π数学杂志7

September 2003http://www.pims.math.ca/pi pi in the Sky is a semi-annual publication of PIMS is supported by the Natural Sciences and Engineer- ing Research Council of Canada, the British Columbia Information, Science and Technology Agency, the Al- berta Ministry of Innovation and Science, Simon Fraser University, the University of Alberta, the University of British Columbia, the University of Calgary, the Univer- sity of Victoria, the University of Washington, the Uni- versity of Northern British Columbia, and the University of Lethbridge. This journal is devoted to cultivating mathematical rea- soning and problem-solving skills and preparing students to face the challenges of the high-technology era. Editor in Chief Ivar Ekeland (University of British Columbia) Tel: (604) 822–3922 E-mail: director@pims.math.ca Editorial Board John Bowman (University of Alberta) Tel: (780) 492–0532 E-mail: bowman@math.ualberta.ca Giseon Heo (University of Alberta) Tel: (780) 492–8220 E-mail: gheo@ualberta.ca Klaus Hoechsmann (University of British Columbia) Tel: (604) 822–5458 E-mail: hoek@pims.math.ca Dragos Hrimiuc (University of Alberta) Tel: (780) 492–3532 E-mail: hrimiuc@math.ualberta.ca Wieslaw Krawcewicz (University of Alberta) Tel: (780) 492–7165 E-mail: wieslawk@shaw.ca David Leeming (University of Victoria) Tel: (250) 721–7441 E-mail: leeming@math.uvic.ca Volker Runde (University of Alberta) Tel: (780) 492–3526 E-mail: runde@math.ualberta.ca Carl Schwarz (Simon Fraser University) Tel: (604) 291–3376 E-mail: cschwarz@stat.sfu.ca Secretary to the Editorial Board Heather Jenkins (PIMS) Tel: (604) 822–0402, E-mail: heather@pims.math.ca Technical Assistant Mande Leung (University of Alberta) Tel: (780) 710–7279, E-mail: mtleung@ualberta.ca Addresses: pi in the Sky pi in the Sky Pacific Institute for Pacific Institute for the Mathematical Sciences the Mathematical Sciences 449 Central Academic Bldg 1933 West Mall University of Alberta University of British Columbia Edmonton, Alberta Vancouver, B.C. T6G 2G1, Canada V6T 1Z2, Canada Tel: (780) 492–4308 Tel: (604) 822–3922 Fax: (780) 492–1361 Fax: (604) 822–0883 E-mail: pi@pims.math.ca http://www.pims.math.ca/pi Contributions Welcome pi in the Sky accepts materials on any subject related to math- ematics or its applications, including articles, problems, cartoons, statements, jokes, etc. Copyright of material submitted to the publisher and accepted for publication remains with the author, with the understanding that the publisher may reproduce it with- out royalty in print, electronic, and other forms. Submissions are subject to editorial revision. We also welcome Letters to the Editor from teachers, stu- dents, parents, and anybody interested in math education (be sure to include your full name and phone number). Cover Page: This picture was created for pi in the Sky by Czech artist Gabriela Novakova. The scene depicted was inspired by the article on mathematical biology written by Jeremy Tatum, “Maths and Moths,” that appears on page 5. Prof. Zmodtwo is again featured on the cover page, this time doing research on moths and butterflies. CONTENTS: Reckoning and Reasoning or The Joy of Rote Klaus Hoechsmann . . . . . . . . . . . . . . . . . . . . . . 3 Maths and Moths Jeremy Tatum . . . . . . . . . . . . . . . . . . . . . . . . . . .5 Shouting Factorials! Byron Schmuland . . . . . . . . . . . . . . . . . . . . . . 10 A Generalization of Synthetic Division Rohitha Goonatilake . . . . . . . . . . . . . . . . . . . . 13 Why Not Use Ratios? Klaus Hoechsmann . . . . . . . . . . . . . . . . . . . . . 17 It’s All for the Best: How Looking for the Best Explanations Revealed the Properties of Light Judith V. Grabiner . . . . . . . . . . . . . . . . . . . . . 20 A.N. Kolmogorov and His Creative Life Alexander Melnikov . . . . . . . . . . . . . . . . . . . . 23 “Quickie” Inequalities Murray S. Klamkin . . . . . . . . . . . . . . . . . . . . .26 Summer Institute for Mathematics at the University of Washington . . . . . . . . . . . . . . 29 Why I Don’t Like “Pure Mathematics” Volker Runde . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Math Challenges . . . . . . . . . . . . . . . . . . . . . . 32 2 This column is an open forum. We welcome opinions on all mathematical issues: research, education, and communication. Please feel free to write. Opinions expressed in this magazine do not necessarily reflect those of the Editorial Board, PIMS, or its sponsors. Reckoning and Reasoning or The Joy of Rote by by Klaus Hoechsmann† You might have heard of this story, but it bears being repeated. In 1992, Lou D’Amore, a science teacher in the Toronto area, sprung a Grade 3 arithmetic test from 1932 on his Grade 9 class, and found that only 25% of his students could do all of the following questions. 1. Subtract these numbers: 9, 864− 5, 947 2. Multiply: 92× 34 3. Add the following: $126.30 + $265.12 + $196.40 4. An airplane travels 360 kilometers in three hours. How far does it go in one hour? 5. If a pie is cut into sixths, how many pieces would there be? 6. William bought six oranges at 5 cents each and had 15 cents left over. How much had he at first? 7. Jane had $2.75. Mary had 95 cents more than Jane. How much did Jane and Mary have to- gether? 8. A boy bought a bicycle for $21.50. He sold it for $23.75. Did he gain or lose and by how much? 9. Mary’s mother bought a hat for $2.85. What was her change from $5? 10. There are 36 children in one room and 33 in the other room in Tom’s school. How much will it cost to buy a crayon at 7 cents each for each child? † Klaus Hoechsmann is a professor emeritus at the Univer- sity of British Columbia in Vancouver, B.C. You can find more information about the author and other interesting articles at: http://www.math.ubc.ca/∼hoek/Teaching/teaching.html. This modest quiz quickly rose to fame as “The D’Amore Test.” Other teachers tried it on their classes, with similar results. There was some improvement in Grades 10 to 12, where 27% of students could get through it, but they tend to be keener anyway since their less ambitious class-mates usually give up on quantitative science after Grade 9. All in all, the chance of acing the D’Amore Test appears to be independent of anything learned in high school. At first glance this seems as it should be, because the test certainly contains no “high school material”. On second thought, however, a strange asymmetry appears: while all students expect to use the first two R’s (Readin’ and Ritin’) throughout their schooling and beyond, they drop the third R (Rithmetic) as soon as they can—if indeed they acquired it at all. Has it always been like this? I doubt it: my grandmother went to school only twice a week (being needed in yard and kitchen) but was later able to handle all the arithmetic in her little grocery store without prior attendance of remedial classes. She did not even have a cash register. To many administrators, think-tankers, etc., this is beside the point, because we now live in the brave new computer age. A highly placed person who has likely never repaired a car engine, and probably knows little about computers, suggested that 20 years ago, “an auto mechanic needed to be good at working with his hands,” whereas now he needs Algebra 11 and 12 to run his array of robots. For a more insights of this kind, you might wish to visit www.geocities.com/Eureka/Plaza/2631/articles.html, where electricians, machinists, tool-and-die makers, and plumbers are also included “among those who need Grade XI or XII algebra.” It doesn’t say what for. Mechanics laugh at this: remember the breaker-point gaps, ignition timing, engine compression, battery charge, alterna- tor voltage, headlight angle, and a multitude of other nu- merical values we had to juggle in our minds and check with fairly simple tools—today’s gadgets make our jobs more rou- tine, they say. But ministerial bureaucrats tend to believe the hype, with a fervour proportional to their distance from “Mathematics 12,” which has gobbled up Algebra 12 in most places I know. Aye, there’s the rub: the third R has morphed into the notorious M. “What’s in a name?,” you ask, “that which we called rithmetic by any other word would sound as meek.” How many times must you be told that M is hard and bor- ing, and hear the refrain “I have never been good at M”? It is the perfect cop-out, acceptable even in the most exclu- sive company—a kind of egalitarian salute by which “nor- mal” members of the species homo sapiens recognize one an- other. How can a teacher of, say, social studies be expected to develop vivid lessons around unemployment, national debt, or global warming—as long as these topics are mired in M? He/she still must mention numbers, to be sure, but can now present them in good conscience as disconnected facts, know- ing that his/her students’ minds will be uplifted in another class, by that lofty but (to him/her) impenetrable M. Ask any marketing expert: labels are not value-free, they attract, repel, or leave you indifferent. Above all, they raise expectations, which, in the case of M, are as manifold and var- ied as the subject itself. Is it conceptualization, exploration, visualization, constructivism, higher-order thinking, problem solving—or all of the above? The guessing and experiment- ing goes on and on, producing bumper crops of learned papers and theses, conferences, surveys, and committees, as well as confused students and teachers. “This is the first time in his- tory that Jewish children cannot learn arithmetic” said an 3 Israeli colleague, referring to the state of Western style edu- cation in his country, where the recent Russian immigrants maintain a parallel school system. Not every country has followed the R to M conversion. In the Netherlands and (what was) Yugoslavia, children still learn rekenen and racˇun, respectively, together with reading and writing. The more weighty M is left for later. Germany clung to Rechnen till the 1960’s, and then rashly followed the American lead, pushing Mathematik all the way down to Kindergarten—with the effect of finding itself cheek-to-jowl with the US (near the end of the list) in international com- parisons. I hear the sound of daggers being honed: what is this guy trying to sell (in this culture we are all vendors), is it “Back to Basics”? Does he hanker for “Drill and Kill,” for “Top Down” at a time when all good men and women aspire to “Bottom Up”? Readers unaccustomed to Educators’ discourse might be puzzled at such extreme positions getting serious attention. They would immediately see middle ground between tyranny and anarchy, boot camp and nature trail, etc. Why do we always argue Black versus White? I really cannot explain it. Maybe it is because we need strident voices and must hold single notes as long as we can, in order to be noticed in this mighty chorus. How did we get here? Although the benefits of planned obsolescence are obvious, they are not often mentioned to justify the present trend to- ward innumeracy. It is the relentless advance of technology which must be seen as the main reason for the retreat of ar- chaic skills. Speech-recognizing computers already exist, and once they are mass-produced, writing will not need to be taught anymore, at least not at public expense. Whatever we now do with our hands and various other body-parts outside the brain will clearly fall into the domain of sports. Only in this spirit does it make sense to climb a mountain top that can be more safely reached by helicopter. Before the advent of electric and later electronic calcula- tors, computations had to follow rigid algorithms that al- lowed the boss or auditor to check them. This was “pro- cedural knowledge” of an almost military kind—justly de- spised and rejected when it became obsolete. Oddly enough it did, however, have an important by-product: by sheer habit, simple calculations were done at lightning speed, and often mentally—of course with a large subconscious component. In many places, this “mental arithmetic” was even practised as a kind of sport, still “procedural,” in some sense, but open to improvisation—more like soccer than like target shooting. Look at the first question of the D’Amore Test: 9, 864 − 5, 947. Abe did it the conventional way and had to “borrow” twice. Beth zeroed in on the last three digits, noting that 947 exceeded 864 by 36+47 = 83, which she subtracted from 4000. Chris topped up the second number by 53 to 6000 and hence had to increase the first one to 9, 864 + 53 = 9, 917. Dan and Edith had yet different ways, but all got 3, 917. On the second question, Abe again used the standard method, since he was a bit lazy but meticulous. Beth looked at the 92 and thought 100− 10 + 2, playing it very safe. Chris spotted one of his favourite short-cuts: 3 × 17 = 51, and reasoned that 9× 34 = 6× 51 = 306, and so on. Dan was attracted to the fact that 92 was twice 46, which lies as far above 40 as 34 lies below it. Therefore 46× 34 was 1600− 36, which had to be doubled to 3200− 72. Edith blurted out the answer 3128 and said she did not remember how she got it. When I was in Grade 7, I knew such kids—and was irked by the fact that many played this mental game as well as they played soccer. Justice was restored when, in Grade 8, they were left in the dust by x and y but continued to outrun me on the playing field. Maybe they never missed the x and y in later life (unlike contemporary plumbers), but I am almost sure their “number sense” often came in handy. Today’s kids are to acquire this virtue by doing brain-teasers and learning to “think like mathematicians,” carefully avoiding “mindless rote.” Whenever I walk by the open door of a mathematician’s work place, I see black or white boards covered with calcu- lations and diagrams. How come they get to indulge in this “rote,” while kids must fiddle with manipulations or puzzle till their heads ache? Could it be that we mathematicians sometimes engage in “mindful rote”—the kind known to mu- sicians and athletes? If so, we ought to step out of the closet and tell the world about the joy of rote. Anyone who has ob- served young children will immediately know what we mean. And while we’re at it, we might reclaim ownership of the M-word, at least suggest that it be kept out of the K-4 world. This does not mean that schools should go back to teaching ’rithmetic—admittedly an awkward label. How about “reck- oning and reasoning,” a third and fourth R to balance the first two? They would be associated with good old common sense, and, as Descartes has pointed out, nobody ever complains of not having enough of that. There are 10 kinds of mathematicians. Those who can think in binary and those who can’t. . . Two math professors are hanging out in a bar. “You know,” the first one complains. “Teaching mathematics nowadays is pearls for swine: the general public is completely clue- less about what mathematics actually is.” “You’re right!” says his colleague. “Look at the waitress. I’m sure she has no clue about any math she doesn’t need to give out correct change—and maybe not even that.” “Well, let’s have some fun and put her to the test,” the first prof replies. He waves the waitress to their table and asks: “Excuse us, but you seem to be an intelligent young woman. Can you tell us what the square of a+ b is?” The girl smiles: “That’s easy: it’s a2 + b2. . . ” The professors look at each another with a barely hidden smirk on their faces, when the waitress adds: “. . . provided that the field under consideration has characteristic two.” Q: What is the difference between a Ph.D. in mathematics and a large pizza? A: A large pizza can feed a family of four. . . A French mathematician’s pick up line: “Voulez–vous Cauchy avec moi?” 4 Maths and Moths Jeremy Tatum† I don’t reveal to many what some might regard as my some- what eccentric hobby of rearing caterpillars and photograph- ing the moths that ultimately emerge. This is my form of relaxation after the day is done, and my mind by then is usually far from mathematics. Yet there is a moth, the Peppered Moth (Biston betularia), that lends itself well to mathematical analysis. It is common in Europe and in North America, including the west coast of Canada and the United States. It is often held to represent one of the fastest known examples of Darwinian evolution by variation and natural selection. A vast literature has ac- cumulated on this moth, both by scientists and, I recently discovered, by creationists. The latter seek to disprove the hypothesis that it is an example of evolution, and their argu- ments do, I suppose, at least keep scientists on their toes to ensure that their evidence is compelling. Figure 1: The normal “peppered” form of Biston betularia. Photographed by the author on Vancouver Island, British Columbia. The normal form of the moth has a “peppered” appearance, shown by the specimen in Figure 1, which I photographed on Vancouver Island. When this normal form rests on a lichen- covered tree trunk it is very difficult to see; it is well protected by its cryptic coloration. There is another form that is almost completely black—the melanic form, illustrated in Figure 2 from a photograph taken in England by Ian Kimber. It is quite conspicuous when resting on a lichen-covered tree trunk, and it is at a grave selective disadvantage. The melanic forms are readily snapped up by hungry birds. † Jeremy Tatum is a former professor in the Department of Physics and Astronomy of the University of Victoria. His E-mail address is universe@uvvm.uvic.ca. Figure 2: The melanic form of Biston betularia. Pho- tographed by Ian Kimber in England. In industrial areas of nineteenth century England, long be- fore modern atmospheric pollution controls, factory chimneys belched out huge quantities of black smoke, which killed the lichens and coated the tree trunks with dirty black grime. Suddenly the “normal” form became conspicuous, and the melanic form cryptic. Within a few generations the popula- tions of these moths changed from almost entirely “normal” to almost entirely “melanic.” This is a situation that cries out for some sort of population growth analysis. We first have to understand a little about genetics—and I hope that professional geneticists will forgive me if I simplify this just a little for the purpose of this article. Figure 3: A melanic, a normal, and an intermediate form of Biston betularia. Photographed by Ian Kimber in England. The colour of the moths’ wings is determined by two genes, which I denote byM for melanic and n for normal. Each moth 5 inherits one gene from each of its parents. Consequently the “genotype” of an individual moth can be one of three types: MM , Mn or nn. MM and nn are described as “homozy- gous,” and Mn is “heterozygous.” An MM moth is melanic in appearance, and an nn moth is normal. What does anMn moth look like? Well, surprisingly, the heterozygous moth isn’t intermediate in appearance; it is melanic. Because of this, we say that the M gene is dominant over the n gene; the n gene is recessive. That is why I have written M as a capital letter and n as a small letter. (Actually the situation is rather more complicated than this, and there are indeed intermediate forms, as shown in Ian Kimber’s photograph in Figure 3—but the purpose of this article is to illustrate some principles of mathematical analysis of natural selection, not to bog ourselves down in detail. So I’ll keep the model simple and, to begin with, I’ll suppose that just the two genes are involved and that one is completely dominant over the other.) One can see now how vulnerable theM gene is in an unpol- luted environment. Not only MM moths but also Mn moths are conspicuous and are easily snapped up by birds; the M gene doesn’t stand a chance. But now blacken the tree trunks. MM a