See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/295910425

The symbol-grounding problem in numerical cognition: A review of theory, evidence, and outstanding questions

Article in Canadian Journal of Experimental Psychology · February 2016 DOI: 10.1037/cep0000070

CITATIONS READS 8 534

2 authors, including:

Daniel Ansari The University of Western Ontario

144 PUBLICATIONS 5,569 CITATIONS

SEE PROFILE

All content following this page was uploaded by Daniel Ansari on 01 March 2016.

The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately. APA NLM tapraid5/cae-cae/cae-cae/cae00415/cae0277d15z xppws Sϭ1 10/1/15 15:45 Art: 2015-1231

Canadian Journal of Experimental Psychology / Revue canadienne de psychologie expérimentale © 2015 Canadian Psychological Association 2015, Vol. 69, No. 4, 000 1196-1961/15/$12.00 http://dx.doi.org/10.1037/cep0000070

The Symbol-Grounding Problem in Numerical Cognition: A Review of Theory, Evidence, and Outstanding Questions

AQ: au Tali Leibovich and Daniel Ansari AQ: 1 The University of Western Ontario

How do numerical symbols, such as number words, acquire semantic meaning? This question, also referred to as the “symbol-grounding problem,” is a central problem in the field of numerical cognition. Present theories suggest that symbols acquire their meaning by being mapped onto an approximate system for the nonsymbolic representation of number (Approximate Number System or ANS). In the present literature review, we first asked to which extent current behavioural and neuroimaging data support this theory, and second, to which extent the ANS, upon which symbolic numbers are assumed to be grounded, is numerical in nature. We conclude that (a) current evidence that has examined the association between the ANS and number symbols does not support the notion that number symbols are grounded in the ANS and (b) given the strong correlation between numerosity and continuous variables in nonsymbolic number processing tasks, it is next to impossible to measure the pure association between symbolic and nonsymbolic numerosity. Instead, it is clear that significant cognitive control resources are required to disambiguate numerical from continuous variables during nonsymbolic number processing. Thus, if there exists any mapping between the ANS and symbolic number, then this process of association must be mediated by cognitive control. Taken together, we suggest that studying the role of both cognitive control and continuous variables in numerosity comparison tasks will provide a more complete picture as to the symbol-grounding problem.

Keywords: numerical cognition, symbol-grounding problem, approximate number system, numerical magnitudes, nonnumerical magnitudes

Whether it is to estimate the distance to the nearest bus stop, to of number that we share with other species (the so-called “Approx- choose the shortest line in the supermarket or to calculate the imate Number System” or ANS). Against this background, the change given to you, we use numbers constantly in our everyday present literature review will examine (a) the extent to which the lives. How we represent and process numbers has been studied available behavioural and neuroscientific evidence supports extensively in humans at different developmental stages, as well as the theory that number symbols are grounded in the ANS and (b) in a variety of animal species. One of the key questions in the the extent to which the purported ANS is numerical in nature. The literature concerns how humans learn the meaning of numerical review will close by outlining questions for future theory and symbols (sometimes referred to as the “symbol-grounding prob- research. lem”), such as Arabic numerals, which are uniquely human cul- tural representation of numerosity that children need to learn over The Number Sense/ANS Account of the the course of development. The present article begins by reviewing Symbol-Grounding Problem the most prominent proposal for the resolution of the symbol- One of the most prominent theories in the field of numerical grounding problem in numerical cognition, which postulates that cognition was put forward by several researchers (e.g., Dehaene, symbols are grounded in a system for the approximate representation 1997; Gallistel & Gelman, 1992). According to this theory, the ability to process numerical quantities (i.e., the total number of items in a set or its numerosity) is a basic, automatic, and innate ability that can be found across species. In other words, humans Tali Leibovich and Daniel Ansari, Numerical Cognition Laboratory, are born with a capacity to process nonsymbolic numerosities (e.g., Department of Psychology & Brain and Mind Institute, The University of dot arrays, groups of objects, number of sounds, etc.). Further- Western Ontario. more, according to this theory, symbolic representations of num- This work was supported by operating grants from the Natural Sciences ber, such as Arabic numerals and number words, which children and Engineering Council of Canada (NSERC), the Canadian Institutes of learn over the course of development, are thought to acquire their Health Research (CIHR), the Canada Research Chairs Program, an E.W.R meaning by being mapped onto the preexisting, nonsymbolic rep- Steacie Memorial Fellowship from the Natural Sciences and Engineering resentations of number. Council of Canada (NSERC) to Daniel Ansari, as well as a Brain & Mind Institute (Western University) Postdoctoral Scholarship to Tali Leibovich. This theory is supported by studies demonstrating that preverbal Correspondence concerning this article should be addressed to Daniel babies and animals are able to spontaneously discriminate between Ansari, Department of Psychology & Brain and Mind Institute, The Uni- numerosities (e.g., Libertus & Brannon, 2009; Nieder, 2005). For versity of Western Ontario, Westminster Hall, Room 325, London, ON example, in a habituation study (repeated presentation of a partic- N6A 3K7, Canada. E-mail: [email protected] ular stimuli followed by the presentation of a stimuli that differs in

1 APA NLM tapraid5/cae-cae/cae-cae/cae00415/cae0277d15z xppws Sϭ1 10/1/15 15:45 Art: 2015-1231

2 LEIBOVICH AND ANSARI

the variable of interest) by Xu and Spelke (2000), infants were habituated to groups of dots in different sizes but that contained the same numerosity. In the test stage, the infants saw two displays— one display with the same numerosity as in the habituation stimuli and another display containing a novel number of dots. The results OC NO of this experiment revealed that infants looked longer toward the LL novel numerosity compared with the numerosity to which they had IO been habituated. Furthermore, it was found that 6-month-old in- NR fants only exhibited the ability to discriminate between the habit- E uated and novel numerosity when the ratio between the two stimuli was 0.5 (e.g., 8 vs. 16 dots), but not when the ratio was higher Figure 1. Basic effect in numerical cognition. (A) The numerical ratio (e.g., 8 vs. 12 dots). These findings have since been replicated in effect. The plot illustrates the relationship between numerical ratio and a number of different labouratories (Coubart, Izard, Spelke, Marie, response time, in a number comparison task. The x-axis describes the numerical ratio: smaller/larger numerosity. The y-axis represents the time & Streri, 2014; Izard, Sann, Spelke, & Streri, 2009). In addition to it takes to respond to the larger number. Task difficulty increases when the work on infants, it has also been demonstrated that a variety of numerical ratio is closer to 1 (i.e., with an increase in numerical ratio). animals are able to discriminate between different nonsymbolic Inside the plot are examples for symbolic and nonsymbolic stimuli: The numerosities (birds, Bogale, Kamata, Mioko, & Sugita, 2011; numerical ratio of 3 and 8 is ϳ0.37, and the numerical distance is 4; the Watanabe, 1998; fish, Agrillo, Dadda, Serena, & Bisazza, 2008, numerical ratio of 7 and 9 is ϳ0.77, and the numerical distance is 2. It is 2009; rodents, Meck & Church, 1983; lions, McComb, Packer, & also true, then, that task difficulty increases with the decrease in numerical Pusey, 1994; and primates, Cantlon & Brannon, 2006; Nieder, distance. (B) Approximate representation of numerosities. Representation Freedman, & Miller, 2002). of numbers is thought to be represented on a logarithmic scale. This Discrimination between different numerosities shows a similar representation is assumed to be approximate and noisy: Larger numbers are response pattern to that of discriminating between different mag- represented more approximately, and the representation of adjacent num- bers overlap. See the online article for the color version of this figure. nitudes, such as brightness, pitch of sound, physical size, and weight (Cantlon, Platt, & Brannon, 2009; but see also Leibovich, Ashkenazi, Rubinsten, & Henik, 2013; Leibovich, Diesendruck, The ANS and the Symbol-Grounding Problem Rubinsten, & Henik, 2013). In all these cases, the ability to detect change (or to understand that two presented magnitudes are dif- To this point, the discussion of number representations has ferent from each other) depends on the ratio between the to-be- focused on nonsymbolic representations of number. This raises compared magnitudes. For example, it is faster and easier to decide questions regarding the relationship between nonsymbolic repre- that 10 dots are more numerous than 3 dots (ratio of 0.3), than sentation of numbers and their symbolic representation (e.g., Ar- deciding that 10 dots are more numerous than 8 dots (ratio of 0.8). abic numerals, number words, etc.): How do symbols acquire their The ratio effect is thought to result from noisy representation of meaning? Once that meaning is acquired, is the representation of F1 numerosities (Figure 1A). It is thought that numerosities are rep- symbolic and nonsymbolic numbers similar? resented in a logarithmic analog format where numerosities that When children first encounter symbolic representations of num- have a larger ratio share more representational overlap and are thus ber, they are meaningless words or visual symbols. Therefore, more easily confused than numerosities that have a smaller nu- children need to connect symbolic representations to their seman- merical ratio and thus share less representational overlap (Figure tic meaning. How symbols, such as numerical symbols, become 1B). representations of semantic referents (such as numerosity) has This ratio dependency is also known as “Weber’s law,” namely, been referred to as the “symbol-grounding problem” (Harnad, that the difference in intensity needed to detect a difference be- 1990, 2003). In the field of numerical cognition, the most widely tween two stimuli (e.g., two different numerosities) is proportional accepted theoretical account to resolve this problem is the notion to the objective intensities of the stimuli (Cantlon, Platt, et al., that symbols are mapped on the preexisting ANS (Dehaene, 2007; 2009). An individual differences measure derived from this law is Piazza, 2010; Stoianov, 2014). the so-called Weber fraction—the minimal differences of the in- In support of this account, it has been demonstrated that, just tensity between stimuli that can still be discriminated. This mea- like nonsymbolic number discrimination, symbolic number dis- sure represents an individuals’ acuity of numerosity representa- crimination is also ratio-dependent, namely, symbolic number tion: Individuals with low Weber fraction scores are able to comparison has been found to obey Weber’s law. This was first discriminate much closer numerosities than are individuals with demonstrated by Moyer and Landauer (1967). In this study, the high Weber fraction scores. Weber’s law is a characteristic of both authors presented two single-digit symbolic numbers and asked human and nonhuman performance and explains the noisy repre- adult participants to decide which number is numerically larger. sentation of numerosities in the ANS (Cantlon & Brannon, 2006). Manipulating the numerical distance (or difference) between the The processing of features that were found to obey Weber’s law in two symbolic numbers, it was found that both response times (RT) classic psychophysical experiments (e.g., loudness, brightness, and accuracy were affected by the numerical distance between the line-length, etc.) is considered very fast and automatic. Because two numbers; responses were faster and more accurate for large numerosity processing was found to obey the same law, it has been numerical distances (e.g., 2 and 9) than for smaller distances (e.g., suggested that numerosity processing is as basic, fast, and innate as 7 and 8). The numerical distance effect is highly correlated with processing of brightness, weight, pitch of sound, and so forth the ratio effect, because when the difference between two numbers (Cantlon et al., 2009; Feigenson, Dehaene, & Spelke, 2004). is larger, the ratio between them is smaller (i.e., closer to 0). The APA NLM tapraid5/cae-cae/cae-cae/cae00415/cae0277d15z xppws Sϭ1 10/1/15 15:45 Art: 2015-1231

SYMBOL-GROUNDING PROBLEM IN NUMERICAL COGNITION 3 numerical ratio, however, takes into account not only the differ- (with some reporting an association between ANS and symbolic ence between two numbers, but also their absolute size, so, in math and others unable to obtain such positive evidence) charac- accordance with Weber’s law, it can explain more variability in terizes the extant literature on the correlations between the ANS performance than the distance effect. The numerical distance (and and symbolic math (De Smedt, Noël, Gilmore, & Ansari, 2013). ratio) effect is a robust finding that has been replicated in numer- Indeed, the review by De Smedt et al. suggests that symbolic ous studies since it was first published (Ansari, Garcia, Lucas, number processing (most typically measured by symbolic number Hamon, & Dhital, 2005; Dehaene, Dupoux, & Mehler, 1990; comparison tasks) is more consistently associated with symbolic Leibovich, Ashkenazi, et al., 2013; Leibovich, Diesendruck, et al., math (typically measured using standardized tests of mental arith- 2013; Tzelgov, Meyer, & Henik, 1992). The similarity in distance and metic) than are measures of nonsymbolic numerical processing. ratio effects across both symbolic and nonsymbolic number processing Taken together, current evidence from studies probing the dom- has been one of the central pieces of evidence behind the suggestion that inant account of the “Symbol-Grounding Problem” in numerical humans come to learn symbolic numbers by mapping them onto their cognition via studies that investigate the correlation between the more primitive and possibility innate nonsymbolic representations of ANS and symbolic math reveal a picture that is far from straight- number. In other words, the similarity of the representational signa- forward. This evidence suggests that associations between the tures (ratio and distance effects) has led to the suggestion that numer- ANS and symbolic math are inconsistent across different investi- ical symbols are grounded in an innate, approximate representation of gations, thereby not supporting the notion that individual differ- number that exhibits both ontogenetic (developmental) and phyloge- ences in the nonsymbolic ANS are strongly predictive of between- netic (evolutionary) continuity. subjects variability in symbolic math. The idea that symbolic and nonsymbolic representations of Now, one might reasonably argue that correlational studies number are tightly connected to one another has also been sub- assessing the relationship between the ANS and symbolic math are stantiated by brain imaging research. Such research has shown that a rather indirect, nonexperimental way to assess the prediction that the intraparietal sulcus (IPS) is engaged during both symbolic and the “symbol-grounding problem” in numerical cognition can be nonsymbolic number processing (Cohen Kadosh, Lammertyn, & resolved by postulating that numerical symbols become semantic Izard, 2008; Dehaene, Piazza, Pinel, & Cohen, 2003; Nieder, representations of numerosity by being mapped onto the ANS. In 2005). Furthermore, brain imaging studies have shown that the other words, correlations with individual differences cannot be activity in the IPS is modulated by the numerical ratio of both used to measure the similarity in the processing mechanisms of symbolic and nonsymbolic numerosities, demonstrating ratio- two systems. This is a general problem because correlating indi- dependent processing of symbolic and nonsymbolic number at vidual processes cannot separate a correlation of state from a both behavioural and brain levels of analysis (Fias, Lammertyn, correlation of process (Cantlon, 2015; Garner & Morton, 1969). Reynvoet, Dupont, & Orban, 2003; Holloway, Price, & Ansari, A decisively more experimental approach to testing the predic- 2010; Kaufmann et al., 2005). tions of the symbol-grounding problem was taken by Lyons et al. (2012). More specifically, the authors tested whether numerical Symbols Grounded in the ANS? symbols are tightly associated with nonsymbolic representations (underpinned by the ANS) by examining the efficiency with which While there is much consensus in the literature for the notion adults can compare symbolic with nonsymbolic representations of that numerical symbols, such as Arabic numerals and number numerosity (henceforth compared with mixed comparisons). In words, acquire their meaning through being mapped onto the ANS, their study (Lyons, Ansari, & Beilock, 2012), adults were asked to there is a surprising dearth of studies that have directly tested this make comparisons both within and across symbolic and nonsym- account of the “symbol-grounding problem” in numerical cogni- bolic representations of numerosity. In other words, participants tion. There are several ways of examining whether numerical were asked to compare (a) which of two Arabic numerals was symbols are grounded in the ANS. One prediction one can make is larger; (b) which of two dot arrays was larger and, in the critical, that, if numerical symbols acquire their meaning by being mapped mixed condition; (c) which of a simultaneously or sequentially onto the ANS, there should be a correlation between individual presented Arabic numeral and dot array was numerically larger. In differences in the ANS and individual differences in symbolic view of the dominant account of the above discussed resolution to number processing, such as arithmetic. Indeed, in recent years the “symbol-grounding problem,” the authors argued that if sym- there has been an explosion of studies examining such relation- bolic and nonsymbolic representations of numerosity are strongly ships by investigating correlations between, on the one hand, connected to one another, then mixed (symbolic–nonsymbolic) individual differences in children’s and adults’ nonsymbolic nu- comparison should be at least as efficient as within format (sym- merosity processing (most typically measured by means of com- bolic or nonsymbolic) comparison. Contrary to this prediction, the parison tasks) and, on the other, symbolic number processing findings revealed that the performance in the mixed condition abilities, which have been most commonly assessed by means of (comparison of symbolic with nonsymbolic) numerosities was by standardized tests of symbolic calculation abilities. Already at the far the most difficult condition. In addition, the authors demon- outset of such investigations, the results of studies across different strated that this mixing cost was specific to the comparison of labouratories were conflicting. While some of the earliest correla- symbolic and nonsymbolic representations of numerical magni- tional studies reported a positive association between individual tude, because the comparison of number words to Arabic numerals differences in the ANS and symbolic math (Halberda, Mazzocco, did not show a similar cost in performance relative to within & Feigenson, 2008), others were unable to uncover such relation- format comparisons (symbolic and nonsymbolic). ships (Holloway & Ansari, 2009). A recent comprehensive review Furthermore, in a recent study, Lyons, Nuerk, and Ansari (2015) of the literature suggests that the initial contradictory evidence demonstrated that only a minority (30%) of kindergarten through APA NLM tapraid5/cae-cae/cae-cae/cae00415/cae0277d15z xppws Sϭ1 10/1/15 15:45 Art: 2015-1231

4 LEIBOVICH AND ANSARI sixth-grade children showed a significant ratio effect in a symbolic In addition to data investigating the mapping between the ANS comparison task, whereas the majority (75%) of children that age and children’s early understanding of the meaning of counting, showed a significant ratio effect in a nonsymbolic comparison there is also a growing body of evidence that speaks against a task. This result is difficult to explain if one assumes that symbols strong mapping between the ANS and symbolic number in older are grounded in the ANS. Indeed, these findings suggest that children. To investigate whether symbolic and nonsymbolic rep- symbolic and nonsymbolic representations of number may be resentations of numerosity are related to one another during the qualitatively different from one another. transition from kindergarten to first grade, Sasanguie, Defever, These findings, therefore, cast significant doubt on the notion Maertens, and Reynvoet (2014) tested children’s nonsymbolic that symbolic and nonsymbolic representations of numerosity are numerosity comparison abilities in kindergarten and their nonsym- strongly associated with one another and undermines the widely bolic and symbolic numerosity comparison abilities 6 months later. suggested resolution to the symbol-grounding problem. However, While children’s performance on the nonsymbolic comparison it is important to point out that the data discussed above were task was found to be correlated across times, the authors failed to obtained from adults and older children, hence it is possible that find a correlation between nonsymbolic numerosity processing in such an association may exist earlier in developmental time and kindergarten and symbolic number comparison abilities 6 months that, as the authors of the study argue, symbols become “es- later. Moreover, the authors were unable to find a significant tranged” from nonsymbolic, ANS-supported representations of correlation between symbolic and nonsymbolic number processing numerosity over the course of developmental time. during the second testing time point. These findings suggest that Indeed, developmental studies do exist that have addressed nonsymbolic number processing abilities, thought to be under- exactly this question. More specifically, there exists a growing pinned by the ANS, do not predict future symbolic number pro- body of studies that have investigated whether children’s devel- cessing competencies. Furthermore, contrary to the prediction that opment of counting abilities, and specifically children’s under- the ANS predicts growth in individual differences in symbolic standing that counting serves to determine the numerosity of sets number processing (because symbolic numbers are grounded in (the “Cardinality Principle”), is scaffolded by the ANS. If chil- the ANS), a recent longitudinal study (Mussolin, Nys, Content, & dren’s developing understanding of the meaning of symbolic num- Leybaert, 2014) with 3- to 4-year-old children has revealed that while early symbolic number processing abilities predict later ber is grounded in the ANS, then there should exist a strong nonsymbolic numerosity processing, the reverse (early nonsym- connection between the ANS and children’s understanding of the bolic predicts later symbolic) is not true. Therefore, contrary to the meaning of counting, because arguably the acquisition of the suggestion that symbols acquire their meaning by being grounded cardinality principle represents the starting point of children’s in the ANS, these data, consistent with those reported by Odic semantic representation of symbolic number. et al. (2015) discussed earlier, show that symbolic scaffolds non- To date, as the case for the previously discussed investigations symbolic, while the reverse is not true. of a correlation between the ANS and symbolic math, the data on While these results speak against the notion that individual the association between the ANS and children’s understanding of differences in the nonsymbolic ANS scaffold symbolic number the meaning of counting are contradictory, with some suggesting processing, there does exist some evidence from training studies to that children acquire an understanding of the meaning of counting suggest that training nonsymbolic numerosity processing has an without the involvement of the ANS (Le Corre & Carey, 2007) and impact on symbolic number processing in both adults and children others suggesting a correlation between children’s understanding (Hyde, Khanum, & Spelke, 2014; Park & Brannon, 2013). How- of the cardinality principle and the ANS (Mussolin, Nys, Leybaert, ever, the precise mechanisms underlying such training-induced & Content, 2012; Wagner & Johnson, 2011). These contradicting transfer effects from nonsymbolic to symbolic number processing results can also be explained by lack of power in some of the remain unclear; data suggest that it is the training-related manip- studies. Recent data, however, suggest that children only map ulation of nonsymbolic numerosities rather than training-induced nonsymbolic to symbolic numbers successfully after they have enhancements of ANS representations that leads to improvements acquired the cardinality principle (Odic, Le Corre, & Halberda, in symbolic number processing following nonsymbolic ANS train- 2015), while at the same time they can map symbolic numbers to ing (Park & Brannon, 2014). nonsymbolic numerosities more successfully. This finding sug- Another line of studies relevant to the symbol-grounding prob- gests that children’s ANS-symbolic number mappings are not fully lem is semantic priming. In semantic priming studies, the target is bidirectional after acquiring the cardinality principle. This finding preceded by a prime stimulus that is presented briefly to the also suggests, at the very least, that the ANS cannot be considered participant. The effect of the prime on the target stimulus is then the sole contributor to children’s development of an understanding measured. For example, Koechlin, Naccache, Block, and Dehaene of the meaning of counting, which is, arguably, their first step (1999) presented a number prime in different notations (verbal, toward an understanding of how symbols (in this case number digit, or nonsymbolic group of dots) and tested the effect on the words) represent numerosity. Thus, contrary to the notion of target number. Participates had to indicate whether the target “symbolic estrangement” (Lyons et al., 2012), these developmen- number was smaller or larger than 5. The results of this study tal studies suggest the early acquisition of a semantic meaning of revealed that performance improved only when both the prime and numerical symbols (number words) is not strongly supported by the target stimulus were of the same notation. When the prime and the ANS. Therefore, even when looking at the very outset of the target had different notations, response was not faster even symbolic number development (children’s understanding of the when both stimuli were smaller or larger than 5. In contrast to meaning of count words), there does not appear to be strong these results, in a different experimental condition, when partici- support for the notion that symbols are grounded in the ANS. pants had to respond to both the prime and the target, performance APA NLM tapraid5/cae-cae/cae-cae/cae00415/cae0277d15z xppws Sϭ1 10/1/15 15:45 Art: 2015-1231

SYMBOL-GROUNDING PROBLEM IN NUMERICAL COGNITION 5 always benefited from the prime and the target leading to the same symbol shapes to those holding nonsymbolic representations of response, even across notations. Accordingly, the authors sug- quantities” (Piazza et al., 2007, p. 293). gested that the internal representation of numerosities dissociate While the findings reported by Piazza et al. (2007) support the into multiple notation-specific subsystems. These results are not in notion of a format-independent representation of numerical quan- line with the notion of a shared representation of symbolic and tity in the parietal cortex, more recent findings have been unable to nonsymbolic numerosities. It is interesting that in a more recent reveal similar evidence. Also using fMRI adaptation to study priming study, Gabay, Leibovich, Henik, and Gronau (2013) used cross-format processing in the brain, Cohen-Kadosh et al. (2011) pictures of small (cat, dog) and large (elephant, horse) animals as found that a change of format in the absence of a change in prime and asked participants to judge whether a target digit rep- numerical quantity (e.g., a change from 12 dots to the symbolic resented an odd or an even number. The authors found that prime representation “12”) led to activation in the same IPS regions that influenced performance: performance was slower when the prime responded to a change in quantity. Furthermore, format changes was a large animal and the target number was smaller than 5, and led to greater activation in the IPS than did quantity changes, performance was faster when both the prime and the target were which is at odds with the existence of a common, amodal quantity either big (e.g., 8 and elephant) or small (e.g., 2 and dog). Ac- code. cordingly, the authors suggested that conceptual size and numer- The use of both conjunction analysis and fMRI adaptation to ical value share an underlying representation, and that symbolic examine commonalities in the neuronal correlates of symbolic and representation of numbers has evolved from, or is rooted in, a more nonsymbolic numerical quantity processing suffers from a number evolutionary ancient system for the representation of magnitudes. of limitations regarding the inferences that can be drawn about the Beyond behavioural data, the symbol-grounding problem can presence or lack of a strong link between symbolic and nonsym- also be assessed by investigating the similarities between symbolic bolic quantity processing in the brain. Because fMRI has a rela- and nonsymbolic representations of numerosity in the brain. To tively coarse spatial resolution and involves averaging the activa- test whether symbolic and nonsymbolic representations of numer- tion across points of observation (called voxels; three-dimensional ical quantity are correlated with activation in similar brain net- pixels), it is entirely possible that the common activation of a brain works, several researchers have used a method referred to as region for symbolic and nonsymbolic numerical quantity process- conjunction analysis. This statistical tool enables researchers to ing does not imply equivalence of processes engaged by each examine which brain regions are activated by both symbolic and condition. In other words, it is possible that the activation of a nonsymbolic representations of numerical quantity. The results brain region by two experimental conditions is driven by two from these studies have revealed that areas of the IPS are activated different underlying mechanisms. by both symbolic and nonsymbolic numerical quantity (Holloway The use of multivariate analytic techniques can help to over- & Ansari, 2010; Holloway, Price, & Ansari, 2010). come some of the aforementioned limitations. In particular, rather Another method to test whether there is an association between than averaging activation across voxels, these new tools analyse nonsymbolic and symbolic representations of numerical quantity the pattern of activation of multiple voxels and compare and in the brain is to examine whether the presentation of one repre- correlate the pattern of activation between conditions. This method sentational format also leads to the activation of the other (i.e., is often referred to as Multi-Voxel Pattern Analysis (MVPA). cross-format activation). Using a method commonly referred to as Using this methodology, Bulthé, De Smedt, and Op de Beeck functional MRI (fMRI) adaptation, Piazza, Pinel, Le Bihan, and (2014) revealed that while it is possible to distinguish the patterns Dehaene (2007) tested whether adaptation to symbolic number of activation elicited by different quantities within formats (e.g., quantities would also activate nonsymbolic representations and distinguishing 4 dots from 6 dots or the numeral 5 from the vice versa. More specifically, participants were first adapted numeral 9), there was no evidence for cross-format classification. (through the repeated presentation of the stimulus of interest) to This evidence points to a lack of overlap in the distributed (across either a symbolic (Arabic numerals) or nonsymbolic (dot arrays) voxels) representations of symbolic and nonsymbolic numerical numerical quantity. Following adaptation (repeated presentation), quantities. they were presented with numerical deviants in the other format Similar evidence was obtained in a study by Damarla and Just (with the hypothesis that deviants will elicit a recovery of the (2013), who also showed good within-format classification of adapted response in neural regions sensitive to the change in the numerical quantities, but poor cross-format classification (i.e., the stimulus between adaptation events and the deviant). Responses to multivoxel patterns of different numbers could be distinguished these deviants were found in a bilateral network of frontal parietal within but not across formats of numerical quantity representa- regions. It is important that the degree of deviant-related response tion). Another study by Eger et al. (2009), using MVPA to classify in the IPS was related to the numerical distance between the the patterns of symbolic and nonsymbolic numerical quantities, number to which the brain was adapted and the deviant. In other revealed modest cross-format classification (just over 50% correct) words, the amount of deviant-related activation increased as the but only in one direction: The multivoxel patterns for dot arrays numerical distance between the adaptation and deviant numbers could be used to predict the activation of corresponding Arabic increased. Critically, this distance effect occurred both when the numerals, but the patterns of activation elicited during the process- change from notation was from symbolic to nonsymbolic and vice ing of the numerals could not be used to predict the patterns versa. This cross-format numerical distance effect suggests that correlated with processing dot arrays. Moreover, through examin- symbolic and nonsymbolic representations of numerical quantity ing the correlation between multivoxel pattern activity (an ap- tap into a common approximate, neural code of quantity represen- proach referred to as Representational Similarity Analysis), Lyons, tation. As the authors put it, the findings: “. . . support the idea that Ansari, and Beilock (2014) found no correlation between the AQ: 3 symbols acquire meaning by linking neural populations coding patterns of individual symbolic and nonsymbolic numbers. APA NLM tapraid5/cae-cae/cae-cae/cae00415/cae0277d15z xppws Sϭ1 10/1/15 15:45 Art: 2015-1231

6 LEIBOVICH AND ANSARI

Taken together, both behavioural and neuroimaging data do not provide strong evidence in support of the notion that symbolic representations of number are grounded in the nonsymbolic ANS. In view of this, it is clear that the symbol-grounding problem in numerical cognition is far from resolved. Clearly, in order for numerical symbols to become semantic representations of numer- osity, they need to be grounded in some nonsymbolic representa- OC tion of number. NO Some have suggested that instead of being grounded in the LL ANS, number symbols become linked to small quantities (ϳ1–4) IO NR that can be enumerated exactly and without counting (a process E referred to as subitizing; Trick & Pylyshyn, 1994). In other words, early in development, the number words 1–4 become associated Figure 2. Nonnumerical variables change with numerosity. (A) Each with nonsymbolic numbers in the subitizing range, but larger array contains six dots. The continuous variables of these arrays, however, numbers are not linked to nonsymbolic representations of numer- are different from one another. (B) Different dot arrays will always have ical quantity (Carey, 2001, 2004). According to this hypothesis, different continuous variables. The lefthand side of each rectangle contains children recognize the link between number words 1–3/4 and sets five dots, and the right-hand side contains 10 dots. As can be seen in the of objects in this numerosity range. From this association, they can different versions, manipulating one variable results in differences in other bootstrap critical principles of the symbolic number sequences variables as well, making it impossible to control for all the continuous such as ordinality (that each number is part of a sequence) and the variables at the same time. See the online article for the color version of successor function (that the next symbol in the sequence is exactly this figure. 1 larger). Another, not mutually exclusive, explanation for the lack of a strong association between numerical symbols and the ANS is that impossible, however, to change the numerosity without changing the the way in which the ANS is typically measured in tasks using continuous variables of the set (Figure 2B). In other words, changing nonsymbolic stimuli may tap into processes other than the repre- the numerosity of a set will always lead to change in the continuous sentation of approximate numerosities. It is this possibility that is variables of this set. In view of this, it is impossible to be sure that discussed in the next section of this review article. participants do not process the continuous variables of a set when performing estimation or comparison tasks of nonsymbolic numer- How Numerical is the ANS? osities. It is unclear, then, to what extent the processing of nonsym- bolic numerosities is purely numerical. This questions the degree to When studying the mapping of symbolic numbers onto the ANS, a which the processing of any nonsymbolic array of items is purely central assumption is that the ANS is numerical; namely, that what we numerical, as is stipulated by the ANS theory. extract from a visually presented set of items is their numerosity and With this theoretical constraint in mind, different studies have that this nonsymbolic numerosity taps into a fundamental numerical manipulated continuous variables in an effort to ensure that such representation. An example for this view is demonstrated in a model variables do not drive the response of participants in nonsymbolic of Dehaene and Changeux (1993). This computational model de- number processing studies. Such approaches have used various strat- scribes a three-step process for exact representation of numerosities. egies, such as (a) manipulating one continuous property at a time First, an input records the location of each item; then, the locations of (Mussolin, Mejias, & Noël, 2010), (b) assigning a random dot size to the items are being mapped topographically. This mapping is not each array (Piazza et al., 2010), (c) using a single array containing two influenced by the continuous variables of the items or the array (such different colours of dots where participants must indicate the colour as the density of the dots, their total area, etc.). In the third step, a of the more numerous dots (Mazzocco, Feigenson, & Halberda, “numerosity detector” sums all the locations, producing a number. 2011), and so forth. The assumption of such studies is that if a Verguts and Fias (2004) provided computational modelling data to continuous variable is not a reliable cue of numerosity, it will be suggest that the same network modules that subserve the nonsymbolic ignored, will not be processed, and thus will not affect perfor- numerosities are also used in the developmental construction of sym- mance. For example, equating the average size of two dot arrays bolic representations of numerosities. creates a correlation between total surface area and numerosity: Theoretical considerations and recent empirical evidence, however, The total area of all the dots will be greater in the array containing reveal a more complicated picture. The stimuli that are used to study more dots. Equating the total surface area of the dots will result in nonsymbolic numerosity processing are sets of items (e.g., dot arrays). a negative correlation between numerosity and average size: The Such sets are characterised, in addition to their numerosity, by differ- average size of a dot in the array containing more dots will be ent nonnumerical, continuous variables: the average size of each item, smaller than the array containing fewer dots. In such a design, the total surface area of all the items, their density, the total occupied some continuous variables are correlated with numerosity and can space of the items and the area between them (i.e., convex-hull), and be used to identify the array containing more dots. If, however, in so forth. In the real world, numerosity and continuous variables are half the trials the average size of the dots will be equated (across often correlated or anticorrelated. Usually, for example, more items different numerosities) and on the other half the total surface area will occupy more area and will be denser than fewer items. In an will be equated, then none of these continuous variables will be a experimental design, it is possible to present the same numerosity reliable cue of numerosity and will not affect performance (for a F2 while changing the continuous variables of a set (Figure 2A). It is review see Leibovich & Henik, 2013). This assumption, however, APA NLM tapraid5/cae-cae/cae-cae/cae00415/cae0277d15z xppws Sϭ1 10/1/15 15:45 Art: 2015-1231

SYMBOL-GROUNDING PROBLEM IN NUMERICAL COGNITION 7 is not supported by recent studies, demonstrating that even when the neural underpinnings of the ANS. To reveal the brain regions that continuous variables are task-irrelevant and not correlated with subserve the ANS, many studies have employed nonsymbolic com- numerosities, they still affect performance in both estimation and parison tasks. In such a task, one theoretically needs to use pairs of dot comparison tasks (Leibovich & Henik, 2014). arrays that are different only in their numerosities. Because this is For example, in comparison tasks, both adults and children in impossible, neuroimaging studies tried to either manipulate continu- various developmental stages were found to be susceptible to the ous variables in the way described previously (so they are not corre- influence of continuous variables. In one such study, Gebuis and lated with numerosity), or to create a control task in which only Reynvoet (2012) asked adults to compare the numerosity of two continuous variables are compared; for example, comparing the area successively presented dot arrays. Continuous variables in this set of of two squares. That way, contrasting the numerosity comparison task stimuli were not correlated with numerosity; namely, different con- and the area comparison task will yield brain areas that are dedicated tinuous variables (total surface area, density, average dot-size, total to numerosity comparison (Chassy & Grodd, 2012). This idea is circumference, etc.) were manipulated all at once (in the same dot problematic for several reasons. First, the two sets of stimuli for the arrays and not in separate dot arrays as done before), so none of these different tasks are visually different from one another; stimuli used in variables could consistently predict numerosity. It is important that the numerosity comparison tasks (e.g., two sets of dots) are composed of authors manipulated the number of continuous variables that were more items than stimuli used in an area comparison task (e.g., one congruent with numerosity. In one condition, only density was con- small and one large square). Second, while the area comparison task gruent with numerosity; namely, the more numerous dots were denser can be solved only by relying on continuous variables, participants than the less numerous dots. In another condition, both density and can use both numerosity and continuous variables in the numerosity total surface area were congruent with numerosity, and so forth. comparison task. Third, one thing that is required only in the numer- Critically, consistent with the notion that continuous variables play a osity task is summation or integration of all the visual properties of the role in nonsymbolic numerosity discrimination, it was found that items in the array. When comparing the numerosity of two sets of accuracy increased with the number of continuous variables that were items, one first needs to extract the numerosity of items in each array congruent with numerosity. Hence, continuous variables affected per- and then compare these numerosities. Such a process requires one to formance even when they were task-irrelevant and could not reliably first sum the dots and then compare them. Such summation is not predict numerosity. These results are important because in many required when comparing the size of two squares (i.e., area compar- studies, the congruity between numerosities and continuous variables ison task). Therefore, when contrasting a numerosity comparison task was manipulated as a way of attempting to prevent participants from with, for example, an area comparison task, the brain areas revealed using continuous variables as cues. Because in such studies congruent by such a contrast might reflect the stage of summing-up the dots, and incongruent trials are not analysed separately (e.g., Halberda, rather than a direct access to the representation of numerosity (see also Mazzocco, & Feigenson, 2008; Odic, Libertus, Feigenson, & Roggeman, Santens, Fias, & Verguts, 2011). Halberda, 2013), it is impossible to evaluate the potential influence Moreover, there exists direct evidence demonstrating that continu- continuous variables had on performance. ous variables affect ANS acuity—a measure that is supposed to rely In addition to the correlation between numerosity and continuous solely on numerosity. Specifically, Tokita and Ishiguchi (2013) mea- magnitudes, these magnitudes have been found to be processed in sured precision (Weber fraction) and accuracy (the point of subjective similar time-windows in event-related potential (ERP) studies; in such equality; when two arrays become indistinguishable to the partici- studies, brain activity is being recorded while participants are engaged pants) of adults and 5- to 6-year-old children in a numerosity com- in an active or passive task. This technique has better temporal parison task. The total area and convex hull of the standard and the resolution than that of fMRI and is being used to answer questions comparison arrays were identical in one condition and different in regarding the time in which different cognitive processes occur. Some the other. It was found that both adults and children overestimated the ERP studies have revealed that continuous magnitudes are processed number of items when their total surface area and convex hull was automatically in a passive viewing task, in the same time window that larger. In other words, Weber fraction was directly affected by dif- was previously attributed to processing numerosities; that is, N2, ferent manipulations of continuous variables. Accordingly, the au- which occurs around 200 ms following the presentation of the stim- thors suggest that numerosity judgments were not solely based on the ulus of interest (Gebuis & Reynvoet, 2013). A more recent ERP study numerical information available, but that continuous variables were also found that processing of numerosities occurs later than process- part of the decision making process. These findings, therefore, show ing of basic visual cues, such as shape, and suggested that numerosity that measures of nonsymbolic numerical acuity are not pure measures is a “higher-level property assembled from naturally correlating per- of numerosity processing but critically depend on the correlation ceptual cues, and hence, it is identified later in the cognitive process- between numerical and continuous variables (Leibovich, Henik & ing stream” (Soltész & Szu˝cs, 2014, p. 203). In contrast, Park, Salti, in press; Leibovich, Vogel, Henik & Ansari, in press). DeWind, Woldorff, and Brannon (2015) reported that numerosities Not only the way in which continuous variables are manipulated, are being processed extremely early (starting at 75 ms from but other factors, such as stimuli presentation time (Inglis & Gilmore, display onset) in the visual stream. In this study too, however, 2013) task context, the range of the tested numerosity, task difficulty continuous magnitudes (and specifically convex hull) could (i.e., the numerical ratio of the to-be-compared numerosities), and have contributed to these early activations. There is, however, even culture, were shown to modulate how much participants might the possibility that what is evident in this time point (N2) is the rely on nonnumerical variables (Leibovich, Henik, & Salti, in press; integration of numerosity and continuous magnitudes. This Leibovich, Vogel, Henik, & Ansari, in press). option should be empirically tested. Related to this, a recent study by Cantrell, Kuwabara, and Smith The problem with the correlation between numerosity and contin- (2015) compared performances of Japanese and U.S preschoolers in a uous variables is also critical in neuroimaging studies aiming to reveal nonsymbolic comparison task. Children first saw a target stimuli APA NLM tapraid5/cae-cae/cae-cae/cae00415/cae0277d15z xppws Sϭ1 10/1/15 15:45 Art: 2015-1231

8 LEIBOVICH AND ANSARI

(array of items) and then were presented with two different arrays and and nonsymbolic numerosity processing be studied when numerosity were asked to choose the array that was most similar to the target. The processing cannot be studied in isolation of continuous variables? array matched the target in either the total area or the numerosity of Next, we raise outstanding questions and suggest future research the dots. Children were found to choose the array similar in numer- directions that will hopefully move the field closer to a resolution of osity to the target array when the number of items was small and when the symbol-grounding problem in numerical cognition. there was a noticeable difference between the numerosity of the target and the test array (i.e., in small numerical ratios). Moreover, children Inhibitory Control Plays a Critical Role in chose the array that was similar in area when the number of dots was Nonsymbolic Numerosity Processing large and the difference between the dots was very small (i.e., in higher numerical ratios). Japanese children relied on total area at To prevent participants from relying on continuous variables dur- relatively smaller set sizes than did U.S. children. Accordingly, ing numerosity comparisons, researchers have varied the degree to it has been suggested that numerosity range, numerical ratio, which numerosity and continuous variables are congruent with one and cultural differences influenced the level of attention (or the another. For example, it might be the case that in half the trials, the weight) given to numerosity as opposed to continuous vari- more numerous dots are denser (i.e., congruent). Conversely, in the ables. These findings seriously question the notion that there other half, the less numerous dots are denser than the more numerous exists a nonsymbolic representation of number that is number- dots (i.e., incongruent). Such a design will mean that density will not specific. Instead, these findings point to a system of magnitude be a reliable cue of numerosity, and thus should not influence per- representation in which continuous and numerical variables formance. There is, however, evidence suggesting that this is not the interact with one another during numerosity processing. case; under these conditions it has been found that participants inhibit In the context of the symbol-grounding problem in numerical the processing of the incongruent continuous variables, resulting in a cognition, such findings raise the distinct possibility that previous congruity effect—faster and more accurate performance when numer- attempts to unveil associations between the ANS and symbolic num- osity and continuous variables are congruent than when incongruent ber did not measure pure nonsymbolic numerosity processing; in- (Clayton & Gilmore, 2014; Gebuis & Reynvoet, 2012; Hurewitz, stead, the process of mapping symbolic numbers to both numerical Gelman, & Schnitzer, 2006; Leibovich, Henik, & Salti, in press; and continuous variables is inherent in nonsymbolic numerosity stim- Leibovich et al., in press; Nys & Content, 2012). These congruity uli. A recent study by Merkley and Scerif (2015) directly tested this effects suggest that nonsymbolic numerosity comparison tasks require notion by means of a training study. In particular, adult participants inhibitory control abilities. were trained to associate abstract symbols with nonsymbolic numer- This problem leads to an impasse: When numerical and continuous osities (i.e., dot arrays). The symbols were associated to dot arrays in variables correlate, there is no conflict and need for inhibition, but which the numerosity and continuous variables were either congruent then one cannot exclude the possibility that processes other than pure or incongruent with one another. During the test phase, participants representation of approximate numerical quantity are being recruited. performed a comparison task of the abstract symbols. Consistent with In other words, it is impossible to be sure that what was measured the evidence discussed previously, the authors found that comparison during nonsymbolic numerosity comparison solely reflects ANS. was modulated by the ratio between the numerosities associated with Against this background, it becomes apparent how ANS acuity and the symbols. Critically, performance was also affected by congruity, inhibition abilities can be confounded with one another. To illustrate because comparisons were also slower and less accurate for symbols this point, consider a recent study by Szu˝cs, Nobes, Devine, Gabriel, that had been associated (during training) with numerosities that were and Gebuis (2013). These authors asked 7-year-old children and incongruent in terms of their continuous variables. This result sup- adults to engage in a typical nonsymbolic numerosity comparison ports the claim that not only numerosity, but also continuous variables task. Weber fractions were calculated separately for congruent and may be associated with the symbolic representation of number. The incongruent trials. This measure, commonly thought to reflect ANS congruity effect found in the symbol comparison task suggests that acuity, was found to be modulated by congruity. Specifically, acuity continuous variables continue to affect performance even after the was worse in incongruent compared with congruent trials. In addition, symbols are learned and associated with numerosity. Overall, the the difference between Weber fractions in congruent and incongruent results of this training study emphasize the important role continuous trials was greater in children than in adults. Accordingly, the authors variables play in the process of associating symbolic and nonsymbolic suggested that Weber fraction is not a pure measure of ANS acuity but representations of number. may also reflect individual differences in inhibitory control abilities. In a similar vein, Gilmore et al. (2013) found that that only incon- Future Directions gruent trials during nonsymbolic numerosity comparisons were cor- related with math abilities. Furthermore, Bugden and Ansari (2015) AQ: 4 While the most dominant theories in the field of numerical cogni- found that children with DD differed from their typically developing tion postulate that the ANS is a system that is number specific (e.g., peers on a nonsymbolic number discrimination task only when the Burr & Ross, 2008; Dehaene & Changeux, 1993; Dehaene, 1997), area and numerosity of the stimuli were incongruent but not when recent evidence suggests that this view should be reconsidered. As they were congruent. discussed previously, it has been shown that continuous variables play In the context of such an impasse, that is, the inability to evaluate a pivotal role in nonsymbolic numerosity processing (Cantrell & pure numerosity representation without confounding it with either Smith, 2013; Leibovich & Henik, 2013; Mix, Huttenlocher, & reliance on continuous variables or inhibition abilities, the existence Levine, 2002). This theoretical shift profoundly affects how the of a correlation between ANS acuity and exact number knowledge symbol-grounding problem can be conceptualised. In other words, should be reevaluated. For example, Mussolin et al. (2012) reported how can the correlation between exact, symbolic number knowledge results supporting the association between exact number knowledge APA NLM tapraid5/cae-cae/cae-cae/cae00415/cae0277d15z xppws Sϭ1 10/1/15 15:45 Art: 2015-1231

SYMBOL-GROUNDING PROBLEM IN NUMERICAL COGNITION 9 and the ANS; in this study, 3–6 year-old children’s performance in a understanding the interplay between these factors will lead to a more numerosity comparison task was positively correlated with exact complete picture of the development of the ability to represent nu- number knowledge. In view of the strong association between nu- merosities, and connect them with exact symbolic representations. merical and continuous variables, however, Negen and Sarnecka We reviewed evidence that Weber fraction, a measure of ANS (2015) repeated the experiment carried out by Mussolin et al. (2012) acuity, is affected by cognitive control abilities that are required to but systematically manipulated continuous variables, to ensure that resolve the conflict between numerical and continuous variables, as these variables were no longer a reliable cue for numerosity. Contrary well as the ratio of the numerosities that are being compared. Hence, to Mussolin et al. (2012), Negen and Sarnecka (2015) found that there the current definition of ANS acuity can be said to confound numer- was no longer a correlation between ANS acuity and exact number osity processing and cognitive control. This confound should not be knowledge when continuous variables were no longer a reliable cue to viewed simply as a side effect of experimental design that should numerosity. Against the background of these data, the authors con- be avoided and overcome. Instead, it is imperative to explore the cluded that the association between ANS acuity and exact number nature of this confound further and to investigate the extent to which abilities may be the result of an “artefact of the procedure used to the resources required to disentangle numerosity from continuous assess ANS acuity in children”(Negen & Sarnecka, 2015, p. 92). variables in nonsymbolic number tasks is key to the way in which Taken together, it is clear that nonsymbolic numerosity processing children learn about nonsymbolic number and how they form asso- cannot be studied in isolation of continuous variables, because nu- ciations between symbolic and nonsymbolic representations of num- merosity processing is confounded by continuous variables and cog- ber. In other words, to understand the meaning of a number and to nitive control abilities. understand that the same numerosity is preserved even if continuous variables change, it is necessary to have the ability to inhibit nonnu- A New Approach to the Symbol-Grounding Problem: merical magnitudes when they do not correlate with numerosity (i.e., The Relationship Between ANS Acuity, Cognitive cognitive control) and to use continuous magnitudes as a cue of Control, and Exact Number Knowledge numerosity when numerical and continuous variables do correlate. We suggest that similar set of abilities might also be required when a The pattern emerging from the literature discussed previously sug- child first learns to connect approximate nonsymbolic numerosities gests that the symbol-grounding problem in numerical cognition can- with exact symbolic representations. To understand that 6 apples and not be resolved by assuming that symbols are mapped onto the ANS. 6 watermelons correspond to the same symbolic representation, it is Specifically, if numerosity is only one of many visual variables necessary to inhibit other properties of these items that usually cor- processed when encountering nonsymbolic numerosity, how can non- relate with numerosity, such as their size. symbolic representations of numerosity be directly linked to symbolic numbers? We contend that cognitive control may be an important Summary and Conclusions component in this mapping process (if indeed it does occur). In other words, cognitive control serves to disentangle numerical and contin- A fundamental issue in the study of numerical cognition concerns uous cues in nonsymbolic numerosity tasks and therefore may play a the way in which numerical symbols, such as number words and critical role in allowing for any connection between nonsymbolic and Arabic numerals, become semantic representations of numerical symbolic representations of number to occur (whether this be to the quantity. At present, a dominant theoretical framework postulates that ANS or the representation of small set sizes). It allows for attention to humans share with other species an approximate system for the numerosity while inhibiting the influence of anti-correlated, nonnu- nonsymbolic representation of number (ANS) and that symbols ac- merical variables. quire their meaning by being mapped onto/associated with this evo- Inhibition abilities (or more generally cognitive control) mature lutionarily ancient and possibly innate system. We examined available with development (e.g., Morton, 2010). Thus, in very young ages, the evidence from both behavioural and neuroimaging studies to evaluate inability to inhibit irrelevant information results in the inability to the extent to which this hypothesis can be supported. properly compare numerosities when numerosities and continuous The overall conclusion we draw from this review of evidence is magnitudes are incongruent. This point was already demonstrated by that the symbol-grounding problem in numerical cognition cannot be Jean Piaget nearly 70 years ago in his famous experiments on chil- simply resolved by postulating that numerical symbols become rep- dren’s conservation abilities. In the conservation task, children are resentations of numerosity by being linked to the nonsymbolic ANS. first presented with the two rows of objects in which the objects are We contend this because our review clearly demonstrates that inves- equally spaced (i.e., same convex-hull) and asked whether the rows of tigations which have directly tested this hypothesis, have failed to find objects contain the same number of items. Then, in a second stage of strong links between symbolic and nonsymbolic number processing the experiment, the experimenter increased the spacing between items in both children and adults. Furthermore, it is becoming increasingly in one of the rows of items, thereby manipulating convex-hull, and clear that any examination of the nonsymbolic ANS involves the then repeated the question. Only children who answered that both processing both numerical and continuous variables. rows contained the same number of items in both stages of the task are Thus, whenever links between symbolic and nonsymbolic num- said, according to Piaget, to be able to conserve number (Piaget, ber processing have been investigated, it is likely that both the 1952), or in other words, to ignore, or inhibit, the changes in contin- processing of the numerosity of nonsymbolic sets as well as uous variables while maintaining a representation of numerical equiv- continuous variables correlating with numerosity were measured. alence. Recent work has demonstrated that not only children but also Put differently, given the strong correlation between numerosity adults are sensitive to continuous variables in conservation tasks and continuous variables in nonsymbolic number processing tasks, (Leroux et al., 2009). In light of the profound connection between it is next to impossible to measure the pure association between numerosity, continuous variables, and cognitive control, we argue that symbolic and nonsymbolic numerosity. Instead, it is clear that APA NLM tapraid5/cae-cae/cae-cae/cae00415/cae0277d15z xppws Sϭ1 10/1/15 15:45 Art: 2015-1231

10 LEIBOVICH AND ANSARI significant cognitive control resources are required in order to Bogale, B. A., Kamata, N., Mioko, K., & Sugita, S. (2011). Quantity disambiguate numerical from continuous variables during non- discrimination in jungle crows, Corvus macrorhynchos. Animal Behav- symbolic number processing. Thus, if there does exist any map- iour, 82, 635–641. http://dx.doi.org/10.1016/j.anbehav.2011.05.025 ping between the ANS and symbolic number, then this process of Bugden, S., & Ansari, D. (2015). Probing the nature of deficits in the association must be mediated by cognitive control. ‘Approximate Number System’ in children with persistent developmen- Taken together, the symbol-grounding problem in numerical tal dyscalculia. Developmental Science. Advanced online publication. cognition has not been resolved, and more questions than answers http://dx.doi.org/10.1111/desc.12324 surround this fundamental theoretical issue in numerical cognition. Bulthé, J., De Smedt, B., & Op de Beeck, H. P. (2014). Format-dependent A future research agenda that embraces the role of cognitive representations of symbolic and non-symbolic numbers in the human cortex as revealed by multi-voxel pattern analyses. NeuroImage, 87, control and continuous variables in nonsymbolic number process- 311–322. http://dx.doi.org/10.1016/j.neuroimage.2013.10.049 ing promises to shed further light on how (and indeed whether) Burr, D., & Ross, J. (2008). A visual sense of number. Current Biology, 18, number symbols are associated with approximate, nonsymbolic 425–428. http://dx.doi.org/10.1016/j.cub.2008.02.052 numerical magnitudes. Alternatively, it remains possible that a Cantlon, J. F. (2015). Analog origins of numerical concepts. In D. Geary, resolution to the symbol-grounding problem of numerical cogni- D. Berch, & K. Mann-Koepke (Eds.), Evolutionary origins and early tion may not require the involvement of the ANS. development of number processing (1st ed., pp. 225–251), London: Academic Press. Résumé Cantlon, J. F., & Brannon, E. M. (2006). Shared system for ordering small and large numbers in monkeys and humans. Psychological Science, 17, Comment les symboles numériques, tels que des mots exprimant 401–406. http://dx.doi.org/10.1111/j.1467-9280.2006.01719.x un nombre, acquièrent-ils leur sens? Cette question, ou « problème Cantlon, J. F., Libertus, M. E., Pinel, P., Dehaene, S., Brannon, E. M., & du fondement des symboles », est au cœur du domaine de la Pelphrey, K. A. (2009). The neural development of an abstract concept cognition numérique. Les théories actuelles suggèrent que les of number. Journal of Cognitive Neuroscience, 21, 2217–2229. http:// symboles acquièrent leur signification lorsqu’ils sont intégrés dans dx.doi.org/10.1162/jocn.2008.21159 un système approximatif pour la représentation non symbolique du Cantlon, J. F., Platt, M. L., & Brannon, E. M. (2009). Beyond the number nombre (système du nombre approximatif; SNA). Dans la présente domain. Trends in Cognitive Sciences, 13, 83–91. http://dx.doi.org/10 revue de littérature, on s’est demandé, premièrement, dans quelle .1016/j.tics.2008.11.007 mesure les données actuelles sur le comportement et les données Cantrell, L., Kuwabara, M., & Smith, L. B. (2015). Set size and culture de neuroimagerie appuient cette théorie, et deuxièmement, dans influence children’s attention to number. Journal of Experimental Child quelle mesure le SNA, dans le cadre duquel il est assumé que Psychology, 131, 19–37. http://dx.doi.org/10.1016/j.jecp.2014.10.010 les nombres symboliques sont ancrés, est de nature numérique. Cantrell, L., & Smith, L. B. (2013). Open questions and a proposal: A Nous concluons que a) les preuves actuelles du rapport entre le critical review of the evidence on infant numerical abilities. Cognition, SNA et les symboles numériques n’appuient pas la notion que 128, 331–352. http://dx.doi.org/10.1016/j.cognition.2013.04.008 ceux-ci sont ancrés dans le SNA et, 2) que vu la forte corrélation Carey, S. (2001). Cognitive foundations of arithmetic: Evolution and entre la numérosité et les variables continues dans la tâche de ontogenisis. Mind & Language, 16, 37–55. http://dx.doi.org/10.1111/ traitement de nombres non symboliques, il est a` peu près impos- 1468-0017.00155 sible de mesurer l’association nette entre la numérosité symbolique Carey, S. (2004). Bootstrapping & the origin of concepts. Daedalus, 133, et non symbolique. Toutefois, il est clair que d’importantes res- 59–68. http://dx.doi.org/10.1162/001152604772746701 Chassy, P., & Grodd, W. (2012). Comparison of quantities: Core and sources cognitives sont requises pour distinguer les variables nu- format-dependent regions as revealed by fMRI. Cerebral Cortex, 22, mériques des variables continues durant le traitement de nombres 1420–1430. http://doi.org/10.1093/cercor/bhr219 non symboliques. Ainsi, s’il existe une mise en correspondance Clayton, S., & Gilmore, C. (2014). Inhibition in dot comparison tasks. entre le SNA et le nombre symbolique, ce processus doit être géré ZDM, 1–12. http://dx.doi.org/10.1007/s11858-014-0655-2 par un contrôle cognitif. Nous suggérons que l’étude a` la fois du Cohen Kadosh, R., Bahrami, B., Walsh, V., Butterworth, B., Popescu, T., rôle du contrôle cognitif et des variables continues dans les tâches & Price, C. J. (2011). Specialization in the human brain: The case of de comparaison de numérosité permettra de dresser un portrait plus numbers. Frontiers in Human Neuroscience, 5, 62. complet du problème du fondement des symboles. Cohen Kadosh, R., Lammertyn, J., & Izard, V. (2008). Are numbers AQ: 7 Mots-clés : cognition des nombres, problème du fondement des special? An overview of chronometric, neuroimaging, developmental symboles, système du nombre approximatif, grandeurs numéri- and comparative studies of magnitude representation. Progress in Neu- robiology, 84, 132–147. http://www.sciencedirect.com/science/article/ ques, grandeurs non numériques. pii/S0301008207002110 Coubart, A., Izard, V., Spelke, E. S., Marie, J., & Streri, A. (2014). Dissociation References between small and large numerosities in newborn infants. Developmental Agrillo, C., Dadda, M., Serena, G., & Bisazza, A. (2008). Do fish count? Science, 17, 11–22. http://dx.doi.org/10.1111/desc.12108 Spontaneous discrimination of quantity in female mosquitofish. Animal Damarla, S. R., & Just, M. A. (2013). Decoding the representation of Cognition, 11, 495–503. http://dx.doi.org/10.1007/s10071-008-0140-9 numerical values from brain activation patterns. Human Brain Mapping, Agrillo, C., Dadda, M., Serena, G., & Bisazza, A. (2009). Use of number 34, 2624–2634. http://dx.doi.org/10.1002/hbm.22087 by fish. PLoS ONE, 4, e4786. http://dx.doi.org/10.1371/journal.pone Dehaene, S., & Changeux, J. P. (1993). Development of elementary nu- .0004786 merical abilities: A neuronal model. Journal of Cognitive Neuroscience, Ansari, D., Garcia, N., Lucas, E., Hamon, K., & Dhital, B. (2005). Neural 5, 390–407. http://dx.doi.org/10.1162/jocn.1993.5.4.390 correlates of symbolic number processing in children and adults. Neu- Dehaene, S. (1997). The number sense: How the mind creates mathemat- roreport, 16, 1769–1773. ics. New York, NY: Oxford University Press. AQ: 8 APA NLM tapraid5/cae-cae/cae-cae/cae00415/cae0277d15z xppws Sϭ1 10/1/15 15:45 Art: 2015-1231

SYMBOL-GROUNDING PROBLEM IN NUMERICAL COGNITION 11

Dehaene, S. (2007). Symbols and quantities in parietal cortex: Elements of numerical magnitude: An fMRI study. NeuroImage, 49, 1006–1017. a mathematical theory of number representation and manipulation. At- http://dx.doi.org/10.1016/j.neuroimage.2009.07.071 tention & Performance XXII. Sensori-Motor, XXII, 527–574. Hurewitz, F., Gelman, R., & Schnitzer, B. (2006). Sometimes area counts Dehaene, S., Dupoux, E., & Mehler, J. (1990). Is numerical comparison more than number. Proceedings of the National Sciences of the United digital? Analogical and symbolic effects in two-digit number compari- States of America, 103, 19599–19604. http://dx.doi.org/10.1073/pnas son. Journal of Experimental Psychology: Human Perception and Per- .0609485103 formance, 16, 626–641. http://dx.doi.org/10.1037/0096-1523.16.3.626 Hyde, D. C., Khanum, S., & Spelke, E. S. (2014). Brief non-symbolic, Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal approximate number practice enhances subsequent exact symbolic arith- circuits for number processing. Cognitive Neuropsychology, 20, 487– metic in children. Cognition, 131, 92–107. http://dx.doi.org/10.1016/j 506. http://dx.doi.org/10.1080/02643290244000239 .cognition.2013.12.007 De Smedt, B., Noël, M-P., Gilmore, C., & Ansari, D. (2013). How do Inglis, M., & Gilmore, C. (2013). Sampling from the mental number line: symbolic and non-symbolic numerical magnitude processing skills relate How are approximate number system representations formed? Cogni- to individual differences in children’s mathematical skills? A review of tion, 129, 63–69. http://dx.doi.org/10.1016/j.cognition.2013.06.003 evidence from brain and behavior. Trends in Neuroscience and Educa- Izard, V., Sann, C., Spelke, E. S., & Streri, A. (2009). Newborn infants tion, 2, 48–55. http://dx.doi.org/10.1016/j.tine.2013.06.001 perceive abstract numbers. Proceedings of the National Academy of Eger, E., Michel, V., Thirion, B., Amadon, A., Dehaene, S., & Klein- Sciences of the United States of America, 106, 10382–10385. http://dx schmidt, A. (2009). Deciphering cortical number coding from human .doi.org/10.1073/pnas.0812142106 brain activity patterns. Current Biology, 19, 1608–1615. http://dx.doi Kaufmann, L., Koppelstaetter, F., Delazer, M., Siedentopf, C., Rhomberg, .org/10.1016/j.cub.2009.08.047 P., Golaszewski, S.,...Ischebeck, A. (2005). Neural correlates of Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. distance and congruity effects in a numerical Stroop task: An event- Trends in Cognitive Sciences, 8, 307–314. http://dx.doi.org/10.1016/j related fMRI study. NeuroImage, 25, 888–898. http://dx.doi.org/10 .tics.2004.05.002 .1016/j.neuroimage.2004.12.041 Fias, W., Lammertyn, J., Reynvoet, B., Dupont, P., & Orban, G. A. (2003). Koechlin, E., Naccache, L., Block, E., & Dehaene, S. (1999). Primed Parietal representation of symbolic and nonsymbolic magnitude. Journal numbers: Exploring the modularity of numerical representations with of Cognitive Neuroscience, 15, 47–56. http://dx.doi.org/10.1162/ masked and unmasked semantic priming. Journal of Experimental Psy- 089892903321107819 chology: Human Perception and Performance, 25, 1882–1905. http:// Gabay, S., Leibovich, T., Henik, A., & Gronau, N. (2013). Size before dx.doi.org/10.1037/0096-1523.25.6.1882 numbers: Conceptual size primes numerical value. Cognition, 129, 18– Le Corre, M., & Carey, S. (2007). One, two, three, four, nothing more: An 23. http://dx.doi.org/10.1016/j.cognition.2013.06.001 investigation of the conceptual sources of the verbal counting principles. Gallistel, C. R., & Gelman, R. (1992). Preverbal and verbal counting and Cognition, 105, 395–438. http://dx.doi.org/10.1016/j.cognition.2006.10.005 computation. Cognition, 44, 43–74. http://dx.doi.org/10.1016/0010- Leibovich, T., Ashkenazi, S., Rubinsten, O., & Henik, A. (2013). Com- 0277(92)90050-R parative judgments of symbolic and non-symbolic stimuli yield different Garner, W. R., & Morton, J. (1969). Perceptual independence: Definitions, patterns of reaction times. Acta Psychologica, 144, 308–315. http://dx models, and experimental paradigms. Psychological Bulletin, 72, 233– .doi.org/10.1016/j.actpsy.2013.07.010 259. http://dx.doi.org/10.1037/h0028024 Leibovich, T., Diesendruck, L., Rubinsten, O., & Henik, A. (2013). The Gebuis, T., & Reynvoet, B. (2012). The interplay between nonsymbolic number importance of being relevant: Modulation of magnitude representations. and its continuous visual properties. Journal of Experimental Psychology: Frontiers in Psychology, 4(June), 369. General, 141, 642–648. http://dx.doi.org/10.1037/a0026218 Leibovich, T., & Henik, A. (2014). Comparing performance in discrete and Gebuis, T., & Reynvoet, B. (2013). The neural mechanisms underlying continuous comparison tasks. Quarterly Journal of Experimental Psychology, passive and active processing of numerosity. NeuroImage, 70, 301–307. 67, 899–917. http://dx.doi.org/10.1080/17470218.2013.837940 http://dx.doi.org/10.1016/j.neuroimage.2012.12.048 Leibovich, T., Henik, A., & Salti, M. (in press). Numerosity Processing is Gilmore, C., Attridge, N., Clayton, S., Cragg, L., Johnson, S., Marlow, N.,... Context Driven Even in the Subitizing Range: An fMRI Study. Neuropsy- Inglis, M. (2013). Individual differences in inhibitory control, not non-verbal chologia. number acuity, correlate with mathematics achievement. PLoS ONE, 8, Leibovich, T., Vogel, S. E., Henik, A., & Ansari, D. (in press). Asymmetric e67374. http://dx.doi.org/10.1371/journal.pone.0067374 processing of numerical and non-numerical magnitudes in the brain: an Halberda, J., Mazzocco, M. M., & Feigenson, L. (2008). Individual dif- fMRI study. Journal of Cognitive Neuroscience. ferences in non-verbal number acuity correlate with math’s achievement. Leibovich, T., & Henik, A. (2013). Magnitude processing in non-symbolic Nature, 455, 665–668. http://dx.doi.org/10.1038/nature07246 stimuli. Frontiers in Psychology, 4, 375. Harnad, S. (1990). The symbol grounding problem. Physica D. Nonlinear Phe- Leroux, G., Spiess, J., Zago, L., Rossi, S., Lubin, A., Turbelin, M.-R.,... nomena, 42, 335–346. http://dx.doi.org/10.1016/0167-2789(90)90087-6 Joliot, M. (2009). Adult brains don’t fully overcome biases that lead to Harnad, S. (2003). Symbol-grounding problem. In L. Nadel (Ed.), Ency- incorrect performance during cognitive development: An fMRI study in clopedia of cognitive science (Vol. 42, pp. 335–346). United Kingdom: young adults completing a Piaget-like task. Developmental Science, 12, AQ: 9 Nature Publishing Group. 326–338. http://dx.doi.org/10.1111/j.1467-7687.2008.00785.x Holloway, I. D., & Ansari, D. (2009). Mapping numerical magnitudes onto Libertus, M. E., & Brannon, E. M. (2009). Behavioral and neural basis of symbols: The numerical distance effect and individual differences in number sense in infancy. Current Directions in Psychological Science, children’s mathematics achievement. Journal of Experimental Child 18, 346–351. http://dx.doi.org/10.1111/j.1467-8721.2009.01665.x Psychology, 103, 17–29. http://dx.doi.org/10.1016/j.jecp.2008.04.001 Lyons, I. M., Ansari, D., & Beilock, S. L. (2012). Symbolic estrangement: Holloway, I. D., & Ansari, D. (2010). Developmental specialization in the Evidence against a strong association between numerical symbols and right intraparietal sulcus for the abstract representation of numerical the quantities they represent. Journal of Experimental Psychology: Gen- magnitude. Journal of Cognitive Neuroscience, 22, 2627–2637. http:// eral, 141, 635–641. http://dx.doi.org/10.1037/a0027248 dx.doi.org/10.1162/jocn.2009.21399 Lyons, I. M., Ansari, D., & Beilock, S. L. (2014). Qualitatively different coding AQ: 11 Holloway, I. D., Price, G. R., & Ansari, D. (2010). Common and segre- of symbolic and nonsymbolic numbers in the human brain. Human Brain gated neural pathways for the processing of symbolic and nonsymbolic Mapping, 36, 475–488. http://dx.doi.org/10.1002/hbm.22641 APA NLM tapraid5/cae-cae/cae-cae/cae00415/cae0277d15z xppws Sϭ1 10/1/15 15:45 Art: 2015-1231

12 LEIBOVICH AND ANSARI

Lyons, I. M., Nuerk, H-C., & Ansari, D. (2015). Rethinking the implica- Park, J., & Brannon, E. M. (2014). Improving arithmetic performance with tions of numerical ratio effects for understanding the development of number sense training: An investigation of underlying mechanism. Cog- representational precision and numerical processing across formats. nition, 133, 188–200. http://dx.doi.org/10.1016/j.cognition.2014.06.011 Journal of Experimental Psychology: General. Advance online publica- Park, J., DeWind, N. K., Woldorff, M. G., & Brannon, E. M. (2015). Rapid tion. http://dx.doi.org/10.1037/xge0000094 and direct encoding of numerosity in the visual Stream. Cerebral Cortex, Mazzocco, M. M. M., Feigenson, L., & Halberda, J. (2011). Impaired 17, 1–16. acuity of the approximate number system underlies mathematical learn- Piaget, J. (1952). Child’s conception of number: Selected works (Vol. 2). ing disability (dyscalculia). Child Development, 82, 1224–1237. http:// New York, NY: Routledge. dx.doi.org/10.1111/j.1467-8624.2011.01608.x Piazza, M. (2010). Neurocognitive start-up tools for symbolic number McComb, K., Packer, C., & Pusey, A. (1994). Roaring and numerical assessment representations. Trends in Cognitive Sciences, 14, 542–551. http://dx.doi in contests between groups of female lions, Panthera leo. Animal Behaviour, .org/10.1016/j.tics.2010.09.008 47, 379–387. http://dx.doi.org/10.1006/anbe.1994.1052 Piazza, M., Facoetti, A., Trussardi, A. N., Berteletti, I., Conte, S., Lucan- Meck, W. H., & Church, R. M. (1983). A mode control model of counting geli, D.,...Zorzi, M. (2010). Developmental trajectory of number and timing processes. Journal of Experimental Psychology: Animal acuity reveals a severe impairment in developmental dyscalculia. Cog- Behavior Processes, 9, 320–334. http://www.ncbi.nlm.nih.gov/pubmed/ nition, 116, 33–41. http://dx.doi.org/10.1016/j.cognition.2010.03.012 6886634. http://dx.doi.org/10.1037/0097-7403.9.3.320 Piazza, M., Pinel, P., Le Bihan, D., & Dehaene, S. (2007). A magnitude Merkley, R., & Scerif, G. (2015). Continuous visual properties of number code common to numerosities and number symbols in human intrapa- influence the formation of novel symbolic representations. Quarterly rietal cortex. Neuron, 53, 293–305. http://dx.doi.org/10.1016/j.neuron Journal of Experimental Psychology, 68, 1860–1870. http://doi.org/10. .2006.11.022 1080/17470218.2014.994538 Roggeman, C., Santens, S., Fias, W., & Verguts, T. (2011). Stages of Mix, K. S., Huttenlocher, J., & Levine, S. C. (2002). Multiple cues for nonsymbolic number processing in occipitoparietal cortex disentangled quantification in infancy: Is number one of them? Psychological Bulle- by fMRI adaptation. The Journal of Neuroscience, 31, 7168–7173. tin, 128, 278–294. http://dx.doi.org/10.1037/0033-2909.128.2.278 http://dx.doi.org/10.1523/JNEUROSCI.4503-10.2011 Morton, J. B., (2010). Understanding genetic, neurophysiological, and Sasanguie, D., Defever, E., Maertens, B., & Reynvoet, B. (2014). The experiential influences on the development of executive functioning: approximate number system is not predictive for symbolic number The need for developmental models. Wiley Interdisciplinary Reviews: processing in kindergarteners. Quarterly Journal of Experimental Psy- Cognitive Science, 1, 709–723. http://dx.doi.org/10.1002/wcs.87 chology, 67, 271–280. http://doi.org/10.1080/17470218.2013.803581 Moyer, R. S., & Landauer, T. K. (1967). Time required for judgements of Soltész, F., & Szu˝cs, D. (2014). Neural adaptation to non-symbolic number numerical inequality. Nature, 215, 1519–1520. http://dx.doi.org/10 and visual shape: An electrophysiological study. Biological Psychology, .1038/2151519a0 103, 203–211. http://dx.doi.org/10.1016/j.biopsycho.2014.09.006 Mussolin, C., Mejias, S., & Noël, M.-P. (2010). Symbolic and nonsymbolic Stoianov, I. P. (2014). Generative processing underlies the mutual enhancement number comparison in children with and without dyscalculia. Cognition, of arithmetic fluency and math-grounding number sense. Frontiers in Psy- 115, 10–25. http://dx.doi.org/10.1016/j.cognition.2009.10.006 chology, 5, 1326. http://dx.doi.org/10.3389/fpsyg.2014.01326 Mussolin, C., Nys, J., Content, A., & Leybaert, J. (2014). Symbolic number Szu˝cs, D., Nobes, A., Devine, A., Gabriel, F. C., & Gebuis, T. (2013). abilities predict later approximate number system acuity in preschool Visual stimulus parameters seriously compromise the measurement of children. PLoS ONE, 9, e91839. http://dx.doi.org/10.1371/journal.pone approximate number system acuity and comparative effects between .0091839 adults and children. Frontiers in Psychology, 4, 444. http://dx.doi.org/ Mussolin, C., Nys, J., Leybaert, J., & Content, A. (2012). Relationships 10.3389/fpsyg.2013.00444 between approximate number system acuity and early symbolic number Tokita, M., & Ishiguchi, A. (2013). Effects of perceptual variables on abilities. Trends in Neuroscience and Education, 1, 21–31. http://dx.doi numerosity comparison in 5–6-year-olds and adults. Frontiers in Psy- .org/10.1016/j.tine.2012.09.003 chology, 4, 431. http://dx.doi.org/10.3389/fpsyg.2013.00431 Negen, J., & Sarnecka, B. W. (2015). Is there really a link between Trick, L. M., & Pylyshyn, Z. W. (1994). Why are small and large numbers exact-number knowledge and approximate number system acuity in enumerated differently? A limited-capacity preattentive stage in vision. young children? British Journal of Developmental Psychology, 33, 92– Psychological Review, 101, 80–102. http://dx.doi.org/10.1037/0033- 105. http://dx.doi.org/10.1111/bjdp.12071 295X.101.1.80 Nieder, A. (2005). Counting on neurons: The neurobiology of numerical Tzelgov, J., Meyer, J., & Henik, A. (1992). Automatic and intentional competence. Nature Reviews Neuroscience, 6, 177–190. http://dx.doi processing of numerical information. Journal of Experimental Psychol- .org/10.1038/nrn1626 ogy: Learning, Memory, and Cognition, 18, 166–179. Retrieved from Nieder, A., Freedman, D. J., & Miller, E. K. (2002). Representation of the http://psycnet.apa.orgjournals/xlm/18/1/166. quantity of visual items in the primate prefrontal cortex. Science, 297, Verguts, T., & Fias, W. (2004). Representation of number in animals and 1708–1711. http://dx.doi.org/10.1126/science.1072493 humans: A neural model. Journal of Cognitive Neuroscience, 16, 1493– Nys, J., & Content, A. (2012). Judgement of discrete and continuous quantity in 1504. http://dx.doi.org/10.1162/0898929042568497 adults: Number counts! Quarterly Journal of Experimental Psychology, 65, Wagner, J. B., & Johnson, S. C. (2011). An association between understanding 675–690. http://dx.doi.org/10.1080/17470218.2011.619661 cardinality and analog magnitude representations in preschoolers. Cognition, Odic, D., Le Corre, M., & Halberda, J. (2015). Children’s mappings 119, 10–22. http://dx.doi.org/10.1016/j.cognition.2010.11.014 between number words and the approximate number system. Cognition, Watanabe, S. (1998). Discrimination of “four” and “two” by pigeons. The 138, 102–121. http://dx.doi.org/10.1016/j.cognition.2015.01.008 Psychological Record. Retrieved from http://opensiuc.lib.siu.edu/tpr/ Odic, D., Libertus, M. E., Feigenson, L., & Halberda, J. (2013). Develop- vol48/iss3/3 mental change in the acuity of approximate number and area represen- Xu, F., & Spelke, E. S. (2000). Large number discrimination in 6-month- tations. Developmental Psychology, 49, 1103–1112. Retrieved from old infants. Cognition, 74, B1–B11. http://dx.doi.org/10.1016/S0010- http://psycnet.apa.orgjournals/dev/49/6/1103. 0277(99)00066-9 Park, J., & Brannon, E. M. (2013). Training the approximate number system improves math proficiency. Psychological Science, 24, 2013– Received May 21, 2015 2019. http://dx.doi.org/10.1177/0956797613482944 Accepted August 6, 2015 Ⅲ

View publication stats