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sp; =
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Math
Department, Mission College, Santa Clara, California
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College Main Page
This paper was written as an assignment f=
or Ian
Walton's Math G - Math for liberal Arts Students - at Mission College. If y=
ou
use material from this paper, please acknowledge it.
To explore other such papers go to the Math G
Projects Page
Born
to Count:
Inherent
Abilities in Mathematics
Pati Brow=
n
![](3D"Pati_files/image005.jpg")
=
Math G, =
Midterm
Dr. Ian =
Walton
26 Octob=
er 2009
Born to Count:
Inherent Abilities in Mathematics
Table of Contents
Introduction. 1<=
span
style=3D'font-size:11.0pt;line-height:200%;font-family:"Calibri","sans-seri=
f";
mso-fareast-font-family:"Times New Roman";mso-no-proof:yes'>
Theories. 2<=
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style=3D'font-size:11.0pt;line-height:200%;font-family:"Calibri","sans-seri=
f";
mso-fareast-font-family:"Times New Roman";mso-no-proof:yes'>
Jean Piaget 2<=
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style=3D'font-size:11.0pt;line-height:200%;font-family:"Calibri","sans-seri=
f";
mso-fareast-font-family:"Times New Roman";mso-no-proof:yes'>
Jerry Fodor 2<=
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style=3D'font-size:11.0pt;line-height:200%;font-family:"Calibri","sans-seri=
f";
mso-fareast-font-family:"Times New Roman";mso-no-proof:yes'>
Brian Butterworth. 3<=
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style=3D'font-size:11.0pt;line-height:200%;font-family:"Calibri","sans-seri=
f";
mso-fareast-font-family:"Times New Roman";mso-no-proof:yes'>
Marian Diamond. 4<=
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style=3D'font-size:11.0pt;line-height:200%;font-family:"Calibri","sans-seri=
f";
mso-fareast-font-family:"Times New Roman";mso-no-proof:yes'>
Veronique Izard. 4<=
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style=3D'font-size:11.0pt;line-height:200%;font-family:"Calibri","sans-seri=
f";
mso-fareast-font-family:"Times New Roman";mso-no-proof:yes'>
Methods, Examples, and Findings. 5<=
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style=3D'font-size:11.0pt;line-height:200%;font-family:"Calibri","sans-seri=
f";
mso-fareast-font-family:"Times New Roman";mso-no-proof:yes'>
Stanislas Dehaene. 5<=
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f";
mso-fareast-font-family:"Times New Roman";mso-no-proof:yes'>
An Experiment with Mimi 7<=
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f";
mso-fareast-font-family:"Times New Roman";mso-no-proof:yes'>
Conclusion. 8<=
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mso-fareast-font-family:"Times New Roman";mso-no-proof:yes'>
Works Cited. 10<=
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mso-fareast-font-family:"Times New Roman";mso-no-proof:yes'>
Images Used. 11<=
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mso-fareast-font-family:"Times New Roman";mso-no-proof:yes'>
Born to Count:
Inherent Abilities in Mathematics
![](3D"Pati_files/image007.jpg")
&nbs=
p;
Are numbers a part of who w=
e 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.
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 everyt=
hing
the infant learned was constructed logically as he aged. Current theories are far more exci=
ting.
Jerry Fodor, an American philosophe=
r/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 abilit=
ies
exist. In this module, specia=
lized
information is processed automatically.&nb=
sp;
For example, cognitive modul=
es
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 b=
asic
component that Fodor believes we are born with.
Conversely, central processes move slowly and operate by will. We aren’t born with them, ra=
ther
we learn these processes. Fod=
or
feels that though we are born with the circuitry to perform tasks like typi=
ng 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 consider=
ed a
function of central processing =
in the
brain. Understanding and manipulating numbers is a much more challenging ta=
sk
than seeing color. In order to
master mathematics, we are required to study – a much more complex br=
ain
activity than what is exercised in =
cognitive
module thinking.
&nbs=
p; Brian
Butterworth, a British cognitive neuroscientist, disputes Fodor’s
findings. He states that numb=
er 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 g=
ood
educational experience and by applying ourselves to learning. Additionally, he believes that num=
erical
abilities could be tied to linguistic and memorization abilities in the bra=
in,
enlisting the area of the brain designed to hold information in our short-t=
erm
working memory and control our reasoning ability.
<=
span
style=3D'mso-tab-count:1'> &=
nbsp; 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 bra=
in 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 wor=
king
with mathematics could create a greater number of neurons in the parietal l=
obe
of the brain.
<=
span
style=3D'mso-tab-count:1'> &=
nbsp; 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” netwo=
rk
that correspond to the same networks in adult
b=
rains. These findings destroy the myth th=
at
changes in infant awareness of object and number are simply a result of a
baby’s getting bored.
Stanislas D=
ehaene is
a French neuroscientist who has recently completed a comprehensive study of
mathematical aptitude in infants, primates, and cultures with limited ̶=
0;number
words.” Dehaene believe=
s 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).
&nbs=
p; In
his studies he breaks down elementary math into two aspects: The first aspe=
ct
is “Symbolic.” Sy=
mbolic
arithmetic corresponds with how numbers and “number words” are
manipulated. “Nonsymbolic” arithmetic relates to how we underst=
and
the amount of items within groups of objects. He believes that the “nonsym=
bolic”
arithmetic is at the root of our mathematical knowledge.
&nbs=
p; 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 sho=
wed
images of 8 trucks at the same repeated intervals. When the image changed f=
rom
ducks to trucks, the researchers found a boost in activity in the temporal
lobe. But very quickly the ba=
bies grew
bored of the images of 8 trucks as well.&n=
bsp;
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 sur=
prised
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” theo=
ry,
indicating that infants have an inherent intuition for numbers. The temporal
lobe registered the “what” while the parietal lobe registered t=
he change
in quantity. While neuroscien=
tists
originally believed the comprehension of changes in objects and amounts sho=
wn
to infants simply showed their ability to grab and hold their attention,
Dehaene’s study shows that there is a designated part of infants̵=
7;
brains that specifically processes quantities.
&nbs=
p; 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 diffe=
rent
than the way an adult connects with numbers. Adults think of numbers in an
incremental sense. The differ=
ence
between 10 and 11 to an adult is very small. While the difference between 10 an=
d 11
is nonexistent to an infant, but the difference between 1 and 2 to them is
immense – they understand that the amount is double.
&nbs=
p; Dehaene
did further studies with Amazonian groups.=
One such group only had number-names up to 5. The experiment required these peop=
le 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:
=
![](3D"Pati_files/image020.gif")
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.&=
nbsp;
You and I would easily say that the midpoint is 5. However the researchers were surpr=
ised
to find that the Amazonian group, without hesitation, said the midpoint was
3. They reasoned that they co=
uld
count by threes: 3 x 3 =3D 9 or 3 is to 1 as 9 is to 3. These people see numbers algorithm=
ically. This shows us that even individual=
s without
formal educations can develop significant and sophisticated mathematical
vocabulary and perform all necessary expressions to work efficiently within=
a
culture.
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 hav=
e a
sense of numbers, however the number sequence seems to be: 1, 2, a
bunch…. I decided to perform this experiment
myself.
I placed about 20 pennies o=
n 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, child=
ren
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 childr=
en at
this age, numbers are just words.
W=
here 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 ne=
uron
connections with which to build on than do adults. Numbers in all cultures and
throughout the ages are invented and foreign, however we are born with brai=
ns
that hold the intuitive capacity to build on numbers in extremely complex w=
ays.
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.
Researchers have proven that
infants react intuitively to their surroundings, these intuitions create ne=
ural
structures that are reliant on environmental stimulation to grow. Once toddlers learn number words a=
nd
counting they begin to build on their skills pulling from their inherent
intuitions. Additionally, var=
ying
neuroscientific theories tell us that number concepts not only live in our
brains, but have strong foundation from the very beginning of life.
<=
o:p>
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 ment=
al
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.=
221;
|
|
![](3D"Pati_files/image022.jpg")
Albert Einstein
|
|
Bower, Bruce. “Tots Who Tote: Babies Show Neural Si=
gns
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/&#=
8204;start.do?prodId=3DGRGM>.
Butterworth, B. “The Development of Arithmetical
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Butterworth, Brian. What Counts: How Every Brain is
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Dehaene, Stanislas. “Numbers.” Interview by J=
ad
Abumrad and Robert Krulwich. =
Radio
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odes/2009/10/09>.
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ins of
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Healy, Jane M., Ph.D. Your Child’s Growing Mind:=
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ide
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Albert Einstein. N.d. www.etsu.edu/math/gardner/einste=
in/person.htm.
www.etsu.edu, n.d. Web. 23 Oct. 2009.
<http://www.etsu.edu/math/gardner/
einstein/einstein-folded-hands.jpg>.<=
/o:p>
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2009.
<http://www.uoregon.edu/~cstein/Brains%20b=
aby.jpg>.
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2009.
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Jean Piaget.
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&=
nbsp; <http://commons.activemath.org/ActiveMath=
2/content/piaget/pics/
&=
nbsp; jean_piaget.jpg>.
Jerry
Fodor. N.d. www.uncommondescent.com. N.p., n.d. Web. 23 Oct. 2009. <=
br>
&=
nbsp; <http://www.uncommondescent.com/images/fo=
dor.jpg>.
Marian Diamond. N.d. sciencematters.berkeley.edu.
N.p., n.d. Web. 23 Oct. 2009.
<http://sciencematters.berkeley.edu/archives/vol=
ume3/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/1Dmxio9DVD=
c/
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>.
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09-numbers.jpg>.
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