π数学杂志3

We’ve Got Your Number June 2001 Sensitive Questions Constructing Fractals pi in the Sky is a semiannual publication of PIMS is supported by the Natural Sciences and En- gineering Research Council of Canada, the British Columbia Information, Science and Technology Agency, the Alberta Ministry of Innovation and Sci- ence, Simon Fraser University, the University of Al- berta, the University of British Columbia, the Uni- versity of Calgary, the University of Victoria, the University of Washington, the University of North- ern British Columbia and the University of Leth- bridge. This journal is devoted to cultivating mathematical reasoning and problem-solving skills and preparing students to face the challenges of the high-technology era. Editors in Chief Nassif Ghoussoub (University of British Columbia) Tel: (604) 822–3922, E-mail: director@pims.math.ca Wieslaw Krawcewicz (University of Alberta) Tel: (780) 492–7165, E-mail: wieslawk@v-wave.com Associate Editors John Bowman (University of Alberta) Tel: (780) 492–0532 E-mail: bowman@math.ualberta.ca Dragos Hrimiuc (University of Alberta) Tel: (780) 492–3532 E-mail: hrimiuc@math.ualberta.ca Editorial Board Peter Borwein (Simon Fraser University) Tel: (640) 291–4376, E-mail: pborwein@cecm.sfu.ca Florin Diacu (University of Victoria) Tel: (250) 721–6330, E-mail: diacu@math.uvic.ca Klaus Hoechsmann (University of British Columbia) Tel: (604) 822–5458, E-mail: hoek@math.ubc.ca Michael Lamoureux (University of Calgary) Tel: (403) 220–3951, E-mail: mikel@math.ucalgary.ca Ted Lewis (University of Alberta) Tel: (780) 492–3815, E-mail: tlewis@math.ualberta.ca Copy Editor Barb Krahn & Associates (11623 78 Ave, Edmonton AB) Tel: (780) 430–1220, E-mail: bkrahn@v-wave.com Addresses: pi in the Sky pi in the Sky Pacific Institute for Pacific Institute for the Mathematical Sciences the Mathematical Sciences 501 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, car- toons, 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 repro- duce them without 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 or anybody interested in math education (be sure to include your full name and phone number). Cover Page: The picture on the cover page was taken at Old Scona Academic High School in Edmonton. Old Scona was established in 1976 to provide motivated students who have demonstrated success and potential for growth with an oppor- tunity to pursue studies that challenge and enrich their learning experience. The school offers the International Baccalaureate program in English, History, and Biology, with instructional focus on university preparation. If you would like to see your school on the cover page of pi in the Sky , please invite us for a short visit to meet your students and staff. CONTENTS: Perils of Modern Math Education Stuart Wachowicz, Edmonton Public Schools . . . . . . . . . . . . . . 3 We’ve Got Your Number Ted Lewis, University of Alberta . . . . . . . . . . . . . . . . . . . . . . .5 Constructing Fractals in Geometer’s SketchPad Michael P. Lamoureux, University of Calgary . . . . . . . . . . . . . .7 Dramatic Story of Algebraic Equations Wieslaw Krawcewicz, University of Alberta . . . . . . . . . . . . . . 12 Have You Used Illegal Drugs Lately? or How to Ask Sensitive Questions Carl Schwarz, Simon Fraser University . . . . . . . . . . . . . . . . . 15 The Top Mathematical Award Florin Diacu, University of Victoria . . . . . . . . . . . . . . . . . . . 17 Some Famous Physicists Anton Z. Capri, University of Alberta . . . . . . . . . . . . . . . . . . 19 The Chancellor of the Exchequer Klaus Hoechsmann, University of British Columbia . . . . . . . . 22 Induction Principle Dragos Hrimiuc, University of Alberta . . . . . . . . . . . . . . . . . 24 Math Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Our Forum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Math Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28 c©Copyright 2001 Zbigniew Jujka This column is an open forum. We welcome opinions on all mathematical issues: research; education; and commu- nication. Please feel free to write us. Opinions expressed in this forum do not necessarily re- flect those of the editorial board, PIMS, or its sponsors. Perils of Modern Math Education by Stuart Wachowicz† “Pride in craftsmanship obligates the mathemati- cians of one generation to dispose of the unfinished business of their predecessors.” -E.T. Bell, The Last Problem The above statement most accurately describes the legacy of one generation of mathematicians to the next. However, on might be tempted to ponder whether this will continue to be possible in North America. The dis- cipline of mathematics, as we have known it, is clearly under threat. The threat is a consequence of allowing cur- riculum writers to change the centuries-old definition of mathematics and what needs to be learned based on utili- tarianism, combined with the current practice of allowing unproven fads to influence pedagogy. A century ago the utilitarian threat was expressed curtly by Andrew Carnegie, who stated, “Schools are a place where children learn how to manufacture!” Today, the same idea is masked in the notion that mathematics, to be of value, must be studied in a way that always gives a “real world” (whatever that means) application. While it is true that students benefit when they see the power of mathematics at work in deriving a solution to a common situation, there is also the broader aspect of the disci- pline of mathematics, one that empowers the individual to appreciate this most elegant and exact language. I am reminded of a statement written by Harold Jacobs in a forward to his book, Mathematics: A Human Endeavour, †Stuart Wachowicz is the Director of Curriculum for Edmonton Public Schools. During the past 25 years, he has also taught various subjects and served as a principal of a high school considered to be one of the highest achieving schools in Alberta. Stuart Wachowicz graduated with degrees in Geography and Education. “Some of the topics in this book may seem of little practical use, but the significance of mathematics does not rest on its practical value. It is hard to believe that someone flying over the Grand Canyon for the first time would remark. “What good is it?” Some people say the very same thing about mathematics. A great mathematician of our cen- tury, G.H. Hardy, said, “A mathematician, like a painter or a poet, is a maker of patterns.” Some of these patterns have immediate and obvious ap- plications; others may never be of any use at all. But, like the Grand Canyon, mathematics has its own beauty and appeal to those who are willing to look.” In a quest for utilitarian value, much of the modern mathematics curriculum in the public school system no longer seriously attempts to inculcate a deep understand- ing of what Newton referred to as “the language of the universe.” Lip service is paid to the goal of helping stu- dents to become more adept at problem solving, but mod- ern curricula fail to place the emphasis on the foundation of problem solving—the mastery of number relationships and the fundamental axioms and postulates upon which mathematical reasoning is based. Virtually all students can and should learn these. The second threat is that of allowing politically expe- dient fads to unduly influence public school pedagogy. If many centuries ago the developers of the abacus were able to market to society the concept that this new technology could remove from students the need to become masters of basic arithmetic calculation, we may have seen mathemat- ics take a different turn. Certainly the argument may have been valid for those societies utilizing very cumbersome number systems, such as in Greece and Rome, but even in areas that adopted Hindu numeration, the abacus did not remove the perception that mastery of hand calcula- tion was still needed by the person who was even partially educated. Even with later technological innovation, no one seriously considered that mastery of hand calculation was no longer necessary. Today, however, given advances in microelectronics (a consequence of traditional rigor in mathematics and science), there are those who promote the replacement of mastery of numerical operations with the use of a calculator. Throughout the U.S. and Canada there exist “educators” who are always seeking something new and innovative. This is seldom connected with quan- tified research to determine if the innovation actually pro- duces a better result, but it can generate a graduate degree and place one on the lucrative speaking circuit. Driven by progressivist ideology, they seek to liberate students from the drudgery of calculation, especially the dreaded long division. 3 Given that calculators of the mid 1970s were incapable of handling fractions, curricula were altered to allow deci- mals to be introduced earlier, pushing aside the antiquated fraction. The role of fractional operations was thus re- duced in elementary school, and it was left to the junior high teacher to address. Alas, calculators grew in intellect and soon learned how to work with fractions. Modern cur- riculum documents (such as the Western Canadian Proto- col Framework for Mathematics) began to include instruc- tions that students would do such calculations with pencil and paper and/or with a calculator. As a consequence, many a teacher, parent, and employer now lament the fact that graduates display an inability to comprehend or work with rational numbers. High school teachers dealing with students who have not internalized number relationships, previously ingrained as a result of mastery of paper and pencil calculation, experience increasing difficulty in de- veloping fluency with rational algebraic expressions. Only a few years ago, calculator manufacturers intro- duced the graphing calculator. The gurus immediately lobbied for more curriculum change to allow this new inno- vation to take math students (already lacking a numerate foundation) to a “new” level of understanding. No longer would they have to labour to calculate the parameters of a hyperbola. Just punch in the coefficients and watch the little lines move on the screen. No longer would stu- dents have to become proficient at completing the square of quadratics, or memorizing the unit circle. Instant re- call from the mind could be replaced with a microchip. Those making these decisions never stopped to ask who was using graphing calculators beyond high school. The fact that there is virtually no application seemed to be missed. The fact that universities do not permit these in- struments on examinations was not considered. Even the concept of memorization, the greatest tool for developing mental capacity, was spurned. There is a logical fallacy at work here that few in public education seem to be willing to expose. New approaches to public school math postulate that without an ingrained knowledge base of number relationships and without flu- ency in calculation, symbolic manipulation and formal training in reasoning, students are still able to grasp al- gebraic, trigonometric, and geometric principles at a level that will enable them to become effective problem solvers in mathematics. The fact that fewer than 10% of the mathematics graduate students in our province received their early education in North American public schools may cast some doubt on this theory, which disregards the collective wisdom of centuries. The truth is that with- out an appreciation for the discipline of mathematics, de- veloped from an early age, mathematical reasoning and potential is impeded. The new approach embraces the notion that technology is inexorably linked with the discipline of mathematics. Technology is but a consequence of mathematics. Real mathematics is in fact an independent form of technol- ogy. Today, however, technology is mindlessly driving ed- ucational philosophy, curriculum design and assessment. Mathematical reasoning is thus impeded and held hostage to this anti-intellectual, technological imperative. While technology may have many positive applications in education such as helping an instructor amplify a con- cept, its current overuse is a problem from both a financial and a pedagogical perspective. The imperative implies that because there is so much information, it is impossible to know it all. Therefore, instead of students becoming knowledge rich, they must become skilled at accessing in- formation. This too is a logical fallacy. One may go so far as to postulate the present imperative constitutes a war on both memory skills and the establishment of a broad knowledge base. Knowledge has been, is, and al- ways will be the raw material of reason, and without an internal knowledge base, process skills become ineffectual. Nowhere is this more true than in mathematics. If we are to have the craftsmen to dispose of the unfin- ished business of our predecessors, as Bell observed, stu- dents in public schools must be given the knowledge and skills that will enable them to do just that. When the Emperor is naked there is a responsibility for those aware of his condition to have the courage to so inform him. c©Copyright 2001 Gabriela Novakova 4 We’ve Got Your Number Ted Lewis† Your life is filled with code numbers. Every commercial product has a 12-digit number called the UPC (the Uni- versal Product Code). The UPC number is written as a barcode so it can be read by scanners and in decimal form so it can be read by humans. Soon, all products will carry a 13-digit barcode number that looks a lot like a UPC—in fact, it is a superset of the UPC called the EAN (Euro- pean Article Number). To order a book, you may have to supply its ISBN (International Standard Book Number). To subscribe to a magazine, you may be asked for its ISSN (International Standard Serial Number). Open your wallet and check your student ID card. It likely has a code number on it. Your driver’s permit has a ‘license number’ and it is probably accompanied by a barcode or a magnetic strip as well. Your credit card has a 16 digit code on it. If you order something over the internet, you will be asked to provide that code. If you make a mistake entering the digits, you will see a message like, “Invalid VISA number! Please check the number and re-enter.” A few years ago, I encountered a similar situation when I was using a computer program to prepare my income tax return. I wanted to see what the tax would be for variety of taxpayers, so I made up some fictitious data for “Richard Richman.” As part of the data, I included my own Social Insurance Number (S.I.N.). The program promptly rejected this because my S.I.N. was already in the small database it was creating on my hard drive, and two people cannot have the same number. So, I made up a completely arbitrary one, but that didn’t get me very far—the program told me that I had entered an invalid †Ted Lewis is a professor in the Department of Mathemat- ical Sciences at the University of Alberta. His web site is http://www.math.ualberta.ca/∼tlewis. number, and it wouldn’t let me continue until I provided an acceptable one. How can an on-line book company tell when you have entered an incorrect VISA number? How did the income tax program know that I was entering a fake S.I.N.? This “magic” is accomplished by using what is called an error- detecting code and, as is true of all magic, the idea behind error detection is quite simple.∗ The IBM scheme For validation, most error-detecting schemes use a check digit. This is usually the rightmost digit of the code. The other digits, the information digits, can be freely chosen, but the check digit is calculated. For Canadian S.I.N.s and for many credit cards, the check digit is computed using a method devised by IBM. Spaces have no significance and are only there to make it easier to read the number. Here is how the IBM scheme is used to validate Iowa Lott’s S.I.N. Beginning with the rightmost check digit, identify the alternate digits. I have put them in boxes. 3 2 4 2 1 7 6 9 4 Add the boxed digits: 3 + 4 + 1 + 6 + 4 = 18. Multiply the other digits by 2: 4, 4, 14, 18. Add the digits of these numbers: 4 + 4 + 1 + 4 + 1 + 8 = 22. Add the two results: 18 + 22 = 40. (Note that in the third step we do not add the numbers; rather, we add the digits of the numbers.) The S.I.N. is considered to be valid if the result is divisible by 10, and so Iowa Lott’s number passes the validation test. ∗When you listen to a CD, the music is brought to you courtesy of an error-correcting code. That code not only detects errors, it also repairs them. The digitally encoded music on a CD has such strong error-correcting capabilities that apparently you can drill a 2.5 mm hole through your CD and it will still play flawlessly. PIMS does not advocate that you try this!) 5 Calculating the check digit The validation procedure tells us how the check digit is found: carry out the calculations with x in the place of the check digit and solve for x. For example, suppose the information digits for your S.I.N. are 22501008. Then your S.I.N. will be 225-010-08x, and x is calculated as follows: Identify the alternate digits: 2 2 5 0 1 0 0 8 x Add the boxed digits: 2 + 5 + 1 + 0 + x = 8 + x. Multiply the other digits by 2: 4, 0, 0, 16. Add the digits of these numbers: 4 + 1 + 6 = 11. Add the two results: 8 + x+ 11 = 19 + x. To make 19+x divisible by 10, the digit x must be 1, and the social insurance number would become 225-010-081. How good is the error detection? The most common errors in entering numbers are re- ported to be: • entering one of the digits incorrectly; or • interchanging two adjacent digits. No error-detection scheme can flag all errors, and so they are designed to catch only the most common ones. At the very least, an error-detection scheme should flag either of the above. The IBM method will detect an error if a single digit is changed. This includes the case where the check digit is changed. To illustrate why, let us see what happens if the digit 7 on Iowa Lott’s credit card is changed to something else (see the picture on page 5). That is, suppose that instead of entering 7, you enter an x, and this is the only error that you make. The credit card number is 4 0 0 2 1 2 6 5 x 0 2 1 0 6 9 3 . The position of the digit xmeans that it is one of the digits that will be multiplied by 2 during the validation process. Depending upon x, the number 2x could be either a single digit or a double digit number. We consider each case separately. If 0 ≤ x < 5 (so 2x is a single digit). Carrying out the IBM validation procedure, the final sum will be 45 + 2x (try it). No matter what the digit x is, this will not be divisible by 10, and so an error will be detected. If 5 ≤ x ≤ 9 (so 2x is a two-digit number). The digits of 2x will be 1 and 2x − 10. Carry out the validation and you will get a final sum of 36 + 2x. Since x 6= 7 and since 5 ≤ x ≤ 9, this sum will also fail to be divisible by 10, and an error will be detected. The IBM method is very good at detecting an error if a single digit is entered incorrectly. Although it is not completely successful in dete