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 Š.' 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.
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