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From: Jerry Becker <[log in to unmask]>
Date: Thu, 7 Mar 2002 07:58:27 -0800
To: [log in to unmask]
Subject: MORE RESPONSES to are math and arithmetic independent of each other

Here are some comments about the discussion on the 'independence' of
arithmetic and math.


In his report on Number Theory, 1897, D. Hilbert said 'I have tried to avoid
long numerical computations, thereby following Riemann's postulate that
proofs should be given through ideas and not voluminous computations'.

The question reminds me of the many times I have had to explain that there
is essentially no correlation between 'algorithmic' computation and math.
Indeed, there is a huge gap between them. This confusion of the algorithmic
with the mathematics is not limited to this part of arithmetic. Algorithmic
'arithmetic' versus arithmetic is only the beginning . For instance, one
often finds 'algorithmic' ways of proving trig. identities, factoring
polynomials, calculating derivative or integrals and so on. Numeric, graphic
and symbolic softwares are actually doing the job faster and more accurately
than our brain. The focus has changed dramatically (as it changed at a lower
pace during the evolution of math, remember the logarithms, the sliding
rules, the calculators); we do not need anywore to teach how to compute but
rather what to compute, why to compute, we need to teach the mechanisms of
computation, their relevance, their standards of execution and not the
mechanics of computation.

I was no more fortunate trying to explain the meaning of math to a parent
complaining about the fact that 'kids, today, do not remember their
multiplication table, they should be drilled over and over until they know
them and until they could perform rapidly and exactly any of these
arithmetic operations.' I started the discussion by asking her if she
remembered her tables. Of course she did, after all, she spent her
elementary school years endlessly repeating multiplication tables. Then I
asked her to tell me the answer to 17x13 and how she would go about
calculating it. She replied, 'Well I need a piece of paper and a pen, watch
me now' and she started '3 times 7 is 21, I write 1 and .' By the time she
was finished, I already wrote the answer. Surprised, she asked how I did it.
When I told her that 17 is 20 - 3 or 13 is 10+3, so, 17x13 =170+30+21=221 or
17x13=260-30-9,  she looked at me as if I had just been parachuted in from
Mars. Of course, basic memorization is necessary, and procedures are useful
tools. But there is great danger in using them exclusively and missing the
very essence of math, particularly with the younger children who are
struggling with making sense of the mathematics.

Interestingly, L.M. Ampere was famous for making basic 'arithmetic'
mistakes. He said when he was using the crude memory part of his brain 17+5
was usually everything but 22. Yet, he had no trouble with 17+3+2=22 or 17.5
x 5 = 50+37.5 = 87.5 or 875 x 650 = 560000 + 8750 = 568750. Actually, for
him, these three operations took the same amount of time for they call for
the same mental process involving a sort of 'short time storage' of results
combined with a deep sense of math.

Similarly, Einstein never used the quadratic relation to solve quadratic
equations: he would first visualize the parabola, anticipated the zeroes
from the values of the coefficients, and decided the accuracy of the result
from the context (it is a waste of time to compute if real zeroes are sought
when the parabola does not intersect the x-axis).  Long term memory and
procedures were not needed, a deep understanding of the problem was the key.

That is not to say that there are not famous mathematicians who did possess
a tremendous long term memory faculty. For example, Aitken was tested the
1930's: 25 words were selected randomly from a dictionary and he had to
repeat them. When interviewed in 1961, he was still able to recite them
correctly. Von Neumann, as another example, once asked to recite the 'Tale
of Two Cities', he started without pause and continued until asked to stop
after about fifteen minutes. Colburn was what we might call an 'uneducated,
calculating prodigy': he was able to calculate mentally the square of 4,395
in less than 2 seconds. Interestingly, his abilities diminished considerably
when he gained an education and started to 'understand'. Was it just a
question of aging? (apparently, we loose 'short term memory capabilities'
and we compensate by finding other ways based of connecting concepts). Was
it the fact that the brain being occupied to perform other tasks was no
longer available for storing data?

This topic is certainly the major challenge to the process of teaching
learning and evaluating mathematics. In an interview a few weeks ago, M.
Serre (Berkeley) said 'with all the tools we have today, memory and
computation skills are becoming exponentially irrelevant, we are condemned
to become intelligent'.  He feels that the cells responsible for memory
might disappear from disuse, while new neural connections are constantly
being developed for the higher level thinking we are being forced to do.
When asked later to comment about how the human brain might evolve, he
replied that if there is a mutation to occur, this is certainly at this
level that it will occur. Since we rely more often to 'external'
memorization devices and algorithms, this part of the human brain becomes
irrelevant.'

Although it may seem trivial or cosmetic to argue about which way is better
arithmetic, the method chosen can have a profound impact on the development
of both a sense of mathematics and a mathematical sense of the world. After
all, arithmetic is the nursery for the development of most of the
foundational mathematical ideas. The metaphors of understanding on which
mathematics is built begin in arithmetic. Mathematics cannot grow out of an
arithmetic that is based on memorization and algorithms. Like most learning
the learning in mathematics is metaphorical in nature. Lakoff and Nunez in
their recent book Where Mathematics Comes From point out that concepts like
zero, infinity, square roots and the empty set are metaphorical in nature,
not procedural. They go on to suggest that the 4 grounding metaphors on
which the body of mathematics is built are learned during the learning of
arithmetic.  In a very simplified form the four metaphors are:
1. arithmetic as object collection (counting pebbles)
2. arithmetic as object construction (wholes made up of parts)
3. arithmetic as measuring stick  (distance measured in segments)
4. arithmetic as motion along a path (distance measured from a central
location)

These are the building blocks of mathematics and they can easily be lost
when we are focused finding the algorithmic way to find the answer.
J.P. leRoy, Ph.D. Nanaimo, BC, Canada
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For 14 years I taught students with learning disabilities in the Cleveland
Municipal Schools.  In many ways, when I began my work, I believed in your
general premise that arithmetic and mathematics are independent.

However, the data I collected with my own students, and with students in
some
suburban and urban schools makes me believe differently.  In essence, my
data
suggests that fluency and accuracy on arithmetic do matter.  the question is
how and why.  My own data suggests that there is an initial hump that
students need to get over, and then it doesn't matter after that point.  The
main hump being the urge to just sit and stare at the page because you feel
so helpless that you don't even want to bother to try.

Dr. Becker observed my session at last year's regional NCTM conference in
Columbus, OH and came away with a positive reaction.

I'm not sure how I can explain more in a short e-mail, but maybe someday we
can have a longer discussion--possibly the NCTM conference in Las Vegas.  It
will be especially easier once my data trail becomes longer, I finish my
dissertation, and I get a few papers published.

Sincerely,

Richard Oldrieve
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