problemas do ensino de matemática

7/25/2019 Problemas do ensino de matemtica

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Basic Issues with Math/Science Education

The following article appeared in a recent issue of the Wall Street Journal.

Study Finds Sharp Math, Science

Skills Help Expand Economy

By SARA MURRAYMarch 3, 2008; Page A2

Increased years of education boost economic growth but only if students cognitiveskills, as measured by math and science tests, are improved as a result, a new studysays.

The study, released in this springs issue of Education Next, an education-policyjournal, concluded that if the U.S. performed on par with the worlds leaders inscience and math, it would add about two-thirds of a percentage point to the grossdomestic product, or the total value of goods and services produced in a nation, everyyear.

Those findings diverge from other research that links economic growth to the numberof years of students education. The problem with that research, say study authorsEric Hanushek, a Stanford University professor, and Ludger Woessmann of the Uni-versity of Munich, is that it assumes that a year of schooling in a country like Ghana,for example, is equivalent to a year in the U.S. Instead, it is more important to em-

phasize what people know, not how long people have sat in the classroom, Mr.Hanushek said.

Weve tried all kinds of things but they havent been very effective. To me it sayswe just need to take this much more seriously.

Nearly two decades ago, the National Governors Association called for U.S. studentsto sharply improve in math and science by 2000. If the U.S. had managed to achievethe goal, and joined world leaders like Finland, Hong Kong and South Korea, GDPwould be two percentage points higher today and 4.5 points higher in 2015, the studycalculated. Had we figured out some way to improve our schools, or do what we

could to improve the learning of our students, we would be a lot better offtoday,said Mr. Hanushek.

The research supports the idea that students performance defines the effectivenessof education. It raises questions about the U.S. approach, which focuses more onpouring resources into the front end such as spending money to reduce class sizes.

The U.S. has had this very naive assumption that if you just pour more inputs intoeducation then youll get more output, said William Easterly, an economics professorat New York University who wasnt involved in the study. But he said it could be a

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leap to conclude that cognitive skills could yield a specific level of economic growth.

Because so many macroeconomic factors affect GDP, it is difficult to determine the

precise impact of any one of them, he said. In the U.S., nearly every state developsits own curriculum, and while math curricula tend to be more standardized, sciencevaries greatly. In some school districts, children rarely encounter science in elemen-tary school, said Gerald Wheeler, executive director of the National Science TeachersAssociation, which is based in Arlington, Va.

Experts dont agree on how to improve students performances. One suggestion isto create national standards for science and math designed by teachers, scientistsand mathematicians, said William Schmidt, a Michigan State University educationprofessor who works on international science testing. But, he said, those standardsmust be enforced in a way that doesnt encourage teachers to base classes on a test

one of the criticisms of the No Child Left Behind law.

This should not be a surprise. After all, math and science are the foundations for ourmodern society, with mathematics playing a much more central roll than one might expect.But actually, the stakes are even higher than we might guess as the following articledescribes

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We can give a further idea of the seriousness of the situation when we look at theage data for the scientists and engineers at NASA. The data below shows the number ofthese employees in each five year age group for the years 1998, 2003 and late 2007 so that

we can easily compare the changes in the age distribution. Note that very few hires havebeen made over the last 10 years of younger people, and that the situation is becomingquite uncertain, as huge numbers of retirements are looming just as NASA ramps up forthe manned Mars mission and returning to the moon. Note that as a government agency,NASA is required to hire only U.S. citizens.

Here is another picture of the situation, taken from U.S. Dept. of Education data.

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In this Contest, China Beats UC-BerkeleyLisa M. Krieger

Mercury News, July 18, 2008

The pipeline into Americas Ph.D. programs used to flow from Berkeley. Now, thehighest number of students pour in from Chinas Tsinghua and Beijing universities.

Trailing third is the University of California-Berkeley, which had been the top sourceof students to the nations doctoral studies for over a decade, according to an analysisby the National Science Foundation. Of the top 10 schools producing future doctoralcandidates, only six are U.S. schools.

In fact, more than one-third of all people in the United States earning doctorates inscience and engineering come from somewhere else. About 13 percent - or 4,236 people -hail from China. Science and engineering account for the bulk of the doctorates in thiscountry.

The startling trend reflects declining interest among American youngsters in science,engineering and math, experts agree. But its also a tribute to Americas doctoral pro-grams, which attract the best from around the world.

The international students are coming very well-prepared - theyre high-quality stu-dents, said Dennis Hengstler, UC-Berkeleys assistant vice chancellor of planning and

analysis.

There has been, over last decade, a strong emphasis in China to produce morescientists and engineers, he said. Why is this a strong interest in China, and less of aninterest for students graduating from U.S. colleges? It is a concern for our nations futurecompetitiveness.

The Survey of Earned Doctorates, administered by the National Science Foundation,is a census of all doctoral recipients in the United States. Its findings were reported in themost recent issue of the journal Science.

The United States needs to improve its K-12 education and better value science ifit wants to regain its position, said Charles Vest, president of the National Academy ofEngineering.

At Tsinghua and Beijing universities, undergraduates are selected through an ex-tremely intense process from across a nation of 1.3 billion people. They are smart, cele-brated and highly motivated, said Vest, former president of the Massachusetts Instituteof Technology.

At Intel, the director of research, Andrew Chien says he increasingly hires foreign-born

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- yet American-educated - Ph.D.s.

The ease of global interchange has fueled the trend, he said.

But there is a tangible decline in the numbers of Americans interested in pursuingadvanced degrees in science and engineering, he said.

Some experts say American students are choosing better-paying fields such as financeand medicine, or more purely creative careers, such as the arts.

But Intels Chien said a doctoral degree holds the keys to the kingdom, in terms ofunleashing what is possible in the world and driving change.

From issues ranging from global warming to ecological sustainability, a deep under-

standing is needed to explore solutions that are technologically feasible, he said.

Many students wrongly assume that a doctorate is required only for a career in long-range research, he said. But at Intel, we expect that depth of understanding and ex-perience in all areas of the organization, such as high-quality manufacturing, design andproduction.

The good news is that the U.S. universities continue to be a magnet for the best studentsfrom overseas.

Chinas schools, which have mastered rigorous rote learning, now are scrambling toreplicate the U.S. doctoral programs in an effort to inspire more creativity in their students.

We weave together teaching and research in a way that keeps the overall learningfresh, cutting-edge, Vest said. Although other countries are adopting the U.S. model,

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teaching and learning are more open and engaging than is traditionally the case in manyother countries, he added.

Many who come for doctoral study decide to stay - and contribute to the nationsinnovation. One recent survey found that 93 percent of all new doctorate recipients holdingpermanent visas and 65 percent of temporary visa holders said they would remain in theUnited States after graduation.

The U.S. educational system continues to be the envy of the world, Chien said.

Detailed data from SRS Study of U.S. Doctoral Awards.

http://www.nsf.gov/statistics/infbrief/nsf08301/

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How Much Mathematics is Hard Wired?

Numbers GuyAre our brains wired for math?

by Jim Holt

The New Yorker Magazine, March 3, 2008.

According to Stanislas Dehaene, humans have an inbuilt number sense capable ofsome basic calculations and estimates. The problems start when we learn mathematicsand have to perform procedures that are anything but instinctive.

One morning in September, 1989, a former sales representative in his mid-forties

entered an examination room with Stanislas Dehaene, a young neuroscientist based inParis. Three years earlier, the man, whom researchers came to refer to as Mr. N, hadsustained a brain hemorrhage that left him with an enormous lesion in the rear half ofhis left hemisphere. He suffered from severe handicaps: his right arm was in a sling; hecouldnt read; and his speech was painfully slow. He had once been married, with twodaughters, but was now incapable of leading an independent life and lived with his elderlyparents. Dehaene had been invited to see him because his impairments included severeacalculia, a general term for any one of several deficits in number processing. When askedto add 2 and 2, he answered three. He could still count and recite a sequence like 2, 4,6, 8, but he was incapable of counting downward from 9, differentiating odd and evennumbers, or recognizing the numeral 5 when it was flashed in front of him.

To Dehaene, these impairments were less interesting than the fragmentary capabilitiesMr. N had managed to retain. When he was shown the numeral 5 for a few seconds, heknew it was a numeral rather than a letter and, by counting up from 1 until he got to theright integer, he eventually identified it as a 5. He did the same thing when asked the ageof his seven-year-old daughter. In the 1997 book The Number Sense, Dehaene wrote, Heappears to know right from the start what quantities he wishes to express, but recitingthe number series seems to be his only means of retrieving the corresponding word.

Dehaene also noticed that although Mr. N could no longer read, he sometimes hadan approximate sense of words that were flashed in front of him; when he was shown the

word ham, he said, Its some kind of meat. Dehaene decided to see if Mr. N still had asimilar sense of number. He showed him the numerals 7 and 8. Mr. N was able to answerquickly that 8 was the larger number - far more quickly than if he had had to identifythem by counting up to the right quantities. He could also judge whether various numberswere bigger or smaller than 55, slipping up only when they were very close to 55. Dehaenedubbed Mr. N the Approximate Man. The Approximate Man lived in a world where ayear comprised about 350 days and an hour about fifty minutes, where there were fiveseasons, and where a dozen eggs amounted to six or ten. Dehaene asked him to add 2 and2 several times and received answers ranging from three to five. But, he noted, he neveroffers a result as absurd as 9.

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In cognitive science, incidents of brain damage are natures experiments. If a lesionknocks out one ability but leaves another intact, it is evidence that they are wired intodifferent neural circuits. In this instance, Dehaene theorized that our ability to learn

sophisticated mathematical procedures resided in an entirely diff

erent part of the brainfrom a rougher quantitative sense. Over the decades, evidence concerning cognitive deficitsin brain-damaged patients has accumulated, and researchers have concluded that we havea sense of number that is independent of language, memory, and reasoning in general.Within neuroscience, numerical cognition has emerged as a vibrant field, and Dehaene,now in his early forties, has become one of its foremost researchers. His work is completelypioneering, Susan Carey, a psychology professor at Harvard who has studied numericalcognition, told me. If you want to make sure the math that children are learning ismeaningful, you have to know something about how the brain represents number at thekind of level that Stan is trying to understand.

Dehaene has spent most of his career plotting the contours of our number sense andpuzzling over which aspects of our mathematical ability are innate and which are learned,and how the two systems overlap and affect each other. He has approached the problemfrom every imaginable angle. Working with colleagues both in France and in the UnitedStates, he has carried out experiments that probe the way numbers are coded in our minds.He has studied the numerical abilities of animals, of Amazon tribespeople, of top Frenchmathematics students. He has used brain-scanning technology to investigate preciselywhere in the folds and crevices of the cerebral cortex our numerical faculties are nestled.And he has weighed the extent to which some languages make numbers more difficult thanothers. His work raises crucial issues about the way mathematics is taught. In Dehaenesview, we are all born with an evolutionarily ancient mathematical instinct. To become

numerate, children must capitalize on this instinct, but they must also unlearn certaintendencies that were helpful to our primate ancestors but that clash with skills neededtoday. And some societies are evidently better than others at getting kids to do this. Inboth France and the United States, mathematics education is often felt to be in a state ofcrisis. The math skills of American children fare poorly in comparison with those of theirpeers in countries like Singapore, South Korea, and Japan. Fixing this state of affairsmeans grappling with the question that has taken up much of Dehaenes career: What isit about the brain that makes numbers sometimes so easy and sometimes so hard?

Dehaenes own gifts as a mathematician are considerable. Born in 1965, he grew up inRoubaix, a medium-sized industrial city near Frances border with Belgium. (His surname

is Flemish.) His father, a pediatrician, was among the first to study fetal alcohol syndrome.As a teen-ager, Dehaene developed what he calls a passion for mathematics, and heattended the Ecole Normale Superieure in Paris, the training ground for Frances scholarlyelite. Dehaenes own interests tended toward computer modelling and artificial intelligence.He was drawn to brain science after reading, at the age of eighteen, the 1983 book NeuronalMan, by Jean-Pierre Changeux, Frances most distinguished neurobiologist. Changeuxsapproach to the brain held out the tantalizing possibility of reconciling psychology withneuroscience. Dehaene met Changeux and began to work with him on abstract models ofthinking and memory. He also linked up with the cognitive scientist Jacques Mehler. Itwas in Mehlers lab that he met his future wife, Ghislaine Lambertz, a researcher in infant

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cognitive psychology.

By pure luck, Dehaene recalls, Mehler happened to be doing research on how num-

bers are understood. This led to Dehaenes first encounter with what he came to charac-terize as the number sense. Dehaenes work centered on an apparently simple question:How do we know whether numbers are bigger or smaller than one another? If you areasked to choose which of a pair of Arabic numerals - 4 and 7, say - stands for the biggernumber, you respond seven in a split second, and one might think that any two digitscould be compared in the same very brief period of time. Yet in Dehaenes experiments,while subjects answered quickly and accurately when the digits were far apart, like 2 and9, they slowed down when the digits were closer together, like 5 and 6. Performance alsogot worse as the digits grew larger: 2 and 3 were much easier to compare than 7 and8. When Dehaene tested some of the best mathematics students at the Ecole Normale,the students were amazed to find themselves slowing down and making errors when asked

whether 8 or 9 was the larger number.

Dehaene conjectured that, when we see numerals or hear number words, our brainsautomatically map them onto a number line that grows increasingly fuzzy above 3 or 4.He found that no amount of training can change this. It is a basic structural property ofhow our brains represent number, not just a lack of facility, he told me.

In 1987, while Dehaene was still a student in Paris, the American cognitive psychol-ogist Michael Posner and colleagues at Washington University in St. Louis published apioneering paper in the journal Nature. Using a scanning technique that can track theflow of blood in the brain, Posners team had detailed how different areas became active

in language processing. Their research was a revelation for Dehaene. I remember verywell sitting and reading this paper, and then debating it with Jacques Mehler, my Ph.D.adviser, he told me. Mehler, whose focus was on determining the abstract organizationof cognitive functions, didnt see the point of trying to locate precisely where in the brainthings happened, but Dehaene wanted to bridge the gap, as he put it, between psychologyand neurobiology, to find out exactly how the functions of the mind - thought, perception,feeling, will - are realized in the gelatinous three-pound lump of matter in our skulls. Now,thanks to new technologies, it was finally possible to create pictures, however crude, ofthe brain in the act of thinking. So, after receiving his doctorate, he spent two yearsstudying brain scanning with Posner, who was by then at the University of Oregon, inEugene. It was very strange to find that some of the most exciting results of the budding

cognitive-neuroscience field were coming out of this small place, the only place where Iever saw sixty-year-old hippies sitting around in tie-dyed shirts he said.

Dehaene is a compact, attractive, and genial man; he dresses casually, wears fash-ionable glasses, and has a glabrous dome of a head, which he protects from the elementswith a chapeau de cowboy. When I visited him recently, he had just moved into a newlaboratory, known as NeuroSpin, on the campus of a national center for nuclear-energyresearch, a dozen or so miles southwest of Paris. The building, which was completed a yearago, is a modernist composition in glass and metal filled with the ambient hums and whirsand whooshes of brain-scanning equipment, much of which was still being assembled. A

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series of arches ran along one wall in the form of a giant sine wave; behind each was aconcrete vault built to house a liquid-helium-cooled superconducting electromagnet. (Inbrain imaging, the more powerful the magnetic field, the sharper the picture.) The new

brain scanners are expected to show the human cerebral anatomy at a level of detail neverbefore seen, and may reveal subtle anomalies in the brains of people with dyslexia andwith dyscalculia, a crippling deficiency in dealing with numbers which, researchers sus-pect, may be as widespread as dyslexia. One of the scanners was already up and running.You dont wear a pacemaker or anything, do you? Dehaene asked me as we entered aroom where two researchers were fiddling with controls. Although the scanner was built toaccommodate humans, inside, I could see from the monitor, was a brown rat. Researcherswere looking at how its brain reacted to various odors, which were puffed in every so often.Then Dehaene led me upstairs to a spacious gallery where the brain scientists working atNeuroSpin are expected to congregate and share ideas. At the moment, it was empty.Were hoping for a coffee machine, he said.

Dehaene has become a scanning virtuoso. On returning to France after his time withPosner, he pressed on with the use of imaging technologies to study how the mind processesnumbers. The existence of an evolved number ability had long been hypothesized, basedon research with animals and infants, and evidence from brain-damaged patients gaveclues to where in the brain it might be found. Dehaene set about localizing this facilitymore precisely and describing its architecture. In one experiment I particularly liked,he recalled, we tried to map the whole parietal lobe in a half hour, by having the subjectperform functions like moving the eyes and hands, pointing with fingers, grasping anobject, engaging in various language tasks, and, of course, making small calculations, likethirteen minus four. We found there was a beautiful geometrical organization to the areas

that were activated. The eye movements were at the back, the hand movements were inthe middle, grasping was in the front, and so on. And right in the middle, we were ableto confirm, was an area that cared about number.

The number area lies deep within a fold in the parietal lobe called the intraparietalsulcus (just behind the crown of the head). But it isnt easy to tell what the neurons thereare actually doing. Brain imaging, for all the sophistication of its technology, yields a fairlycrude picture of whats going on inside the skull, and the same spot in the brain mightlight up for two tasks even though different neurons are involved. Some people believethat psychology is just being replaced by brain imaging, but I dont think thats the case atall, Dehaene said. We need psychology to refine our idea of what the imagery is going

to show us. Thats why we do behavioral experiments, see patients. Its the confrontationof all these different methods that creates knowledge.

Dehaene has been able to bring together the experimental and the theoretical sides ofhis quest, and, on at least one occasion, he has even theorized the existence of a neurologicalfeature whose presence was later confirmed by other researchers. In the early nineteen-nineties, working with Jean-Pierre Changeux, he set out to create a computer model tosimulate the way humans and some animals estimate at a glance the number of objectsin their environment. In the case of very small numbers, this estimate can be made withalmost perfect accuracy, an ability known as subitizing (from the Latin word subitus,

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meaning sudden). Some psychologists think that subitizing is merely rapid, unconsciouscounting, but others, Dehaene included, believe that our minds perceive up to three orfour objects all at once, without having to mentally spotlight them one by one. Getting

the computer model to subitize the way humans and animals did was possible, he found,only if he built in number neurons tuned to fire with maximum intensity in responseto a specific number of objects. His model had, for example, a special four neuron thatgot particularly excited when the computer was presented with four objects. The modelsnumber neurons were pure theory, but almost a decade later two teams of researchersdiscovered what seemed to be the real item, in the brains of macaque monkeys that hadbeen trained to do number tasks. The number neurons fired precisely the way Dehaenesmodel predicted - vindication of theoretical psychology. Basically, we can derive thebehavioral properties of these neurons from first principles, he told me. Psychology hasbecome a little more like physics.

But the brain is the product of evolution - a messy, random process - and thoughthe number sense may be lodged in a particular bit of the cerebral cortex, its circuitryseems to be intermingled with the wiring for other mental functions. A few years ago,while analyzing an experiment on number comparisons, Dehaene noticed that subjectsperformed better with large numbers if they held the response key in their right hand butdid better with small numbers if they held the response key in their left hand. Strangely,if the subjects were made to cross their hands, the effect was reversed. The actual handused to make the response was, it seemed, irrelevant; it was space itself that the subjectsunconsciously associated with larger or smaller numbers. Dehaene hypothesizes that theneural circuitry for number and the circuitry for location overlap. He even suspects thatthis may be why travellers get disoriented entering Terminal 2 of Pariss Charles de Gaulle

Airport, where small-numbered gates are on the right and large-numbered gates are onthe left. Its become a whole industry now to see how we associate number to space andspace to number, Dehaene said. And were finding the association goes very, very deepin the brain.

Last winter, I saw Dehaene in the ornate setting of the Institut de France, across theSeine from the Louvre. There he accepted a prize of a quarter of a million euros fromLiliane Bettencourt, whose father created the cosmetics group LOreal. In a salon hungwith tapestries, Dehaene described his research to a small audience that included a for-mer Prime Minister of France. New techniques of neuroimaging, he explained, promise toreveal how a thought process like calculation unfolds in the brain. This isnt just a matter

of pure knowledge, he added. Since the brains architecture determines the sort of abilitiesthat come naturally to us, a detailed understanding of that architecture should lead tobetter ways of teaching children mathematics and may help close the educational gap thatseparates children in the West from those in several Asian countries. The fundamentalproblem with learning mathematics is that while the number sense may be genetic, exactcalculation requires cultural tools - symbols and algorithms - that have been around foronly a few thousand years and must therefore be absorbed by areas of the brain thatevolved for other purposes. The process is made easier when what we are learning harmo-nizes with built-in circuitry. If we cant change the architecture of our brains, we can atleast adapt our teaching methods to the constraints it imposes.

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For nearly two decades, American educators have pushed reform math, in whichchildren are encouraged to explore their own ways of solving problems. Before reformmath, there was the new math, now widely thought to have been an educational disaster.

(In France, it was called les maths modernes, and is similarly despised.) The new math wasgrounded in the theories of the influential Swiss psychologist Jean Piaget, who believedthat children are born without any sense of number and only gradually build up theconcept in a series of developmental stages. Piaget thought that children, until the age offour or five, cannot grasp the simple principle that moving objects around does not affecthow many of them there are, and that there was therefore no point in trying to teach themarithmetic before the age of six or seven.

Piagets view had become standard by the nineteen-fifties, but psychologists have sincecome to believe that he underrated the arithmetic competence of small children. Six-month-old babies, exposed simultaneously to images of common objects and sequences of

drumbeats, consistently gaze longer at the collection of objects that matches the number ofdrumbeats. By now, it is generally agreed that infants come equipped with a rudimentaryability to perceive and represent number. (The same appears to be true for many kindsof animals, including salamanders, pigeons, raccoons, dolphins, parrots, and monkeys.)And if evolution has equipped us with one way of representing number, embodied in theprimitive number sense, culture furnishes two more: numerals and number words. Thesethree modes of thinking about number, Dehaene believes, correspond to distinct areas ofthe brain. The number sense is lodged in the parietal lobe, the part of the brain thatrelates to space and location; numerals are dealt with by the visual areas; and numberwords are processed by the language areas.

Nowhere in all this elaborate brain circuitry, alas, is there the equivalent of the chipfound in a five-dollar calculator. This deficiency can make learning that terrible quartet -Ambition, Distraction, Uglification, and Derision, as Lewis Carroll burlesqued them - achore. Its not so bad at first. Our number sense endows us with a crude feel for addition,so that, even before schooling, children can find simple recipes for adding numbers. Ifasked to compute 2 + 4, for example, a child might start with the first number and thencount upward by the second number: two, three is one, four is two, five is three, six isfour, six. But multiplication is another matter. It is an unnatural practice, Dehaeneis fond of saying, and the reason is that our brains are wired the wrong way. Neitherintuition nor counting is of much use, and multiplication facts must be stored in thebrain verbally, as strings of words. The list of arithmetical facts to be memorized may

be short, but it is fiendishly tricky: the same numbers occur over and over, in differentorders, with partial overlaps and irrelevant rhymes. (Bilinguals, it has been found, revertto the language they used in school when doing multiplication.) The human memory,unlike that of a computer, has evolved to be associative, which makes it ill-suited toarithmetic, where bits of knowledge must be kept from interfering with one another: ifyoure trying to retrieve the result of multiplying 7 6, the reflex activation of 7 + 6 and7 5 can be disastrous. So multiplication is a double terror: not only is it remote fromour intuitive sense of number; it has to be internalized in a form that clashes with theevolved organization of our memory. The result is that when adults multiply single-digitnumbers they make mistakes ten to fifteen per cent of the time. For the hardest problems,

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like 7 8, the error rate can exceed twenty-five per cent.

Our inbuilt ineptness when it comes to more complex mathematical processes has

led Dehaene to question why we insist on drilling procedures like long division into ourchildren at all. There is, after all, an alternative: the electronic calculator. Give acalculator to a five-year-old, and you will teach him how to make friends with numbersinstead of despising them, he has written. By removing the need to spend hundreds ofhours memorizing boring procedures, he says, calculators can free children to concentrateon the meaning of these procedures, which is neglected under the educational status quo.This attitude might make Dehaene sound like a natural ally of educators who advocatereform math, and a natural foe of parents who want their childrens math teachers to goback to basics. But when I asked him about reform math he wasnt especially sympathetic.The idea that all children are different, and that they need to discover things their ownway - I dont buy it at all. he said. I believe there is one brain organization. We see

it in babies, we see it in adults. Basically, with a few variations, were all travelling onthe same road. He admires the mathematics curricula of Asian countries like China andJapan, which provide children with a highly structured experience, anticipating the kindof responses they make at each stage and presenting them with challenges designed tominimize the number of errors. Thats what were trying to get back to in France, hesaid. Working with his colleague Anna Wilson, Dehaene has developed a computer gamecalled The Number Race to help dyscalculic children. The software is adaptive, detectingthe number tasks where the child is shaky and adjusting the level of difficulty to maintainan encouraging success rate of seventy-five per cent.

Despite our shared brain organization, cultural differences in how we handle numbers

persist, and they are not confined to the classroom. Evolution may have endowed uswith an approximate number line, but it takes a system of symbols to make numbersprecise - to crystallize them, in Dehaenes metaphor. The Munduruku, an Amazon tribethat Dehaene and colleagues, notably the linguist Pierre Pica, have studied recently, havewords for numbers only up to five. (Their word for five literally means one hand.)Even these words seem to be merely approximate labels for them: a Munduruku who isshown three objects will sometimes say there are three, sometimes four. Nevertheless, theMunduruku have a good numerical intuition. They know, for example, that fifty plusthirty is going to be larger than sixty, Dehaene said. Of course, they do not know thisverbally and have no way of talking about it. But when we showed them the relevant setsand transformations they immediately got it.

The Munduruku, it seems, have developed few cultural tools to augment the inbornnumber sense. Interestingly, the very symbols with which we write down the countingnumbers bear the trace of a similar stage. The first three Roman numerals, I, II, and III,were formed by using the symbol for one as many times as necessary; the symbol for four,IV, is not so transparent. The same principle applies to Chinese numerals: the first threeconsist of one, two, and three horizontal bars, but the fourth takes a different form. EvenArabic numerals follow this logic: 1 is a single vertical bar; 2 and 3 began as two and threehorizontal bars tied together for ease of writing. (Thats a beautiful little fact, but I dontthink its coded in our brains any longer, Dehaene observed.)

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Today, Arabic numerals are in use pretty much around the world, while the wordswith which we name numbers naturally differ from language to language. And, as Dehaeneand others have noted, these differences are far from trivial. English is cumbersome. There

are special words for the numbers from 11 to 19, and for the decades from 20 to 90. Thismakes counting a challenge for English-speaking children, who are prone to such errorsas twenty-eight, twenty-nine, twenty-ten, twenty-eleven. French is just as bad, withvestigial base-twenty monstrosities, like quatre-vingt-dix-neuf (four twenty ten nine) for99. Chinese, by contrast, is simplicity itself; its number syntax perfectly mirrors thebase-ten form of Arabic numerals, with a minimum of terms. Consequently, the averageChinese four-year-old can count up to forty, whereas American children of the same agestruggle to get to fifteen. And the advantages extend to adults. Because Chinese numberwords are so brief - they take less than a quarter of a second to say, on average, comparedwith a third of a second for English - the average Chinese speaker has a memory span ofnine digits, versus seven digits for English speakers. (Speakers of the marvellously efficient

Cantonese dialect, common in Hong Kong, can juggle ten digits in active memory.)

In 2005, Dehaene was elected to the chair in experimental cognitive psychology atthe College de France, a highly prestigious institution founded by Francis I in 1530. Thefaculty consists of just fifty-two scholars, and Dehaene is the youngest member. In hisinaugural lecture, Dehaene marvelled at the fact that mathematics is simultaneously aproduct of the human mind and a powerful instrument for discovering the laws by whichthe human mind operates. He spoke of the confrontation between new technologies likebrain imaging and ancient philosophical questions concerning number, space, and time.And he pronounced himself lucky to be living in an era when advances in psychology andneuroimaging are combining to render visible the hitherto invisible realm of thought.

For Dehaene, numerical thought is only the beginning of this quest. Recently, hehas been pondering how the philosophical problem of consciousness might be approachedby the methods of empirical science. Experiments involving subliminal number primingshow that much of what our mind does with numbers is unconscious, a finding that has ledDehaene to wonder why some mental activity crosses the threshold of awareness and somedoesnt. Collaborating with a couple of colleagues, Dehaene has explored the neural basisof what is known as the global workspace theory of consciousness, which has elicited keeninterest among philosophers. In his version of the theory, information becomes consciouswhen certain workspace neurons broadcast it to many areas of the brain at once, makingit simultaneously available for, say, language, memory, perceptual categorization, action-

planning, and so on. In other words, consciousness is cerebral celebrity, as the philosopherDaniel Dennett has described it, or fame in the brain.

In his office at NeuroSpin, Dehaene described to me how certain extremely longworkspace neurons might link far-flung areas of the human brain together into a singlepulsating circuit of consciousness. To show me where these areas were, he reached intoa closet and pulled out an irregularly shaped baby-blue plaster object, about the size ofa softball. This is my brain! he announced with evident pleasure. The model thathe was holding had been fabricated, he explained, by a rapid-prototyping machine (asort of three-dimensional printer) from computer data obtained from one of the many

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MRI scans that he has undergone. He pointed to the little furrow where the numbersense was supposed to be situated, and observed that his had a somewhat uncommonshape. Curiously, the computer software had identified Dehaenes brain as an outlier, so

dissimilar are its activation patterns from the human norm. Cradling the pastel-coloredlump in his hands, a model of his mind devised by his own mental efforts, Dehaene pausedfor a moment. Then he smiled and said, So, I kind of like my brain.

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The Whole Numbers

The most basic objects in mathematics are the natural or whole numbers, 0, 1, 2, . . ..In mathematics instruction in the high achieving countries such as the former iron curtaincountries, China, Singapore, Korea, and Japan, instruction starts in their first lessons inthe first grade with an introduction of these numbers and their basic properties.

Instruction in the United States also starts with the whole numbers. But the focusturns out to be completely different.

Here is the first part of the introduction to the book for the first Methods Course inthe Hungarian Mathematics Education curriculum:

When a parent takes his child to first grade he is usually happy to tell the teacher thatlittle Tommy can count till 100 already. Pride is a good sign, because it signals: itsimportant for the parent to see his child develop not only physically, but intellectuallyas well. Sometimes it soon turns out (and the sooner the better) that the correctlynamed numbers by this same child have no content; the child knows the numerals,but not the concept of numbers. He can say the words in order: one, two, three ...thirty eight, thirty nine, forty, forty one ..., but he cannot really tell which one ismore: 5 apples or 7 apples, or have a clear picture about the order of size, equalitiesor the contents of the numerals.

We need to add quickly that its not a problem: building the concept of numbers

is the task of the school. If a child doesnt possess the concept of numbers whenstarting school, then he will learn it with the teachers help; its not too late in thefirst grade. The only thing is that the child who is ahead, will need to do somethingelse, than the one who is just learning. (Its not an easy job to adapt to differentneeds, but developing starts with learning about the different levels: its importantfor the conscious developmental process.) Concept building has many components:it consists of forming many thoughts that can be separated in theory, whereas inpractice they appear together, reinforcing each other.

Main contents of the forming of the concept of natural numbers:

Connection between numbers and reality;

Writing and reading numbers; place value form and numerical system form ofnumbers;

Magnitude of numbers

The many different names of numbers;

Properties of numbers, relations of numbers.

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The forming of the concept of numbers begins way before the age of 6 and doesntend in lower grades. We will now examine the different content components from thechilds first experiences till the end of grade 4.

In sharp contrast to this, here are the introductory remarks in the chapter on DevelopingEarly Number Concepts and Number Sensein one of the best selling mathematics methodstexts in the United States:

Number is a complex and multifaceted concept. A rich understanding of number,a relational understanding, involves many different ideas, relationships, and skills.Children come to school with many ideas about number. These ideas shoud be builtupon as we work with children and help them develop new relationships. It is sad tosee the large number of students in grades 5 and above who essentially know littlemore about number than how to count. It takes time and lots of experiences for

children to develop a full understnading of number that will grow and enhance all thefurther number-related concepts of the school year.

This chapter looks at the development of number ideas for number up to about 20.These founcational ideas can all be extended to larger number, operations, basic facts,and compuation.

Then a list of three big ideasis given:

1. Counting tells how many things are in a set. When counting a set of objects, the lastword in the counting sequence names the quantity for that set.

2. Numbers are related to each other through a variety of number relationships. Thenumber 7, for example, is more than 4, two less than 9, composed of 3 and 4 ans wellas 2 and 5, is three away from 10, and can be quickly recognized in several patternedarrangements of dots.

3. Number concepts are intimately tied to the world around us. Application of numberrelationships to the real world marks the beginning of making sense of the world in amathematical manner.

Note that in the U.S. book there is a clearchild centeredapproach: These ideas should be

built upon as we work with children. It is clearly implied that whatever the concepts ofnumber children come to school with, they are valid. Contrast this with the perspective inthe Hungarian text Sometimes it turns out (and the sooner the better) that the correctlynamed number by this same child have no content, togther with the statement thatbuilding the concept of numbers is the task of the school.

Additionally, it would appear that whole numbers are regarded as somewhat myste-rious objects in the United States imbued with profound depths, ideas and relationships.But the big ideas about number focus first on counting, then there are unnamed rela-tionships, and finally there are unnamed connections to the world around us. By contrast,

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the introduction to the Hungarian book is very clear about what a number is and thatcounting properly is only a use of numbers, not a property of numbers. This distinctionwill turn out to be profoundly important.

In this chapter we will first give a rigorous definition of the whole numbers as theyare understood by mathematicians. Then we will present the first lessons in the RussianFirst Grade books from the 1980s. The vocabulary used in the rigorous definitions is not,of course, available to first grade students, yet, by comparing the material in the actuallessons with the Peano axioms, it should be clear that using grade appropriate language these early lessons are setting forth the same properties, and in much the same way.

The Peano axioms define the properties of whole numbers, usually represented as aset0. The first four axioms describe the equality relation

(1) For every whole number x,x= x. That is, equality is reflexive.

(2) For all whole numbersxand y , ifx= y, then y= x. That is, equality is symmetric.

(3) For all whole numbersx, y and z, ifx= y and y =z , then x= z . That is, equalityis transitive.

(4) For all x and y, ifx is a whole number and x = y, then y is also a whole number.That is, the whole numbers are closed under equality.

The first axiom says any whole number is equal to itself. The second says that if a whole

number a is equal to a whole number b, then b is equal to a as well. The third can berephrased, using the second axiom, as saying that if two whole numbers are equal to athird, then they are equal to each other.

Later, when we talk about relations, particularly equivalence relations we will see thefirst three axioms again, in a different guise that will help us see why they are the keyproperties of equality.

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The remaining five axioms define the key properties of the whole numbers. We startby assuming that there is at least one whole number, the number 0.

(5) 0 is a whole number.

Then we introduce a special mapSfrom the whole numbers to the whole numbers calledthe successor function, that can be thought of as adding 1. Later in the course, when wetalk further about the underlying foundations for school mathematics we will discuss functions and their

properties in much more depth. For now, we note that a function or map from the setX to the setY issimply a rule that determines a unique element y Y for each elementx X. We usually writef(x)for they Ythat is associated to x, and sometimes we writef(x) =y ory = f(x).

(6) For every whole number x,S(x) is a whole number.

We obtain 1 as S(0), 2 as S(1), 3 as S(2) and so on. But to make this sensible we have tolist two properties that the successor function Smust satisfy.

(7) For every whole number x, S(x)= 0. That is, there is no whole number whosesuccessor is 0.

(8) For all whole numbersxandy, ifS(x) =S(y), then x = y. That is, S is one-to-one.But note that Axiom (7) shows that Sis not onto since 0 is not in the image ofS.

These two axioms together imply that the set of natural numbers has the propertythat it is in one-to-one correspondence with a proper subset, since Sis one-to-one and 0 is

not in the image ofS. We can see intuitively that if the set of whole numbers were finitethis cannot happen. Any one-to-one mapping of a finite set to itself must be onto theentire set. Consequently, our intuition is that the set of whole numbers must be infinite.

However, in mathematics, intuition is only an aid. All our terms must be preciselydefined, so we DEFINEthe property of being infinite as follows:

Definition. A set Wis infinite if and only if there is a one-to-one correspondence betweenthe setWand a proper subset ofW.

Now, we can unambiguously state that Peanos axioms (7) and (8) imply that the set

of whole numbers is infinite.

These terms may not be familiar to everyone. One-to-one correspondence is a map or function froma setXto a setY that has the property thatf(x) =f(x)if and only ifx= x. The functionx 2xis one-to-one on the whole numbers but the function x2 4x+ 6 is not one-to-one since the values forx= 1 andx= 3 are both3. The image of a function f is the set of those elementsy Y so that thereis somex X withf(x) = y. The image ofx2 4x+ 5 as a map from the whole numbers to thewhole numbers is all the whole numbers but0 and1.

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There is one more Peano axiom for the set of whole numbers. So far, the axioms givea picture of the following kind

0 S(0) S(S(0)) S(S(S(0)))) . . .

A S(A) S(S(A)) S(S(S(A)))) . . .B S(B) S(S(B)) S(S(S(B)))) . . ....

......

... . . .

together with a number of circular sequences V, S(V),S(S(V)), etc. with

S(S(S( S(V) ))) =V

for some iterate ofS. The axioms to this point do not specify how many elements thereare in the whole numbers that are not in the image of the successor function, nor thenumber of circles, just that Sis one-to-one, which means each element not in the imageofSleads to an infinite tower as sketched above, and the towers or circles for different

elements do not intersect. The final axiom, Axiom (9), specifies that 0 is the only wholenumber that is not in the image ofSand that there are no circles.

(9) IfK is a set so that

0 is contained in K and

for every whole number x, ifx K, then S(x) K,

thenK contains every whole number.

This axiom is usually called the axiom of induction because one of its consequences isthat ifT is a property that can be true or false for whole numbers then if it true for 0,and if we can show that the truth of the statement for any whole number n implies thatthe statement is true for S(n), then the conclusion is that the statement is true for everywhole number.

Here is an example. We note that 1 = 12, 1 + 3 = 22, 1 + 3 + 5 = 32, and we wonderif the sum of the first k+ 1 odd whole numbers is the square (k+ 1)2. We can verify thatthis is true for all whole numbers if we can verify that it is true for the first which it issince 1 = 12 and verify that if it is true for k then it is also true for k+ 1. In this casethe argument is that the truth for k is exactly the statement that

1 + 3 + 5 + + (2k+ 1) = (k+ 1)2

is true. Then the statment for k+ 1 is

(1 + 3 + 5 + + (2k+ 1)) + 2(k+ 1) + 1 = (k+ 2)2.

We have to show that the truth of the statement for k implies the truth for k +1. Becausewe assume the truth of the statement for k we can rewrite the statement for k + 1 as

(k+ 1)2 + 2(k+ 1) + 1 = (k+ 1 + 1)2

= (k+ 2)2.

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But this is a true statement since we know, from algebra, that (a+b)2 =a2 + 2ab+b2,and substituting k+ 1 for a and 1 for b gives (k+ 1 + 1)2 = (k+ 1)2 + 2(k+ 1) + 1.

There is one more concept that is tied in to these axioms. We say that the wholenumber n is greater than the whole number m if and only ifn is not equal to m, but isthe image ofm under some iterate ofS. We write n > m as a shorthand notation for nis greater than m.

Likewise, we say m is less than n if and only ifn is greater than m, and we writem < n as a shorthand notation for mis less than n.

From the definitions, if m and n are any two whole numbers then they will satisfyone and only one of the three statements m = n,m < norm > n.

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The First Lessons in the First Grade Russian Text

Here the basic notion of successor is introduced in the guise of more or less. A numberis more than another number if and only if it is in the image of an iteration of the

successor map on the original number. Then names are introduced.

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The symbols for greater than, less than, and equalsare introduced as well as the namesof the first three numbers and the sums and differences for them.

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The numbers through five are introduced and the students are asked to do a number ofproblems. Note that the focus is on the mathematical concepts. There are virtually nowords present except for essential vocabulary. It is also worth noting that all the keyconcepts are introduced using very small numbers - numbers that are almost hardwiredin our brains.

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After these problems, the next lessons introduce the rest of the numbers between 6 and 10in a very similar way. Note that subtraction is developed for the same number set at thesame time as addition is for the numbers to 5. Subtraction is also handled simultaneously

with addition in the lessons to 10.

Mathematically, this makes perfect sense because subtraction is defined in terms ofaddition: a b = c is a true statement if and only ifb+c = a. So subtraction is justaddition where one of the addends is not necessarily known.

Of course, we have not really defined addition for whole numbers to this point. How-ever, this can be accomplished by using the successor function and induction. The keyinductive statement is that 0 +n= n for each nas the start of an inductive definition ofaddition. Then, if we assume that the value ofk+n is known for each n we define thevalue ofS(k) + n as S(k+n). Using this definition we have to show that the standard

properties for addition hold. These are

(1) k+n= n+k for all whole numbers n and

(2) k and k+ (s+n) = (k+s) +n for all whole numbers k ,sand n.

Exercise.

(1) Can you prove statement (1) above in the case wheren= 0?

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Corresponding Lessons in a Solid U.S. Text

These and those on the next page are from the Kindergarten program.

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These are from the first grade program.

Note that the mathematical content is much less. There is no building up of arithmeticstructure for these numbers. Mostly, the effort seems to be focused on listing the numbersand learning number words in all these lessons for both Kindergarten and first grade.

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The Introduction of Addition and Subtraction

The Formal Definition of Addition in Terms of the Axioms. We define a func-

tion from the Cartesian product of the whole numbers to the whole numbers,

f :0 00that we call addition via an inductivedefinition. First, set

f(0, n) = n

for any whole number n, and now, assuming that f(k, n) is defined for 0 k msetf(S(m), n) = S(f(m, n)).

In words, think of this as the requirement that the sum of (m + 1) and n is the sum ofmandn augmented by one (or (m+1)+ n= (m+ n)+1). From the axiom of induction, thisdefines the addition function for all elements (m, n) 0 0, and we use the notationf(m, n) =m+n for the image of the addition function.

Addition is commutative. We want to show that addition - as defined above - iscommutative. So we want to prove from the axioms and the definition off, thatf(n, m) =f(m, n) for all (m, n) 0 0. We begin with a preliminary result:

Lemma. For all(m, n) 0 0 we have that f(m, S(n)) = f(S(m), n) = S(f(m, n)).

Proof. The second equality holds because that is the definition off(m, n) for m in theimage ofS. Hence it suffices to prove thatf(m, S(n)) = f(S(m), n) for all (m, n) 00.

Note that f(0, S(n)) = S(n) by definition, so f(0, S(n)) = S(f(0, n)) = f(S(0), n).As a result, the first equality holds for m= 0.

We now assume it holds for f(k, n) with n any whole number and k a fixed wholenumber. We show that it must also hold for f(S(k), n). By assumption f(k, S(n)) =f(S(k), n) for any whole number n. But from the definition of the addition functionf(S(k), S(n)) = S(f(k, S(n)). On the other hand, we can rewrite this as S(f(S(k), n)),and again applying the definition of the addition function S(f(S(k), n) = f(S(S(k)), n).

Thus we have shown that f(k, S(n)) = f(S(k), n) for all whole numbers, n, im-plies that the same statement holds for the successor of k and all n: f(S(k), S(n)) =f(S(S(k)), n), and the lemma follows.

Corollary. The addition function f is commutative. That is to sayf(m, n) =f(n, m)for any(m, n) 0 0.

Proof. We first verify the statement forf(0, n). We know thatf(0, n) = n sof(0, 0) = 0,and the statement is true for n = 0. Suppose that the statement is true for n - so

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f(0, n) =f(n, 0). We need to show that this assumption leads to the conclusion that thestatement is also true forS(n). Butf(0, S(n)) = f(S(0), n) =S(f(0, n)) from the lemma.On the other hand, our assumption gives that f(0, n) =f(n, 0) so

f(0, S(n)) = S(f(n, 0) =f(S(n), 0),

and the corollary is true for m= 0.

Now suppose the corollary is true form, so for all whole numbern we havef(m, n) =f(n, m). We need to show that this implies it is true for S(m). As was the case withm= 0 we have

f(S(m), n) = S(f(m, n))

= S(f(n, m))

= f(S(n), m)

= f(n, S(m))

with the last statement being a direct application of the lemma.

Addition is associative. In an entirely similar way we prove that addition is associa-tive. This is the same as the statement that for all (m,n,p) 0 0 0 we havef(f(m, n), p) =f(m, f(n, p)) or ((m+n) +p) = (m+ (n+p)).

Here is how we proceed. First we prove the result for m = 0. So we show thatf((0, n), p) =f(0, f(n, p)) for any pair of whole numbers (n, p). Then we will assume thatthe result is true for m and any pair (n, p) and show that this implies the truth of theresult for S(m) and any pair (n, p).

For m = 0 we have f(0, n) = n so f(f(0, n), p) = f(n, p). But f(n, p) is a wholenumber so it is also f(0, f(n, p)), and this is the statement for m= 0.

Now we assume that for any pair (n, p) we have f(f(m, n), p) = f(m, f(n, p)). Wehave

f(f(S(m), n), p) = f(f(m, S(n)), p)

= f(m, f(S(n), p))

= f(m, S(f(n, p)))

= f(S(m), f(n, p))

using the lemma two times, and the result follows.

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Here are the basic lessons in the Russian first grade program that relate to additionand subtraction. They basically starts with the successor function:

Note that the focus is on small numbers as before, and that subtraction is presented atthe same time as addition. From the beginning, it is given as the inverse to addition -not as a totally separate operation. Also, addition is presented in terms of the functiona

a+ 1 that can also be interpreted as the successor function:

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and then the function a a+ 2 is presented in terms of iterating the successor function:

Adding 2 is then iterated and it is pointed out that +2 increases magnitude

Next the focus turns to the function a a 2 and it is pointed out that2 decreases

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magnitude.

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Then as similar, but not quite as detailed discussion is provided for a a+ 3.

Finally, they include the commutative and associative rules for addition in an age appro-priate way:

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The Introduction of Multiplication and Division

The Peano axioms allow us to define the multiplication of whole numbers in terms of

addition. There is a functiong :0 00called multiplication and written g(m, n) =m n or sometimes m n, that has the following inductive definition -

g(0, n) = 0 for all n 0.

g(S(m), n) = f(g(m, n), n) or in more standard notation S(m) n= (m n) +n.

Multiplication is Commutative. If we apply this definition when n = 0 we obtainS(m) 0 =m 0 + 0. In particular, 0 0 = 0, so S(0) 0 = (0 0) + 0 = 0, and we canuse induction to show that m 0 = 0 for anym 0.

Indeed, we know 0 0 = 0. Now assume that m 0 = 0. ThenS(m) 0 = (m 0) + 0

= 0 + 0

= 0

som 0 = 0 for all m 0. Thusg(m, 0) = 0 for all whole numbers m, and we have thatg(m, S(n)) = g(m, n) +m

is true for all whole numbers n ifm = 0, since 0 + 0 = 0.

Lemma. For all pairs of whole numbers(m, n) 00we haveg(m, S(n)) =g(m, n)+m,or the same,m S(n) = (m n) +m, or using the fact that S(n) =n+ 1 we see that thisis exactly the statementsm (n+ 1) =m n+m.

Proof. Since the statement is true for all whole numbers n ifm = 0, we use inductiononm. Assumeg(m, S(n)) = g(m, n) + mfor all whole numbers n. We want to prove thatthis impliesg(S(m), S(n)) = g(S(m), n) +S(m). We have

g(S(m), S(n)) = g(m, S(n)) +S(n)

= (g(m, n) +m) +S(n)

= g(m, n) + (m+S(n))

= g(m, n) + (S(m) +n)

= g(m, n) + (n+S(m))

= (g(m, n) +n) +S(m)

= g(S(m), n) +S(m)

which is exactly what we wanted to show. Note that the intermediate steps used theassociative and commutative properties of addition extensively.

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Using this lemma, as was the case with addition we can prove that multiplication ofwhole numbers is commutative.

Corollary. For all pairs(m, n) of whole numbers we havem n= n m.

Proof. We have already seen that when m= 0 the corollary is true for all whole numbersn, sog(0, n) = g(n, 0) = 0. We now assume that the corollary is true for the whole numbermand all whole numbers n, so g(m, n) =g(n, m) for all n.

We need to verify that the assumption g(m, n) = g(n, m) for all whole numbers nimplies that g(S(m), n) = G(n, S(m)) for all whole numbers n. We have g(S(m), n) =g(m, n) +n from the definition ofg . Now our assumption gives g(m, n) =g(n, m) so

g(S(m), n) = g(n, m) +n

and from the lemma, g(n, m) +n= g(n, S(m)) and

g(S(m), n) =g(n, S(m))

as was to be proved.

Multiplication is distributive. We now wish to show that (m+n) p= (m p) +(n p)for any triple of whole numbers (m,n,p). In functional notation this can be written

g(f(m, n), p) =f(g(m, p), g(n, p)).

We verify the truth of this formula when m = 0 and (n, p) are any two whole numbers.f(0, n) =n, so g(f(0, n), p) =g(n, p). Also g(0, p) = 0 and f(0, g(n, p)) =g(n, p) so bothsides of the equation above are equal to g(n, p). Consequently the equation is true for allpairs of whole numbers (n, p) as desired.

Now we are in the position to induct on m. Thus we assume that

g(f(m, n), p) =f(g(m, p), g(n, p))

is true form and all pairs of whole numbers (n, p). We want to show that this implies thetruth of the equation for S(m). We have

g(f(S(m), n), p) = g(S(f(m, n)), p)

= g(f(m, n), p) +p

= (g(m, p) +g(n, p)) +p

= g(m, p) + (g(n, p) +p)

= g(m, p) + (p+g(n, p))

= (g(m, p) +p) +g(n, p)

= g(S(m), p) +g(n, p)

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The result follows.

Multiplication is associative. We first note that g(g(0, n), p) = g(0, p) = 0 while

g(0, g(n, p)) = 0 as well, so when m= 0 we haveg(g(m, n), p) = g(m, g(n, p)) for all pairsof whole numbers (n, p). We now apply induction on m. So we assume that

g(g(m, n), p) =g(m, g(n, p))

for m and all pairs of whole numbers (n, p), and we want to show that this implies thesame statement for S(m) in place ofm. We have

g(g(S(m), n), p) = g(g(m, n) +n), p)

= f(g(g(m, n), p), g(n, p))

= f(g(m, g(n, p)), g(n, p))

= g(S(m), g(n, p)).

and this is what we wanted to show.

Exercise.

(1) Justify each step in the argument above for associativity.

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The introduction of multiplication and division in the Russian grade 2 text.Multiplication and division are introduced after a short review of addition and subtractionat the beginning of the Russian second grade text. Almost immediately both are defined:

The are taught together as inverse operations, and their most important properties areset forth:

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Then they note the distributive rule for multiplication

Throughout, as you can see from these sample pages, the students are presented withchallenging problems that get to the heart of what they are supposed to be learning.

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Introduction of the Number Line

A very important means of visualizing the whole numbers and their properties is thenumber line. Here is the introduction to the number line in the first volume of the grade1 Singapore Mathematics Series:

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In the first volume of the grade 2 Singapore Mathematics Series the number line is repre-sented by meter sticks and tape measures laid out in centimeters.

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Basics about Place Value Notation

Base 10 place value notation is a shorthand notation for labeling the whole numbers.

We assume that multiplication and addition for the whole numbers are understood, andwe note that a convenient way of representing the whole numbers is via equally spacedmarks on thenumber linewhere we putS(n) at the next position on the number line aftern.

...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................

......

................

......

................

......

................

......

................

......

.....

0 1 2 3 4 5 6 7 8 9

Now, there is a way of skip counting by replacing S by S2 so we put a special markevery 2 places:

...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

......

......

...

......

..........................

......

..........................

......

..........................

......

..........................

......

..........................

0 1 2 3 4 5 6 7 8 9

or every three places:

...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

......

......

...

......

................

......

......

......

...

......

................

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0 1 2 3 4 5 6 7 8 9

or every five places:

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0 1 2 3 4 5 6 7 8 9

or more. If we skip count in this way we may not reach every whole number - for exampleskip counting by 2 will miss all the odd numbers - but sooner or later we will reach or

passany number we choose. This must be the case since a certain number of iterations ofSwill arrive at any number, and the same number of iterations ofSm will get quite a bitfurther.

Suppose we choose as our skip counting numbers the numbers 1, 10, 102 = 10 10,103 = 10 10 10, 104, 105 and so on. Given any whole number, there will be a biggestnumber among these powers of 10 that is smaller than or equal to the given number. (Thisstatement actually involves an inductive argument, but it is relatively direct.) Call thisnumber 10m. In particular, the next power of 10, 10m+1, will be further to the right onthe number line than the given number. So we have trapped the given number between10m and 10m+1.

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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.

10m n 10m+1

If 10m =n then we can denote n as 10m. Whether or not this is the case, since 10m+1 =10 10m we can subdivide the interval between 10m and 10m+1 into exactly 10 equalintervals, each of length 10m, and hence will be able to mark the points 2 10m, 3 10m,. . ., 9 10m, and there will be a greatest number k1 10m 1k19 so that k1 10mis less than or equal to nand (k1+ 1) 10m is greater thann. (At worst, k1+ 1 = 10 so10 10m = 10m+1 is obtained - a number we already know is past n.) Ifn = k1 10m

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we can denote n as k1 10m. If not we have thatn lies properly between k1 10m and(k1+ 1) 10m.

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k1 10m n (k1+ 1) 10

m

Since 10m = 1010m1, assuming that m 1, we have that k1 10m + 0 10m1,k1 10

m + 10m1,k1 10m + 2 10m1, k1 10

m + 3 10m1,. . ., k1 10m + 9 10m1,

k1 10m + 10 10m1 = k1 10

m + 10m = (k1+ 1) 10m

decompose the interval between k 10m and (k + 1) 10m into 10 equal length sub-intervals.As before there is a unique k2 with 0 k2 9 so that

k1 10m +k2 10

m1

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Key Issues with Place Value Notation

Perhaps the biggest mistake made in mathematics instruction in this country is the

confounding of the place value notation for whole numbers with the numbers themselves.We often see questions like Identify the value of 3 in the number 713,125 on stateassessments. And lessons like the following are very common:

This type of approach focuses far more on the language involved in reading the shorthandnotation for the expanded form than it does on the connection between the notation andthe actual number. We should not lose site of the key paragraph quoted earlier:

Today, Arabic numerals are in use pretty much around the world, while the words withwhich we name numbers naturally differ from language to language. And, as Dehaeneand others have noted, these differences are far from trivial. English is cumbersome.There are special words for the numbers from 11 to 19, and for the decades from

20 to 90. This makes counting a challenge for English-speaking children, who areprone to such errors as twenty-eight, twenty-nine, twenty-ten, twenty-eleven. Frenchis just as bad, with vestigial base-twenty monstrosities, like quatre-vingt-dix-neuf(four twenty ten nine) for 99. Chinese, by contrast, is simplicity itself; its numbersyntax perfectly mirrors the base-ten form of Arabic numerals, with a minimum ofterms. Consequently, the average Chinese four-year-old can count up to forty, whereasAmerican children of the same age struggle to get to fifteen. And the advantagesextend to adults. Because Chinese number words are so brief - they take less than aquarter of a second to say, on average, compared with a third of a second for English -the average Chinese speaker has a memory span of nine digits, versus seven digits for

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English speakers. (Speakers of the marvelously efficient Cantonese dialect, commonin Hong Kong, can juggle ten digits in active memory.)

In fact, what should be focused on in the initial introduction of base 10 place-value notationis the connection between the usual way of writing numbers, 123, 600 for example, and theunderlying expanded form, in this case

1 105 + 2 104 + 3 103 + 6 102 + 0 101 + 0 1,

with very little attention being paid to the way they are read verbally. From this focus, itbecomes natural to understand how we can determine whenn > mor m > n. nis greaterthan m if the number of digits in the base 10 place-value notation for m is larger than

those in the corresponding expansion for n, and, in case the number of digits involved isthe same for both, then - working from the left - the first time the corresponding digitsdiffer, the digit for m is larger than the digit for n. Here is how these objectives areprepared for in the first grade Russian program:

Then, in second grade this is continued. In the following two pages note, first the consistentmeasures. Then note the sophisticated problems that the students are asked to solve thattie in place value to all the basic properties of the numbers, comparison, addition and

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subtraction.

A few pages later, this is made abundantly clear with the following lessons. Pay particularattention to the exercises here:

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Contrast the Russian development of place value with the following typical U.S. thirdgrade lessons:

Note the sort of one dimensional (linear) measure of 10s, two dimensional (area) measure

of 100s, the 3 dimensional measure of 1000s, so there is no consistent numericalmeasureof 10s 100s and 1000s. Rather, students are given the impression that they are differentkinds of measures. The exercises on the second page are not too bad. They do try toreinforce the relationship between place-value notation and expanded form, but they donot correct for the misconception implicit in the lesson that 1000s, 100s, 10s and 1s aredifferent kinds of measures.

Tied in to this is the important exploration that should be done of the order ofmagnitude of pure powers of 10. Students should be aware of how big numbers like 1080

are. For example, 1080 is about the number of atoms in the entire observable universe,while the distance that light travels in one year is about 6 1012 miles, and the distancefrom the earth to the edge of the visible universe is about 5 10

10

light-years.

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Grade 2: Russian Development of Algorithms

The development starts by carfully settng up the theoretical underpinnings of thealgorithms for addition and subtraction:

and setting up oral practice situations that focus on expanded form

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At the end of the second grade book, when numbers to 1000 are considered, the additionand subtraction algorithms are finally developed:

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At the same time that the theoretical development for the addition and subtractionalgorithms are being developed, there is also work being done at the theoretical level onmultiplication and division.

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Finally, towards the end of the second grade book division with remainder is intro-duced.

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Grade 3: Russian Development of Algorithms

The focus in grade three is on developing the foundation material necessary for thebase 10 place value standard multiplication and division algorithms. The material startswith multiplying multidigit numbers by one digit numbers and immediately continues withdividing by one digit numbers:

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Then separate aspects of what is needed are developed:

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The next lesson on how quotients change when the dividend is changed is typical of the carethat is taken in setting up proper foundations for understanding why the multiplication

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and division algorithms work.

The next step is to develop the algorithms for multiplication when the multiplier hasmore than one digit and as usual, paired with the multiplication algorithm is the division

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algorithm when the divisor has more than one digit.

And dividing by two or more digits is continued, after which mastery of long division for

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whole numbers is expected.

and finally

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Basic Terminology and Numbers

Numbers: A set of formal numbers(or anumber setfor short, is a set of objectsN with

two special elements, the first denoted 1 and the second denoted 0, and two operations(maps)NNNdenoted + andwith the usual properties:

1) Both + andare commutative (+)(a, b) = (+)(b, a) and ()(a, b) = ()(b, a) for all(a, b) NN

2) Both + andare associative (+)(a, (+)(b, c)) = (+)((+)(a, b), c), ()(a, ()(b, c)) =()(()(a, b), c) for all (a,b,c) NNN

3) The distributive law holds ()(a, (+)(b, c)) = (+)(()(a, b), ()(a, c)) for all (a,b,c) N

N

N

4) (+)(0, a) =a, ()(1, a) = a for all a N

5) Cancellation holds for +. If we have a, b, c N and (+)(a, b) = (+)(a, c) thenb= c.

6) Cancellation holds for. Ifa = 0 and ()(a, b) = ()(a, c) then b= c.

It is customary to use the shorthand notationa+bfor (+)(a, b), and the shorthand notationa b for ()(a, b). So, for example the distributive rule is usually writtena (b+c) =(a

b) + (a

c).

Remark: Given a set X, a map X XX such as + and above is called a binaryoperation on X.

Remark: The cancellation laws are equivalent to the statements that

(a) For everyn N, the operation m m+n is one-to-one,

(b) For everyn = 0 with n N, the operation m m n is one-to-one.

These last two properties are usually direct to verify in specific examples.

Examples of Number Sets.

The examples we know are the set of whole numbers,

0 = {0, 1, 2, 3, 4, . . . , }the set of integers 0, +1, 1, +2, 2, +3, 3, . . ., the set of rational numbers a/b where aand b are integers with b = 0.

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But there are many other sets of numbers. One very small one is the set with justthe two elements 0 and 1, where 0 + 0 = 0, 0 + 1 = 1 and 1 + 1 = 0, while 0 0 = 0,1 0 = 0 and 1 1 = 1.

Remark: This set of numbers satisfies all the rules above, but to this point weve notseen any situations that require numbers like these. However, the set of numbers aboveis actually the working set of numbers in calculators, computers, cell phones, and DVD

players. From these numbers we can represent our usual numbers and all their properties,but the details of how this is done are topics that will only appear in more advancedcourses in mathematics, logic, and even electrical engineering.

For now, what is most important is that we understand the key sets of numbers thatcontain0, such as the integers, rationals, and reals.

Subtraction.

IfNis a number set, then we define subtraction by setting a b to be that numberc N, if it exists, so that b+c = a. Note that a b may not always exist in N. Alsonote that there is no reason that a b is the same as b a, since the first is that c, if itexists, so that b+c= a, while the second is that d, if it exists, so that a +d= b.

If there is c N so that a b= c, then c is the only element in N with thisproperty. Sinceb + d= a and b + c= a impliesd = c, using cancellation, it follows thatif the difference, a minusbora b exists in Nthen it is unique.

Examples. In the whole numbers 3 1 exists and is equal to 2 since 1 + 2 = 3. But 1 3is not defined in the whole numbers, since 3 +c 3 for any whole number c, and 1< 3.

In the two element number setNdefined above, 1 1 = 0, 0 1 = 1, 0 0 = 0 and1 0 = 1, so subtraction is defined for any pair, and a b= b a for any pair. But thisis very special.

Division.

IfN is a number set, then we define division by setting a/b for b= 0, to be thatnumber c N, if it exists, so that b c= a. Again, cancellation shows that ifa/b existsinNthen it is unique. This is where we need the condition that b = 0. Ifb= 0, then wewill show in the next section, that 0 c= 0 for all c N. As a result, ifb were allowedto be 0 then the only a for which a/0 could be defined is 0. However, since 0 c= 0 forall cN, if we allowed divison by 0, then the answer to 0/0 would be all cN, not asingle element ofN, and a/0 would not exist in N for anya = 0.

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Consequences of the Definition ofN

Properties (1) through (6) above allow us to make a number of further conclusions.

For example we can verify

For anya Nwe havea 0 = 0.

Note that 0 + 0 = 0 so a 0 = a (0 + 0). So using the distributive rulea 0 =(a 0) + (a 0). We also have a 0 + 0 =a 0 using (4). So

(a 0) + 0 = (a 0) + (a 0)

and using (5) we have that 0 = a 0 as desired.

Here is another important consequence of (1) through (6). Suppose thata is a non-zero element ofNSuppose also that a b= 0. Then we have a b= a 0, so b= 0 by(6). Consequently we have that if both aand b are non-zero then a b cannot be 0, and

The product of any two non-zero elements ofN is non-zero.

Example: Properties (5) and (6) above and their consequences are quite restrictive.Suppose thatN1 andN2 are number sets. Then we can define + andon the Cartesianproduct N1 N2 via the rules

(a, b) + (c, d) = (a+c, b+d)

(a, b) (c, d) = (a c, b d)

and these binary operations satisfy properties (1) through (4) and (6) above, with the zeroinN1 N2 being (0, 0) and the 1 being (1, 1). However they do not satisfy (6) since

(1, 0) (0, 1) = (1 0, 0 1) = (0, 0).

Quotients and their sums and products in N.

Notation: Suppose that a b = c in N for a= 0. Then we sometimes writeb in theform c

a, read c divided by a. Since a b= c and a e= c for n= 0 implies that b= e,

there is at most one element ca N for each pair (a, c) with a = 0.

Lemma: If ca

and de

are both contained in N then

(a) ca

+ de

= ce+adae

inN

(b) ca d

e = cd

ae inN

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(c) For anyh = 0 in Nwe have that chah

exists in N and chah

= ca

.

Proof. We verify (a) as follows. If we multiply ca

+ de

byae, the distributive rule shows

the product is the sum ca ae

+

d

e ae

.

But ca ae= ce and d

e ae=ad. It follows that

ca ae

+

d

e ae

= ce+ad

so, from the definition

c

aae +

d

eae =

ce+ad

ae

.

We show (b) in a similar way.c

ad

e

ae =

ca a

d

e e

= c d

so ca d

e = cd

aeas claimed. The proof of (c) is equally direct.

The following result is also very useful.

Lemma. Let ba

and de

be contained

problemas do ensino de matemática

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