tag:blogger.com,1999:blog-2516188730140164076Mon, 28 May 2018 01:35:07 +0000mathematical thinkingCommon CoreMOOCmathematics educationFibonacciLeonardo of PisaJo Boaleralgebraonline educationCourseraDragonBoxKhan AcademyMOOCsPisaSteve Jobseducational video gamesgame-based learningproblem solvingAlpine Road TrailApple MacintoshHindu-arabic arithmeticJonathan BorweinJordan EllenbergKeith DevlinMultiplicationPi DayStanford Universityapplications of mathematicsarithmeticcomputers and mathematicsconceptual understandingeducational startupsgolden ratioinfinitymathematical modelingmountain bikingnumber sensepiproofsquantitative literacyscience of patternstransition courseAdolf ZeisingAlex GibneyAndrew HackerAndrew NgArea of a circle formulaAronszajn treeAsa ButterfieldAspen Ideas FestivalBedtime MathBrexitCarl SaganCreateSpaceDan MeyerDaphne KollerDavid LeonhardtDerek MullerDowns paradoxETSEdge questionEinstein’s energy equationEric MazurErlwangerEuler’s IdentityEuler’s equationEuropean CommunityExploding dotsFibonacci sequenceFinnish educationGOFMGeorg CantorGeorge BooleGrant SandersonGregory’s SeriesH-STARHerman MehtaHilbert's HotelHypercubesIMOITSInquiry-Based LearningIs algebra necessaryJames Paul GeeJames TantonJohn BrockmanJon BarwiseK-12 educationKarim AniKen RobinsonKool AidKönig’s LemmaLaura OverdeckLearnLabLet’s Make a DealLiber abbaciLiping MaM.C. 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Mathematician Keith Devlin is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guyon NPR's Weekend Edition.http://devlinsangle.blogspot.com/noreply@blogger.com (Mathematical Association of America)Blogger82125tag:blogger.com,1999:blog-2516188730140164076.post-8134008208204677690Wed, 02 May 2018 12:00:00 +00002018-05-02T09:50:12.923-04:00Common Corecomputers and mathematicsJonathan Borweinmathematics educationCalculation was the price we used to have to pay to do mathematicsB<span style="background-color: white; color: #222222; display: inline; float: none; font-family: "georgia" , "utopia" , "palatino linotype" , "palatino" , serif; font-size: 13.2px; font-style: normal; font-weight: 400; letter-spacing: normal; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;">y Keith Devlin</span><br /><div class="MsoNormal" style="-webkit-text-stroke-width: 0px; background-color: white; color: #222222; font-family: Georgia, Utopia, "Palatino Linotype", Palatino, serif; font-size: 13.2px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; line-height: normal; margin-bottom: 0in; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><br /></div><div class="MsoNormal" style="-webkit-text-stroke-width: 0px; background-color: white; color: #222222; font-family: Georgia, Utopia, "Palatino Linotype", Palatino, serif; font-size: 13.2px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; line-height: normal; margin-bottom: 0in; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span style="color: black;">You can follow me on Twitter </span><span id="goog_646323682"></span><a href="https://www.blogger.com/goog_646323681" style="text-decoration: none;"><span style="color: #3d85c6;">@profkeithdevlin</span><span id="goog_646323683"></span></a></div><div class="MsoNormal" style="-webkit-text-stroke-width: 0px; background-color: white; color: #222222; font-family: Georgia, Utopia, "Palatino Linotype", Palatino, serif; font-size: 13.2px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; line-height: normal; margin-bottom: 0in; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span style="color: black;"><span style="color: #3d85c6;"><br /></span></span></div><div class="MsoNormal" style="-webkit-text-stroke-width: 0px; background-color: white; color: #222222; font-family: Georgia, Utopia, "Palatino Linotype", Palatino, serif; font-size: 13.2px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; line-height: normal; margin-bottom: 0in; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span style="color: black;"></span><br /></div>Ever since mathematics got properly underway around 3,000 years ago, there was only one way to achieve access to the field. You had to spend many years developing a fairly extensive calculation skillset. In the first instance, to pass the graduation and entrance examinations to gain initial access to the field. Then, once accepted into the world of mathematics, calculation of one kind or another was what all mathematicians spent the bulk of their mathematical time doing. Arguably, for most of mathematics history, the subject really was, to a large extent, primarily about calculation of one form or another. Newton, Leibniz, Bernoulli (any of them), Fermat, Euler, Riemann, Gauss, and the other greats of times past, were all superb masters of calculation. (We should also include Boole, since his famous Boolean algebra is also a calculation system.) <br /><br />But whereas most laypersons seem to think that calculation is all there is to mathematics, surely none of the greats did. Calculation was an important tool (more accurately, a set of tools) you needed to do mathematics, they must have realized, but the essence of mathematics is much more, a plateau of knowledge that transcends all the calculation techniques.<br /><br />In the 19th Century, that somewhat tacit understanding became explicit. The increasing complexity of the problems mathematicians tackled led to a series of results that defied the human intuition. (Several of them were referred to as “paradoxes”.) This led to an intense period of mathematical introspection, where the primary focus was not performing a calculation or computing an answer, but formulating and understanding abstract concepts and relationships. In other words, a shift in emphasis from doing to understanding. What had previously been implicit, became full-on explicit. <br /><br />Mathematical objects were no longer thought of as given primarily by formulas, but rather as carriers of conceptual properties. Proving something was no longer a matter of transforming terms in accordance with rules, but a process of logical deduction from concepts. Mathematics was reconceptualized as “thinking in concepts” (<i>Denken in Begriffen</i>). <br /><br />This was, in every sense, a mathematical revolution, with the primary revolutionaries being<br />leading mathematicians such as Lejeune Dirichlet, Richard Dedekind, Augustin-Louis Cauchy, Karl Weierstrass, and Bernhard Riemann. <br /><br />To give just one instance of the shift, prior to the nineteenth century, mathematicians were used to the fact that a formula such as y = x<b><span style="font-size: x-small;"><span style="font-family: "arial";"><span style="color: black;"><sup style="line-height: 0.9;">2</sup></span></span></span></b> + 3x – 5 specifies a function that produces a new number y from any given number x. Then the revolutionary Dirichlet came along and said, forget the formula and concentrate on what the function does in terms of input-output behavior. A function, according to Dirichlet, is any rule that produces new numbers from old. The rule does not have to be specified by an algebraic formula. In fact, there's no reason to restrict your attention to numbers. A function can be any rule that takes objects of one kind and produces new objects from them.<br /><br />This definition legitimized functions such as the one defined on real numbers by the rule:<br />If x is rational, set f(x) = 0; if x is irrational, set f(x) = 1. <br /><br />Of course, you cannot draw a graph of such a monster. Instead, mathematicians began to study the properties of <i>abstract functions</i>, specified not by some formula but by their behavior. For example, you can investigate questions such as, is the function one-one, injective, surjective, continuous, differentiable, etc.?<br /><br />For any mathematician alive today, mathematics is a subject that studies formally-defined concepts, with a focus on the establishment of truth (based on accepted axioms), with various forms of calculation (numerical, algebraic, set-theoretic, logical, etc.) being tools developed and used in the pursuit of those goals. That’s the only kind of mathematics we have known.<br /><br />Except, that is, when we were at school. By and large, the 19th Century revolution in mathematics did not permeate the world’s school systems, which remained firmly in the “mathematics is about calculation” mindset. The one attempt to bring the school system into the modern age (in the US, the UK, and a few other countries), was the 1960s “New Math”. Though well-intentioned, its rollout was disastrous, in large part because very few teachers understood what it was about – and hence could not teach it well. The confusion caused to parents (other than mathematician parents) was nicely encapsulated by the satirical songwriter and singer Tom Lehrer (who taught mathematics at Harvard, and did understand New Math), in his hilarious, and pointedly accurate, song <a href="https://www.youtube.com/watch?v=UIKGV2cTgqA" target="_blank">New Math</a>. <br /><br />As a result of the initial chaos, the initiative was quickly dropped, and school math remained largely unchanged while real-world uses of mathematics kept steadily changing, leaving the schools increasingly separated from the way people did math in their jobs. Eventually, the separation blew up into a full-fledged divorce. That occurred in the late 1980s. The divorce was finalized on June 23, 1988. That was the date when Steve Wolfram released his mammoth software package <i>Mathematica</i>. Within a few short years of that release, if not on the release-date, <i>Mathematica</i> (and a similar package released a few months later in Canada, <i>Maple</i>) could answer pretty well any school or university math exam question with at least a grade B+, and very often an A.<br /><br />The days when calculation (of pretty well any kind, not just numerical) was the price humans had to pay to do mathematics were over. <br /><br />Given that thirty years have passed since that initial epochal moment, and most of the world has still not woken up to the fact that the entire mathematical world has changed dramatically and forever, let me repeat the core of that statement in caps.<br /><br />THE DAYS WHEN CALCULATION WAS THE PRICE HUMANS HAD TO PAY TO DO MATHEMATICS ARE OVER. <br /><br />To be sure, after that symbolic 1988 date, it took a few years for the change to percolate through the system, gain momentum, and eventually reach critical mass. Three further developments were also hugely significant: the birth of the World Wide Web in 1989 and the browser in 1993, and the launch of <i>Wolfram Alpha</i> in 2009. (Others might want to add other factors. I’m being selective here.)<br /><br />Talking about being selective, I’ve mentioned Wolfram products twice now. Though I was a member of Wolfram’s <i>Mathematica </i>Advisory Board in the first few years, I have no stake in or involvement with the company. While both <i>Mathematica</i> and <i>Alpha</i> were indeed major players in changing the way mathematics is done – particularly in applied settings – I am citing those particular products largely as icons, using two specific products to represent a range of new digital tools that were being developed around the world at that time. While Wolfram’s systems were ones I myself made early use of in my work, other mathematicians were also active in that digital mathematical revolution, using different systems. Still, <i>Mathematica</i> was the system that caught the public attention.<br /><br />Since the turn of the new Millennium, I doubt if anyone making professional use of mathematics in their job, or indeed any adult using mathematics in their everyday lives, has taken out paper-and-pencil and followed a classical algorithm to add, subtract, multiply or divide numbers in an array of real-life size, or perform complex algebraic reasoning to solve systems of equations, or solve problems using calculus, or any other established mathematical procedure. Not only would it now be a waste of valuable human time and energy doing something a cheap machine can do in far less time with no possibility of error, but many of the problems that people encounter in their careers and lives have simply too much data for the human mind to handle. Those same digital tools that have made the execution of mathematical procedures unnecessary have also come to dominate and drive our world, so many of the problems that require mathematics in their solution are now simply beyond human capacity. That’s why Amazon Web Services has become such a behemoth for data storage and processing.<br /><br />But that does not mean humans no longer need to have some mathematical skills. On the contrary, they are as crucial as ever – unless you are willing to be totally reliant on others, but personally, I have never felt comfortable doing that with things that are part of my life every day. What has changed are the specific mathematical skills required today. There are plenty of things computers cannot do or do poorly. Genuinely creative thinking and analogical reasoning are two obvious ones – though with today’s massive cloud computing resources we can use systems that provide an approximation often adequate for the purpose, and on occasion can be better than humans.<br /><br />Mostly, however, where you need humans is going from a real-world challenge situation to formulating one or mathematical tasks that can help you make progress. Sometimes, progress means solving a real-world problem in the sense of getting a specific answer (say, a number), but much more commonly it’s about finding a better method, where “better” can mean faster, cheaper, safer, or whatever other criterion is important, and where the change may involve developing a new method or improving an existing one.<br /><br />This way of using mathematics was the focus of that mini-course I gave at a California school (Nueva School) in January of this year, that I wrote about in the February, March, and April posts to this column.<br /><br />Though several mathematicians and mathematics education scholars expressed agreement with what I wrote, my articles brought some critiques from teachers and parents. The critiques all made reference to my asides about the Common Core State Standards in the first two of the posts. Since “Devlin’s Angle” no longer seems to be a target for the CCSS social media trolls (likely because the yield of issues to react to relative to the length and substance of most of my posts makes it less rewarding to them), I made some efforts to find out what exactly it is about the CCSSs that they objected to. As far as I could ascertain, the issue was inevitably (and predictably) to do with particular implementations of the Standards in specific curricula or (and this seems to be the most common occurrence) claims that a particular homework exercise was a “Common Core exercise”, which of course it cannot be since the CCSS are, as the name indicates, purely a set of standards to attain, not in any way a curriculum or curriculum content. <br /><br />More generally, in fact, pretty well all critiques of the CCSS are due to a complete misunderstanding of <b>what</b> they are, <b>why</b> they are, and what they <b>say</b>. The issue was nicely dealt with in this <a href="http://hechingerreport.org/content/common-core-math-problem-hard-supporters-common-core-respond-problematic-math-quiz-went-viral_15361/" target="_blank">2014 article</a> in the Hechinger Report.<br /><br />My reason for bringing the Common Core into my series of posts was to point out that the standards were developed precisely to help guide school districts, schools, and teachers in the tricky task of updating K-12 mathematics education to adequately prepare tomorrow’s citizens for life and work in a world where calculation is no longer a central pillar of mathematics.<br /><br />Having said that, I should point out that the above statement in no way implies that we should drop the teaching of basic arithmetic and algebra from the school system. As I discussed in some length in the third of my Nueva-inspired articles, the change that is required in K-12 math education is not so much in the mathematical topics but the <b>reason</b> they are now being taught and, in consequence, the <b>way</b> they should be taught. <br /><br />Teaching for execution is no longer the primary driver, since no one using mathematics in the real world does that anymore. What is now of cru<b></b>cial importance is teaching for understanding. Digital systems outperform humans to an insane degree when it comes to execution. But they don’t understand; people have to supply that.<br /><br />I leave you with an image I pulled from one of those Common Core social media rants some time ago. (I no longer remember the exact source.)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-QWpknbeizNI/WunBg6gBtOI/AAAAAAAALIQ/PEafeEbWxjkuN4fHBLeCXvmdzQIyt-WVgCLcBGAs/s1600/CommonCoreProblem.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1072" data-original-width="1600" height="267" src="https://3.bp.blogspot.com/-QWpknbeizNI/WunBg6gBtOI/AAAAAAAALIQ/PEafeEbWxjkuN4fHBLeCXvmdzQIyt-WVgCLcBGAs/s400/CommonCoreProblem.png" width="400" /></a></div><div style="text-align: center;"><span style="background-color: transparent; color: black; display: inline; float: none; font-family: "times new roman"; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;">Typical social media posts about Common Core mathematics.</span></div><div class="separator" style="clear: both; text-align: left;"><a href="https://web.stanford.edu/~kdevlin/" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;" target="_blank"></a></div><div style="text-align: left;"><a href="https://web.stanford.edu/~kdevlin/" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;" target="_blank"></a></div><div class="separator" style="clear: both; text-align: left;">I have three comments about the post on the left. First, the mathematics in the bottom left is not some fancy new algorithm, it is what a child wrote down in reasoning (sensibly) about a particular arithmetic problem. Second, if you are unable to follow what the child is doing, you would have trouble making effective use of mathematics in today’s world. It’s pretty basic. (Your kid just did it, right?) Third, if you are a parent and you don’t see why it is important that today’s school students acquire those math reasoning skills, please don’t communicate your skepticism to your children. Doing so would be a great disservice, to your child, to your child’s math teacher, and to society. The mathematical world has changed significantly. That occurred over twenty years ago. It is not going to change back. Sit back, relax, be encouraging, and let the kids take over. They do just fine with it.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">REFERENCE: During the period when the computer revolutionized how mathematics is done, I edited the American Mathematical Society’s “Computers and Mathematics” section of their monthly notices publication, sent to all members. I wrote about the column and that period in general in a paper that I submitted to the Proceedings of the Jon Borwein Commemorative Conference, held in 2017. Borwein, who died tragically young in 2016, was a leading pioneer in bringing digital technologies into mathematics. You can access a preprint of the paper <a href="https://web.stanford.edu/~kdevlin/Papers/BorweinCommemorativePaper.pdf" target="_blank">HERE</a>.</div><br /><br /><br />http://devlinsangle.blogspot.com/2018/05/calculation-was-price-we-used-to-have.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-2787071718529576557Wed, 04 Apr 2018 14:54:00 +00002018-04-04T11:21:06.361-04:00How today’s pros solve math problems: Part 3 (The Nueva School course)<span style="background-color: white; color: #222222; display: inline; float: none; font-family: "georgia" , "utopia" , "palatino linotype" , "palatino" , serif; font-size: 13.2px; font-style: normal; font-weight: 400; letter-spacing: normal; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;">By Keith Devlin</span><br /><div class="MsoNormal" style="-webkit-text-stroke-width: 0px; background-color: white; color: #222222; font-family: Georgia, Utopia, "Palatino Linotype", Palatino, serif; font-size: 13.2px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; line-height: normal; margin-bottom: 0in; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><br /></div><div class="MsoNormal" style="-webkit-text-stroke-width: 0px; background-color: white; color: #222222; font-family: Georgia, Utopia, "Palatino Linotype", Palatino, serif; font-size: 13.2px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; line-height: normal; margin-bottom: 0in; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span style="color: black;">You can follow me on Twitter </span><span style="color: black;"><span id="goog_646323682"></span><a href="https://www.blogger.com/goog_646323681"><span style="color: blue;">@profkeithdevlin</span><span id="goog_646323683"></span></a></span><br /><span style="color: #b00000;"><span style="color: blue;"></span><br /></span></div><div class="MsoNormal" style="-webkit-text-stroke-width: 0px; background-color: white; color: #222222; font-family: Georgia, Utopia, "Palatino Linotype", Palatino, serif; font-size: 13.2px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; line-height: normal; margin-bottom: 0in; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span style="color: black;"><a href="https://www.blogger.com/"></a><br /></span></div><i>NOTE: This article is the final installment of a four-episode mini-series posted here starting in mid-January. In writing it, I have assumed my readers have read those three earlier pieces.</i><br /><br />At the end of last month’s post, I left readers with a (seemingly) simple arithmetic problem. I prefaced the problem with the following two instructions:<br /><br />1. Solve it as quickly as you can, in your head if possible. Let your mind jump to the answer.<br /><br />2. Then, and only then, reflect on your answer, and how you got it.<br /><br />The goal here, I said, is not to get the right answer, though a great many of you will. Rather, the issue is how do our minds work, and how can we make our thinking more effective in a world where machines execute all the mathematical procedures for us?<br /><br />Here is the problem.<br /><br />PROBLEM: A bat and a ball cost $1.10. The bat costs $1 more than the ball. How much does the ball cost on its own? (There is no special pricing deal.)<br /><br />What answer did you get? And what did you learn from the subsequent reflection?<br /><br />Before I continue, I should note that the use of this problem (which you can find in many puzzle books and on countless websites) in the context of trying to maximize the human mind’s innate abilities in order to become good 21st Century mathematical thinkers, is due to Gary Antonick, with whom I co-taught a Stanford Continuing Studies adult education course last fall. It was in that course that I gave the second iteration of the UPS problem I subsequently based my Nueva School course on. The discussion of the bat-and-ball problem that follows is the one Antonick presented in our course.<br /><br />Now to the problem itself. The most common answer people give instantly to this problem is that the ball costs 10¢. It’s wrong (and many realize that is the case soon after their mind has jumped to that wrong number). What leads many astray is that the problem is carefully worded to run afoul of what under normal circumstances is an excellent strategy. (So if you got it wrong, you probably did so because you are a good thinker with some well-developed problem-solving strategies— problem-solving heuristics is the official term, and I’ll get to those momentarily. So take heart. You are well placed to do just fine in 21st Century mathematical thinking. You simply need to develop your heuristics to the next level.)<br /><br />Here is, most likely, what your mind did to get to that 10¢ answer. As you read through the problem statement and came to that key phrase “cost more,” your mind said, “I will need to subtract.” You then took note of the data: those two figures $1.10 and $1. So, without hesitation, you subtracted $1 from $1.10 (the smaller from the larger, since you knew the answer has to be positive), getting 10¢.<br />Notice you did not really perform any calculation. The numbers are particularly simple ones. Almost certainly, you retrieved from memory the fact that if you take a dollar from a dollar-ten, you are left with 10¢. You might even have visualized those amounts of money in your hand.<br /><br />Notice too that you understood the mathematical concepts involved. Indeed, that was why the wording of the problem led you astray!<br /><br />What you did is apply a heuristic you have acquired over many financial transactions and most likely a substantial number of arithmetic quiz questions in elementary school. In fact, the timed tests in schools actively encourage such a “pattern recognition” approach. For the simple reason that it is fast and usually works!<br /><br />We can, therefore, formulate a hypothesis as to why you “solved’ the problem the way you did. You had developed a heuristic (identify the arithmetic operation involved and then plug in the data) that is (a) fast, (b) requires no effort, and (c) usually works. This approach is a smart one in that it uses something the human brain is remarkably good at—pattern recognition—and avoids something our minds find difficult and requiring effort to master (namely, arithmetic calculation).<br /><br />Of course, primed by the context in which I presented this particular problem, you probably expected there to be a catch. So, after letting your mind jump to the 10¢ answer, you likely took a second stab at it (or, if you were anxious about “getting a wrong answer,” made this your first solution) by applying an algorithm you had learned at school. Namely, you reasoned as follows:<br /><br />Let x = cost of bat and y = cost of the ball. Then, we can translate the problem into symbolic<br />form as x + y = 1.10 , x = y + 1<br /><br />Eliminate x from the two equations by algebra, to give<br />1.10 – y = y + 1<br /><br />Transform this by algebra to give<br />0.10 = 2y<br /><br />Thus, dividing both sides by 2, you conclude that<br />y = 5¢.<br /><br />And this time, you get the correct answer.<br /><br />You may, in fact, have been able to carry out this procedure in your head. When I was at school, I could do algebraic manipulations far more complicated than this in my head, at speed. But, truth be told, since I started outsourcing arithmetic to machines many decades ago, I have lost that skill, and now have to write down the equations and solve them on paper. (This is a confirmation, if any were needed, that arithmetic calculations do not come naturally to the human brain. Over the years, as my mental arithmetic skills have declined, my pattern recognition abilities have not diminished, but on the contrary have dramatically improved, as I learned—automatically, through exposure—to recognize ever more fine-grained distinctions.)<br /><br />Whether or not you can do the calculation in your head, it is of course entirely formulaic and routine. Unlike the first method I looked at (a <b><i>heuristic</i></b> that is <i><b>fast</b></i> and <i><b>usually right</b></i>), this method is an <b><i>algorithmic procedure</i></b>, it is <b><i>slow</i></b> (much slower than the first method, even when the algebraic reasoning is carried out in your head), but it <i><b>always works</b></i>. It is also an approach that can be executed by a machine. True, for such a simple example, it’s quicker to do it by hand on the back of an envelope, but as a general rule, it makes no sense to waste the time of a human brain following an algorithmic procedure, not least because, even with simple examples it is familiarly easy to make a small error that leads to an incorrect answer.<br /><br />But there is another way to solve the problem. It’s the way I addressed it, and, according to Antonick, who has given it to many professional mathematicians and asked them to vocalize their solutions, the way many math pros solve it. Like the first method we looked at, it is a heuristic, hence instinctive and fast, but unlike the first heuristic method, it <b>always </b>works.<br /><br />This third method requires looking beyond the words, and beyond the symbols in the case of a problem presented symbolically, to the <b><i>quantities</i></b> represented. Though I (and likely other mathematicians) don’t visualize it quite this way (in my case it is more of a vague sense-of-size), the following image captures what we do.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://www.stanford.edu/~kdevlin/" target="_blank"><img alt="http://www.stanford.edu/~kdevlin/" border="0" data-original-height="580" data-original-width="796" height="145" src="https://4.bp.blogspot.com/-q4QAYbGkFhk/Wr6R5f7AptI/AAAAAAAALFQ/sGXesiCiRwguSp7dNV6RGNJTOWVu2UiLwCLcBGAs/s200/Bat%2526ball_solution.jpg" width="200" /></a></div><br />As we read the problem, we form a mental sense of the two quantities, the cost of the ball-on-its-own and the cost of the bat-plus-ball, together with the stated relation between them, namely that the latter is $1 more than the former. From that mental image, where we see the $1.10 total consists of three pieces, one of which has size $1 and the other two of which are equal, we simply “read off” the fact that the ball costs 5¢. No calculation, no algorithm. Pure pattern recognition.<br /><br />This solution is an example of Number Sense, the critical 21st Century arithmetic skill I wrote about in the January 1, 2017 Huffington Post <a href="https://www.huffingtonpost.com/entry/number-sense-the-most-important-mathematical-concept_us_58695887e4b068764965c2e0" target="_blank"><span style="color: blue;">companion piece</span></a> to the article I published on the same day as my article about all my math skills becoming obsolete, which I referred to in my last post here on Devlin’s Angle.<br /><br />It is, I suggest, hard to imagine how a computer system could solve the problem that way. (Of course, you could write a program so it can perform that particular pattern recognition, but the essence of number sense is that you can apply it to many numerical problems you come across.)<br /><br />Those three ways to solve the bat-and-ball problem I just outlined are examples of what the famous Australian (pure) mathematician Terrence Tao<a href="https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/" target="_blank"><span style="color: blue;"> has called (in his blog)</span></a>, respectively, <b><i>pre-rigorous</i></b> thinking, <i><b>rigorous</b></i> thinking, and <i><b>post-rigorous</b></i> thinking. You can also listen to him explain these three categories in a <a href="https://www.youtube.com/watch?v=48Hr3CT5Tpk" target="_blank"><span style="color: blue;">short video</span></a> in the <i>Numberphile</i> series.<br /><br />Post-rigorous heuristic thinking is how today’s math pros use mathematics to solve real-world problems. In fact, as Tao makes clear, post-rigorous thinking is what the pros use most of the time to solve abstract problems in pure math. The formal, symbolic, rigorous stuff comes primarily at the end, to check that the solution is logically correct, or at various intermediate points to make those checks along the way.<br /><br />In the case of solving real-world problems, the pros almost always turn to technology to handle any algebraic deductions. In contrast, though pure mathematicians sometimes do use those technology products as well, they often find it much quicker, and perhaps more fruitful in terms of gaining key insights, to do the algebraic work by hand.<br /><br />So, one of the big question facing math teachers today is, how do we best teach students to be good post-rigorous mathematical thinkers?<br /><br />In the days when the only way to acquire the ability to use mathematics to solve real-world problems involved mastering a wide range of algorithmic procedures, becoming a mathematical problem solver frequently resulted in becoming a post-rigorous thinker automatically.<br /><br />But with the range of tools available to us today, there is a good reason to assume that, with the right kinds of educational experiences, we can significantly shorten (though almost certainly not eliminate) the learning path from pre-rigorous, through rigorous thinking, to post-rigorous mathematical thinking. The goal is for learners to acquire enough effective heuristics.<br /><br />To a considerable extent, those heuristics are not about “doing math” as such. Rather, they are focused on making efficient and effective use of the many sources of information available to us today. But before you throw away your university-level textbooks, you need to be aware that the intermediate step of mastering some degree of rigorous thinking is likely to be essential. Post-rigorous thinking is almost certainly something that <b><i>emerges</i></b> from <b><i>repeated</i></b> practice at rigorous thinking. Any increased efficiency in the education process will undoubtedly come from teaching the formal methods in a manner<i><b> optimized for understanding</b></i>, as opposed to optimized for attaining procedural efficiency, as it was in the days when we had to do everything by hand. See Daniel Willingham’s excellent book <a href="https://www.amazon.com/Why-Dont-Students-Like-School/dp/047059196X" target="_blank"><span style="color: blue;">Why Don’t Students Like School?</span></a> for a good, classroom-oriented look at what it takes to achieve mastery in a discipline.<br /><br />Now to that UPS routing problem that was the focus of my Nueva School course. [You will find it discussed <a href="http://devlinsangle.blogspot.com/2018/02/" target="_blank"><span style="color: blue;">here</span></a>.] Here are some of the hints and suggestions about solving the problem I made to the students in the three courses where I used it. Whether they followed my advice was entirely up to them. The purpose of the course was not to solve the problem unaided—even an entire semester would not be enough time for that with students who had never approached a problem the way the pros do. Rather, it was to give them an experience of the method.<br /><br />First, they had to work in teams of three to five. I let them select the teams, but said it would be good if at least one person on each team felt they were “good at math.”<br /><br />Then, start out by using Google to find out what you can about the problem domain, and any attempts made by others to solve it.<br /><br />Whenever you come across a reference to a concept, an approach, or a method that you suspect might be relevant, use general Web resources like Wikipedia to get an initial understanding of what they are and what they can do.<br /><br />Follow any leads your search brings up to solutions of problems that look similar. Note what methods were used to solve them.<br /><br />If you come across references to others who have worked on the problem, or a similar one, send them a brief email. You may not get a reply, but occasionally you will, and it could be invaluable. (I receive such emails all the time. Mostly I do not have time to respond, but occasionally one lands in my inbox when I have a spare moment, and I happen to know something that might help, so I shoot back a brief reply, often just a reference to a particular source.)<br /><br />When you get to a point where you need to perform a specific calculation, perhaps because you have found a solution to a very similar problem someone else has obtained and published, but your data is different, use Wolfram Alpha. It is structured so you can use pattern recognition (of formulas) to identify the appropriate technique and then edit the example provided to be the one you want to solve.<br /><br />Reinforce your use of Wolfram Alpha by using YouTube to find suitable videos that provide you with quick tutorials on the technique.<br /><br />The resources I just mentioned are all listed on that chart of “Important Mathematical Technology Tools” I published with the first two articles in this series.<br /><br />As it turns out, with the UPS routing problem, the sequence of steps I have outlined so far quickly leads to identification of a small number of possible solution techniques for which there are many very accessible YouTube videos, and in fact, for this problem there is no need to go much further into my list of tools, if at all.<br /><br />You should, though, check out the various other resources on my list, to see what they offer. Each new problem has to be approached afresh, in its own terms. Twitter is on my list because it is my list, and I have sufficiently many math-expert Twitter followers that a quick tweet can often yield just the information I need, saving me having to send out a slew of emails to people I think might be able to help. LinkedIn is also idiosyncratic to me, since I have a good network of mathematics and technology professionals I can contact. But the other resources are pretty generic.<br /><br />Ideally, everything goes much more smoothly if you can avail yourself the services of a math consultant to assist you in negotiating the various resources. (I was that consultant to the teams in the three courses I gave.)<br /><br />Interestingly, in the final meeting of my Princeton class (which was the fist time I used the UPS problem in a course), after having the student teams present their solutions, I gave the solution I had obtained, at the end of which two individuals came up to me to say they hoped I had not minded their sitting in on the class. (It was an experimental course, and there had been strangers sitting in for one or two sessions throughout the semester, so I had not paid them any attention.) They were, they said, postdocs working with Professor X, who was a math consultant for UPS and had worked on the algorithm the class and I had been trying to reverse engineer. Hence their curiosity-driven attendance on the last day! Unbeknownst to me, my final lecture had been my oral exam!<br /><br />“How did I do?”, I asked. “You got it pretty well right,” they replied.<br /><br />Which was nice for me, but it would not have mattered if I had followed a different track. What was important from an educational standpoint was the process.<br /><br />Something else I suggested to the class was to come up with a solution—<b><i>any</i></b> solution—as soon as you can. “Don’t worry if it is optimal or even right,” I said. “Just check it by computation, perhaps in the form of a spreadsheet simulation. Once you have some solution that you can check (in the case of my UPS problem, check against the shipping data I supplied, or any other UPS data you can find on the Web), you can iterate to find a better one. It might turn out that your first solution, or your first three or four, won’t even get you to first base, but in the process of formulating and checking those initial attempts, you will inevitably gain insight into the problem you are trying to solve. Remember, computation is cheap, fast, and essentially limitless.”<br /><br />If you are not familiar with this way of solving math problems, it may not seem like an approach that will work. But it does. It is, in fact, how all of today’s pros do it!<br /><br />If you have not already done so, now is a good time to check out the dictionary definition of the word heuristic! Here is Wikipedia’s (at the time of writing):<br /><span style="background-color: yellow;"><br /></span> “A heuristic technique (from the ancient Greek for “find” or “discover”), often called simply a heuristic, is an approach to any problem-solving, learning, or discovery that employs a practical method not guaranteed to be optimal or perfect, but sufficient for the immediate goals. Where finding an optimal solution is impossible or impractical, heuristic methods can be used to speed up the process of finding a satisfactory solution. Heuristics can be mental shortcuts that ease the cognitive load of making a decision. Examples of this method include using a rule of thumb, an educated guess, an intuitive judgment, guesstimate, stereotyping, profiling, or common sense.”<br /><br />Without an expert consultant, the heuristics approach to solving real-world problems can work, but it definitely goes a lot faster, and with a far great likelihood of success, if you have a math expert you can call on. Not to “do any math.” Computer systems handle those parts. Rather to help you negotiate the vast array of resources at your disposal and select the most promising one(s) to try next. For that is what using mathematics to solve a real-world problem really boils down to these days: managing resources.<br /><br />And managing resources is something humans are innately good at. Natural selection always favors those creatures which are best able to manage the available resources. We are here as present-day humans because as a species we are good at doing that. What is new in the case of mathematical problem solving is that pieces of mathematics (formulas, equations, procedures, algorithms, techniques) are now among the “intellectual Lego pieces” (freely accessible on the Web) we can use as we assemble a solution.<br /><br />As the students in my three courses could, in principle, attest, you don’t need vast expertise in mathematics to work this way. You just need to be a good thinker able to work in a small team. I say, “in principle,” since I think it highly likely most of not all the students felt they did not do much at all by way of using math to solve a problem. But that, I would say, is because they have a conception of “using math to solve a problem” rooted in the Nineteenth Century, if not the Fourth Century BCE. From my perspective, they absolutely were able to do what I just said they did.<br /><br />Of course, they were not as good at it as I am. I’ve been at this game a lot longer, and, make no mistake about it, experience counts. (I think it is close to being the only thing that counts.)<br /><br />What does not count, at least to any extent even remotely approaching the prohibitive degree it used to, is the ability to “do the math.” You just need to be able to select (hopefully, with help from someone with experience) the right pieces from the available online mathematical smorgasbord, and stitch them together in an appropriate way.<br /><br />This kind of problem-solving doesn’t feel like math (as we all came to love or hate), that’s for sure. In fact, it doesn’t even feel like work. Once they got into the swing of it, even the students who declared they were not good at math or did not enjoy it, found they were having a good time, working in teams in a creative, explorative way. For the fact is, properly approached, humans enjoy problem-solving. (That’s another consequence of natural selection— problem-solving, particularly group problem solving, is one of our species’ key survival advantages.)<br /><br />In fact, another way to look at the recent revolution in how we “use math” to solve real-world problems, is that it has brought “using math” into the mainstream of human group activities we naturally find enjoyable. At heart, mathematical thinking is little more than formalized common sense. It always has been. Which means it is something we can all do. (In my 2000 book<span style="color: blue;"> </span><span style="color: blue;"><a href="https://www.amazon.com/Math-Gene-Mathematical-Thinking-Evolved/dp/0465016197" target="_blank">The Math Gene</a></span>, I presented an evolutionary explanation for the human brain’s acquisition of the ability to do mathematics, which implied that mathematical capacity is in the h<span style="background-color: transparent; color: blue; font-family: "times new roman"; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;"><a href="https://www.amazon.com/Math-Gene-Mathematical-Thinking-Evolved/dp/0465016197" target="_blank"></a></span>uman gene pool, and hence available for all of us to “switch on.”) What caused many people problems over the centuries was that, before we had technologies that could handle the formal symbol-manipulation stuff, the only way to employ our innate capacity for mathematical thinking was to train the brain to do those manipulations. But manipulating algebraic symbols with logical precision is most definitely not something our brains evolved to do. (Our early ancestors’ lives on the savannas did not present much by way of a need for algebra.) So we find it very hard. Only with great effort over several years can we train our brains to do such work. And even then, we are error-prone.<br /><br />Incidentally, practically everything I have said in this article applies to the way 21st Century coders work. In coding as in mathematics, the days are long gone when it was all about writing thousands of lines of instructions. The modern-day mathematician’s Web resource MathOverflow (on my chart of useful math tools) was modeled on, and named after, the coding world’s StackOverflow. Both groups of professionals use heuristics. In today’s world, highly regarded math problem solvers and good coders have simply acquired a richer and more effective set of heuristics than the ones who are less highly ranked. And for the most part, developing heuristics is a result of reflective experience, not some innate talent.<br /><br />And there you have it. The primary goal in 21st Century mathematics-education-for-all is the development of a good repertoire of heuristics.<br /><br />I’ll leave you with a graphical summary of Tao’s categorization of the three kinds of mathematical thinking we can bring to problem-solving. I introduced this categorization above to provide a perspective on the three phases each one of us has to go through to become proficient mathematical (real-world) problem solvers. But it also provides an excellent summary of three historical stages of mathematical thinking as it has evolved over the past ten thousand years or so, from the invention of numbers in Sumeria, where the mathematical thinking of the time was accessible to all, through three millennia of formal mathematics development, where many people were never able to make effective use of it, and now into the third phase, where, because of technology, mathematical thinking can once again be accessible to all.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://www.stanford.edu/~kdevlin/" target="_blank"><img alt="http://www.stanford.edu/~kdevlin/" border="0" data-original-height="900" data-original-width="1600" height="225" src="https://3.bp.blogspot.com/-ykMJbheU-G8/Wr6SPNlXLhI/AAAAAAAALFU/9k1kvdQkyh0UuCUnm1x1pKo0b73y1cN1wCLcBGAs/s400/ThreeKinds.jpg" width="400" /></a></div><br />To be sure, we do not know the degree to which people have to master rigorous thinking to become good post-rigorous thinkers. As I already noted, I don’t for a second imagine that stage can be by-passed. (See the Willingham book I cited.) But, given today’s technological toolkit, including search, social media, online resources like Wolfram Alpha and Khan Academy, and a wide array of online courses, it is absolutely possible to master most of the rigorous thinking you need “on the job,” in the course of working on meaningful, and hence motivational and rewarding, real-world problems.<br /><br />This is not to say there is no further need for teachers. Far from it. Very few people are able to become good mathematical thinkers on their own. Newtons and Ramanujans, who achieved great things with just a few books, are extremely rare. The vast majority of us need the guidance and feedback of a good teacher.<br /><br />What the inevitable transition to 21st Century math learning requires is that mathematics teachers operate very differently than in the past. The days where you need a live person to deliver information are largely over. Today, teaching is much more a matter of being a coach and mentor. To be sure, you can occasionally find such teaching on the Internet, but it works only if you can be one-on-one with that teacher. I expect there will be change, but I don’t expect an economy of scale. If I had to make a guess, I would predict that in due course you will find your (specialist) math teacher by going online to a Math-Teacher-Match.com website, where you will be paired with a practicing 21st Century math professional who spends part of each day coaching and mentoring students.<br /><br /><br />LABELS: mathematical thinking, problem-solving, rigorous thinking, pre-rigorous thinking, post-rigorous thinking, Terrence Tao, social media in mathematicshttp://devlinsangle.blogspot.com/2018/04/how-todays-pros-solve-math-problems.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-5550417883470394165Fri, 09 Mar 2018 17:19:00 +00002018-03-12T08:48:13.966-04:00How Today’s Pros Solve Math Problems: Part 2By Keith Devlin<br /><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">You can follow me on Twitter @profkeithdevlin</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span><br /><span style="color: black; font-family: "" "calibri" "" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">CHANGE OF PLAN:<i style="color: black;"> When I wrote last month’s post, I said I would conclude the description of my <a href="http://devlinsangle.blogspot.com/2018/01/deja-vu-all-over-again.html" target="_blank">Nueva School Course</a></i></span><i><span style="color: black; font-family: "" "calibri" "" , serif;"> this time. But when I sat down to write up that concluding piece, I realized it would require not one but two further posts. The course itself was the third iteration of an experiment I had tried out on a university class of non-science majors and an Adult Education class. This series of articles is my first attempt to try to describe it and articulate the thinking behind it. As is often the case, when you try to describe something new (at least it was new to me), you realize how much background experience and unrecognized tacit knowledge you have drawn upon. In this post, I’ll try to capture those contextual issues. Next month I’ll get back to the course itself. </span></i><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="separator" style="clear: both; text-align: center;"></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">We all know that mathematics is not always easy. It requires practice, discipline and patience, as do many other things in life. And if learning math is not easy, it follows that teaching math is not easy either. But it can help both learner and teacher if they know what the end result is supposed to be. </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">In my experience, many learners and teachers don’t know that. In both cases, the reason they don’t know it is that no one has bothered to tell them. There is a general but unstated assumption that everyone knows why the teaching and learning of mathematics is obligatory in every education system in the world. But do they really?</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">There are two (very different) reasons for teaching and learning mathematics. </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">One reason is that it is a way of thinking that our species has developed over several thousand years, that provides wonderful exercise for the mind, and yields both challenging intellectual pleasure and rewarding aesthetic beauty to many who can find their way sufficiently far into it. In that respect, it is like music, drama, painting, philosophy, natural sciences, and many other intellectual human activities. This is a perfectly valid reason to provide everyone with an opportunity to sample it, and make it possible for those who like what they see to pursue it as far as they desire. What it is not, is a valid reason for making learning math obligatory throughout elementary, middle, and high school education.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">The argument behind math’s obligatory status in education is that it is useful; more precisely, it is useful in the practical, everyday world. This is the view of mathematics I am adopting in the short series of “Devlin’s Angle” essays of which this is the third. (There will be one more next month. See episode 1 <a href="http://devlinsangle.blogspot.com/2018/01/deja-vu-all-over-again.html" target="_blank">here</a> </span><span style="color: black; font-family: "" "calibri" "" , serif;">and episode 2 <a href="http://devlinsangle.blogspot.com/2018/02/how-todays-pros-solve-math-problems.html" target="_blank">here</a>.</span><span style="color: black; font-family: "" "calibri" "" , serif;">) </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">Indeed, mathematics <b>is </b>useful in the everyday practical world. In fact, we live in an age where mathematics is more relevant to our lives than at any previous time in human history. </span><br /><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">It is, then, perfectly valid to say that we force each generation of school students to learn math because it is a useful skill in today’s world. True, there are plenty of people who do just fine without having that skill, but they can do so only because there are enough other people around who do have it.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">But let’s take that argument a step further. How do you teach mathematics so that it prepares young people to use it in the world? Clearly, you start by looking at the way people currently use math in the world, and figure out how best to get the next generation to that point. (Accepting that by the time those students finish school, the world’s demands may have moved on a bit, so those new graduates may have a bit of catch up and adjustment to make.)</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">If the way the professionals use math in the world changes, then the way we teach it should change as well. Don’t you think?</span><span style="color: black; font-family: "times new roman" , serif;"> </span>That’s certainly what has happened in the past.</div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">For instance, in the ninth century, the Arabic-Persian speaking traders around Baghdad developed a new, and in many instances more efficient, way to do arithmetic calculations at scale, by using logical reasoning rather than arithmetic. Their new system, which quickly became known as <i>al-jabr</i>after one of the techniques they developed to solve equations, soon found its way into their math teaching. </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">When Hindu-Arabic arithmetic was introduced into Europe in the thirteenth century, the school systems fairly quickly adopted it into their arithmetic teaching as well. (It took a few decades, but knowledge moved no faster than the pace of a packhorse back then. I tell the story of that particular mathematics-led revolution in my 2011 book <a href="https://www.amazon.com/Man-Numbers-Fibonaccis-Arithmetic-Revolution/dp/0802778127/ref=sr_1_1?ie=UTF8&qid=1306990867&sr=8-1" target="_blank">The Man of Numbers</a></span><span style="color: black; font-family: "" "calibri" "" , serif;">.)</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">The development of modern methods of accounting and the introduction of financial systems such as banks and insurance companies, which started in Italy around the same time, also led to new techniques being incorporated into the mathematical education of the next generation.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">Later, when the sixteenth century French mathematician <span style="background: white;">François Viète introduced symbolic algebra, it too became part of the educational canon. </span></span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="background: white; color: black; font-family: "" "calibri" "" , serif;">In each case, those advances in mathematics were introduced to make mathematics more easy to use and to increase its application. There was never any question of “What is this good for?” People eagerly grabbed hold of each new development and made everyday use of it as soon as it became available.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="background: white; color: black; font-family: "" "calibri" "" , serif;">The rise of modern science (starting with Galileo in the seventeenth century) and later the Industrial Revolution in the nineteenth century, led to still more impetus to develop new mathematical concepts and techniques, though some of those developments were geared more toward particular groups of professionals. (Calculus, for example.)</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">To make it possible for an average student or worker to make use of each new mathematical concept or technique, sets of formal calculating rules (<i>algorithmic procedures</i>) were developed and refined. Once mastered, these made it possible to make use of the new mathematics to handle—in a practical way—the tasks and problems of the everyday world for which those concepts and techniques had been developed to deal with in the first place. </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">As a result of all those advances, by the time the Baby Boomers came onto the educational scene in the 1950s, the curriculum of mathematical algorithms that were genuinely important in everyday life was fairly large. It was no longer possible for a student to understand all the underlying mathematical concepts and techniques behind the algorithms and procedures they had to learn. The best that they could do was master, by repetitive practice, the algorithmic procedures as quickly as possible and move on. [A few of us had difficulty doing that. We wanted to understand what was going on. By and large, we frustrated our teachers, who seemed to think we were simply troublesome slow learners. Some of us eventually learned to “play the mindless algorithm game” in class to pass the test, but kept struggling on our own to understand what was going on, setting us on a path to becoming mathematics professors in the 1970s.] </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">It was while that Boomer generation was going through the school system that mathematics underwent the first step of a seismic shift that within a half of a century would completely revolutionize the way mathematics was done. Not the pure mathematics practiced by a few specialists as an art—though that too would be impacted by the revolution to some extent. Rather, it was mathematics-as-used-in-the-world that would be radically transformed.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">The first step of that revolution was the introduction of the electronic desktop calculator in 1961. Although, mechanical desktop calculators had been available since the turn of the Twentieth Century, by and large their use was restricted to specialists—often called “computers” in businesses. [I actually had a summer-job with British Petroleum as such a specialist in my last three years at high school, and it was in my final year in that job that the office I worked in acquired its first electronic desktop calculator and the British Petroleum plant bought its first digital computer, both of which I learned to use.] But with the increasing availability of electronic calculators, and in particular the introduction of pocket-sized versions in the early 1970s, their use in the workplace rapidly became ubiquitous. Mathematics underwent a major change. Humans no longer needed to do arithmetic calculations themselves, and professionals using arithmetic in their work no longer did.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">It was not too many years later that, one by one, electronic systems were developed that could execute more and more mathematical procedures and techniques, until, by the late 1980s, there were systems that could handle all the mathematical procedures that constituted the bulk of not only the school mathematics curriculum, but the entire undergraduate math curriculum as well. The final nail in the coffin of humans needing to execute mathematical procedures was the release of the mathematics system <i>Mathematica</i> in 1988, followed soon after by the release of <i>Maple</i>.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">In the scientific, industrial, engineering, and commercial worlds, each new tool was adopted as soon as it became available, and since the early 1990s, professionals using mathematical techniques to carry out real-world tasks and solve real-world problems have done so using tools like <i>Mathematica</i>, <i>Maple</i>, and a host of others that have been developed. </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">Simultaneously, colleges and universities quickly incorporated the use of those new tools into their teaching. And while the cost of the more extensive tools put their use beyond most schools, the graphing calculator too was quickly brought into the upper grades of the K-12 system, after its introduction in 1990.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">Yet, while the pros in the various workplaces changed over to the new <i>human-machine-symbiotic</i>way of doing math with little hesitation, most educators, exhibiting very wise instincts, proceeded with far more caution. The first wave of humans to adopt the new, machine-aided approach had all learned mathematics in an age when you had to do everything yourself. Back then, “computers” were people. For them, it was easy and safe to switch to executing a few keystrokes to make a computer run a procedure they had carried out by hand many times themselves. But how does a young person growing up in this new, digital-tools-world learn how to use those new tools safely and effectively?</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">To some extent, the answer is (and was) obvious. You teach not for smooth, proficient, accurate execution of procedures, but for broad, general understanding of the underlying mathematics. The downplay of execution and increased emphasis on understanding are crucial. Computers outperform us to ridiculous degrees (of speed, accuracy, size of dataset, and information storage and retrieval) when it comes to execution of an algorithm. But they do not understand mathematics. They do not understand the problem you are working on. They do not understand the world. They don't understand anything. </span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">People, on the other hand, can understand, and have a genetically inherited desire to do so.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">But just how <b><i>do</i></b> you go about teaching for the kind of understanding and mastery that is required for students to transition into worlds and workplaces dominated by a wide array of new mathematical tools, where they will encounter work practices that involve very little by way of hand execution of algorithms? </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">We know so little about how people learn (though we do know a whole lot more than we did just a few decades ago), that most of us with a stake in the education business are rightly concerned about making any change that would effectively be a massive experiment on an entire generation. So we can, and should, expect small steps, particularly in systemic education.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">In the U.S., the mathematicians who developed the mathematical guidelines for the Common Core State Standards made a good first attempt at such a small step. True, it quickly ran into difficulties when it came to implementing the guidelines in a large and complex public educational system that is answerable to the public. But that is surely a temporary hiccup. Most of the problems at launch came from a lack of effective ways to assess the new kind of learning. Those problems can be and are being fixed. Which is just as well. For, although it’s possible to argue for tinkering with specific details of the Common Core State Standards guidelines, in terms of setting out a broad set of educational goals to aim for, there is no viable alternative first step. The pre-1970s educational approach is no longer an option.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">In the meantime, individual teachers at some schools (particularly, but not exclusively, private schools) have been trying different approaches, in some cases sharing their experiences on the <a href="https://mtbos.org/" target="_blank">MTBOS</a></span><span style="color: black; font-family: "" "calibri" "" , serif;"> (Math Twitter Blog-O-Sphere), making use of another technological tool (social media) now widely available. [For a quick overview of one global initiative to support and promote such innovations, the OECD’s <i>Innovative Pedagogies for Powerful Learning</i> project (IPPL), <a href="https://www.brookings.edu/blog/education-plus-development/2018/02/08/innovation-in-everyday-teaching-no-more-waiting-for-superman/" target="_blank">see this recent article</a> </span><span style="color: black; font-family: "" "calibri" "" , serif;">from the Brookings Institution.] </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">The mini-course I gave at Nueva School in the San Francisco Bay Area last January, which I talked about in the first of this short series of essays, is one such experiment in teaching mathematics in a way that best prepares the next generation for the world they will live and work in after graduation. I tested it first with a class of non-science majors in Princeton in the fall of 2015 and then again with an Adult Education class at Stanford in the fall of 2017. The Nueva School class was its third outing.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">With the above backstory now established, next month I will describe that course</span><span style="color: black; font-family: "times new roman" , serif;"> </span>and talk about how today’s pros “do the math”. (Again, let me stress, I am not talking here about “pure math”, the academic discipline carried out by professional mathematicians in universities and a few think tanks. My focus here is on using math in the everyday world.)</div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">In the meantime, I’ll leave you with a simple arithmetic problem that I will discuss in detail next time. </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">It comes with two instructions:</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><ol start="1" style="margin-top: 0in;" type="1"><li class="MsoNormal" style="line-height: normal; margin-bottom: 0in; vertical-align: baseline;"><span style="font-family: "" "calibri" "" , serif;">Solve it as quickly as you can, in your head if possible. Let your mind <b><i>jump</i></b> to the answer.</span></li><li>Then, <b><i>and only then</i></b>, reflect on your answer, and <b><i>how you got it</i></b>.</li></ol><ol start="1" style="margin-top: 0in;" type="1"></ol><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">The goal here is not to get the right answer, though a great many of you will. Rather, the issue is how do our minds work, and how can we make our thinking more effective in a world where machines execute all the mathematical procedures for us?</span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;"><br /></span><span style="color: black; font-family: "" "calibri" "" , serif;">Ready for the problem? Here it is. </span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><br /></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "" "calibri" "" , serif;">PROBLEM: A bat and a ball cost $1.10. The bat costs $1 more than the ball. How much does the ball cost on its own? (There is no special pricing deal.)</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"></div><br />http://devlinsangle.blogspot.com/2018/03/how-todays-pros-solve-math-problems_9.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-32802275784570121Wed, 07 Feb 2018 13:00:00 +00002018-02-08T11:57:32.295-05:00How today’s pros solve math problems: Part 1Last month, I wrote about my recent experience teaching a three-day mini-course in the <a href="http://www.nuevaschool.org/" target="_blank">Nueva School</a> January electives “Intersession” program. What I left out was a description of the course itself. I ended with the below diagram as a teaser. I said that, when reading in the usual left-right-down reading order, these were the technology tools that I typically turn to when I start to work on solving a new problem.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-lWbPzXlaJkw/WnSegR9V4LI/AAAAAAAALCI/3dzh7U6HtbQuDdhBDfGhqYHrAXiT0ZzgwCLcBGAs/s1600/Math_toolsFeb_edited.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="669" data-original-width="1361" height="196" src="https://3.bp.blogspot.com/-lWbPzXlaJkw/WnSegR9V4LI/AAAAAAAALCI/3dzh7U6HtbQuDdhBDfGhqYHrAXiT0ZzgwCLcBGAs/s400/Math_toolsFeb_edited.jpg" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div>A number of mathematicians commented on social media that their list would be almost identical to mine. That did not surprise me. My chart simply captures the way today’s pros approach new problems. A number of math teachers expressed puzzlement. That too did not surprise me. The current mathematics curriculum is still rooted in a conception of “doing math” that developed to meet society’s needs in the 19th Century.<br /><br />Actually, I should point out that the diagram above is not exactly the one I published last month. I have added an icon for a spreadsheet. A mathematician in Austria emailed me to say I should have included it. The two of us had corresponded in the past about the use of spreadsheets in mathematics, both in problem solving and in teaching, and we were (and are) very much on the same page as to their usefulness in a wide variety of circumstances. My excuse for overlooking it the first time round was that it was only the second technological tool I brought into my mathematics arsenal, so far back in my career that I had long ago stopped thinking of it as something new. (The first piece of “new tech” I adopted was the electronic calculator, and that too did not appear in my chart.) I suspect that almost all math teachers, and indeed, pretty well all of society, make frequent use of calculators and spreadsheets, not only in their professional activities but in their social and personal lives as well. Still, the spreadsheet is such a powerful, ubiquitous mathematics tool, I should have included it, and now I have. (Its use definitely figured in the guidance I gave to the Nueva School class.) I have placed it in the position in my list that, on reflection, I find I turn to in order of frequency. <br /><br />Some of the responses I received from teachers indicated that I need to clarify that, by “solving a mathematical problem”, I mean using mathematics to solve a real-world problem. The problem we worked on at Nueva School was one UPS worked on not long ago: “What is the most efficient way to route packages from place to place?” More on that later. A simpler example in the same vein is when we ask ourselves “Which kind, model, and hardware configuration of mobile phone best meets my needs within my current budget?”—an example where, for most of us, the item’s cost is high enough for us to weigh the (many) options fairly carefully. <br /><br />This is clearly not the same as “solving a math problem” in a typical math textbook. For example, “What are the roots of the equation <i>x</i><sup>2</sup> + 3<i>x</i> – 5 = 0?” Those kinds of questions are, of course, designed to provide practice in using various specific, sharply focused, mathematics techniques, procedures, formulas, or algorithms. <br /><br />Those techniques, procedures, etc. are the basic building blocks for using mathematics to solve problems in real life, but they don’t really present much of a <i>problem</i>, in the sense the word is used outside the math class. Indeed, the reason it can be valuable to master those basic techniques, etc. is that being able to use them fluidly means they <i>won’t</i> be a problem (in the sense of an <i>obstacle</i>) that gets in the way of solving what really is a mathematical <i>problem</i> (e.g., which phone to buy). That, of course, is why we call them basic skills. But having mastery of a range of <i>basic</i> skills does not make a person a good problem solver any more than being a master bricklayer makes someone an architect or a construction engineer. <br /><br />My focus then, is on using math to solve real-world problems. That’s where things are very different from the days when I first learned mathematics. Back in the 1950s and 60s, when I went through the school system, we spent a huge amount of time mastering algorithms and techniques for performing a variety of different kinds of numerical and symbolic calculations, geometric reasoning, and equation solving. We had to. In order to solve any real-world problem, we had to be able to crank the algorithmic and procedural handles. <br /><br />Not any more. These days, that smartphone in your pocket has access to cost-free cloud resources that can execute any mathematical procedure you need. What’s more, it will do it with a dataset of any real-world size, with total accuracy to whatever degree you demand, and in the majority of cases in a fraction of a second. <br /><br />To put it another way, all those algorithms, techniques, and procedures I spent years mastering, all the way through to earning my bachelors degree in mathematics, became obsolete within my lifetime, an observation I wrote about <a href="https://www.huffingtonpost.com/entry/all-the-mathematical-methods-i-learned-in-my-university_us_58693ef9e4b014e7c72ee248" target="_blank">in an article</a> in the <i>Huffington Post</i> in January of last year. <br /><br />So, does that mean all that effort was wasted? Not at all. Discounting the fact that in my case, I was able to make good use of those skills and knowledge for several decades before the march of technology rendered them obsolete, the one thing that I gained as a result of all that procedural learning that is as valuable today as it was back then, was the ability to think mathematically. I wrote about one aspect of that “mathematical thinking” mental ability, number sense, in a simultaneously published <a href="https://www.huffingtonpost.com/entry/number-sense-the-most-important-mathematical-concept_us_58695887e4b068764965c2e0" target="_blank">follow-up piece</a> to that<i> Huffington Post</i> article. <br /><br />In today’s world, all the algorithmic, computational, algebraic, geometric, logical, and procedural skills that used to take ten years of effort to master can now be bought for $699. At least, that amount (the price of an iPhone 8, which I chose for illustration) is all it costs to give you access to all those skills. Making effective use of that vast powerhouse of factual knowledge and procedural capacity requires considerable ability. Anyone who mastered mathematics the way I did acquired that ability as an automatic by-product of mastering the basic skills. But what does it take to acquire it in an age when all those new tools are widely available? <br /><br />The answer, <i>of course</i> (though not everyone involved in the mathematics education system thinks it is obvious, or even true), is that the educational focus has to shift from procedural mastery to <i>understanding</i>. Which is precisely the observation that guided the Common Core initiative in the United States. Yes, I know that the current leadership of the US Department of Education <a href="https://www.edsurge.com/news/2018-01-16-betsy-devos-touts-personalized-learning-slams-common-core-and-reform-efforts" target="_blank">believes that the Common Core is a bad idea</a>, but that is an administration that also believes the future of energy lies in fossil fuels, not renewables, and the highly qualified, career-professional contacts I have in the Department of Education have a very different view. <br /><br />How do you acquire that high-level skill set? The answer is, the same way people always did: through lots of practice. <br /><br />But be careful how you interpret that observation. What need to be practiced are <i>the kinds of activities that you would use as a professional</i>—or at least a <i>competent user</i> of mathematics—in the circumstances of the day. In my school days, that meant we had to practice with highly constrained, “toy” problems. But with today’s technologies, we can practice on <i>real-world problems</i> using <i>real-world</i> data. <br /><br />Almost inevitably, when you do that, you find you frequently need to drop down to suitably chosen “toy problem” variants of your task in order to understand how a particular online tool (say) works and what it can (and cannot) do. But today, the purpose of, say, inverting a few 2x2 or 3x3 matrices is not (as it was in my day) so you can become fluent at doing so, and certainly not because you will actually invert by hand that 100x100 matrix that has just reared its ugly head in your real-world problem. No, you just need to get a good understanding of what it means to invert a matrix, why you might do so, and what problems can arise.<br /><br />And you know what? That’s rarely a problem. Once you have identified a mathematical technique you need to understand, the chances are high you will find not one but a dozen or more YouTube videos that explain it to you. <br /><br />These new tools certainly don’t solve the problem for you. [Well, sometimes they may do, but in that case it wasn’t a problem that required the time of a mathematician. Better to move on and put your efforts into a problem that cannot be solved by an app in the Cloud!] All that these fancy new tools have done is change the level at which we humans operate. <br /><br />At heart, that shift is no different from the level-shift introduced in the 9th Century when traders in and around Baghdad developed techniques for doing routine arithmetic calculations at scale, by performing operations not on specific numbers but on classes of numbers. One of the techniques they developed was called, <i>al-jabr</i>, a term that ended up giving the name we use today to refer to that new kind of calculation procedure: algebra. <br /><br />Throughout mathematics’ history, mathematicians have calculated and reasoned logically with the basic building blocks of the time. Today’s procedures (that have to be executed) turn into tomorrow’s basic entities (<i>on which you operate</i>). A classic example is differential calculus, where functions are no longer viewed as rules that you execute to yield new numbers from old numbers, but higher-level objects <i>on which you operate</i> to produce new functions from old functions. <br /><br />So (finally), what exactly did we do in that Nueva School mini-course to illustrate the way today’s pros use math to solve a problem? The problem, remember, was this: Reverse engineer the core algorithm than UPS uses to route packages from origin to destination? <br /><br />To start the class off—they worked in small teams of three or four—I provided a small amount of information to get them started:<br /><ol><li>Tracking information for a fairly large, heavy case, including a partially dismantled bicycle, I had shipped from Petaluma, California to Fair Haven, New Jersey, in 2015. See image below.</li><li>I told them I sent the case by “three day select.”</li><li>I reported that my package went by plane from Louisville, Kentucky, to the UPS facility in Newark, where it was immediately loaded onto a truck, and was delivered to the intended Fair Haven destination with just a few hours to spare within the three-day period guaranteed.</li></ol>That information, I told the class, was enough to figure out how the routing algorithm worked. [This itself is useful information that I did not have when I first solved the problem, but they had to figure it out by the end of the course, so I was happy to give them additional information.] In solving this problem, they could elicit my help as their “math consultant,” to call on with specific questions when required. But they had to carry out the key steps. <br /><br />They could, of course, use the various tools in my “modern math tools” chart, and any others they could find. (Since the UPS routing algorithm is an extremely valuable trade secret, they would not find that online, of course.) <br /><br />Next month, I’ll tell you how they got on. In the meantime, you might like to see how far you can get with it. Happy problem solving! Happy mathematical thinking! <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-SX-pwAkG6FY/WnN-tTM1kyI/AAAAAAAALBo/HmqKtPNEQwscQb0J7XwSCT2LG4YTs3SfACLcBGAs/s1600/Bike_data.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="768" data-original-width="1024" height="480" src="https://4.bp.blogspot.com/-SX-pwAkG6FY/WnN-tTM1kyI/AAAAAAAALBo/HmqKtPNEQwscQb0J7XwSCT2LG4YTs3SfACLcBGAs/s640/Bike_data.jpg" width="640" /></a></div><i>Part 2 will appear next month.</i><br /><br />http://devlinsangle.blogspot.com/2018/02/how-todays-pros-solve-math-problems.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-1116466154367577955Tue, 23 Jan 2018 15:31:00 +00002018-01-23T11:13:35.475-05:00Déjà vu, all over againI gave a short course at a local high school recently. Three days in a row, two hours a day, to fifteen students. To my mind, it was a huge success. By the end of the course, the students had successfully reverse-engineered UPS’s core routing/scheduling algorithm. In fact, they spent the last half hour brainstorming how UPS might improve their efficiency. (My guess is the company had long ago implemented, or at least considered, the ideas the kids came up with, but that simply serves to illustrate how far they had come in just six hours of class-time.) <br /><br />To be sure, it was not an average class in an average high school. <a href="http://www.nuevaschool.org/" target="_blank">Nueva School</a>, located in the northern reaches of Silicon Valley, is private and expensive (tuition runs at $36,750 for an 8th gader), and caters to students who have already shown themselves to be high achievers. Many Silicon Valley tech luminaries send their children there, and some serve on the board. They have an excellent faculty. Moreover, the fifteen students in my class had elected to be there, as part of their rich, January, electives learning experience called “Intersession”. <br /><br />I was familiar with the school, having been invited to speak at their annual education conference on a couple of occasions, but this was the first time I had taught a class. <br /><br />Surprisingly, the experience reminded me of my own high school education, back in the UK in the early 1960s. My high school was a state run, selective school in the working class city of Hull, a major industrial city and large ocean fishing and shipping port. Socially and financially, it was about as far away as you could get from Nueva School on the San Francisco Peninsula, and my fellow students came from very different backgrounds than the students at Nueva. <br /><br />What made my education so good was a highly unusual set of historical circumstances. Back then, Hull was a fiercely socialist city that, along with the rest of the UK, was clawing its way out of the ravages of the Second World War. For a few short years, the crippling English class system broke down, and an entire generation of baby boomers entered the school system determined to make better lives for themselves—and everyone else. (“Me first” came a generation later.) <br /><br />We had teachers who had either just returned from fighting the war (the men on the battlefields, the women in the factories or in military support jobs), or were young men and women just starting out on their teaching careers, having received their own school education while the nation was at war. There was a newly established, free National Health Service, an emerging new broadcasting technology (television) run by a public entity, a rapidly growing communications systems (a publicly funded telephone service), and free education, including government-paid- for university education for the 3 percent or so able to pass the challenging entrance exams. <br /><br />We were the generation that the nation was dependent on to rebuild, making our way through the education system in a social and political environment where the class divisions that had been a part of British life for centuries had been (temporarily, it turned out) cast aside by the need to fight a common enemy across the English Channel. The result was that, starting in the middle of the 1960s, a “British Explosion” of creative scientific, engineering, and artistic talent burst forth onto the world. Within our individual chosen domains, we all felt we could do anything we set our minds to. And a great many of us did just that. About half my high school class became highly successful people. That from a financially impoverished, working class background. <br /><br />It was short lived, lasting but a single generation. I was simply lucky to be part of it. <br /><br />What brought it all back to me was finding myself in a very similar educational environment in my three days at Nueva School. The circumstances could hardly be more different, of course. But talking and working with those students, I sensed the same thirst to learn, the same drive to succeed (in terms they set for themselves), and the same readiness to keep trying I had experienced two generations earlier. It felt comfortingly—and encouragingly—familiar. <br /><br />But I digress. In fact, I’ve done more than digress. I’ve wandered far from my intended path. Or have I? The point I want to get across is that when it comes to learning, success is about 5 percent talent, 35 percent the teachers and students around you, and 60 percent desire and commitment. (I just made up those figures, but they represent more or less how I see the landscape, having been an education professional for half a century.) <br /><br />It turns out that, in today’s world, given those ingredients, in roughly those proportions, it is possible for a small group of people, in the space of just a few days, to make significant progress in solving a major problem of massive societal importance. (If you can figure out how UPS performs its magic, you can do the same thing with many other large organizations, Walmart, Amazon, United Airlines, and so on.) <br /><br />How can it be possible to take a small group of students, still in high school, and make solid progress on a major mathematical problem like that? It would not have been possible in my school days. The answer is, in today’s world, everyone has access to the same rich toolset the professionals use. Moreover, most of those tools—or at least, enough of them—are free to anyone with access to a smartphone or a personal computer. You just have to know how to make effective use of them. <br /><br />Next month, I will describe how my Nueva class went about the UPS project. (I had done it once before, with a non-science majors undergraduate class at Princeton University. Doing it with high school students confirmed my belief that a group with less academic background could achieve the same result, in the process providing me with some major-league ammunition to back up my oft-repeated—and oft-ignored or disputed—claim that K-12 mathematics education is in need of a major (and I mean MAJOR) makeover. (After the invention of the automobile, it made more sense to teach people how to drive than how to look after a horse. I feel the math ed argument should end with that razor-sharp analogy, but it rarely does.) <br /><br />As I say, that discussion is for next month. But let me leave you with a teaser. Actually, two teasers. One is my January 1, 2017 opinion piece in the <i>Huffington Post</i>, "<a href="https://www.huffingtonpost.com/entry/all-the-mathematical-methods-i-learned-in-my-university_us_58693ef9e4b014e7c72ee248" target="_blank">All The MathematicalMethods I Learned In My University Math Degree Became Obsolete In My Lifetime</a>." The other teaser is the diagram I will end with. It summarizes some of the most useful tools that a professional mathematician today uses when starting to work on a new problem. (Note: I’m talking about using math to solve real-world problems here. Pure mathematics is very different, although all the tools I will mention <i>can</i> be of use to a pure mathematician.) <br /><br />This is my set of “most useful tools,” I should note, and reading the diagram left-to- right, top to bottom, the tools I list are roughly in the order I have used them in working on various projects over the past fifteen years. Other mathematicians might produce different collections and different orders. But they won’t be <i><b>that</b></i> much different, and I’ll bet they all begin with the same first tool. <br /><br />If you find this diagram in any way surprising, you likely have not worked in today’s world of mathematical problem solving. If you find it surprising <b><i>and</i></b> are in mathematics education, I respectfully point out that this is the mathematical toolset that your students will need to master in order to make use of math in the world they will inhabit after graduation. You may or may not like that. If you don’t like it, then that is unfortunate. Mathematical problem solving is simply done differently today. It just is. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-VvrMRmX2ZKg/WmdO8FapoaI/AAAAAAAALBI/iqA9kgfwvIcIRTSPT3-Z-sO0c6s0oNhcwCLcBGAs/s1600/Math_tools%2B%25281%2529.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="644" data-original-width="1340" height="191" src="https://2.bp.blogspot.com/-VvrMRmX2ZKg/WmdO8FapoaI/AAAAAAAALBI/iqA9kgfwvIcIRTSPT3-Z-sO0c6s0oNhcwCLcBGAs/s400/Math_tools%2B%25281%2529.jpg" width="400" /></a></div><br /><br /><br /><br />http://devlinsangle.blogspot.com/2018/01/deja-vu-all-over-again.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-4646632644831926229Thu, 14 Dec 2017 20:37:00 +00002017-12-14T15:37:14.152-05:00mathematical cognitionpercentagespie chartsClash of representations<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-ZhAuLxqHGCk/Wi7xCCFyiiI/AAAAAAAAK_c/oWSNle3TL5IZbcWyEsU8cmZ_n25JO2N4gCLcBGAs/s1600/Original_tweet.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1326" data-original-width="960" height="400" src="https://2.bp.blogspot.com/-ZhAuLxqHGCk/Wi7xCCFyiiI/AAAAAAAAK_c/oWSNle3TL5IZbcWyEsU8cmZ_n25JO2N4gCLcBGAs/s400/Original_tweet.jpg" width="289" /></a></div><br />The pie chart in the above tweet jumped out of the page when it appeared in my twitter feed on September 14. My initial shock at seeing the figure 1% attached to a region of the pie chart that was evidently almost 25% of the total area of the disk did not last long, of course, since the accompanying text made it clear what the diagram was intended to convey. The 1% label referred to the section of the population being discussed, whereas the pie-chart indicated the share of taxes paid by that group. Indeed, the image was an animated GIF; when I clicked on it, the region labeled “1%” shrank, culminating with the chart on the right in the image shown below:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-n1mBg8RGILw/Wi7yQdbx0AI/AAAAAAAAK_0/WzRnrTYoETYWdxVcKRW2mOmXsCRNAcGswCLcBGAs/s1600/Pie_GIF.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="474" data-original-width="851" height="221" src="https://3.bp.blogspot.com/-n1mBg8RGILw/Wi7yQdbx0AI/AAAAAAAAK_0/WzRnrTYoETYWdxVcKRW2mOmXsCRNAcGswCLcBGAs/s400/Pie_GIF.jpg" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"></div>But here’s the thing. Even after I had figured out what the chart was intended to convey, I still found it confusing. I wondered if a lay-reader, someone who is not a professional mathematician, would manage to parse out the intended meaning. It was not long before I found out. The image below shows one of the tweets that appeared in response less than an hour later:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-g7-5kLuiETo/Wi7xl0FPZ2I/AAAAAAAAK_o/haf-Gqyva8MP-61rhIEnwiUHsDMPjEoWwCLcBGAs/s1600/Reply.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="169" data-original-width="925" height="71" src="https://2.bp.blogspot.com/-g7-5kLuiETo/Wi7xl0FPZ2I/AAAAAAAAK_o/haf-Gqyva8MP-61rhIEnwiUHsDMPjEoWwCLcBGAs/s400/Reply.jpg" width="400" /></a></div>As I had suspected, a common reaction was to dismiss the chart as yet another example of a bad data visualization created by an innumerate graphics designer. Indeed, that had been my initial reaction. But this particular example is more interesting. Yes, it is a bad graphic, for the simple reason that it does not convey the intended message. But not because of the illustrator’s innumeracy. In fact, numerically, it appears to be as accurate as you can get with a pie chart. The before and after charts do seem to have regions whose areas correspond to the actual data on the tax-payer population. <br /><br />This example was too good to pass up as an educational tool: asking a class to discuss what the chart is intended to show, could lead to a number of good insights into how mathematics can help us understand the world, while at the same time having the potential to mislead. I was tempted to write about it in my October post, but wondered if I should delay a couple of months to avoid an example that was at the heart of a current, somewhat acrimonious party-political debate. As it turned out, the September 30 death of the game-show host Monty Hall resolved the issue for me—I had to write about that—and then November presented another “must do” story (the use of mathematics in election jerrymandering). So this month, with the background political, tax votes now a matter of historical record, I have my first real opportunity to run this story. <br /><br />The two-month delay brought home to me just how problematic this particular graphic is. Even knowing in advance what the issue is, I still found I had to concentrate to “see” the chart as conveying the message intended. That “1%” label continued to clash with the relative area of the labeled region. <br /><br />It’s a bit like those college psychology-class graphics that show two digits in different font sizes, and ask you to point to the digit that represents the bigger integer. If the font sizes clash with the sizes of the integers, you take measurably longer to identify the correct one, as shown below:<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-yx8NR9Vkn7A/Wi7x4Gw6ioI/AAAAAAAAK_s/YsP1uhGBpuslAZ2iCfYGR5xqwVwxoNn2ACLcBGAs/s1600/Stroop.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="393" data-original-width="759" height="206" src="https://1.bp.blogspot.com/-yx8NR9Vkn7A/Wi7x4Gw6ioI/AAAAAAAAK_s/YsP1uhGBpuslAZ2iCfYGR5xqwVwxoNn2ACLcBGAs/s400/Stroop.jpg" width="400" /></a></div>For me, the really big take-home lesson from the tax-proposal graphic is the power of two different mathematical representations of proportions: pie charts and numerical percentages. Each, on its own, is instant. In the case of the pie chart, the representation draws on the innate human cognitive ability to judge relative areas in simple, highly symmetrical figures like circular disks or rectangles. With percentages, there is some initial learning required—you have to understand percentages—but once you have done that, you know instantly what is meant by figures such as “5%” or “75%." <br /><br />But how do you get that understanding of the meaning of numerical percentages? For most of us (I suspect all of us), it comes from being presented (as children) with area examples like pie charts and subdivided rectangles. This sets us up to be confused, bigly, by examples where those two representations are used in the same graphic but with the percentage representing something other than the area of the segment (or what that area is intended to represent). <br /><br />The message then, from this particular example—or at least the message I got from it—is that powerful graphics are like any powerful tool, their power for good depends on using them wisely; if used incorrectly, they can confuse and mislead. And make no mistake about it, numbers are incredibly powerful tools. Their invention <b><i>alone</i></b> is by far the greatest mathematical invention in human history. That’s why in every nation in the world, math is the only mandated school subject apart from the native language. <br /><br />http://devlinsangle.blogspot.com/2017/12/clash-of-representations.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-1529472912946980232Thu, 16 Nov 2017 21:40:00 +00002017-12-11T15:52:16.423-05:00electionsgerrymanderingredistrictingSupreme CourtvotingWisconsin state electionsMathematics and the Supreme CourtAmerican courts have never appeared to be very receptive to mathematical arguments, in large part, some (including me) have assumed, because many judges do not feel confident evaluating mathematical reasoning and, in the case of jury trials, no doubt because they worry that unscrupulous, math-savvy lawyers could use formulas and statistics to fool jury members. There certainly have been some egregious examples of this, particularly when bogus probability arguments have been presented. Indeed, one classic misuse of conditional probability is now known as the “<a href="https://en.wikipedia.org/wiki/Prosecutor%27s_fallacy" target="_blank">prosecutor’s fallacy</a>."<br /><br />Another example where the courts have trouble with probability is in cases involving DNA profiling, particularly Cold Hit cases, where a DNA profile match is the only hard evidence against a suspect. I myself have been asked to provide expert testimony in some such cases, and I wrote about the issue in this column in <a href="https://www.maa.org/external_archive/devlin/devlin_09_06.html" target="_blank">September</a> and <a href="https://www.maa.org/external_archive/devlin/devlin_10_06.html" target="_blank">October</a>of 2006. <br /><br />In both kinds of case, the courts have good reason to proceed with caution. The prosecutor’s fallacy is an easy one to fall into, and with Cold Hit DNA identification there is a real conflict between frequentist and Bayesian probability calculations. In neither case, however, should the courts try to avoid the issue. When evidence is presented, the court needs to have as accurate an assessment as possible as to its reliability or veracity. That frequently has to be in the form of a probability estimate. <br /><br />Now the courts are facing another mathematical conundrum. And this time, the case has landed before the US Supreme Court. It is a case that reaches down to the very foundation of our democratic system: How we conduct our elections. Not how we use vote counts to determine winners, although that is also mathematically contentious, as I wrote about in this column in <a href="https://www.maa.org/external_archive/devlin/devlin_11_00.html" target="_blank">November</a> of 2000, just before the Bush v Gore Presidential Election outcome ended up before the Supreme Court. Rather, the issue before the Court this time is how states are divided up into electoral districts for state elections. <br /><br />How a state carves up voters into state electoral districts can have a huge impact on the outcome. In six states, Alaska, Arizona, California, Idaho, Montana, and Washington, the apportioning is done by independent redistricting commissions. This is generally regarded—at least by those who have studied the issue—as the least problematic approach. In seven other states, Arkansas, Colorado, Hawaii, Missouri, New Jersey, Ohio, and Pennsylvania, politician commissions draw state legislative district maps. In the remaining 37 states, the state legislatures themselves are responsible for state legislative redistricting. And that is where the current problem arises. <br /><br />There is, of course, a powerful temptation for the party in power to redraw the electoral district maps to favor their candidates in the next election. And indeed, in the states where the legislatures draw the maps, both major political parties have engaged in that practice. One of the first times this occurred was in 1812, when Massachusetts governor Elbridge Gerry redrew district boundaries to help his party in an upcoming senate election. A journalist at the <i>Boston Gazette</i> observed that one of the contrived districts in Gerry’s new map looked like a giant salamander, and gave such partisan redistricting a name, combining <i>Gerry</i> and <i>mander</i> to create the new word <i>gerrymander</i>. Though Gerry lost his job over his sleight-of- hand, his redistricting did enable his party to take over the state senate. And the name stuck. <br /><br />Illegality of partisan gerrymandering is generally taken to stem from the 14th Amendment, since it deprives the smaller party of the equal protection of the laws, but it has also been argued to be, in addition, a 1st Amendment issue—namely an apportionment that has the purpose and effect of burdening a group of voters’ representational rights. <br /><br />In 1986, the Supreme Court issued a ruling that partisan gerrymandering, if extreme enough, is unconstitutional, but it has yet to throw out a single redistricting map. In large part, the Supreme Court’s inclination to stay out of the redistricting issue is based on a recognition that both parties do it, and over time, any injustices cancel out, as least numerically. Historically, this was, generally speaking, true. Attempts to gerrymander have tended to favor both parties to roughly the same extent. But in 2012, things took a dramatic turn with a re-districting process carried out in Wisconsin. <br /><br />That year, the recently elected Republican state legislature released a re-districting map generated using a sophisticated mathematical algorithm running on a powerful computer. And that map was in an altogether new category. It effectively guaranteed Republican majorities for the foreseeable future. The Democrat opposition cried foul, a Federal District Court agreed with them, and a few months ago the case found its way to the Supreme Court. <br /><br />That the Republicans come across as the bad actors in this particular case is likely just an accident of timing; they happened to come to power at the very time when political parties were becoming aware of what could be done with sophisticated algorithms. If history is any guide, either one of the two main parties would have tried to exploit the latest technology sooner or later. In any event, with mathematics at the heart of the new gerrymandering technique, the only way to counter it may be with the aid of equally sophisticated math. <br /><br />The most common technique used to gerrymander a district is called “packing and cracking." In packing, you cram as many of the opposing party’s voters as possible into a small number of “their” districts where they will win with many more votes than necessary. In cracking, you spread opposing party’s voters across as many of “your” districts as possible so there are not enough votes in any one of those districts to ever win there. <br /><br />A form of packing and cracking arises naturally when better-educated liberal-leaning voters move into in cities and form a majority, leaving those in rural areas outnumbered by less-educated, more conservative-leaning voters. (This is thought to be one of the factors that has led to the increasing polarization in American politics.) Solving that problem is, of course, a political one for society as a whole, though mathematics can be of assistance by helping to provide good statistical data. Not so with partisan gerrymandering, where mathematics has now created a problem that had not arisen before, for which mathematics may of necessity be part of the solution. <br /><br />When Republicans won control of Wisconsin in 2010, they used a sophisticated computer algorithm to draw a redistricting map that on the surface appeared fair—no salamander-shaped districts—but in fact was guaranteed to yield a Republican majority even if voter preferences shifted significantly. Under the new map, in the 2012 election, Republican candidates won 48 percent of the vote, but 60 of the state’s 99 legislative seats. The Democrats’ 51 percent that year translated into only 39 seats. Two years later, when the Republicans won the same share of the vote, they ended up with 63 seats—a 24-seat differential. <br /><br />Recognizing what they saw as a misuse of mathematics to undermine the basic principles of American democracy, a number of mathematicians around the country were motivated to look for ways to rectify the situation. There are really two issues to be addressed. One is to draw fair maps—a kind of “positive gerrymandering.” The other is to provide reliable evidence to show that a particular map has been intentionally drawn to favor one party over another, if such occurs, and moreover to do so in a way that the courts can understand and accept. Neither issue is easy to solve, and without mathematics, both are almost certainly impossible. <br /><br />For the first issue, a 2016 Supreme Court ruling gave a hint about what kind of fairness measure it might look kindly on: one that captures the notion of “partisan symmetry,” where each party has an equal opportunity to convert its votes into seats. The Wisconsin case now presents the Supreme Court with the second issue. <br /><br />When, last year, a Federal District Court in Wisconsin threw out the new districting map, they cited both the 1st and 14th Amendments. It was beyond doubt, the court held, that the new maps were “designed to make it more difficult for Democrats, compared to Republicans, to translate their votes into seats.” The court rejected the Republican lawmakers’ claim that the discrepancy between vote share and legislative seats was due simply to political geography. The Republicans had argued that Democratic voters are concentrated in urban areas, so their votes have an impact on fewer races, while Republicans are spread out across the state. But, while that is true, geography alone does not explain why the Wisconsin maps are so skewed. <br /><br />So, how do you tell if a district is gerrymandered? One way, that has been around for some time, is to look at the geographical profile. The <i>gerrymandering score</i>, G, is defined by:<br /><i>G</i> = <i>gP/A</i>, where<br />g: the district’s boundary length, minus natural boundaries (like coastlines and rivers)<br />P: the district’s total perimeter<br />A: the district’s area<br />The higher the score, the wilder is the apportionment as a geographic region, and hence the more likely to have been gerrymandered. <br /><br />That approach is sufficiently simple and sensible to be acceptable to both society and the courts, but unfortunately does not achieve the desired aim of fairness. And, more to the point in the Wisconsin case, use of sophisticated computer algorithms can draw maps that have a low gerrymandering score and yet are wildly partisan. <br /><br />The Wisconsin Republicans’ algorithm searched through thousands of possible maps looking for one that would <b><i>look reasonable</i></b> according to existing criteria, but would favor Republicans <i><b>no matter what the election day voting profile might look like</b></i>. As such, it would be a statistical <i><b>outlier</b></i>. To find evidence to counter that kind of approach, you have to look at the results the districting produces when different voting profiles are fed into it. <br /><br />One promising way to identify gerrymandering is with a simple mathematical formula suggested in 2015, called the “<a href="https://www.brennancenter.org/sites/default/files/legal-work/How_the_Efficiency_Gap_Standard_Works.pdf" target="_blank">efficiency gap</a>." It was the use of this measure that caused, at least in part, the Wisconsin map to be struck down by the court. It is a simple idea—and as I noted, simplicity is an important criterion, if it is to stand a chance of being accepted by society and the courts. <br /><br />You can think of a single elector’s vote as being “wasted” if it is cast in a district where their candidate loses or it is cast in a district where their candidate would have won there anyway. The efficiency gap measures those “wasted” votes. For each district, you total up the number of votes the winning candidate receives in excess of what it would have taken to elect them in that district, and you total up the number of votes the losing candidate receives. Those are the two parties’ “wasted votes” for that district. <br /><br />You then calculate the difference between those “wasted-vote” totals for each of the two parties, and divide the answer by the total number of votes in the state. This yields a single percentage figure: the <b><i>efficiency gap</i></b>. If that works out to be greater than 7%, the systems developers suggest, the districting is unfair. <br /><br />By way of an example, let’s see what the efficiency gap will tell us about the last Congressional election. In particular, consider Maryland’s 6 th Congressional district, which was won by the Democrats. It requires 159K votes to win. In the last election, there were 186K Democrat votes, so 186K – 159K = 26K Democrat votes were “wasted,” and 133K Republican votes, all of which were “wasted.” <br /><br />In Maryland as a whole, there were 510K Democrat votes “wasted” and 789K Republican votes “wasted.” So, statewide, there was a net “waste” of 789K – 510K = 279K Republican votes. <br /><br />There were 2,598M votes cast in total. So the efficiency gap is 279K/2598K = 10.7% in favor of the Democrats. <br /><br />I should note, however, that the gerrymandering problem is currently viewed as far more of a concern in state elections than in congressional races. Last year, two social scientists published <a href="http://www-personal.umich.edu/~jowei/gerrymandering.pdf" target="_blank">the results</a> they obtained using computer simulations to measure the extent of intentional gerrymandering in congressional district maps across most of the 50 states. They found that on the national level, it mostly canceled out between the parties. So banning only intentional gerrymandering would likely have little effect on the partisan balance of the U.S. House of Representatives. The efficiency gap did, however, play a significant role in the Wisconsin court’s decision. <br /><br />Another approach, developed by a team at <a href="https://services.math.duke.edu/projects/gerrymandering/" target="_blank">Duke University</a>, takes aim at the main idea behind the Wisconsin redistricting algorithm—searching through many thousands of possible maps looking for ones that met various goals set by the creators, any one of which would, of necessity, be a <b><i>statistical outlier</i></b>. To identify a map that has been obtained in this way, you subject it to many thousands of random tweaks. If the map is indeed an outlier, the vast majority of tweaks will yield a fairly unremarkable map. So, you compare the actual map with all those thousands of seemingly almost identical, and apparently reasonable, variations you have generated from it. If the actual map produces significantly different election results from all the others, when presented with a range of different statewide voting profiles, you can conclude that it is indeed an “outlier” — a map that could only have been chosen to deliberately subvert the democratic process. <br /><br />And this is where we—and the Supreme Court—are now. We have a problem for our democracy created using mathematics. Mathematicians looking for mathematical ways to solve it, and there are already two candidate “partisan gerrymandering test” in the arena. Historically, the Supreme Court has proven resistant to allowing math into the courtroom. But this time, it looks like they may have no choice. At least as long as state legislatures continue to draw the districting maps. Maybe the very threat of having to deal with mathematical formulas and algorithms will persuade the Supreme Court to recommend that Congress legislates to enforce all states to use independent commissions to draw the districting maps. Legislation under pain of math. We will know soon enough. <br /><br />http://devlinsangle.blogspot.com/2017/11/mathematics-and-supreme-court.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-3498933541202426241Wed, 11 Oct 2017 21:45:00 +00002017-10-12T09:37:29.227-04:00Carl SaganLet’s Make a DealMonty HallMonty Hall Problemprobability theoryMonty Hall may now rest in peace, but his problem will continue to frustrate<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-uMMlSwZYqjw/Wd6Q2LQyGXI/AAAAAAAAK7w/bNey7uxYHNQh1ducy7jasljp7VEP915PwCLcBGAs/s1600/MontyHallDoors.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="312" data-original-width="516" height="241" src="https://1.bp.blogspot.com/-uMMlSwZYqjw/Wd6Q2LQyGXI/AAAAAAAAK7w/bNey7uxYHNQh1ducy7jasljp7VEP915PwCLcBGAs/s400/MontyHallDoors.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Monty Hall with a contestant in <i>Let's Make a Deal</i>.</td></tr></tbody></table>The news that American TV personality Monty Hall died recently (<a href="https://www.nytimes.com/2017/09/30/obituaries/monty-hall-dead-lets-make-a-deal.html?hp&amp;action=click&amp;pgtype=Homepage&amp;clickSource=story-heading&amp;module=second-column-region&amp;region=top-news&amp;WT.nav=top-news&amp;_r=0" target="_blank">The New York Times, September 30</a>) caused two groups of people to sit up and take note. One group, by far the larger, was American fans of television game shows in the 1960s and 70s, who tuned in each week to his show “Let’s Make a Deal.” The other group include lovers of mathematics the world over, most of whom, I assume, have never seen the show. <br /><br />I, and by definition all readers of this column, are in that second category. As it happens, I have seen a key snippet of one episode of the show, which a television documentary film producer procured to use in a mathematics program we were making about probability theory. Our interest, of course, was not the game show itself, but the famous — indeed infamous — “Monty Hall Problem” it let loose on an unsuspecting world. <br /><br />To recap, at a certain point in the show, Monty would offer one of the audience participants the opportunity to select one of three doors on the stage. Behind one, he told them, was a valuable prize, such as a car, behind each of the other two was a booby prize, say a goat. The contestant chose one door. Sometimes, that was the end of the exchange, and Monty would open the door to reveal what the contestant had won. But on other occasions, after the contestant had chosen a door, Monty would open one of the two unselected doors to reveal a booby prize, and then give them the opportunity to switch their selection. (Monty could always do this since he knew exactly which door the prize was hidden behind.) <br /><br />So, for example, if the contestant first selects Door 2, Monty might open Door 1 to reveal a goat, and then ask if the contestant wanted to switch their choice from Door 2 to Door 3. The mathematical question here is, does it make any difference if the contestant switches their selection from Door 2 to Door 3? The answer, which on first meeting this puzzler surprises many people, is that the contestant doubles their chance of winning by switching. The probability goes up from an original 1/3 of Door 2 being the right guess, to 2/3 that the prize is behind Door 3. <br /><br />I have discussed this problem in <i>Devlin’s Angle</i> on at least two occasions, the most recent being <a href="https://www.maa.org/external_archive/devlin/devlin_12_05.html" target="_blank">December 2005</a>, and have presented it in a number of articles elsewhere, including national newspapers. That on each occasion I have been deluged with mail saying my solution is obviously false was never a surprise; since the problem is famous precisely because it presents the unwary with a trap. That, after all, is why I, and other mathematics expositors, use it! What continues to amaze me is how unreasonably resistant many people are to stepping back and trying to figure out where they went wrong in asserting that switching doors cannot possibly make any difference. For such reflection is the very essence of learning. <br /><br />Wrapping your mind around the initially startling information that switching the doors doubles the probability of winning is akin to our ancestors coming to terms with the facts that the Earth is not flat or that the Sun does not move around the Earth. In all cases, we have to examine how it can be that what our eyes or experience seem to tell us is misleading. Only then can we accept the rock-solid evidence that science or mathematics provides. <br /><br />Some initial resistance is good, to be sure. We should always be skeptical. But for us and society to continue to advance, we have to be prepared to let go of our original belief when the evidence to the contrary becomes overwhelming. <br /><br />The Monty Hall problem is unusual (though by no means unique) in being simple to state and initially surprising, yet once you have understood where your initial error lies, the simple correct answer is blindingly obvious, and you will never again fall into the same trap you did on the first encounter. Many issues in life are much less clear-cut. <br /><br />BTW, if you have never encountered the problem before, I will tell you it is not a trick question. It is entirely a mathematical puzzle, and the correct mathematics is simple and straightforward. You just have to pay careful attention to the information you are actually given, and not remain locked in the mindset of what you initially <i><b>think</b></i> it says. Along the way, you may realize you have misunderstood the notion of probability. (Some people maintain that probabilities cannot change, a false understanding that most likely results from first encountering the notion in terms of the empirical study of rolling dice and selecting colored beans from jars.) So reflection on the Monty Hall Problem can provide a valuable lesson in coming to understand the hugely important concept of mathematical probability. <br /><br />As it happens, Hall’s death comes at a time when, for those of us in the United States, the system of evidence-based, rational inquiry which made the nation a scientific, technological, and financial superpower is coming under dangerous assault, with significant resources being put into a sustained attempt to deny that there are such things as scientific facts. For scientific facts provide a great leveler, favoring no one person or one particular group, and are thus to some, a threat. <br /><br />The late Carl Sagan warned of this danger back in 1995, in his book <i><a href="https://www.amazon.com/Demon-Haunted-World-Science-Candle-Paperback/dp/B00EQBY4TW/ref=sr_1_2?ie=UTF8&qid=1506948921&sr=8-2&keywords=the+demon+haunted+world" target="_blank">The Demon-Haunted World:Science as a Candle in the Dark</a></i>, writing:<br /><blockquote>“I have a foreboding of an America in my children’s or my grandchildren’s time — when the United States is a service and information economy; when nearly all the key manufacturing industries have slipped away to other countries; when awesome technological powers are in the hands of a very few, and no one representing the public interest can even grasp the issues; when the people have lost the ability to set their own agendas or knowledgeably question those in authority; when, clutching our crystals and nervously consulting our horoscopes, our critical faculties in decline, unable to distinguish between what feels good and what’s true, we slide, almost without noticing, back into superstition and darkness. ...”</blockquote>Good scientists, such as Sagan, are not just skilled at understanding what is, they can sometimes extrapolate rationally to make uncannily accurate predictions of what the future might bring. It is chilling, but now a possibility that cannot be ignored, that a decade from now, I could be imprisoned for writing the above words. Today, the probability that will happen is surely extremely low, albeit nonzero. But that probability could change. As mathematicians, we have a clear responsibility to do all we can to ensure that Sagan’s words do not describe the world in which our children and grandchildren live. <br /><br /><br /><br /><br /><br />http://devlinsangle.blogspot.com/2017/10/monty-hall-may-now-rest-in-peace-but.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-2543952525288897833Wed, 20 Sep 2017 18:02:00 +00002017-09-20T14:22:37.383-04:00computers and mathematicsexperimental mathematicsJon BarwiseJonathan BorweinThe Legacy of Jonathan Borwein<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-eabNsYY8UW4/WcKs4cjposI/AAAAAAAAK6A/wXD7oNry-8MJx2A2RprinA6kiFXemNzIwCEwYBhgL/s1600/keith%2Bdevlin.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="891" data-original-width="1600" height="222" src="https://4.bp.blogspot.com/-eabNsYY8UW4/WcKs4cjposI/AAAAAAAAK6A/wXD7oNry-8MJx2A2RprinA6kiFXemNzIwCEwYBhgL/s400/keith%2Bdevlin.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Keith Devlin and Jonathan Borwein talk to host Robert Krulwick on stage at the World Science Festival in 2011.</td></tr></tbody></table><br />At the end of this week I fly to Australia to speak and participate in the <a href="https://carma.newcastle.edu.au/meetings/jbcc/">Jonathan Borwein Commemorative Conference</a> in Newcastle, NSW, Borwein’s home from 2009 onwards, when he moved to the Southern hemisphere after spending most of his career at various Canadian universities. Born in Scotland in 1951, Jonathan passed away in August last year, leaving behind an extensive collection of mathematical results and writings, as well as a long list of service activities to the mathematical community. [For a quick overview, read the brief <a href="http://experimentalmath.info/blog/2016/08/jonathan-borwein-dies-at-65/">obituary</a>written by his long-time research collaborator David Bailey in their joint blog Math Drudge. For more details, check out his <a href="https://en.wikipedia.org/wiki/Jonathan_Borwein">Wikipedia entry</a>.] <br /><br />Jonathan’s (I cannot call him by anything but the name I always used for him) career path and mine crossed on a number of occasions, with both of us being highly active in mathematical outreach activities and both of us taking an early interest in the use of computers in mathematics. Over the years we became good friends, though we worked together on a project only once, co-authoring an expository book on experimental mathematics, titled <i><a href="https://www.amazon.com/Computer-Crucible-Introduction-Experimental-Mathematics/dp/1568813430/ref=asap_bc?ie=UTF8" target="_blank">The Computer as Crucible</a></i>, published in 2008. <br /><br />Most mathematicians, myself included, would credit Jonathan as the father of experimental mathematics as a recognized discipline. In the first chapter of our joint book, we defined experimental mathematics as “the use of a computer to run computations—sometimes no more than trial-and- error tests—to look for patterns, to identify particular numbers and sequences, to gather evidence in support of specific mathematical assertions that may themselves arise by computational means, including search.” <br /><br />The goal of such work was to gather information and gain insight that would eventually give rise to the formulation and rigorous proof of a theorem. Or rather, I should say, that was Jonathan’s goal. He saw the computer, and computer-based technologies, as providing new tools to formulate and prove mathematical results. And since he gets to define what “experimental mathematics” is, that is definitive. But that is where are two interests diverged significantly. <br /><br />In my case, the rapidly growing ubiquity of ever more powerful and faster computers led to an interest in what I initially called “soft mathematics” (see my 1998 book<i> <a href="https://www.amazon.com/Goodbye-Descartes-Logic-Search-Cosmology/dp/0471251860/ref=asap_bc?ie=UTF8" target="_blank">Goodbye Descartes</a></i>) and subsequently referred to as “mathematical thinking,” which I explored in a number of articles and books. The idea of mathematical thinking is to use a mathematical approach, and often mathematical notations, to gather information and gain insight about a task in a domain that enables improved performance. [A seminal, and to my mind validating, example of that way of working was thrust my way shortly after September 11, 2001, when I was asked to join a team tasked with improving defense intelligence analysis.] <br /><br />Note that the same phrase “gather information and gain insight” occurs in both the definition of experimental mathematics and that of mathematical thinking. In both cases, the process is designed to lead to a specific outcome. What differs is the nature of that outcome. (See my 2001 book <i><a href="https://www.amazon.com/Infosense-Turning-Information-Into-Knowledge/dp/0716741644/ref=asap_bc?ie=UTF8" target="_blank">InfoSense</a></i>, to get the general idea of how mathematical thinking works, though I wrote that book before my Department of Defense work, and before I adopted the term “mathematical thinking.”) <br /><br />It was our two very different perspectives on the deliberative blending of mathematics and computers that made our book <i>The Computer as Crucible</i> such a fascinating project for the two of us. <br /><br />But that book was not the first time our research interests brought us together. In 1998, the American Mathematical Society introduced a new section of its ten-issues- a-year <i>Notices</i>, sent out to all members, called “Computers and Mathematics,” the purpose of which was both informational and advocacy. <br /><br />Though computers were originally invented by mathematicians to perform various numerical calculations, professional mathematicians were, by and large, much slower at making use of computers in their work and their teaching than scientists and engineers. The one exception was the development of a number of software systems for the preparation of mathematical manuscripts, which mathematicians took to like ducks to water. <br /><br />In the case of research, mathematicians’ lack of interest in computers was perfectly understandable—computers offered little, if any, benefit. (Jonathan was one of a very small number of exceptions, and his approach was initially highly controversial, and occasionally derided.) But the writing was on the wall—or rather on the computer screen—when it came to university teaching. Computers were clearly going to have a major impact in mathematics education. <br /><br />The “Computers and Mathematics” section of the AMS <i>Notices</i> was intended to be a change agent. It was originally edited by the Stanford mathematician Jon Barwise, who took care of it from the first issue in the May/June 1988 Notices, to February 1991, and then by me until we retired the section in December 1994. It is significant that 1988 was the year Stephen Wolfram released his mathematical software package Mathematica. And in 1992, the first issue of the new research journal <i>Experimental Mathematics</i> was published. <br /><br />Over its six-and- a-half years run, the column published 59 feature articles, 19 editorial essays, and 115 reviews of mathematical software packages — 31 features 11 editorials, and 41 reviews under Barwise, 28 features, 8 editorials, and 74 reviews under me. [The <i>Notices</i>website has a <a href="http://www.ams.org/notices/199502/devlinsixyear.pdf" target="_blank">complete index</a>.] One of the feature articles published under my watch was “Some Observations of Computer Aided Analysis,” by Jonathan Borwein and his brother Peter, which appeared in October 1992. Editing that article was my first real introduction to something called “experimental mathematics.” For the majority of mathematicians, reading it was their introduction. <br /><br />From then on, it was clear to both of us that our view of “doing mathematics” had one feature in common: we both believed that for some problems it could be productive to engage in mathematical work that involved significant interaction with a computer. Neither of us was by any means the first to recognize that. We may, however, have been among the first to conceive of such activity as constituting a discipline in its own right, and each to erect a shingle to advertise what we were doing. In Jonathan’s case, he was advancing mathematical knowledge; for me it was about utilizing mathematical thinking to improve how we handle messy, real-world problems. In both cases, we were engaging in mental work that could not have been done before powerful, networked computers became available. <br /><br />It’s hard to adjust to Jonathan no longer being among us. But his legacy will long outlast us all. I am looking forward to re-living much of that legacy in Australia in a few days time. <br /><br />http://devlinsangle.blogspot.com/2017/09/the-legacy-of-jonathan-borwein.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-1953274983910512263Mon, 07 Aug 2017 18:44:00 +00002017-09-07T16:41:07.060-04:00universitiesuniversity administrationuniversity educationuniversity researchWhat are universities for and how do they work?Exactly 30 years ago, I and my family arrived in the U.S. from the U.K. to take up a one-year visiting position in the mathematics department at Stanford University. (We landed on July 28, 1987.) That one year was subsequently extended to two, and in the end we never returned to the U.K. A very attractive offer of a newly endowed chair in mathematics at Colby College in Maine provided the pull. But equally significant was a push from the U.K. <br /><br />The late 1980s were a bad time for universities in Britain, as Prime Minister Margaret Thatcher launched a full-scale assault on higher education, motivated in part by a false understanding of what universities do, and in part by personal vindictiveness stemming from her being criticized by academics for her poor performance as Minister for Education some years earlier. My own university, Lancaster, where I had been a regular faculty member since 1977, had been a source of some of the most vocal criticisms of the then Minister Thatcher, and accordingly was dealt a particularly heavy funding hit when Prime Minister Thatcher started to wield her axe. A newly appointed vice chancellor (president), with a reputation for tough leadership as a dean, was hired from the United States to steer the university through the troubled waters ahead. <br /><br />One of the first decisions the new vice chancellor made was to cut the mathematics department faculty by roughly 50%, from around 28 to 14. (I forget the actual numbers.) The problem he faced in achieving that goal was that in the British system at the time, once a new Lecturer (= Assistant Professor) had passed a three-year probationary period, they had tenure for life. The only way to achieve a 50% cut in faculty was to force out anyone who could be “persuaded” to go. That boiled down to putting pressure on those whose reputation was sufficiently good for them to secure a position elsewhere. (So, a strategy of “prune from the top,” arguably more productive in the garden than a university.) <br /><br />In my case, the new vice chancellor made it clear to me soon after his arrival that my prospects of career advancement at Lancaster were low, and I could expect ever increasing teaching loads that would hamper my research, and lack of financial support to attend conferences. As a research mathematician early in my career, with my work going well and my reputation starting to grow, that prospect was ominous. Though I was not sure whether he would ever actually follow through with his threat, it seemed prudent to start thinking in terms of a move, possibly one that involved leaving the U.K. <br /><br />Then, just as all of this was going on, out of the blue I got the invitation from Stanford. (I had started working on a project that aligned well with a group at Stanford who had just set up a new research center to work on the same issues. As a result, I had gotten to know some of them, mostly by way of an experimental new way to communicate called “e-mail,” which universities were just starting to use.) <br /><br />In my meeting with the vice chancellor to request permission to accept the offer and discuss the arrangements, I was told in no uncertain terms that I would be wise not to return after my year in California came to an end. The writing was on the wall. Lancaster wanted me gone. In addition, other departmental colleagues were also looking at opportunities elsewhere, so even if I were to return to Lancaster after my year at Stanford, it might well be to a department that had lost several of its more productive mathematicians. (It would have been. The vice chancellor achieved his 50% departmental reduction in little more than two years.) <br /><br />Yes, these events were all so long ago, in a different country. So why am I bringing the story up now? The answer, is that, as is frequently observed, history can provide cautionary lessons for what may happen in the future. <br /><br />Those of us in mathematics are deeply aware of the hugely significant role the subject plays in the modern world, and have seen with every generation of students how learning mathematics can open so many career doors. We also know sufficient mathematics to appreciate the enormous impact on society that new mathematical discoveries can have—albeit in many cases years or decades later. To us, it is inconceivable that a university—an institution having the sole purpose of advancing and passing on new knowledge for the good of society—would ever make a conscious decision to cut down (especially from the top), or eliminate, a mathematics department. <br /><br />But to people outside the universities, things can look different. Indeed, as I discovered during my time as an academic dean (in the U.S.), the need for mathematics departments engaged in research is often not recognized by faculty in other departments. Everyone recognizes the need for each new generation of students to be given some basic mathematics instruction, of course. But mathematics research? That’s a much harder sell. In fact, it is an extremely hard sell. Eliminating the research mathematicians in a department and viewing it as having a solely instructional role can seem like an attractive way to achieve financial savings. But it can come at a considerable cost to the overall academic/educational environment. Not least because of the message conveyed to the students. <br /><br />As things are, students typically graduate from high school thinking of mathematics as a toolbox of formulas and procedures for solving certain kinds of problems. But at university level, they should come to understand it as a particular way of thinking. To that end, they should be exposed to an environment where tasks can be approached on their own terms, with mathematicians being one of any number of groups of experts who can bring a particular way of thinking that may, or may not, be effective. <br /><br />The educational importance of having an active mathematics research group in a university is particularly important in today’s world. As I noted in <a href="http://www.huffingtonpost.com/entry/all-the-mathematical-methods-i-learned-in-my-university_us_58693ef9e4b014e7c72ee248" target="_blank">an article</a> in <i>The Huffington Post</i> in January, pretty well all the formulas and procedures that for many centuries have constituted the heart of a university mathematics degree have now been automated and are freely available on sites such as <a href="https://www.wolframalpha.com/examples/Math.html" target="_blank">Wolfram Alpha</a>. Applying an implemented, standard mathematical procedure to solve, say, a differential equation, is now in the same category as using a calculator to add up a column of numbers. Just enter the data correctly and the machine will do the rest. <br /><br />In particular, a physicist or an engineer (say) at a university can, for the most part, carry out their work without the need for specialist mathematical input. (That was always largely the case. It is even more so today.) But one of the functions of a university is to provide a community of experts who are able to make progress when the available canned procedures do not quite fit the task at hand. The advance of technology does not eliminate the need for creative, human expertise. It simply shifts the locus of where such expertise is required. Part of a university education is being part of a community where that reliance on human expertise is part of the daily activities; a community where all the domain specialists are experts in their domains, and able to go beyond the routine. <br /><br />It is easy to think of education as taking place in a classroom. But that’s just not what goes on. What you find in classrooms is instruction, maybe involving some limited discussion. Education and learning occur primarily by way of interpersonal interaction in a community. That’s why we have universities, and why students, and often their parents, pay to attend them. It’s why “online universities” and MOOCs have not replaced universities, and to my mind never will. The richer and more varied the community, the better the education. <br /><br />Lest I have given the impression that my focus is on topline research universities, stocked with award winning academic superstars, let me end by observing that nothing I have said refers to level of achievement. Rather it is all about the attitude of mind and working practices of the faculty. As long as the mathematics faculty love mathematics, and enjoy doing it, and are able to bring their knowledge to bear on a new task or problem, they contribute something of real value to the environment in which the students learn. It’s a human thing. <br /><br />A university that decides to downgrade a particular discipline to do little more than provide basic instruction is diminishing its students educational experience, and is no longer a bona fide university. (It may well, of course, continue to provide a valuable service. The university, my focus in this essay, is just one form of educational institution among many.) <br /><br /><br />http://devlinsangle.blogspot.com/2017/08/what-are-universities-for-and-how-do.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-8036995707127677168Thu, 13 Jul 2017 16:18:00 +00002017-07-13T12:51:06.793-04:00Exploding dotsHindu-arabic arithmeticJames Tantonmathematics educationplace-value arithmeticvisual representationsThe Power of Simple RepresentationsThe great mathematician Karl Freidrich Gauss is frequently quoted as saying “What we need are notions, not notations.” [In “About the proof of Wilson's theorem,” <i>Disquisitiones Arithmeticae</i> (1801), Article 76.] <br /><br />While most mathematicians would agree that Gauss was correct in pointing out that concepts, not symbol manipulation, are at the heart of mathematics, his words do have to be properly interpreted. While a <i>notation</i> does not matter, a <i>representation</i> can make a huge difference. The distinction is that developing or selecting a representation for a particular mathematical concept (or <i>notion</i>) involves deciding which features of the concept to capture. <br /><br />For example, the form of the ten digits 0, 1, … , 9 does not matter (as long as they are readily distinguishable), but the usefulness of the Hindu-Arabic number system is that it embodies base- 10 place-value representation of whole numbers. Moreover, it does so in a way that makes both learning and using Hindu-Arabic arithmetic efficient. <br /><br />Likewise, the choice of 10 as the base is optimal for a species that has highly manipulable hands with ten digits. Although the base-10 arithmetic eventually became the standard, other systems were used in different societies, but they too evolved from the use of the hands and sometimes the feet for counting: base-12 (where finger-counting used the three segments of each of the four fingers) and base-20 where both fingers and toes were used. Base-12 arithmetic and base-20 arithmetic both remained in regular use in the monetary system in the UK when I was a child growing up there, with 12 pennies giving one shilling and 20 shillings one pound. And several languages continue to carry reminders of earlier use of both bases — English uses phrases such as “three score and ten” to mean 70 (= 3x20 + 10) and French articulates 85 as “quatre-vingt cinq (4x20 + 5). <br /><br />Another number system we continue to use today is base-60, used in measuring time (seconds and minutes) and in circular measurement (degrees in a circle). Presumably the use of 60 as a base came from combining the finger and toes bases 10, 12, and 20, allowing for all three to be used as most convenient. <br /><br />These different base-number representation systems all capture features that make them useful to humans. Analogously, digital computers are designed to use binary arithmetic (base 2), because that aligns naturally with the two states of an electronic gate (open or closed, on or off). <br /><br />In contrast, the <i>shapes</i> of the Hindu-Arabic numerals is an example of a superfluous feature of the representation. The fact that it is possible to draw the numerals in a fashion whereby each digit has the corresponding number of angles, like this<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-YVDUy7CSxgo/WWeRZ91JdiI/AAAAAAAAK2c/FLqZuX8IVZogvX3K5mJN1OT1d89oYXWUwCLcBGAs/s1600/H-A_numerals.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="210" data-original-width="1558" height="43" src="https://3.bp.blogspot.com/-YVDUy7CSxgo/WWeRZ91JdiI/AAAAAAAAK2c/FLqZuX8IVZogvX3K5mJN1OT1d89oYXWUwCLcBGAs/s320/H-A_numerals.png" width="320" /></a></div>may be a historical echo of the evolution of the symbols, but whether or not that is the case (and frankly I find it fanciful), it is of no significance in terms of their use—the <i>form</i> of the numerals is very much in Gauss’s “unimportant notations” bucket. <br /><br />On the other hand, the huge difference a <i>representation system</i> can make in mathematics is indicated by the revolutionary change in human life that was brought about by the switch from Roman numerals and abacus-board calculation to Hindu-Arabic arithmetic in Thirteenth Century Europe, as I described in my 2011 book <i><a href="https://www.amazon.com/Man-Numbers-Fibonaccis-Arithmetic-Revolution/dp/0802778127/ref=sr_1_1?ie=UTF8&qid=1306990867&sr=8-1" target="_blank">The Man of Numbers</a></i>. <br /><br />Of course, there is a sense in which representations do not matter to mathematics. There is a legitimate way to understand Gauss’s remark as a complete dismissal of how we represent mathematics on a page. The notations we use provide mental gateways to the abstract notions of mathematics that live in our minds. The notions themselves transcend any notations we use to denote them. That may, in fact, have been how Gauss intended his reply to be taken, given the circumstances. <br /><br />But when we shift our attention from mathematics as a body of eternal, abstract structure occupying a Platonic realm, to an activity carried out by people, then it is clear that notations (i.e., a representation system) are important. In the early days of Category Theory, some mathematicians dismissed it as “abstract nonsense” or “mere diagram chasing”, but as most of us discovered when we made a serious attempt to get into the subject, “tracing the arrows” in a commutative diagram can be a powerful way to approach and understand a complex structure. [Google “the snake lemma”. Even better, watch actress Jill Clayburgh <a href="https://www.youtube.com/watch?v=etbcKWEKnvg" target="_blank">explain it</a> to a graduate math class in an early scene from the 1980s movie <i>It’s My Turn</i>.] <br /><br />A well-developed mathematical diagram can also be particularly powerful in trying to understand complex real-world phenomena. In fact, I would argue that the use of mathematical representations as a tool for highlighting hidden abstract structure to help us understand and operate in our world is one of mathematics most significant roles in society, a use that tends to get overlooked, given our present day focus on mathematics as a tool for “getting answers.” Getting an answer is frequently the end of a process of thought; gaining new insight and understanding is the start of a new mental journey. <br /><br />A particularly well known example of such use are the <a href="https://en.wikipedia.org/wiki/Feynman_diagram" target="_blank">Feynmann Diagrams</a>, simple visualizations to help physicists understand the complex behavior of subatomic particles, introduced by the American physicist Richard Feynmann in 1948. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-mOGUeuoP5MA/WWeR0IIEQbI/AAAAAAAAK2g/pzXCt-qIDssO22H9ykXN7mqbf5v3vN_aQCLcBGAs/s1600/Feynmann_Diagram.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="366" data-original-width="574" height="204" src="https://4.bp.blogspot.com/-mOGUeuoP5MA/WWeR0IIEQbI/AAAAAAAAK2g/pzXCt-qIDssO22H9ykXN7mqbf5v3vN_aQCLcBGAs/s320/Feynmann_Diagram.png" width="320" /></a></div><br /><br />A more recent example that has proved useful in linguistics, philosophy, and the social sciences is the “completion diagram” developed by the American mathematician Jon Barwise in collaboration with his philosopher collaborator John Perry in the early 1980s, initially to understand information flow. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-fEXr7O3UZjc/WWeR6upRpsI/AAAAAAAAK2k/oxW8sjjYlYw-C2FCa4s8GHSnpfNjJfJNgCLcBGAs/s1600/CompletionDiagram.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="558" data-original-width="892" height="250" src="https://1.bp.blogspot.com/-fEXr7O3UZjc/WWeR6upRpsI/AAAAAAAAK2k/oxW8sjjYlYw-C2FCa4s8GHSnpfNjJfJNgCLcBGAs/s400/CompletionDiagram.png" width="400" /></a></div><br />A discussion of one use of this diagram can be found in a survey article I wrote in the volume <i>Handbook of the History of Logic</i>, Volume 7, edited by Dov Gabbay and John Woods (Elsevier, 2008, pp.601-664), a <a href="https://web.stanford.edu/~kdevlin/Papers/HHL_SituationTheory.pdf" target="_blank">manuscript version</a> of which can be found on my Stanford homepage. That particular application is essentially the original one for which the diagram was introduced, but the diagram itself turned out be to be applicable in many domains, including improving workplace productivity, intelligence analysis, battlefield command, and mathematics education. (I worked on some of those applications myself; some <a href="https://web.stanford.edu/~kdevlin/papers.html" target="_blank">links to publications</a> are on my homepage.) <br /><br />To be particularly effective, a representation needs to be simple and easy to master. In the case of a representational diagram, like the Commutative Diagrams of Category Theory, the Feynmann Diagram in physics, and the Completion Diagram in social science and information systems development, the representation itself is frequently so simple that it is easy for domain experts to dismiss them as little more than decoration. (For instance, the main critics of Category Theory in its early days were world famous algebraists.) But the mental clarity such diagrams can bring to a complex domain can be highly significant, both for the expert and the learner. <br /><br />In the case of the Completion Diagram, I was a member of the team at Stanford that led the efforts to develop an understanding of information that could be fruitful in the development of information technologies. We had many long discussions about the most effective way to view the domain. That simple looking diagram emerged from a number of attempts (over a great many months) as being the most effective. <br /><br />Given that personal involvement, you would have thought I would be careful not to dismiss a novel representation I thought was too simple and obvious to be important. But no. When you understand something deeply, and have done so for many years, you easily forget how hard it can be for a beginning learner. That’s why, when the MAA’s own James Tanton told me about his “Exploding Dots” idea some months ago, my initial reaction was “That sounds cute," but I did not stop and reflect on what it might mean for early (and not so early) mathematics education. <br /><br />To me, and I assume to any professional mathematician, it sounds like the method simply adds a visual element on paper (or a board) to the mental image of abstract number concepts we already have in our minds. In fact, that is exactly what it does. But that’s the point! “Exploding Dots” does nothing for the expert. But for the learner, it can be huge. It does nothing for the expert because it represents on a page what the expert has in their mind. <i>But that is why it can be so effective in assisting a learner arrive at that level of understanding!</i> All it took to convince me was to watch Tanton’s <a href="https://vimeo.com/204368634" target="_blank">lecture video</a> on Vimeo. Like Tanton, and I suspect almost all other mathematicians, it took me <i>many years of struggle</i> to go beyond the formal symbol manipulation of the classical algorithms of arithmetic (developed to enable people to carry our calculations efficiently and accurately in the days before we had machines to do it for us) until I had created the mental representation that the exploding dots process capture so brilliantly. Many learners subjected to the classical teaching approach never reach that level of understanding; for them, basic arithmetic remains forever a collection of incomprehensible symbolic incantations. <br /><br />Yes, I was right in my original assumption that there is nothing new in exploding dots. But I was also wrong in concluding that there was nothing new. There is no contradiction here. Mathematically, there is nothing new; it’s stuff that goes back to the first centuries of the First Millennium—the underlying idea for place-value arithmetic. Educationally, however, it’s a big deal. A very big deal. Educationally explosive, in fact. Check it out! <br /><br />http://devlinsangle.blogspot.com/2017/07/the-power-of-simple-representations.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-1313249873137848776Wed, 14 Jun 2017 20:53:00 +00002017-06-14T16:53:23.620-04:00Classroom Clickers Are Good; Except When They Are NotMany math instructors use clickers in their larger lecture classes, and can cite numerous studies to show that they lead to more student attention and better learning. A recent research paper on clicker use devotes a page-long introductory section to a review of some of that literature. (<a href="http://www.sciencedirect.com/science/article/pii/S0360131517300726" target="_blank">Shapiro et al, Computers & Education 111 (2017), 44–59</a>) But the paper—by clicker aficionadas, I should stress—is not all good news. In fact, its main new finding is that when clickers are used in what may be the most common way, they actually have a negative effect on student learning. This finding was sufficiently startling that <i>EdSurge</i> put out a <a href="https://www.edsurge.com/news/2017-05-25-study-finds-classroom-response-clickers-can-impede-conceptual-understanding" target="_blank">feature article</a> on the paper on May 25, which is how I learned of the result. <br /><br />The most common (I believe) use of clickers is to provide students with frequent quiz questions to check that they are retaining important facts. (The early MOOCs, including my own, used simple, machine-graded quizzes embedded in the video lectures to achieve the same result.) And a lot of that research I just alluded to showed that the clickers achieve that goal. <br /><br />So too does the latest study. All of which is fine and dandy if the main goal of the course is retention of facts. Where things get messy is when it comes to conceptual understanding of the material—a goal that almost all mathematicians agree is crucial. <br /><br />In the new study, the researchers looked at two versions of a course (physics, not mathematics), one fact-focused, the other more conceptual and problem solving. In each course, they gave one group fact-based clicker questions and a second group clicker questions that concentrated on conceptual understanding in addition to retention of basic facts. <br /><br />As the researchers expected, both kinds of questions resulted in improved performance in fact- based questions on a test administered at the end. <br /><br />Neither kind of question led to improved performance in a problem-based test questions that required conceptual understanding. <br /><br />The researchers expressed surprise that the students who were given the conceptual clicker questions did not show improvement in conceptual questions performance. But that was not the big surprise. That was, wait for it: students who were given only fact-based clicker questions actually performed <b><i>worse</i></b> on conceptual, problem solving questions. <br /><br />To those of us who are by nature heavy on the conceptual understanding, not showing improvement as a result of enforced fact-retention comes as no big surprise. But a negative effect! That’s news. <br /><br />By way of explanation, the researchers suggest that the fact-based clicker questions focus the student’s attention on retention of what are, of course, surface features, and do so <b><i>to the detriment of acquiring the deeper understanding required to solve problems.</i></b><br /><br />If this conclusion is correct—and is certainly seems eminently reasonable—the message is clear. Use clickers, but do so with questions that focus on conceptual understanding, not retention of basic facts. <br /><br />The authors also recommend class discussions of the concepts being tested by the clicker questions, again something that comes natural to we concepts matter folks. <br /><br />I would expect the new finding to have implications for game-based math learning, which regular readers will know is something I have been working on for some years now. The games I have been developing are entirely problem-solving challenges that require deep understanding, and university studies have shown they achieve the goal of better problem-solving skills. (See the <a href="http://devlinsangle.blogspot.com/2015/12/life-inside-impossible-escher-figure.html">December 4, 2015</a> <i>Devlin’s Angle</i> post.) The majority of math learning games, in contrast, focus on retention of basic facts. Based on the new clickers study, I would hypothesize that, even if a game were built on math concepts (many are not), unless the gameplay involves active, problem-solving engagement with those concepts, the result could be, not just no conceptual learning, but a <i><b>drop</b></i> in performance on a problem solving test. <br /><br />Both clickers and video games set up a feedback cycle that can quickly become addictive. With both technologies, regular positive feedback leads to improvement in what the clicker- questions or game-challenges ask for. Potentially more pernicious, however, that positive feedback will result in the students thinking they are doing just fine overall—and hence have no need to wrestle more deeply with the material. And that sets them up for failure once they have to go beneath the surface fact they have retained. Thinking you are winning all the time seduces you to ease off, and as a result is the path to eventual failure. If you want success, the best diet is a series of challenges— that is to say, challenges in coming to grips with the essence of the material to be learned—where you experience some successes, some failures from which you can recover, and the occasional crash-and- burn to prevent over-confidence. <br /><br />That’s not just the secret to learning math. It’s the secret to success in almost any walk of life. <br /><br />http://devlinsangle.blogspot.com/2017/06/classroom-clickers-are-good-except-when.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-6323129377743866240Mon, 22 May 2017 19:19:00 +00002017-06-01T11:34:58.430-04:00gifted and talentedHerman MehtaJo Boalermathematics educationmind setThe Math Gift MythMy May post is more than a little late. The initial delay was caused by a mountain of other deadlines. When I did finally start to come up for air, there just did not seem to be any suitable math stories floating around to riff off, but I did not have enough time to dig around for one. That this has happened so rarely in the twenty years I have been writing Devlin’s Angle (and various other outlets going back to the early 1980s in the UK), that it speaks volumes against the claim you sometimes hear that nothing much happens in the world of mathematics. There is always stuff going on.<br /><br />Be that as it may, when I woke up this morning and went online, two fascinating stories were waiting for me. What’s more, they are connected – at least, that’s how I saw them. <br /><br />First, my Stanford colleague Professor Jo Boaler sent out a group email pointing to a <i>New York Times</i> article that quoted her, and which, she noted, she helped the author to write. Titled "<a href="https://www.nytimes.com/2017/05/15/well/family/trying-to-add-up-girls-and-math.html?_r=1" target="blank’">No Such Thing as a Math Person</a>," it summarizes the consensus among informed math educators that mathematical ability is a spectrum. Just like any other human ability. What is more, the basic math of the K-8 system is well within the capacity of the vast majority of people. Not easy to master, to be sure; but definitely within most people’s ability. It may be defensible to apply terms such as “gifted and talented” to higher mathematics (though I will come back to that momentarily), but basic math is almost entirely a matter of wanting to master it and being willing to put in the effort. People who say otherwise are either (1) education suppliers trying to sell products, (2) children who for whatever reason simply do not want to learn and find it reassuring to convince themselves they just don’t have the gift, or (3) mums and dads who want to use the term as a parental boast or an excuse. <br /><br />Unfortunately, the belief that mathematical ability is a “gift” (that you either have or you don’t) is so well established it is hard to get rid of. Part of the problem is the way it is often taught, as a collection of rules and procedures, rather than a way of thinking (and a very simplistic one at that). Today, this is compounded by the rapid changes in society over the past few decades, that have revolutionized the way mathematics needs to be taught to prepare the new generation for life in today’s – and tomorrow’s – world. (See my January 1 article in <i>The Huffington Post</i>, "<a href="http://www.huffingtonpost.com/entry/all-the-mathematical-methods-i-learned-in-my-university_us_58693ef9e4b014e7c72ee248" target="blank’">All The Mathematical Methods I Learned In My University Math Degree Became Obsolete In My Lifetime</a>," and its follow up article (same date), "<a href="http://www.huffingtonpost.com/entry/number-sense-the-most-important-mathematical-concept_us_58695887e4b068764965c2e0" target="blank’">Number Sense: the most important mathematical concept in 21st Century K-12 education</a>.") <br /><br />With many parents, and not a few teachers, having convinced themselves of the “Math Gift Myth,” attempts over the past several decades to change that mindset have met with considerable resistance. If you have such a mindset, it is easy to see what happens in the educational world around you as confirming it. For instance, one teacher commented on <i>The New York Times</i> article: <br /><br />“Excuse me? I'm a teacher and I refute your assertion. I have seen countless individuals who have problems with math – and some never get it. The same goes for English. But, unless you've spent years in the classroom, it takes years to fully accept that observation. The article's writer is a doctor, not a teacher; accomplishment in one field does not necessarily translate readily to another.” <br /><br />Others were quick to push back against that comment, with one pointing out that her final remark surely argues in favor of everyone in the education world keeping up with the latest scientific research in learning. We are all liable to seek confirmation of our initial biases. And both teachers and parents are in powerful positions to pass on those biases to a new generation of math learners. <br /><br />In her most recent book, <i><a href="https://www.amazon.com/Mathematical-Mindsets-Unleashing-Potential-Innovative/dp/0470894520/ref=sr_1_1?ie=UTF8&qid=1495502636&sr=8-1&keywords=mathematical+mindsets" target="blank’">Mathematical Mindsets: Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching</a></i>, Prof Boaler lays out some of the considerable evidence against the Math Gift Myth, and provides pointers to how to overcome it in the classroom. The sellout audiences Boaler draws for her talks at teachers conferences around the world indicates the hunger there is to provide math learning that does not produce the math-averse, and even math-phobic, citizens we have grown accustomed to. <br /><br />And so to that second story I came across. Hemant Mehta is a former National Board Certified high school math teacher in the suburbs of Chicago, where he taught for seven years, who is arguably best known for his blog <i>The Friendly Atheist</i>. His post on May 22 was titled "<a href="http://www.patheos.com/blogs/friendlyatheist/2017/05/22/years-later-the-mother-who-audited-an-evolution-exhibit-reflects-on-the-viral-response/" target="blank’">Years Later, the Mother Who 'Audited' an Evolution Exhibit Reflects on the Viral Response</a>." Knowing Mehta’s work (for the record, I have also been <a href="http://www.patheos.com/blogs/friendlyatheist/2015/09/20/friendly-atheist-podcast-episode-73-dr-keith-devlin-mathematics-communicator-and-author/" target="blank’">interviewed by him</a> on his education-related podcast), that title hooked me at first glance. I could not resist diving in. <br /><br />As with <i>The New York Times</i> article I led off with, Mehta’s post is brief and to the point, so I won’t attempt to summarize it here. Like Mehta, as an experienced educator I know that it requires real effort, and courage, to take apart ones beliefs and assumptions, when faced with contrary evidence, and then to reason oneself to a new understanding. So I side with him in not in any way trying to diminish the individual who made the two videos he comments on. What we can do, is use her videos to observe how difficult it can be to make that leap from interpreting seemingly nonsensical and mutually contradictory evidence <b><i>from within our (current!) belief system</i></b>, to seeing it from a new viewpoint from which it all makes perfect sense – to rise above the trees to view the forest, if you will. The video lady cannot do that, and assumes no one else can either. <br /><br />Finally, what about my claim that post K-12 mathematics may be beyond the reach of many individuals’ innate capacity for progression along that spectrum I referred to? Of course, it depends on what you mean by “many”. Leaving that aside, however, if someone, for whatever reason, develops a passionate interest in mathematics, how far can they go? I don’t know. Based on a sample size of one, me, we can go further than we think. I look at the achievement of mathematicians such as Andrew Wiles or Terrence Tao and experience the same degree of their being from a different species as the keen-amateur- cyclist-me feels when I see the likes of Tour de France winner Chris Froome or World Champion Peter Sagan climb mountains at twice the speed I can sustain. <br /><br />Yet, on a number of occasions where I failed to solve a mathematics problem I had been working on for months and sometimes years, when someone else did solve it, my first reaction was, “Oh no, I was so close. If only I had tried just a tiny bit harder!” Not always, to be sure. Not infrequently, I was convinced I would never have found the solution. But I got within a hairsbreadth on enough occasions to realize that with more effort I could have done better than I did. (I have the same experience with cycling, but there I do not have a particular desire to aim for the top.) <br /><br />In other words, all my experience in mathematics tells me I do not have an absolute ability limit. Nor, I am sure, do you. Mathematical proficiency is indeed a spectrum. We can all do better – <b><i>if we want to</i></b>. That, surely is the message we educators should be telling our students, be they in the K-8 classroom or the postgraduate seminar room. <br /><br />Gifted and talented? Time to recognize that as an educational equivalent of the Flat Earth Belief. Sure, we are surrounded by seemingly overwhelming daily experience that the world is flat. But it isn’t. And once you accept that, guess what? From a new perspective, you start to see supporting evidence for the Earth being spherical. http://devlinsangle.blogspot.com/2017/05/the-math-gift-myth.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-114325093655230072Wed, 05 Apr 2017 17:24:00 +00002017-04-05T13:25:09.912-04:00Adolf ZeisingFibonacciFibonacci sequencegolden ratioFibonacci and Golden Ratio MadnessThe first reviews of my new book <i><a href="https://www.amazon.com/Finding-Fibonacci-Rediscover-Forgotten-Mathematical/dp/0691174865/" target="_blank">Finding Fibonacci</a></i> have just come out, and I have started doing promotional activities to try to raise awareness. As I expected, one of the first reviews I saw featured a picture of the Nautilus shell (no connection to Fibonacci or the Golden Ratio), and media interviewers have inevitably tried to direct the conversation towards the many fanciful—but for the most part totally bogus—claims about how the Golden Ratio (and hence the Fibonacci sequence) are related to human aesthetics, and can be found in a wide variety of real-world objects besides the Nautilus shell. [Note: the Fibonacci sequence absolutely <i><b>is</b></i> mathematically related to the Golden Ratio. That’s one of the few golden ratio claims that is valid! There is no evidence Fibonacci knew of the connection.] <br /><br />For some reason, once a number has been given names like “Golden Ratio” and “Divine Ratio”, millions of otherwise sane, rational human beings seem willing to accept claims based on no evidence whatsoever, and cling to those beliefs in the face of a steady barrage of contrary evidence going back to 1992, when the University of Maine mathematician George Markovsky published a seventeen- page paper titled "<a href="https://www.goldennumber.net/wp-content/uploads/George-Markowsky-Golden-Ratio-Misconceptions-MAA.pdf" target="_blank">Misconceptions about the Golden Ratio</a>" in the MAA’s <i>College Mathematics Journal</i>, Vol. 23, No. 1 (Jan. 1992), pp. 2-19. <br /><br />In 2003, mathematician, astronomer, and bestselling author Mario Livio weighed in with still more evidence in his excellent book <i><a href="https://www.amazon.com/Golden-Ratio-Worlds-Astonishing-Number/dp/0767908163/ref=sr_1_1?ie=UTF8&amp;qid=1490933233&amp;sr=8-1&amp;keywords=phi+astonishing" target="_blank">The Golden Ratio: The Story ofPHI, the World's Most Astonishing Number</a></i>. <br /><br />I first entered the fray with a Devlin’s Angle post in June 2004 titled "Good Stories Pity They’re Not True" [the MAA archive is not currently accessible], and then again in May 2007 with "The Myth That Will Not Go Away" [ditto]. <br /><br />Those two posts gave rise to a number of articles in which I was quoted, one of the most recent being "<a href="https://www.fastcodesign.com/3044877/the-golden-ratio-designs-biggest-myth" target="_blank">The Golden Ration: Design’s Biggest Myth</a>," by John Brownlee, which appeared in <i>Fast Company</i> <i>Design</i> on April 13, 2015. <br /><br />In 2011, the Museum of Mathematics in New York City invited me to give a public lecture titled "<a href="https://www.youtube.com/watch?v=JuGT1aZkPQ0" target="_blank">Fibonacci and the Golden Ratio Exposed: Common Myths andFascinating Truths</a>," the recording of which was at the time (and I think still is) the most commented-on MoMath lecture video on YouTube, largely due to the many Internet trolls the post attracted—an observation that I find very telling as to the kinds of people who hitch their belief system to one particular ratio that does not quite work out to be 1.6 (or any other rational number for that matter), and for which the majority of instances of those beliefs are supported by not one shred of evidence. (File along with UFOs, Flat Earth, Moon Landing Hoax, Climate Change Denial, and all the rest.)<br /><br />Needless to say, having been at the golden ratio debunking game for many years now, I have learned to expect I’ll have to field questions about it. Even in a media interview about a book that, not only flatly refutes all the fanciful stuff, but lays out the history showing that the medieval mathematician known today as Fibonacci left no evidence he had the slightest interest in the sequence now named after him, nor had any idea it had several cute properties. Rather, he simply included among the hundreds of arithmetic problems in his seminal book <i>Liber abbaci</i>, published in 1202, an ancient one about a fictitious rabbit population, the solution of which is that sequence.<br /><br />What I have always found intriguing is the question, how did this urban legend begin? It turns out to be a relatively recent phenomenon. The culprit is a German psychologist and author called <a href="https://en.wikipedia.org/wiki/Adolf_Zeising" target="_blank">Adolf Zeising</a>. In 1855, he published a book titled: <i>A New Theory of the proportions of the human body, developed from a basic morphological law which stayed hitherto unknown, and which permeates the whole nature and art, accompanied by a complete summary of the prevailing systems.</i><br /><br />This book, which today would likely be classified as “New Age,” is where the claim first appears that the proportions of the human body are based on the Golden Ratio. For example, taking the height from a person's naval to their toes and dividing it by the person's total height yields the Golden Ratio. So, he claims, does dividing height of the face by its width. <br /><br />From here Zeising leaped to make a connection between these human-centered proportions and ancient and Renaissance architecture. Not such an unreasonable jump, perhaps, but it was, and is pure speculation. After Zeising, the Golden Ratio Thing just took off. <br /><br />Enough! I can’t bring myself to continue. I need a stiff drink. <br /><br />For more on Zeising and the whole wretched story he initiated, see the article by writer Julia Calderone in business Insider, October 5, 2015, "<a href="http://www.businessinsider.com/the-golden-ratio-fibonacci-numbers-mathematics-zeising-beauty-2015-9" target="_blank">The one formula that's supposed to 'prove beauty' is fundamentally wrong</a>."<br /><br />See also the <a href="https://misfitsarchitecture.com/tag/adolf-zeising/" target="_blank">blogpost on Zeising</a> on the blog <a href="https://misfitsarchitecture.com/" target="_blank"><i>misfits’ architecture</i></a>, which presents an array of some of the battiest claims about the Golden Ratio. <br /><br />That’s it. I’m done. <br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />http://devlinsangle.blogspot.com/2017/04/fibonacci-and-golden-ratio-madness.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-4731062327114722175Wed, 08 Mar 2017 16:48:00 +00002017-03-08T11:51:20.847-05:00FibonacciLeonardo of Pisamathematics outreachFinding Fibonacci<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-9rmd71_ydyU/WL8io9bBagI/AAAAAAAAKyg/mGPU2s-sj_oocaNcZU-ZES4uz9vPKgrkgCLcB/s1600/Keith_Leonardo_statue.jpeg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://3.bp.blogspot.com/-9rmd71_ydyU/WL8io9bBagI/AAAAAAAAKyg/mGPU2s-sj_oocaNcZU-ZES4uz9vPKgrkgCLcB/s320/Keith_Leonardo_statue.jpeg" width="240" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Devlin makes a pilgrimage to Pisa to see the<br />statue of Leonardo Fibonacci in 2002.</td></tr></tbody></table>In 1983, I did something that would turn out to have a significant influence on the direction my career would take. Frustrated by the lack of coverage of mathematics in the weekly science section of my newspaper of choice, <i>The Guardian</i>, I wrote a short article about mathematics and sent it to the science editor. A few days later, the editor phoned me to explain why he could not to publish it. “But,” he said, “I like your style. You seem to have a real knack for explaining difficult ideas in a way ordinary people can understand.” He encouraged me to try again, and my second attempt was published in the newspaper on May 12, 1983. Several more pieces also made it into print over the next few months, eliciting some appreciative letters to the editor. As a result, when <i>The Guardian</i> launched a weekly, personal computing page later that year, it included my new, twice-monthly column MicroMaths. The column ran without interruption until 1989, when my two-year visit to Stanford University in California turned into a permanent move to the US.<br /><br />Before long, a major publisher contracted me to publish a collection of my MicroMaths articles, which I did, and following that Penguin asked me to write a more substantial book on mathematics for a general audience. That book, <a href="https://www.amazon.com/Mathematics-New-Golden-Keith-Devlin/dp/023111639X/ref=asap_bc?ie=UTF8" target="_blank"><i>Mathematics: The NewGolden Age</i></a>, was first published in 1987, the year I moved to America. <br /><br />In addition to writing for a general audience, I began to give lectures to lay audiences, and started to make occasional appearances on radio and television. From 1991 to 1997, I edited <i>MAA FOCUS</i>, the monthly magazine of the Mathematical Association of America, and since January 1996 I have written this monthly <i>Devlin’s Angle </i>column. In 1994, I also became the <i>NPR </i>Math Guy, as I describe in my latest article in the <i><a href="http://www.huffingtonpost.com/entry/how-i-became-the-npr-math-guy_us_58bb4169e4b0fa65b844b419" target="_blank">Huffington Post</a></i>. <br /><br />Each new step I took into the world of “science outreach” brought me further pleasure, as more and more people came up to me after a talk or wrote or emailed me after reading an article I had written or hearing me on the radio. They would tell me they found my words inspiring, challenging, thought-provoking, or enjoyable. Parents, teachers, housewives, business people, and retired people would thank me for awakening in them an interest and a new appreciation of a subject they had long ago given up as being either dull and boring or else beyond their understanding. I came to realize that I was touching people’s lives, opening their eyes to the marvelous world of mathematics. <br /><br />None of this was planned. I had become a “mathematics expositor” by accident. Only after I realized I had been born with a talent that others appreciated—and which by all appearances is fairly rare—did I start to work on developing and improving my “gift.”<br /><br />In taking mathematical ideas developed by others and explaining them in a way that the layperson can understand, I was following in the footsteps of others who had also made efforts to organize and communicate mathematical ideas to people outside the discipline. Among that very tiny subgroup of mathematics communicators, the two who I regarded as the greatest and most influential mathematical expositors of all time are Euclid and Leonardo Fibonacci. Each wrote a mammoth book that influenced the way mathematics developed, and with it society as a whole. <br /><br />Euclid’s classic work <i>Elements</i> presented ancient Greek geometry and number theory in such a well-organized and understandable way that even today some instructors use it as a textbook. It is not known if any of the results or proofs Euclid describes in the book are his, although it is reasonable to assume that some are, maybe even many. What makes <i>Elements</i> such a great and hugely influential work, however, is the way Euclid organized and presented the material. He made such a good job of it that his text has formed the basis of school geometry teaching ever since. Present day high school geometry texts still follow <i>Elements </i>fairly closely, and translations of the original remain in print. <br /><br />With geometry being an obligatory part of the school mathematics curriculum until a few years ago, most people have been exposed to Euclid’s teaching during their childhood, and many recognize his name and that of his great book. In contrast, Leonardo of Pisa (aka Fibonnaci) and his book <i>Liber abbaci</i> are much less well known. Yet their impact on present-day life is far greater. <i>Liber abbaci</i> was the first comprehensive book on modern practical arithmetic in the western world. While few of us ever use geometry, people all over the world make daily use of the methods of arithmetic that Leonardo described in<i> Liber abbaci</i>. <br /><br />In contrast to the widespread availability of the original Euclid’s <i>Elements</i>, the only version of Leonardo’s <i>Liber abbaci</i> we can read today is a second edition he completed in 1228, not his original 1202 text. Moreover, there is just one translation from the original Latin, in English, published as recently as 2002. <br /><br />But for all its rarity, <i>Liber abbaci </i>is an impressive work. Although its great fame rests on its treatment of Hindu-Arabic arithmetic, it is a mathematically solid book that covers not just arithmetic, but the beginnings of algebra and some applied mathematics, all firmly based on the theoretical foundations of Euclid’s mathematics. <br /><br />After completing the first edition of <i>Liber abbaci</i>, Leonardo wrote several other mathematics books, his writing making him something of a celebrity throughout Italy—on one occasion he was summonsed to an audience with the Emperor Frederick II. Yet very little was written about his life. <br /><br />In 2001, I decided to embark on a quest to try to collect together what little was known about him and bring his story to a wider audience. My motivation? I saw in Leonardo someone who, like me, devoted a lot of time and effort trying to make the mathematics of the day accessible to the world at large. (Known today as “mathematical outreach,” very few mathematicians engage in that activity.) He was the giant whose footsteps I had been following. <br /><br />I was not at all sure I could succeed. Over the years, I had built up a good reputation as an expositor of mathematics, but a book on Leonardo would be something new. I would have to become something of an archival scholar, trying to make sense of Thirteenth Century Latin manuscripts. I was definitely stepping outside my comfort zone. <br /><br />The dearth of hard information about Leonardo in the historical record meant that a traditional biography was impossible—which is probably why no medieval historian had written one. To tell my story, I would have to rely heavily on the <i>mathematical </i>thread that connects today’s world to that of Leonardo—an approach unique to mathematics, made possible by the timeless nature of the discipline. Even so, it would be a stretch. <br /><br />In the end, I got lucky. Very lucky. And not just once, but several times. As a result of all that good fortune, when my historical account <i><a href="https://www.amazon.com/Man-Numbers-Fibonaccis-Arithmetic-Revolution/dp/0802778127/ref=sr_1_1?ie=UTF8&amp;qid=1306990867&amp;sr=8-1" target="_blank">The Man of Numbers: Fibonacci’s Arithmetic Revolution</a></i> was published in 2011, I was able to compensate for the unavoidable paucity of information about Leonardo’s life with the first-ever account of the seminal discovery showing that my medieval role-model expositor had indeed played the pivotal role in creating the modern world that most historians had hypothesized. <br /><br />With my Leonardo project such a new and unfamiliar genre, I decided from the start to keep a diary of my progress. Not just my findings, but also my experiences, the project's highs and lows, the false starts and disappointments, the tragedies and unexpected turns, the immense thrill of holding in my hands seminal manuscripts written in the thirteenth and fourteenth centuries, and one or two truly hilarious episodes. I also encountered, and made diary entries capturing my interactions with, a number of remarkable individuals who, each for their own reasons, had become fascinated by Fibonacci—the Yale professor who traced modern finance back to Fibonacci, the Italian historian who made the crucial archival discovery that brought together all the threads of Fibonacci's astonishing story, and the remarkable widow of the man who died shortly after completing the world’s first, and only, modern language translation of<i> Liber abbaci</i>, who went to heroic lengths to rescue his manuscript and see it safely into print. <br /><br />After I had finished the <i>Man of Numbers</i>, I decided that one day I would take my diary and turn it into a book, telling the story of that small group of people (myself included) who had turned an interest in Leonardo into a passion, and worked long and hard to ensure that Leonardo Fibonacci of Pisa will forever be regarded as among the very greatest people to have ever lived. Just as <i>The Man of Numbers</i> was an account of the writing of <i>Liber abbaci</i>, so too <i>Finding Fibonacci</i> is an account of the writing of <i>The Man of Numbers</i>. [So it is a book about a book about a book. As Andrew Wiles once famously said, “I’ll stop there.”] <br /><br /><i>This post is adapted from the introduction of Keith Devlin’s new book</i> <a href="https://www.amazon.com/Finding-Fibonacci-Rediscover-Forgotten-Mathematical/dp/0691174865/"><i>Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World</i></a><i>, published this month by Princeton University Press.</i>http://devlinsangle.blogspot.com/2017/03/finding-fibonacci.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-8233984482849369276Wed, 08 Feb 2017 14:01:00 +00002017-02-08T09:01:01.742-05:00Hans Rosling, July 27, 1948 – February 7, 2017<div style="text-align: center;"><span style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 12.8px;"><a href="https://www.youtube.com/watch?v=RUwS1uAdUcI" target="_blank">The power of numbers to help us understand our world.</a></span></div><div style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/RUwS1uAdUcI/0.jpg" src="https://www.youtube.com/embed/RUwS1uAdUcI?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div><div style="text-align: center;"><br /></div>http://devlinsangle.blogspot.com/2017/02/hans-rosling-july-27-1948-february-7.htmlnoreply@blogger.com (Mathematical Association of America)1tag:blogger.com,1999:blog-2516188730140164076.post-4240772258053024859Fri, 06 Jan 2017 05:02:00 +00002017-01-09T15:56:49.437-05:00analytic continuationanalytic functionscomplex numbersGrant Sandersoninfinite seriesinfinite sumsmathematics visualizationNumberphileRiemann’s HypothesisZeta functionSo THAT’s what it means? Visualizing the Riemann HypothesisTwo years ago, there was a sudden, viral spike in online discussion of the Ramanujan identity <br /><br />1 + 2 + 3 + 4 + 5 + . . . = –1/12 <br /><br />This identity had been lying around in the mathematical literature since the famous Indian mathematician Srinivasa Ramanujan included it in one of his books in the early Twentieth Century, a curiosity to be tossed out to undergraduate mathematics students in their first course on complex analysis (which was my first exposure to it), and apparently a result that physicists made actual (and reliable) use of. <br /><br />The sudden explosion of interest was the result of a <a href="https://www.youtube.com/watch?v=w-I6XTVZXww" target="_blank">video</a> posted online by Australian video journalist Brady Haran on his excellent <a href="https://www.youtube.com/user/numberphile" target="_blank">Numberphile</a> YouTube channel. In it, British mathematician and mathematical outreach activist James Grime moderates as his physicist countrymen Tony Padilla and Ed Copeland of the University of Nottingham explain their “physicists’ proof” of the identity. <br /><br />In the video, Padilla and Copeland manipulate infinite series with the gay abandon physicists are wont to do (their intuitions about physics tends to keep them out of trouble), eventually coming up with the sum of the natural numbers on the left of the equality sign and –1/12 on the right. <br /><br />Euler was good at doing that kind of thing too, so mathematicians are hesitant to trash it, rather noting that it “lacks rigor” and warning that it would be dangerous in the hands of a lesser mortal than Euler. <br /><br />In any event, when it went live on January 9, 2014, the video and the result (which to most people was new) exploded into the mathematically-curious public consciousness, rapidly garnering hundreds of thousands of hits. (It is currently approaching 5 million in total.) By February 3, interest was high enough for <i>The New York Times</i> to run a <a href="https://www.nytimes.com/2014/02/04/science/in-the-end-it-all-adds-up-to.html?_r=0" target="_blank">substantial story</a> about the “result”, taking advantage of the presence in town of Berkeley mathematician Ed Frenkel, who was there to promote his new book <i>Love and Math</i>, to fill in the details. <br /><br />Before long, mathematicians whose careers depended on the powerful mathematical technique known as <i>analytic continuation</i> were weighing in, castigating the two Nottingham academics for misleading the public with their symbolic sleight-of- hand, and trying to set the record straight. One of the best of those corrective attempts was another <a href="https://www.youtube.com/watch?v=0Oazb7IWzbA" target="_blank">Numberphile video</a>, published on March 18, 2014, in which Frenkel give a superb summary of what is really going on. <br /><br />A year after the initial flair-up, on January 11, 2015, Haran published a <a href="http://www.bradyharanblog.com/blog/2015/1/11/this-blog-probably-wont-help" target="_blank">blogpost</a> summarizing the entire episode, with hyperlinks to the main posts. It was quite a story.<br /><br />[[ASIDE: The next few paragraphs may become a bit too much for casual readers, but my discussion culminates with a link to a really cool video, so keep going. Of course, you could just jump straight to the video, now you know it’s coming, but without some preparation, you will soon get lost in that as well! The video is my reason for writing this essay.]] <br /><br />For readers unfamiliar with the mathematical background to what does, on the face of it, seem like a completely nonsensical result, which is the MAA audience I am aiming this essay at (principally, undergraduate readers and those not steeped in university-level math), it should be said that, as expressed, Ramanujan’s identity is nonsense. But not because of the -1/12 on the right of the equals sign. Rather, the issue lies in those three dots on the left. Not even a mathematician can add up infinitely many numbers. <br /><br />What you can do is, under certain circumstances, assign a meaning to an expression such as <br /><br />X<sub>1</sub> + X<sub>2</sub> + X<sub>3</sub> + X<sub>4</sub> + … <br /><br />where the X<sub>N</sub> are numbers and the dots indicate that the pattern continues for ever. Such expressions are called <i>infinite series</i>. <br /><br />For instance, undergraduate mathematics students (and many high school students) learn that, provided X is a real number whose absolute value is less than 1, the infinite series <br /><br />1 + X + X<sup>2 </sup>+ X<sup>3</sup> + X<sup>4 </sup>+ … <br /><br />can be assigned the value 1/(1 – X). Yes, I meant to write “can be assigned”. Since the rules of real arithmetic do not extend to the vague notion of an “infinite sum”, this has to be defined. Since we are into the realm of definition here, in a sense you can define it to be whatever you want. But if you want the result to be meaningful and useful (useful in, say, engineering or physics, to say nothing of the rest of mathematics), you had better define it in a way that is consistent with that “rest of mathematics.” In this case, you have only one option for your definition. A simple mathematical argument (but not the one you can find all over the web that involves multiplying the terms in the series by X, shifting along, and subtracting—the rigorous argument is a bit more complicated than that, and a whole lot deeper conceptually) shows that the value has to be 1/(1 – X). <br /><br />So now we have the identity <br /><br />(*) 1 + X +X<sup>2 </sup>+ X<sup>3</sup> + X<sup>4 </sup>+ … = 1/(1 – X) <br /><br />which is valid (by definition) whenever X has absolute value less than 1. (That absolute value requirement comes in because of that “bit more complicated” aspect of the rigorous argument to derive the identity that I just mentioned.) <br /><br />“What happens if you put in a value of X that does not have absolute value less than 1?” you might ask. Clearly, you cannot put X = 1, since then the right-hand side becomes 1/0, which is totally and absolutely forbidden (except when it isn’t, which happens a lot in physics). But apart from that one case, it is a fair question. For instance, if you put X = 2, the identity (*) becomes <br /><br />1 + 2 + 4 + 8 + 16 + … = 1/(1 – 2) = 1/(–1) = –1 <br /><br />So you could, if you wanted, make the identity (*) the definition for what the infinite sum <br /><br />1 + X + X<sup>2 </sup>+ X<sup>3</sup> + X<sup>4</sup> + … <br /><br />means for any X other than X = 1. Your definition would be consistent with the value you get whenever you use the rigorous argument to compute the value of the infinite series for any X with absolute value less than 1, but would have the “benefit” of being defined for all values of X apart from one, let us call it a “pole”, at X = 1. <br /><br />This is the idea of analytic continuation, the concept that lies behind Ramanujan’s identity. But to get that concept, you need to go from the real numbers to the complex numbers. <br /><br />In particular, there is a fundamental theorem about differentiable functions (the accurate term in this context is <i>analytic functions</i>) of a single complex variable that says that if any such function has value zero everywhere on a nonempty disk in the complex plane, no matter how small the diameter of that disk, then the function is zero everywhere. In other words, there can be no smooth “hills” sitting in the middle of flat plains, or even one small flat clearing in the middle of a “hilly” landscape—the quotes are because we are beyond simple visualization here. <br /><br />An immediate consequence of this theorem is that if you pull the same continuation stunt as I just did for the series of integer powers, where I extended the valid formula (*) for the sum when X is in the open unit interval to the entire real line apart from one pole at 1, but this time do it for analytic functions of a complex variable, then if you get an answer at all (i.e., a formula), <i><b>it will be unique</b></i>. (Well, no, the formula you get need not be unique, rather the function it describes will be.) <br /><br />In other words, if you can find a formula that describes how to compute the values of a certain expression for a disk of complex numbers (the equivalent of an interval of the real line), and if you can find another formula that works for all complex numbers and agrees with your original formula on that disk, then your new formula tells you <i><b>the</b></i> right way to calculate your function for any complex number. All this subject to the requirement that the functions have to be analytic. Hence the term “<b><i>analytic</i></b> continuation.' <br /><br />For a bit more detail on this, check out the <a href="https://en.wikipedia.org/wiki/Analytic_continuation" target="_blank">Wikipedia explanation</a> or the one on <a href="http://mathworld.wolfram.com/AnalyticContinuation.html" target="_blank">Wolfram Mathworld</a>. If you find those explanations are beyond you right now, just remember that this is not magic and it is not a mystery. It is mathematics. The thing you need to bear in mind is that the complex numbers are very, very regular. Their two-dimensional structure ties everything down as far as analytic functions are concerned. This is why results about the integers such as Fermat’s Last Theorem are frequently solved using methods of Analytic Number Theory, which views the integers as just special kinds of complex numbers, and makes use of the techniques of complex analysis. <br /><br />Now we are coming to that video. When I was a student, way, way back in the 1960s, my knowledge of analytic continuation followed the general path I just outlined. I was able to follow all the technical steps, and I convinced myself the results were true. But I never was able to visualize, in any remotely useful sense, what was going on. <br /><br />In particular, when our class came to study the (famous) <a href="https://en.wikipedia.org/wiki/Riemann_zeta_function" target="_blank">Riemann zeta function</a>, which begins with the following definition for real numbers S bigger than 1: <br /><br />(**) Zeta(S) = 1 + 1/2<sup>S</sup> + 1/3<sup>S</sup> + 1/4<sup>S</sup> + 1/5<sup>S</sup> + … <br /><br />I had no reliable mental image to help me understand what was going on. For integers S greater than 1, I knew what the series meant, I knew that it summed (converged) to a finite answer, and I could follow the computation of some answers, such as Euler’s <br /><br />Zeta(2) = π<sup>2</sup>/6 <br /><br />(You get another expression involving π for S = 4, namely π<sup>4</sup>/90.) <br /><br />It turns out that the above definition (**) will give you an analytic function if you plug in any complex number for S for which the real part is bigger than 1. That means you have an analytic function that is rigorously defined everywhere on the complex plane to the right of the line x = 1. <br /><br />By some deft manipulation of formulas, it’s possible to come up with an analytic continuation of the function defined above to one defined for all complex numbers except for a pole at S = 1. By that basic fact I mentioned above, that continuation is unique. Any value it gives you can be taken as <i><b>the right answer</b></i>. <br /><br />In particular, if you plug in S = –1, you get <br /><br />Zeta(–1) = –1/12 <br /><br />That equation is totally rigorous, meaningful, and accurate. <br /><br />Now comes the tempting, but wrong, part that is not rigorous. If you plug in S = –1 in the original infinite series, you get <br /><br />1 + 1/2<sup>-1</sup> + 1/3<sup>-1</sup> + 1/4<sup>-1</sup> + 1/5<sup>-1</sup> + … <br /><br />which is just <br /><br />1 + 2 + 3 + 4 + 5 + … <br /><br />and it seems you have shown that <br /><br />1 + 2 + 3 + 4 + 5 + . . . = –1/12 <br /><br />The point is, though, you can’t plug S = –1 into that infinite series formula (**). That formula is not valid (i.e., it has no meaning) unless S > 1. <br /><br />So the only way to interpret Ramanujan’s identity is to say that there is a unique analytic function, Zeta(S), defined on the complex plane (apart from at the real number 1), which for all real numbers S greater than 1 has the same values as the infinite series (**), which for S = –1 gives the value Zeta(–1) = –1/12. <br /><br />Or, to put it another way, more fanciful but less accurate, if the sum of all the natural numbers were to suddenly find it had a finite answer, <i><b>that answer could only be</b></i> –1/12. <br /><br />As I said, when I learned all this stuff, I had no good mental images. But now, thanks to modern technology, and the creative talent of a young (recent) Stanford mathematics graduate called <a href="http://www.3blue1brown.com/" target="_blank">Grant Sanderson</a>, I can finally see what for most of my career has been opaque. On December 9, he uploaded <a href="https://www.youtube.com/watch?v=sD0NjbwqlYw" target="_blank">this video</a> onto YouTube.<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/sD0NjbwqlYw/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/sD0NjbwqlYw?feature=player_embedded" width="320"></iframe></div><br /><br />It is one of the most remarkable mathematics videos I have ever seen. Had it been available in the 1960s, my undergraduate experience in my complex analysis class would have been so much richer for it. Not easier, of that I am certain. But things that seemed so mysterious to me would have been far clearer. Not least, I would not have been so frustrated at being unable to understand how Riemann, based on hardly any numerical data, was able to formulate his famous hypothesis, finding a proof of which is agreed by most professional mathematicians to be <i><b>the</b></i> most important unsolved problem in the field. <br /><br />When you see (in the video) what looks awfully like a gravitational field, pulling the zeros of the Zeta function towards the line x = 1/2, and you know that it is the only such gravitational field there is, and recognize its symmetry, you have to conclude that the universe could not tolerate anything other than all the zeros being on that line. <br /><br />Having said that, it would, however, be <i><b>really</b></i> interesting if that turned out not to be the case. Nothing is certain in mathematics until we have a rigorous proof. <br /><br />Meanwhile, do check out some of Grant’s other videos. There are some real gems! http://devlinsangle.blogspot.com/2017/01/so-thats-what-it-means-visualizing.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-2042228397204354363Tue, 13 Dec 2016 21:14:00 +00002016-12-14T11:35:37.149-05:00You can find the secret to doing mathematics in a tubeless bicycle tire<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-A9Nsf6YrkN8/WFBkEN1G73I/AAAAAAAAKuQ/_o6sZqb8xq4jK3E0galcpn1N3Y3YglUKACLcB/s1600/CountryView.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="300" src="https://2.bp.blogspot.com/-A9Nsf6YrkN8/WFBkEN1G73I/AAAAAAAAKuQ/_o6sZqb8xq4jK3E0galcpn1N3Y3YglUKACLcB/s400/CountryView.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">The author climbing the locally-notorious Country View Road just south of San Jose, CA</td></tr></tbody></table><br />As regular readers may know, one of my consuming passions in life besides mathematics is cycling. Living in California, where serious winters were wisely banned many years ago, on any weekend throughout the year you are likely to find me out on a road- or a mountain bike. <br /><br />Being also a lover of well-designed technology, I long ago switched to using tubeless tires on my road bike. Actually, it’s bikes, in the plural—my road bikes number four, all with different riding conditions in mind, but all having in common the same kind of ultra- narrow saddle that non-cyclists think must be excruciatingly painful, but is in fact engineered to be the only thing comfortable enough to sit on for many hours at a stretch. [Keep going; I am working my way to making a mathematical point. In fact, I am heading towards THE most significant mathematical point of all: What is the secret to doing math?] <br /><br />Road tubeless tires have several advantages over the more common type of tire, which requires an airtight innertube. One advantage is that you need inflate them only to 80 pounds per square inch, as opposed to the 110 psi or more for a tubed tire, which provides even more comfort over those many hours in the saddle.<br /><br />You need tire pressures 3 or more times that of a car tire because of the extremely low volume in a road-bike tire, which sits on a 700 cm diameter wheel with a rim whose width is between 21 mm and 25 mm. It is that high pressure that made the manufacture of tubeless wheels and tires for bicycles such a significant challenge. How can you ensure an almost totally airtight fit when the tire is inflated, and it still be possible for an average person to remove and mount a deflated tire with their bare hands. (Tire levers can easily damage tubeless wheels and tires.) We are almost to the secret to doing math. Hang in there. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-UEpD5QbsJ5o/WFBkSqYwhiI/AAAAAAAAKuU/W8n8OrKbUgo8KU-zr_Cg1W5PZhqNCJjigCLcB/s1600/Tubeless_Rims.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="152" src="https://3.bp.blogspot.com/-UEpD5QbsJ5o/WFBkSqYwhiI/AAAAAAAAKuU/W8n8OrKbUgo8KU-zr_Cg1W5PZhqNCJjigCLcB/s400/Tubeless_Rims.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Clever design: Tubeless rims and tires on a road bike wheel.</td></tr></tbody></table><br />The airtight fit is possible precisely because of that relatively high pressure inside the tire—80 psi is over five times the air pressure outside the tire. (An automobile tire is inflated to roughly twice atmospheric pressure, much lower.) The cross-sectional photo on the left shows how a tubeless tire has a squared-off ridge that fits into a matching notch in the rim. The more air pressure there is in the tire, the tighter that ridge binds to the rim, increasing the air seal. <br /><br />The problem is, as I mentioned, getting the tire on and off the rim. The tire ridge that fits into the rim-notch has a steel wire running through it, and its squared-off shape is designed to make it difficult for the tire to separate from the rim—that, after all, is the point. To solve the mounting/removal problem, the wheel has a channel in the middle, as shown more clearly in the photo on the right. <br /><br />To mount the tire, you push the two tire-rims into that channel, one after the other. By the formula for the circumference of a circle, when a tire rim is in that center channel, you have just over 3 times the depth of the channel of superfluous tire length to play with, roughly 12mm of tire looseness. The idea is to use that “looseness” to work your way around the wheel, pushing (actually rolling) first one tire edge over the wheel rim and into the channel, then the other. Once the tire is seated on the rim, inflating it with a hand pump forces the tire rims out of the channel into the notches. To remove the tire after it is deflated, you push the two tire rims into the channel and reverse the process. <br /><br />That, at least, is the theory. Putting theory into practice turns out to be quite a challenge. When I first started to use road tubeless tires, several years ago, I read several online manuals and watched a number of YouTube videos demonstrating how to do it,<i><b> and could never do it</b></i>. I usually ended up taking the wheel and tire to my local bike shop, where the mechanic would do it for me with seeming ease before my eyes. “Fifteen dollars, please.” <br /><br />But what would happen if I had a flat on one of the remote rides I regularly do in the mountains that surround Silicon Valley, where I live? One major advantage of tubeless tires is that, even if they puncture, usually the air leaks out only very slowly, and can generally be stopped by inflating the tire from a small pressure-can of air and liquid latex you carry in your back pocket, which seals the hole. Which is how I was always able to get to a bike shop where someone else could solve the problem for me. But a major puncture in the remote, with no cell phone access, could leave me dangerously stranded. Clearly, I had to learn how to do it myself. <br /><br />From now on, when I say “change a tubeless tire”, you can interpret it as “do mathematics”. The secret is coming up. <i>Moreover, it is coming with a moral that those of us in mathematics education ignore at our students’ peril</i>. <br /><br />What I find cool is that, for me I somehow stumbled on the secret to doing math fairly early in life, before math had become such a problem that I felt I could never do it. But taking up cycling later in life, when I had a fully developed set of metacognitive skills, I approached the problem of changing a tubeless tire in much the same way as many people—including, I suspect, the mechanics in my local bicycle shop—see math. Namely, people like me (and that smart kid sitting in the front row in the school math class) make doing math look effortless, but many people feel they could never master it in a million years. <br /><br />Nothing, surely, can look less requiring of skill or expertise than putting a tire on a bicycle wheel. (This is why I think this is such a great example.) Surely, you just need to read an instruction manual, or perhaps have someone demonstrate to you. But no matter how many times I read the instructions, no matter how many times I viewed—and re-viewed—those how-to YouTube videos, and no matter how many times I stood alongside the bike shop mechanic and watched as he quickly and effortlessly put the tire onto the wheel, I could never do it. <br /><br />Just think about that for a moment. For some tasks, <b><i>instruction (on its own) just does not work</i></b>. Not even for the seemingly simple task of changing a bicycle tire. And yet we think that forcing kids to sit in the math class while we force-feed <b><i>instruction</i></b> will result in their being able to do math! Dream on. <br /><br />What does work, in fact what is absolutely necessary, both for changing tubeless tires and doing math, is that the learner has to learn to <b><i>see things the way the expert does</i></b>. And, since instruction does not work, that key step has to be made by the learner. All that a good teacher can do, then, is find a way to help the learner make that key leap. [That short initial word “all” belies the human expertise required to do this.] <br /><br />Clearly, when I was, yet again, standing in the bicycle repair shop, watching the mechanic change my tire, what he was doing—more precisely, what he was <i><b>experiencing</b></i>—was very different from what I was doing and experiencing when I tried and failed. What was I not getting? <br /><br />My big breakthrough finally came the one time when the mechanic, holding the wheel horizontally pressed to his stomach, while manipulating the tire with both hands, told me what he was <b><i>really</i></b> doing. “You have to think of the tire as alive,” he said. “It wants to be sitting firmly on the rim” [that, after all, is what it was—expensively—designed for], “but it is not very disciplined. It’s like a small child. It moves around and resists your attempts to force it. You have to understand it, and be aware, through your hands, of what it is doing. Work with it—be constantly aware of what it is trying to do—so you both get what you want: the tire gets onto the wheel, where it belongs, and you can inflate it and get back on your bike (where you want to be).” <br /><br />Fanciful? Maybe. But it worked. And it continues to work. As a result, not only can I now change my tubeless tires, it has for me become “mindless and automatic,” as effortless (to me) as Picasso drawing a simple doodle on a restaurant napkin to pay the bill for his meal was to him. (I thought that if you got this far, you deserved a second example with greater cultural overtones.) <br /><br />It took many years for Picasso to learn to draw the way he did (and for the marketplace to assign high value to his work), but that does not mean his work was not skillful; rather, he simply routinized part of it. When I watch a film of him at work, I see superficially how he created, and it looks routine and effortless, but I do not see his canvas as he did, and I could not draw as he did. <br /><br />Likewise, my skill in fitting a tubeless tire, now effortless and automatic, is a result of my now seeing and understanding what earlier had been opaque. <br /><br />I admit that it is far easier to learn to mount a tubeless tire on a road bike wheel than to draw like Picasso. But I am less sure the difference is so great between changing a tubeless tire and being able to solve any one particular kind of math problem. Still, no matter how great the difference in the degree of skill required, it is possible to learn from the analogy. <br /><br />Given what I have said here, will reading this essay mean you can go out and immediately be able to change a tubeless tire? Have I just made a case for instruction working after all? It’s possible—for changing bicycle tires, but surely not for painting like Picasso. Instruction can and does work, and it is an important part of learning. But my guess is you would find my words are not enough. I think that the reason that one piece of bike-shop instruction was so instantly transformative for me was that I had spent an aggregate of many hours struggling to change my tire and failing. I had reached a stage where the effective key was <b><i>to get inside the mind of an expert</i></b>. But the ground had to be prepared for that simple revelation to work. <br /><br />In education, as in so many parts of life, there are no silver bullets. But given enough of the right preparation—enough experience acquired through repeated trying and failing—an ordinary lead bullet will do the job. <br /><br />---- <br /><br />This month’s column is loosely adapted from a passage of my forthcoming book <a href="https://www.amazon.com/Finding-Fibonacci-Rediscover-Forgotten-Mathematical/dp/0691174865/"><i>Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World</i></a>, due out in March. <br /><br />http://devlinsangle.blogspot.com/2016/12/you-can-find-secret-to-doing.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-8966259372490164051Fri, 04 Nov 2016 14:01:00 +00002016-11-04T13:51:18.978-04:00Downs paradoxelection mathgolden ratioKool Aidmathematical modelingmilkparadox of votingMathematical Milk and the U.S. Presidential Election<div style="text-align: right;"></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-9BWK4dkWBOA/WBzGZEjPp-I/AAAAAAAAKsU/AvrdcRR9-1YXKMNWmQ3r2AUFLYOIrMkmACLcB/s1600/Devlin_voting.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="300" src="https://1.bp.blogspot.com/-9BWK4dkWBOA/WBzGZEjPp-I/AAAAAAAAKsU/AvrdcRR9-1YXKMNWmQ3r2AUFLYOIrMkmACLcB/s400/Devlin_voting.JPG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Keith Devlin mails his completed election ballot. What does math have to say about his act?</td></tr></tbody></table>With the United States is the final throes of a presidential election, my mind naturally turned to the decidedly tricky matter of election math. Voting provides a great illustration of how mathematics – which rules supreme, yielding accurate and reliable answers to precise questions, in the natural sciences and engineering – can lead us astray when we try to apply it to human and social activities.<br /><br />A classic example is how we count votes in an election, the topic of an earlier <i><a href="https://www.maa.org/external_archive/devlin/devlin_11_00.html" target="_blank">Devlin’sAngle</a> </i>post, in November, 2000. In that essay, I looked at how different ways to tally votes could affect the imminent Bush v. Gore election, at the time blissfully unaware of how chaotic would be the process of counting votes and declaring a winner on that particular occasion. The message there was, particularly in the kinds of tight race we typically see today, the different ways that votes can be tallied can lead to very different results.<br /><br />Everything I said back then remains just as valid and pertinent today (mathematics is like that), so this time I’m going to look at another perplexing aspect of election math: why do we make the effort to vote? After all, while elections are sometimes decided by a small number of votes, it is unlikely in the extreme that an election on the scale of a presidential election will hang on the decision of a single voter. Even if it did, that would be well within the range of procedural error, so it makes no difference if any one individual votes or not.<br /><br />To be sure, if a large number of people decide to opt out, that can affect the outcome. But there is no logical argument that takes you from that observation to it being important for a single individual to vote. This state of affairs is known as the Paradox of Voting, or sometimes Downs Paradox. It is so named after Anthony Downs, a political economist whose 1957 book <i><a href="https://www.amazon.com/Economic-Theory-Democracy-Anthony-Downs/dp/0060417501/ref=sr_1_1?ie=UTF8&amp;qid=1477921620&amp;sr=8-1&amp;keywords=an+economic+theory+of+democracy" target="_blank">An Economic Theory of Democracy</a></i> examined the conditions under which (mathematical) economic theory could be applied to political decision-making.<br /><br />On the face of it, Downs’ analysis does lead to a paradox. Economic theory tells us that rational beings make decisions based on <a href="https://en.wikipedia.org/wiki/Cost%E2%80%93benefit_analysis" target="_blank">expected benefit</a> (a notion that can be made numerically precise). That approach works well for analyzing, say, why people buy insurance year after year, even though they may never submit a claim. The theory tells you that the expected benefit is greater than the cost; so it is rational to buy insurance. But when you adopt the same approach to an election, you find that, because the chance of exercising the pivotal vote in an election is minute compared to any realistic estimate of the private individual benefits of the different possible outcomes, the expected benefits of voting are less than the cost. So you should opt out. [The same observation had in fact been made much earlier, in 1793, by Nicolas de Condorcet, but without the theoretical backing that Downs brought to the issue.] <br /><br />Yet, many otherwise sane, rational citizens do not opt out. Indeed, society as a whole tends to look down on those who do not vote, saying they are not "doing their part." (In fact, many countries make participation in a national election obligatory, but that is a separate, albeit related, issue.) <br /><br />So why do we (or at least many of us) bother to vote? I can make the question even more stark, and personal. Suppose you have intended to "do your part" and vote. You wake up on election morning with a sore throat, and notice that it is raining heavily. Being numerically able (as all <i>Devlin’s Angle</i> readers must be), you say to yourself, "It cannot possibly affect the result if I just stay at home and nurse my throat. I was <i><b>intending</b></i> to vote, after all. Changing my mind about voting <i><b>at the last minute</b></i> cannot possibly influence anyone else. Especially if I don’t tell anyone." The math and the logic, surely, are rock solid. Yet, professional mathematician as I am, I would struggle out and cast my vote. And I am sure many <i>Devlin’s Angle</i> readers would too – most of them, I would suspect. <br /><br />So what is going on? We can do the math. We are good logical thinkers. Why don’t we act according to that reasoning? Are we conceding that mathematics actually isn’t that useful? [SPOILER: Math is useful; but only when applied with a specific purpose in mind, and chosen/designed in a way that makes it appropriate for that purpose.] <br /><br />Which brings me to my main point. To make it, let me switch for a moment from elections to the Golden Ratio. In April 2015, the magazine <i>Fast Company Design</i> published an article titled "<a href="https://www.fastcodesign.com/3044877/the-golden-ratio-designs-biggest-myth" target="_blank">The Golden Ratio: Design’s Biggest Myth</a>," in which I was quoted at length. (The author also drew heavily on a <a href="https://www.maa.org/external_archive/devlin/devlin_05_07.html" target="_blank"><i>Devlin’s Angle</i> post</a> of mine from May 2007.) <br /><br />With a readership much wider than <i>Devlin’s Angle</i>, over the years the <i>Fast Company Design</i> piece has generated a fair amount of correspondence from people beyond mathematics academia, often designers who have not been able to overcome drinking Golden Ratio Kool-Aid during their design education. One recent email came, not from a designer but a high school math teacher, who objected to a statement the article quoted me (accurately) as saying, “Strictly speaking, it's impossible for anything in the real-world to fall into the golden ratio, because it’s an irrational number.” The teacher had, it was at once clear to me, drunk not just Golden Ratio Kool-Aid, but Math Kool-Aid as well. <br /><br />In the interest of full disclosure, let me admit that, in the early part of my career as a mathematics expositor, I was as guilty as anyone of distributing both Golden Ratio Kool-Aid and Math Kool-Aid, to whoever would drink it. But, as a committed scientist, when presented with evidence to the contrary, I re-examined my thinking, admitted I had been wrong, and started to push better, more honest products, which I will call Golden Ratio Milk and Mathematical Milk. I described Golden Ratio Milk in my 2007 MAA post and peddled it more in that <i>Fast Company Design</i> interview. Here I want to talk about Mathematical Milk. <br /><br />The reason why the Golden Ratio’s irrationality prevents its use in, say architecture, is that the issue at hand involves measurement. Measurement requires fixing a unit of measure – a scale. It doesn’t matter whether it is meters or feet or whatever, but once you have fixed it, that is what you use. When you measure things, you do so to an agreed degree of accuracy. Perhaps one or two decimal places. Almost never to more than maybe twenty decimal places, and that only in a few instances in subatomic physics. So it terms of actual, physical measurement, or manufacturing, or building, you never encounter objects to which a numerical measurement has more than a few decimal places. You simply do not need a number system that has fractions with denominator much greater than, say, 1,000,000, and generally much less than that. <br /><br />Even if you go beyond physical measurement, to the theoretical realm where you imagine having an unlimited number decimal places available, you will still be in the domain of the rational numbers. Which means the Golden Ratio does not arise. Irrational numbers arise to meet mathematical needs, not the requirements of measurement. They live not in the physical world but in the human imagination. (Hence my <i>Fast Company Design</i> quote.) It is important to keep that distinction clear in our minds. <br /><br />The point is, when we abstract from our experiences of the world around us, to create mathematical models, two important things happen. A huge amount of information is lost; and there is a significant gain in precision. The two are not independent. If we want to increase the precision, we lose more information, which means that our model has less in common with the real world it is intended to represent. Moreover, when we construct a mathematical model, we do so with a particular question, or set of questions in mind. <br /><br />In astronomy and physics, and related domains such as engineering, all of this turns out to be not too problematic. For example, the simplistic model of the Solar System as a collection of point-masses orbiting around another, much heavier, point-mass, is extremely useful. We can formulate and solve equations in that model, and they turn out to be very useful. At least they turn out to be useful in terms of the goal questions, initially in this case predicting where the planets will be at different times of the year. The model is not very helpful in telling us what the color of each planet’s surface is, or even if it has a surface, both of which are certainly precise, scientific questions. <br /><br />When we adopt a similar approach to model money supply or other economic phenomena, we can obtain results that are, mathematically, just as precise and accurate, but their connection to the real world is far more tenuous and unreliable – as has been demonstrated several times in recent years when those mathematical results have resulted in financial crises, and occasionally disasters. <br /><br />So what of the paradox of voting? The paradox arises when you start by assuming that people vote to choose, say, a president. Yes, we all say that is what we do. But that’s just because we have drunk Election Kool-Aid. We don’t actually behave in accordance with that statement. If we did, then as rational beings we would indeed stay at home on election day. <br /><br />Time to throw out the Kool-Aid and buy a gallon jug of far more beneficial Election Milk: (Presidential) elections are about <b><i>a society</i></b> choosing a president. Where that purpose impacts the individual voter is not who we vote for, but in providing social pressure <i><b>to be an active member of that society</b></i>. <br /><br />That this is what is actually going on is illustrated by the fact that U.S. society created, and millions of people wear, "I have voted" badges on election day. The focus, and the personal reward, is not "Who I voted for" but "I participated in the process." [For an interesting perspective on this, see the recent article in the <i>Smithsonian</i> <i>Magazine</i>, "<a href="http://www.smithsonianmag.com/smart-news/why-women-bring-their-i-voted-stickers-susan-b-anthonys-grave-180958847/?no-ist" target="_blank">WhyWomen Bring Their “I Voted” Stickers to Susan B. Anthony’s Grave</a>."] <br /><br />To be sure, you can develop mathematical models of group activities, like elections, and they will tend to lead to fewer problems (and "paradoxes") than a single-individual model will, but they too will have limitations. All mathematical models do. Mathematics is not reality; it is just a model of reality (or rather, it is a whole, and constantly growing, collection of models). <br /><br />When we develop and/or apply a mathematical model, we need to be clear what questions it is designed to help us answer. If we try to apply it to a different question, we may get lucky and get something useful, but we may also end up with nonsense, perhaps in the form of a "paradox."<br /><br />With both measurement and the election, as is so often the case, one benefit we get from trying to apply mathematics to our world and to our lives is we gain insight into what is really going on. <br /><br />Attempting to use the real numbers to model the acts of measuring physical objects leads us to recognize the dependency on the <b><i>physical activity of measurement</i></b>. <br /><br />Likewise, grappling with Downs Paradox leads us to acknowledge what elections are really about – and to recognize that choosing a leader is a <i><b>societal</b></i> activity. In a democracy, <b><i>who</i></b> each one of us votes for is inconsequential; <b><i>that</i></b> we vote is crucial. That’s why I did not just spend a couple of hours yesterday making my choices and filling in my ballot and leaving it at that. I also went out earlier today – in light rain as it happens (and without a sore throat) – and put my ballot in the mailbox. Yesterday I acted as an individual, motivated by my felt societal obligation to participate in the election process. Today I acted as a member of society. <br /><br />As a professional set theorist, I am familiar with the relationship between, and the distinction between, a set and its members. When we view a set in terms of its individual members, we say we are treating it extensionally. When we consider a set in terms of its properties as a single entity, we say we are treating in intensionally. In an election, we are acting intensionally (and intentionally) – at the set level, not as an element of a set. <br /><br />* A <a href="http://www.huffingtonpost.com/dr-keith-devlin/mathematical-milk-and-the_b_12740266.html" target="_blank">shorter version</a> of this article was published simultaneously in <i>The Huffington Post</i>.<br /><br /><br /><br />http://devlinsangle.blogspot.com/2016/11/mathematical-milk-and-us-presidential.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-5680996641846894034Wed, 12 Oct 2016 17:26:00 +00002016-10-12T16:42:07.758-04:00mathematical thinkingmathematical writingIt was Twenty Years Ago TodayThe title of the famous Beatles song does not exactly apply to <i>Devlin’s Angle</i>. The online column (now run on a blog platform, but unlike most blogs, still subject to an editor’s guiding hand) is in its twentieth year, but it actually launched on January 1, 1996. <br /><br />In <a href="http://devlinsangle.blogspot.com/2016/09/then-and-now-devlins-angle-turned.html" target="_blank">last month’s column</a>, I looked back at the very first post. It was a fascinating exercise to try to put myself back in the mindset of how the world looked back then, which was about the time when the World Wide Web was just starting to find its way onto university campuses, but had not yet penetrated the everyday lives of most of the world’s population. <br /><br />That period of intense technological and societal change – looking back, it is clear it was just beginning, in the first half of the 1990s being more evolutionary rather than the revolutionary that was soon to follow – and the strong sensation of change both underway and pending, is reflected in some of the topics I chose to write about each month in that first year. Here is a list of those first twelve posts, with hyperlinks.<br /><br /><ul><li>"<a href="http://www.maa.org/external_archive/devlin/devlin.html" target="_blank">Good Times</a>," January, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlinfeb.html" target="_blank">Base Considerations</a>," February, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/deepblue.html" target="_blank">Deep Blue</a>," March, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlinangle_april.html" target="_blank">Are Mathematicians Turning Soft?</a>," April, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlin_may.html" target="_blank">The Five Percent Solution</a>," May, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlin_june.html" target="_blank">Laws of Thought</a>," June, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlin_july.html" target="_blank">Tversky's Legacy Revisited</a>," July, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlin_aug.html" target="_blank">Of Men, Mathematics, and Myths</a>," August, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlin_sept96.html" target="_blank">Dear New Student</a>," September, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/october.html" target="_blank">Wanted: A New Mix</a>," October, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlin_nov96.html" target="_blank">Spreading the Word</a>," November, 1996</li><li>"<a href="http://www.maa.org/external_archive/devlin/devlin_12_96.html" target="_blank">Moment of Truth</a>," December, 1996</li></ul><br />Along with essays you might find in a mathematics magazine for students (February, June, July, August, November, December), there are reflections on where mathematics and its role in the world might be heading in the next few years.<br /><br />January’s post, about the growth of computer viruses in the digital domain, was clearly in that Brave New World vein, as I noted last month, and in February I focused on another aspect of the rapid growth of the digital world, with a look at the ongoing debate about the future of Artificial Intelligence. Though that field has undoubtedly made many advances in the ensuing two decades, the core argument I summarized there seems as valid today as it did then. Digital devices still do not “think” in anything like a human fashion (though these days it can sometimes be harder to tell the difference).<br /><br />The posts for April, May, and October looked at different aspects of the “Where is mathematics heading?” question. Of course, I was not claiming then, nor am I suggesting now, that the core of pure mathematics is going to change. (Though the growth of <a href="https://www.amazon.com/Computer-Crucible-Introduction-Experimental-Mathematics/dp/B00EQCA1VG/ref=sr_1_sc_2?ie=UTF8&amp;qid=1475680338&amp;sr=8-2-spell&amp;keywords=computer+as+cricuble" target="_blank">Experimental Mathematics</a> in the New Millennium was a new direction, one I addressed in a <a href="https://www.maa.org/external_archive/devlin/devlin_03_09.html" target="_blank"><i>Devlin’s Angle</i> post</a> in March 2009.) Rather, I was taking a much broader view of mathematics, stepping outside the mathematics department of colleges and universities and looking at the way mathematics is used in the world. <br /><br />The October post, in particular, turned out to be highly prophetic for my own career. Shortly after the terrorist attack on the World Trade Center on September 11, 2001, I was contacted by a large defense contractor, asking if I would join a large team they were putting together to bid for a Defense Department contract to find ways to improve intelligence analysis. I accepted the offer, and worked on that project for the next several years. (From my perspective, that project and the work that followed did not end uniformly well, as I lamented in an <i>AMS Notices</i> <a href="http://www.ams.org/notices/201406/rnoti-p623.pdf" target="_blank">opinion piece</a> in 2014.) When that project ended, I did similar work for a large contractor to the US Navy and another project for the US Army. In all three projects, I was living in the kind of world I portrayed in that October, 1996 column.<br /><br />In fact, my professional life as a mathematician for the entire life of <i>Devlin’s Angle</i> has been in that world – a way of using mathematics I started to refer to as “mathematical thinking.” In a <i>Devlin’s Angle</i> <a href="http://devlinsangle.blogspot.com/2012/08/what-is-mathematical-thinking.html" target="_blank">post in 2012</a>, I tried to articulate what I mean by that term. (The term is used by others, sometimes with different meanings, though I see strong overlaps and general agreements among them all.) That same year, I launched the world’s first mathematical MOOC on the newly established online course platform <a href="https://www.coursera.org/" target="_blank">Coursera</a>, with the title “Introduction to Mathematical Thinking”, and published <a href="https://www.amazon.com/Introduction-Mathematical-Thinking-Keith-Devlin/dp/0615653634/ref=sr_1_5?ie=UTF8&amp;qid=1342652878&amp;sr=8-5&amp;keywords=devlin+mathematical+thinking" target="_blank">a book</a> with the same title.<br /><br />With the world as it is today, in particular the pervasive (though largely hidden) role played by mathematics and mathematical ideas in almost every aspect of our lives, I would hazard a guess that there are far more people using “mathematical thinking” than there are people doing mathematics in the traditional sense.<br /><br />If so, that would make the professions of mathematician and mathematics educator two of the most secure careers in the world. For there is one thing in particular you need in order to engage in (effective) mathematical thinking about a real world problem: an adequate knowledge of, and conceptual understanding of, mathematics. In fact, that need was ever so, but it often tended to be overlooked in the pre-digital eras, when doing mathematics meant engaging in a lot of paper-and- pencil, symbolic computations, which meant that the bulk of mathematics instruction focused on computation, with wide ranging knowledge and conceptual understanding often getting short shrift.<br /><br />But those days are gone. Today, we carry around in our pockets devices that give us instant access to pretty well all of the world’s mathematical information and computational procedures we might need to use. (Check out <a href="https://www.wolframalpha.com/examples/Math.html" target="_blank">Wolfram Alpha</a>.) But the thinking still has to be done where it always has: in our heads. <br /><br /><br />http://devlinsangle.blogspot.com/2016/10/it-was-twenty-years-ago-today.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-5188734630296998525Tue, 13 Sep 2016 19:16:00 +00002016-09-19T09:24:15.108-04:00computer virusesMAAonline magazinesWorld Wide WebThen and Now: Devlin’s Angle Turned Twenty This Year<i>Devlin’s Angle</i> turned 20 this year. The first post appeared on January 1, 1996, as part of the MAA’s move from print to online. I was the editor of the MAA’s regular print magazine <i>MAA FOCUS</i> at the time, continuing to act in that capacity until December 1997. (See the last edition of MAA FOCUS that I edited <a href="http://www.maa.org/sites/default/files/pdf/pubs/focus/past_issues/FOCUS_17_6.pdf" target="_blank">here.</a>)<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-Q5eh9C13eeE/V9hPlcIMF7I/AAAAAAAAKnU/Cx6pHvg1bEgiNd31AeDzlNS5Qnb7CxDkACLcB/s1600/oregon_math_summit%2B1997.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="242" src="https://2.bp.blogspot.com/-Q5eh9C13eeE/V9hPlcIMF7I/AAAAAAAAKnU/Cx6pHvg1bEgiNd31AeDzlNS5Qnb7CxDkACLcB/s400/oregon_math_summit%2B1997.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Keith Devlin at a mathematical exposition summit in Oregon in 1997. L to R: Ralph Abraham (Univ of California at Santa Cruz), Devlin, Roger Penrose (Univ of Oxford, UK), and Ivars Peterson (past MAA Director of Publications for Journals and Communications).</td></tr></tbody></table><br />One of the innovations I made when I took over as <i>MAA FOCUS</i> editor in September 1991 was the inclusion of an editorial (written by me) in each issue. Though my ten-times-a-year essays were very much my own personal opinion, they were subject to editorial control by the organization's Executive Director, supported by an MAA oversight committee, both of which had approved my suggestion to do this. Over the years, the editorials generated no small amount of controversy, sometimes based on a particular editorial content, and other times on the more general principle of whether an editor’s personal opinion had a proper place in a professional organization's newsletter. <br /><br />As to the latter issue, I am not sure anyone’s views changed over the years of my editorial reign, but the consensus at MAA Headquarters was that it did result in many more MAA members actually picking up <i>MAA FOCUS</i> when it arrived in the mail and reading it. That was why I was asked to write a regular essay for the new <i>MAA Online</i>. Though blogs and more generally social media were still in the future, the MAA leadership clearly had it right in thinking that an online newsletter was very much an organ in which informed opinion had a place. <br /><br />And so <i>Devlin’s Angle</i> was born. When I realized recently that the column turned twenty this year — in its early days we thought of it very much an online “column”, with all that entailed in the world of print journalism — I was curious to remind myself what topic I chose to write about in my <a href="http://www.maa.org/external_archive/devlin/devlin.html" target="_blank">very first post</a>. <br /><br />Back then, I would have needed to explain to my readers that they could click on the highlighted text in that last sentence to bring up that original post. For the <a href="https://en.wikipedia.org/wiki/World_Wide_Web" target="_blank">World Wide Web</a> was a new resource that people were still discovering, with 1995-96 seeing its growth in academia. Today, of course, I can assume you have already looked at that first post. The words I wrote then (when I might have used the term “penned”, even though I typed them at a computer keyboard) provide an instant snapshot of how the present-day digital world we take for granted looked back then.<br /><br />A mere twenty years ago. http://devlinsangle.blogspot.com/2016/09/then-and-now-devlins-angle-turned.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-8046285082487204651Mon, 01 Aug 2016 16:44:00 +00002016-08-01T12:44:01.903-04:00Alex Gibneycomputer viruscyberwarfarePLCStuxnetMathematics and the End of Days<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/--YZr0rH4uC8/V5ZQhG7ygQI/AAAAAAAAKkM/ess6v-TpFcEqrZW1STrL_Cbqc7cYyTCCgCLcB/s1600/zerodays.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="225" src="https://4.bp.blogspot.com/--YZr0rH4uC8/V5ZQhG7ygQI/AAAAAAAAKkM/ess6v-TpFcEqrZW1STrL_Cbqc7cYyTCCgCLcB/s400/zerodays.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">A scene from <i>Zero Days</i>, a Magnolia Pictures resease. Photo courtesy of Magnolia Pictures</td></tr></tbody></table><br />The new documentary movie<a href="http://www.zerodaysfilm.com/" target="_blank"> <i>Zero Days</i></a>, written and directed by Alex Gibney, is arguably the most important movie of the present century. It is also one of particular relevance to mathematicians for its focus is on the degree to which mathematics has enabled us to build our world into one where a few algorithms could wipe out all human life within a few weeks.<br /><br />In theory, we have all known this since the mid 1990s. As the film makes clear however, this is no longer just a hypothetical issue. We are there.<br /><br />Ostensibly, the film is about the creation and distribution of the computer virus <a href="https://en.wikipedia.org/wiki/Stuxnet" target="_blank">Stuxnet</a>, that in 2011 caused a number of centrifuges in Iran’s nuclear program to self-destruct. And indeed, for the first three-quarters of the film, that is the main topic.<br /><br />Most of what is portrayed will be familiar to anyone who followed that fascinating story as it was revealed by a number of investigative journalists working with commercial cybersecurity organizations. What I found a little odd about the treatment, however, was the degree to which the U.S. government intelligence community appeared to have collaborated with the film-makers, to all intents and purposes confirming on camera that, as was widely suspected at the time but never admitted, Stuxnet was the joint work of the United States and Israel.<br /><br />The reason for the unexpected degree of openness becomes clear as the final twenty minutes of the movie unfold. Having found themselves facing the very real possibility that small pieces of computer code could constitute a human Doomsday weapon, some of the central players in contemporary cyberwarfare decided it was imperative that there be an international awareness of the situation, hopefully leading to global agreement on how to proceed. As one high ranking contributor notes, awareness that global nuclear warfare would (as a result of the ensuing nuclear winter) likely leave no human survivors, led to the establishment of an uneasy, but stable, equilibrium, which has lasted from the 1950s to the present day. We need to do the same for cyberwarfare, he suggests.<br /><br />Mathematics has played a major role in warfare for thousands of years, going back at least to around 250 BCE, when Archimedes of Syracuse designed a number of weapons used to fight the Romans.<br /><br />In the 1940s, the mathematically-driven development of weapons reached a terrifying new level when mathematicians worked with physicists to develop nuclear weapons. For the first time in human history, we had a weapon that could bring an end to all human life.<br /><br />Now, three-quarters of a century later, computer engineers can use mathematics to build cyberwarfare weapons that have at least the same destructive power for human life.<br /><br />What makes computer code so very dangerous is the degree to which our lives today are heavily dependent on an infrastructure that is itself built on mathematics. Inside most of the technological systems and devices we use today are thousands of small solid-state computers called Programmable Logic Controllers (PLCs), that make decisions autonomously, based on input from sensors.<br /><br />What Stuxnet did was embed itself into the PLCs that controlled the Iranian centrifuges and cause them to speed up well beyond their safe range to the point where they simply broke apart, all the while sending messages to the engineers in the control room that the system was operating normally.<br /><br />Imagine now a collection of similar pieces of code that likewise cause critical systems to fail: electrical grids, traffic lights, water supplies, gas pipeline grids, hospitals, the airline networks, and so on. Even your automobile – and any other engine-driven vehicle – could, in principle, be completely shut off. There are PLCs in all of these devices and networks.<br /><br />In fact, imagine that the damage could be inflicted in such a catastrophic and interconnected way that it would take weeks to bring the systems back up again. With no electricity, water, transportation, or communications, it would be just a few days before millions of people start to die, starting with thousands of airplanes, automobiles, and trains crashing, and soon thereafter doubtless accompanied by major rioting around the world.<br /><br />To be sure, we are not at that point, and the challenge of a malicious nation being able to overcome the difficulty of bringing down many different systems would be considerable – though the degree to which they are interdependent could mitigate that “safety” factor to some extent. Moreover, when autonomous code gets released, it tends to spread in many directions, as every computer user discovers sooner or later. So the perpetrating nation might end up being destroyed as well.<br /><br />But Stuxnet showed that such a scenario is a realistic, if at present remote, possibility. (Not just Stuxnet, but the Iranian response. See the movie to learn about that.) If you can do it once (twice?), then you can do it. The weapon is, after all, just a mathematical structure; a piece of code. Designing it is a mathematical problem. Unlike a nuclear bomb, the mathematician does not have to hand over her results to a large, well-funded organization to build the weapon. She can create it herself at a keyboard.<br /><br />That raw power has been the nature of mathematics since our ancestors first began to develop the subject several thousand years ago. Those of us in the mathematics profession have always known that. It seems we have now arrived at a point where that power has reached a new level, certainly no less awesome than nuclear weapons. Making a wider audience more aware of that power is what Gibney’s film is all about. It’s not that we face imminent death by algorithm. Rather that we are now in a different mathematical era.http://devlinsangle.blogspot.com/2016/08/mathematics-and-end-of-days.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-8968782239150488048Fri, 15 Jul 2016 13:06:00 +00002016-07-20T16:20:44.467-04:00BrexitEuropean Communitynumber senseSimon's ParadoxStatisticsUnited KingdomWhat Does the UK Brexit Math Tell Us?The recent (and in many respects ongoing) Brexit vote in the United Kingdom provides a superb example of poor use of mathematics. Regardless of your views on the desirability or otherwise of the UK remaining a member of the European Community (an issue on which this column is appropriately agnostic), for a democracy to base its decision on a complex issue based on a single number displays a woeful misunderstanding of numbers and mathematics.<br /><br />Whenever there is an issue that divides a population more or less equally, making a decision based on a popular vote certainly provides an easy decision, but in terms of accurately reflecting “the will of the people”, you might just as well save the effort and money it costs to run a national referendum and decide the issue by tossing a coin—or by means of a penalty shootout if you want to provide an illusion of human causality.<br /><br />Politicians typically turn to popular votes to avoid the challenge of wrestling with complex issues and having to accept responsibility for the decision they make, or because they believe the vote will turn out in a way that supports a decision they have already made. Unfortunately, with a modicum of number sense, and a little more effort, it’s possible to take advantage of the power that numbers and mathematics offer, and arrive at a decision that actually <i><b>can</b></i> be said to “reflect the will of the people”.<br /><br />The problem with reducing any vaguely complex situation to a single number is that you end up with some version of what my Stanford colleague Sam Savage has referred to as the <a href="https://www.amazon.com/Flaw-Averages-Underestimate-Risk-Uncertainty/dp/1118073754">Flaw of Averages</a>. At the risk of over-simplifying a complex issue (and in this of all articles I am aware of the irony here), the problem is perhaps best illustrated by the old joke about the statistician whose head is in a hot oven and whose feet are in a bucket of ice who, when asked how she felt, replies, “On average I am fine.”<br /><br />Savage takes this ludicrous, but in actuality all-too- common, absurdity as the stepping-off point for using the power of modern computers to run large numbers of simulations to better understand a situation and see what the best options may be. (This kind of approach is used by the US Armed Forces, who run computer simulations of conflicts and possible future battles all the time.)<br /><br />A simpler way to avoid the Flaw of Averages that is very common in the business world is the well-known SWOT analysis, where instead of relying on a single number, a team faced with making a decision lists issues in four categories: strengths, weaknesses, opportunities, and threats. To make sense of the resulting table, it is not uncommon to assign numbers to each category, which opens the door to the Flaw of Averages again, but with four numbers rather than just one, you get some insight into what the issues are.<br /><br />Notice I said “insight” there; not “answer”. For insight is what numbers can give you. Applications of mathematics in the natural sciences and engineering can give outsiders a false sense of the power of numbers to decide issues. In science (particularly physics and astronomy) and engineering, (correctly computed) numbers essentially have to be obeyed. <b><i>But that is almost never the case</i></b> in the human or social realm.<br /><br />When it comes to making human decisions, including political decisions, the power of numbers is even less reliable than the expensively computed numbers that go into producing the daily weather forecast. And surely, no politician would regard the weather forecast as being anything more than a guide—information to help make a decision.<br /><br />One of mathematicians’ favorite examples of how single numbers can mislead is known as Simpson’s Paradox, in which an average can indicate <i><b>the exact opposite</b></i> of what the data actually says.<br /><br />The paradox gets its name from the British statistician and civil servant Edward Simpson, who described it in a technical paper in 1951, though the issue had been observed earlier by the pioneering British statistician Karl Pearson in 1899. (Another irony in this story is that the British actually led the way in understanding how to make good use of statistics, obtaining insights the current UK government seems to have no knowledge of.)<br /><br />A famous illustration of Simpson’s Paradox arose in the 1970s, when there was an allegation of gender bias in graduate school admissions at the University of California at Berkeley. The fall 1973 figures showed that of the 9,442 men and 4,321 women who applied, 44% of men were admitted but only 35% of women. That difference is certainly too great to be due to chance. But was there gender bias? On the face of it, the answer is a clear “Yes”.<br /><br />In reality, however, when you drill down <b><i>just one level </i></b>into the data, from the school as a whole to the individual departments, you discover that, not only was there no gender bias in favor of men, there was in actuality a statistically significant bias in favor of women. The School was going out of its way to correct for an overall male bias in the student population. Here are the figures.<br /><br /><div class="separator" style="clear: both; text-align: center;"> <a href="https://1.bp.blogspot.com/-EiYe8rIgGmo/V4PXUzbHgWI/AAAAAAAAKiI/IBTslKG3OPYWKVuRwUvj-kqVPX5wuh-lgCLcB/s1600/Devlins%2BAngle%2BBrexit.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="138" src="https://1.bp.blogspot.com/-EiYe8rIgGmo/V4PXUzbHgWI/AAAAAAAAKiI/IBTslKG3OPYWKVuRwUvj-kqVPX5wuh-lgCLcB/s400/Devlins%2BAngle%2BBrexit.PNG" width="400" /></a></div><br />In Departments A, B, D, and F, a higher proportion of women applicants was admitted, in Department A significantly so.<br /><br />There was certainly a gender bias at play, but not on the part of University Admissions. Rather, as a result of problems in society as a whole, women tended to apply to very competitive departments with low rates of admission (such as English), whereas men tended to apply to less-competitive departments with high rates of admission (such as Engineering). We see a similar phenomenon in the recent UK Brexit vote, though there the situation is much more complicated. British Citizens, politicians, and journalists who say that the recent referendum shows the “will of the people” are, either though numerically informed malice or basic innumeracy, plain wrong. Just as the UC Berekeley figures did not show an admissions bias against women (indeed, there was a bias in favor of women), so too the Brexit referendum does not show a national will for the UK to leave the EU.<br /><br />Britain leaving the EU may or may not be their best option, but in making that decision the government would do well to drill down at least one level, as did the authorities at UC Berkeley. When you do that, you immediately find yourself with some much more meaningful numbers. Numbers that tell more of the real story. Numbers on which elected representatives of the people can base an informed discussion as how best to proceed—which is, after all, what democracies elect governments to do.<br /><br />Much of that <a href="http://www.bbc.com/news/uk-politics-36616028" target="_blank">“one level down” data</a> was collected by the BBC and published on its website. It makes for interesting reading.<br /><br />For instance, it turned out that among 18-24 years old voters, a massive 73% voted to remain in the UK, as did over half of 25-49 years of age voters. (See Table.) So, given that the decision was about the <i>future</i> of the UK, the result seems to provide a strong argument to remain in the EU. Indeed, it is only among voters 65 or older that you see significant numbers (61%) in favor of leaving. (Their voice matters, of course, but few of them will be alive by the time any benefits from an exit may appear.)<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-6d_dLAJcDRc/V4PXfTffNPI/AAAAAAAAKiM/fu6OM4xjfCITfnTqGrdmPFSGfDig_XYRQCLcB/s1600/Table.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="262" src="https://4.bp.blogspot.com/-6d_dLAJcDRc/V4PXfTffNPI/AAAAAAAAKiM/fu6OM4xjfCITfnTqGrdmPFSGfDig_XYRQCLcB/s400/Table.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Source: <a href="http://www.bbc.com/news/uk-politics-%2036616028" target="_blank">http://www.bbc.com/news/uk-politics- 36616028</a></td></tr></tbody></table><br />You see a similar Simpson’s Paradox effect when you break up the vote by geographic regions, with London, Scotland, and Northern Ireland strongly in favor of remaining in the UK (Scotland particularly so).<br /><br />It’s particularly interesting to scroll down through the long chart in the section headed “Full list of every voting area by Leave”, which is ordered in order of decreasing Leave vote, with the highest Leave vote at the top. I would think that range of numbers is extremely valuable to anyone in government.<br /><br />There is no doubt that the British people have a complex decision to make, one that will have a major impact on the nation’s future for generations to come. Technically, I am one of the “British people,” but having made the US my home thirty years ago, I long ago lost my UK voting rights. My interest today is primarily that of a mathematician who has made something of a career arguing for improved public understanding of the sensible use of my subject, and railing against the misuse of numbers.<br /><br />My emotional involvement today is in the upcoming US presidential election, where there is also an enormous amount of misuse of mathematics, and many lost opportunities where the citizenry could take advantage of the power numbers provide in order to make better decisions.<br /><br />But for what it’s worth, I would urge the citizens of my birth nation to drill down one level in your referendum data. For what you have is a future-textbook example of Simpson’s Paradox (albeit with many more dimensions of variation). To my mathematician’s eye (trained as such in the UK, I might add), the referendum provides very clear numerical information that enables you to form a well-informed, reasoned decision as to how best to proceed.<br /><br />Deciding between the “will of the older population” and the “will of the younger population” is a political decision. So too is deciding between “the will of London, Scotland, and Northern Ireland” and “the will of the remainder of the UK”. What would be mathematically irresponsible, and to my mind politically and morally irresponsible as well, would be to make a decision based on a single number. Single numbers rarely make decisions for us. Indeed, single numbers are usually as likely to mislead as to help. A range of numbers, in contrast, can provide valuable data that can help us to better understand the complexities of modern life, and make better decisions.<br /><br />We humans invented numbers and mathematics to understand our world (initially physical and later social), and to improve our lives. But to make good use of that powerful, valuable gift from our forbearers, we need to remember that numbers are there to serve us, not the other way round. Numbers are just tools. We are the ones with the capacity to make decisions.<br /><br /><i>* A version of this blog post was also published on The Huffington Post.</i><br /><br />http://devlinsangle.blogspot.com/2016/07/what-does-uk-brexit-math-tell-us.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-5470449280026104921Tue, 07 Jun 2016 20:45:00 +00002016-06-08T09:12:33.013-04:00Aronszajn treecountable setHilbert's HotelinfinityKönig’s Lemmatreesuncountable setInfinity and IntuitionOn May 30, Gary Antonick’s always interesting Numberplay section in the New York Times featured a<a href="http://wordplay.blogs.nytimes.com/2016/05/30/frenkel-cantor/?_r=0"> contribution</a> by Berkeley mathematician Ed Frenkel on the difficulties the human mind can encounter when trying to come to grips with infinity. If you have not yet read it, you should.<br /><br />Infinity offers many results that are at first counter-intuitive. A classic example is Hilbert's Hotel, which has infinitely many rooms, each one labeled by a natural number printed on the door: Room 1, Room 2, Room 3, etc., all the way through the natural numbers. One night, a traveler arrives at the front desk only to be told be the clerk that the hotel is full. "But don't worry, sir," says the clerk, "I just took a mathematics course at my local college, and so I know how to find you a room. Just give me a minute to make some phone calls." And a short while later, the traveler has his room for the night. What the clerk did was ask every guest to move to the room with the room number the next integer. Thus, the occupant of Room 1 moved into Room 2, the occupant of Room 2 into Room 3, etc. Everyone moved room, no one was ejected from the hotel, and Room 1 became vacant for the newly arrived guest.<br /><br />This example is well known, and I expect all regular readers of <i>MAA Online</i> will be familiar with it. But I expect many of you will not know what happens when you step up one level of infinity. No sooner have you started to get the hang of the countable infinity (cardinality aleph-0), and you encounter the first uncountable infinity (cardinality aleph-1) and you find there are more surprises in store.<br /><br />One result that surprised me when I first came across it concerns trees. Not the kind the grow in the forest, but the mathematical kind, although there are obvious similarities, reflected in the terminology mathematicians use when studying mathematical trees.<br /><br />A tree is a partially ordered set (T,<) such that for every member x of T, the set {y in T : y < x} of elements below x in the tree is well ordered. This means that the tree has a preferred direction of growth (often represented as upwards in diagrams), and branching occurs only in the upward direction. It is generally assumed that a tree has a unique minimum element, called the root. (If you encounter a tree without such a root, you can simply add one, without altering the structure of the remainder of the tree.)<br /><br />Since each element of a tree lies at the top of a unique well ordered set of predecessors, it has a well defined height in the tree - the ordinal number of the set of predecessors. For each ordinal number k, we can denote by T_k the set of all elements of the tree of height k. T_k is called the k'th level of T. T_0 consists of the root of the tree, T_1 is the set of all immediate successors of the root, etc.<br /><br />Thus, the lower part of a tree might look something like this:<br /><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-V6yTO9KD2dA/V1czATl5u3I/AAAAAAAAKhc/2rbB4C0uGcADNvAltnT6YoEyFDrqqN3OQCLcB/s1600/tree.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-V6yTO9KD2dA/V1czATl5u3I/AAAAAAAAKhc/2rbB4C0uGcADNvAltnT6YoEyFDrqqN3OQCLcB/s1600/tree.jpg" /></a></div><br /><br />(It could be different. There is no restriction on how many elements there are on each level, or how many successors each member has.)<br /><br />A classical result of set theory, sometimes called König's Lemma, says that if T is an infinite tree, and if each level T_n, for n a natural number, is finite, then T has an infinite branch, i.e., an infinite linearly ordered subset.<br /><br />It's easy to prove this result. You define a branch {x_n : n a natural number} by recursion. To start, you take x_0 to be the root of the tree. Since the tree is infinite, but T_1 is finite, there is at least one member of T_1 that has infinitely many elements above it. Let x_1 be one such element of T_1. Since x_1 has infinitely many elements above it and yet only finitely many successors on T_2, there is at least one successor of x_1 on T_2 that has infinitely many elements above it. Let x_2 be such an element of T_2. Now define x_3 in T_3 analogously so it has infinitely many elements of the tree above it, and so on. This simple process clearly defines an infinite branch {x_n : n a natural number}.<br /><br />Having seen why König's Lemma holds, it's tempting to argue by analogy that if you have an uncountable tree T (i.e., a tree whose cardinality is at least aleph-1) and if every level T_k, for k a countable ordinal, is countable, then T has an uncountable branch, i.e., a linearly ordered subset that meets level T_k for every countable ordinal k.<br /><br />But it turns out that this cannot be proved. It is possible to construct an uncountable tree, all of whose levels T_k, for k a countable ordinal, are countable, for which there is no uncountable branch. Such trees are called Aronszajn trees, after the Russian mathematician who first constructed one.<br /><br />Here is how to construct an Aronszajn tree. The members of the tree are strictly increasing (finite and countably transfinite), bounded sequences of rational numbers. The tree ordering is sequence extension. It is immediate that such a tree could not have an uncountable branch, since its limit (more precisely, its set-theoretic union) would be an uncountable strictly increasing sequence of rationals, contrary to the fact that the rationals form a countable set.<br /><br />You build the tree by recursion on the levels. T_0 consists of the empty sequence. After T_k has been constructed, you get T_(k+1) by taking each sequence s in T_k and adding in every possible extension of s to a strictly increasing (k+1)-sequence of rationals. That is, for each s in T_k and for each rational number q greater than or equal to the supremum of s, you put into T_(k+1) the result of appending q to s. Being the countable union of countably many sets, T_(k+1) will itself be countable, as required.<br /><br />In the case of regular recursion on the natural numbers, that would be all there is to the definition, but with a recursion that goes all the way up through the countable ordinals, you also have to handle limit ordinals - ordinals that are not an immediate successor of any smaller ordinal.<br /><br />To facilitate the definition of the limit levels of the tree, you construct the tree so as to satisfy the following property, which I'll call the Aronszajn property: for every pair of levels T_k and T_m, where k < m, and for every s in T_k and every rational number q that exceeds the supremum of s, there is a sequence t in T_m which extends s and whose supremum is less than q.<br /><br />The definition of T_(k+1) from T_k that I just gave clearly preserves this property, since I threw in EVERY possible sequence extension of every member of T_k.<br /><br />Suppose now that m is a limit ordinal and we have defined T_k for every k < m. Given any member s of some level T_k for k < m, and any rational number q greater than the supremum of s, we define, by integer recursion, a path (s_i : i a natural number) through the portion of the tree already constructed, such that its limit (as a rational sequence) has supremum q.<br /><br />You first pick some strictly increasing sequence of rationals (q_i : i a natural number) such that q_0 exceeds the supremum of s and whose limit is q.<br /><br />You also pick some strictly increasing sequence (m_i : i a natural number) of ordinals less than m that has limit m and such that s lies below level m_0 in the tree.<br /><br />You can then use the Aronszajn property to construct the sequence (s_i : i a natural number) so that s_i is on level m_i and the supremum of s_i is less than q_i.<br /><br />Construct one such path (s_i : i a natural number) for every such pair s, q, and let T_k consist of the limit (as a sequence of rationals) of every sequence so constructed. Notice that T_k so defined is countable.<br /><br />It is clear that this definition preserves the Aronszajn property, and hence the construction may be continued.<br /><br />And that's it.<br /><br />NOTE: <i>The above article first appeared in Devlin’s Angle in January 2006. Seeing Frenkel’s Numberplay article prompted me to revive it and give it another airing.</i>http://devlinsangle.blogspot.com/2016/06/infinity-and-intuition.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-4125738874907900223Wed, 04 May 2016 19:32:00 +00002016-05-04T15:33:15.004-04:00algebraCommon CoreDragonBoxITSLearnLabmathematical mindsetmathematics educationrepresentations of mathematicsAlgebraic Roots – Part 2What does it mean to “do algebra”? In Part 1, published here last month, I described how algebra (from the Arabic <i>al-Jabr</i>) began in 9th Century Baghdad as a way to approach arithmetical problems in a systematic way that scales. It was a way of thinking, using logical reasoning rather than (strictly speaking, in addition to) arithmetical calculation, and the first textbook on the subject explained how to solve problems that way using ordinary language, not symbolic expressions. Symbolic algebra was introduced later, in 16th Century France.<br /><br />Just as the formal algorithms of Hindu-Arabic arithmetic make it possible to do arithmetic in a purely procedural, rule-following way (without the need for any thought), so too symbolic algebra made it possible to solve algebraic problems by manipulating symbolic expressions using formal rules, again without the need for any thought.<br /><br />Over the ensuing centuries, schools focused more and more exclusively on the formal, procedural rules of arithmetic and symbolic algebra, driven in part by the needs of industry and commerce to have large numbers of people who could carry out computations for them, and in part for the convenience of the school system.<br /><br />Today, however, we have digital devices that carry out arithmetical and algebraic procedural calculations for us, faster and with greater accuracy, shifting society’s needs back to <i>arithmetical</i> and <i>algebraic thinking</i>. This is why you see the frequent use of those terms in educational circles these days, along with <i>number sense</i>. (All three terms are so common that definitions of each are easily found on the Web by searching on the name, as is also the case for the more general term <i>mathematical thinking</i>.)<br /><br />As more (and hopefully better) technological aids are developed, the nature of the activity involved in solving an arithmetical or algebraic problem changes, both for learning and for application. The fluent and effective use of arithmetical calculators, graphing calculators (such as <i>Desmos</i>), spreadsheets, computer algebra systems (such as <i>Mathematica</i> or <i>Maple</i>), and <i>Wolfram Alpha</i>, are now marketable skills and important educational goals. Each of these tools, and others, provides a different representation of numbers, numerical problems, and algebraic problems.<br /><br />One consequence of this shift that seemed to take an entire generation of parents off guard, is that mastery of the “traditional algorithms” for solving arithmetic and algebraic problems, which were developed to optimize human computations and at the same time create an audit trail, and which used to be the staple of school mathematics instruction, became a much less important educational goal. Instead, it is evidently far more valuable for today’s students to spend their time working with algorithms optimized to develop good arithmetical and algebraic thinking skills, that will (among other things) support fluent and effective use of the new technologies.<br /><br />I said “evidently” above, since to those of us in the education business, it was just that. With hindsight, however, it seems clear that in rolling out the Common Core State Standards, those in charge should have put much more effort into providing that important background context <i>that was evident to them</i> but, clearly, <i>not</i> evident to many people <i>not</i> working in mathematics education.<br /><br />I was not involved in the CCSS initiative, by the way, but I doubt I would have done any better. I still find it hard to wrap my mind round the fact that the “evident” (to me) need to modify mathematics education to today’s world is actually not at all evident to many of my fellow citizens—even though we all live and work in the same digital world. I guess it is a matter of the <i>educational perspective</i> those of us in the math ed business bring to the issues.<br /><br />But even those of us in the education business can sometimes overlook just how much, and how fast, things have changed. The most recent example comes from a highly respected learning research center, LearnLab in Pittsburgh (formerly called the Pittsburgh Science of Learning Center), funded by the National Science Foundation.<br /><br />The tweet shown below caught my eye a few weeks ago.<br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-EDt6JE0oojk/Vyoqv7wKDgI/AAAAAAAAKgs/YqCv_omwGB8uqA_STta-h6h8Ty0cCjbTwCLcB/s1600/Dragonbox.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="333" src="https://4.bp.blogspot.com/-EDt6JE0oojk/Vyoqv7wKDgI/AAAAAAAAKgs/YqCv_omwGB8uqA_STta-h6h8Ty0cCjbTwCLcB/s400/Dragonbox.PNG" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"></div><br />The tweet got my attention because I am familiar with DragonBox, and include it in the (very small) category of math learning apps I usually recommend. (I also know the creator, and have given occasional voluntary feedback on their development work, but I have no other connection to the company.)<br /><br />“Ineffective”? “#dragonboxfail”? Those are the words used in the tweet. <i>But neither can possibly be true</i>. DragonBox provides <i>an alternative representation</i> for linear equations in one unknown. Anyone who completes the game (for want of a better term) has demonstrated mastery of algebraic thinking for single variable linear problems. Period. (There is a separate issue of the representation that I will come to later.)<br /><br />Indeed, since the mechanics in DragonBox are essentially isomorphic to the rules of classical symbolic algebra (as taught in schools for the last four hundred years), completing the game demonstrates mastery of those mechanics too. From a logical perspective then, the tweet made no sense. All very odd for an official tweet from a respected, federally-funded research institute. Suspecting what must be going on, I looked further.<br /><br />The tweet was in response to a <a href="https://www.edsurge.com/news/2016-03-13-enter-the-dragonbox-can-a-game-really-teach-third-graders-algebra">review</a> of DragonBox, published by EdSurge. I recognized the name of the reviewer, Brady Fukumoto, a former game developer I had meet a few times. It was a well analyzed review. Overall, I agreed with everything Brady said. In particular, he spent some time comparing “doing algebra in the DragonBox representation” to “doing algebra using the traditional symbolic equations representation”, pointing out how much richer is the latter—but noting too that the former can result in higher levels of student engagement. Hardly the “promote” of a product that LearnLab accused him of. Indeed, Brady correctly summarized, and referenced (with a link) the Carnegie Mellon University <a href="http://www.cs.cmu.edu/~ylong/papers/Long&Aleven_ITS2014.pdf">study</a> the LearnLab tweet implicitly referred to.<br /><br />I recommend you read Brady’s review. It gets at many aspects of the “what does it mean to do algebra?” issue. As does playing DragonBox itself, which toward the end gradually replaces its initial “game representation” with the standard symbolic equation representation on a touch screen (a process often referred to as deconcretization).<br /><br />Unlike the tweet, the CMU paper was careful in stating its conclusion. The authors say, and Brady quotes, that they found DragonBox to be “ineffective in helping students acquire skills in solving algebra equations, <i>as measured by a typical test of equation solving</i>.” (The emphasis is mine.)<br /><br />Now we are at the root of that odd tweet. (One should not make too much of a tweet, of course. Twitter is an instant medium. But, rightly or wrongly, tweets in the name of an organization or a public figure are generally viewed as PR, presenting an authoritative, public stance.) The folks at LearnLab, their knowledge of educational technology notwithstanding, are assuming a perspective in which one particular representation of algebra is privileged; namely, the traditional symbolic one. (Which is the representation they adopt in developing their own algebra instruction app, an Intelligent Tutoring System called <i>Lynnette</i>.) But as I pointed out last month, that representation became the dominant one entirely by virtue of what was at that time the best available distribution technology: the printing press.<br /><br />With newer technologies, in particular the tablet computer (“printed paper on steroids”), other representations are possible, some better suited to learning, others to applications. To be sure, there are learning benefits to be gained from mastering symbolic algebra, perhaps even from doing so using paper-and-pencil, as Brady points out in his review. But at this stage in the representational technology development, we should adopt a perspective of all bets being off when it comes to how to best represent algebra in different contexts. I think it highly unlikely that we will ever again view algebra as something you learn or do exclusively by using a pen to pour symbols onto a page.<br /><br />Indeed, with his background in video game design, Brady ends his review by rating DragonBox according to three metrics:<br /><br /><b>Fun Factor – A:</b> I collected all 1,366 stars available in DragonBox 1 and 2 and had a great time.<br /><br /><b>Academic Value – B:</b> I worry that many will underestimate the effort needed to transfer DragonBox skills to practical algebra proficiency.<br /><br /><b>Educational Value – A+:</b> Anytime a kid leaves a game with thoughts like, “algebra is fun!” or “hey, I’m really good at math!” that is a huge win.<br /><br />The LearnLab researchers are locked into the second perspective: what he calls Academic Value. (So too is Brady, to some extent, with his use of the phrase “practical algebra proficiency” to mean “symbolic algebra proficiency.”)<br /><br />Make no mistake about it, transfer from mastery in an interactive engagement on a tablet to paper-and-pencil math is not automatic, as both Brady and the CMU researchers observe. To modify the old horse aphorism, DragonBox takes its players right to the water’s edge and dips their feet in, but still the players have difficulty drinking. (My best guess is that, for most learners it takes a good teacher to facilitate transfer.)<br /><br />I note in passing that initially I had difficulty playing DragonBox. My problem was, classical, symbolic algebra is a second language to me that I have been fluent in since childhood and use every day. I found it difficult mastering the corresponding actions in DragonBox. Transfer is difficult in both directions.<br /><br />At the present moment in time, those of us in education (or learning research) should absolutely not assume any one representation is privileged. Particularly so when it comes to learning. In that respect, Brady is right to note that DragonBox’s success in terms of his third metric (essentially, attitude and engagement) is indeed “a huge win.”<br /><br />In the world in which our students will live their lives, arithmetic, algebra, and many other parts of mathematics, should be learned, and will surely be applied, in multimedia environments. All the evidence available today suggests that mastery of the traditional symbolic representation will be a crucial ingredient in becoming proficient at arithmetic and algebra. But the more effective practitioners are likely to operate with the aid of various technological tools. Indeed, for some future practitioners, mastery of the traditional symbolic representation (which is, remember, just a user interface to a certain kind of thinking) may turn out to be primarily just a key step in the cognitive process of achieving conceptual understanding, not used directly in applications, which may all be by way of mathematical reasoning tools.<br /><br />Exactly when, in the initial learning process, it is best to introduce the classical symbolic representation is as yet unclear. What the evidence of countless generations of students-turned-parents makes abundantly clear, however, is that teaching only the classical symbolic approach is a miserable failure. That much is affirmed every time a parent posts on social media that they are unable to understand a Common Core math question that requires nothing more than understanding the place-value representation of integers. (Which is true of most of the ones I have seen posted.)<br /><br />There is some evidence (see for example Jo Boaler’s <a href="http://www.amazon.com/Mathematical-Mindsets-Unleashing-Potential-Innovative/dp/0470894520/ref=sr_1_1?ie=UTF8&qid=1462196234&sr=8-1&keywords=mathematical+mindsets">new book</a>) that a more productive approach is to use learning technologies to develop and sustain student engagement and develop a growth mindset, and provide learning environments for safe, productive failure, with the goal of developing number sense and general forms of creative problem solving (mathematical thinking), bringing in symbolic representations and specific techniques as and when required.<br /><br />**Full declaration: I should note that my own work in this area, some of it through my startup company <a href="http://www.brainquake.com/">BrainQuake</a>, adopts this philosophy. The significant <a href="http://documents.brainquake.com/backed-by-science/Stanford-Pope_Mangram_SUMMARY.pdf">learning gains </a> obtained with our <a href="http://wuzzittrouble.com/">first app</a> were in number sense and creative problem solving for a novel, complex performance task. Acquisition of traditional “basic skills” with our app comes about (intentionally, by design) as a valuable by-product. The improvement we see in the basic skills category is much more modest, and may well be better achieved by a tool such as LearnLab’s ITS. In a world where we have multiple representations, it is wise to make effective use of them all, according to context. It is not a case of an interface “fail”; to say that (with or without a hashtag) is to remain locked in past thinking. Easy to do, even for experts. Rather, in an era when algebra is being forced to return to its roots of being a way of thinking to help us solve practical problems, using all available representations in unison can provide us with a major win.http://devlinsangle.blogspot.com/2016/05/what-does-it-mean-to-do-algebra-in-part.htmlnoreply@blogger.com (Mathematical Association of America)0