Evolution and development of maths
- From: Lance <LanceGary@xxxxxxxxx>
- Date: Wed, 6 Feb 2008 06:04:07 -0800 (PST)
The Evolutionary and Developmental Foundations of Mathematics
Michael J. Beran
Understanding the evolutionary precursors of human mathematical
ability is a
highly active area of research in psychology and biology with a rich
and
interesting history. At one time, numerical abilities, like language,
tool use,
and culture, were thought to be uniquely human. However, at the turn
of the 20th
century, scientists showed more interest in the numerical abilities of
animals.
The earliest research was focused on whether animals could count in
any way that
approximated the counting skills of humans, though many early studies
lacked the
necessary scientific controls to truly prove numerical abilities in
animals. In
addition, both the public and many in the scientific community too
readily
accepted cases of "genius" animals, including those that performed
amazing
mathematical feats. One such animal still lends its name to the
phenomenon of
inadvertent cuing of animals by humans: Clever Hans. Hans was a horse
that
seemed to calculate solutions to all types of numerical problems. In
reality,
the horse was highly attuned to the subtle and inadvertent bodily
movements that
people would make when Hans had reached the correct answer (by tapping
his hoof)
and should have stopped responding. One consequence of this
embarrassing
realization was a backlash for the better part of the 20th century
against the
idea that animals could grasp numerical concepts. The second, more
positive
consequence, however, was that future researchers would include
appropriate
controls to account for such cues.
A resurgence of interest in animal numerical abilities in the early
1980s
followed closely on the heels of a landmark book on children's number
learning
that outlined the critical principles that children must master to
become
proficient in counting. This resurgence also was the result of a
general
increase in studies of animal cognition and intelligence. New research
programs
provided compelling evidence that animals are sensitive to the
numerical
properties of various kinds of things, even if they do not quite reach
the level
of human counting abilities. For instance, multiple research teams
showed that
chimpanzees could learn the meanings of Arabic numerals when taught to
collect
sets of items to match the values of numerals, to label sets of items
with the
correct numeral, and even to add the values of those numerals. Animal
numerical
competence had reclaimed the spotlight, and it remains a highly
visible area of
animal cognition research. Today, we know just how important quantity
and number
concepts are for a great variety of animals such as salamanders, rats,
various
types of birds, dolphins, monkeys, and apes.
What consistently emerges from these kinds of studies is that animals
are
mathematically inclined, but only to a certain degree. Their
performance usually
is constrained by an objective measure of task difficulty that relates
to
well-known psychophysical phenomena. Namely, animals seem to use
approximate
representations of number. For example, in comparison tasks,
performance can
nearly always be predicted very well by knowing the relation of the
two sets to
each other. As the difference (or quantitative distance) between sets
becomes
smaller, the task of choosing the correct set becomes harder (for
example,
comparing 4 to 6 is easier than comparing 4 to 5). When the difference
between
sets is held constant, then the task becomes harder as both sets
increase in
their overall magnitude (for example, comparing 4 to 6 is easier than
comparing
6 to 8). What is interesting is that when adult humans are prevented
from
counting, they also show similar evidence that they have access to a
system of
noisy magnitudes that they use as a form of nonverbal representation,
and even
societies without language-based numerical systems show evidence of
nonverbal
number approximation. In fact, when directly compared, monkeys often
show highly
similar patterns of performance to young human children and human
adults. The
latest example of such similarities in performance comes from a paper
in PLoS
Biology by Cantlon and Brannon. Researchers tested humans and monkeys
in a task
where two sets of dots were shown in succession on a computer screen,
and
participants had to add the sets and then find a match option that had
the same
total number of dots. Humans did better than monkeys, but the
important finding
was that both species were constrained in their performance based on
how closely
the two options resembled each other. The researchers concluded that
monkeys and
humans share components of a mathematical tool kit that can be applied
to
various types of problems. The novel aspect of this research is that
it shows
that monkeys, like humans, can add sets together and remember the
total number
of items. Most likely, monkeys are not the only animals to share these
abilities
with humans.
Given these behavioral similarities, one wonders whether monkeys and
humans
might not only perform these tasks at similar levels, but also in
truly similar
ways. Hypothetical models for numerical representation can account for
the
distance and magnitude effects described above, and environmental
pressures seem
to place a premium on an approximate "number sense" for nonhuman
animals as well
as humans. Research with neuropsychological patients and from
functional
neuroimaging studies indicates that two brain areas, the intraparietal
sulcus
(IPS) and the prefrontal cortex (PFC), seem intricately linked to
number skills
in humans (for overviews, see). We now know that animal brains also
are attuned
to numerical properties. There is evidence that there are distinct
neural
populations and processing stages within the IPS for different
quantity
presentation types (e.g., sequential versus simultaneous) in monkeys,
although
abstract representations that occur later in processing may subsume
these
distinct stages. Single neurons in the ventral intraparietal region
(VIP) and
PFC in the macaque brain have been found to be attuned to specific
numerical
values. Thus, a neuron in VIP responds maximally to one value, and the
firing
rate decreases with distance from this preferred value. However, a
recent paper
by Roitman, Brannon, and Platt published in PLoS Biology shows that
another
region in the parietal cortex, the lateral intraparietal region,
encodes numbers
differently than the VIP. These neurons increase or decrease in
activity based
on the number of elements in a visual array, suggesting that they
serve to
represent accumulated magnitude, an important part of the formal
counting
routine that provides cardinal (exact) values of numerosity. They may
even
provide the magnitude information necessary for other brain regions to
discern
cardinal numerical representations as the brain moves from summing and
estimating magnitude to representing exact numerical information.
Thus, these
data are exciting because they provide a link between theoretical
models of
number processing and actual brain/behavior relations.
Human mathematical abilities, of course, are highly dependent on
symbolic
representations of number. A recent paper by Diester and Nieder
published in
PLoS Biology shows that brain areas critical to processing symbolic
and analogue
numerosities in humans also support numerical processing in monkeys.
After
monkeys learned to associate Arabic numerals with specific numbers of
items, the
researchers recorded from single neurons in the PFC and IPS when
monkeys judged
whether two successive analog arrays were the same in number or
whether an
analog array matched a numeral in a pairing. PFC neurons were
selectively
responsive to given numerical values, presented in either analog or
symbolic
formats. In other words, the PFC in monkeys seems to be involved in
the
association between symbols and numerical concepts, and it builds upon
the
capacities of the IPS to encode approximate numerical information
early in
quantity processing. By four years of age, the IPS in human children
is already
responsive to changes in the numerosity of visual arrays, but the
parietal
cortex shows a more protracted developmental trajectory for the
representation
of symbolic numbers. Specifically, children who have not yet become
proficient
with numerals show elevated PFC activity in response to numerals,
whereas
parietal areas seemingly take over as proficiency with symbols
emerges. In adult
humans, representation of numerical information across many formats
(numerals,
analog stimuli, number words) relies substantially on parietal areas.
Source: PLoS Biology [Open Access]
http://biology.plosjournals.org/perlserv/?request=get-document&doi=10.1371/journ\
al.pbio.0060019
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