South African Mathematics Foundation: Bloghttps://www.samf.ac.za/BlogFri, 07 May 2021 07:01:55 Zurn:store:1:blog:post:20https://www.samf.ac.za/en/going-beyond-the-curriculum-samf-olympiad-programmesGoing beyond the curriculum: SAMF Olympiad Programmes<div>
<p><em>By Ellie Olivier</em></p>
<p>There is a famous anecdote about a renowned mathematician, John von Neumann. A friend of Von Neumann gave him a problem to solve: two cyclist A and B, at a distance of 20 miles apart, were approaching each other, each going at a speed of 10 miles per hour. A bee flew back and forth between A and B at a speed of 15 miles per hour, starting with A and back to A after meeting B, then back to B after meeting A, and so on. By the time the two cyclists met, how far had the bee travelled? To his friend’s astonishment von Neumann gave the correct answer in a flash by summing an infinite series.</p>
<p>Prof Man Keung Siu, in his paper, <em>The good, the bad and the pleasure (not pressure!) of mathematics</em> <em>competitions </em>apply the story to explain that there are several ways to go about solving a mathematical problem. The process of first calculating when the cyclists met is slick and captures the key point of the problem. The other way of summing the infinite series is slower and goes about systematically solving the problem.</p>
<p>The first method, which has been going on in the classroom of most schools, is to present the subject in a consistently organised and cautiously designed structure complete with exercises and problems. Another approach, which goes on more predominantly in the training of mathematics competitions, is to confront learners with various kinds of problems and to train them to look for multiple ways to solve the problem, thereby acquiring a host of strategies and techniques. Each approach has its separate merit and complements each other.</p>
<p>Learners who shine in mathematical Olympiads and competitions are not all-natural problem-solvers, but exceed because they have good teachers/mentors who expose them to problem-solving over some time. The problems in a mathematics competition are more complicated and stimulating than the usual exercises of the classroom, giving promising young mathematicians chances to expand their horizons and show their potential.</p>
<p>Since mathematics provides the foundation for many desired career paths, the improvement of mathematical problem-solving skills is essential for our country’s economic growth. We have to empower learners to become independent, creative and critical thinkers to interpret and critically analyse everyday situations and to solve problems and develop future leaders and influencers in the science, engineering, technology, economic, financial, and management sciences space.</p>
<p>It is pleasing to see that many teachers understand and recognise the significance of mathematical competitions giving the fact that more than 176,000 primary and high school learners participated in the first rounds of the Old Mutual South African Mathematics Olympiad (SAMO) and the Nestlé Nespray South African Mathematics Challenge (SAMC). More than 86,000 Grade 8-12 learners from 1,200 schools participated in the SAMO and more than 90,000 Grade 4-7 learners from 1,096 schools participated in the SAMC.</p>
<p>We want to thank teachers for allowing their learners to go beyond the curriculum and the provincial education departments, especially Western Cape, Kwa-Zulu Natal and Mpumalanga, for encouraging schools to participate. Learners who achieved at least 50% in the first rounds will advance to the second rounds in May.</p>
<p>Learners who develop their mathematical problem-solving skills over some time have an advantage over learners who only write the different rounds of the competitions. We, therefore, invite learners who have qualified for the second round of the SAMO/SAMC to register for our Olympiad Training Programme - <a href="https://www.samf.ac.za/en/olympiad-training-programme-2">https://www.samf.ac.za/en/olympiad-training-programme-2</a>. The ideal outcome is that you foster a genuine interest in mathematics as a subject, but that you should also enjoy the extracurricular activity. I conclude with a poem by Prof Siu:</p>
<p> </p>
<p><em>The good, the bad and the pleasure of mathematics competitions</em></p>
<p><em>Are to which we should pay our attention.</em></p>
<p><em>Benefit from the good; avoid the bad;</em></p>
<p><em>And soak in the pleasure.</em></p>
<p><em>Then we will find for ourselves satisfaction!</em></p>
</div>urn:store:1:blog:post:19https://www.samf.ac.za/en/2019-nsw-maths-puzzle-62019 NSW: Maths Puzzle 6<p><strong>6th puzzle by Dr Harry Wiggins (Reverse fun)</strong></p>
<p>Did you know 2178 x 4 = 8712?</p>
<p>Find all ten-digit numbers which becomes 9 times bigger if the order of the digits is reversed.</p>urn:store:1:blog:post:18https://www.samf.ac.za/en/2019-nsw-maths-puzzle-52019 NSW: Maths Puzzle 5<p><strong>5th puzzle by Dr Harry Wiggins (All in the family)</strong></p>
<p>Nxolo has four children with different ages. The two daughters are Cindy and Letshego and the two sons have names Thabo and Zuma. Once, at a dinner table, the following conversation happened:</p>
<p>Cindy: The sum of my oldest brother and mom is 53.</p>
<p>Letshego: Six years ago, Thabo was double my age.</p>
<p>Thabo: Four times Cindy’s age plus 6 times my age is 100.</p>
<p>Zuma: The sum of all our five ages is 77.</p>
<p>Thabo: Triple my age plus my oldest sister is 45.</p>
<p>What is the age of the mom? Who is the youngest child? (Take into account that every statement is true and every person’s age is a positive whole number.)</p>
<p> </p>
<p><img src="/Content/Images/uploaded/puzzle5[8339].jpg" alt="" width="225" height="225" /></p>urn:store:1:blog:post:17https://www.samf.ac.za/en/2019-nsw-maths-puzzle-42019 NSW: Maths Puzzle 4<p><strong>4th puzzle by Dr Harry Wiggins (Money, money, money)</strong></p>
<p>Two choices today. Solve either puzzle (a) or (b) and post your interesting comments.</p>
<p>(a) If in South Africa, you only get two types of coins: R19 and R20 coins. What is the least number of coins you need to pay a bill of R2019? </p>
<p>(b) A billionaire is splitting his one billion rand fortune amongst 100 relatives. The first relative gets 1% of his fortune. The second relative gets 2% of the remaining fortune. Then the third relative gets 3% of the remaining fortune and so on, until the last relative gets 100% of the rest. Which relative will inherit the most money and what chunk of the original fortune will it be?</p>
<p><img src="/Content/Images/uploaded/puzzle4[8338].jpg" alt="" width="300" height="300" /></p>urn:store:1:blog:post:16https://www.samf.ac.za/en/2019-nsw-maths-puzzle-32019 NSW: Maths Puzzle 3<p><strong>3rd puzzle by Dr Harry Wiggins (Complete the grid)</strong></p>
<p>Fill in each space of the grid with one of the numbers 1, 2, . . . , 30, using each number once. For 1 ≤ n ≤ 29, the two spaces containing n and n + 1 must be in either the <strong>same row or the same column</strong>. Some numbers have been given to you. What will the entries be in the first row of this grid when completed?</p>
<p><img src="/Content/Images/uploaded/puzzle3[8262].png" alt="" width="375" height="310" /></p>
<p><img src="file:///C:/Users/Alan/Desktop/Alan/2019.jpg" alt="" /><img src="file:///C:/Users/u04564996/Desktop/2019.jpg" alt="" /></p>urn:store:1:blog:post:15https://www.samf.ac.za/en/2019-nsw-maths-puzzle-22019 NSW: Maths Puzzle 2<p><strong>2nd puzzle by Dr Harry Wiggins (Helping Hercules)</strong></p>
<p>Hydra is a serpentine water monster that lives in the lake of Lerna. It is Hercules's task to kill this monster (the second task of his Twelve Labors). The Hydra possesses 100 heads and can only be killed by cutting of ALL his heads. It is know that you can cut off exactly 10, 20, 30 or 40 heads at a time. However with each of these 40, 2, 50 or 4 new heads will re-appear in each case. </p>
<p>How can Hercules defeat the monster? What is the least number of cuts Hercules needs?</p>
<p><img src="/Content/Images/uploaded/puzzle2[8261].jpg" alt="" width="212" height="238" /></p>
<p> </p>
<p> </p>urn:store:1:blog:post:14https://www.samf.ac.za/en/2019-nsw-maths-puzzle-12019 NSW: Maths Puzzle 1<p><strong>1st puzzle by Dr Harry Wiggins (Avoiding 2019)</strong></p>
<p>Thabo writes down all the positive integers in order but he always avoids writing down the digits 2, 0, 1 and 9. So Thabo writes down 3, 4, 5, 6, 7, 8, 33, 34, 35, etcetera. Thus the 7th number he writes dow<img src="C:\Users\Alan\Desktop\Alan\2019.jpg" alt="" />n is 33. What is the 2019th number that Thabo would write down?</p>
<p><img src="file:///C:/Users/Alan/Desktop/Alan/2019.jpg" alt="" /><img src="file:///C:/Users/u04564996/Desktop/2019.jpg" alt="" /><img src="/Content/Images/uploaded/puzzle1[8260].jpg" alt="" width="293" height="172" /></p>urn:store:1:blog:post:7https://www.samf.ac.za/en/does-society-need-imo-medalistsDoes society need IMO Medalists?<p>In this Blog, the invited presentation of Siu Man Keung at the IMO Forum during IMO 2016 in Hong Kong, is reproduced here with his permission. We trust readers will find his title and presentation provocative, and the three problems in the APPENDIX stimulating. (The appendix is available for downloading separately at the provided link).</p>
<p><span style="font-size: 12pt;"><strong>Does society need IMO Medalists?</strong></span></p>
<p>(Invited talk at the IMO Forum at the Hong Kong Polytechnic University on 11 July 2016)</p>
<p>Siu Man Keung</p>
<p>Department of Mathematics</p>
<p>University of Hong Kong</p>
<p><strong>Abstract</strong></p>
<p>The title of this talk that sounds provocative is not chosen with any intention to embarrass the organizers and participants of the event of IMO (International Mathematical Olympiad). It should be seen as the sharing of some thoughts on this activity, or more generally on mathematical competitions, by a teacher of mathematics who had once helped in the coaching of the first Hong Kong Team to take part in the 29<sup>th</sup> IMO held in Canberra in 1988 and in the coordination work of the 35<sup>th</sup> IMO held in Hong Kong in 1994. The speaker tries to look at the issue in its educational context and more broadly in its socio-cultural context.</p>
<p><strong>About the speaker</strong></p>
<p>SIU Man Keung obtained his BSc from the Hong Kong University and went on to earn a PhD in mathematics from Columbia University. Like the Oxford cleric in Chaucer’s <em>The Canterbury Tales</em>, “and gladly would he learn, and gladly teach” for more than three decades until he retired in 2005, and is still enjoying himself in doing that after retirement. He has published some research papers in mathematics and computer science, some more papers of a general nature in history of mathematics and mathematics education, and several books in popularizing mathematics. In particular he is most interested in integrating history of mathematics with the teaching and learning of mathematics and has been participating actively in an international community of History and Pedagogy of Mathematics since the mid-1980s. He has devoted much of his time in offering a course titled <em>Mathematics: A Cultural Heritage</em> in the tradition of liberal studies for undergraduates from various Faculties of the Hong Kong University for a decade during the 2000s as well.<strong> <br /></strong></p>
<p><span style="font-size: 12pt;"><strong>Does society need IMO Medalists?</strong></span></p>
<p>Does society need IMO Medalists? No, society does not “need” IMO Medalists. Society does not even “need” mathematicians. Does this mean my 30-minute talk will end here? Stopping here and now would amount to an admission of wrong choice of my profession in all these years, so I should go on talking. You will notice that I put the word “need” in quotation marks. Society does not “need” (in quotation marks!) IMO Medalists or mathematicians, but society needs (no quotation mark!) MTR maintenance workers, garbage collectors, street cleaners, plumbers, electricians, etc. Now, perhaps you know what I mean. Let us get back to IMO.</p>
<p>After 22 years IMO comes back to Hong Kong as the host. Hong Kong hosted the 35<sup>th</sup> IMO in the summer of 1994. I like to mention two Medalists in that particular IMO. One is Maryam Mirzakhani of Iran, who became the first female mathematician to receive a Fields Medal at the International Congress of Mathematicians in 2014. The other one is Subash Ajit Khot of India, who was awarded the Nevanlinna Prize at the same Congress.</p>
<p>In a talk I gave in 2012 with the title “The good, the bad and the pleasure (not pressure) of mathematics competitions” I outlined certain good and bad points of mathematics competitions. Allow me to repeat them here in summary. [For a more detailed discussion, see: M.K. Siu, Some reflections of a coordinator on the IMO, <em>Mathematics Competitions</em>, 8 (1) (1995), 73-77; M.K. Siu, The good, the bad and the pleasure (not pressure!) of mathematics competitions, <em>Mathematics Competitions</em>, 26(1) (2013), 41-58.]</p>
<p>Good points: Nurturing of (1) clear and logical presentation, (2) tenacity and assiduity, (3) “academic sincerity”; moreover, arouse a passion for and pique the interest in mathematics.</p>
<p>Bad points: (1) competition problems <em>versus </em>research, (2) over-training?</p>
<p>We further ask: Is the passion for the subject of mathematics itself genuine? Can the interest be sustained?</p>
<p>Let me further explain the point about competition problems <em>versus</em> research through three examples (see <strong>APPENDIX</strong> available as a PDF at: <span style="text-decoration: underline;"><a href="http://dynamicmathematicslearning.com/appendix-MKSiu-IMO2016.pdf" target="_blank">http://dynamicmathematicslearning.com/appendix-MKSiu-IMO2016.pdf</a></span> ).</p>
<p><strong>What do we see from these three examples? </strong>It makes me think that there are two approaches in doing mathematics. To give a military analogue one is like positional warfare and the other guerrilla warfare. The first approach, which has been going on in the classrooms of most schools and universities, is to present the subject in a systematically organized and carefully designed format supplemented with exercises and problems. The other approach, which goes on more predominantly in the training for mathematics competitions, is to confront students with various kinds of problems and train them to look for points of attack, thereby accumulating a host of tricks and strategies.</p>
<p>Each approach has its separate merit and they supplement and complement each other. Each approach calls for day-to-day preparation and solid basic knowledge. Just as in positional warfare flexibility and spontaneity are called for, while in guerrilla warfare careful prior preparation and groundwork are needed, in the teaching and learning of mathematics we should not just teach tricks and strategies to solve special type of problems or just spend time on explaining the general theory and working on problems that are amenable to routine means. We should let the two approaches supplement and complement each other in our classrooms. In the biography of the famous Chinese general and national hero of the Southern Song Dynasty, Yue Fei (1103-1142) we find the description: “<em>(Setting up the battle formation is the routine of the art of war. Maneuvering the battle formation skillfully rests solely with the mind.)</em>”</p>
<p>Sometimes the first approach may look quite plain and dull, compared with the excitement acquired from solving competition problems by the second approach. However, we should not overlook the significance of this seemingly bland approach, which can cover more general situations and turns out to be much more powerful than an <em>ad hoc</em> method which, slick as it is, solves only a special case. Of course, it is true that frequently a clever <em>ad hoc</em> method can develop into a powerful general method or can become a part of a larger picture. A classic case in point is the development of calculus in history. In ancient time, only masters in mathematics could calculate the area and volume of certain geometric objects, to name just a couple of them, Archimedes (c. 287 B.C.E. – c. 212 B.C. E.) and LIU Hui ( 3<sup>rd</sup> century). In hindsight their formula for the area of a circle, <em>A</em> = (1/2) x <em>C x r </em>, embodies the essence of the Fundamental Theorem of Calculus. With the development of calculus since the seventeenth century and the eighteenth century, today even an average school pupil who has learnt calculus will be able to handle what only great mathematicians of the past could have resolved.</p>
<p>Since many mathematics competitions aim at testing the contestants’ ability in problem solving rather than their acquaintance with specific subject content knowledge, the problems are set in some general areas which can be made comprehensible to youngsters of that age group, independent of different school syllabi in different countries and regions. That would cover topics in elementary number theory, algebra, combinatorics, sequences, inequalities, functional equations, plane and solid geometry and the like. Gradually the term “Olympiad mathematics” is coined to refer to this conglomeration of topics. One question that I usually ponder over is this: why can’t this type of so-called “Olympiad mathematics” be made good use of in the school classroom as well? If one aim of mathematics education is to let students know what the subject is about and to arouse their interest in it, then interesting non-routine problems should be able to play their part well when used to supplement the day-to-day teaching and learning.</p>
<p>Let us get back to the question in the title of the talk: <strong>Does society need IMO Medalists?</strong> No, society does not “need” IMO Medalists. Society does not even “need” mathematicians. But society needs “friends of mathematics”. A “friend of mathematics” may not know a lot of mathematics but would understand well what mathematics is about and appreciate well the role of mathematics in the modern world.</p>
<p>The mathematician Paul Halmos once said, “<em>It saddens me that educated people don’t even know that my subject exists</em>.” Allen Hammond, editor of <em>Science</em>, once described mathematics as “the invisible culture”. On the other hand, perhaps it is a blessing to remain not that visible! Two months ago I read in the news (Associated Press, May 7, 2016): “Ivy League Professor Doing Math Equation on Flight Mistaken for Terrorist”. An American Airline passenger seated next to Guido Menzio of UPenn suspected the unfamiliar writings of the professor were a code for a bomb. It led to Professor Menzio being taken away from the plane to be interrogated!</p>
<p>In ancient China the third-century mathematician LIU Hui said, “ <em>(The subject [mathematics] is not particularly difficult by using methods transmitted from generation to generation, like the compasses [gui] and gnomon [ju] in measurement, which are comprehensible to most people. However, nowadays enthusiasts for mathematics are few, and many scholars, much erudite as they are, are not necessarily cognizant of the subject.)</em>”</p>
<p><strong>Why is it like that?</strong></p>
<p>Here is a passage taken from a book: “Central to my argument is the idea that ***** distinguished by a self-conscious attention to its own<em> ***** </em>language. Its claim to function <em>as art </em>derives from its peculiar concern with its own materials and their formal patterning, aside from any considerations about its audience or its social use.” Can you guess what the missing words are?</p>
<p>This passage is taken from the book by Julian Johnson, <em>Who Needs Classical Music? Cultural Choice and Musical Value </em>(2002). The missing words are “classical music” and “musical”. However, the passage would ring equally true if “classical music” is replaced by “mathematics”!</p>
<p>In the same book the author says, “… that it [meaning classical music] relates to the immediacy of everyday life but not immediately. That is to say, it takes aspects of our immediate experience and reworks them, reflecting them back in altered form. In this way, it creates for itself a distance from the everyday while preserving a relation to it.” <strong>Mathematics is also like that.</strong> This explains why it is not easy to bring mathematics to the general public. To become a “friend of mathematics” one needs to be brought up from school days onward in an environment where mathematics is not only enjoyable but also makes good sense. In the preface to a textbook [<em>Alice in Numberland: A Students’ Guide to the Enjoyment of Higher Mathematics</em> (1988)] the authors, John Baylis and Rod Haggarty, remark, “The professional mathematician will be familiar with the idea that entertainment and serious intent are not incompatible: the problem for us is to ensure that our readers will enjoy the entertainment but not miss the mathematical point, […]”</p>
<p>My good friend, Tony Gardiner, an experienced four-time UK IMO team leader, once commented that I should not blame the negative aspects of mathematics competitions on the competition itself. He went on to enlighten me on one point, namely, a mathematics competition should be seen as just the tip of a very large, more interesting, iceberg, for it should provide an incentive for each country to establish a pyramid of activities for masses of interested students. It would be to the benefit of all to think about what other activities besides mathematics competitions can be organized to go along with it. These may include the setting up of a mathematics club or publishing a magazine to let interested youngsters share their enthusiasm and their ideas, organizing a problem session, holding contests in doing projects at various levels and to various depth, writing book reports and essays, producing cartoons, videos, softwares, toys, games, puzzles, … .</p>
<p>Finally the question boils down to one in an even more general context: <strong>Does society need ME?</strong> We frequently hear about the cliché “No one is indispensable!” But please bear in mind that everyone has his or her worth and can do his or her part to make this world a better place to live in. An IMO medalist is no exception!</p>
<p>Thank you!</p>
<p> </p>urn:store:1:blog:post:6https://www.samf.ac.za/en/who-needs-mathematicsWho Needs Mathematics?<p>The world we are living in is faced with many problems some of which are pollution, global warming, over-population, starvation, traffic congestion, crime, corruption, terrorism and war. In combating and addressing some of these problems, mathematics as the language of modern science generally, can play a big role. The applications of mathematics pervade our society: from the listening to digital music to the prediction of weather; from estimating fish populations to analyzing the spread of an epidemic, from politics to the arts, from medicine to sport, from optimizing the running of a business to minimizing the effect of an earthquake or tornado, from encryption for cyber banking to spying on a business competitor, etc. The list simply goes on, and on!</p>
<p>Despite this pervasive use of mathematics in almost every field of human endeavor, learners at school, still frequently ask their mathematics teacher at every level: “Who needs mathematics?” or “When will I use math?” A non-profit website that tries to help answer this question is: <span style="text-decoration: underline;"><a href="http://weusemath.org/">http://weusemath.org/</a></span></p>
<p>This non-profit website describes the importance of mathematics and many rewarding career opportunities available to students who study mathematics. It firstly has an introductory video that shows short snippets by various professionals who use mathematics extensively. Then the website has a Blog section which has short, regular articles on interesting mathematics and its applications, people using mathematics, as well as the occasional posting of an interesting problem.</p>
<p>Also listed on the website are 45 different careers ranging from Actuary to Urban Planner, giving additional information on each of these professions. In the “Did You Know’ section there are subsections on “Math in Real Life’, ‘Math Titbits’, ‘New Discoveries’ and ‘Unsolved Problems’. Finally, there is a section on Math Resources for the Teacher’.</p>
<p>Another useful column for teachers to show their learners to illustrate the fascinatingly wide range of fields that use mathematics is the regular ‘Math in the Media’ column at the American Mathematical Society (AMS) website at: <span style="text-decoration: underline;"><a href="http://www.ams.org/news/math-in-the-media/math-in-the-media">http://www.ams.org/news/math-in-the-media/math-in-the-media</a></span> The AMS ‘Features’ (Monthly Essays on Mathematical Topics) Column for July also describe an interesting application of statistics to try and reduce the losses American bombers were suffering during World War II. Find this at: <span style="text-decoration: underline;"><a href="http://www.ams.org/samplings/feature-column/fc-2016-06">http://www.ams.org/samplings/feature-column/fc-2016-06</a></span></p>
<p>For those learners more inclined and attracted to the Creative Arts, a visit to the AMS ‘Mathematical Imagery’ column at <span style="text-decoration: underline;"><a href="http://www.ams.org/mathimagery/">http://www.ams.org/mathimagery/</a></span> will be a true visual delight and inspiration. The connection between mathematics and art goes back thousands of years. Mathematics has been used in the design of Gothic cathedrals, Rose windows, oriental rugs, mosaics and tilings. Mathematicians and artists continue to create stunning works in all media and to explore the visualization of mathematics--origami, computer-generated landscapes, tesselations, fractals, anamorphic art, and more.</p>
<p>The Mathematical Association of America (MAA) also has a section on ‘Careers’ at <span style="text-decoration: underline;"><a href="http://www.maa.org/careers">http://www.maa.org/careers</a></span> that learners might find useful. The most popular careers listed there Teaching, Actuarial Science, Computer Science, Operations Research, Biomathematics, Cryptography, and Finance. Of special interest is that in 2014, <em>CareerCast</em> announced its annual ranking of the 10 best jobs and nine out of the top 10 jobs are in the STEM (science, technology, engineering, and math) category with statistician ranked third and actuary ranked fourth. CareerCast ranks the 200 most populated jobs based on four factors: environment, income, outlook, and stress.</p>
<p>In relation to the above ranking of careers, another useful site to visit is the ‘Why Study Math?’ link of Duke University at: <span style="text-decoration: underline;"><a href="http://math.duke.edu/undergraduate/why-math-major">http://math.duke.edu/undergraduate/why-math-major</a></span></p>
<p>The South African Mathematics Foundation (SAMF) has published a series on <em>Careers in Mathematics</em>; visit their page <span style="text-decoration: underline;"><a href="http://www.samf.ac.za/careers-in-mathematics">http://www.samf.ac.za/careers-in-mathematics</a></span> for more information.</p>
<p>Lastly, this site <span style="text-decoration: underline;"><a href="http://www.thecompleteuniversityguide.co.uk/courses/mathematics/5-reasons-why-you-should-study-mathematics/">http://www.thecompleteuniversityguide.co.uk/courses/mathematics/5-reasons-why-you-should-study-mathematics/</a></span> gives the following 5 reasons for studying mathematics, namely:</p>
<p>1. <em>Humanity needs Maths</em></p>
<p>2. <em>Potential for Joint courses</em></p>
<p>3. <em>Graduate Prospects</em></p>
<p>4. <em>Transferable skills</em></p>
<p>5. <em>Salary advantage</em></p>
<p>Above all, I think a major reason for studying mathematics is the personal intellectual satisfaction one achieves by being challenged and learning not only to appreciate the conquests of those mathematicians and scientists of the past, but also those of the present, in constantly shaping and reshaping both the fascinating world of abstract mathematical thought as well as our natural and technological world through application.</p>
<p> </p>
<p>- Michael de Villiers, Professor Extraordinaire: Mathematics Education, University of Stellenbosch.</p>urn:store:1:blog:post:5https://www.samf.ac.za/en/a-critique-of-the-school-mathematics-curriculumA Critique of the School Mathematics Curriculum<p><strong>A Critique of the School Mathematics Curriculum</strong></p>
<p>Written in 2002, “A Mathematician’s Lament” by a mathematician by the name of Paul Lockhart was published online in 2008 at the Devlin’s Angle on the website of the Mathematical Association of America. It immediately caused a widespread stir as it gave a scathing attack on the traditional mathematics curriculum at school. The purpose of this Blog is to give a brief overview and comment on some of his criticisms. Readers are invited to download and read his full paper directly from (click on link):</p>
<p><span style="text-decoration: underline;"><a href="http://dynamicmathematicslearning.com/LockhartsLament.pdf">http://dynamicmathematicslearning.com/LockhartsLament.pdf</a></span></p>
<p>In addition, it is hoped that readers, whether they agree or disagree with Lockhart, will give feedback by commenting and discussing some of the important issues raised by Lockhart in his provocative paper.</p>
<p>Lockhart’s paper starts off by describing a nightmarish, educational situation experienced by a musician where suddenly children are not allowed:</p>
<p>‘… <em>to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school</em>.’ (p. 1)</p>
<p>After describing a similar ludicrous situation for painting, he draws an analogy between these nightmarish situations and the traditional mathematics curriculum as follows (p. 2):</p>
<p>‘<em>Sadly, our present system of mathematics education is precisely this kind of nightmare. In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul- crushing ideas that constitute contemporary mathematics education</em>.’</p>
<p>Lockhart then goes on to describe mathematics as an art and that it is about ‘<em>wondering, playing, amusing yourself with your imagination</em>’ and that ‘<em>inspiration, experience, trial and error, dumb luck’ all have their role to play in mathematics</em>’ (p. 4). He criticizes the mathematics curriculum (and associated textbooks) for focusing largely only on the learning of facts and manipulative procedures and calculations instead of the creative spirit underlying mathematical thinking as follows:</p>
<p>‘<em>By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject …</em></p>
<p><em>By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself.</em>’ (p. 5)</p>
<p>It is interesting that in the above quotation (and also on p. 8), Lockhart is making the ‘process-product’ distinction that was made popular in mathematics education in the late 1970’s and 1980’s by Hans Freudenthal, Alan Bell, and many others. This distinction tried to draw attention to the processes of mathematical creativity that lead to the eventual, finished products. In other words, to focus more in our teaching on the processes of discovery and creativity that lead to theorems and their proofs, as well as those of algorithmitizing, classifying, defining and axiomatizing that respectively lead to algorithms, definitions and axioms. Noteworthy in the above quotation is also that he assigns an ‘explanatory’ function to proof (argument) and not just traditional one of the ‘verification’ of results.</p>
<p>On p. 9, Lockhart pleads for the connection of the school mathematics curriculum with the historical (and philosophical) development of the subject, and refers to the work of Eudoxus and Archimedes. He strongly argues that problems (and problem-solving) should be made central to mathematics teaching as follows (also compare with p. 16):</p>
<p>‘<em>Technique in mathematics, as in any art, should be learned in context. The great problems, their history, the creative process— that is the proper setting. Give your students a good problem, let them struggle and get frustrated. See what they come up with. Wait until they are dying for an idea, then give them some technique</em>.’ (p. 9)</p>
<p>‘<em>So how do we teach our students to do mathematics? By choosing engaging and natural problems suitable to their tastes, personalities, and level of experience. By giving them time to make discoveries and formulate conjectures. By helping them to refine their arguments and creating an atmosphere of healthy and vibrant mathematical criticism. By being flexible and open to sudden changes in direction to which their curiosity may lead. In short, by having an honest intellectual relationship with our students and our subject</em>.” (p. 10)</p>
<p>In this regard, he argues very similarly to George Polya and others about the importance and centrality of problems and problem-solving not only historically, but also in current mathematical research. Though Lockhart doesn’t say a mathematics teacher has to be a professional mathematician, but that they should not be mere ‘data transmitters’. In essence, they should at least be mathematically active and good problem solvers themselves to be able to engage their learners in genuine mathematical activity (p. 11).</p>
<p>On p. 15, he strongly criticizes the early introduction of ‘a priori’ definitions of terminology and concepts if the need for those definitions have not yet been developed in some relevant problem setting. In particular, on p. 22, he gives a lengthy criticism of the way definitions are handled in the curriculum:</p>
<p>“<em>Even the traditional way in which definitions are presented is a lie. In an effort to create an illusion of “clarity” before embarking on the typical cascade of propositions and theorems, a set of definitions are provided so that statements and their proofs can be made as succinct as possible. On the surface this seems fairly innocuous; why not make some abbreviations so that things can be said more economically? The problem is that definitions matter. They come from aesthetic decisions about what distinctions you as an artist consider important. And they are problem-generated. To make a definition is to highlight and call attention to a feature or structural property. Historically this comes out of working on a problem, not as a prelude to it.</em></p>
<p><em>The point is you don’t start with definitions, you start with problems. Nobody ever had an idea of a number being “irrational” until Pythagoras attempted to measure the diagonal of a square and discovered that it could not be represented as a fraction. Definitions make sense when a point is reached in your argument which makes the distinction necessary. To make definitions without motivation is more likely to cause confusion.</em></p>
<p><em>This is yet another example of the way that students are shielded and excluded from the mathematical process. Students need to be able to make their own definitions as the need arises— to frame the debate themselves. I don’t want students saying, “the definition, the theorem, the proof,” I want them saying, “my definition, my theorem, my proof</em>.” (p. 22)</p>
<p>Clearly, his comments about the poor teaching of definitions remind one strongly of similar criticisms already given by Freudenthal, Krygowska, and others in the 1970’s. However, despite these, hardly any changes in the mathematics curricula around the world have been affected.</p>
<p>In line with the above, Lockhart describes the typical High School Geometry course at school as an ‘instrument of the devil’ if it is introduced in a highly formalized manner as follows (p. 18):</p>
<p>‘<em>The student-victim is first stunned and paralyzed by an onslaught of pointless definitions, propositions, and notations, and is then slowly and painstakingly weaned away from any natural curiosity or intuition about shapes and their patterns by a systematic indoctrination into the stilted language and artificial format of so-called “formal geometric proof.”</em>’</p>
<p>His viewpoint in this regard is in correspondence with a learning theory perspective such as the Van Hiele theory, which argues that learners ought to develop through at least two levels of Recognition and Properties of Shapes before the Deductive Ordering of Shapes would become meaningful to learners.</p>
<p>Lockhart further argues on p. 20 that ‘formal, rigorous proof’ should not be a student’s first introduction to logical argument, and that it only becomes important when there are crises or paradoxes that have developed naturally. He opposes the two—column proof method often used by teachers in the classroom saying that it is artificial, ugly and unnatural. Specifically, on p. 21 he writes: “<em>No mathematician works this way. No mathematician has ever worked this way. This is a complete and utter misunderstanding of the mathematical enterprise. Mathematics is not about erecting barriers between ourselves and our intuition, and making simple things complicated. Mathematics is about removing obstacles to our intuition, and keeping simple things simple</em>.”</p>
<p>Continuing in the same mode (p. 22), Lockhart concludes: “<em>The problem with the standard geometry curriculum is that the private, personal experience of being a struggling artist has virtually been eliminated. The art of proof has been replaced by a rigid step-by step pattern of uninspired formal deductions. The textbook presents a set of definitions, theorems, and proofs, the teacher copies them onto the blackboard, and the students copy them into their notebooks. They are then asked to mimic them in the exercises. Those that catch on to the pattern quickly are the “good” students</em>.”</p>
<p>Lockhart finishes of his critique by giving a short global critique of each phase of the whole school mathematics curriculum starting at Lower and Middle School, and proceeding through Algebra, Geometry through to Pre-Calculus and Calculus.</p>
<p>- Michael de Villiers, Professor Extraordinaire: Mathematics Education, University of Stellenbosch. </p>urn:store:1:blog:post:3https://www.samf.ac.za/en/the-work-of-teachingThe Work of Teaching<p><iframe src="http://www.youtube.com/embed/nrwDM4ejNqs" width="613" height="440"></iframe></p>
<p> </p>
<p> </p>
<p> </p>
<p> </p>
<p> </p>
<p> </p>
<p> </p>
<p> </p>
<p> </p>