It's hard enough for Darwinists to explain why humans are hardwired for
speech. But if this American Scientist article is true, now they have to explain
why humans are hardwired for mathematics too!
As Barrow muses:
"But how strange this is. Our minds are the products of the laws of Nature;
yet they are in a position to reflect upon them. How fortuitous that our
minds (or at least the minds of some) should be poised to fathom the
depths of Nature's secrets....A more interesting problem is the extent to
which the brain is qualitatively adapted to understand the Universe. Why
should its categories of thought and understanding be able to cope with the
scope and nature of the real world? Why should the Theory of Everything
be written in a 'language' that our minds can decode? Why has the process
of natural selection so over-endowed us with mental faculties that we can
understand the whole fabric of the Universe far beyond anything required
for our past and present survival? (Barrow J.D., "Theories of Everything:
The Quest for Ultimate Explanation", 1992, pp172-173).
But as Phil Johnson points out, it is an "absurdity" to claim that "the
scientific mind itself was designed by natural selection", which "rewards
only superiority at reproduction":
"Another absurdity is that the scientific mind itself was designed by natural
selection, a force that rewards only superiority at reproduction and by
whose standards the mind of the cockroach is every bit as effective as the
mind of Einstein. On the contrary, the rationality and reliability of the
scientific mind rests on the fact that the mind was designed in the image of
the mind of the Creator, who made both the laws and our capacity to
understand them." (Johnson P.E., "The Wedge: Breaking the Modernist
Monopoly on Science", Touchstone, July/August 1999, Vol. 12, No. 4,
p23).
*Only* an Intelligent Designer could create *both* the universe with an
interior structure based on mathematics and the human brain with the
interior structure of understanding that universe.
Steve
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http://www.sigmaxi.org/amsci/bookshelf/Leads99/Sherman.html
September-October 1999
Scientists' Bookshelf
Author Argues that Everyone Is Born with a Head for Numbers
Malcolm J. Sherman
What Counts: How Every Brain is Hardwired for Math. Brian
Butterworth. 320 pp. Free Press, 1999. $26.
Brian Butterworth, a British cognitive neuropsychologist and founding
editor of the journal Mathematical Cognition, has summarized several lines
of evidence pointing to the conclusion that the normal human brain
contains a "number module"--a highly specialized set of neural circuits that
enable us to categorize small collections of objects in terms of their so-
called numerosities. When we see three brown cows our brains immediately
tell us both that there are three of them and that they are brown. Just as we
see colors automatically and involuntarily and without being taught the
concept of color, so we immediately recognize and distinguish small
numerosities without being taught the meaning of number. In order to
communicate, we need to learn the words "brown" and "three," but our
perception of small numerosities is as innate and as automatic as is our
perception of color.
Of course, through instruction and practice we can greatly extend the
capacities of our number module, just as we can similarly improve our
ability to read, type or play the piano. But reading, typing and piano
playing are not based on hard-wired, specialized, genetically created neural
circuits ("cognitive modules"); they depend instead on the slow, purposeful
development of general-purpose brain circuits ("central processes").
Butterworth disagrees with the influential Swiss psychologist Jean Piaget,
who denied that "any a priori or innate cognitive structures exist in man."
For Piaget, a child's understanding of number was founded on years of
sensorimotor interactions with physical and temporal realities. Before
children can understand number, they must master, for example, transitive
inference (if a < b and b < c, then a < c) and must be capable of separating
number from the sensory properties of objects. Piaget concluded that the
concept of number cannot be understood by children below the age of four
or five.
Some of the disagreement between Piaget and Butterworth stems from
Piaget's more stringent criterion for what it means to understand number.
Piaget did not deny that toddlers recognize the difference between two and
three. But Piaget did not regard the ability to make this distinction as proof
of an understanding of number. In Piaget's famous experiments, children
were shown two identical collections, and then the objects in one collection
were moved farther apart. The children were then prone to say that the
more spread-out collection had more objects. Piaget regarded the failure to
realize that number is conserved when objects are moved as a failure to
understand number.
More recent experiments, described in Stanislas Dehaene's 1997 book The
Number Sense (cited by Butterworth), seem to show that Piaget's subjects
knew perfectly well that number is conserved when objects are moved; the
problem was that they did not understand the questions they were being
asked. According to Dehaene, if children are asked to choose between four
pieces of candy spread apart and five pieces close together, they are
unlikely to be fooled. Piaget may have underestimated children's early
understanding of number, but he is probably still correct in claiming, for
example, that children below the age of five or six cannot count two sets of
objects and compare them unless the collections are simultaneously
present. Such capacities for abstract or symbolic representation may
plausibly depend on more than the number module. Butterworth's view of
the origins of our mathematical abilities is analogous to linguist Noam
Chomsky's thesis that the logic of grammar is built into our brain--that
spoken language depends on an innate cognitive structure. (Piaget, of
course, denies that such structures exist.) Books and conferences have
been devoted to attempts to reconcile Piaget's and Chomsky's views of the
foundations of cognition.
Although Chomsky and Butterworth have similar theoretical perspectives
about knowledge, they disagree about math. Chomsky sees the number
concept as a special aspect of language, whereas Butterworth argues
(citing, for example, studies of the cognitive consequences of injuries to
various parts of the brain) that math and language use different regions of
the brain. Butterworth also disagrees with those who explain math as a
combination of language, general intelligence and spatial ability.
Modern cognitive science and physical investigations of brain structure may
someday resolve or clarify an ancient philosophical issue: Does knowledge
have a large innate component (as Kant, Chomsky and various religious
philosophers would argue), or is the mind a tabula rasa whose contents are
determined by the social and physical environment (Locke and Piaget)?
What are the implications for math education of various cognitive
perspectives? Because Piaget believed that number itself was dependent on
abstract and logical thought, Piagetians are prone to deduce that premature
exposure to mathematics will lead to rote learning without understanding
and to disabling confusions and anxieties. "Developmentally appropriate
practice" has become a shibboleth in U.S. schools of education--largely
reflecting the Piagetian belief in fixed stages of cognitive development.
France severely deemphasized the early teaching of numerals and counting
words, believing that such instruction was useless or harmful. Even if
Piaget's theories are right, it is an empirical issue whether it is helpful or
harmful to teach children to memorize counting words before they can
abstractly link these words to collections of objects.
Although Butterworth sensibly rejects Piagetian-based pessimism about
what children are capable of learning at various ages, he is implausibly
optimistic about our mathematical potential. Although his central thesis, the
number module, is genetic, he argues that the main sources of individual
differences in developed math ability are environmental: "provided [that]
the basic Number Module has developed normally ... differences in
mathematical ability ... are due solely to acquiring the conceptual tools
provided by our culture. Nature, courtesy of our genes provides the piece
of specialist equipment, the Number Module. All else is training. To
become good at numbers, you must become steeped in them." Butterworth
denies that there is any "essential and innate difference between children ...
who find maths [the British usage for math] really easy and those who find
it a struggle. There may have been differences in their capacity for
concentrated work or in what they found interesting ... but there was no
difference in their innate capacity specifically for maths."
Butterworth cites international comparisons that show large differences in
performance (for instance, a test on which the average score of Iranian
children is equal to that of the lowest 5 percent of children in Singapore).
Cultural resources and pedagogy clearly matter. But it does not follow that
all individual differences in developed math ability are due to
temperamental and environmental factors (ability to concentrate, ambition,
interest and time devoted to math). Intelligence, verbal aptitude and spatial
ability are also likely to be important for math.
According to Piaget, children must discover or construct for themselves
certain regularities about the world (that objects continue to exist even
when we can't see them, for example). Some constructivists go beyond
Piaget, claiming that all genuine knowledge must be gained through a
process of discovery. Although infants and toddlers do need to learn basic
facts and distinctions (hard versus soft, solid versus liquid) through their
experiences with external objects, it does not follow that more advanced
material must be learned by recapitulating the original process of discovery.
If so, the potential for human progress in science and other areas would be
severely limited.
Although Butterworth rejects Piaget's theoretical framework, he agrees
with most Piagetians in advocating discovery learning. Butterworth argues
that schools limit children's potential for growth when they insist that there
is a preferred way to do math problems, then drill students in approved
methods. He reasons that since we all have a number module, we all have
the capacity to work out our own approach. Butterworth approvingly
quotes educational researcher Lauren Resnick:
The failure of much of our present teaching to make a cognitive connection
between children's own math-related knowledge and the school's version
of math feeds a view held by many children that what they know does not
count as mathematics. This devaluing of their own knowledge is especially
exaggerated among children from families that are traditionally alienated
from schools, ones in which parents did not fare well in school and do not
expect-however much they desire--their children to do well, either. In the
eyes of these children, math is what is taught in school. But a large body of
empirical evidence (not cited by Butterworth) shows that discovery
learning is ineffective with all but the most basic material. Few children will
discover for themselves efficient ways of multiplying three-digit numbers,
and virtually none will discover Archimedes' law by experimenting with
floating bodies. To be sure, children may work out ways of doing simple
arithmetic problems. Butterworth cites as an example an untutored
Brazilian coconut seller, who calculated the price of 10 coconuts without
understanding decimal place notation. But the ad hoc methods children
discover for themselves are most unlikely to be suitable building blocks for
more advanced knowledge. Even when successful, discovery learning is
inefficient, taking time that could be better devoted to practice.
Butterworth correctly relates that great mathematicians all steeped
themselves in mathematics. Yet he disparages school practice and
somehow regards it as antithetical to understanding.
Butterworth sees the international comparisons he cites as proof that
children can learn more math than they typically do. But the best countries
(such as Singapore) are the ones that emphasize direct instruction and drill,
not the student-centered discovery methods he advocates. Butterworth's
findings and views of mathematical cognition may well be sound. But the
existence of a number module does not in and of itself establish the relative
soundness of various educational methods. Doing so would require an
evaluation of empirical research in educational settings, and this
Butterworth has not done. Cognitive theorists are too prone to jump from
models of cognition to classroom practice without empirical testing under
realistic classroom circumstances.
---Malcolm J. Sherman is a professor in the Department of Mathematics and Statistics at the University at Albany, State University of New York, where his primary interests include statistics and mathematics education.============================================================--------------------------------------------------------------------"It is not difficult to imagine how feathers, once evolved assumed additional functions, but how they arose initially presumably from reptilian scales, defies analysis." (Stahl B.J., "Vertebrate history: Problems in Evolution", Dover: New York, 1985, p349)Stephen E. Jones | sejones@iinet.net.au | http://www.iinet.net.au/~sejones--------------------------------------------------------------------