By studying brain-damaged patients who have lost basic number skills, Dr. Dehaene and others have tentatively traced this arithmetical module to an area of the brain called the inferior parietal cortex, a poorly understood location where visual, auditory and tactile signals converge. Scientists are intrigued by clues that this region is also involved in language processing and in distinguishing right from left. Mathematics is, after all, a kind of language intimately involved with using numbers to order space.
The inferior parietal cortex also seems to be important for manual dexterity, and arithmetic begins with counting on the hands. Imaging experiments, in which people's brains are monitored as they calculate, point to the same region as a primitive number processor. If this neurological calculator has indeed been bequeathed by evolution, then traces of it should be found in other species. In making his argument, Dr. Dehaene draws on experiments over the last few decades suggesting that even rats have a rudimentary number sense. The animals were taught to press lever A four times and then lever B to get food, or to press lever A when they heard a two-tone sequence and lever B when they heard an eight-tone sequence.
To insure that the rats were responding to the number of signals and not just to their duration, the two-tone sequence sometimes lasted longer than the eight-tone one. Even more striking were later experiments in which rats were first trained to associate lever A with two tones and lever B with four tones. Then they were taught to associate A with two flashes of light and B with four flashes.
If the rats heard two tones and saw two flashes they learned to push B, not A. They seemed to have comprehended the notion that two plus two equals four.
The rats were not precise. Trained to press one lever four times, they often pressed it five or six times, expecting to be rewarded just the same, or they confused a seven-tone sequence with an eight-tone one. But the experiments support the notion of a primitive neurological number processor, even in rodents. In other experiments, chimpanzees seemed to learn simple arithmetic. Given a choice between one tray with a pile of three chocolate chips and another pile of four and a second tray with piles of two and three chips, they chose the first tray with the most candy.
But when the totals on the trays differed by only one chip, the chimps were less likely to make the discrimination. The number sense is approximate, not exact. More recent experiments on infants, using Mickey Mouse toys instead of chocolate chips, found signs of the same kind of rough numerical ability in babies less than 5 months old. Dehaene says this instinct is innate, as singing is for songbirds or spinning webs is for spiders.
Numbers are not Platonic ideals but neurological creations, artifacts of the way the brain parses the world. In that sense they are like colors. Red apples are not inherently red. They reflect light at wavelengths that the brain, as it was wired by evolution, interprets as red.
While people are born with an understanding of the rudiments of arithmetic, he contends, going beyond that requires learning and creativity. Multiplication, division and the whole superstructure of higher mathematics -- from algebra and trigonometry, to calculus, fractal geometry and beyond -- are a beautiful improvisation, the work of human culture. The ability to weave simple ideas, like two plus two equals four, into the tapestries of higher mathematics, he suggests, is not unlike the human skill for language.
People take a relatively small collection of words and, using a few simple rules of grammar and syntax, create literature. At the University of California at Berkeley, Dr. Lakoff, a linguist and cognitive scientist, and Dr. Nunez, a developmental psychologist, contend that the source of mathematics lies not just in the brain but in the human body and the physical world.
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People favor number systems based on 10 because they have 10 fingers and 10 toes. But that is just the beginning of the story. Training is what you do when you learn to operate a lathe or fill out a tax form. It means you learn how to use or operate some kind of machine or system that was produced by people in order to accomplish specific tasks. People often go to training institutes to become certified to operate a machine or perform certain skills. Then they can get jobs that directly involve those specific skills.
Education is very different. Education is not about any particular machine, system, skill, or job. Education is both broader and deeper than training. An education is a deep, complex, and organic representation of reality in the student's mind. It is an image of reality made of concepts, not facts.
Concepts that relate to each other, reinforce each other, and illuminate each other. Yet the education is more even than that because it is organic: it will live, evolve, and adapt throughout life. Education is built up with facts, as a house is with stones. But a collection of facts is no more an education than a heap of stones is a house.
An educated guess is an accurate conclusion that educated people can often "jump to" by synthesizing and extrapolating from their knowledge base. People who are good at the game "Jeopardy" do it all the time when they come up with the right question by piecing together little clues in the answer. But there is no such thing as a "trained guess. No subject is more essential nor can contribute more to becoming a liberally educated person than mathematics.
Become a math major and find out! Some people may understand all that I've said above but still feel a bit uneasy. After all, there are bills to pay.
What Is Mathematics Really? by Reuben Hersh - Penguin Books Australia
If mathematics is as I've described it, then perhaps it is no more helpful in establishing a career than, say, philosophy. Here we mathematicians have the best of both worlds, as there are many careers that open up to people who have studied mathematics. Real Mathematics, the kind I discussed above. See the Careers web page for a sampling. About twenty years ago when personal computers were becoming more common in small businesses and private homes, I was having lunch with a few people, and it came up that I was a mathematician.
One of the other diners got a funny sort of embarrassed look on her face. I steeled myself for that all too common remark, "Oh I was never any good at math. It turned out that she was thinking that with computers becoming so accurate, fast, and common, there was no longer any need for mathematicians! She was feeling sorry me, as I would soon be unemployed! Apparently she thought that a mathematician's work was to crank out arithmetic computations. Nothing could be farther from the truth. Thinking that computers will obviate the need for mathematicians is like thinking 90 years ago when cars replaced horse drawn wagons, there would be no more need for careful drivers.
On the contrary, powerful engines made careful drivers more important than ever. Today, powerful computers and good software make it possible to use and concretely implement abstract mathematical ideas that have existed for many years. For example, the RSA cryptosystem is widely used on secure internet web pages to encode sensitive information, like credit card numbers.
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It is based on ideas in algebraic number theory, and its invulnerability to hackers is the result of very advanced ideas in that field. Finally, here are a few quotes from an essay well worth reading by David R. Garcia on a similar topic:.
- The Economist - 15 September 2001?
- Blood Feud: The Hatfields and the McCoys: The Epic Story of Murder and Vengeance.
- Managing built heritage: the role of cultural values and significance.
- Fumetti Di Cinema.
Americans like technology but seldom have a grasp of the science behind it. And the mathematics that is behind the science is regarded as even more mysterious, like an inner sanctum into which only initiates may gain entry. They see the rich and nourishing technological fruit on this tree of knowledge, but they see no deeper than the surface branches and twigs on which these fruits grow.
To them, the region behind this exterior of the tree, where the trunk and limbs grow, is pointless and purposeless. What is it that would cause us to focus only on this external fruit of material development and play down the antecedent realms of abstraction that lie deeper? It would be good to find a word less condemning than "superficiality", but how else can this tendency be described in a word? Perhaps facing up to the ugly side of this word can stir us into action to remedy what seems to be an extremely grave crisis in Western education.
The first step toward [progress in crucial social problems] is to recognize the deceptive illusions bred by seeing only the surface of issues, of seeing only a myriad of small areas to be dealt with by specialists, one for each area. Piecemeal superficiality won't work. Teaching is not a matter of pouring knowledge from one mind into another as one pours water from one glass into another. It is more like one candle igniting another. Each candle burns with its own fuel. The true teacher awakens a love for truth and beauty in the heart--not the mind--of a student after which the student moves forward with powerful interest under the gentle guidance of the teacher.
Isn't it interesting how the mention of these two most important goals of learning--truth and beauty--now evokes snickers and ridicule, almost as if by instinct, from those who shrink from all that is not superficial. These kinds of teachers will inspire love of mathematics, while so many at present diffuse a distaste for it through their own ignorance and clear lack of delight in a very delightful subject. Around December 19, , this essay was "discovered" somehow and attracted an enormous amount of attention. For many hours, the web page was getting a hit every second!
This is very gratifying, and I am grateful for the overwhelmingly positive response. Many people said it is the best thing they have ever read on the subject of mathematics education. Let me address a point that has come up. This essay was first developed around and appeared online in By January of it contained all of the "parables" except "Cargo Cult Education", which was added around , and "Step High", which was added around The reference to the essay by David Garcia was added sometime around Therefore, any similarity to more recent essays by others is either convergent evolution, or must be explained by those others.
Also, someone pointed out to me that Richard Feynman mentions "cargo-cult science" in one of his books. I am not aware of ever reading that passage in his books, which I greatly admire. The first time I ever read a clear description of training vs. I believe that appeared in the mid s. The comments that were emailed to me by readers can be found on my site.
Key phrases: Mathematics education, improving mathematics education, improving math education, high school math education, misconceptions about mathematics. What is Mathematics? Robert H. Lewis Professor of Mathematics, Fordham University For more than two thousand years, mathematics has been a part of the human search for understanding. Why do so many people have such misconceptions about mathematics?
What is mathematics really like? Scaffolding When a new building is made, a skeleton of steel struts called the scaffolding is put up first. Ready for the Big Play Professional athletes spend hours in gyms working out on equipment of all sorts. The hostile party goer. Reuben Hersh. He is a winner of the Chauvenet Prize and the Ford Prize. He is the author with Philip J.
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