tag:blogger.com,1999:blog-2516188730140164076Sat, 24 Jun 2017 06:37:45 +0000mathematical thinkingCommon CoreMOOCmathematics educationFibonacciLeonardo of PisaJo Boaleralgebraonline educationCourseraDragonBoxKhan AcademyMOOCsPisaSteve Jobseducational video gamesgame-based learningproblem solvingAlpine Road TrailApple MacintoshJordan EllenbergKeith DevlinMultiplicationPi DayStanford Universityapplications of mathematicsarithmeticconceptual understandingeducational startupsgolden ratioinfinitymathematical modelingmountain bikingnumber sensepiproofsquantitative literacyscience of patternstransition courseAdolf ZeisingAlex GibneyAndrew HackerAndrew NgArea of a circle formulaAronszajn treeAsa ButterfieldAspen Ideas FestivalBedtime MathBrexitCreateSpaceDan MeyerDaphne KollerDavid LeonhardtDerek MullerDowns paradoxETSEdge questionEinstein’s energy equationEric MazurErlwangerEuler’s IdentityEuler’s equationEuropean CommunityFibonacci sequenceFinnish educationGOFMGeorg CantorGeorge BooleGrant SandersonGregory’s SeriesH-STARHerman MehtaHilbert's HotelHindu-arabic arithmeticHypercubesIMOITSInquiry-Based LearningIs algebra necessaryJames Paul GeeJohn BrockmanK-12 educationKarim AniKen RobinsonKool AidKönig’s LemmaLaura OverdeckLearnLabLiber abbaciLiping MaM.C. EscherMAAMTT2KMarilyn BurnsMath and musicMath moviesMonument Valley gameMoore MethodNSANumberphileOECDPIAACPLCPaul CohenPeer InstructionPhil DaroPolyaPythagoras’ equationR L MooreRiccardiana LibraryRiemann’s HypothesisSERPSanfordSebastian ThrunSergio CorreaSimon's ParadoxStandard algorithmStatisticsSteven StrogatzStuxnetSugata MitraTIMSSTeachingTest FlightUdacityUnited KingdomUniversity of ChicagoUri TreismanVeritasiumWikipedia contributionWorld Science FestivalWorld Wide WebWuzzit TroubleYouCubedZeta functionabstractional-Khwarizmiamygdalaan approach that sucksanalytic continuationanalytic functionsanswer gettingappreciation of mathematicsbeauty in mathematicsboolean algebraboolean logiccalculuscollaborative workingcomplex numberscomputer science educationcomputer viruscomputer virusescountable setcyberwarfarecyclingdistance learningdot diagramseducation researcheducational appseducational strandseducational tsunamieducational videoselection mathelementary arithmeticequityexceptional performancefailureforcinggeometrygifted and talentedgooglehigher educationinfinite seriesinfinite sumsinstructioninstructional videointelligence analysislarge cardinalslearning outcomeslearning scienceliteracymachine gradingmath and dancemath appsmath competitionsmath learningmath learning appmath learning video gamemath on TVmath video gamesmathematical mindsetmathematical modelmathematical proofsmathematical writingmathematicsmathematics and societymathematics outreachmathematics transition coursemathematics visualizationmemesmilkmind setmultiple-choice questionsmusicnature of mathematicsnumeracyonline learningonline magazinesparadox of votingpeer evaluationpop quizzespovertyprocedural masteryproof techniquespublish-on-demandquantitative reasoning requirementrepeated additionrepeated addition (not)representations of mathematicsscience videoself-publishingset theorysoft mathematicstalent searchtextbookstreesuncountable setDevlin's AngleDevlin's Angle is a monthly column sponsored by the Mathematical Association of America. 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)Blogger71125tag: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)0tag:blogger.com,1999:blog-2516188730140164076.post-7982488435981996827Mon, 04 Apr 2016 19:33:00 +00002016-04-04T15:54:32.484-04:00al-KhwarizmialgebraCommon Coremathematics educationAlgebraic roots – Part 1<div class="separator" style="clear: both; text-align: center;"></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://3.bp.blogspot.com/--sdywWmDWlA/VwKtXmK9fKI/AAAAAAAAKfo/XdGmL94Fr2Y9_nPd-OzAqsnUkrz2BEt2Q/s1600/Fig1.jpeg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="228" src="https://3.bp.blogspot.com/--sdywWmDWlA/VwKtXmK9fKI/AAAAAAAAKfo/XdGmL94Fr2Y9_nPd-OzAqsnUkrz2BEt2Q/s400/Fig1.jpeg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Fig. 1: A problem from the first ever algebra textbook.</td></tr></tbody></table>The first ever algebra text book was written in Baghdad around 830CE, by the Persian mathematician Muhammad ibn Musa al-Khwarizmi, our modern word “algebra” coming from the Arabic term al-Jabr, a technique for balancing an equation, described in the book. If you were a student – or a teacher – back then, the problem shown above (Figure 1) is the kind of thing you would be faced with in your math class. It is a direct translation from the original Arabic of a problem in al-Khwarizmi’s book.<br /><br />Most modern readers, on seeing this for the first time and being told it is an algebra problem, are surprised that there are no symbols. Yet it is clearly not an “algebra word problem” in the usual sense. It’s just about numbers. It is, in fact, a quadratic equation problem. Figure 2 below is the same problem as we would present it in an algebra textbook today.<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/-p5thdoqU-9I/VwKtnx4kZ5I/AAAAAAAAKfs/qp0hTRwi6GQn6qM852TwXSa60iPd0srtg/s1600/Fig2.jpeg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="196" src="https://4.bp.blogspot.com/-p5thdoqU-9I/VwKtnx4kZ5I/AAAAAAAAKfs/qp0hTRwi6GQn6qM852TwXSa60iPd0srtg/s320/Fig2.jpeg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Fig. 2: Al-Khwarizmi's quadratic equation in modern notation.</td></tr></tbody></table><br /><i>Symbolic </i>algebra, as we understand it today, was not introduced until the Sixteenth Century, when the French mathematician François Viète took what until then had been a discipline presented in prose, and turned it into the symbolic process we are familiar with today.<br /><br />This is not to say that mathematicians back in Ninth Century Persia did not use symbolic expressions in their work. They surely did. The issue is how they presented it in textbook form. In the days when books were handwritten and duplicated by hand-copying, the author of a mathematics book was faced with a problem that other writers did not have to worry about: faithful copying. Copying of manuscripts was largely done by monks in monasteries. While masters of the written word – they did, after all, “live by a book” — few monks mastered mathematics, and hence could not be relied upon to create an accurate copy of anything other than prose. Aware of this issue, authors of mathematics books wrote everything in prose.<br /><br />With the introduction of the printing press in the Fifteenth Century, however, everything changed. Indeed, one of the first printed books published after Gutenberg printed his famous edition of the Bible was an Italian book on practical arithmetic. True, to handle a symbolic textbook, you have first to master the linguistic rules for reading, writing, and manipulating symbolic expressions, but once you do, algebra becomes a whole lot easier to do, as a line-by-line comparison of Figures 1 and 2 makes abundantly clear. (Actually, it’s lines-by-line!)<br /><br />Notice, however, that the two presentations of the quadratic problem specify the same problem, and both solutions are, from a logical deduction point of view, the same. To some extent, the al-Khwarizmi’s prose version describes what goes on in your head when you solve the problem. At least — <i>and this is where I am going with this</i> — it does if you solve the problem by thinking about it. <br /><br />With the symbolic presentation, it is possible to reduce the solution of an algebra problem to the <i>mindless </i>(literally), algorithmically-specified manipulations of symbols. Ever since the invention of the printing press, generations of students quickly discovered that you can pass an algebra test by mastering a collection of symbolic-manipulation rules.<i> No understanding necessary</i>. Moreover, when taught this way, the teacher’s job became immeasurably easier. It is easier to teach rules to be followed than to develop thinking skills, and it is easy to evaluate a student’s work if the goal is simply to check that it accords with the rules and arrives at the correct answer. (Indeed, teachers soon realized that the quickest way to grade a student’s work was to first see if the answer is correct, and only if it is not look at the symbolic working.)<br /><br />While the student in Ninth Century Baghdad solved (linear and quadratic) equations by performing essentially the same steps as a student would today, with the problem presented in words, and the solution written out (presumably) in words, it can’t be carried out in a mindless fashion. The human mind can learn to follow rules for manipulating symbols, without knowing what they mean, but words are so much an integral part of human thinking that we cannot use them without their having meaning (albeit possibly a meaning other than the one intended by the author of an algebra book).<br /><br />There is, then, a potential loss in taking algebra from a prose presentation to a symbolic one: namely, the student can lose the appreciation that algebra is a <i>powerful way of thinking</i> with countless uses in the everyday world. Instead of algebra being a codification of human logical thinking that emerges from within, it becomes a set of externally imposed, and often arbitrary-seeming rules to be mastered by repetitive practice. The natural, relevant, and empowering becomes the artificial, pointless, and tedious. (Those of us who like symbolic algebra see beyond the rules.)<br /><br />“When will I ever use algebra?” today’s student justifiably asks. In terms of rule-based, symbol manipulation, the answer is, for most people (not all – and this is educationally significant), “Never.” But in terms of algebra, that codified way of thinking that has evolved and developed considerably since al-Khwarizmi’s day, the answer is, “All the time.” (Whenever you use a spreadsheet, for example.)<br /><br />In the introduction to his algebra book, al-Khwarizmi declared that he was presenting<br /><br /><i> “... what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.”</i><br /><br />This was cutting edge stuff back then. It doesn’t get much more practical than that! <br /><br />As al-Khwarizmi explains, he was asked to write his book by the Caliph, who recognized the importance — for trade and engineering in particular (both of which were crucial to the regional society at the time) — of making those new methods of calculation widely available. The Caliph’s reasoning was as sound and significant then as it would be today. When a society reaches a state of development where trade and commercial and financial activity go beyond two people engaging in one-off transactions, it needs a more efficient tool than basic arithmetic. What is required is <i>arithmetic-at-scale</i>. When you boil it down to its essence, that is what algebra is. Al-Khwarizmi’s book codifies and formalizes the numerical reasoning that people use in their daily personal and professional lives in a fashion that enables them to operate at scale.<br /><br />In the years since the printing press made it possible to produce algebra textbooks that used symbolic representations, the focus in the algebra class has gradually shifted from being about sophisticated reasoning about numbers to an often mindless game of symbol manipulation. For several centuries that could be justified on the grounds that the only effective way for society to be able to handle the arithmetic-at-scale required to advance was to train lots of people to carry out the necessary calculations. And for that, the most efficient way is to use rule-based, symbolic manipulation. The people carrying out those calculations no more had to understand what they were doing than the electronic calculator on your iPhone has to understand what it is doing. All that matters it that it – the human symbolic-algebraist or the calculator app — gets the right answer.<br /><br />But now that those of us in more advanced societies (and in a great many less advanced societies, for that matter) do have ready access to those powerful calculating devices, devices that in addition to performing numerical calculations can also solve algebraic problems (arithmetic-at-scale, aka the electronic spreadsheet), the once-important societal need for many human symbolic calculators has gone away. What is required today is that people can make effective use of those new tools. That has shifted the emphasis back from symbolic-rule-mastery to the kind of formalized, rigorous thinking about quantitative matters that, thanks to al-Khwarizmi, we call algebra. Only now, we are back to the realm, not of symbol manipulation, but codified, logical, rigorous thinking about issues in our lives and in the world we inhabit.<br /><br />To be sure, symbolic algebra is not going away. It is way too powerful to ignore. But whereas it used to be possible to provide a rationale for teaching algebra as pure, rule-based symbolic manipulation (albeit a societal rationale that views people as fodder for industry), it makes no sense to teach it that way today.<br /><br />Which is why the Common Core now directs the focus not on the symbolic rules that dominated math instruction for centuries past, but on sophisticated mathematical thinking skills that develop and require a deeper understanding of numbers. This is why there is now so much talk of “number sense” and why Mary and Johnny are coming home from school with homework questions that their parents often find strange and occasionally incomprehensible. <br /><br />In other words, algebra has returned to its roots. (Pun intended.)<br /><br /><b>END OF PART 1</b><br />In Part 2 of this commentary, to be published here next month, I will look at how those same digital technologies that have rendered obsolete much of what used to constitute K-12 algebra education, have provided new ways to teach the subject that are ideally suited to the way we use — and will increasingly use even more — algebra. After all, if the printing press turned algebra from prose to symbolic expressions, what will algebra look like now that the digital computer, and in particular the tablet device, has largely replaced the printing press?<br /><br /><b>NOTE</b>: I realize that there is little in this month’s post that is new to MAA members. But as I know from emails and comments I receive, <i>Devlin’s Angle</i> posts find their way to a wide variety of readers, occasionally onto the desks of governors, education administrators, and others who play a role in the nation’s education system. With so much media attention currently being given to a mathematics education proposal being made by an individual having little knowledge of mathematics or current mathematics education (see last month’s <a href="http://devlinsangle.blogspot.com/2016/03/the-math-myth-that-permeates-math-myth.html">column</a>), I thought it timely to bring us back to an appreciation of algebra (i.e., algebraic thinking) that was apparent to a Ninth Century Caliph in Baghdad, and which is even more relevant to our lives today than it was back then. <br /><br />I cannot avoid ending by observing that 2016 will surely go down as the year when the US media devoted more media space and time to individuals pontificating on topics they knew almost nothing about, than they did to experts, of which the United States has large numbers with global reputations. I think many editors would benefit from a (good) course in algebraic thinking.http://devlinsangle.blogspot.com/2016/04/algebraic-roots-part-1.htmlnoreply@blogger.com (Mathematical Association of America)3tag:blogger.com,1999:blog-2516188730140164076.post-5688277157532213592Tue, 01 Mar 2016 15:11:00 +00002016-03-02T10:53:47.183-05:00algebraAndrew HackerCommon CoreIs algebra necessarymathematics educationquantitative literacyThe Math Myth that permeates “The Math Myth”March 1 saw the publication of the book <i>The Math Myth: And Other STEM Delusions</i>, by Andrew Hacker. MAA members are likely to recognize the author’s name from an opinion piece he published in the New York Times in 2012, with the arresting headline "<a href="http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html">Is Algebra Necessary?</a>"<br /><br />Yes, I thought you’d remember it! It’s almost up there with John Lennon’s murder in terms of knowing where you were at the time you first heard of it. But just to be sure we are all on the same page, let me recap that, in that essay, Hacker, a retired college professor of political science who over the years had taught some non-majors math courses, laid out a case for dropping algebra as a required course in K-12 and college.<br /><br />Before I dive into Hacker’s new book, you would be advised to refresh your memory of the case he presented in that article, since his book is essentially an extension of what he said then, expanded to cover the entire Common Core Mathematical Standards. Prior to writing this review, I wrote an <a href="http://www.huffingtonpost.com/dr-keith-devlin/andrew-hacker-and-the-cas_b_9339554.html">article for the Huffington Post</a> in which I summarized, with my commentary, his 2012 article, together with a <a href="http://chronicle.com/article/The-Case-Against-Mandating/235500">recent interview</a> he gave to the Chronicle of Higher Education.<br /><br />In my article, I noted that Hacker has no idea what algebra really is. His focus is entirely on school algebra as it is very often taught, as a collection of rules for manipulating symbolic expressions. What his argument actually establishes, with sound arguments and good examples, are two conclusions I would agree with:<br /><ol><li>Algebra as typically taught in the school system is presented as a meaningless game with arbitrary rules that does more harm than good.</li><li>There are strong arguments for teaching algebra as it was originally developed and how professional mathematicians today view it.</li></ol>I’ll leave you to read my HuffPost piece for more of the gory details. For the benefit of lay readers who may come to this site, I should though repeat here the brief summary I gave in that article of the difference between algebra (as mathematicians understand and practice it) and the rule-based-manipulation-of-symbolic-expressions that so often passes for algebra in our schools.<br /><br />First codified by the Persian mathematician al-Khwarizmi in his book <i>The Compendious Book on Calculation by Completion and Balancing</i> (balancing = <i>al-Jabr</i>), written in Bhagdad around CE 820, algebra is a powerful method for solving numerical problems more efficiently than by arithmetic. It does so by introducing two new ways of handling numerical problems.<br /><br />First, algebra provides methods for handling entire classes of numbers, rather than specific ones. (That’s where those x’s, y’s, and z’s come in, but that’s just an implementation detail introduced in France several centuries later.)<br /><br />Second, it provides a way to find numerical answers not by computing, which is often very difficult, but by reasoning logically to hone-in on the answer, using whatever information is available. Thus, whereas in arithmetic you work forwards, starting with numbers and computing with them to arrive at an answer, in algebra you work backwards, starting by postulating an answer and reasoning logically to figure out what it is. True, this powerful application of human logical reasoning capacity frequently gets boiled down to mastering various symbolic procedures to “Solve for x,” but again that’s just a particular implementation. Numerical forensics would be a sexier, and more descriptive, term for the real thing.<br /><br />The familiar symbolic expressions calculus usually taught in schools as “algebra” was a particular implementation of al-Khwarizmi’s ways of thinking introduced by the French mathematician François Viète in the 16th Century (700 years after algebra first began) to streamline paper-and-pencil problem solving. A more recent implementation of algebra is the computer spreadsheet.<br /><br />Since his new book follows the same line of attack as his 2012 opinion piece, but with his sights widened from school algebra to the Common Core, instead of crafting another analytic essay, I will do what Hacker himself does, and list a number of examples to make my case. More precisely, I’ll select some of the 20 instances (in a book of just over 200 pages) where I found a claim that is either plain wrong, wildly misleading, or otherwise problematic, and ask where he went wrong. In marking 20 pages, it’s likely I missed some. There were so many wild and inaccurate claims, I frequently found myself skimming through.<br /><br />First though, I should repeat what I said in my HuffPost article about his algebra piece. Just as his essay actually amounted to a strong argument in favor of teaching algebra to all students (albeit not the rule-based manipulations of formulas so often presented in place of algebra), so too his book includes a strong argument in favor of Common Core Math. In the same way that Hacker mischaracterized algebra in 2012, so too his portrayal of the CCSSM (Common Core State Standards for Mathematics) is totally at odds with the real thing—though not quite so far off if you turn your attention from the Standards themselves to some <i>implementations</i> of the CC.<br /><br />One of the book’s flaws is that Mr Hacker seems to get carried away with the flow of his rhetoric, since for the most part his argument consists of the erection of a series of straw men which he then, in time-honored tradition, proceeds to attack.<br /><br />“It’s a waste of time forcing kids to master azimuths and asymptotes,” he cries [not an exact quote] as early as page 2.<br /><br />I had to look up the word azimuth, since in my entire career as a mathematician and mathematics educator, I had never come across it. According to Wikipedia, azimuth is a “concept used in navigation, astronomy, engineering, mapping, mining and artillery.” I ran a search for the word on the entire, 93-page CCSSM document and, as I expected, it did not turn up. Straw man.<br /><br />Asymptotes are a different matter, of course, since a general sense of asymptotic behavior of functions is useful in many walks of life. The word is mentioned, but just once, in the CCSSM, in the section on Interpreting Functions (F-IF), where it says:<br /><br /><i>Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.</i><br /><br />That’s it. One mention, buried towards the end of the document, in the section that says the student should:<br /><ul><li>Understand the concept of a function and use function notation</li><li>Interpret functions that arise in applications in terms of the context</li><li>Analyze functions using different representations</li></ul>From the overall thrust of Hacker’s argument, I think it’s clear he believes this kind of knowledge is indeed important for everyone to have. But it’s also clear it is not a central pillar of the CC, to be used on page 2 to set the scene for what his book is about.<br /><br />Unfortunately, this example is indeed a good characterization of his overall argument: to knock down straw men.<br /><br />“<i>We’re told that if our nation is to stay competitive, on a given morning all four million of our fifteen-year-olds will be studying azimuths and asymptotes</i>,” he writes. (I am still on page 2, with over 200 more pages to go.) He provides no citation regarding who, exactly, is making this proclamation for the nation’s future. It’s not just disingenuously misleading, it’s about as far from reality as you could imagine, and not because of those azimuths. (See momentarily for the real story.)<br /><br />He continues, “<i>Then, to graduate from high school, they will face tests on radical notations and elliptical equations</i>.”<br /><br />To be sure, you will find mention of the word radical in the CCSSM, in the context of “Work with radicals and integer exponents” in the Section on Expressions and Equations (8.EE), which provides the helpful illustration,<br /><br />“<i>For example, estimate the population of the United States as 3 × 10<sup>8</sup> and the population of the world as 7 × 10<sup>9</sup>, and determine that the world population is more than 20 times larger</i>.”<br /><br />Again, this is exactly the kind of thing Hacker says (towards the end of his book) students should be able to do! And it is entirely reasonable that they be asked to demonstrate that ability on a test.<br /><br />“Elliptical equations” is another straw man.<br /><br />The point is, what Hacker keeps attacking are straw men. The CCSS is just what its name implies, a set of standards. It is not a curriculum, nor does it specify anything remotely like a daily, or even weekly timetable. How and when teachers across the land cover the various standards is for them, or perhaps their school district, to decide. As far as the CCSS are concerned, teachers can operate fluidly, depending on how their class progresses. (And no one will even suggest that they mention azimuths, let alone force the class to master them.)<br /><br />I would hazard a guess that Hacker has never looked at the CCSS document. Nor sat in on many math classes, as I have, and observed what actually goes on in today’s schools.<br /><br />Caveat: I get to see classes to which, for one reason or another, I have been invited to visit. Likely they are some of the best, since their teachers invite me along so their students can talk for a while with someone who has devoted a career to mathematical research. I hear enough stories to be prepared to believe things are often a lot worse. Perhaps even as bad as Hacker says. But his book is purported to be about educational policy, not what you can actually find in good or bad classrooms.<br /><br />Not only does Hacker give no indication he is familiar with the Common Core—the real one, not the azimuth-strewn, straw-man version he creates—he gives every indication he does not understand mathematics as it is practiced today. (He also does not know that pi is irrational, but I’ll come to that later.)<br /><br />Certainly, the examples he selects to illustrate the irrelevancy (in today’s world) of some of the test problems students are asked to solve simply demonstrate that he is lacking the basic, every-day, number sense he is arguing for. Let me give just three examples.<br /><br />On page 48, Hacker presents a question he took from an MCAT paper. It provides some technical data and asks what happens to the ratio of two inverse-square law forces between charges of given masses when the distance between them is halved. The context Hacker provides for this question is that medical professionals needs to be able to read and understand the mathematics used in technical papers. His claim is that this requirement does not extend to the physics of electrical and gravitational forces. In that, he is surely correct. But anyone with a grain of number sense will recognize at once that the setting is totally irrelevant. It’s a simple question about what happens to a ratio when the underlying scale is changed. The answer, of course, is nothing happens. It’s a ratio. The changes to the numerator and denominator cancel out. The ratio remains the same.<br /><br />What this question is asking for is, <i>Do you understand what a ratio is</i>? Surely that is something that any medical professional who will have to read and understand journal articles would need to know. Hacker completely misses this simple observation, and presents the question as an example of baroque mathematical testing run amok.<br /><br />On page 70, he presents a question from an admissions test for selective high schools. A player throws two dice and the same number comes up on both. The question asks the student to choose the probability that the two dice sum to 9 from the list 0, 1/6, 2/9, 1/2, 1/3. Hacker’s problem is that the student is supposed to answer this in 90 seconds. Now, I share Hacker’s disdain for time-limited questions, but in this case the answer can only be 0. It’s not a probability question at all, and no computation is required. It just requires you to recognize that you can never get a sum of 9 when two dice show the same number. As with the MCAT question, the question is simply asking, <i>Do you understand numbers</i>? In this case, do you recognize that the sum of two equal numbers can never be odd.<br /><br />Finally, on page 101, Hacker presents a list of mathematics requirements high school students must meet in order to study at Harvard and similar universities. The list includes the names of various kinds of analytic functions. As usual, Hacker seems overwhelmed by the technical terms, or worries that the students will be, but all the list is asking for is that students can read graphs and charts and know what they represent in terms of growth and change. An essential skill, surely, for anyone in today’s information-rich world, not just students at elite universities.<br /><br />You get the pattern surely? Hacker’s problem is he is unable to see through the surface gloss of a problem and recognize that in many cases it is just asking the student if she or he has a very basic grasp of number, quantity, and relationships. Yet these are precisely the kinds of abilities he argues elsewhere in the book are crucial in today’s world. He is, I suspect, a victim of the very kind of math teaching he rightly decries—one that concentrates on learning rules and mastering formal manipulations, with little attention to understanding.<br /><br />This, surely, explains why he would write (page 96), “<i>Reasoning mathematically may be a nice skill, but it is not relevant to most of life. We reason about many things: parenting, marriage, careers. Do we learn how to reason about these things by learning algebra?</i>”<br /><br />If he had asked instead if we learn such reasoning in a typical school algebra class, I would agree with his implied answer of “No.” But algebra arose by codifying the everyday reasoning people carried out—and still carry out today—about the numerical or quantity aspects of any human activity that involves them. (Trade, commerce, and civil engineering were the original applications.)<br /><br />From that historical perspective, it is absolutely clear that learning algebra can help us master such reasoning. It helps by providing an opportunity to carry out that kind of reasoning free of the complexities a problem generally brings with it when it arises in a real world context.<br /><br />The tragedy of <i>The Math Myth</i> is that Hacker is actually arguing for exactly the kind of life-relevant mathematics education that I and many of my colleagues have been arguing for all our careers. (Our late colleague Lynn Steen comes to mind.) Unfortunately, and I suspect because Hacker himself did not have the benefit of a good math education, his understanding of mathematics is so far off base, he does not recognize that the examples he holds up as illustrations of bad education only seem so to him, because he misunderstands them.<br /><br />The real myth in <i>The Math Myth</i> is the portrayal of mathematics that forms the basis of his analysis. It’s the same myth you see propagated in Facebook posts from frustrated parents about Common Core math homework their children bring home from school.<br /><br />In the interests of their overall cardiovascular health, I have to recommend that math educators do not read <i>The Math Myth</i>. But if you do, perhaps you should start with the final chapter, titled “Numeracy 101.” Here, at least, you will find things you are likely to agree with, as he lays out what he believes would be a good quantitative literacy course for college students.<br /><br />But even there, where all seems warm and friendly and positive, you will be jolted by Hacker’s fundamental lack of knowledge of mathematics. He writes,<br /><br />“<i>Along with phenomena like earthquakes and cyclones, nature also has some numbers that control or explain how the world works. One of them is pi, whose</i> 3.14159<i> goes on indefinitely, at least as far as we know</i>.”<br /><br />Yes, you read that last part correctly.<br /><br />“Few people writing today … can make more sense of numbers” proclaims the <i>Wall Street Journal</i> on the cover of Hacker’s book. Well, if that’s the view of the newspaper that purports to have the expertise to cover the nation’s financial markets, it is only a matter of time before we have another financial meltdown.http://devlinsangle.blogspot.com/2016/03/the-math-myth-that-permeates-math-myth.htmlnoreply@blogger.com (Mathematical Association of America)9tag:blogger.com,1999:blog-2516188730140164076.post-7574208066658684828Thu, 11 Feb 2016 15:49:00 +00002016-02-11T16:40:24.816-05:00exceptional performanceHypercubesmathematical modelingTheorem: You are exceptional“Everyone excels at something.” We hear it all the time, usually said to console someone who is miserable after underperforming at something. Parents, in particular, often fall back on it with their children. What few people realize, though, is that the statement can be mathematically verified. You need only consider a collection of 200 essentially independent human performance characteristics for 98% of people to measure as exceptional in at least one of them, where exceptional is defined as being in the top or bottom 1%. (The mathematics gives extremal values; if you want to effectively guarantee being in the top 1%, you need more characteristics. The phenomenon is asymptotic.) <br /><br />This result is a consequence of a rather surprising, but little known, observation about high-order hypercubes: as the dimension increases, the proportion of points in the interior (i.e., not on the bounding shell) decreases without limit.<br /><br />Here is how you can prove to your child, spouse, student, best friend, or whoever, that they—or you, as the circumstances may require—can or will excel at something.<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="http://2.bp.blogspot.com/-5aVnO2xvjz0/VrokChM7TsI/AAAAAAAAKc0/8EX50jzeub8/s1600/Fig1.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="118" src="https://2.bp.blogspot.com/-5aVnO2xvjz0/VrokChM7TsI/AAAAAAAAKc0/8EX50jzeub8/s200/Fig1.jpg" width="200" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small; text-align: start;">Fig 1. The bell curve (normal distribution)</span></td></tr></tbody></table><br />Everyone is familiar with the bell curve (normal distribution) showing the typical distribution of performance measures of a single characteristic across a sufficiently large population. This graph captures the fact that the scores for the majority of the population cluster around an “average”, middling value, with only a few individuals at either end (exceptionally poor or exceptionally good).<br /><br />For the purposes of the multi-dimensional computation, we can start with a geometrically simpler model, namely the closed interval [0,100], as in Figure 2. We define the <i>exceptional</i> points to be those in the unit intervals at each end. In this model, for a single characteristic, only 2% of the population are exceptional. The remaining 98% are “normal."<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/-tCymcfRBOkU/VryqRi4oAHI/AAAAAAAAKds/DJQjtHJYhWw/s1600/Devlin%2Bnew%2Bfig.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://2.bp.blogspot.com/-tCymcfRBOkU/VryqRi4oAHI/AAAAAAAAKds/DJQjtHJYhWw/s1600/Devlin%2Bnew%2Bfig.png" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small; text-align: start;">Fig 2. A simple model of exceptionality in one characteristic</span><br /><div><br /></div></td></tr></tbody></table>Now consider two characteristics, X and Y (assumed to be independent). The distribution then is represented by a 100x100 square with an inner 98x98 cube, as in Figure 2.<br /><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/-lYwIYcaoWWY/VryqmDLkSZI/AAAAAAAAKdw/94ISWc-PVoA/s1600/Devlin%2Bnew%2Bfigure.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://3.bp.blogspot.com/-lYwIYcaoWWY/VryqmDLkSZI/AAAAAAAAKdw/94ISWc-PVoA/s1600/Devlin%2Bnew%2Bfigure.png" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small; text-align: start;">Fig 3. A simple model of exceptionality in two characteristics</span></td></tr></tbody></table><br />An individual’s X measure is shown by the x-coordinate, their Y measure by the y-coordinate. The ordinary individuals are represented by points in the inner square; the exceptional individuals by points in the outer perimeter region.<br /><br />The total number of points is 100x100. The number of normal points is 98x98. So the number of exceptional points is 10,000 – 9,604 = 396.<br /><br />The proportion of exceptional points is thus 396/10,000 = 0.0396, i.e., 3.96%. Thus, more individuals are classified as exceptional when you consider two characteristics (3.96% as opposed to 2%).<br /><br />Going to three characteristics, X, Y, and Z, the model will be a 100x100x100 cube with an inner 98x98x98 cube, as in Figure 4.<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="http://2.bp.blogspot.com/-sNY53gvcS9M/Vrok_bk8b6I/AAAAAAAAKdc/Xu-qb4bYA_U/s1600/Devlin%2Bfig4.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://2.bp.blogspot.com/-sNY53gvcS9M/Vrok_bk8b6I/AAAAAAAAKdc/Xu-qb4bYA_U/s1600/Devlin%2Bfig4.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small; text-align: start;">Fig 4. A simple model of exceptionality in three characteristics</span></td></tr></tbody></table><br />The volume of the outer cube (representing the total population) is 1,000,000. The volume of the inner cube (representing the normal individuals) is 941,192. Thus the volume of the perimeter-region (representing the exceptional individuals) = 1,000,000 – 941,192 = 58,808. Hence, the proportion of exceptional individuals = 58,808/1,000,000 = 0.0588, i.e. 5.88%.<br /><br />So far, everything seems fairly straightforward and reasonable. Going beyond three characteristics, the model is a hypercube of four or more dimensions, and we can no longer provide meaningful illustrations. But by now we have grown familiar with the idea that the model represents exceptional individuals by points in the outer 1% shell. To see what this entails, let’s jump to 10 characteristics, X<sub>1</sub>,…,X<sub>10</sub>. In that case, our model will represent the situation as a 100<sup>10</sup> hypercube with an inscribed 98<sup>10</sup> hypercube.<br /><br />The volume of the outer hypercube (~ total population) = 100<sup>10</sup>. The volume of the inner hypercube (~ normal individuals) = 98<sup>10</sup>. Thus, the volume of the perimeter-region (~ exceptional individuals) = 100<sup>10</sup> – 98<sup>10</sup>, and the proportion of exceptional individuals = (100<sup>10</sup> – 98<sup>10</sup>)/100<sup>10</sup>. At this point, it’s time to bring in Wolfram Alpha to do the calculation. This gives the result that, with 10 characteristics, 18.29% of the population is exceptional.<br /><br />With 100 characteristics, X<sub>1</sub>,…,X<sub>100</sub>, our model gives: Volume of hypercube (~ total population) = 100<sup>100</sup>. Volume of inner hypercube (~ normal individuals) = 98<sup>100</sup>. Volume of perimeter- region (~ exceptional individuals) = 100<sup>100</sup> – 98<sup>100</sup>. Proportion of exceptional individuals = (100<sup>100</sup> – 98<sup>100</sup>)/100<sup>100</sup>. Calling on Wolfram Alpha again, we compute that with 100 characteristics, 86.74% of the population is exceptional.<br /><br />With 200 characteristics, X<sub>1</sub>,…,X<sub>200</sub>, our model gives: Volume of hypercube (~ total population) = 100<sup>200</sup>. Volume of inner hypercube (~ normal individuals) = 98<sup>100</sup>. Volume of perimeter-region (~ exceptional individuals) = 100<sup>200</sup> – 98<sup>200</sup>. Proportion of exceptional individuals = (100<sup>200</sup> – 98<sup>200</sup>)/100<sup>200</sup>. So with 200 characteristics, 98.24% of the population is exceptional. (Once again calling on the services of the normally unflappable Wolfram Alpha.)<br /><br />And there’s our result.<br /><br />Of course, we have been working with a model. As always, that entails making various assumptions and simplifications. If the final result surprises you, you have two choices. Either go back and modify your initial assumptions and generate another model. Or accept the result and modify the prejudices that led to your surprise. <br /><br />In this case, we have to accept that in higher dimensions, almost all the material in an equal-sided, rectangular, <i>solid</i> (!) box is on the outer shell. The (solid) inside is almost empty.<br /><br />When we consider more dimensions to a situation, the math can sometimes lead us to a counter-intuitive—but correct—conclusion we did not expect. Not everyone can accept that.<br /><br />Yes, in this US Election Season, this is a story with a moral. <a href="https://1.bp.blogspot.com/-vUhjWsgB9yc/Vryrb7op4BI/AAAAAAAAKeE/Kmjb2tYa-f4/s1600/simely.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://1.bp.blogspot.com/-vUhjWsgB9yc/Vryrb7op4BI/AAAAAAAAKeE/Kmjb2tYa-f4/s1600/simely.jpg" /></a><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br />http://devlinsangle.blogspot.com/2016/02/theorem-you-are-exceptional.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-3603082345084297799Fri, 01 Jan 2016 13:30:00 +00002016-01-28T11:01:05.113-05:00an approach that sucksBedtime MathLaura Overdeckmath appsmath learningStanford UniversityUniversity of ChicagoWuzzit TroubleDo your kids find learning math hard? There may be an app for that!<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-CV53K7QV8Es/VnHsJZHXpwI/AAAAAAAAKcE/U9gPiFFjWH4/s1600/BedtimeMath.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-CV53K7QV8Es/VnHsJZHXpwI/AAAAAAAAKcE/U9gPiFFjWH4/s320/BedtimeMath.png" width="213" /></a></div>If you are like me, you probably sigh and switch off when you read an article with a title claiming kids’ math scores show significant improvement after using some great new app for a few minutes each day. <br /><br />In which case, you may have paid little attention when a<a href="http://news.sciencemag.org/brain-behavior/2015/10/bedtime-problems-boost-kids-math-performance"> news article</a> came out in <i>Science</i> magazine recently, reporting a new study showing that after just one year of parents using a bedtime-story-telling app called<i> <a href="https://itunes.apple.com/us/app/bedtime-math/id637910701?mt=8">Bedtime Math</a></i> with their young children, those kids were three months ahead of fellow students whose parents were using an app to provide non-mathematical stories. In fact, children of math-anxious parents showed even greater improvement, ending up six months ahead. <br /><br />If you happened to see the article, you likely assumed it was essentially a piece of marketing, where a bogus “study” was carried out to produce the “results” the marketing folks wanted. After all, there are no magic bullets in the math ed world, right? <br /><br />My reaction was very different. I immediately wanted to know more. The reason being, as I recounted in <a href="http://devlinsangle.blogspot.com/2015/12/life-inside-impossible-escher-figure.html">last month’s column</a>, the same thing had happened a few months earlier with a math learning app I had created, a mobile game called <i><a href="https://itunes.apple.com/us/app/wuzzit-trouble/id600190128?mt=8">Wuzzit Trouble</a></i>. (Actually, I should have written “we” there. Paul McCartney may have sung that he had “got by with a little help from my friends,” but the reality was it took a lot of work by all four Beatles to make them a global phenomenon, and in the apps business it usually takes a whole team of highly talented people to create a great product. My team are listed <a href="http://www.brainquake.com/core/">here</a>.) A Stanford classroom study led by Prof Jo Boaler had found significant math learning after just two hours play of <i>Wuzzit Trouble</i> spread over four weeks.<br /><br />The <i>Bedtime Math</i> study was unlikely to have been a bogus marketing “study”, I felt, since it had been carried out at the University of Chicago, which is a great university. True, as the <i>Science</i> article noted, the study was funded by an entity called the Overdeck Family Foundation, whose chair, Laura Overdeck, a former astrophysics major at Princeton, established the nonprofit Bedtime Math Foundation, which created and supports the app. Some might read that and smell a rat – as some did when they first read of the Stanford <i>Wuzzit Trouble</i> study.<br /><br />But to my mind such a reaction says more about the reader than the researchers. We are talking about an educational app made and distributed for free by a nonprofit organization. Why would anyone want to fake data? Really!<br /><br />In fact, even if the app were for sale – say for a mind boggling $4.99 – the idea that seven researchers at a major university would fake a study about a small children’s app is simply not credible. As a number of news articles have made clear, the price for a university researcher faking a scientific study is dismissal from the university and the end of their career. When it does happen, the motivation is invariably massive career prestige and fame, or a huge follow-up research grant (or both). Neither of which are likely to result from an at best encouraging, small scale classroom study of <i>Bedtime Math</i>, <i>Wuzzit Trouble</i>, or any other kids’ app.<br /><br />If a foundation or a company wanted to run a fake study for marketing purposes, they could simply do it themselves, or else farm it out to an unscrupulous, individual researcher. Such people are to be found, sometimes associated with universities. (Google “intelligent design” or “climate change denial” for examples.) <br /><br />Certainly, James Stigler, a well known educational psychologist at UCLA, is not skeptical. <i>Science </i>quotes him as saying, "I think it's a fantastic study. But it is just the beginning." <br /><br />Another respected scholar, Andee Rubin, a mathematician and computer scientist at the nonprofit education R&D company TERC in Cambridge, Mass, has a similar reaction. <i>Science</i> quotes him as observing, "I'm interested in teasing it apart and seeing what makes this effective." <br /><br />Those are pretty much the same as the reaction I had to Stanford’s <i>Wuzzit Trouble</i> results, which prompted me to draw up the list of possible explanatory factors I published in my last column.<br /><br />With both the full paper and a cover article available in <i>Science</i>, all I will do here is provide a brief summary of the cover article.<br /><br />The Chicago team recruited 587 first-graders from 22 schools in the Chicago metropolitan area. The parents of each child were given a tablet computer with which to read to the child at bedtime. 420 families were told to use it to work through word problems related to counting, shapes, arithmetic, fractions, and probability using <i>Bedtime Math</i>. Another 167 families were instructed to use a reading app. With a standardized test, the researchers assessed all the subjects' mathematics performance at the beginning and end of the school year.<br /><br />As you would expect, use of the reading app made little difference to the children's math performance. In contrast, children who used the math app two or more times per week outpaced peers whose family rarely used it, ending up three months ahead.<br /><br />Perhaps most important, use of the app brought students whose parents said they were anxious about math up to par with those whose parents were at ease with the subject. Among children whose family rarely used the math app, those with math-phobic parents made only half as much progress as the children of parents comfortable with math. <br /><br />The researchers make some suggestions as to what may be going on. My own best guess, based on several months reflections on the Stanford and (subsequent) Finnish studies of <i>Wuzzit Trouble</i>, are consistent with what they think. Namely:<br /><br />We have created a system where learning is walled off from everyday life. Particularly in math. The “math classroom” operates according to its own rules. Even with a truly great teacher – and I have met many – there are many restrictions on what can be done. Not least because of an incessant rhythm of performance testing. <br /><br />Go into most math classrooms and what you see will most likely bring to mind a room full of clerks in the pre-computer age when companies employed large numbers of numerically-able people to crunch their numbers. (Young people will have to rely on old photographs or depictions in movies.) Which was, of course, what the system was set up to provide.<br /><br />The classroom certainly does not look remotely like a room full of professional mathematicians at work. The first words that might come to mind if you were to walk into such a room would be “fun”, “engagement”, “argument”, “passion”, “social interaction”. <br /><br />Nor does it look like the human activity that hundreds of thousands of years of natural selection have inbred into us to maximize learning in the young: play. (Some wise person once said that “play is the work of the child.” I agree.) <br /><br />And there is something else that evolution hard-wired into is: our love of stories. Effective political speeches are usually laden with stories of individual people, and for very good reason. Because they are powerful.<br /><br />At which point, it’s probably a good idea to do what we math instructors tell our student to do: look for patterns. Well, what do we see when we look at professional mathematicians at work and kids in their ideal learning activity?<br /><br />Fun, engagement, argument, passion, social interaction, play, stories. <br /><br />Those are all, I would argue, <b><i>essential</i></b> ingredients for good learning. Yet you would have difficulty finding <b><i>any</i></b> in many math classrooms. <br /><br />Indeed, society seems to have gotten into a mindset that these items are distractions that you have to eliminate to achieve good math learning. Even when good teachers do their best to inject some of those valuable features into their classes, they have to operate within a system that disapproves. And everyone knows that, most of all the kids.<br /><br />No wonder then, that when you have a well designed, engaging app – a game, a puzzle, a family-supporting bedtime story provider, or whatever – you will likely get good results. Because apps, if properly designed, create their own environment.<br /><br />Though I labored long and hard to create <i>Wuzzit Trouble</i>, and I am sure Laura Overdeck worked equally hard on <i>Bedtime Math</i>, all to good effect for sure, my strong suspicion is our apps work as well as they do <b><i>in such a short time</i></b> primarily because of <b><i>what they are not</i></b>. Namely they are not the typical school classroom approach to mathematics education.<br /><br />How else can you explain dramatic results after a few hours engaging with an app, other than it unlocking what had hitherto been shackled by the chains of an Industrial Revolution conception of mathematics learning? Learning something that is genuinely new takes time and a lot of effort. Freeing something that is already there – if only in embryonic form – is much quicker. If so, then this means that our received wisdom that it takes a lot of hard work, repetitive practice, frustration, tears, and pathological levels of anxiety to achieve competency in mathematics may simply be very strong evidence that our approach sucks.http://devlinsangle.blogspot.com/2016/01/do-your-kids-find-learning-math-hard.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-883903209776583603Fri, 04 Dec 2015 19:54:00 +00002015-12-16T17:40:34.303-05:00game-based learningJames Paul GeeJo BoalerM.C. EscherMonument Valley gameLife inside an impossible Escher figure<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><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-JomIwhHa-6o/VmHfWQuhX5I/AAAAAAAAKb0/8aQUlnZ9rkc/s1600/MonumentValley2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="175" src="http://3.bp.blogspot.com/-JomIwhHa-6o/VmHfWQuhX5I/AAAAAAAAKb0/8aQUlnZ9rkc/s400/MonumentValley2.jpg" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"></div><span id="goog_2140169023"></span><span id="goog_2140169024"></span><br />When the M.C. Escher inspired puzzle video-game<i> <a href="http://www.monumentvalleygame.com/" target="_blank">Monument Valley</a> </i>came out last year, I knew I had to check it out. The more so when it started getting rave reviews and winning awards. But with so many other things to hold my attention, I never managed to get round to it. The recent decision of the creators to make a version available for free prompted me to finally take a quick look. Not that it had been expensive. Rather, a tweet about the new free version happened to come when I had an hour or so of free time on my hands. <br /><br />That free hour got immediately swallowed up, as did many more hours after that. I was hooked from the getgo. <br /><br />Anyone who is intrigued by those impossible figures that Escher made famous, even those of you with little or no interest in puzzle (video-) games, will surely be captivated by <i>Monument Valley</i>, where the solution to many of the puzzles involves orienting the figure to create an illusion of a continuous object. For when the player views the object as continuous, characters in the game that traverse the figures can move along it. Impossible chasms that prevented a character’s progress suddenly disappear as you rotate the entire figure just the right way.<br /><br />It’s not a learning game. I don’t see a player learning any new mathematics. But what it does is provide a rich, immersive experience of complex geometrical spaces <i>from the inside</i>. As a player, your task is to assist the princess on her quest, which involves finding her way through a fantasy world of Escher-like structures, the geometry of which you can sometimes change as you progress. By projecting yourself into the princess, you get a sense of what it would be like to live in such a world. <br /><br />And a beautiful world it is. The creators, based in the UK and operating under the name UsTwo, have crafted a series of truly gorgeous fantasy worlds, which you encounter one after another. It is not so much a game as a collection of interactive pieces of art where you play with, and experience, geometric shapes.<br /><br />In fact, it is the artistic creation that the developers bring to the work. The idea of taking Escher worlds and turning them into a puzzle game goes back to a 2007 video game called <a href="https://www.youtube.com/watch?v=QfICeBtVv8U" target="_blank">Echodrome</a>, designed for the Sony Play Station 3 by the Japanese designer Jun Fujiki. By all accounts it was fiendishly difficult, and never broke out beyond a small group of hard-core puzzle aficionados. <br /><br /><i>Monument Valley</i> shows the huge difference presentation can make. If you want to hold people’s attention, you often need to think carefully about the medium. The message on its own may not be enough. That holds in the math class or the math lecture hall as much as in a video game.<br /><br />Regular readers of this MAA blog or my other blog profkeithdevlin.org will know that I have a long-standing interest in video games, particularly so as an educational medium, where I am professionally active as a player, a learning researcher, and an entrepreneur. <br /><br />In fact, much of my career has involved looking for ways to use different media to make mathematics accessible to as many people as possible. I have authored many “<a href="http://www.amazon.com/Keith-Devlin/e/B000APRPC6/ref=ntt_athr_dp_pel_1" target="_blank">popular mathematics books</a>”, written for newspapers (MAA compilation of some of my articles<a href="http://www.amazon.com/All-Math-thats-Fit-Print/dp/0883855151" target="_blank"> here</a>), worked on television programs (including <a href="https://vimeo.com/127338218" target="_blank"><i>A Mathematical Mystery Tour</i></a>, BBC-tv 1984; <a href="http://www.amazon.com/Life-Numbers-Danny-Glover/dp/B000FVQM10" target="_blank"><i>Life by the Numbers</i></a>, PBS 1998; and <i><a href="http://www.amazon.com/Numb3rs-The-Complete-First-Season/dp/B000ERVJKE" target="_blank">NUMB3RS</a></i>, CBS, 2005-2008), and of course there is my regular <a href="http://profkeithdevlin.com/MathGuy.html" target="_blank"><i>Math Guy</i></a> radio gig for NPR, which started in 1994. More recently, in 2012, I launched the first ever <a href="https://www.coursera.org/course/maththink" target="_blank">math MOOC</a> on Coursera (the seventh session just ended). I even made a foray into using music, song, and dance, with the 2007 show <a href="http://profkeithdevlin.com/HE.html" target="_blank"><i>Harmonius Equations</i></a>. To me, video games are one more medium to carry mathematical content.<br /><br />In fact, when it comes to K-12 mathematics, video games are in many ways the most effective medium we currently have to provide good math learning, as I tried to articulate in a <a href="http://www.amazon.com/Mathematics-Education-New-Era-Learning/dp/1568814313/ref=sr_1_1?ie=UTF8&s=books&qid=1299701150&sr=8-1" target="_blank">book</a> I wrote in 2011, and a presentation I gave at the big Teaching and Learning 2014 conference in Washington D.C. last year, a 20-min video summary of which is available <a href="https://vimeo.com/89583763" target="_blank">here</a>.<br /><br />Until recently, there was relatively little research available to put any flesh onto educators’ beliefs/hopes/suspicions that video games could yield good math learning outcomes. That is starting to change. (Reports from two classroom studies, one of I was involved in, are due to be published in the <i><a href="http://journal.seriousgamessociety.org/" target="_blank">International Journal of Serious Games</a></i> this month. Preprints are available<a href="http://documents.brainquake.com/backed-by-science/Stanford-Pope-Mangram.pdf" target="_blank"> here</a> and <a href="http://profkeithdevlin.com/Papers/Kiili-Devlin_2015.pdf" target="_blank">here</a>.)<br /><br />Certainly, the results obtained in those two papers raise more questions than they answer. (Moreover, pending further, and substantially larger, studies, the results themselves have to be viewed as tentative.) What we are seeing is that, for mathematics in the K-8 range, significant learning outcomes can be observed after a video-game intervention of as little as two hours play spread over a month or so. (Some measures show an increase of 20% over a comparison group.)<br /><br />I’d seen reports earlier that made similar claims, and dismissed them as product- marketing masquerading as research. It was only when the first of the two particular studies I cited above came out in late 2014, carried out by Prof <a href="https://www.youcubed.org/" target="_blank">Jo Boaler</a>’s research group at Stanford University’s Graduate School of Education, using my own math learning video game <i><a href="http://www.brainquake.com/our-platform/" target="_blank">Wuzzit Trouble</a></i> as the intervention, that I sat up and really took notice. <br /><br />In fact, I did more than that. Together with a research colleague from Tampere University in Finland, Prof Kristian Kiili, who, like me, has founded a math-learning video game company, and who was spending the year at Stanford, we carried out our own study. (The second of the two papers I cited.) Kiili was developing a fractions learning game, <a href="http://kristiankiili.com/semideus/" target="_blank"><i>Semideus</i></a>, and wanted to see how well it could serve as an evaluation tool. So we repeated essentially the same study Boaler’s team had done, with <i>Wuzzit Trouble</i> as the intervention, but instead of a written pre- and post-test (which the Boaler team used), we used <i>Semideus</i>. The results were very similar to those obtained in the previous study. (With the added twist that this time we found transfer — in a game context — from the whole number arithmetic of <i>Wuzzit Trouble</i> to the fractional reasoning of <i>Semideus</i>.)<br /><br />Something is going on, that’s for sure. But what? It did not take long to come up with a fairly long list of possible factors. Among the many things that a (well designed) math learning game can offer, all which are known to have a positive impact on learning, are:<br /><br /><ul><li><a href="http://www.teachinglearning2014.org/media/blog/breaking-the-symbol-barrier" target="_blank">Breaking the Symbol Barrier</a> – human-friendly representation (not the traditional abstract symbols of math textbooks).</li><li>Focus on developing number sense and problem solving ability.</li><li>High level of engagement.</li><li>Instant feedback (both positive and negative).</li><li>Steady flow of dopamine – known to have positive impact on memory formation and consolidation.</li><li>Learning through failure – in a playful, safe environment.</li><li>“Failure” treated – and regarded – as “not yet succeeded”.</li><li>Constant sense of “I can do this on the next try.”</li><li>Lots of repetition – but at the demand of the student/player.</li><li>Student/player is in control.</li><li>Student/player has ownership.</li><li>Growth Mindset – good games encourage and develop this. (This is the important notion <a href="http://mindsetonline.com/whatisit/about/" target="_blank">Carol Dweck</a> is famous for.)</li><li>Fluid intelligence (Gf) – games require and develop this. (Loosely speaking, this is the ability to hold several pieces of information in the mind at the same time and reason fluidly with them.)</li><br />I have written about many (not all) of these factors in my series of video game learning articles in my blog <a href="http://profkeithdevlin.org/" target="_blank">profkeithdevlin</a>. (See also the many writings and videos on games and learning by <a href="http://www.jamespaulgee.com/" target="_blank">Prof James Paul Gee</a>.)<br /><br />My current guess is that all of these factors, and likely others, are at play in those dramatic learning outcomes. The only way to find out for sure, of course, is to do more research. A lot more. Prof Kiili, now back in Finland, is already hard at work on that, as am I and some of my colleagues at Stanford. And we are by no means alone. The field is wide open. Stay tuned. (Even better, get involved.) Truly, it’s an exciting time to be involved in mathematics education.<br /><br />Meanwhile, I have to sign off. <i>Monument Valley</i> is calling.<br /></ul>http://devlinsangle.blogspot.com/2015/12/life-inside-impossible-escher-figure.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-5710359148963460476Mon, 02 Nov 2015 14:42:00 +00002015-11-02T10:08:08.328-05:00boolean algebraboolean logicGeorge BooleToday is George Boole’s 200th Birthday<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-8ujtPLDmUDk/Vjd1_NUXN-I/AAAAAAAAKaQ/KHAsRw9lO4I/s1600/Google.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="233" src="http://1.bp.blogspot.com/-8ujtPLDmUDk/Vjd1_NUXN-I/AAAAAAAAKaQ/KHAsRw9lO4I/s400/Google.png" width="400" /></a></div><br />Today, November 2, 2015, marks the 200th anniversary of the birth of George Boole, one of the most influential mathematicians of all time – though it would be long after his death that his influence would manifest itself, when the growth of the modern digital age made significant aspects of our lives boolean. (To the degree that adjectival use of his name is no longer capitalized nor in need of italicization.)<br /><br />Born in England, Boole spent the major part of his mathematical career as a professor at Queen’s College Cork, and the Irish mathematical community has been actively celebrating Boole’s life, work, and legacy throughout this year. Of particular note, is an Irish ballad, “The Mathematician - The Bould Georgie Boole”, specially written for the occasion and performed by the Arthur Céilí Band, which you can hear, with visual biographic accompaniment about Boole, on YouTube and Vimeo:<br /><a href="https://www.youtube.com/watch?v=05IMBfkpn_Mhttps://vimeo.com/143768018" target="_blank">https://www.youtube.com/watch?v=05IMBfkpn_M</a><br /><a href="https://vimeo.com/143768018">https://vimeo.com/143768018</a>. <br /><br />The lyric and a download link to the song are available at:<br /><a href="https://arthurceiliband.bandcamp.com/releases">https://arthurceiliband.bandcamp.com/releases</a>. <br /><br />In a more academic vein, University College Cork has created a video biography available at:<br /><a href="https://www.youtube.com/watch?v=y-eav8-EEY4">https://www.youtube.com/watch?v=y-eav8-EEY4</a>. <br /><br />And US-based Irish mathematician Colm Mulcahy has a <a href="http://blogs.scientificamerican.com/guest-blog/the-bicentennial-of-george-boole-the-man-who-laid-the-foundations-of-the-digital-age/" target="_blank">celebratory article</a> in Scientific American.<br /><br />There is a lot more available on the Boole Bicentennial that digital search technology (part of Boole’s legacy) makes easy to find, so I’ll keep this post short and let you explore on your own.<br /><br />Be sure to log on to Google today. The company logo for the day is an active demonstration of boolean algebra using colors.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-_oR0H_8RErY/Vjd2Gp6TmiI/AAAAAAAAKaY/Y1hNI0T5mHw/s1600/Google2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="263" src="http://4.bp.blogspot.com/-_oR0H_8RErY/Vjd2Gp6TmiI/AAAAAAAAKaY/Y1hNI0T5mHw/s400/Google2.png" width="400" /></a></div><br />http://devlinsangle.blogspot.com/2015/11/today-is-george-booles-200th-birthday.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-7730381965233577730Fri, 02 Oct 2015 15:14:00 +00002015-10-02T12:10:00.474-04:00appreciation of mathematicsbeauty in mathematicscalculusLetter to a calculus student – The Sequel<br />Devlin’s Angle for July 2006 was titled <a href="https://www.maa.org/external_archive/devlin/devlin_06_06.html" target="_blank">Letter to a calculus student</a>. In it, I tried to describe, as briefly but as effectively as I could, the deep beauty there is in calculus, a beauty that arises from the depth of human brilliance that it took for the human mind to find a way to tame the infinite, and bend it to our use, a beauty made the more so by the enormous impact calculus has had on life on Earth.<br /><br />In my essay, I acknowledged that there was little chance any calculus student would be able to understand what I was trying to convey. I wrote:<br /><br /><i>“Those techniques [of calculus] are so different from anything you have previously encountered in mathematics, that it will take you every bit of effort and concentration simply to learn and follow the rules. Understanding those rules and knowing why they hold can come only later, if at all. Appreciation of the inner beauty of the subject comes later still. Again, if at all.</i><br /><br /><i>I fear, then, that at this stage in your career there is little chance that you will be able to truly see the beauty in the subject. Beauty - true, deep beauty, not superficial gloss - comes only with experience and familiarity. To see and appreciate true beauty in music we have to listen to a lot of music - even better we learn to play an instrument. To see the deep underlying beauty in art we must first look at a great many paintings, and ideally try our own hands at putting paint onto canvas. It is only by consuming a great deal of wine - over many years I should stress - that we acquire the taste to discern a great wine. And it is only after we have watched many hours of football or baseball, or any other sport, that we can truly appreciate the great artistry of its master practitioners. Reading descriptions about the beauty in the activities or creations of experts can never do more than hint at what the writer is trying to convey.</i><br /><br /><i>My hope then is not that you will read my words and say, "Yes, I get it. Boy this guy Devlin is right. Calculus is beautiful. Awesome!" What I do hope is that I can at least convince you that I (and my fellow mathematicians) can see the great beauty in our subject (including calculus). And maybe one day, many years from now, if you continue to study and use mathematics, you will remember reading these words, and at that stage you will nod your head knowingly and think, "Yes, now I can see what he was getting at. Now I too can see the beauty."</i><br /><br />I then proceeded to describe, as articulately as I could, the beauty there is to be seen in calculus, or at least the beauty I see in it, taking the reader on a guided tour of the standard definition of the derivative, but from the perspective of how it takes advantage of what the human brain can do, while circumventing what it cannot.<br /><br />I ended my essay by quoting poet William Blake’s <i>Auguries of Innocence</i>, saying:<br /><br /><i>That's what [the derivative limit formula] asks you to do: to hold infinity in the palm of your hand. To see an infinite (and hence unending) process as a single, completed thing. Did any work of art, any other piece of human creativity, ever demand more of the observer? And to such enormous consequence for Humankind? If ever any painting, novel, poem, or statue can be thought of as having a beauty that goes beneath the surface, then the definition of the derivative may justly claim to have more beauty by far. </i><br /><br />As I noted above, I was really writing for my fellow mathematicians. I knew then, as I still acknowledge today, that what I had written was true: it is impossible to experience the beauty in many human creations until one has sufficient experience.<br /><br />It was then, with great pleasure, that I received the following email a few weeks ago (on August 17), which I reproduce in its entirety, unedited, with the permission of the sender. I hope you enjoy it to. And, if you are a math instructor at a college or university, maybe print off this blog post and pin it somewhere on a corridor in the department as a little seed waiting to germinate.<br /><br /><div style="text-align: center;">* * * * * *</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="http://3.bp.blogspot.com/-oSGU6TMzAos/Vg6MkQyb53I/AAAAAAAAKZo/9Yen-npdpbM/s1600/WWU.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="237" src="http://3.bp.blogspot.com/-oSGU6TMzAos/Vg6MkQyb53I/AAAAAAAAKZo/9Yen-npdpbM/s400/WWU.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Western Washington University, in Bellingham, WA</td></tr></tbody></table><br />Hi Dr. Devlin,<br /><br />My name is Murray Pendergrass. I am a math student at Western Washington University, a small public liberal arts college in the Pacific Northwest where I am pursuing a BS in Mathematics.<br /><br />Sometime around 2006 you authored a post on Devlin's Angle titled "Letter to a calculus student" and I suppose someone in the math department at my school enjoyed it because it has been tacked to a bulletin board on the math floor for quite sometime. I would have only been going into the 8th grade when it was originally posted, with absolutely no idea that I would ever become interested in mathematics. I did take a calculus course my junior year of high school, but I don't think I could even briefly explain what a derivative was by the time the course was over (time well spent, obviously).<br /><br />I must have first seen your article either my sophomore or junior year of college, 2014 most likely. I would have either been in precalculus or calculus I (differential calculus), and still completely unaware that I would end up declaring a math major. At that time I would have still been a member of the business school. I was probably waiting outside a professor's office for office hours when the title caught my eye,<br /><br />" 'Letter to a calculus student' … Hmm, maybe I should read this."<br /><br />However being the impatient person that I am, I believe I started in and thought "ok this is boring, I'll check the next page and see if it gets better,<br /><br />"Nope, second page is boring too. Oh well."<br /><br />And I have to admit, it was not until last night that I actually read the whole thing through for the first time.<br /><br />But not long after that first initial and brief encounter with the letter my passion for mathematics truly began to develop and I realized that you can actually major in math without being a child prodigy (yes I actually thought this for quite sometime). It would have been shortly after this time, less than a year ago, that I realized I wanted to major in math. Since changing majors, very few hours have been spent not working on math.<br /><br />I was studying at school late last evening when I decided to take a break and cruise up and down the hallway when for the second time in my life I noticed the letter tacked to the bulletin board. I must walk past it every single day but it was not until last night that it caught my eye again and I thought "I've seen this before! Oh wow I should give it a shot now that I am passionate about math."<br /><br />In the very first sentence you open with a quote by Bertrand Russell, someone I have taken great interest in over the last year since mathematical logic has become a particular interest of mine. I immediately knew this was going to be a whole different experience reading this letter, and I was right.<br /><br />What provoked me to feel the need to write you this letter was that I feel I am a precise example of the reader you are mentioning when you say,<br /><br />"And maybe one day, many years from now, if you continue to study and use mathematics, you will remember reading these words, and at that stage you will nod your head knowingly and think, 'Yes, now I can see what he was getting at. Now I too can see the beauty.' "<br /><br />Just as you predicted, the first time I made an attempt to read the letter "there was little chance I could see the true beauty in math", a statement so true that I could not only fail to see the beauty in math but I could not even read a letter about someone else promising me that even though I couldn't see the beauty, it was there.<br /><br />It was quite a shock to me to read the letter last night and realize what a strange coincidental experience it was to randomly come across it a year after diving head first into the world of mathematics. It felt like a testament to myself of the progress I have made in math over the last year, a type of progress that cannot be explained or noticed through grades or high marks but by reading and truly relating to a mathematicians admiration for the beauty in math.<br /><br />Before college I lived a bit of a bumpy life, it was a long and interesting road getting to where I am now. I will spare you the details as this letter has already turned out to be longer than I expected but I can truly say that finding math has been the best thing that has ever happened to me. In a lot of ways it has set me free. I am very grateful to have the opportunity to study math at a university, to study something I am passionate about, and to reflect on how my relationship with math has evolved. I also must note that I hope I don't sound naive! I know I have only been doing math for a little over a year, which might sound like child's play to a Doctor of Philosophy in Mathematics. I am ecstatic that I have reached the point where I can appreciate mathematical beauty and I am also confident that math will continue to fascinate me and reveal its beauty for many years to come. Like most things, math is a journey not a destination.<br /><br />Overall, I just felt the need to write to you because I thought you might enjoy knowing that even 9 years after you wrote it there are still students thinking for the first time:<br /><br />"Yes, now I can see what he was getting at. Now I too can see the beauty".<br /><br />Thank you,<br /><br />Murray Pendergrass<br /><br />http://devlinsangle.blogspot.com/2015/10/letter-to-calculus-student-sequel.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-5955268167176696663Tue, 01 Sep 2015 13:52:00 +00002015-09-09T09:37:29.298-04:00Asa ButterfieldIMOmath competitionsMath moviesA Brilliant Young Mind: The IMO goes to the movies<div class="separator" style="clear: both; text-align: center;"></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="http://3.bp.blogspot.com/-csyetufhC1k/VeS5Nvk3eXI/AAAAAAAAKYU/RpPAwQiiByo/s1600/ABYM6.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="266" src="http://3.bp.blogspot.com/-csyetufhC1k/VeS5Nvk3eXI/AAAAAAAAKYU/RpPAwQiiByo/s400/ABYM6.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Jo Yang as Zhang Mei and Asa Butterfield as Nathan Ellis in <i>A Brilliant Young Mind</i>. <i>Credit: Samuel Goldwyn Films</i></td></tr></tbody></table><br />Mainstream movies about mathematicians used to be a rarity, but are now fairly common. <i>Good Will Hunting</i>, <i>Proof</i>, <i>A Beautiful Mind</i>, <i>Pi</i>, <i>Cube</i>, <i>The Bank</i>, <i>Travelling Salesman</i>, <i>The Imitation Game</i> come immediately to mind. So too does <i>Stand and Deliver</i> if you include mathematics teaching. <br /><br />The titles I just listed are such good movies, there is now a high bar to success in this growing genre. In particular, the movie has to have a good story, a strong cast, and it needs to get the math right – and moreover do so in a way that intrigues the audience but does not detract from the pace of the story. The new movie <i>A Brilliant Young Mind</i>, by British director Morgan Matthews, meets that standard. <br /><br />Due to be released in the US on September 11, <i>A Brilliant Young Mind</i>, starring the hugely talented young British Actor Asa Butterfield (who starred as Ender in <i>Ender’s Game</i>, opposite Harrison Ford) was first screened in the UK last year, originally under the title “X + Y”, on which more later. <br /><br />The film focuses on the <a href="https://www.imo-official.org/">International Mathematical Olympiad</a> (IMO), the competition held annually around the world, where national teams of six pre-collegiate students compete for individual and team medals. The movie follows one particular British student as he goes through the grueling process of preparing for and taking the test to qualify for team pre-selection in the British National Mathematical Competition, going off to a training camp in Taiwan, where the final team of six is selected in a mock IMO competition, and then heading to Cambridge, England, for the international competition itself.<br /><br />Both the mathematics and the mathematics competitions are handled well. (More later.) Mathematicians will not be disappointed on that score. <br /><br />I have to admit that, on first viewing, I felt that the romantic thread between the Asa Butterfield character (Nathan Ellis) and the young female Chinese math whiz he meets at the training camp, played by Jo Yang, was a crude injection to create a movie with mainstream audience appeal. In particular, I thought the dramatic ending (you’ll have to watch the movie to find out if it is a happy or sad ending) was way over the top. <br /><br />But then I watched the original BBC documentary that <i>A Brilliant Young Mind</i> director Matthews made back in 2006, on which he based the movie, and guess what? The story in <i>A Brilliant Young Mind</i> stays pretty close to real life! Right down to what at first viewing of the movie I thought were syrupy shots included purely for cinematic romantic effect. (Cue the rainbow in the background as the British and Chinese math whizzes travel by train through the British countryside. Taken right out of real life!) <br /><br />So if my jaded-by-Hollywood mathematical colleagues find themselves, like me, lamenting to themselves, “Why do movie makers spoil the real story with all that romantic mush?” you should suppress that reaction at once. What the movie gives you is a dramatic (and dramatized) recreation of real life.<br /><br />That, by the way, is why I like the movie’s original title in the UK: “X + Y”, a nicely succinct way to link the mathematics problem solving and the romantic engagement. Still the title <i>A Brilliant Young Mind</i> does convey the idea of a young version of the brilliant (but, like Butterworth’s Nathan Ellis, mentally troubled) John Nash in the movie <i>A Beautiful Mind</i>, and Matthews himself pulled on the same association with the title of his earlier documentary.<br /><br />Despite taking his basic storyline straight out of real life, Matthews does (of course) take plenty of dramatic license in order to give us a watchable movie. He is, after all, telling a fictional story, albeit one based (unusually closely) on real life. Few in the audience will have much interest in the mathematics, or even math competitions (besides, perhaps, being surprised that such things exist), but everyone likes a good story about people. And that is what <i>A Brilliant Young Mind</i> delivers.<br /><br />In particular, I suspect many of my fellow mathematicians will also balk at the portrayal of several of the British IMO team selectees as exhibiting various forms of autism. (In real life, good mathematicians of all ages run the full spectrum of human characteristics, with <i>the vast majority</i> of math whizzes being just like everyone else.) But that aspect too is what you will find in the documentary. (But see my postscript comments at the end.)<br /><br />The IMO team members who the Butterfield character interacts with in the movie are also clearly based on real-life counterparts in the documentary. In particular, the student who has made his way onto the team by learning a lot of mathematical facts and procedures that he can regurgitate and apply at speed, but falters when it comes to having to apply original thinking. (Both the US and the UK math education systems encourage and reward that approach, which is why they do so poorly in the international PISA tests, which look for original thought. My Stanford colleague <a href="http://joboaler.com/">Jo Boaler</a> has a new book on that misguided, and sad, state of affairs, <i>Mathematical Mindsets</i>, coming out in the Fall.)<br /><br />The only complaint you could make about Matthews is the choices he made in selecting the footage he shot for the original documentary. But that is true for any documentary film. Matthews followed the UK 2006 IMO team through the entire competition process, and then told a story based on what he had captured.<br /><br />Interestingly enough, another documentary film maker, George Csicery, followed the US IMO team at the same time. You can compare the two documentaries. Matthews’ BBC documentary is <a href="http://topdocumentaryfilms.com/beautiful-young-minds/">available online</a> for free streaming. Csicery’s film <i>Hard Problems</i>, is <a href="http://www.hardproblemsmovie.com/">available for purchase</a> from the MAA ($19.95 to members).<br /><br />Enough of all these words. We’re talking about a movie, after all. Time to watch some movie clips.<br /><br />You can watch the official trailer for <i>A Brilliant Young Mind</i> <a href="http://www.rottentomatoes.com/m/a_brilliant_young_mind/">here</a>.<br /><br />Mathematicians will particularly like the one short sequence where the movie shows a brilliant mathematical mind in action, solving a problem, which you can see in isolation in this officially sanctioned YouTube <a href="https://www.youtube.com/watch?v=mYAahN1G8Y8">video</a>. It is a superbly chosen (and acted) example. No fake numerical mumbo jumbo here. Genuine mathematical thinking in action. And good mathematical thinking to boot. Any math instructor would surely love to have a student produce such a solution for the class.<br /><br />During the year the two documentaries were made, the IMO was held in Ljubljana, Slovenia (not Cambridge, England, as in the movie). You can see the actual problems the competitors faced <a href="https://www.imo-official.org/problems/IMO2006SL.pdf">here</a>. (With sample solutions.)<br /><br />Finally, for long lists of scenes in movies that feature a mathematician or a math problem, see <a href="http://www.math.harvard.edu/~knill/mathmovies/" target="_blank">here</a> and <a href="http://www.qedcat.com/moviemath/" target="_blank">here</a>.<br /><br />Enjoy the film!<br /><br /><b>EDITORIAL POSTSCRIPT</b><br /><br />Both the movie and the BBC documentary raise some issues that concern me as a mathematician. The main danger of any documentary or movie is if viewers (and for a film like <i>A Brilliant Young Mind</i>, the audience may well include young kids showing an early interest in mathematics) get the impression that what they see is representative of the field. This of course, is true for pretty well any movie, be the topic crime detection, politics, business, law, the military, sports, or whatever. I think it is particularly worrying in mathematics, because most people have a very impoverished, and often completely erroneous perception of mathematics. Both of Matthews’ films trouble me on that count.<br /><br />These thoughts are in no way a criticism of the movie I am reviewing. It is what it is. I think it is a good movie and I like it. (Though as I noted, in my case much of my positive evaluation comes from knowing that the things in the movie that initially turned me off as being unrealistic and contrived turned out to be true!) Rather, the issues I raise are general ones about the public perception of mathematics. In making both his documentary and the movie, Morgan Matthews set out to make good films. His goal was not to improve the public understanding of mathematics. That, on the other hand, is something I have spent a great deal of my career focusing on. Hence this postscript, separate from my review.<br /><br />First, it has to be said that competition mathematics is in many respects a very different activity than the professional mathematics that most of us in the business pursue. For one thing, competition math requires speed, whereas many good mathematicians are slow thinkers. (I certainly am.) <br /><br />There is also something very unusual about the kinds of problems that the IMO presents. Of necessity, they have to be solvable in at most an hour, and in many cases, the way to go about solving them is to be very conscious of that time limitation. They have to depend on seeing a particular insight or trick. Some people are naturally good at that kind of problem solving, but it can also be to some extent taught – which is what goes on at those IMO training camps. On the other hand, most mathematics problems that the pros grapple with are very different. In many cases, no one has any idea if there is a solution at all, or how long it may be.<br /><br />Moreover, the connection between being good at competition math and having the aptitude to succeed in professional mathematics is not at all clear-cut. Certainly, some IMO medal winners have gone on to pursue mathematics at university level, but not all of them have gone on to lead a successful career in mathematics. (Some have.) And many of the best mathematicians in the world have never in their lives had any interest in competition math. Though the two domains do have abstract mathematics in common, they are in many ways very different activities.<br /><br />So to a child or the parent of a child who shows aptitude toward mathematics, I would say this. If you fancy the idea of competition math, give it a try. If you do well, enjoy the experience. It will certainly show that you have some abilities that could help you succeed in a mathematical career. But if you find you do not enjoy it, or if you like it but do not do well, that in no way means you could not grow up to be a top rank mathematician.<br /><br />Another unintended message that math competitions tend to convey is that you have to have a special talent for mathematics (a “math gene” if you like). This notion that mathematics is something for the “gifted and talented” is pervasive in many cultures, and it is total BS. The two most important factors in achieving success in mathematics are wanting to do math and growing up in a supportive, educationally rich, sociocultural environment. Not only is the world of mathematics replete with examples of world class mathematicians who will tell you flat out how many hours of effort it took them to get to that point, and how others helped them on their way, there is also a growing body of evidence from nueroscience studies to support the hypothesis that mathematics, like pretty well any other human endeavor, is 5% inspiration and 95% perspiration.<br /><br />Society would do well to banish that term “gifted and talented” once and for all, and replace it with something more accurate. “Motivated and bloody hard working” is my nomination. (Individual mileages do, of course, vary.)<br /><br />My final editorial remark is something I touched upon in my review. The movie, focuses on a small group of mathematics students, and one of them in particular, who exhibit various forms and degrees of autism. True, the same was true of the young students in the director’s earlier documentary, but that clearly reflects the particular perceptual lens the director brought to the project. That lens is made dramatically clear by the opening scenes in the documentary where we meet one of the competitors, Jos. Matthews set out to portray IMO participants as being unusual and different. And he found some.<br /><br />Cleary, getting to represent their country in an international competition in of itself makes them different. But presenting them as intrinsically different is a definite editorial decision.<br /><br />Contrast Matthews’ documentary with Csicery’s. In the latter, focusing on the US team at the same IMO, the director sets out to convey the very ordinariness of the participants, highlighting not what is different about them (they love math being the main thing) but how much they are just like any other kids of that age. <br /><br /><a href="http://www.zalafilms.com/about.html">Csicery</a>, as many readers of this column will know, makes documentary films about mathematicians and mathematics as a profession, and he makes them primarily for the mathematics and mathematics education professions. So he strives to inform an audience who are interested in mathematics. This, of course, is very different from <a href="http://www.imdb.com/name/nm2051728/">Matthews</a>, who sets out to make movies that intrigue viewers who do not necessarily have any interest in the topic, whatever it is. Csicery succeeds with his audience by being as accurate and representative as he can, while also managing to tell a story. Matthews has to tell a good human interest story that hangs on some strong characters, with everything else revealed during the film. They are doing different things.<br /><br />I found Matthews’ documentary fascinating and highly engaging, and I’m really glad it inspired him to turn it into a movie. You should watch both. <br /><br />But if what you want is to get a good overall sense of the world of competitive math, you should watch Csicery’s documentary. The two documentaries provide very different perceptions of the same IMO competition.<br /><br />http://devlinsangle.blogspot.com/2015/09/a-brilliant-young-mind-imo-goes-to.htmlnoreply@blogger.com (Mathematical Association of America)1tag:blogger.com,1999:blog-2516188730140164076.post-8022703553387304016Sat, 01 Aug 2015 04:03:00 +00002015-08-01T00:03:00.121-04:00game-based learningmath video gamesHard fun – video games creep into the math classroom<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-mzOCEgJUkRM/Vbuc763Q00I/AAAAAAAAKX4/NVqaMj4z-qk/s1600/video%2Bgames.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="http://1.bp.blogspot.com/-mzOCEgJUkRM/Vbuc763Q00I/AAAAAAAAKX4/NVqaMj4z-qk/s320/video%2Bgames.jpg" width="209" /></a></div>This month’s musings were inspired by the appearance of Greg Toppo’s excellent new book <a href="http://www.amazon.com/Game-Believes-You-Digital-Smarter/dp/1137279575/ref=sr_1_1?ie=UTF8&qid=1438312811&sr=8-1&keywords=greg+toppo" target="_blank"><i>The Game Believes in You: How Digital Play Can Make Our Kids Smarter</i></a>. In it, Toppo, who is USA Today's national K-12 education writer, does an excellent job of not only surveying the current scene in educational video games, he also exhibits a deep understanding of, and appreciation for, the educational potential of well designed video games. I have gone on record as saying it will likely turn out to be the most influential book on the role of video games in education since James Paul Gee’s 2003 classic <a href="http://www.amazon.com/Video-Games-Learning-Literacy-Second-ebook/dp/B00OFL6RDE/ref=sr_1_1?s=books&ie=UTF8&qid=1438313160&sr=1-1&keywords=video+games+gee" target="_blank"><i>What Video Games Have to Teach Us About Learning and Literacy</i></a>.<br /><br />Like it or loath it, video games are slowly finding their way into the nation’s math classes, as teachers and parents increasingly see video games as a valuable educational resource. For instance, according to a recently published survey designed by the Joan Ganz Cooney Center, 55% of teachers report having their students play video games at least once a week, with 47% of teachers saying low-performing students benefited most from the use of games. (<i>Games and Learning</i>, 2015)<br /><br />Well-designed educational video games offer meaningful learning experiences based on principles of situated learning, exploration, immediate feedback, and collaboration. The power of experiential learning in engaging contexts that have meaning for learners has been demonstrated in several studies (e.g. Lave, 1988; Nunes et al, 1993, Shute & Ventura 2013).<br /><br />But when it comes to education, not all games are equal. Of the many mathematics education video games (or gamified apps) available today (Apple’s App Store lists over 20,000), the majority focus on traditional drill to develop mastery of basic skills, particularly automatic recall of fundamental facts such as the multiplication tables. They require repetition under time pressure. Such games make no attempt to teach mathematics, to explore mathematical concepts, or to help students learn how to use mathematical thinking to solve real world problems. Their purpose is purely to make repetitive drill more palatable to students.<br /><br />The proliferation of such games is in large part a consequence of the mathematics education many Americans have experienced: teacher and textbook instruction emphasizing isolated facts, procedures, memorization, and speed.<br /><br />So widespread is this educational model in the US, that many American parents, teachers, and game developers think that this is the nature of mathematics, a perception that can result in underdeveloped <i>mathematical proficiency</i>. (See, for example, Boaler 2002; Boaler 2008; or Fosnot & Dolk 2001.)<br /><br />While command of basic computation skills was a valuable asset to previous generations, in an era where fast, accurate computation is cheaply and readily available (in our pockets and briefcases, and on our desks), the crucial ability has shifted to what is often called mathematical proficiency: the ability to solve a novel problem that requires creative, multi-step reasoning, making appropriate use of computational technology as and when required.<br /><br />The National Research Council’s recognized this significant change in the nation’s mathematical needs in its seminal 2001 recommendations for the future of US K-12 mathematics education, which laid out the case for the promotion of mathematical problem solving ability, built on <i>number sense</i>. Number sense involves being mathematically proficient with numbers and computations. It moves beyond the basics to developing a deep understanding about properties of numbers, and thinking flexibly about operations with numbers.<br /><br />The last few years have seen the emergence of a tiny handful of video games designed to meet the educational requirements laid out by the National Research Council. Games such as Mind Research Institute’s K-5 focused <i>Jiji</i> games, <i>Motion Math</i>, <i>DragonBox</i>, <i>Refraction</i>, <i>Slice Fractions</i>, and my own <i>Wuzzit Trouble</i>. These games represent mathematics in a fashion native to the game medium (not the “symbolic” representations developed for the static page). They present the player with conceptually deep, complex problem solving tasks wrapped up in a game mechanic.<br /><br />As such, these games leverage the representational power of personal computers and touch-screen devices to provide students with a means to interact with mathematical concepts in a direct fashion, not mediated through a symbolic representation, thereby facilitating exploration and learning through interactive problem solving.<br /><br />In this context, it is worth reminding ourselves that the dominance of the printed symbol in the systemic mathematics education world is itself a product of the then-available technology, namely the invention of printing press in the 15th Century. Modern devices allow us to greatly expand on the symbolic interface, which for many people is a <i>known barrier</i> to mathematics learning (Nunes et al 1993, Devlin 2011).<br /><br /><b>References</b><br />Boaler, <i>Experiencing school mathematics: Traditional and reform approaches to teaching and their impact on student learning</i>, Revised and Expanded Edition. Mahwah, N.J. : L. Erlbaum, 2002.<br /><br />Boaler, “Promoting ‘relational equity’ and high mathematics achievement through an innovative mixed‐ability approach,” <i>British Educational Research Journal</i>, vol. 34, no. 2, pp. 167–194, Apr. 2008.<br /><br />Fosnot & Dolk, <i>Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction.</i> Portsmouth, NH: Heinemann, 2001.<br /><br />Devlin, <i>Mathematics education for a new era: video games as a medium for learning.</i> CRC Press, 2011.<br /><br /><i>Games and Learning</i> report, 2015. <a href="http://www.gamesandlearning.org/2014/06/09/teachers-on-using-games-in-class/#" target="_blank">http://www.gamesandlearning.org/2014/06/09/teachers-on-using-games-in-class/#</a><br /><br />Lave, 1988. <i>Cognition in Practice: Mind, Mathematics and Culture in Everyday Life (Learning in Doing)</i>, Cambridge University Press.<br /><br />National Research Council, <i>Adding It Up: Helping Children Learn Mathematics</i>. Washington, DC: National Academies Press: National Academy Press, 2001, pp. 1–462.<br /><br />Nunes, Carraher, & Schliemann, 1993. <i>Street Mathematics and School Mathematics</i>, Cambridge University Press.<br /><br />Pope, Boaler, & Milgram 2015. “Wuzzit Trouble: The Influence of a Digital Math Game on Student Number Sense”, submitted to International Journal of Serious Games.<br /><br />Shute & Ventura, 2013. <i>Stealth Assessment: Measuring and Supporting Learning in Video Games</i>, MIT Press.<br /><br />http://devlinsangle.blogspot.com/2015/08/hard-fun-video-games-creep-into-math.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-7432777528338159738Wed, 08 Jul 2015 17:04:00 +00002015-07-08T13:04:00.854-04:00Aspen Ideas FestivalDavid LeonhardtJo BoalerJordan EllenbergKhan AcademySteven StrogatzIs Math Important?This month’s column is short on words, because I want to give you time to watch a <a href="http://video.pbs.org/video/2365521689/" target="_blank">great video</a> (1 hr 18 min in length) from the recent <a href="http://www.aspenideas.org/" target="_blank">Aspen Ideas Festival</a>. It’s a panel discussion (actually, two discussions, back-to-back) hosted by New York Times journalist David Leonhardt. The topic is the question that I have chosen as the title for this post: Is math important? What makes this particularly worth watching is the selection of speakers and the views they express.<br /><br />From the mathematical world there are Steven Strogatz of Cornell University and Jordan Ellenberg of the University of Wisconsin, and from mathematics education research there is Jo Boaler of Stanford University. They are joined by David Coleman, President of the College Board, education writer Elizabeth Green, author of the recent book <i>Building a Better Teacher</i>, Pamela Fox, a computer scientist working with Khan Academy, and financier Steve Rattner.<br /><br />The conversation is lively and informative, and moves along at a brisk, engaging pace, with each speaker given time to provide in-depth answers (a refreshing antidote to the idiotic “received wisdom” that today’s viewers are not capable of watching a video longer than two-and-a-half minutes, a Big Data statistic that almost certainly says more about the abysmal engagement quality of most videos than about audience attention span).<br /><br />That’s it from me. <a href="http://video.pbs.org/video/2365521689/" target="_blank">Here</a> is the video.<br /><br />http://devlinsangle.blogspot.com/2015/07/is-math-important.htmlnoreply@blogger.com (Mathematical Association of America)0tag:blogger.com,1999:blog-2516188730140164076.post-6147616842470664912Tue, 02 Jun 2015 17:40:00 +00002015-06-02T13:40:37.771-04:00ETSliteracynumeracyOECDPIAACPisaproblem solvingPIACC – PISA for grown-ups<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-GHZ76RaEwJc/VW3evpfCGzI/AAAAAAAAKWg/0oGkSfnN4hU/s1600/ETS_cover.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="http://1.bp.blogspot.com/-GHZ76RaEwJc/VW3evpfCGzI/AAAAAAAAKWg/0oGkSfnN4hU/s320/ETS_cover.png" width="247" /></a></div>Greetings from 37,000 feet. As I write these words, I am on my way from San Francisco, California, to Boston, Massachusetts, to participate in a two-day workshop at Harvard, sponsored by the OECD (Organisation for Economic Co- operation and Development), to look at what should go into the math tests that will be administered to children around the world for PISA 2021.<br /><br />PISA, the Programme for International Student Assessment, gets such extensive press coverage each time one of its reports is published, that it really needs no introduction. Americans have grown used to the depressing fact that US school children invariably perform dismally, ranked near the bottom of the international league tables, with countries like Japan and Finland jostling around at the top.<br /><br />But chances are you have not heard of PIACC – the Programme for the International Assessment of Adult Competencies. The OECD introduced this new program a few years ago to investigate the nation-based adult skillsets that are most significant to national prosperity in a modern society: literacy, numeracy, and problem solving in a technology-rich environment (PS-TRE).<br /><br />Whereas the PISA surveys focus on specific age-groups of school students, PIAAC studied adults across the entire age range 16 to 65.<br /><br />The first report based on the PIAAC study was published in fall 2013: <a href="http://dx.doi.org/10.1787/9789264204256-en" target="_blank"><i>OECD Skills Outlook 2013: First Results from the Survey of Adult Skills</i></a>.<br /><br />A subsequent OECD report focused on PIACC data for US adults. The report’s title, <i><a href="http://dx.doi.org/10.1787/9789264204904-en" target="_blank">Time for the U.S. to Reskill</a></i>, gives the depressing-for-Americans headline that warns you of its contents. The skill levels of American adults compared to those of 21 other participating OECD countries were found to be dismal right across the board. The authors summarized US performance as “weak on literacy, very poor on numeracy,” and slightly below average on PS-TRE.<br /><br />“Broadly speaking, the weakness affects the entire skills distribution, so that the US has proportionately more people with weak skills than some other countries and fewer people with strong skills,” the report concluded.<br /><br />I have not read either OECD report. As happened when I never was able to watch the movie Schindler’s List, it is one of those things I feel I ought to read but cannot face the depression it would inevitably lead to. Rather, for airplane reading on my flight from Stanford to Harvard, I took with me a recently released (January 2015) report from the Princeton, NJ-based Educational Testing Service (ETS), titled <a href="http://www.ets.org/millennials" target="_blank">AMERICA’S SKILLS CHALLENGE: Millennials and the Future</a>.<br /><br />The ETS report disaggregates the PIAAC US data for millennials—the generation born after 1980, who were 16–34 years of age at the time of the assessment.<br /><br />The millennial generation has attained more years of schooling than any previous cohort in American history. Moreover, America spends more per student on primary through tertiary education than any other OECD nation. Surely then, this report would not depress me? I would find things to celebrate. <br /><br />Did I? Read on.<br /><br />This month’s column is distilled from the notes I made as I read through the ETS report. (These are summarizing notes. I did not bother to quote exactly, or even to use quotation marks when lifting a passage straight from the report. The originals of all the reports cited here are all freely available on the Web, so please go to the source documents to see what was originally written.)<br /><br />A central message emerging from the ETS report is that, despite all the costly and extensive education, US millennials on average demonstrate relatively weak skills in literacy, numeracy, and problem solving in technology-rich environments, compared to their international peers. Sigh.<br /><br />And this is not just true for millennials overall, it also holds for our best performing and most educated young adults, for those who are native born, and for those from the highest socioeconomic background. Moreover, the report’s findings indicate a decrease in literacy and numeracy skills for US adults when compared with results from previous adult surveys.<br /><br />Some of the data highlights: <br /><ul><li>In literacy, US millennials scored lower than 15 of the 22 participating countries.</li><li>In numeracy, US millennials ranked last.</li><li>In PS-TRE, US millennials also ranked last.</li><li>The youngest segment of the US millennial cohort (16- to 24-year-olds), who will be in the labor force for the next 50 years, ranked last in numeracy and among the bottom countries in PS-TRE.</li></ul>Even worse for those of us in higher education, this dismal picture holds for those with higher education:<br /><ul><li>US millennials with a four-year bachelor’s degree scored third from bottom in numeracy.</li><li>US millennials with a master’s or research degree were fourth from bottom.</li></ul>All very depressing. I fear that this state of affairs will continue all the time US education continues to be treated as a political football, with our nation’s children and their teachers treated as pawns while various groups fight political battles, and make decisions, based not on learning research (of which there is now a copious amount, much of it generated in US universities) but on uninformed beliefs and political ideology. [You were surely waiting for me to throw in my two cents worth of opinion. There it is.]<br /><br />To finish on a high note, we Americans famously like winners. So let’s raise a glass to the nations that came out on top in the rankings (in order, top first):<br /><br />Literacy: Japan, Finland, Netherlands<br /><br />Numeracy: Japan, Finland, Belgium<br /><br />PS-TRE: Japan, Finland, Australia<br /><br />In their own way, each of these countries seems to be doing education better than we are.<br /><br />Yet here’s the fascinating thing. I’ve spent time in all of those countries. They each have a lot to offer, and I like them all. I also was born and grew up in the UK, moving to the US as an adult in 1987. I am a lifelong educator. But for all its faults (and its education system is just one of a legion of things America does poorly) I’d rather live where I do now, in the USA, with Italy in second place. But that’s another story. A complicated story. (If you think California is a separate nation, and in many ways it is, then my preference statement needs further parsing.) Doing well on global tests of educational attainment is just one factor that we can use to measure quality of life.http://devlinsangle.blogspot.com/2015/06/piacc-pisa-for-grown-ups.htmlnoreply@blogger.com (Mathematical Association of America)0