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Born
to Count:

Inherent
Abilities in Mathematics

Pati Brown

Math G, Midterm

Dr. Ian Walton

26 October 2009

Born to Count:

Inherent Abilities in Mathematics

Table of Contents

Introduction. 1

Theories. 2

Jean Piaget 2

Jerry Fodor 2

Brian Butterworth. 3

Marian Diamond. 4

Veronique Izard. 4

Methods, Examples, and Findings. 5

Stanislas Dehaene. 5

An Experiment with Mimi 7

Conclusion. 8

Works Cited. 10

Images Used. 11

Born to Count:

Inherent Abilities in Mathematics

Introduction

Are numbers a part of who we are,
how our brains are wired?  Or do we
learn mathematics as a part of our survival skills?  Many could argue that performing
calculations is as anxiety provoking as having a root canal.  These same people would also argue that
an intuition and aptitude for math was never part of the “standard
equipment” of their brains.
Fortunately, psychologists and neuroscientists are doing research to
prove to the math-phobic masses as well as the educational community that an
intuition for numbers does indeed exist as a part of a core knowledge
pre-programmed into the healthy human brain.[1]

Theories

Jean Piaget

In the earlier part of the 20th
century, Swiss psychologist Jean Piaget believed that all infants were born
with a brain that is a “blank slate.”  He believed that absolutely everything
the infant learned was constructed logically as he aged.  Current theories are far more exciting.[2]

Jerry Fodor

Jerry Fodor, an American philosopher/cognitive
scientist, who has done considerable research on human cognition, believes that
there are two separate sets of cognitive abilities.  He calls the first set the
“cognitive module” in which highly specialized cognitive abilities
exist.  In this module, specialized
information is processed automatically.
For example, cognitive modules
allow us to see color.  We
don’t need to learn to see color; this ability comes to us naturally. And
though we have to learn languages and alphabet, work out how to process the
letters, and string them into words to find meaning; reading is part of a basic
component that Fodor believes we are born with.

Conversely, central processes move slowly and operate by will.  We aren’t born with them, rather
we learn these processes.  Fodor
feels that though we are born with the circuitry to perform tasks like typing and
painting we still have to decide to develop and use these skills.  They aren’t automatic brain
tasks.  He also believes that memory
and reasoning are central processes.  We decide what to remember and how to
reason.

Since reasoning and memory are
critical to performing mathematical calculations, these skills are considered a
function of central processing in the
brain. Understanding and manipulating numbers is a much more challenging task
than seeing color.  In order to
master mathematics, we are required to study – a much more complex brain
activity than what is exercised in cognitive
module thinking.[3]

Brian Butterworth

Brian
Butterworth, a British cognitive neuroscientist, disputes Fodor’s
findings.  He states that number use
is so widespread that we must all be born with a “Number Module” or
a “Mathematical Brain.”
Butterworth believes that we are all born with the ability to see and
process our worlds in numbers, expanding our mathematical capacity with a good
educational experience and by applying ourselves to learning.  Additionally, he believes that numerical
abilities could be tied to linguistic and memorization abilities in the brain,
enlisting the area of the brain designed to hold information in our short-term
working memory and control our reasoning ability.[4]

Marian Diamond

Several
years ago researchers rediscovered Einstein’s brain.  One of the researchers was a noted
neuroanatomist from U. C. Berkeley, Dr. Marian Diamond.  When studying Einstein’s brain she
found that Einstein’s left parietal lobe had more densely packed glial
cells than an average brain.  It was
originally thought that some people were genetically blessed with a greater
number of cells in the parietal lobe; however Dr. Diamond believes that this
increased ratio of neurons was a result of Einstein’s judicious use of
his reasoning and thinking skills. This gives fuel to the argument that working
with mathematics could create a greater number of neurons in the parietal lobe
of the brain.[5]

Veronique Izard

Harvard
neuroscientist, Veronique Izard, recently discovered that infants as young as
3-monthsof age show signs of very distinct neural pathways that allow them to
distinguish the number of objects and the identity of objects.  She believes that there is a neural
“where and how” network and a separate “what” network
that correspond to the same networks in adult

brains.  These findings destroy the myth that
changes in infant awareness of object and number are simply a result of a
baby’s getting bored.[6]

Methods, Examples, and Findings

Stanislas Dehaene

Stanislas Dehaene is
a French neuroscientist who has recently completed a comprehensive study of
mathematical aptitude in infants, primates, and cultures with limited “number
words.”  Dehaene believes that
we show early signs of number intuition as infants and it is from this
intuition that our mathematical knowledge is built. Dehaene believes that
“the term ‘intuition’…is fast, automatic, and
inaccessible to introspection.” (233).[7]

In
his studies he breaks down elementary math into two aspects: The first aspect
is “Symbolic.”  Symbolic
arithmetic corresponds with how numbers and “number words” are
manipulated. “Nonsymbolic” arithmetic relates to how we understand
the amount of items within groups of objects.  He believes that the “nonsymbolic”
arithmetic is at the root of our mathematical knowledge.[8]

Dehaene
proved his theory with infants between the ages of 8 and 12 weeks old. Each infant
was hooked up to a brain-monitoring device.  While sitting on his mother’s lap
in front of a computer screen, the baby viewed images of 8 ducks.  These images were shown at repeated
intervals until the monitoring device indicated the baby was becoming
bored.  Next, the computer showed
images of 8 trucks at the same repeated intervals. When the image changed from
ducks to trucks, the researchers found a boost in activity in the temporal
lobe.  But very quickly the babies grew
bored of the images of 8 trucks as well.
For the final step of the experiment, truck images flashed before the
infants, however the number of trucks changed from 8 to 16.  This time the researchers were surprised
to see that a spike in activity in the parietal lobe occurred as a result of
the change of quantity.  This
experiment proved Dehaene’s “nonsymbolic arithmetic” theory,
indicating that infants have an inherent intuition for numbers. The temporal
lobe registered the “what” while the parietal lobe registered the change
in quantity.  While neuroscientists
originally believed the comprehension of changes in objects and amounts shown
to infants simply showed their ability to grab and hold their attention,
Dehaene’s study shows that there is a designated part of infants’
brains that specifically processes quantities.[9]

Not
only does this prove the intuition for numbers, but also proves that babies
possess a sense of logarithms and connect with quantities in a logarithmic
sense.  This is entirely different
than the way an adult connects with numbers.  Adults think of numbers in an
incremental sense.  The difference
between 10 and 11 to an adult is very small.  While the difference between 10 and 11
is nonexistent to an infant, but the difference between 1 and 2 to them is
immense – they understand that the amount is double.[10]

Dehaene
did further studies with Amazonian groups.
One such group only had number-names up to 5.  The experiment required these people to
find the midpoint between the numbers 1 and 9.  To do this Dehaene showed the study
participants a picture that looked something like this:

What is the midpoint between 1 and 9?

He then asked the participants to tell him what number
represented the midpoint between 1 and 9.
You and I would easily say that the midpoint is 5.  However the researchers were surprised
to find that the Amazonian group, without hesitation, said the midpoint was
3.  They reasoned that they could
count by threes: 3 x 3 = 9 or 3 is to 1 as 9 is to 3.  These people see numbers algorithmically.  This shows us that even individuals without
formal educations can develop significant and sophisticated mathematical
vocabulary and perform all necessary expressions to work efficiently within a
culture.

An Experiment with Mimi

Finally, Harvard Psychology
Professor, Susan Carey performed a simple test with a two-year-old.  In this test she gives the child a bunch
of pennies and asks them to give her specific amounts.  She finds that toddlers indeed have a
sense of numbers, however the number sequence seems to be: 1, 2, a
bunch….[11]   I decided to perform this experiment
myself.

I placed about 20 pennies on a
table and asked my 22-month-old friend, Mimi, to give me one penny.  She quickly picked up one penny and
handed it to me.  Then I asked her
to give me two pennies, and she handed me two pennies.  But when I asked her to hand me 3
pennies she grabbed a handful of pennies and proudly handed them all to
me.  According to Carey, children
usually grasp the concept of numbers over “1” at about the age of 18
months.  By 3½ years of age
they often begin counting, but only because adults count to them; to children at
this age, numbers are just words.

Conclusion

Where we
once thought infants’ brains were “blank slates,” we now know
that the average brain comes equipped with an intuition for quantity.  In fact, babies have vastly more neuron
connections with which to build on than do adults.[12]   Numbers in all cultures and
throughout the ages are invented and foreign, however we are born with brains
that hold the intuitive capacity to build on numbers in extremely complex ways.
We know that infants can respond to amounts of objects in their world long
before they can count, communicate verbally, reason, or even manipulate
objects.[13]
Researchers have proven that
infants react intuitively to their surroundings, these intuitions create neural
structures that are reliant on environmental stimulation to grow.  Once toddlers learn number words and
counting they begin to build on their skills pulling from their inherent
intuitions.  Additionally, varying
neuroscientific theories tell us that number concepts not only live in our
brains, but have strong foundation from the very beginning of life.[14]

Perhaps Albert Einstein described our mathematical intuition best in this
introspective quote about his own thinking processes:

“Words
of language, whether spoken or written, do not seem to play any role in my
thinking mechanisms.  The mental
entities that serve as elements of my thought are certain signs or images, more
or less clear, that can ‘at will’ be reproduced or combined.”[15]

Albert Einstein

Works Cited

Bower, Bruce. “Tots Who Tote: Babies Show Neural Signs
of Budding Number Sense.” Science News 9 Feb. 2008: 84-85. Gale.  Mission College Library. Web. 4 Oct.
2009. <http://0-find.galegroup.com.library.wvmccd.cc.ca.us/‌gtx/‌start.do?prodId=GRGM>.

Butterworth, B. “The Development of Arithmetical
Abilities.” Journal of Child Psychology and Psychiatry and Allied
Disciplines (2009): n. pag. PubMed - indexed for MEDLINE. Web. 4
Oct. 2009. <http://www.ncbi.nlm.nih.gov/‌pubmed/‌15660640>.

Butterworth, Brian. What Counts: How Every Brain is
Hardwired for Math. New York: The Free Press, 1999. Print.

Dehaene, Stanislas. “Numbers.” Interview by Jad
Abumrad and Robert Krulwich.  Radio
Lab. WNYC. 30 Sept. 2009. WNYC -- Radio Lab . Web. Transcript. 7
Oct. 2009. <http://www.wnyc.org/‌shows/‌radiolab/‌episodes/‌2009/‌10/‌09>.

“Origins of
Mathematical Intuitions:  The Case
of Arithmetic.” Annals of the New York Academy of Sciences (2009):
232-259. Unicog.org. Web. 16 Oct. 2009. <http://www.unicog.org/‌publications/‌Dehaene
S (2009) Origins of mathematical intuitions...The
YearInNeuroscienceNYAS.pdf>.

Healy, Jane M., Ph.D. Your Child’s Growing Mind: A
Practical Guide to Brain Development and Learning from Birth to Adolescence.
New York: Doubleday, 1987, 1994. Print.

Kluger, Jeffrey. “Language and the Baby Brain.” TIME,
Your Brain:  A User’s Guide
2009: 26-30. Print.

Images Used

Albert Einstein. N.d. www.etsu.edu/math/gardner/einstein/person.htm.

www.etsu.edu, n.d. Web. 23 Oct. 2009.
<http://www.etsu.edu/math/gardner/

einstein/einstein-folded-hands.jpg>.

Baby with Glasses. N.d. www.uoregon.edu, n.d. Web. 23 Oct.
2009.

<http://www.uoregon.edu/~cstein/Brains%20baby.jpg>.

Brian Butterworth. N.d. www.mathematicalbrain.com.
N.p., n.d. Web. 23 Oct.

2009.
<http://www.mathematicalbrain.com/images/bbpor.JPG>.

Jean Piaget.
N.d. www.commons.activemath.org. N.p., n.d. Web. 23 Oct. 2009.

<http://commons.activemath.org/ActiveMath2/content/piaget/pics/

jean_piaget.jpg>.

Jerry
Fodor. N.d. www.uncommondescent.com. N.p., n.d. Web. 23 Oct. 2009.

<http://www.uncommondescent.com/images/fodor.jpg>.

Marian Diamond. N.d. sciencematters.berkeley.edu.
N.p., n.d. Web. 23 Oct. 2009.

<http://sciencematters.berkeley.edu/archives/volume3/issue20/images/

diamond1.jpg>.

Math Baby. N.d. math2mind.blogspot.com/. N.p., n.d. Web.
21 Oct. 2009.

<http://2.bp.blogspot.com/_ySjExcU0kyU/SQz2MJzCnI/AAAAAAAAIxs/1Dmxio9DVDc/
LOVE+MATH+BABY.jpg>.

Stanislas Dehaene. N.d. www.edge.org/3rd_culture/bios/dehaene.html.
N.p., n.d.

Web. 23 Oct. 2009.
<http://www.edge.org/3rd_culture/bios/images/

dehaene200.jpg>.

Veronique Izard. N.d. www.news.harvard.edu. Harvard
University, n.d. Web. 23

Oct. 2009.
<http://news.harvard.edu/gazette/wp-content/uploads/2009/08/

09-numbers.jpg>.

[1] Stanislas Dehaene. “Origins of Mathematical
Intuitions:  The Case of
Arithmetic.” Annals of the New York Academy of Sciences (2009):
232-259. Unicog.org. Web. 16 Oct. 2009. <http://www.unicog.org/‌publications/‌Dehaene
S (2009) Origins of mathematical intuitions...The
YearInNeuroscienceNYAS.pdf>.

[2] Healy, Jane M., Ph.D. Your Child’s Growing
Mind: A Practical Guide to Brain Development and Learning from Birth to
Adolescence. New York: Doubleday, 1987, 1994. Print.

[3] Brian Butterworth. What Counts: How Every Brain is
Hardwired for Math. New York: The Free Press, 1999. Print.

[4] Brian Butterworth. What Counts: How Every Brain is
Hardwired for Math. New York: The Free Press, 1999. Print.

[5] Jane M. Healy. Ph.D. Your Child’s Growing
Mind: A Practical Guide to Brain Development and Learning from Birth to
Adolescence. New York: Doubleday, 1987, 1994. Print.

[6] Bruce Bower. “Tots Who Tote: Babies Show Neural
Signs of Budding Number Sense.” Science News 9 Feb. 2008: 84-85. Gale.  Mission College Library. Web. 4 Oct.
2009. <http://0-find.galegroup.com.library.wvmccd.cc.ca.us/‌gtx/‌start.do?prodId=GRGM>.

[7] Stanislas Dehaene. “Origins of Mathematical
Intuitions:  The Case of
Arithmetic.” Annals of the New York Academy of Sciences (2009):
232-259. Unicog.org. Web. 16 Oct. 2009. <http://www.unicog.org/‌publications/‌Dehaene
S (2009) Origins of mathematical intuitions...The
YearInNeuroscienceNYAS.pdf>.

[8] Stanislas Dehaene. “Numbers.” Interview
by Jad Abumrad and Robert Krulwich.
Radio Lab. WNYC. 30 Sept. 2009. WNYC -- Radio Lab . Web.
Transcript. 7 Oct. 2009. <http://www.wnyc.org/‌shows/‌radiolab/‌episodes/‌2009/‌10/‌09>.

[9] Stanislas Dehaene. “Origins of Mathematical
Intuitions:  The Case of
Arithmetic.” Annals of the New York Academy of Sciences (2009):
232-259. Unicog.org. Web. 16 Oct. 2009. <http://www.unicog.org/‌publications/‌Dehaene
S (2009) Origins of mathematical intuitions...The
YearInNeuroscienceNYAS.pdf>.

[10] Bruce Bower. “Tots Who Tote: Babies Show Neural
Signs of Budding Number Sense.” Science News 9 Feb. 2008: 84-85. Gale.  Mission College Library. Web. 4 Oct.
2009. <http://0-find.galegroup.com.library.wvmccd.cc.ca.us/‌gtx/‌start.do?prodId=GRGM>.

[11] Stanislas Dehaene. “Numbers.” Interview
by Jad Abumrad and Robert Krulwich.
Radio Lab. WNYC. 30 Sept. 2009. WNYC -- Radio Lab . Web.
Transcript. 7 Oct. 2009. <http://www.wnyc.org/‌shows/‌radiolab/‌episodes/‌2009/‌10/‌09>.

[12] Kluger, Jeffrey. “Language and the Baby
Brain.” TIME, Your Brain:
A User’s Guide 2009: 26-30. Print.

[13] B. Butterworth. “The Development of
Arithmetical Abilities.” Journal of Child Psychology and Psychiatry
and Allied Disciplines (2009): n. pag. PubMed - indexed for MEDLINE.
Web. 4 Oct. 2009. <http://www.ncbi.nlm.nih.gov/‌pubmed/‌15660640>.

[14] Jane M. Healy, Ph.D. Your Child’s Growing
Mind: A Practical Guide to Brain Development and Learning from Birth to
Adolescence. New York: Doubleday, 1987, 1994. Print.

[15] Stanislas Dehaene. “Origins of Mathematical
Intuitions:  The Case of
Arithmetic.” Annals of the New York Academy of Sciences (2009):
232-259. Unicog.org. Web. 16 Oct. 2009. <http://www.unicog.org/‌publications/‌Dehaene
S (2009) Origins of mathematical intuitions...The
YearInNeuroscienceNYAS.pdf>.