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16
Math, Monkeys, and
the Developing Brain
JESSICA F. CANTLON
Thirty thousand years ago, humans kept track of numerical quantities by
carving slashes on fragments of bone. It took approximately 25,000 years
for the first iconic written numerals to emerge among human cultures
(e.g., Sumerian cuneiform). Now, children acquire the meanings of verbal
counting words, Arabic numerals, written number words, and the proce-
dures of basic arithmetic operations, such as addition and subtraction, in
just 6 years (between ages 2 and 8). What cognitive abilities enabled our
ancestors to record tallies in the first place? Additionally, what cognitive
abilities allow children to rapidly acquire the formal mathematics knowl-
edge that took our ancestors many millennia to invent? Current research
aims to discover the origins and organization of numerical information
in humans using clues from child development, the organization of the
human brain, and animal cognition.
T
his review traces the origins of numerical processing from “primi-
tive” quantitative abilities to math intelligence quotient (IQ). “Prim-
itive” quantitative abilities are those that many animals use to
estimate the value of an object or event, for instance its distance, length,
duration, number, amplitude, saturation, or luminance (among others).
Department of Brain and Cognitive Sciences, University of Rochester, Rochester, NY 14627.
E-mail: jcantlon@bcs.rochester.edu.
293
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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 law 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
represents 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 consider the psychological
and neural relationship between “number” and other quantitative dimen-
sions. Similar constraints on processing across different quantities have
been interpreted as evidence that they have a common 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 and 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 under-
standing how our formal mathematical abilities originated.
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 abil-
ity to appreciate quantities, such as the approximate number of objects 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,
children, 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
representations 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 humans, 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 regions involved in analog numerical processing are related to the
development of 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 com-
plex and sophisticated math.
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Math, Monkeys, and the Developing Brain / 295
OLDEST NUMBERS IN THE WORLD
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? To
identify the inherent constraints on humans’ ability to process numerical
information, it is helpful to consider the evolutionary history of numerical
thought. We can look for clues to the evolutionary precursors of numerical
cognition by comparing human cognition with nonhuman primate cogni-
tion. The degree to which humans and nonhuman primates share numeri-
cal abilities is evidence that those abilities might derive from a common
ancestor, in the same way that common morphology like the presence of
10 fingers and toes in two different primate species points to a common
morphological heritage.
So far, there is evidence that nonhuman primates share three essential
numerical processing mechanisms with modern humans: an ability to rep-
resent 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 algorithms 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].
Representation
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: accuracy 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 (Fig. 16.2).
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 increases with the objective
number of items that are coded by that representation. Consequently, the
probability of confusion (i.e., the overlap between distributions) between
any two objective numbers increases as their value increases. This means
that the probability of having an accurate subjective representation of a
numerical value decreases with its objective value. This relationship can
be succinctly quantified by the ratio between the numerical values being
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FIGURE 16.1 Accuracy on a nu-
merical 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 and so does the probability that they will be confused
(leading to more errors). For example, one might be 80% accurate at choos-
ing the larger number when the numerical choices are 45 vs. 70 (45/70 =
a 0.64 ratio) but might perform at chance when the choices are 45 vs. 50
(45/50 = a 0.9 ratio). This effect is known as Weber’s law. The curves in 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.
FIGURE 16.2 An analog represen-
tation of numerical value repre-
sents an objective numerical value
with a probability distribution that
scales with the size of the objective
numerical value. From Cantlon et
al. (2009a). Reprinted with per-
mission from the American As-
sociation for the Advancement of
Science.
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Math, Monkeys, and the Developing Brain / 297
The empirical data from monkeys and humans and the fit of the ana-
log model demonstrate that although humans have a means of represent-
ing numerical values precisely using words and Arabic numerals, they still
have an approximate, analog numerical system that functions essentially
in the same way as in monkeys.
Comparison
The ratio effect, described by Weber’s law, indicates that numerical
values can be represented in an analog format. However, that does not
tell us anything about the process by which two numerical values are
compared. We have identified a signature of mental comparison in mon-
keys 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 whenever 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 the smaller of a small pair
of items (e.g., ant vs. rat) than they are to choose the larger of that pair.
However, for pairs of large items (e.g., horse vs. cow), people are faster
to choose the larger item than the smaller item. This effect suggests that
the physical size of the animal interacts with the “size” of the question
(whether “Which is larger?” or “Which is smaller?”) in subjects’ judg-
ments. In humans, the semantic congruity effect is observed for judgments
of many dimensions, including judgments of numerical values, from Ara-
bic numerals. We found that this effect is also observed in monkeys when
they compare numerical values from arrays of dots. Monkeys performed
a task in which they had to choose the larger numerical value from two
visual arrays when the background color of the computer screen was
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 larger 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 reference 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 accrue in the comparison of each item to the reference point (Holyoak,
1977). In humans the semantic congruity effect is observed for a variety
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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 smaller 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 comparison process
that yields the semantic congruity effect is a primitive, generalized, non-
verbal mental comparison process for judging quantities and other one-
dimensional properties.
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In fact, the ability to compare quantities, and the proposed algorithm
underlying that ability, could be so primitive that it extends to nonpri-
mate animals. A recent study by Scarf et al. (2011) showed 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 numerical task is comparable to that of monkeys
tested on an identical task (Brannon and Terrace, 1998).
Arithmetic
Arithmetic is the ability to mentally combine values together to create
a new value without having directly observed that new value. 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
earlier. 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 arithmetic in monkeys’ performance: the problem
size effect. Adult humans typically exhibit a problem size effect wherein
performance worsens as the problem outcome value increases (Campbell,
2005). Like humans, monkeys exhibited a problem size effect in their addi-
tion accuracy (even when controlling for the ratio effect).
However, there are also important and potentially informative differ-
ences 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
practiced (Campbell, 2005). Monkeys do not show a practice effect for
specific problems. This was 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, which has been observed primarily in
symbolic arithmetic performance in humans. Presumably, discrete sym-
bols are necessary for humans to encode arithmetic problems in a format
that is amenable to memorization, which is why monkeys do not show a
practice effect.
The overarching conclusion from this line of research is that the abili-
ties to represent, compare, and perform arithmetic computations reflect a
cognitive system for numerical reasoning that is primitive and based on
analog magnitude representations. However, if analog numerical cogni-
tion is truly “primitive” and homologous across primate species, then it
should be rooted in the same physical (neural) system in monkeys and
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No Practice Effect
100%
90%
80%
Accuracy
70%
60%
Boxer
50%
Feinstein
FIGURE 16.4 The lack of a practice 40%
May-06
May-07
May-08
Jan-06
Jan-07
Jan-08
Jan-09
Sep-07
Sep-06
Sep-08
effect in monkeys’ addition per-
formance over 3 years. Data from
Cantlon and Brannon (2007a). Timepoint
humans. 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).
In monkeys who are trained to match visual arrays of dots accord-
ing to number, single neurons along the intraparietal sulcus (IPS) will
FIGURE 16.5 Monkeys, human adults, and human children exhibit similar ac-
tivation 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).
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Math, Monkeys, and the Developing Brain / 301
respond maximally to a preferred numerical value, and their firing rate
decreases as the number that is presented gets numerically farther from
that preferred value (Nieder and Miller, 2004). This neural firing pattern
has been linked to the behavioral ratio effect and is thought to reflect ana-
log numerical tuning in the IPS. A similar pattern of numerical tuning has
been observed with functional MRI in the human IPS. Manuela Piazza et
al. (2004) found a neural adaptation effect for numerical values in the IPS
that depended on the ratio between the adapted 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 rep-
resentation that bridges species as well as stages of human development
and is thus independent of language and formal mathematics experience.
These neural data support the conclusion derived from the behavioral data
that there is continuity between humans and nonhuman animals in the
mechanisms underlying analog numerical representations.
THEN THERE WERE SYMBOLS
A long history of studies with preverbal human infants has shown
that they too possess an ability to quantify objects with approximate,
analog representations (Feigenson et al., 2004). Thus, there is general
agreement that the analog system for numerical reasoning is primitive in
human development. A fundamental question is how a child’s develop-
ing understanding of numerical symbols interfaces with preverbal analog
representations of number. Of particular interest is how children initially
map numerical meanings 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 chil-
dren’s verbal counting. However, regardless of how this initial mapping
transpires, behavioral evidence suggests that as children learn words in
the counting sequence, they map them to approximate, analog represen-
tations 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, if 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,
however, 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 corresponds without counting, sug-
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gesting that number words are attached to the analog numerical code
as soon as they are learned. These data have been taken to indicate that
analog numerical representations are used to assign semantic meanings
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
arithmetic (Gilmore et al., 2007). 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 ani-
mals) provide a scaffolding for verbal counting in early childhood.
IS “NUMBER” ALONE?
The data from the development of counting in early childhood make
the case that a primitive numerical system is conceptually transformed
into a system for symbolic numbers. However, how do we know that ana-
log numerical representations are the sole precursors of formal, symbolic
numerical cognition? Currently, we do not. Although numerical reasoning
seems to be primitive in the sense that it is shared among primate spe-
cies, other quantitative abilities are 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 human 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 many 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 cog-
nition 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 from those data that the processes subserving numerical
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Math, Monkeys, and the Developing Brain / 303
knowledge are domain specific [see Dehaene et al. (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 semantic
and linguistic knowledge has also been demonstrated (Warrington, 1982;
Cipolotti et al., 1991). Moreover, in cases of developmental dyscalculia,
mathematical reasoning can become selectively impaired over develop-
ment (without impairments to other aspects of reasoning). Furthermore,
developmental dyscalculia is coupled with atypical anatomy and func-
tional responses in the IPS (Molko et al., 2003; Price et al., 2007). The fact
that focal brain injuries and developmental impairments, perhaps espe-
cially 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 other psychophysical and conceptual
domains. Most previous neuropsychological and neuroimaging studies
controlled for many nonnumerical abilities (eye movements, spatial atten-
tion, memory, semantic knowledge), but they did not test performance on
continuous dimensions other than number (length, area, brightness, etc.).
Thus we cannot know whether other quantitative abilities were simultane-
ously impaired in many of those neuropsychological patients.
Recently, Marco Zorzi et al. (2002) found that representations of spatial
and numerical continua can be jointly impaired in patients with right pari-
etal lesions and hemispatial neglect; patients not only neglect the left visual
field 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
bisection 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).
The degree to which “space” (e.g., size, height, or length) interacts
with numerical information is currently being investigated with a range of
methods [see Walsh (2003), Cantlon et al. (2009c), and Lourenco and Longo
(2011) for reviews]. One view is that space and number have a biologically
privileged psychological relationship (Dehaene et al., 2008; de Hevia and
Spelke, 2009, 2010). Evidence for this view comes from developmental
studies of number and space representation (de Hevia 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, 2009). In addition, infants
spontaneously map number onto space when habituated to positively
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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 and number.
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 widespread among
human cultures. The Mundurucu, an Amazonian people who lack a rich
linguistic system for discrete number words or symbols, can place sets of
objects that vary in numerical 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 gen-
erally use such symbols. However, this finding does not necessarily indi-
cate the presence of an innate bias to map numbers to space in humans,
but may represent an analogical relation between the ordinal properties
of the stimuli or the primacy of “space” alone (Cantlon et al., 2009a). In
support of those alternatives, there is evidence that a similar mapping to
space is made with representations of pitch in typical adults from Western
cultures (Rusconi 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, loudness, 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 dimen-
sions. In their review of behavioral data from humans and other animals,
Gallistel and Gelman (2000) argued that although number is objectively
a discrete property, it should be represented with an analog magnitude
code. They argued that animals must combine discrete number with con-
tinuous 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, combining information from these incom-
patible formats necessitates conversion to a common analog format. The
same argument could be applied to “density,” which integrates informa-
tion about number and surface area. This idea implicates the possibility of
common representations and shared computations for multiple quantities.
Studies in young children provide evidence that different quantitative
representations have a common foundation, in the sense that they develop
together. As described earlier, numerical discriminations are modulated
by the ratio between the values, as per Weber’s law. In human infants,
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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. Similarly, 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,
it is likely that either the same mechanism underlies the different abili-
ties or that different mechanisms are subject to the same constraints. 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 these
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 association 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 (Srinivasan 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 between 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 note,
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].
Perhaps the best evidence for early-developing psychological relations
among quantities is that infants at 4 mo of age spontaneously prefer to
look at a ball that is bouncing congruently with the pitch of an auditory
stimulus (the ball goes up when the pitch goes up) compared with a ball
that is bouncing incongruently with pitch. In addition, they prefer to look
at a shape that is getting sharper as 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
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match high-pitched sounds to smaller and brighter balls in a categoriza-
tion task (Mondloch and Maurer, 2004). Those data show that magnitude
dimensions beyond the canonical “privileged” dimensions of space and
number can be mapped onto each other early in development.
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 processing. 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 one 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 stimuli 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, including functional MRI stud-
ies of adults, have led some researchers 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). Sim-
ply put, different types of magnitude representation, including size, num-
ber, and time (and possibly others such as brightness), share some neural
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 theoreti-
cal frameworks for how quantitative relations might originate (Bonn and
Cantlon, 2012). 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 developmen-
tal 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 arise from expo-
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sure to correlations in the environment. Under this account, relations
among magnitudes 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 space are correlated). In this way, empirical cor-
relations between different quantities can be absorbed through experience.
Analogical Reasoning
Another possibility is that conceptual alignment of relational infor-
mation, termed “structural similarity,” mediates mapping among mag-
nitude dimensions (Gentner and Medina, 1998). On this view, cross-
dimensional mapping could be a form of analogy. Relations between
magnitudes could develop through conceptual knowledge 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 dimensions.
Amodal Representations
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 rep-
resentation of intensity or amount of stimulation is present from birth
or very early in infancy. On her view, amodal representations 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 conceptualization of magnitude
representation within this framework, redundancy of information would
be the main source of representational overlap. For example, a bright
light could be mapped to a loud tone because they both evoke an amodal
representation of relatively high intensity.
Automatic Cross-Activation
A fourth hypothesis is suggested by evidence that infants experi-
ence something akin to synesthesia of sensory representations near birth
[reviewed in Spector and Maurer (2009)]. A strong version of this hypoth-
esis claims that a percept experienced in one modality automatically
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stimulates a percept in another modality. Over the course of the first year
of life, these associated percepts become weaker as overabundant neu-
ral connections between different functional areas of the brain become
pruned or inhibited. Magnitudes, under a similar conceptualization, might
be related via automatic cross-activation of dimension representations.
This could 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 associations among magnitudes might not strengthen
straightforwardly over development.
Evolutionary History
A final possibility is that relations among magnitudes derive from their
evolutionary history rather than solely from developmental processes that
unfold within an individual lifespan. On this view, one quantitative dimen-
sion 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 with
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 quanti-
ties could come to be related in the mind and brain. The five hypotheses
sketched above address different levels of influence ranging from ontog-
eny to phylogeny. They also address different levels 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 cogni-
tion in humans. However, although questions remain as to how primitive
numerical representations are organized with respect to other types of
quantities (e.g., size, time, loudness), it is clear that human children use
those primitive numerical representations to learn the process of verbal
counting early in development. Verbal counting (discussed earlier) is the
first formal cognitive step toward acquiring the uniquely human capacity
for complex symbolic math. In the next section we discuss how the “primi-
tive” analog numerical abilities are related to symbolic math in humans.
ORIGINS OF MATH IQ
A further issue central to understanding the taxonomy of primitive
numerical cognition is the extent to which analog numerical abilities bear
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Math, Monkeys, and the Developing Brain / 309
FIGURE 16.6 Childhood math IQ (mea-
sured by the TEMA-2) is correlated
with the precision of analog numerical
discrimination (measured by subjects’
Weber fractions). A higher Weber frac-
tion reflects worse discrimination. Re-
drawn from Halberda et al. (2008).
a neural relationship with full-blown formal mathematics IQ. Researchers
have begun to examine, in humans, how formal math intelligence may
be modulated by developments in the “primitive” analog numerical sys-
tem 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 performance on math IQ tests and that
formal math ability is more closely correlated with analog numerical abili-
ties than it is with other formal abilities, such as reading. For example, in
Fig. 16.6, adolescents’ analog numerical ability (measured by the Numeri-
cal 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
full-blown math skills. Other studies highlight the role of executive func-
tion 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 a common foundation for analog numerical abilities and formal
math by testing whether a common neural substrate underlies both fac-
ulties. As described above, analog quantity judgments recruit regions of
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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 adults process
the symbolic numbers that are unique to human culture (e.g., numerals,
number words). Several studies suggest that they do: regions of the IPS
exhibit activity that is greater for numerical symbols compared with con-
trol 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; Ansari, 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 the
same subjects (Piazza et al., 2007). Together, these results implicate neural
overlap in the substrates underlying symbolic and nonsymbolic (analog)
numerical representations in humans.
In humans, a second brain region is often recruited during symbolic
numerical tasks: the prefrontal 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 pre-
frontal cortex is thought to be unique in primates compared with other
mammals (Preuss, 2007). In humans the prefrontal 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 develop-
ment of semantic representations, symbols, and rules [see Nieder (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 symbolic numerical pro-
cessing 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
representations. 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 associ-
ating analog numerical values with arbitrary symbols at the level of single
neurons in monkeys (Diester and Nieder, 2007).
Findings that highlight mutual involvement of the IPS and prefrontal
cortex in basic numerical tasks have led to the hypothesis that interac-
tions between frontal and parietal regions are important for the develop-
ment of uniquely human numerical cognition, such as symbolic coding.
Specifically, it has been proposed that the IPS computes “primitive” ana-
log numerical representations and the prefrontal cortex facilitates links
between those analog numerical computations and symbolic number
representations in humans (Cantlon et al., 2009b; Nieder, 2009). If this
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hypothesis is correct then network-level neural synchrony between fron-
tal and parietal regions should predict formal mathematics development
in humans. That is, individual variability in the strength of correlations
between neural responses in frontal and parietal regions, or “functional
connectivity,” should be related to individual variability in mathematics
performance. We have recently tested this hypothesis and found that
number-specific functional connectivity of the fronto-parietal network
does predict children’s math IQ test scores (independently of their ver-
bal 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 that
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.
CONCLUSION
The goal of this review has been to examine the origins and organiza-
tion of numerical abilities ranging from analog quantification to formal
arithmetic. The general hypothesis is that the uniquely human ability 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 and nonhuman animals possess a com-
mon system for making numerical judgments via analog representations.
Throughout development, analog numerical representations 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 process-
ing support this conclusion and provide independent confirmation that
these are in fact related systems. Questions remain regarding the precise
taxonomy of the development and organization of numerical informa-
tion, and its relationship to other domains, such as “space.” However,
the general nature of the relationship between “primitive” and modern
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.
ACKNOWLEDGMENTS
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.
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