other mammals. A male raccoon was trained to select the groups of three objects – grapes or small metal bells – within the range of four alternatives: 1 vs. 3, 2 vs. 3, 4 vs. 3, and 5 vs. 3. The subjects showed the capacity to select the reinforced numerosity in all of these contrasts (Davis, 1984). Vonk and Beran (2012) reported the first evidence of numerical abilities in black bears. In order to get a food reward (fruits and vegetables), subjects were trained to choose the larger or smaller arrays of dots in the range of 1–12. Results showed that black bears can be trained to use numerical information, showing a performance that aligns with that of primates tested with similar training procedures. Concerning aquatic mammals, Jaakkola et al. (2005) investigated numerical competence in dolphins trained to select the smaller group of dots to get a food reward in the comparisons of 2 vs. 6, 1 vs. 3, and 3 vs. 7. Subjects learned all discriminations, proving their ability to generalize the numerical rule also to novel numerical contrasts in the range of 1–8. Yaman et al. (2012) expanded our knowledge regarding dolphins’ numerical abilities. Initially, the animals were trained to discriminate between one and five. As soon as they reached the learning criterion, a discrimination showing a reduced numerical distance was presented (1 vs. 4, 1 vs. 3, and 1 vs. 2). In the following test phase, more difficult contrasts were presented, in the range of 2–10. Subjects showed a significant discrimination in 4 vs. 5 (85%) and a significant but lower discrimination in 5 vs. 6 (65%). One of the earliest evidences of counting behavior in animals is provided by Skinner’s experiments on fixed-ratio schedule of reinforcement (Ferster and Skinner, 1957). In these experiments, rats or pigeons were trained to press a lever a specific number of times in order to obtain food. The variation in their pattern of response while approaching the requested number of responses on the lever suggested that subjects could count quite accurately the number of lever presses. One of the main problems with this procedure is that the delivery of the reward signals conclusion of the press sequence and this represents a potential confounding factor. To overcome this limitation, Mechner (1958) introduced a two-lever variation of this procedure: rats were trained to press the lever B to obtain food after the completion of a specific number of consecutive responses on lever A. The results showed that rats were able to gauge the requested number responses (4, 8, 12 or 16 responses) but that their accuracy decreased with increasing numerosity.




In another study, rats were trained to use ordinal information to get a food reward (Davis and Bradford, 1986). In details, rats were inserted into a runway where different tunnels were available. Food was only provided at the end of the sixth tunnel. In this type of study, a different pool of continuous quantities must be taken into account: the ordinal position of the tunnels co-varies with the overall distance from the starting point, or the overall time spent in reaching the reward. For this reason, the experimenters varied the overall distance among the tunnels so that only ordinal information could be used to enter the correct tunnel. Subjects proved able to solve the task, selecting the sixth tunnel of the sequence more than chance (Davis and Bradford, 1986). Suzuki and Kobayashi (2000) extended the conclusion of this study showing that trained rats can also learn how to use ordinal information up to 18 units. Trained numerical abilities are not restricted to mammals, however. Alex, the parrot trained by Irene Pepperberg, proved able to learn Arabic numbers, vocally producing the label that represented the exact result of an addition (reviewed in Pepperberg, 2012). Pigeons (Brannon et al., 2001) and crows (Bogale et al., 2011) also showed numerical abilities after extensive training, with a performance that approximates those observed in primates, such as 6 vs. 7 discrimination (Emmerton and Delius, 1993). The similarity between pigeons and monkeys has also been reported in a recent study where the same stimuli and procedures were used in the two species (Scarf et al., 2011). In addition, domestic chicks can be successfully trained to peck at the sixth position in a series of ten identical locations, even when the distance among the alternative targets are experimentally modified (Rugani et al., 2007). Training procedures have been used to study numerical abilities in fish with two types of rewards: social and food. Agrillo et al. (2010) trained mosquitofish to discriminate between 4 and 8 objects. Subjects were singly inserted in an unfamiliar environment provided with two doors, one associated with four objects and the other associated with eight objects, placed at two opposite corners. To re-join social companions, subjects were required to discriminate between the two numerosities and select the door associated with the reinforced numerosity. Results showed that mosquitofish could also learn this discrimination when continuous quantities were controlled for. The capacity of trained fish seemed to have no upper limit of numerosity, since they were as accurate in 4 vs. 8, 15 vs. 30, and 100 vs. 200. When presented with numerical contrasts having different numerical ratios, their accuracy decreased with an increase in the numerical ratio and the discrimination was significant in 7 vs. 14 (0.50 ratio) and 8 vs. 12 (0.67), but not 9 vs. 12 (0.75 ratio). In another study, guppies were singly housed in rectangular tanks where two monitors were placed at the two short ends. Two stimuli (groups of dots differing in numerosity) were repeatedly presented on each monitor and food was delivered near the numerosity that was to be reinforced. For instance, in a 2 vs. 4 discrimination, food could be provided only in correspondence with the larger numerosity. As a measure of discrimination, this study used the time spent near positive stimulus in probe trials without reward. Guppies proved able to discriminate both small (2 vs. 4) and large (6 vs. 12) numerosities. In addition, they showed a higher numerical acuity when items were in motion, being able to also discriminate between 3 vs. 4 objects (Agrillo et al., 2014). The capacity to learn a numerical rule is not confined to adult fish. A recent study using a similar training procedure to that adopted with adult fish found that one-week-old guppies successfully learned to discriminate between 7 and 14 objects (Piffer et al., 2013). 4. A comparison of the two methodologies As readers can easily observe, the two methodological approaches differ in many respects and might reveal different

aspects of numerical abilities. In this section, we will directly compare the procedure previously described and try to summarize its pros and cons, outlining some theoretical and practical issues concerning the use of these two methodologies. First, the two methodologies differ in the ecological validity of results. In spontaneous choice tests, animals are thought to exhibit their natural behavioral repertoire in the presence of biologically-relevant stimuli. As they receive no special training, the performance of monkeys involved in a 2 vs. 3 apple discrimination is likely to reflect the cognitive functions they would activate when a similar problem occurs in nature. In this sense, results obtained with spontaneous choice tests can help us to understand in which contexts animals might be advantaged by possessing numerical abilities, indirectly revealing the selective pressures that led to the development of these capacities. On the contrary, results obtained after an extensive training do not provide direct insight into the practical use of numerical abilities in nature. In training procedures, animals are required to discriminate between quantities of neutral stimuli in order to get a reward (e.g., Biro and Matsuzawa, 2001), a condition that can be hardly compared to most of the problems faced in nature. Second, one may wonder if results obtained with these two methodologies reflect only the activation of neuro-cognitive systems involved in numerical discrimination. While it is undeniable that spontaneous choice tests reveal the normal repertoire of animal numerical abilities, it is unclear as to whether or not training procedures lead to an exclusive recruitment of neuro-cognitive systems devoted to number processing or instead engage other neural circuits usually not involved in numerical tasks. In our species, it is widely accepted that expertise may determine exceptional performance and the consistent modification of cognitive systems, including some that are not directly involved in the specific domain of expertise (Campitelli et al., 2007; Gauthier et al., 2000). Professional musicians, for instance, show better performance in symbolic (Cheek and Smith, 1999; Schmithorst and Holland, 2004) and non-symbolic numerical abilities (Agrillo and Piffer, 2012) – abilities that are not directly related to musical training. Musical training also causes functional and structural rearrangement in the brain, with musicians differing from non-musicians for an increased left-sided asymmetry of the planum temporale, a brain area that is usually involved in verbal language (Schlaug et al., 1995). In this sense, expertise may lead to the recruitment of other neural networks in order to accommodate for the extensive requirements of a specific cognitive task. Even though there is a lack in the number of studies of neural correlates supporting similar conclusions in non-human animals, the possibility remains – as outlined by other authors (Barnard et al., 2013; Hauser and Spelke, 2004; Hauser et al., 2000) – that the performance exhibited by animals after extensive training may be the result of the recruitment of neuro-cognitive systems usually not involved in number processing. Nonetheless, the use of training procedures presents some advantages. For instance, in spontaneous choice tests, motivation plays a key-role and null results do not necessarily imply a lack of discrimination. For instance, if a monkey does not show any preference for the larger quantity in the presence of 20 and 30 apples, this might be due to the fact that the numerical ratio presented exceeds its numerical acuity or that the monkey is simply not motivated to select the larger group as both groups are large enough to satisfy it. There is no way to disentangle these two hypotheses unless we adopt training procedures. Similar problems may arise when social stimuli are presented. For example, fish could discriminate a group of 20 from a group of 30 individuals and still not manifest a preference since both groups provide enough protection from predators.




Even the commonly observed decrease in accuracy with increasing numerical ratio – that is normally attributed to limitations in numerical acuity – might actually depend on the fact that the benefit of selecting the most favorable option decreases with decreasing numerical distance (i.e., increasing the ratio) between the two options. Animals might also exhibit individual difference in preferences. These may vary according to sex (the motivation of selecting one group may differ as a function of a subject’s sex and sex ratios among the groups), age (individuals at different ages may exhibit a different degree of sociality), individual features of the stimuli (subjects may be attracted by a single stimulus because of single traits, such as color, body size, etc). In training procedures, reward is not represented by the possibility of obtaining the stimuli. As a consequence, most of these problems can be avoided. After all, it is no coincidence that trained animals can perform a large number of trials in a few experimental sessions. In training studies, monkeys, pigeons, and dolphins can perform up to one thousand trials (Beran, 2008; Jaakkola et al., 2005; Brannon et al., 2001). The lack of discrimination exhibited by animals after extensive training is likely to reflect a true limit in their ability to process numerical information instead of being the result of concomitant factors, such as motivation or individual preferences. In addition, while spontaneous choice tests are necessarily limited to the natural preference of reaching the larger quantity of food or social companions, training procedures can permit us to also study the capacity to select the smaller numerosity, giving us the possibility to assess whether the ability to select the smaller and larger numerosities requires a similar cognitive effort. The potential recruitment of broader neural circuits, together with the possibility of having higher levels of motivation during extensive training, might explain why several animals show better performance in training tasks. Trained chimpanzees can easily discriminate 0.80 ratio (Tomonaga, 2008), a capacity hardly reported in spontaneous choice tests (Beran, 2001). Similarly, trained rhesus monkeys showed a more accurate performance in the 0.80 ratio (Cantlon and Brannon, 2007a) as compared to that exhibited in a study using spontaneous choice tests (Hauser et al., 2000). Trained pigeons can discriminate between up to 6 vs. 7 items (Emmerton and Delius, 1993) and match the performance of primates (Scarf et al., 2011), well above the performance commonly observed with untrained birds (e.g., parrots: 2 vs. 3 – see Al Aïn et al., 2009; New Zealand robins: 3 vs. 4 – see Hunt et al., 2008). One-week-old fish can be trained to discriminate between 7 and 14 items (Piffer et al., 2013), a capacity displayed in spontaneous choice tests only after 20–40 days (Bisazza et al., 2010). As said before, however, in all of these cases the possibility remains that the observed outstanding performances reflect cognitive capacities that would never be displayed in natural conditions. Another issue speaks in favor of training procedures. Even though several experimental procedures have been set up to control for continuous quantities, such controls remain difficult in spontaneous choice tests. For instance, despite all attempts to control for olfactory cues (e.g., by keeping the subjects far from the point in which the experimenter inserts the fruit in the containers), the possibility remains that the animals can be attracted by the magnitude of olfactory cues as soon as they get close to the two alternatives. Similarly, when potential prey or social companions are presented, the group containing more stimuli had a higher probability of having at least one stimulus active, thus increasing the probability that it would be detected by the subjects, regardless of the numerosity of the items, as outlined by Uller et al. (2003). The stimuli used in training procedures are usually bi-dimensional, static objects presented on a monitor, thus avoiding any problems related to olfactory cues and uncontrollable motions of the items. Today, specific software can generate

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groups of bi-dimensional objects accurately controlled for all continuous quantities (see Gebuis and Reynvoet, 2011; Piazza et al., 2004). Hence, the experimenters can present several hundred different stimulus pairs that have in common only the numerosity of the items so that, if an animal reaches the learning criterion, one can be fairly confident that continuous quantities have played no role in discrimination. Again, training procedures may be more suitable for an interspecific comparison. Spontaneous choice tests require the use of the most appropriate stimuli for the species model investigated. Salamanders, for instance, show little or no sociality and, using spontaneous numerical preferences, their performance cannot be easily compared with that of fish (Buckingham et al., 2007; Gómez-Laplaza and Gerlai, 2011a,b, 2012, 2013; Piffer et al., 2012). However, salamanders are very sensitive to moving prey (Uller et al., 2003; Krusche et al., 2010) which may not happen in other species (Vonk and Beran, 2012). To avoid the possibility that different results might be due to different types of stimuli, the use of a training procedure presenting the same experimental material (i.e., black dots on a white background) appears to be the best option. Actually, most of the published inter-specific studies adopted training procedures (Agrillo et al., 2010, 2012a; Beran, 2006; Cantlon and Brannon, 2006, 2007a; Scarf et al., 2011). On the other hand, inter-specific differences observed in experiments with training procedures could be due to general differences in learning abilities rather than reflecting true differences in numerical abilities among species. In other words, if we find that two species differ in numerical acuity after being trained in a Skinner box, the possibility exists that this procedure is less suitable for one of the two species, or that one species has better learning abilities than another. In a recent study that compared numerical abilities of five different teleost fish with the same procedure, zebrafish exhibited a lower performance than the other four species. A control experiment requiring to make a shape discrimination showed that zebrafish had a similar lower performance also in this non-numerical task, suggesting that the observed difference resulted from the zebrafish’s difficulty in learning this procedure rather than from a cross-species variation in the numerical domain (Agrillo et al., 2012a). Other practical issues might be considered prior to deciding which methodology to adopt. As previously stated, in spontaneous choice tests motivation is critical and may decrease throughout the experiment. In a food discrimination task, animals can learn that they will get a certain quantity of food in any case (also through selecting the smaller group), thus decreasing the attention trial after trial. Similarly, in a shoal choice test, the environment may become more and more familiar to the fish if different tests are planned for the same subjects, reducing the motivation to join the larger group. Between-subjects design would be more appropriate, especially if easy-to-collect species are investigated. As a consequence, a large sample size is usually required in spontaneous choice tests. For instance, Hauser et al. (2000) tested more than 200 rhesus monkeys, Rugani et al. (2009) tested 52 chicks, Uller et al. (2003) tested 210 salamanders, and Agrillo et al. (2012c) tested 340 guppies. In training procedures, a few individuals may be enough – in some cases, even one individual – through the assumption that, if at least one individual can achieve the task, the brain of that species is equipped with neuro-cognitive systems able to potentially support the resolution of the task (Pepperberg and Brezinski, 1991). Brannon and Terrace (1998) trained two rhesus monkeys, Vonk and Beran trained two bears (2012), and Yaman et al. (2012) trained one dolphin. MacPherson and Roberts (2013) tested the spontaneous numerical abilities of 27 dogs but trained a single dog; Pepperberg trained one parrot in several studies (reviewed in Pepperberg, 2012), and Agrillo et al. (2011) trained six mosquitofish per experiment.




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Table 1 Summary of the main pros and cons involved in using spontaneous choice tests or training procedures.

Major confounds Number vs. continuous quantities

Use of sensory modalities other than that investigated Motivation

Theoretical issues Ecological validity

Spontaneous choice tests

Training procedures

Despite the fact that several procedures are available to prevent the use of continuous quantities, the control of these variables is always difficult with biologically-relevant stimuli.

The use of artificial inanimate stimuli permits a fine-grained manipulation of stimuli, increasing the possibility to dissociate between numerical abilities and continuous quantity discrimination. Reward is not represented by the possibility of obtaining the stimuli and thus this problem normally does not apply. Animals are constantly motivated by rewarding the correct choice, regardless of the numerosity of the items and the type of stimuli.

Since the stimulus to discriminate coincides with the reward (e.g. pieces of food), it is necessary to exclude that other sensory modalities (e.g., olfactory cues) provide non-numerical clues. Motivation may play a key role. Null results can be due to ceiling effects in food choice or may reflect different individual preferences when social stimuli are presented. If animals show a spontaneous ability to discriminate between quantities of biologically-relevant stimuli in the laboratory, they are likely to also make use of this capacity in nature when facing similar problems.

Natural repertoire of numerical abilities?

As animals exhibit a spontaneous behavior, the neuro-cognitive systems recruited in these tests are likely to be same involved in similar tasks under natural conditions.

Inter-specific studies

As biologically relevant stimuli differ across species, different results might be partially due to the use of different stimuli.

Practical issues Sample size

Between vs. within subject design Time consumed per subject

A large sample size is usually collected to assess whether animals can solve the task at the population level.

As motivation/accuracy tends to decrease with an increase in the number of trials, between-subjects design should be preferred. A subject is usually tested in a relatively short amount of time.

On the other hand, discrimination in learning procedures generally requires that each subject undergo a time-consuming training procedure. Testing a single fish in a spontaneous choice test might require 15–20 min (Agrillo et al., 2008a, 2012c; Gómez-Laplaza and Gerlai, 2011a) while almost a month may be required to train a fish in similar numerical contrasts (Agrillo et al., 2014). In Table 1, we have summarized the pros and cons of both methodologies. It is clear that the two methods are complementary. Some studies actually adopted an integrated methodology that combines the features of both training and spontaneous choice. In this procedure, subjects are initially trained to discriminate between an easy numerical ratio (i.e., 5 vs. 10) with the only purpose being to teach the subjects a numerical rule. As soon as the subjects reach the learning criterion, novel numerosities are presented in probe trials without reward (e.g., 6 vs. 9 and 6 vs. 8). In this way, although the experiment involves an initial training phase, what we observe in probe trials is the spontaneous capacity to discriminate between numerosities. Using this methodology, Brannon and Terrace (1998) trained rhesus monkeys to respond to 35 different stimulus sets in the range of 1–4. The task required touching the groups in ascending numerical order. Subjects were subsequently tested without reward on their ability to order novel numerosities in the range of 5–9. Monkeys proved to be able to spontaneously represent the numerosity of novel stimuli, extrapolating an ordinal rule to numerosities never reinforced during the experiment. Similarly, mosquitofish were initially trained to discriminate between two groups of bi-dimensional figures having a 0.50 ratio (5 vs. 10 and 6 vs. 12). After reaching the learning criterion, they were presented with novel numerosities in non-reinforced trials. In this way, it was possible to assess the natural limit of numerical discrimination when groups differ in numerical ratio or in total set size (Agrillo et al., 2012b). Results showed that the numerical ratio affected the performance – in agreement with previous literature (Barnard et al., 2013; Perdue et al., 2012; Agrillo et al., 2012c; Gómez-Laplaza and

Feral animals are often required to associate new, arbitrary stimuli to a specific response but learning an association between the numerosity of arbitrary stimuli and a reward, as in typical laboratory experiments, is probably much rarer in nature. Extensive training might lead to extraordinary numerical abilities through the recruitment of neuro-cognitive systems that are not normally involved in numerical cognition, an event that is very unlikely to occur under natural conditions. The use of the same stimuli even with distantly related species may permit a fine inter-specific comparison. A reduced sample size is enough. If a few individuals can achieve the task then it is assumed that the species is equipped of neuro-cognitive systems potentially able to learn that numerical rule. Within-subjects design is usually preferred as animals can normally undergo thousands of trials. Each subject may require a long amount of time (e.g., months) before reaching the learning criterion.

Gerlai, 2011a,b). The total set size was irrelevant in fish numerosity judgments, with mosquitofish being able to discriminate between 4 vs. 8 as well as 100 vs. 200. This finding matched the results previously obtained using a training procedure, thus confirming that this ability is indeed part of the natural cognitive repertoire of the species (Agrillo et al., 2010). This procedure has the potential to combine the advantages of the two methods we have discussed in this review, and, although so far it has been adopted much less frequently than the other two methodologies, we believe this procedure will become more and more used in the near future.

Acknowledgments This work was funded by FIRB grant (RBFR13KHFS) from “Ministero dell’Istruzione, Università e Ricerca” (MIUR, Italy) to Christian Agrillo and “Progetto Strategico NEURAT” from University of Padova to Angelo Bisazza.

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Varieties of numerical abilities.

Numerical and area comparison abilities in Down syndrome.

Extensive training extends numerical abilities of guppies.

Numerical modeling of human mastication, a simplistic view to design foods adapted to mastication abilities.

Numerical magnitude processing in abacus-trained children with superior mathematical ability: an EEG study.

Electrostatic interaction between nonuniformly charged colloids: experimental and numerical study.

Numerical and sensory categorization of serial information by animals.

Is numerical comparison digital? Analogical and symbolic effects in two-digit number comparison.

Obsessional cognition: performance on two numerical tasks.

Numerical and experimental study of the nonlinear interaction between a shear wave and a frictional interface.

Mathematical and numerical model to study two-dimensional free flow isoelectric focusing.

Simple test statistics for major gene detection: a numerical comparison.

Comparison of Subcutaneous Regular Insulin and Lispro Insulin in Diabetics Receiving Continuous Nutrition: A Numerical Study.

Long-Term Creep Behavior of the Intervertebral Disk: Comparison between Bioreactor Data and Numerical Results.

Assessing the Approximate Number System: no relation between numerical comparison and estimation tasks.

Failure of a numerical quality assessment scale to identify potential risk of bias in a systematic review: a comparison study.

The relationship between working memory for serial order and numerical development: a longitudinal study.

Numerical Study on Droplet Sliding across Micropillars.

Numerical study of fractional nonlinear Schrödinger equations.

Brain Structural Integrity and Intrinsic Functional Connectivity Forecast 6 Year Longitudinal Growth in Children's Numerical Abilities.

Effects of Non-Symbolic Approximate Number Practice on Symbolic Numerical Abilities in Pakistani Children.

Numerical calculation of interaction forces between paramagnetic colloids in two-dimensional systems.

Distribution of aerosolized particles in healthy and emphysematous rat lungs: comparison between experimental and numerical studies.

Comparison of cephalic and extracephalic montages for transcranial direct current stimulation--a numerical study.