... The constraints on how human and animal minds process these different quantities are similar (Gallistel and Gelman, 1992). For example, all of these quantities show cognitive processing limitations that can be predicted by Weber’s law. Weber’s ... states that quantity discrimination is determined by the objective ratio between their values. This ratio-based psychological and neural signature of quantity processing indicates that many quantities are represented in an analog format, akin to the way in which a machine ... intensities in currents or voltages (Gallistel and Gelman, 1992). I discuss the types of constraints that influence quantity discrimination, using “number” as the initial example, and then ... the psychological and neural relationship between “number” and other quantitative dimensions. Similar constraints on processing across different quantities have been interpreted as evidence that they have a ... evolutionary and/or developmental origin and a common foundation in the mind and brain (Zorzi et al., 2002; Walsh, 2003; Pinel et al., 2004; Feigenson, 2007; Ansari, 2008; Cohen Kadosh et al., 2008; Cantlon et al., 2009c; de Hevia ... Spelke, 2009; Lourenco and Longo, 2011; Bonn and Cantlon, 2012). The resolution of these issues is important for understanding the inherent organization of our most basic conceptual faculties. The issue is also important for understanding how our formal mathematical ...

... Primitive quantitative abilities play a role in how modern humans learn culture-specific, formal mathematical concepts (Gallistel and Gelman, 1992). Preverbal children and nonhuman animals possess a primitive ability to appreciate quantities, such as the approximate number of ... in a set, without counting them verbally. Instead of counting, children and animals can mentally represent quantities approximately, in an analog format. Studies from our group and others have shown that human adults, ... , and nonhuman primates share cognitive algorithms for encoding numerical values as analogs, comparing numerical values, and arithmetic (Meck and Church, 1983; Gallistel, 1989; Feigenson et al., 2004; Cantlon et al., 2009c). Developmental studies indicate that these analog numerical ... interact with children’s developing symbolic knowledge of numbers and mathematics (Gelman and Gallistel, 1978; Feigenson et al., 2004). Furthermore, the brain regions recruited during approximate number representations are shared by adult ... , nonhuman primates, and young children who cannot yet count to 30 (Dehaene et al., 2003; Nieder, 2005; Ansari, 2008). Finally, it has recently been demonstrated that neural ... math IQ (Halberda et al., 2008). Taken together, current findings implicate continuity in the primitive numerical abilities that are shared by humans and nonhumans, as well as a degree of continuity in human numerical abilities ranging from primitive approximation to complex and sophisticated math....

... The fact that humans have been recording tallies with sticks and bones for 30,000 years is impressive, but the critical issue is this: what cognitive abilities enabled them to encode quantities in the first place? ... clues to the evolutionary precursors of numerical cognition by comparing human cognition with nonhuman primate cognition. The degree to which humans and nonhuman primates share numerical abilities is evidence that those abilities might derive from a common ancestor, in the same way that common ... like the presence of 10 fingers and toes in two different primate species points to a common morphological heritage....

... that nonhuman primates share three essential numerical processing mechanisms with modern humans: an ability to represent numerical values (Brannon and Terrace, 1998; Nieder, 2005; Cantlon and Brannon, 2006, 2007b), a general mechanism for mental comparison (Cantlon and Brannon, 2005), and arithmetic ... for performing addition and subtraction (Beran and Beran, 2004; Cantlon and Brannon, 2007a). These findings compliment and extend a long history of research on the numerical abilities of nonhuman animals [see Emmerton (2001) for review]....

... When adult humans and monkeys are given a task in which they have to rapidly compare two visual arrays and touch the array with the smaller numerical value (without counting the dots), their performance reliably yields the pattern shown in Fig. 16.1: ... decreases as the ratio between the numerical values in the two arrays approaches 1 [Cantlon and Brannon (2006); see Dehaene (1992) and Gallistel and Gelman (1992) for review]. The explanation of this performance pattern is that both groups are representing the numerical values in an analog format (...

... In an analog format, number is represented only approximately, and it is systematically noisy (Dehaene, 1992; Gallistel and Gelman, 1992]. More precisely, the probability of noise (i.e., the spread of the distributions) in the subjective representation of a number ...

... FIGURE 16.1 Accuracy on a numerical discrimination task for monkeys and humans plotted by the numerical ratio between the stimuli. From Cantlon and Brannon (2006)....

... compared. Two different pairs of numerical values that have the same ratio (e.g., 2 and 4, 4 and 8) have the same amount of overlap, or the same probability of confusion. As numerical pairs get larger and closer together, their ratio increases ... Fig. 16.1 [from Cantlon et al. (2009c)] represent predicted data from a model of number representation under Weber’s law (Pica et al., 2004), and they show that the predictions of this analog numerical model fit the data well....

... The empirical data from monkeys and humans and the fit of the analog model demonstrate that although humans have a means of representing numerical values precisely using words and Arabic numerals, ...

... of mental comparison in monkeys that is commonly observed when adult humans make judgments of magnitudes: the semantic congruity effect (Cantlon and Brannon, 2005; Holyoak, 1977). The semantic congruity effect is a response time effect that is observed in adult humans’ response times ... they have to compare things along a single dimension. For instance, when people are presented with pairs of animal names and asked to identify the larger or smaller animal from memory, they show a semantic congruity effect in their response time: people are faster to choose ... blue, but when the screen background was red, they had to choose the smaller numerical value of the two arrays. As shown in Fig. 16.3 [from Cantlon and Brannon (2007a)], both monkeys showed a crossover pattern of faster response times when choosing the smaller of two small values compared with the ... of two small values, and the opposite pattern for large values. The semantic congruity effect is the signature of a mental comparison process wherein context-dependent mental ... points are established (e.g., 1 for “choose smaller” and 9 for “choose larger”), and reaction time is determined by the distance of the test items from the reference points; this has been modeled as the time it takes for evidence to ...

... FIGURE 16.3 The semantic congruity effect in the response times of two different monkeys (Feinstein and Mikulski) on a numerical comparison task where they sometimes chose the larger numerical value from two arrays (dark line) and other times chose the ... value (light line). The cross-over pattern reflects the effect of semantic congruity. From Cantlon and Brannon (2005)....

... of mental comparisons from both perceptual and conceptual stimuli: brightness, size, distance, temperature, ferocity, numerals, etc. Our data from nonhuman primates indicate that the mental ... process that yields the semantic congruity effect is a primitive, generalized, nonverbal mental comparison process for judging quantities and other one-dimensional properties....

... In fact, the ability to compare quantities, and the proposed algorithm underlying that ability, could be so primitive that it extends to nonprimate animals. A recent study by Scarf et al. (2011) ... that pigeons can compare numerical values, and in doing so they represent an abstract numerical rule that can be applied to novel numerical values. Pigeons’ accuracy on that ordinal ... task is comparable to that of monkeys tested on an identical task (Brannon and Terrace, 1998)....

.... We have found that monkeys possess a capacity for basic, nonverbal addition that parallels human nonverbal arithmetic in a few key ways (Cantlon and Brannon, 2007a). First, monkeys and humans show a ratio effect when performing rapid nonverbal addition, similar to the ratio effect described ... . Monkeys’ and humans’ accuracy during arithmetic depends on the ratio between the values of the choice stimuli. We also observed a classic signature of human ...

... However, there are also important and potentially informative differences between the performance of humans and monkeys. Adult humans and young children show a practice effect in their arithmetic performance wherein performance on a specific problem improves the more that it is ... the case even over 3 years of practice on a specific problem (Fig. 16.4 shows performance for two monkeys, over 3 years of testing on 1 + 1, 2 + 2, and 4 + 4). Nonhuman primate arithmetic thus parallels human nonverbal arithmetic in the ratio and problem size effects but not the practice effect, ...

... The overarching conclusion from this line of research is that the abilities to represent, compare, and perform arithmetic computations reflect a cognitive system for numerical reasoning that is primitive and based on analog magnitude representations. ... , if analog numerical cognition is truly “primitive” and homologous across primate species, then it should be rooted in the same physical (neural) system in monkeys and...

... FIGURE 16.4 The lack of a practice effect in monkeys’ addition performance over 3 years. Data from Cantlon and Brannon (2007a)....

.... In fact, there is evidence from multiple sources that analog numerical processing recruits a common neural substrate in monkeys, adult humans, and young children (Fig. 16.5)....

... FIGURE 16.5 Monkeys, human adults, and human children exhibit similar activation in the IPS during analog numerical processing. Redrawn from Nieder and Miller (2004), Piazza et al. (2004) (reprinted with permission from Elsevier, Copyright 2004), and Cantlon et al. (2006)....

... numerical value and a deviant numerical value. Our group also observed neural adaptation in the IPS for numerical values ranging from 8 to 64 in preschool children who could not yet verbally count to 30 (Cantlon et al., 2006). Together, these studies reflect a common neural source for analog numerical ...

... that the analog system for numerical reasoning is primitive in human development. A fundamental question is how a child’s developing understanding of numerical symbols interfaces with preverbal analog representations of number. Of particular interest is how children initially map numerical ... to the first few symbolic number words (Gelman and Gallistel, 1978; Wynn, 1990; Gelman and Butterworth, 2005; Le Corre and Carey, 2007; Piazza, 2010). There is currently a debate over the types of preverbal numerical representations that form the initial basis of children& ... suggests that as children learn words in the counting sequence, they map them to approximate, analog representations of number (Wynn, 1992; Lipton and Spelke, 2005; Gilmore et al., 2007). Lipton and Spelke (2005) found that 4-year-old children could look at a briefly presented array of 20 dots and, ... they could count to 20, they could verbally report (without counting) that there were 20 dots in the array, and their errors were systematically distributed around 20 (i.e., their errors exhibited a numerical ratio effect). If they could not yet count to 20, ... , they responded with random number labels. Thus, as soon as children learn a particular verbal count word in the sequence, they know the approximate quantity to which it ...

... to numerical symbols over human development. There is also evidence that children who have learned to count verbally, but have not yet learned to add and subtract, psychologically “piggyback” on analog arithmetic representations as they transition to an understanding of exact symbolic ... ). The general conclusion that then emerges is that the cognitive faculties that children initially use for nonsymbolic, analog numerical operations (and which they share with nonhuman animals) provide a scaffolding for verbal counting in early childhood....

... just as widespread. For instance, the abilities to judge nonnumerical intensities such as size, time, brightness, height, weight, velocity, pitch, and loudness are as common among animal species as the ability to judge numerical values. Furthermore, all of these quantities can be discriminated by ... infants, and discriminations among instances from those continua bear many of the same properties and signatures as numerical discrimination [e.g., ordinality, Weber’s law, the semantic congruity effect, arithmetic transformations; see Feigenson ... (2007) for review]. In adults, all of these dimensions are effortlessly mapped to numerals. For example, adult humans can represent loudness, handgrip pressure, time, size, and brightness as numerical values. Finally, evidence from the semantic congruity effect (described earlier) suggests that ... different quantitative dimensions are mentally compared by a common process. The modularity and taxonomy of analog numerical representations is a central issue for understanding the development and origins of numerical and mathematical cognition. ... Here I discuss relations between numerical cognition and other quantitative dimensions, such as size, length, duration, brightness, pitch, and loudness....

... Until recently, the cognitive and neural mechanisms of numerical cognition were considered to be specialized processes. Neuropsychological and neuroimaging studies of adult humans have shown that numerical knowledge dissociates from other forms of semantic knowledge, and it has been argued ...

... (2003) for review]. For example, individuals with semantic dementia, resulting from left temporal lobe atrophy, exhibit severe impairments on picture and word naming tasks but can be spared for number tasks (Cappelletti et al., 2001). The opposite disorder of impaired numerical cognition but spared ... and linguistic knowledge has also been demonstrated (Warrington, 1982; Cipolotti et al., 1991). Moreover, in cases of developmental dyscalculia, ... over development (without impairments to other aspects of reasoning). Furthermore, developmental dyscalculia is coupled with atypical anatomy and functional responses in the IPS (Molko et al., 2003; Price et al., 2007). The fact that focal brain injuries and developmental impairments, perhaps ... to the IPS, specifically impair numerical reasoning indicates that at some level of cognitive and neural processing, numerical computation is independent. However, it remains unclear what aspects of numerical processing operate independently of ... psychophysical and conceptual domains. Most previous neuropsychological and neuroimaging studies controlled for many nonnumerical abilities (eye movements, spatial attention, memory, semantic knowledge), but they did not test ...

... Recently, Marco Zorzi et al. (2002) found that representations of spatial and numerical continua can be jointly impaired in patients with right parietal lesions and hemispatial neglect; patients not only neglect the left visual ... and place the midpoint of a line right of center in a line bisection task, but they also overestimate the middle value of two numbers in a numerical ... task. The patients thus neglect both the left side of a line and the left side of their mental representation of the numerical continuum. This finding and several others have led to proposals that concepts of & ... ;space” and “number” are interrelated (Walsh, 2003; Pinel et al., 2004)....

... and Spelke, 2009, 2010). In line-bisection tasks, incidental displays of dot arrays presented at the endpoints of the line systematically distort preschoolers’ perception of the line’s midpoint; subjects bisect the line asymmetrically toward the larger number of dots (de Hevia and Spelke, ...

... correlated number/line-length pairs (de Hevia and Spelke, 2010). The fact that infants map number onto space within the first months of life has been used to argue for an innate bias to relate space ...

... Biologically privileged relations between space and number are also indicated by the universality of their association (Dehaene et al., 2008). The ability to map numbers onto space (number lines) is ... value onto horizontal lines in numerical order (just as Western subjects do). That finding supports the conclusion that mapping between space and number is not culturally determined by reading and reciting numerical symbols, because Mundurucu do not generally use such symbols. However, this ... et al., 2006). If pitch shows the same kind of relation to space as number does, then a biologically “privileged” relation between space and number seems less likely. One possibility is that the relationship is ubiquitous among any of a number of dimensions (e.g.., pitch, number, length, ... , etc.). Alternatively, number and space and pitch and space could be related because of a privileged representation of space alone, which grounds a number of quantitative representations....

... Several researchers have suggested deep psychological interactions not just between number and space but among many quantitative dimensions. In their review of behavioral data from humans and other animals, Gallistel and Gelman (2000) argued ... must combine discrete number with continuous quantities in making decisions. For example, they observed that animals need to combine estimated time and amount of potential food in making foraging decisions (i.e., for “rate”). Because natural numbers are discrete and time is continuous, ... conversion to a common analog format. The same argument could be applied to “density,” which integrates information about number and surface area. This idea implicates the possibility of common representations and shared computations for multiple quantities....

... the ratio effects for judgments of size, time, and number are refined at a similar rate of development (Brannon et al., 2006; vanMarle and Wynn, 2006; Feigenson, 2007). Infants’ discriminations of size, time, and number improve by approximately 30% between 6 and 9 mo of age. ... , in children, the precision of numerical discrimination improves from ages 6 to 8 years, and the discrimination of luminance, duration, and length systematically follow the same developmental trajectory (Holloway and Ansari, 2008; Volet et al., 2008). Because they develop at the same rate, ... . The developmental trajectories of the discrimination of other quantities, such as loudness, pitch, pressure, temperature, density, motion, and saturation, have not been tested. However, there is evidence that young children and even infants can form compatible representations across many of ... different dimensions (Smith and Sera, 1992; Gentner and Medina, 1998; Mondloch and Maurer, 2004; Walker et al., 2010)....

... As mentioned earlier, the dimensions of space and number can be related to one another already in infancy (de Hevia and Spelke, 2010). One recent study showed that 9-mo-olds were equally likely to transfer an arbitrary, experimentally learned magnitude-to-texture ... from one dimension (e.g., number) to another dimension (size or duration) (Lourenco and Longo, 2010). In addition, 9-mo-olds can readily learn pairs of positively (but not negatively) correlated line lengths and tone durations ( ... and Carey, 2010), suggesting that infants at least can represent an abstract “more-than” and “less-than” representation that applies to both dimensions. However, 9-mo-old infants do not show equal sensitivity to monotonic pairings ... the dimensions of loudness and space as they do for pairing of space and time (Srinivasan and Carey, 2010). Those findings suggest that there may be an asymmetry between magnitudes in their intrinsic ordinal associations. It is important to ... , however, that asymmetries in relations between magnitudes could arise either through a biologically privileged psychological mapping (de Hevia and Spelke, 2009) or through correlational and statistical learning [see Bonn and Cantlon (2012) for discussion]....

... the pitch of the auditory stimulus gets higher than the reverse (Walker et al., 2010). Infants are thus capable of aligning the dimensions of pitch and space (height) as well as pitch and shape (sharpness) early in development. Similarly, 3-year-olds reliably...

... match high-pitched sounds to smaller and brighter balls in a categorization task (Mondloch and Maurer, 2004). Those data show that magnitude dimensions beyond the canonical “privileged” dimensions of space and number can be mapped ...

... Relations among different quantities also have been found at the neural level in adult humans and nonhuman primates. As mentioned above, individuals with spatial neglect resulting from damage to parietal cortex can exhibit impaired numerical ... . Single-neuron data from neurophysiology studies of monkeys broadly indicate that regions of parietal cortex represent space, time, and number (Tudusciuc and Nieder, 2007). Moreover, some data even suggest that a single parietal neuron can represent more than one type of magnitude. In ... study (Tudusciuc and Nieder, 2007), monkeys were trained to perform a line-length matching task and a numerical matching task. During stimulus presentation as well as a subsequent delay, single neurons in the IPS responded selectively to visual ... according to their numerosity or length. Although some neurons responded only to numerosity and others only to line length, a subset of cells (~20%) responded to both magnitudes of line length and numerical value. These and other studies, ... to argue for a “distributed but overlapping” representation of different magnitudes at the neural level (Pinel et al., 2004; Tudusciuc and Nieder, 2007; Cantlon et al., 2009c). Simply put, different types of magnitude representation, including size, number, and time (and possibly others ... resources in parietal cortex but not others. The next section discusses some possible explanations of the origin of the relationship between number and other quantitative dimensions....

... HOW IS NUMBER LINKED TO OTHER QUANTITIES?...

... How do different quantitative dimensions become related in the mind and brain in the first place? We have recently reviewed existing theoretical frameworks for how quantitative relations might originate (Bonn and Cantlon, ... ). Here, I briefly sketch five mechanisms for how different quantities could become related in the mind. These hypotheses are not mutually exclusive and may even be complementary....

... Correlational and Statistical Associations...

... Learning via association and correlation is the classic developmental account of the origins of abstract percepts and concepts [e.g., Piaget (1952)]. On this view, integrated representations of information coming from separate senses, modalities, or cognitive domains ...

... would arise from the strength of their correlations in the natural environment. For example, it takes a long time to walk a great distance (time and space are correlated), and a large number of a particular object tends to take up more surface area than a small number of that object (number and ...

... that conceptual alignment of relational information, termed “structural similarity,” mediates mapping among magnitude dimensions (Gentner and Medina, 1998). On this view, cross-dimensional mapping could be a form of analogy. Relations between magnitudes could develop through conceptual ... of how those dimensions are structured (Srinivasan and Carey, 2010). For example, knowledge of the conceptual fact that time and number are ordinal and monotonic dimensions (they are organized from small/short to large/long) could serve as the cognitive basis for identifying relations among those ...

... A third conceptual framework that could be useful for understanding relations among magnitudes derives from the literature on cross-modal sensory perception. Gibson (1969) argued that an abstract, amodal ... can take one of two forms: (i) intersensory redundancy (e.g., timing information about hammer strikes can be sampled from both the auditory and visual modalities), and (ii) relative intensity [e.g., “sharpness, bluntness, and jerkiness”; Gibson (1969, p. 219)]. Under a ...

... hypothesis is suggested by evidence that infants experience something akin to synesthesia of sensory representations near birth [reviewed in Spector and Maurer (2009)]. A strong version of this hypothesis claims that a percept experienced in one modality automatically...

... imply that patterns of associations (mappings) between many magnitudes are initially strong in infancy, then get weaker during the first year(s), and then return to a strong state later in development. Generally speaking, the developmental data from cross-modal perception indicate that patterns of ...

... that unfold within an individual lifespan. On this view, one quantitative dimension evolved from another, inheriting functional similarities and potentially mutual dependencies in neural and computational operations. For example, many magnitude representations could have emerged from descent ... modification of the functional substrates that code for space, resulting in a common psychological and neural code for dimensions such as space, number, time, loudness, brightness, and pitch (Bonn and Cantlon, 2012)....

... Clearly there is a dense set of possibilities for how different quantities could come to be related in the mind and brain. The five hypotheses sketched above address different levels of influence ranging from ontogeny to phylogeny. They also address different ... of psychological functioning ranging from basic representations of psychophysical values to abstract perceptual and conceptual relations. Different levels of analysis will be important for understanding the full taxonomy of numerical cognition in humans. However, ...

... A further issue central to understanding the taxonomy of primitive numerical cognition is the extent to which analog numerical abilities bear...

... may be modulated by developments in the “primitive” analog numerical system that is shared by nonhuman primates, adult humans, and children. These studies have largely hinged on analyses of individual differences in numerical and mathematical abilities....

... Individual differences in math IQ are predicted by differences in analog numerical sensitivity (Bull and Scerif, 2001; Halberda et al., 2008; Holloway and Ansari, 2009). Studies with children indicate that analog numerical ability correlates with ... on math IQ tests and that formal math ability is more closely correlated with analog numerical abilities than it is with other formal abilities, such as reading. For ... Weber Fraction) correlates with their math IQ from early childhood [measured by the Test of Early Mathematics Ability (TEMA)-2 test score]. This and similar findings indicate that the “primitive” ability to estimate numerical values from sets of objects is related to the development of ... -blown math skills. Other studies highlight the role of executive function and working memory in the development of formal mathematical reasoning (Bull and Scerif, 2001; Mazzocco et al., 2006; Mazzocco and Kover, 2007). Together, ... these studies indicate a need to understand the relative contributions of domain-specific and domain-general processes to formal mathematical skill....

... Behavioral data, like those described earlier, provide evidence of a relationship between the skills required for analog numerical processing and those that are used in formal mathematics by children. Neuroimaging studies of children can provide an independent source of data on whether there is ... common foundation for analog numerical abilities and formal math by testing whether a common neural substrate underlies both faculties. As described above, analog quantity judgments recruit regions of...

... the IPS in adult humans, human children, and nonhuman primates. One issue is whether the same neural patterns that are evoked during analog numerical processing are observed when children and ... ). Several studies suggest that they do: regions of the IPS exhibit activity that is greater for numerical symbols compared with control stimuli, and those IPS regions also exhibit the numerical distance and ratio effects in their neural responses (Cohen Kadosh et al., 2007; Piazza et al., 2007; ... , 2008; Cantlon et al., 2009b; Holloway and Ansari, 2010). Research further suggests that the same neural response patterns are elicited for both symbolic and nonsymbolic (analog) numbers in ... same subjects (Piazza et al., 2007). Together, these results implicate neural overlap in the substrates underlying symbolic and nonsymbolic (analog) numerical representations in humans....

... cortex, particularly the inferior frontal gyrus, bordering insular cortex (Ansari et al., 2005; Piazza et al., 2007; Cantlon et al., 2009b; Emerson and Cantlon, 2012). Structurally, the prefrontal cortex is thought to be unique in primates compared with other mammals (Preuss, 2007). In humans the ... cortex responds during many types of abstract judgments (Miller et al., 2002), and several studies have noted a unique involvement of the prefrontal cortex in the development of semantic representations, symbols, and rules [see ... (2009) for review]. A pattern of greater activation of prefrontal sites in children compared with adults has also been observed for numerical and basic mathematical tasks (Ansari et al., 2005; Rivera et al., 2005; Cantlon et al., 2009b). The role of prefrontal cortex in children’s ... numerical processing is related to performance factors such as response time, or “time on task” [Emerson and Cantlon (2012); see also Schlaggar et al. (2002)], which could reflect the nascent state of children’s abstract, symbolic numerical ... . Studies with nonhuman primates have suggested that they too engage prefrontal cortex during numerical processing [see Nieder (2009) for review] and that prefrontal regions play a unique role in associating analog numerical values with arbitrary symbols at the level of single neurons in monkeys ( ... and Nieder, 2007)....

... Findings that highlight mutual involvement of the IPS and prefrontal cortex in basic numerical tasks have led to the hypothesis that interactions between frontal and parietal regions are important for the ... cognition, such as symbolic coding. Specifically, it has been proposed that the IPS computes “primitive” analog numerical representations and the prefrontal cortex facilitates links between those analog numerical computations and symbolic number representations in humans (Cantlon et al., ...

... hypothesis is correct then network-level neural synchrony between frontal and parietal regions should predict formal mathematics development in humans. That is, individual variability in the strength of correlations between ... responses in frontal and parietal regions, or “functional connectivity,” should be related to individual variability in mathematics performance. We have recently ... this hypothesis and found that number-specific functional connectivity of the fronto-parietal network does predict children’s math IQ test scores (independently of ... verbal IQ test scores) (Emerson and Cantlon, 2012). The implication is that number-specific changes in the interactions between frontal and parietal regions are related to the development of symbolic, formal math concepts in children. This general conclusion is in line with the hypothesis ... interactions between the “primitive” numerical operations of the IPS and the abstract, symbolic operations of frontal cortex give rise to formal mathematics concepts in humans....

... The goal of this review has been to examine the origins and organization of numerical abilities ranging from analog quantification to formal arithmetic. The general hypothesis is that the uniquely human ... to perform complex and sophisticated mathematics can be traced back to a simpler computational system that is shared among many animals: the analog numerical system. Humans ... interact with the uniquely human ability to represent numerical values symbolically, suggesting a relationship between “primitive” and modern numerical systems in humans. Data from neural analyses of numerical processing support this conclusion and provide independent confirmation ... these are in fact related systems. Questions remain regarding the precise taxonomy of the development and organization of numerical information, and its relationship to other domains, such as “space.” However, the general nature of the relationship between “primitive” and ... numbers seems to derive from evolutionary constraints on the structure of numerical concepts in the mind and brain as well as the conceptual and neural foundation that evolution has provided for the development of numerical thinking in humans....

... I thank Brad Mahon and Vy Vo for comments. Support was received from National Institute of Child Health and Human Development Grant R01HD064636 and the James S. McDonnell Foundation....

... A Hierarchical Model of the Evolution of Human Brain Specializations...

..., in part because of disputes about the form such adaptations might take. Many psychologists assume that adaptations come in two kinds, specialized and general-purpose. Specialized mechanisms are typically thought of as innate, domain-specific, and isolated from other brain systems, whereas ... mechanisms are developmentally plastic, domain-general, and interactive. However, if brain mechanisms evolve through processes of descent with modification, they are likely to be heterogeneous, rather than ... in just two kinds. They are likely to be hierarchically organized, with some design features widely shared across brain systems and others specific to particular processes. Also, they are likely to be largely developmentally plastic and interactive with other brain systems, rather ... canalized and isolated. This chapter presents a hierarchical model of brain specialization, reviewing evidence for the model from evolutionary developmental ... , genetics, brain mapping, and comparative studies. Implications for the search for uniquely human traits are discussed, along with ways in which conventional views of modularity ...

... Center for Behavior, Evolution, and Culture and FPR-UCLA Center for Culture, Brain and Development, Department of Anthropology, University of California, Los Angeles, CA 90095. E-mail: barrett@anthro.ucla.edu....