tag:blogger.com,1999:blog-25161887301401640762024-03-12T22:39:37.410-04:00Devlin'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.Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.comBlogger90125tag:blogger.com,1999:blog-2516188730140164076.post-3249741693231692382019-01-03T14:50:00.000-05:002019-01-03T14:50:20.577-05:00New Devlin's Angle posts can be found on the Mathematical Association of America's <a href="https://www.mathvalues.org/devlinsangle" target="_blank">Math Values</a> blog. This site will remain live as an archive for all previous posts.Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-36360580797974085172018-11-30T15:36:00.000-05:002018-11-30T15:36:46.899-05:00To Boldly Go …This month, it will be exactly 22 years since the MAA first went online. After its initial release in 1994, the web browser <i>Netscape</i> had, by 1996, started to acquire users rapidly, in the process turning the new World Wide Web from a scientists' communications platform into a citizens' global network. Like many organizations, the MAA was quick to establish a presence on the new communication medium. In December 1996, the Association launched <i>MAA Online</i>. It’s the platform on which you are reading these words, though the word “Online” was eventually dropped, when it no longer made sense to call out its online nature!<br />
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At the time, I was the editor of the MAA’s flagship, members (print-) magazine FOCUS, which was sent to all members six times a year. I had become the editor in September 1991, and would continue through until December 1997. As such, I was involved in the process of getting the Association’s new online presence off the ground—or more precisely, into the (ethernet) cable.<br />
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With FOCUS being the primary way the Association informed members of its activities, it fell to me to get the word out that there was a new kid on the block. I reprint below the FOCUS editorial that I used to spread the news. If you are under forty, this might provide some insight into how the “world of online” looked back then.<br />
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Note in particular that I went to some lengths to reassure members that the new medium would be <i>an optional addition</i> to the Association’s existing offerings. There was a general feeling among the MAA officers that not every member would leap to adopt the new technology. Indeed, many of them did not have access to a computer, let alone own one. Note also that I gave assurances that FOCUS would not go away. And indeed, the magazine remains with us to this day. (Though most of the advantages I listed for a print magazine have long been obliterated by technology.)<br />
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For the rest of us, it can provide a short trip down memory lane. Enjoy the ride!<br />
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FOCUS, December 1996 Editorial<br />
<b><br />Spreading the Word, at 186,000 miles per second</b><br />
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Here at FOCUS we put in heroic efforts to ensure that your bimonthly MAA news magazine reaches you as rapidly as possible. But for all our efforts, almost two months elapse between the moment we stop accepting copy and the mailing out of your copy of FOCUS.<br />
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Things move much faster for my colleague Fernando Gouvêa, the editor of<i> MAA Online</i>. If necessary, he can even beat the New York Times in getting the news out. While FOCUS moves at the speed of overnight delivery during the production stage and the speed of second class U.S. mail for distribution, <i>MAA Online</i> travels at the speed of light through optical fiber and electrons through copper wire. Corrections can be made at any time, in an instant.<br />
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There is no doubt then that if you want up-to-the-minute information about the MAA, you would be advised to consult <i>MAA Online</i>. If you are reluctant to do so because you prefer the professional magazine look of FOCUS that you have become used to, think again. <i>Online</i> is no text-only database. It’s a full-color, professionally laid-out, typeset magazine, with masthead, photographs, and illustrations. Just like FOCUS, in fact, only with full colors.<br />
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And what’s more, where FOCUS often abbreviates articles or entirely omits important stories, items, and reports, due to limitations of space, <i>MAA Online</i> gives you the whole thing—all the MAA news that’s fit to print. Care to look at that long report the Association just put out? You’ll find it in <i>MAA Online</i>. Want to know the current members of the Board of Governors? That’s on <i>Online</i> as well.<br />
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In short, with the arrival of <i>MAA Online</i>, the whole news reporting structure of the MAA has changed. Or at least, it is in the process of changing. Aware of the fact that many members do not yet have full access to the World Wide Web, FOCUS is still carrying all the really important news stories—or at least as many of them as it always has. But the writing is on the wall—or more accurately on the computer screen. As far as news and the full reporting of committees are concerned, <i>MAA Online</i> is where tomorrow’s MAA member will turn.<br />
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What place then for FOCUS?<br />
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Well, ultimately that is a question not for me but for the Association as a whole, as represented through its Board of Governors and the appropriate elected committees. But I can give you my thoughts.<br />
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I don’t see the growth of <i>MAA Online</i> as heralding the end of FOCUS any more than the arrival of radio brought an end to newspapers or the introduction of television brought an end to the cinema. I suspect I share the view of most MAA members that there is something very significant—indeed symbolic—about receiving our copy of FOCUS every two months. Its very physical tangibility makes it a “badge of membership.” Receiving FOCUS, which for many members is the only MAA publication they receive regularly, is a significant part of what it means to be a member of the Association. Apart from renewing your membership once a year, all that is required of you to obtain the latest issue of FOCUS is to empty your mailbox. You don’t have to remember to log on to your computer, launch Netscape, and bookmark into http://www.maa.org. FOCUS may take its time to reach you, but it does so reliably, like an old friend. And what’s more, you can take it with you to read in bed, on the train, bus, or plane, in the coffee room, in the garden, or wherever.<br />
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Launched by MAA Executive Director Marcia Sward in 1981, FOCUS is now a part of the very identity of the MAA. Over the years, it has grown and developed in response to the changing needs and expectations of the membership. And that is as it should be. Of course, it will continue to change and evolve, and one of the forces that will guide its change is the newly arrived presence of <i>MAA Online</i>. That too is as it should be.<br />
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One change you will notice from this issue onward is that FOCUS will carry pointers to articles and reports in <i>Online</i>, with just a brief summary or extract appearing in the hard copy magazine you hold in front of you. No doubt further changes will follow.<br />
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In the meantime, from the editor of FOCUS, let me say a formal “Welcome” to our new sibling, <i>MAA Online</i>.<br />
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<i>The above opinions are those of the FOCUS editor and do not necessarily represent the official view of the MAA.</i>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-73671681843435497972018-11-02T09:01:00.000-04:002018-11-02T10:43:46.823-04:00T-assessment: a bold suggestion modestly advanced I sometimes use this column to float an idea I think deserves attention. Not on a whim, but after considerable thought and discussion with others expert in the relevant domain(s). This is one of those times. I already set the scene with last month’s post. Here is a nuanced, bullet-point summary of what I wrote then: <b></b><br />
<ol>
<li>The heart of learning mathematics is mastering a particular way of thinking – what I (and some others) call “mathematical thinking,” sometimes also described as “thinking like a mathematician.”</li>
<li>You can master mathematical th<b></b>inking by focusing on any <b><i>one</i></b> branch of mathematics – arithmetic, geometry, algebra, trigonometry, calculus, etc. – and going fairly deep.</li>
<li>Once you have mastered mathematical thinking, you can fairly quickly acquire an <b><i>equivalent</i></b> mastery of <b><i>any</i></b> branch of mathematics with relatively shallow coverage. (There are limits to this. There is a complexity and abstraction hierarchy of branches of mathematics. But for K-12 mathematics, that is not an issue.)</li>
<li>Thus, to learn mathematics effectively, it suffices to (i) master mathematical thinking by the study of one branch of the subject, and (ii) acquire some breadth by branching out to a few other areas.</li>
<li>For the same reason, it meets society’s need for assessment of mathematics learning if we (i) assess mathematical thinking restricted to one branch, and (ii) measure the individual’s <b><i>knowledge</i></b> in a number of other branches.</li>
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If you buy this picture – and if you keep reading, I am going to try to sell it to you – then it means we can make use of modern technologies to be far more efficient, in terms of teacher and student time and of money spent, for both learning and assessment. I propose calling this approach T-learning and T-assessment, on account of the visualization of the model shown below.<br />
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The vertical of the T denotes the topic that is studied <b><i>in depth</i></b> to build mathematical thinking capacity. In last month’s post, I discussed a suggestion by mathematics learning expert Liping Ma that school arithmetic is the best subject for doing that, so I have put that down for the vertical. Read last month’s post to see my summary of her argument, and follow the link I gave there if you want to know more. But school arithmetic is not the only choice. Euclidean geometry could also work. In both cases (or with any other choice), it would be important to teach the T-vertical the right way, so as to bring out the general <b><i>mathematical thinking</i></b> patterns.<br />
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The horizontal bar of the T represents the collection of topics chosen to provide <b><i>breadth</i></b>. Each of the topics on the bar can be covered relatively quickly, once mathematical thinking has been mastered.<br />
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In my <a href="https://www.coursera.org/learn/mathematical-thinking" target="_blank">Introduction to Mathematical Thinking</a> MOOC on Coursera, which has been running regularly since 2013, I used the structure of everyday language as the T-vertical, and some topics in elementary number theory for the bar of the T. I only needed one branch of mathematics on the bar, since the goal was to teach mathematical thinking itself, and for that, one application domain was enough. (I chose number theory since you need nothing more than arithmathetic to get into the early parts of the subject.) If the goal is to cover everything in the Common Core State Standards, you’d need a number of branches of mathematics (though only a handful).<br />
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So much for the diagram. Before I launch into my efficiencies sales pitch, let me make a few remarks about the itemized list above.<br />
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<b><i>Item 1</i></b>. I fear many readers will not really understand what I mean by this. Mathematicians surely will. But a sad consequence of the way mathematics has typically been taught, as a smorgasbord of definitions and facts to learn, tricks to remember, and procedures to practice, is that relatively few people survive their math education long enough to realize that the entire discipline revolves around a very small collection of thought patterns. I discussed this tragedy at some length in my last month’s post, citing some research results of my Stanford colleague Prof. Jo Boaler that show just how great a tragedy it can be.<br />
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Dr. Boaler is a former school teacher, education system administrator, and more recently a world-renowned mathematics education scholar of many years standing. She is one of a number of mathematics pedagogy experts I work and/or consult with. I mention that because my primary expertise is in mathematics, a discipline I have worked in for half a century. I do have a fair amount of knowledge of mathematics pedagogy, but purely as a result of studying the subject fairly extensively. I have not and do not engage in original research into mathematics pedagogy. I cannot, therefore, claim to be an expert in that domain. The suggestions I make here are, as always, in my capacity as a mathematician.<br />
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<b><i>Item 2</i></b>. As Boaler described, many students come away from the math class in despair, complaining that there are way too many things to remember. Yet, if you think back to your school days, there were probably one or two kids in the class for whom it seemed effortless. What was their secret? Were they simply math geniuses? Were they, as some parents like to say, “gifted”? I’ll answer those three questions in reverse order.<br />
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Very rarely, in various areas of human endeavors, exceptional individuals come along. That’s simply a feature of distributions with an element of randomization. But for the most part, children classified as “gifted” are simply the offspring of relatively well-off, educated parents who provide their children with excellent early role models and an educationally stimulating start in life. That is their “gift.” And indeed it is a gift; they were <b><i>given</i></b> it – by their parents. I point this out because, as research by Boaler and others has shown, labeling a child as “gifted” frequently turns out to have crippling consequences for that child. Having been told they were “gifted,” the child assumes that whatever level of effort leads to initial success in the math class will continue to do so – they rely on their “gift.<span style="background-color: transparent; color: black; display: inline; float: none; font-family: "times new roman"; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;">” </span>But in mathematics, as in many walks of life, the further you get into something, the harder it gets. While a child who accepts struggle and failure in math as part of the learning process will often keep trying, the “gifted” child may well (and often does) give up when they are no longer acing all the tests, perhaps claiming that they simply no longer found it interesting, in order to mask (from themselves as much as anyone else) the devastating consequence to their self-esteem that results from their having built up no tolerance of failure. (We learn when we get something wrong and figure out why. Getting something right simply gives us reassurance and maybe makes us feel good for a while.)<br />
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The “math geniuses” question. Unlike some of my colleagues, I don’t mind that term being used, as long as it is understood to refer to an individual who (a) was born with a brain particularly well suited to mathematical thinking, (b) found mathematics totally fascinating (for whatever reason, perhaps a desire to escape a miserable childhood environment by retreating into the mental world of mathematics), and (c) devoted thousands of childhood hours working on mathematics. For those are the three ingredients it takes to produce an individual who could merit being called a “genius.” There are very few such math geniuses in the world. In contrast, I suspect (on numerical grounds) there are a great many children born with a brain suited to mathematical thinking, who never pursue, or show prowess in, mathematics. The term “born genius,” which you sometimes come across, strikes me as idiotic.<br />
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The “What was their secret?” question. Those kids in your class who seemed to find math easy were the ones who, for whatever reason, managed to recognize that, for all that math was presented to them as a jumble of tricks and techniques, there was method to the seeming madness. Not just method, but a fairly simple method. Mathematics, they realized, was a theme-and-variations affair. There was no need to <b><i>memorize</i></b> anything beyond the basic multiplicative number bonds (the “times tables” as they were called when I was a lad growing up in the UK) – which even in today’s computation-rich environment are extremely useful to have at your mental fingertips.<br />
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ASIDE: Fortunately, the multiplicative number bonds can be committed to memory by using numbers often enough in meaningful contexts. But to my mind, since they can be mastered by rote (or even better, by playing one of several cheap, first-person-shooter, multiplication video games), you might as well get them out of the way as quickly as possible by a repetitive memorization process. Moreover, there is mathematical thinking mileage to be gained by this approach, when kids discover that there are various patterns that can be used to avoid actively memorizing most of the multiplication facts (x5, x10, and commutativity are three such time-saving patterns), leaving only a handful that have to be actually learned (6 x 7 and 7 x 8 are two such – though to this day I don’t have instant recall of 6 x 7, but rely on commutativity and instant recall of 7 x 6 – don’t ask!).<br />
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But I digress. The point is, that kid on the front row who annoyingly seemed to remember everything almost certainly <b><i>remembered almost nothing</i></b>; they worked out most of their answers on the fly. As I did with my answers for 6 x 7 and 8 x 9. (Okay, I guess I was one of those annoying kids.)<br />
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The point is, the crucial importance of approaching math learning as a process of acquiring a particular way of thinking does not just apply in the elementary grades – where many kids do manage to get by with pure memorization. The same is true all the way up into the more advanced parts of the subject, where memorization becomes impossible. Yet there is<b> <i>no need to memorize</i> </b>much of anything. Ever. Just a few key concepts, facts, and procedural details in each new branch of mathematics. In fact, if you continue in mathematics, after a while you realize that <b><i>all</i></b> the different branches of mathematics share what is essentially the same structure.<br />
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And the really nice thing is, mastering mathematical thinking to an adequate degree is like learning to ride a bike or to swim. Once you have it, you never lose it.<br />
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So much for the first two items on my initial list. But in elaborating on those, I’ve essentially covered <i><b>Item 3</b>, <b>Item 4</b></i>, and <i><b>Item 5</b> </i>as well. So we are done with that.<br />
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In fact, we’ve got so much useful stuff on the table now, it’s pretty straightforward to make that efficiencies sales pitch I promised you.<br />
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Current systemic assessments rely to a very high degree on digital technology, where students take a test presented and answered on a computer, which automatically grades their answers. To fit that format, questions are restricted to multiple-choice questions, questions that require a entry of single number as an answer, or some minor variant of one of these question types. (Earlier assessments used multiple-choice tests printed on paper that the student filled in with a pencil, with the completed test-paper then optically scanned into a computer.) This is fine for assessing what a student has learned on the horizontal (breadth) bar of the T. But on its own, the results of such a test are <b><i>essentially useless</i></b>. They measure either facts memorized or shallow (and often brittle) procedural manipulations based on memorized facts. They say nothing about an individual’s ability to think mathematically.<br />
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That is why the better systemic assessment systems on the market also present students with open-ended questions where the student has to solve a problem using paper and pencil, with the solutions for a whole class, school, or district being sent out for grading by trained human evaluators, who follow an evaluation rubric. Though this process does bring in an element of graders’ subjectivity – even with a well-thought-out and clearly expressed rubric, the graders are still faced with an often formidable interpretation task – it works remarkably well. But it is both time-consuming and expensive. It tends to be used only for major, summative assessments at the end of a unit or a school year. The time-delay alone makes it unsuitable for formative assessments intended to provide feedback to students about their progress and to alert teachers to the need for individual-student interventions or changes in the rest of the course.<br />
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With the T-model, only the core subject chosen to constitute the T-vertical needs to be assessed this way, of course. Even with existing assessment methods, making that restriction could lead to <b><i>some</i></b> cost reduction. But<b><i> substantial savings</i></b>, in personnel, time, and money, would be obtained if the subject in the T-vertical could be assessed automatically. With today’s technologies, it can.<br />
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To obtain good assessments of mathematical thinking, educators typically present students with what are known as “complex performance tasks” (or “rich performance tasks”), requiring multi-step reasoning. CPTs often (though not always) have more than one “correct” answer, with some answers being better than others. Even when there is a unique answer, there is frequently more than one solution (= sequence of reasoning steps) that gets to that answer. CPTs can range from very basic tasks, perhaps requiring only one or two individual steps (though with a period of reflective thought required in order to start) to the fiendishly difficult.<br />
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Some kinds of CPT (particularly in subjects such as arithmetic, geometry, and algebra) can be implemented as digital puzzles, where the student has to manipulate objects or symbols on a computer screen in order to find the solution. When deployed in this format, such CPTs can be used as systemic assessment tools. Not all mathematical subjects or topics lend themselves to this kind of presentation, so it is not a feasible approach for systemic assessment <b><i>as currently conceived</i></b>. But for T-assessment, it can work just fine. Simply specify the T-vertical to consist of mathematical topics that can be assessed using digitally-implemented CPTs. Because the assessment is conducted on the computer, the student’s entire solution to the CPT is captured and can be analyzed by an algorithm. In real-time. At no incremental cost. At whatever scale is required.<br />
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Of course, the key requirement here is to have a mathematical topic, or set of topics, and a set of CPTs in that area, that is collectively sufficient to demonstrate mathematical thinking ability. For that, remember, is what the T-vertical is all about.<br />
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Such digital assessment tools already exist. (Full disclosure: I am a member of one team developing and testing such tools.) So far, they have been subjected to limited testing on a small scale. The results have been encouraging. Conducting large-scale trials is clearly a necessary first step before they can be deployed in the manner I am suggesting. Moreover, to be useful, mathematics education has to be configured according to the T-model, where an in-depth study of one part of mathematics is used to develop the key capacity of mathematical thinking, coupled with much more shallow experiences in a number of other parts of math to achieve breadth.<br />
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That’s my suggestion. In putting it out, I might hear back that others have thought about, or advocated, something very much along the same lines. (In fact, Liping Ma essentially did just that in the article I discussed in my last post, albeit not in terms of the use of digital puzzles to provide automated assessments.)<br />
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I may also hear from psychometricians who will instantly recognize difficulties that would need to be overcome to put my proposal into practice. In fact, having talked with psychometricians, I am already aware of some issues that would need to be taken into account. Psychometrics is another of those disciplines of which I have some superficial knowledge but in which I have no expertise. But I have not yet encountered any reason why my suggestion cannot be made to work. (If I had, you would not be reading this article.)<br />
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To my mind, the really challenging obstacle is for the mathematics education establishment to accept, and then adopt, the T-model. Fifty years experience as a professional mathematician (the first fifteen or so in abstract pure mathematics, the remainder in various applied fields) has left me in no doubt that the T-model is not only perfectly viable, it is far superior to the “broad curriculum” approach we currently use, often referred to (derisively, but justifiably) as “a mile wide and an inch deep” education. But I am not in a position to mandate educational change. Nor, frankly, have I ever wanted to work my way into a position where I could have such influence. I like doing and teaching math too much! Instead, I am using what platform I have to put this suggestion out there in the hope that those who do have influence might take up the idea and run with it.<br />
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Of course, I can keep repeating my message. In fact, you can count on me doing that. :)<br />
<br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-79205134086618927502018-10-05T09:01:00.001-04:002018-10-05T10:15:50.351-04:00It's high time to re-focus systemic mathematics education - and change the way we assess it“In math you have to remember, in other subjects you can think about it.” That statement by a female high-school student, was quoted by my Stanford colleague Prof Jo Boaler in her 2009 book <a href="https://www.amazon.com/Whats-Math-Got-Do-Teachers/dp/0143115715/ref=sr_1_1?ie=UTF8&s=books&qid=1272973115&sr=1-1" target="_blank">What's Math Got To Do With It?</a> I took it as the title of my June 2010 <a href="https://www.maa.org/external_archive/devlin/devlin_06_10.html" target="_blank">Devlin’s Angle post</a>, which was in part a review of Boaler’s book. In a discussion peppered with quotations similar to that one, Boaler describes the conception of mathematics expressed by the students in the schools where she conducted her research. To those students, math was a seemingly endless succession of (mostly meaningless) rules to be learned and practiced. Among the remarks the students made, are (the highlighting is mine):<br />
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<i>“We're usually set a task first and <b>we're taught the skills needed to do the task</b>, and then we get on with the task and we ask the teacher for help.”</i><br />
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<i>“You're just set the task and then you go about it ... you explore the different things, and they help you in doing that ... so <b>different skills are sort of tailored to different tasks</b>.”</i><br />
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<i>“In maths, <b>there's a certain formula to get to, say from A to B, and there's no other way to get it.</b> Or maybe there is, <b>but you've got to remember the formula, you've got to remember it</b>.”</i><br />
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Given that is their experience of mathematics, there is no surprise that many students that are taught that way give up and bail out at the first opportunity. In fact, a more natural question is, “Why do a few students enjoy math and do well in it, answering questions at the board with seeming ease.”<br />
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The answer is, the students who do well in math and enjoy it, are <b><i>doing something very different from the activity described in the above quotes</i></b>. Indeed, one of the things that attracts students to math is that it is the subject where <b><i>you have to learn the least number of facts or methods</i></b>, and can spend most of your time in <b><i>creative thinking</i></b>. Those “good” students have discovered that, in the math class, there is relatively little you have to remember; most of the time, you can wing it, and you’ll do just fine. It’s not about learning a wide range of formulas and special techniques, the trick is to learn to <b><i>think a certain way</i></b>.<br />
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For the few who know the “one big trick” professional mathematicians rely on, math class is an engaging and enjoyable creative experience. How those few get to that point seems to be exposure to an inspiring teacher at some point, hopefully before the rot sets in and the student has been completely turned off math, or perhaps some other fortuitous event. Absent such a stimulus, though, it’s no surprise that when fed a steady diet of math classes focused on mastering one concept, formula, or special technique after another, the majority sooner-or-later give up, and simply endure it (in bored frustration) until they are through with it.<br />
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Which brings me to the mathematics <a href="http://www.corestandards.org/Math/" target="_blank">Common Core State Standards</a>, rolled out in 2009 to guide developments in education required to meet the changing environment and needs of the 21st Century.<br />
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If you go onto the CCSS website, you will find a large database of specific standards items, one such (which I picked at random) being<br />
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CCSS.MATH.CONTENT.5.MD.C.5. Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.<br />
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Parsing out the reference code for this particular standard, it relates to Grade 5, Measurement & Data, Geometric measurement: understand concepts of volume, item 5.<br />
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It is important to realize that the Common Core is not a curriculum, nor does it stipulate how any topic should be taught. It is exactly what its name indicates: a set of <b><i>standards</i></b> that educators should aim to meet at each stage. But it is tempting to view it as a list of specific topics that a teacher should cover one after another. (Tempting, but not easy if you are working from the website since it isn’t presented as a list. I suspect that is deliberate.) If you do that, then there is an obvious danger that the result will be a continuation of the approach to math education that students experience as a process of learning one little trick after another, and you are back with the situation Boaler catalogued.<br />
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But those individual CC items are the terminal-nodes on a branching tree that has a regular structure, and it is in that structure that you see not only order but just a handful of basic principles. It is those basic principles that should guide math instruction. There are just eight of them. They are called the Common Core State Standards for <a href="http://www.corestandards.org/Math/Practice/" target="_blank">Mathematical Practice</a>. Here they are:<br />
<ol>
<li><b>Make sense of problems and persevere in solving them.</b></li>
<li><b>Reason abstractly and quantitatively.</b></li>
<li><b>Construct viable arguments and critique the reasoning of others.</b></li>
<li><b>Model with mathematics.</b></li>
<li><b>Use appropriate tools strategically.</b></li>
<li><b>Attend to precision.</b></li>
<li><b>Look for and make use of structure.</b></li>
<li><b>Look for and express regularity in repeated reasoning.</b></li>
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Those eight principles (the website elaborates on each one) constitute the core of the mathematics Common Core. They encapsulate the key features of mathematics learning essential for anyone living or working in today’s world. Notice that there is nothing about having to learn specific facts, formulas, or techniques. The focus is entirely on <b><i>thinking</i></b>.</div>
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The same is true of the specific items you will find in the rest of the Common Core website. When you drill down, you will find targets to aim for in order to <b><i>develop thinking</i></b>, following those eight principles at each grade level. </div>
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When mathematics is taught as a way of thinking, along the lines specified in those eight Common Core principles, then along the way, a student will in fact pick up a whole range of facts, and meet and learn to use a variety of formulas and techniques. But the human brain does that naturally, as an automatic consequence of lived experience. We are hardwired that way! </div>
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In contrast, learning becomes hard when presented as a sequence of items to be learned and practiced one by one, each in isolation, based on the false premise that you must first learn the “basics” before you can “put them together” to form the whole. The moment you realize that mathematics is about <b><i>process</i></b> rather than content, about doing rather than knowing, the absurdity of the “must master the basics first” philosophy becomes apparent.</div>
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Notice I am not saying “the basics” are irrelevant. Rather, they are picked up far more easily, and in a robust fashion that will last a lifetime, by <b><i>using</i></b> them as part of living experience. For sure, a good teacher can speed the process up by helping a student recognize the used-all-the-time basics, and maybe provide instruction on how they can be used in other contexts. But the focus at all times should be on the thinking process. Because that’s what mathematics is!</div>
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If the above paragraph sounds a bit like learning to ride a bike, then all to the good. A child learning to ride a bike will acquire a good understanding of gravity, friction, mechanical advantage, and a host of other physics basics. An understanding that a physics teacher can use to motivate and exemplify lessons in those notions. But no one would say that you cannot learn to ride a bike until you have mastered those basics! Think of doing math as a mental equivalent to riding a bike. (I wrote about this parallel in Devlin’s Angle before, in my <a href="http://devlinsangle.blogspot.com/2014/03/how-mountain-biking-can-provide-key-to.html" target="_blank">March 2014 post</a>. My final point there was somewhat speculative, but as a mathematician who also rides bikes, I claim that the overall parallel between the two activities is very strong and illuminating.) </div>
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Now comes the point where I part company with the CCSS, and indeed much of the focus in present-day American mathematics education and standardized math assessment.</div>
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Let me ease myself in by way of my cycling. I learned to ride a bike as a child and used a bike to get around throughout my entire childhood up to graduation from high school. I then hardly ever got on a bike again until I was 55 years old and my knees gave out after a quarter century of serious running, and I bought my first (racing-style) road bike. For the first twenty minutes or so, I felt a bit unsteady on my new recreational toy, but I did not need to seek instruction or help in order to get on it that first time and ride. The basic bike-riding skill I had mastered as a small child was still there, available instinctively, albeit a bit rusty and in need of a bit of adjusting for the first few minutes. </div>
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Moreover, when I started riding with a local club, my fellow riders gave me lots of tips and advice that made me able to ride more safely and at higher speeds. Some of what I learned was not obvious, and I had to practice. It was not the same as riding a city bike at low speed.</div>
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Likewise, when I bought, first, a mountain bike, and then a gravel bike, I had to take my basic bike-riding ability and transfer it to a different device and different kinds of terrain, and, in each case, once again learn from experts how to make good, safe use of my new machine.</div>
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The point is, they were all bicycles and it was all cycling. So too with mathematics. Once someone has mastered — truly mastered — one part of mathematics, it is relatively easy to master another. Yes, you will need to learn some new things, including a new vocabulary, some new techniques, and likely a new ontology, and yes you will almost certainly benefit from (and possibly need) help, guidance, and advice from experts in that new area of math. But you already have the one key, crucial ability: <b><i>you can think like a mathematician</i></b>.</div>
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In terms of learning mathematics, what this means is that it is enough to devote considerable effort to genuinely mastering <b><i>just one topic </i></b>— say elementary arithmetic — and then spending some time going through the process of branching out from that one area to a number of others (perhaps algebra, geometry, trigonometry, and probability theory).</div>
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In terms of mathematics assessment, it means that it is enough to test students’ <b><i>mathematical thinking ability</i></b> focused on just one topic, and then test to see if they have <b><i>sufficient knowledge</i></b> of a number of other topics. The advantage of approaching assessment this way is that, at least with current assessment methods, testing thinking is time-consuming and expensive, since it requires a small army of trained human assessors to grade solutions to open-ended questions, often “complex performance tasks,” whereas assessing breadth of knowledge can be done with a variety of machine-grades tests. So there are significant savings in cost and time if assessing thinking ability is done separately on just one topic. Which is absolutely all that is required, since the ability to think mathematically is just like the ability to ride a bike — once someone can ride one kind of bike, they can, with perhaps some adjustment, ride any kind. There is absolutely no need to test for that. As a result of natural selection of many thousands of years, humans can all do it.</div>
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The only question that remains is what mathematical topic should we focus on to develop the ability to think mathematically — including, I should add, an understanding of the importance of the precise use of language, the ability to handle abstraction, the need for formal definitions, and the nature and significance of proof.</div>
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Well, why not once again take our cue from how most of us learn to ride a bike. What is the equivalent of our first child’s bicycle? Elementary arithmetic. </div>
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What’s that you say? “There isn’t enough meat in elementary arithmetic to learn all you need to know about thinking mathematically, with all those bells and whistles I just mentioned.” Think again. Alternatively, check out the article written by mathematics educator Liping Ma in the article she published in the November 2013 issue of the Notices of the American Mathematical Society, titled <a href="https://www.ams.org/notices/201310/fea-ma.pdf" target="_blank">A Critique of the Structure of U.S. Elementary School Mathematics</a>. </div>
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Based on her experience with mathematics education in China, Ma argues forcefully, and effectively, that there is more than enough depth and breadth in “school arithmetic” (as she calls it) to fully develop the ability to think mathematically. True, in the West we don’t teach elementary arithmetic that way; indeed, we present it as a series of basic number facts to be memorized and algorithms to be practiced, as in the Boaler critique. We do so, at some speed I should add, in large part because we are in so much of a hurry to move on to all the other mathematical topics that someone at some time in the past declared were “essential” to learn in school. But as many have pointed out over several decades, the result is that our mathematics curriculum is “a mile wide and an inch deep”, resulting in students leaving school believing that “In math you have to remember, in other subjects you can think about it.” </div>
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In the June 2010 Devlin’s Angle post I referred to earlier, where I talked about Boaler’s then-new book, I mentioned Ma, and said I agreed with her argument about using school arithmetic as the topic to develop the ability to think mathematically. I still do.</div>
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I also think school arithmetic provides the one topic you need to assess mathematical thinking ability — regardless of whether you are assessing student learning, teacher performance, or district system performance. Given that, assessment of whatever breadth is required can be done relatively easily and cheaply. Because the thinking part is essentially the same, the assessment of the breadth can focus on what is <i><b>known</b></i> (rather than what can be done with that knowledge).</div>
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And (of course), one really valuable benefit of focusing on school arithmetic is that it provides as level a playing field as you can hope for, with elementary arithmetic the one mathematical topic that everyone is exposed to at an early age.</div>
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In a future post, I’ll take this topic further, looking at the implications for teaching, the educational support infrastructure (including textbooks), the effective use of modern technologies, and the educational implications of those technologies. </div>
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Also, as the title makes clear, the focus of this article has been <b><i>systemic</i></b> mathematics education, the mathematics that states decide is essential for all future citizens to learn in order to survive and prosper and contribute to society. There is a whole other area of mathematics education, where the focus is on the subject as an important part of human culture. That’s actually the area where I have devoted most of my efforts over the years, writing books and articles, giving public talks, and participating in radio and television programs. So I’ll leave that for other times and other places.</div>
<br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-33138018356832723462018-09-04T11:28:00.000-04:002018-09-04T11:28:10.442-04:00Is math really beautiful?<div class="separator" style="clear: both; text-align: center;">
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The above tweet caught my eye recently. The author is a National Board Certified mathematics teacher in New York City who has an active social media presence. Is his claim correct? Not surprisingly, a number of other mathematics educators responded, and in the course of the exchange, the author modified his claim to include the word “just”, as in “It isn’t just about beauty …” In which case, I think he is absolutely correct.<br />
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Like many mathematicians who engage in public outreach, I have frequently discussed the inherent elegance and beauty of mathematics, the wonder of its purity, and the power of its abstraction. And as a body of human knowledge, I maintain (as do pretty well all other mathematicians) that such descriptions of the subject known as pure mathematics are totally justified. (Cue: for the standard quotation, Google “Bertrand Russell mathematical beauty”.) Anyone who is unable to recognize it as such surely has not (yet) understood what (pure) mathematics is truly about.<br />
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In contrast, the <b>activity</b> of doing mathematics is indeed “messy,” as Pershan claims. That is the case not only for the activity of using mathematics to solve problems in the real world, but also the activity of engaging in pure mathematics research. The former activity is messy because the world is. The latter is messy because the logical elegance and beauty of (many) mathematical theories and proofs are characteristics of the finished product, not the process of development.<br />
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And there, surely, we have the motivation for Pershan’s comment. When we teach mathematics to beginners, we don’t do them any service by making claims about beauty and elegance if what they are experiencing is anything but. With good teaching of a well-designed curriculum, we can ensure that they are exposed to the beauty, of course, and perhaps experience the elegance. But it’s surely better to let them know that the messiness, the uncertainty, the repeated stumbles, and the blind allies they are encountering are part of the package of <b>doing</b> mathematics that the pros experience all the time, whether the doing is trying to prove a theorem or using mathematics to solve a real-world problem.<br />
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By chance, the same day I read that tweet, I came across an <a href="https://towardsdatascience.com/decision-makers-need-more-math-ed4d4fe3dc09" target="_blank">excellent online article</a> on <i>Medium</i> about the huge demand for mathematical thinking in today’s data-rich and data-driven world. Like me, the author is a pure mathematician who, later in his career, became involved in using mathematics and mathematical thinking in working on complex real-world problems. I strongly recommend it. Not only does it convey the inherent messiness of real-world problems, it convincingly makes the case that without at least one good mathematical thinker on the team, management decisions based on numerical data can go badly astray. As the author states in a final footnote, he takes pleasure in the process of applying the rigor of mathematics to the complex messiness of real-world problems.<br />
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To my mind, therein lies another kind of mathematical beauty: the beauty of making productive use of the interplay between the abstract purity of formal rigor and the messy stuff of everyday life. Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-90080499323324266882018-08-08T10:28:00.000-04:002018-08-08T10:47:57.203-04:00How a Fields Medal led to a mathematical roller-coaster journeyBy Keith Devlin<br />
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You can follow me on Twitter <a href="https://twitter.com/profkeithdevlin" target="_blank">@profkeithdevlin</a><br />
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First, congratulations to Caucher Birkar, Alessio Figalli, Peter Scholze, and Akshay Venkatesh on being awarded the Fields Medal, an award that for regular “Devlin’s Angle” readers needs neither introduction nor description. (If it does, use Google.)<br />
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With Fields Medals awarded only (at most) once every four years to mathematicians who produce truly exceptional mathematics before they turn forty, few of us who enter the field come close to getting one. (Indeed, in some ways they are more akin to Olympic Gold Medals than the Nobel Prizes with which they are usually compared. Few club athletes will get one of those either.) On the other hand, many of us earn our doctorates, or build our careers, by understanding a new approach or mastering a new technique that led to a Fields Medal. Just as Bill Gates copied the groundbreaking Macintosh interface to create Windows, so too it can pay off handsomely, and often quickly, for a young mathematician to “reverse-engineer” a Fields-Medal-winning new result and try to use it to solve a different – though often related – problem.<br />
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In fact, sometimes, the medal-winning breakthrough has such broad applicability that is initiates an entire new subfield of mathematics. That was exactly how I began my mathematical career almost a half a century ago. An interest in computing, initiated by a high-school summer internship writing software for British Petroleum (using the very first digital computer delivered to the city I grew up in), stayed with me throughout my undergraduate years, culminating with me interviewing for a job at IBM on graduation. But I was put off by the strong corporate culture and the lack of intellectual freedom I feared would come with joining Big Blue. Instead, I decided to go for a PhD in the general area of computing. Unfortunately, this was before Computer Science was a recognized discipline, and though it was possible to pursue graduate research related to computing, mathematically speaking there was not much of a “there” there back then. <br />
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The one mathematically-intriguing little “there” was a relatively new subject called Automata Theory that I had come across references to. Moreover, one of the pioneers of that field, John Shepherdson, was a professor of mathematics at the University of Bristol, just over a hundred miles from London, where I had just graduated. As chair of department, Shepherdson had built up a strong research team of experts in different branches of Mathematical Logic, the subfield of mathematics that provided the mathematical tools for Automata Theory. And so it was then that I applied to do a PhD at Bristol. That was in the fall of 1968. <br />
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Once I arrived in Bristol, everything changed. Among the mathematics graduate-student community at Bristol University, all the buzz – and there was a lot of it – was about an emerging new field called Axiomatic Set Theory. Actually, the field itself was not new. But as a result of a Fields Medal winning new result, it had recently blossomed into an exciting new area of research. <br />
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Not long after Georg Cantor’s introduction of abstract Set Theory in the late 19th century, Bertrand Russell came up with his famous paradox, concerning the set of all sets that are not members of themselves. To escape from the paradox – more accurately, to rescue the appealing, natural notion of using abstract sets as the basic building block out of which to construct all mathematical objects – Ernst Zermelo formulated a seemingly simple set of axioms to legislate the formation of sets. With an important addition from Abraham Fraenkel in 1925, that axiom system seemed to provide an adequate basis for the construction of all the objects of mathematics, while avoiding Russell’s Paradox. <i>Zermelo-Fraenkel Set Theory</i>, as it became known, rapidly came to be regarded as the "Grand Unified Theory" of mathematics, the basic system on which everything else is built. It was generally referred to as ZFC, the “C” denoting the Axiom of Choice, a basic principle Zermelo included but which was sufficiently controversial that its use was often acknowledged explicitly.<br />
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While the ZFC axiom system was indeed sufficient to ground all of mathematics, there were a small number of seemingly-simple questions about sets that no one could answer using just those axioms. The most notorious by far went back to Cantor himself: <i>Cantor’s Continuum Problem</i> asks how many real numbers there are? Of course, one answer is that there are an infinite number of such. But the ZFC axioms allow the construction of an entire system of infinite numbers of increasing size, together with an arithmetic, that can be used to provide a “count” of any set whatsoever. The smallest such infinite number, aleph-0, is the number of natural numbers. After aleph-0, the next infinite number is aleph-1. Then aleph-2, and so on. (It’s actually a lot more complicated than that, but let’s leave that to one side for now.)<br />
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Cantor showed that the real continuum, the set of all real numbers, has an infinite size strictly larger than aleph-0, so it must be at least aleph-1. But which aleph exactly was it? It’s tempting (on the grounds of pure laziness) to assume it’s aleph-1, an assumption known as the <i>Continuum Hypothesis</i> (CH). But there is no known evidence to support such an assumption.<br />
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In 1940, Kurt Goedel contructed a set-theoretic model of ZFC in which CH is true, thereby demonstrating that CH could never be proved false. But that does not imply that it is true. Maybe it was possible to construct another model in which CH was false. If so, then CH would be completely <i>undecidable</i>, based on the ZFC axioms. This would mean that the ZFC axioms are not sufficient to answer all reasonable questions about sets.<br />
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In 1963, Paul Cohen, a young mathematician at Stanford University, found such a model. Using an ingenious new method for constructing models of set theory that he called <i>forcing</i>, Cohen was able to create a model of ZFC in which CH is false. That result earned him the Fields Medal in 1966.<br />
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By 1968, when I went to the University of Bristol to commence my doctoral work, Cohen’s new method of forcing had been shown to have wide applicability, making it possible to prove that a number of long-standing, unanswered mathematical questions were in fact undecidable in the ZFC system. This opened up an exciting new pathway to getting a PhD. Learn how to use the method of forcing and then start applying it to unsolved mathematical problems, of which there was no shortage. Large numbers of beginning graduate students did just that, and by the time I joined a group of them, a few months after arriving at Bristol, the field was red hot. My interest in computation did not go away, but it would be over two decades before I would pick it up again. At 21 years of age, with a newly minted bachelors degree in mathematics under my belt, I had a mathematical research career to build, and axiomatic set theory was by far the most exciting field to do it in. I jumped onto the roller coaster and joined in the fun.<br />
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Working in my newly chosen field was just like working in any other branch of mathematics. Each day, you woke up and attempted to prove various mathematical statements using logically rigorous reasoning. To an observer looking over your shoulder, doing that involved scribbling formulas on paper and manipulating them in an attempt to construct a proof, just like any other branch of mathematics. The “rules of the game” were exactly the same as in any other branch of mathematics as well. The only difference was the nature of the answers you obtained – on the rare occasion when you did so. (Mathematics research is 95% failure. Actually, the failure rate may be higher than that; we have a far worse batting average than any professional baseball hitter.) In what those of us in this new field called “classical mathematics,” the goal was to prove statements about mathematical objects were true or false. In the new mathematics of undecidability proofs, the goal was to prove that statements about mathematical objects were undecidable (in the ZFC system). In both cases, the result was a rigorous mathematical statement (a <i>theorem</i>) justified by a rigorous mathematical argument (a <i>proof</i>).<br />
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From the perspective of mathematics as a whole, this meant that, thanks to Cohen, mathematicians had a new way to answer a mathematical question. Classically, there had been just two possibilities: true and false. If you can do neither, you had failed to find an answer. Now, there was a third possibility: (provably) undecidable. What had previously been failure could now become success. Absence of a definite answer could be replaced by getting a definitive answer. Lack of knowledge could be replaced by knowledge. <br />
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The two decades following Cohen saw a whole range of unsolved mathematical problems proved undecidable, as a whole army of us jumped into the fray. Some results were easy. Success came quickly to those smart enough or lucky enough (or both) to find an unsolved problem that yielded relatively easily to the forcing technique. Others took much longer to resolve, and a few resisted all attempts (and have done so to this day). But by the start of the 1980s, the probability of success had dropped to that in other areas of mathematics. From then on, for most young mathematicians, getting a PhD by solving an undecidability problem meant finding some relatively minor variant of a result someone else had already obtained. The party was over.<br />
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Looking back, I realize that I was simply very lucky to be starting my mathematical career when a productive new subfield was just starting up. By going to the University of Bristol to do a PhD in Automata Theory, I found myself in the right place at the right time to jump ship and have the time of my life. When the field started to settle down and slowdown in the 1980s, I started to lose interest. Not in mathematics, just in that particular area as my main research focus. My main interest shifted elsewhere as my attention was caught by some new mathematics being developed at Stanford (as with forcing, Stanford turned out to be the place that generated the new ideas that caught my attention). That new mathematics was closely intertwined with my earlier high school interest in computing. But that’s another story.<br />
<br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-24458333499139396622018-07-05T12:49:00.000-04:002018-07-05T12:49:13.940-04:00By Keith Devlin<br />
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You can follow me on Twitter <a href="https://twitter.com/profkeithdevlin" target="_blank">@profkeithdevlin</a><br />
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<b>21<sup>st</sup> Century Math: The Movie </b>
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All my <i>Devlin’s Angle</i> posts this year so far have studied the dramatic shift that took place over the past twenty-five years in the way professional mathematicians “do the math” in order to solve real-world problems. There have been parallel changes in the way pure mathematicians work as well, but those changes are somewhat less visible, and not as dramatic. In any case, I have been focusing on mathematics in the wild.
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Those changes in how math is done have put pressure on global education systems to catch up. In previous posts, I addressed these changing educational needs, but overall, there has been a considerable lag. In the United States, many of the better, selective, private schools have adjusted, but little has changed in the math classrooms of most state-funded schools. There are a number of reasons for that lack of action, some educationally valid, others resulting from Americans’ proclivity to treat mathematics education as a political football. But that is another story.
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The fact is, however, the mathematical world has changed significantly, it is not going to change back, and sooner or later the educational system must catch up. Hopefully sooner, given that today’s students will enter a world and a workforce where no one does calculations anymore – where by “calculation” I mean performing any form of algorithmic procedure.
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In May, I participated in <a href="http://curriculumredesign.org/maths-for-the-21st-century-conference-strongly-supports-critical-necessary-changes-of-oecds-pisa-maths-for-2021-4dedu/" target="_blank">Maths for the 21<sup>st</sup> Century</a>, a global mathematics education summit in Geneva, Switzerland, organized to discuss the new way mathematics is being done and how best to prepare students to live and work in such a world. Both the United States Department of Education and the OECD’s (Organization for Economic Cooperation and Development’s) PISA educational testing organization were represented at the summit.
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The half hour talk I gave at the summit is in many ways a summary (absent all the details) of my series of posts for the MAA. So I thought I would wrap up the series, at least for now, by pointing you to the <a href="https://www.youtube.com/watch?v=qBOnWZyq468&feature=youtu.be" target="_blank">video</a> of my presentation. The main summit page, linked above, also provides a link to a lightly abridged PDF version of the deck I used to accompany my talk.
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My experience in giving public talks on this topic over the past several years has been that it evokes two very different reactions. Engineers and scientists in the audience, for the most part, nod along in agreement with everything I say. I am, after all, just describing the way they have been working for twenty years. In contrast, teachers, or at least a great many of them, often show surprise, confusion, and not infrequently hostility. Many parents react similarly.
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Why is that? Well, to repeat an arguably over-used quotation from the great Paul Newman movie <a href="https://www.youtube.com/watch?v=452XjnaHr1A" target="_blank"><i>Cool Hand Luke</i></a>, “What we have here is failure to communicate.” After my talks, I am often left feeling like the Paul Newman character, Luke, in that clip. However, for the analogy to work, Luke has to represent not me but the entire mathematics community.
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Teachers are taken aback to be told that calculation is less relevant in today’s world. I believe this is because no one in the mathematics business – that is, the business of using mathematics to solve real-world problems – has taken the trouble to inform teachers that the entire game has changed, and in what ways. It’s time we bring better communication to this issue. My series of <i>Devlin’s Angle</i> posts this year is one of my latest attempts to do just that. The Geneva video I am directing you to is another.
Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-41145907838422758782018-06-06T10:05:00.002-04:002018-06-11T14:57:44.816-04:00Cycling can be such a drag – and math can tell you exactly how muchBy Keith Devlin<br />
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You can follow me on Twitter <a href="https://twitter.com/profkeithdevlin" target="_blank">@profkeithdevlin</a><br />
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Last month, my two greatest passions collided: mathematics and cycling. On my way back from a short biking trip to the Californian Central Coast (gorgeous in the late spring, when the grass is still green and the wildflowers are in full bloom), I stopped off in Morgan Hill (home of the <a href="https://aimath.org/about/futurehome/" target="_blank">American Institute of Mathematics</a>) to watch part of Stage 4 of the seven stage, Amgen Tour of California. Wandering around the start area, where the teams warm up on stationary trainers and the big vendors show off their wares, I noticed a couple of guys in one corner, promoting a curious looking bike.
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Actually, it wasn’t the bike itself that looked unusual, it was the strange, overly large fenders over the two wheels. Had the bike been in a regular bike shop, I would have thought they were splash guards for image-conscious, athletic-leaning commuters to use on rainy days. But the exhibits at a professional cycle racing event are aimed at hard-core cyclists, whose passion revolves around razor-thin saddles, aerodynamic bike design, and low bicycle weight. Whatever those fenders were for, they were not for keeping a rider dry. It had to be about performance. But modern, racing-bikes are designed to be as light and thin as possible – the wheels have tires just a tad over 20mm wide – so those bulky-looking fenders seemed completely out of place. I could not resist asking for an explanation.
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The conversation soon got very mathy, and I quickly sensed an opportunity for a Devlin’s Angle post showing the power of thinking mathematically about everyday activities—be those activities work- or leisure- related. It would be a natural continuation of my recent series of posts (starting in January) on how mathematics education needs to change to prepare people for life in the 21st century.
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The strange looking fenders were designed by Garth Magee, a former aerospace engineer from Southern California, who like me is a keen cyclist. I had already gotten some clue as to what the fenders were for from the company name on a poster: Null Winds Technology. The fenders must have something to do with reducing drag. (If so, then we need to start calling them “fairings”.) That would also explain Magee’s presence at the Time Trial stage of the Tour of California. Modern professional time-trialing is all about aerodynamic bicycle design, with all the major international bicycle manufacturers spending small fortunes on computer-aided designs and hours of testing in wind tunnels. Magee’s main purpose in Morgan Hill was likely to raise interest from some of the other manufacturers present at the event, I surmised. In addition, professional time-trialing is heavily regulated to ensure competitive cycling’s “purity” and “fairness”, and any kind of wheel fairings are banned in competition, so Magee’s product was surely not aimed at professional racers. He was likely seeking to sell to amateur riders, like myself.
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[Disclaimer: Magee told me he had formed his company in 2012 to turn his discovery/invention into a product people can use. I have no involvement with the company, I don’t own any of their products, and I am not actively promoting their products. My focus here is on the really cool application of mathematical thinking.]
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From past research into what was known about the mathematics of cycling, I knew it is complicated, uses advanced techniques, and has yet to fully explain the physics of cycling. You can get a sense of just how advanced and complicated it is by perusing <a href="https://en.wikipedia.org/wiki/Bicycle_performance" target="_blank">the description</a> on Wikipedia.
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As any experienced cyclist will tell you, at anything beyond very low speeds, the greatest resistance to forward motion is due to the air. That’s why professional cycle racers stay close together in tight packs (“pelotons”) or long lines (“pacelines”), where the few rides at the front sacrifice their chance of winning in order to shield their teammates from the wind. In still air, the headwind is caused entirely by the cyclist’s own forward motion (at speeds up to 30 mph on the flat). If there is a headwind, the resistance is greater. Moreover, it increases with <i><b>the cube</b></i> of the rider’s speed relative to the air. That’s why riding a bike into a strong head wind is so hard. The Wikipedia article gives you the <a href="https://en.wikipedia.org/wiki/Bicycle_performance#Air_drag" target="_blank">key formula</a>, which I reproduce below.
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Another nice summary of the relevant math you can access online can be found <a href="https://www.sheldonbrown.com/rinard/aero/formulas.htm" target="_blank">here</a>.
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What the math tells you is that wind drag is a huge problem that every cyclist encounters when trying to go faster. At a speed of approximately 7 mph, overcoming air resistance takes about half of your effort (with ground friction on the tires and mechanical resistance in the drive train accounting for the other half). As you go faster, that cubic growth starts to flex its muscles, which means you have to increasingly flex <b><i>your</i></b> muscles in order to overcome air resistance that demands a larger and larger proportion of your total effort. At around 15 mph, approximately 70% of your effort is being used to overcome air resistance; at 20 mph it takes roughly 85% of your effort. At the top end, a typical average speed for a flat stage in the Tour de France is about 29 mph. At that speed, over 90% of the effort needed to maintain this speed is used to overcome air resistance.
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Needless to say, the bicycle manufacturing industry has put in a lot of effort over the years to try to minimize the effect of headwind drag. The results of those efforts, explained for cyclists rather than mathematicians, are nicely summarized in two articles that you will find online at:
<a href="https://tunedintocycling.com/2014/06/28/aerodynamics-part-1-air-resistance/" target="_blank">https://tunedintocycling.com/2014/06/28/aerodynamics-part-1-air-resistance/</a>
and
<a href="https://tunedintocycling.com/2014/07/25/aerodynamics-part-2-small-things-that-reduce-air-resistance-and-drag/" target="_blank">https://tunedintocycling.com/2014/07/25/aerodynamics-part-2-small-things-that-reduce-air-resistance-and-drag/</a>.
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With so much research put into the problem of headwind drag, you would think the industry had done as much as could be done. But as Magee showed, there were still more efficiencies to be obtained. His observation is an excellent illustration of the power of mathematical thinking.
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The mathematics I’ve summarized so far treats the bicycle as a single item, not an assembly of components. (An instance of the mathematician’s standard approach, as encapsulated in the quip “Consider a spherical cow not subject to frictional forces.”) Magee focused on the effect of headwind on the wheels.
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To be sure, he was not the first to do that. Most of the racers at the Morgan Hill time-trial rode bicycles with solid rear wheels. Though considerably heavier than the more common spoked wheels, a solid wheel creates far less rotational drag than does a regular wheel, where the spokes create turbulence as they cut through the air. (The only reason the pros don’t use a solid wheel at the front as well is because it would make the bicycle highly unstable, with a slight crosswind likely to send the bike and the cyclist out of control. In indoor races on banked tracks, you do see bikes with two solid wheels.)
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And now, as that last paragraph should indicate (note the word<i> turbulence</i>), we are deep into aerodynamics. The field Magee worked in for many years. What Magee did, that no one had previously done (at least to the point of taking out a patent on a design), was observe that, with wind resistance increasing with the cube of the speed through the air, the resistance increases rapidly as you go up the wheel from the axle to the top of the wheel (where the top of the tire is moving at twice the speed of the bicycle), and as you go down from the axle to the ground it decreases to 0 relative to the ground. What would happen, Magee asked, if a shield were to keep the wind from hitting the fast-moving top portion of the wheel? Sure, it would add some weight, but with that cubic function to contend with, it seemed likely the drag reduction could be significant.
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In particular, what portion of the wheel should be protected to optimize any gain due to overall reduced wind resistance on the wheel? This is where a bit of good-old-fashioned math comes into play. The chart below is the key to Magee’s fairing design.<br />
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According to the math, there would be little or no benefit when riding in still air (or with a tailwind). But in a headwind, the difference should be noticeable to the rider. The stronger the headwind, the greater the benefit. To test his invention out, he turned to his friend Robert Keating, a former bike racer and triathlete who teaches triathlon and works at a local Triathlon shop in Los Angeles. (I met Keating at the Morgan Hill event, where he and Magee were jointly demoing the new device.) A number of road tests showed that the idea worked as the math said it should. You can see a video of one recent test <a href="https://www.youtube.com/watch?time_continue=64&v=rYeeFT9N0cM" target="_blank">here</a>. This, folks, is the power of mathematics in action.
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Just as with the UPS routing problem I had the students at Nueva School look at in January (see my previous posts for January 2018 onwards), Magee’s problem was all about optimization. Not unique right answers; rather better performance.
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Yes, there was a fair amount of sophisticated calculation to be done. But the key was to approach the problem in the right way, so that mathematical power could do its work.
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Note, too, that all the background mathematics needed to solve the design problem can be found on the internet. Indeed, in writing this article, I simply used Google to locate suitable sources. (Remember, Google was the very first modern tool on that chart of modern tools to do mathematics I presented in my February post.)
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As I keep saying, in today’s world, in using mathematical thinking to help solve a problem, you (usually) don’t need to re-invent the wheel. Those days are largely gone. Today, you mostly need to understand mathematics in a fundamental, conceptual way so you can make an existing wheel do the work for you.
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In Magee’s case, of course, that was true both literally as well as in my original metaphorical sense.
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All in all, it’s a superb example of 21st century mathematical problem solving.
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Finally, I note that I was originally motivated to reverse-engineer the UPS routing algorithm because of the strange movements (and non-movements) of a bicycle I was shipping from California to Princeton. So might it be that math is particularly well suited to problems involving bicycles. Not at all. Bicycles figure in both examples because they interest me. After all, I was only at the Tour of California Time Trial because I am passionate about cycling, and it was my mathematical bent that prompted me to approach the Null Winds display and ask for an explanation. (Remember also my 2014 <a href="http://devlinsangle.blogspot.com/2014/03/how-mountain-biking-can-provide-key-to.html">Devlin’s Angle post</a> on mountain biking and proving theorems.)
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The real lesson is that mathematical thinking can be applied to almost anything, particularly if the question is “Can we make it better (in some way)?” You are not interested in biking? Fine. Think about something that really does interest you. The chances are high—very high—that mathematics could be used to make it better in some way.
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Once you have decided <b><i>what</i></b> to optimize, use the wide range of tools that are now freely available to start to find out how you might do it. Go as far as you can, then seek help from a mathematician. Your passion, experience, and domain knowledge coupled with the mathematician’s experience at using math to solve problems can make a powerful team.
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Of course, to proceed this way, as I discussed in my earlier posts this year, you do need to understand what mathematics (really) is and how it can be used. But that is really all you need. Yes, it’s a big “all”. But it’s THE “all”. That’s why it needs to be the focus of mathematics education in the 21st century.
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LABELS: mathematical thinking, problem solving, aerodynamics, wind resistance, bicycle mathematics.Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-81340082082046776902018-05-02T08:00:00.000-04:002018-05-02T09:50:12.923-04:00Calculation was the price we used to have to pay to do mathematicsB<span style="background-color: white; color: #222222; display: inline; float: none; font-family: "georgia" , "utopia" , "palatino linotype" , "palatino" , serif; font-size: 13.2px; font-style: normal; font-weight: 400; letter-spacing: normal; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;">y Keith Devlin</span><br />
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Ever since mathematics got properly underway around 3,000 years ago, there was only one way to achieve access to the field. You had to spend many years developing a fairly extensive calculation skillset. In the first instance, to pass the graduation and entrance examinations to gain initial access to the field. Then, once accepted into the world of mathematics, calculation of one kind or another was what all mathematicians spent the bulk of their mathematical time doing. Arguably, for most of mathematics history, the subject really was, to a large extent, primarily about calculation of one form or another. Newton, Leibniz, Bernoulli (any of them), Fermat, Euler, Riemann, Gauss, and the other greats of times past, were all superb masters of calculation. (We should also include Boole, since his famous Boolean algebra is also a calculation system.) <br />
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But whereas most laypersons seem to think that calculation is all there is to mathematics, surely none of the greats did. Calculation was an important tool (more accurately, a set of tools) you needed to do mathematics, they must have realized, but the essence of mathematics is much more, a plateau of knowledge that transcends all the calculation techniques.<br />
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In the 19th Century, that somewhat tacit understanding became explicit. The increasing complexity of the problems mathematicians tackled led to a series of results that defied the human intuition. (Several of them were referred to as “paradoxes”.) This led to an intense period of mathematical introspection, where the primary focus was not performing a calculation or computing an answer, but formulating and understanding abstract concepts and relationships. In other words, a shift in emphasis from doing to understanding. What had previously been implicit, became full-on explicit. <br />
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Mathematical objects were no longer thought of as given primarily by formulas, but rather as carriers of conceptual properties. Proving something was no longer a matter of transforming terms in accordance with rules, but a process of logical deduction from concepts. Mathematics was reconceptualized as “thinking in concepts” (<i>Denken in Begriffen</i>). <br />
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This was, in every sense, a mathematical revolution, with the primary revolutionaries being<br />
leading mathematicians such as Lejeune Dirichlet, Richard Dedekind, Augustin-Louis Cauchy, Karl Weierstrass, and Bernhard Riemann. <br />
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To give just one instance of the shift, prior to the nineteenth century, mathematicians were used to the fact that a formula such as y = x<b><span style="font-size: x-small;"><span style="font-family: "arial";"><span style="color: black;"><sup style="line-height: 0.9;">2</sup></span></span></span></b> + 3x – 5 specifies a function that produces a new number y from any given number x. Then the revolutionary Dirichlet came along and said, forget the formula and concentrate on what the function does in terms of input-output behavior. A function, according to Dirichlet, is any rule that produces new numbers from old. The rule does not have to be specified by an algebraic formula. In fact, there's no reason to restrict your attention to numbers. A function can be any rule that takes objects of one kind and produces new objects from them.<br />
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This definition legitimized functions such as the one defined on real numbers by the rule:<br />
If x is rational, set f(x) = 0; if x is irrational, set f(x) = 1. <br />
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Of course, you cannot draw a graph of such a monster. Instead, mathematicians began to study the properties of <i>abstract functions</i>, specified not by some formula but by their behavior. For example, you can investigate questions such as, is the function one-one, injective, surjective, continuous, differentiable, etc.?<br />
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For any mathematician alive today, mathematics is a subject that studies formally-defined concepts, with a focus on the establishment of truth (based on accepted axioms), with various forms of calculation (numerical, algebraic, set-theoretic, logical, etc.) being tools developed and used in the pursuit of those goals. That’s the only kind of mathematics we have known.<br />
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Except, that is, when we were at school. By and large, the 19th Century revolution in mathematics did not permeate the world’s school systems, which remained firmly in the “mathematics is about calculation” mindset. The one attempt to bring the school system into the modern age (in the US, the UK, and a few other countries), was the 1960s “New Math”. Though well-intentioned, its rollout was disastrous, in large part because very few teachers understood what it was about – and hence could not teach it well. The confusion caused to parents (other than mathematician parents) was nicely encapsulated by the satirical songwriter and singer Tom Lehrer (who taught mathematics at Harvard, and did understand New Math), in his hilarious, and pointedly accurate, song <a href="https://www.youtube.com/watch?v=UIKGV2cTgqA" target="_blank">New Math</a>. <br />
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As a result of the initial chaos, the initiative was quickly dropped, and school math remained largely unchanged while real-world uses of mathematics kept steadily changing, leaving the schools increasingly separated from the way people did math in their jobs. Eventually, the separation blew up into a full-fledged divorce. That occurred in the late 1980s. The divorce was finalized on June 23, 1988. That was the date when Steve Wolfram released his mammoth software package <i>Mathematica</i>. Within a few short years of that release, if not on the release-date, <i>Mathematica</i> (and a similar package released a few months later in Canada, <i>Maple</i>) could answer pretty well any school or university math exam question with at least a grade B+, and very often an A.<br />
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The days when calculation (of pretty well any kind, not just numerical) was the price humans had to pay to do mathematics were over. <br />
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Given that thirty years have passed since that initial epochal moment, and most of the world has still not woken up to the fact that the entire mathematical world has changed dramatically and forever, let me repeat the core of that statement in caps.<br />
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THE DAYS WHEN CALCULATION WAS THE PRICE HUMANS HAD TO PAY TO DO MATHEMATICS ARE OVER. <br />
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To be sure, after that symbolic 1988 date, it took a few years for the change to percolate through the system, gain momentum, and eventually reach critical mass. Three further developments were also hugely significant: the birth of the World Wide Web in 1989 and the browser in 1993, and the launch of <i>Wolfram Alpha</i> in 2009. (Others might want to add other factors. I’m being selective here.)<br />
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Talking about being selective, I’ve mentioned Wolfram products twice now. Though I was a member of Wolfram’s <i>Mathematica </i>Advisory Board in the first few years, I have no stake in or involvement with the company. While both <i>Mathematica</i> and <i>Alpha</i> were indeed major players in changing the way mathematics is done – particularly in applied settings – I am citing those particular products largely as icons, using two specific products to represent a range of new digital tools that were being developed around the world at that time. While Wolfram’s systems were ones I myself made early use of in my work, other mathematicians were also active in that digital mathematical revolution, using different systems. Still, <i>Mathematica</i> was the system that caught the public attention.<br />
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Since the turn of the new Millennium, I doubt if anyone making professional use of mathematics in their job, or indeed any adult using mathematics in their everyday lives, has taken out paper-and-pencil and followed a classical algorithm to add, subtract, multiply or divide numbers in an array of real-life size, or perform complex algebraic reasoning to solve systems of equations, or solve problems using calculus, or any other established mathematical procedure. Not only would it now be a waste of valuable human time and energy doing something a cheap machine can do in far less time with no possibility of error, but many of the problems that people encounter in their careers and lives have simply too much data for the human mind to handle. Those same digital tools that have made the execution of mathematical procedures unnecessary have also come to dominate and drive our world, so many of the problems that require mathematics in their solution are now simply beyond human capacity. That’s why Amazon Web Services has become such a behemoth for data storage and processing.<br />
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But that does not mean humans no longer need to have some mathematical skills. On the contrary, they are as crucial as ever – unless you are willing to be totally reliant on others, but personally, I have never felt comfortable doing that with things that are part of my life every day. What has changed are the specific mathematical skills required today. There are plenty of things computers cannot do or do poorly. Genuinely creative thinking and analogical reasoning are two obvious ones – though with today’s massive cloud computing resources we can use systems that provide an approximation often adequate for the purpose, and on occasion can be better than humans.<br />
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Mostly, however, where you need humans is going from a real-world challenge situation to formulating one or mathematical tasks that can help you make progress. Sometimes, progress means solving a real-world problem in the sense of getting a specific answer (say, a number), but much more commonly it’s about finding a better method, where “better” can mean faster, cheaper, safer, or whatever other criterion is important, and where the change may involve developing a new method or improving an existing one.<br />
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This way of using mathematics was the focus of that mini-course I gave at a California school (Nueva School) in January of this year, that I wrote about in the February, March, and April posts to this column.<br />
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Though several mathematicians and mathematics education scholars expressed agreement with what I wrote, my articles brought some critiques from teachers and parents. The critiques all made reference to my asides about the Common Core State Standards in the first two of the posts. Since “Devlin’s Angle” no longer seems to be a target for the CCSS social media trolls (likely because the yield of issues to react to relative to the length and substance of most of my posts makes it less rewarding to them), I made some efforts to find out what exactly it is about the CCSSs that they objected to. As far as I could ascertain, the issue was inevitably (and predictably) to do with particular implementations of the Standards in specific curricula or (and this seems to be the most common occurrence) claims that a particular homework exercise was a “Common Core exercise”, which of course it cannot be since the CCSS are, as the name indicates, purely a set of standards to attain, not in any way a curriculum or curriculum content. <br />
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More generally, in fact, pretty well all critiques of the CCSS are due to a complete misunderstanding of <b>what</b> they are, <b>why</b> they are, and what they <b>say</b>. The issue was nicely dealt with in this <a href="http://hechingerreport.org/content/common-core-math-problem-hard-supporters-common-core-respond-problematic-math-quiz-went-viral_15361/" target="_blank">2014 article</a> in the Hechinger Report.<br />
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My reason for bringing the Common Core into my series of posts was to point out that the standards were developed precisely to help guide school districts, schools, and teachers in the tricky task of updating K-12 mathematics education to adequately prepare tomorrow’s citizens for life and work in a world where calculation is no longer a central pillar of mathematics.<br />
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Having said that, I should point out that the above statement in no way implies that we should drop the teaching of basic arithmetic and algebra from the school system. As I discussed in some length in the third of my Nueva-inspired articles, the change that is required in K-12 math education is not so much in the mathematical topics but the <b>reason</b> they are now being taught and, in consequence, the <b>way</b> they should be taught. <br />
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Teaching for execution is no longer the primary driver, since no one using mathematics in the real world does that anymore. What is now of cru<b></b>cial importance is teaching for understanding. Digital systems outperform humans to an insane degree when it comes to execution. But they don’t understand; people have to supply that.<br />
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I leave you with an image I pulled from one of those Common Core social media rants some time ago. (I no longer remember the exact source.)<br />
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I have three comments about the post on the left. First, the mathematics in the bottom left is not some fancy new algorithm, it is what a child wrote down in reasoning (sensibly) about a particular arithmetic problem. Second, if you are unable to follow what the child is doing, you would have trouble making effective use of mathematics in today’s world. It’s pretty basic. (Your kid just did it, right?) Third, if you are a parent and you don’t see why it is important that today’s school students acquire those math reasoning skills, please don’t communicate your skepticism to your children. Doing so would be a great disservice, to your child, to your child’s math teacher, and to society. The mathematical world has changed significantly. That occurred over twenty years ago. It is not going to change back. Sit back, relax, be encouraging, and let the kids take over. They do just fine with it.</div>
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REFERENCE: During the period when the computer revolutionized how mathematics is done, I edited the American Mathematical Society’s “Computers and Mathematics” section of their monthly notices publication, sent to all members. I wrote about the column and that period in general in a paper that I submitted to the Proceedings of the Jon Borwein Commemorative Conference, held in 2017. Borwein, who died tragically young in 2016, was a leading pioneer in bringing digital technologies into mathematics. You can access a preprint of the paper <a href="https://web.stanford.edu/~kdevlin/Papers/BorweinCommemorativePaper.pdf" target="_blank">HERE</a>.</div>
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<br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-27870717185295765572018-04-04T10:54:00.000-04:002018-04-04T11:21:06.361-04:00How today’s pros solve math problems: Part 3 (The Nueva School course)<span style="background-color: white; color: #222222; display: inline; float: none; font-family: "georgia" , "utopia" , "palatino linotype" , "palatino" , serif; font-size: 13.2px; font-style: normal; font-weight: 400; letter-spacing: normal; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;">By Keith Devlin</span><br />
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<i>NOTE: This article is the final installment of a four-episode mini-series posted here starting in mid-January. In writing it, I have assumed my readers have read those three earlier pieces.</i><br />
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At the end of last month’s post, I left readers with a (seemingly) simple arithmetic problem. I prefaced the problem with the following two instructions:<br />
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1. Solve it as quickly as you can, in your head if possible. Let your mind jump to the answer.<br />
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2. Then, and only then, reflect on your answer, and how you got it.<br />
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The goal here, I said, is not to get the right answer, though a great many of you will. Rather, the issue is how do our minds work, and how can we make our thinking more effective in a world where machines execute all the mathematical procedures for us?<br />
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Here is the problem.<br />
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PROBLEM: A bat and a ball cost $1.10. The bat costs $1 more than the ball. How much does the ball cost on its own? (There is no special pricing deal.)<br />
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What answer did you get? And what did you learn from the subsequent reflection?<br />
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Before I continue, I should note that the use of this problem (which you can find in many puzzle books and on countless websites) in the context of trying to maximize the human mind’s innate abilities in order to become good 21st Century mathematical thinkers, is due to Gary Antonick, with whom I co-taught a Stanford Continuing Studies adult education course last fall. It was in that course that I gave the second iteration of the UPS problem I subsequently based my Nueva School course on. The discussion of the bat-and-ball problem that follows is the one Antonick presented in our course.<br />
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Now to the problem itself. The most common answer people give instantly to this problem is that the ball costs 10¢. It’s wrong (and many realize that is the case soon after their mind has jumped to that wrong number). What leads many astray is that the problem is carefully worded to run afoul of what under normal circumstances is an excellent strategy. (So if you got it wrong, you probably did so because you are a good thinker with some well-developed problem-solving strategies— problem-solving heuristics is the official term, and I’ll get to those momentarily. So take heart. You are well placed to do just fine in 21st Century mathematical thinking. You simply need to develop your heuristics to the next level.)<br />
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Here is, most likely, what your mind did to get to that 10¢ answer. As you read through the problem statement and came to that key phrase “cost more,” your mind said, “I will need to subtract.” You then took note of the data: those two figures $1.10 and $1. So, without hesitation, you subtracted $1 from $1.10 (the smaller from the larger, since you knew the answer has to be positive), getting 10¢.<br />
Notice you did not really perform any calculation. The numbers are particularly simple ones. Almost certainly, you retrieved from memory the fact that if you take a dollar from a dollar-ten, you are left with 10¢. You might even have visualized those amounts of money in your hand.<br />
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Notice too that you understood the mathematical concepts involved. Indeed, that was why the wording of the problem led you astray!<br />
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What you did is apply a heuristic you have acquired over many financial transactions and most likely a substantial number of arithmetic quiz questions in elementary school. In fact, the timed tests in schools actively encourage such a “pattern recognition” approach. For the simple reason that it is fast and usually works!<br />
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We can, therefore, formulate a hypothesis as to why you “solved’ the problem the way you did. You had developed a heuristic (identify the arithmetic operation involved and then plug in the data) that is (a) fast, (b) requires no effort, and (c) usually works. This approach is a smart one in that it uses something the human brain is remarkably good at—pattern recognition—and avoids something our minds find difficult and requiring effort to master (namely, arithmetic calculation).<br />
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Of course, primed by the context in which I presented this particular problem, you probably expected there to be a catch. So, after letting your mind jump to the 10¢ answer, you likely took a second stab at it (or, if you were anxious about “getting a wrong answer,” made this your first solution) by applying an algorithm you had learned at school. Namely, you reasoned as follows:<br />
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Let x = cost of bat and y = cost of the ball. Then, we can translate the problem into symbolic<br />
form as x + y = 1.10 , x = y + 1<br />
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Eliminate x from the two equations by algebra, to give<br />
1.10 – y = y + 1<br />
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Transform this by algebra to give<br />
0.10 = 2y<br />
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Thus, dividing both sides by 2, you conclude that<br />
y = 5¢.<br />
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And this time, you get the correct answer.<br />
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You may, in fact, have been able to carry out this procedure in your head. When I was at school, I could do algebraic manipulations far more complicated than this in my head, at speed. But, truth be told, since I started outsourcing arithmetic to machines many decades ago, I have lost that skill, and now have to write down the equations and solve them on paper. (This is a confirmation, if any were needed, that arithmetic calculations do not come naturally to the human brain. Over the years, as my mental arithmetic skills have declined, my pattern recognition abilities have not diminished, but on the contrary have dramatically improved, as I learned—automatically, through exposure—to recognize ever more fine-grained distinctions.)<br />
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Whether or not you can do the calculation in your head, it is of course entirely formulaic and routine. Unlike the first method I looked at (a <b><i>heuristic</i></b> that is <i><b>fast</b></i> and <i><b>usually right</b></i>), this method is an <b><i>algorithmic procedure</i></b>, it is <b><i>slow</i></b> (much slower than the first method, even when the algebraic reasoning is carried out in your head), but it <i><b>always works</b></i>. It is also an approach that can be executed by a machine. True, for such a simple example, it’s quicker to do it by hand on the back of an envelope, but as a general rule, it makes no sense to waste the time of a human brain following an algorithmic procedure, not least because, even with simple examples it is familiarly easy to make a small error that leads to an incorrect answer.<br />
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But there is another way to solve the problem. It’s the way I addressed it, and, according to Antonick, who has given it to many professional mathematicians and asked them to vocalize their solutions, the way many math pros solve it. Like the first method we looked at, it is a heuristic, hence instinctive and fast, but unlike the first heuristic method, it <b>always </b>works.<br />
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This third method requires looking beyond the words, and beyond the symbols in the case of a problem presented symbolically, to the <b><i>quantities</i></b> represented. Though I (and likely other mathematicians) don’t visualize it quite this way (in my case it is more of a vague sense-of-size), the following image captures what we do.<br />
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As we read the problem, we form a mental sense of the two quantities, the cost of the ball-on-its-own and the cost of the bat-plus-ball, together with the stated relation between them, namely that the latter is $1 more than the former. From that mental image, where we see the $1.10 total consists of three pieces, one of which has size $1 and the other two of which are equal, we simply “read off” the fact that the ball costs 5¢. No calculation, no algorithm. Pure pattern recognition.<br />
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This solution is an example of Number Sense, the critical 21st Century arithmetic skill I wrote about in the January 1, 2017 Huffington Post <a href="https://www.huffingtonpost.com/entry/number-sense-the-most-important-mathematical-concept_us_58695887e4b068764965c2e0" target="_blank"><span style="color: blue;">companion piece</span></a> to the article I published on the same day as my article about all my math skills becoming obsolete, which I referred to in my last post here on Devlin’s Angle.<br />
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It is, I suggest, hard to imagine how a computer system could solve the problem that way. (Of course, you could write a program so it can perform that particular pattern recognition, but the essence of number sense is that you can apply it to many numerical problems you come across.)<br />
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Those three ways to solve the bat-and-ball problem I just outlined are examples of what the famous Australian (pure) mathematician Terrence Tao<a href="https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/" target="_blank"><span style="color: blue;"> has called (in his blog)</span></a>, respectively, <b><i>pre-rigorous</i></b> thinking, <i><b>rigorous</b></i> thinking, and <i><b>post-rigorous</b></i> thinking. You can also listen to him explain these three categories in a <a href="https://www.youtube.com/watch?v=48Hr3CT5Tpk" target="_blank"><span style="color: blue;">short video</span></a> in the <i>Numberphile</i> series.<br />
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Post-rigorous heuristic thinking is how today’s math pros use mathematics to solve real-world problems. In fact, as Tao makes clear, post-rigorous thinking is what the pros use most of the time to solve abstract problems in pure math. The formal, symbolic, rigorous stuff comes primarily at the end, to check that the solution is logically correct, or at various intermediate points to make those checks along the way.<br />
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In the case of solving real-world problems, the pros almost always turn to technology to handle any algebraic deductions. In contrast, though pure mathematicians sometimes do use those technology products as well, they often find it much quicker, and perhaps more fruitful in terms of gaining key insights, to do the algebraic work by hand.<br />
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So, one of the big question facing math teachers today is, how do we best teach students to be good post-rigorous mathematical thinkers?<br />
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In the days when the only way to acquire the ability to use mathematics to solve real-world problems involved mastering a wide range of algorithmic procedures, becoming a mathematical problem solver frequently resulted in becoming a post-rigorous thinker automatically.<br />
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But with the range of tools available to us today, there is a good reason to assume that, with the right kinds of educational experiences, we can significantly shorten (though almost certainly not eliminate) the learning path from pre-rigorous, through rigorous thinking, to post-rigorous mathematical thinking. The goal is for learners to acquire enough effective heuristics.<br />
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To a considerable extent, those heuristics are not about “doing math” as such. Rather, they are focused on making efficient and effective use of the many sources of information available to us today. But before you throw away your university-level textbooks, you need to be aware that the intermediate step of mastering some degree of rigorous thinking is likely to be essential. Post-rigorous thinking is almost certainly something that <b><i>emerges</i></b> from <b><i>repeated</i></b> practice at rigorous thinking. Any increased efficiency in the education process will undoubtedly come from teaching the formal methods in a manner<i><b> optimized for understanding</b></i>, as opposed to optimized for attaining procedural efficiency, as it was in the days when we had to do everything by hand. See Daniel Willingham’s excellent book <a href="https://www.amazon.com/Why-Dont-Students-Like-School/dp/047059196X" target="_blank"><span style="color: blue;">Why Don’t Students Like School?</span></a> for a good, classroom-oriented look at what it takes to achieve mastery in a discipline.<br />
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Now to that UPS routing problem that was the focus of my Nueva School course. [You will find it discussed <a href="http://devlinsangle.blogspot.com/2018/02/" target="_blank"><span style="color: blue;">here</span></a>.] Here are some of the hints and suggestions about solving the problem I made to the students in the three courses where I used it. Whether they followed my advice was entirely up to them. The purpose of the course was not to solve the problem unaided—even an entire semester would not be enough time for that with students who had never approached a problem the way the pros do. Rather, it was to give them an experience of the method.<br />
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First, they had to work in teams of three to five. I let them select the teams, but said it would be good if at least one person on each team felt they were “good at math.”<br />
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Then, start out by using Google to find out what you can about the problem domain, and any attempts made by others to solve it.<br />
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Whenever you come across a reference to a concept, an approach, or a method that you suspect might be relevant, use general Web resources like Wikipedia to get an initial understanding of what they are and what they can do.<br />
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Follow any leads your search brings up to solutions of problems that look similar. Note what methods were used to solve them.<br />
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If you come across references to others who have worked on the problem, or a similar one, send them a brief email. You may not get a reply, but occasionally you will, and it could be invaluable. (I receive such emails all the time. Mostly I do not have time to respond, but occasionally one lands in my inbox when I have a spare moment, and I happen to know something that might help, so I shoot back a brief reply, often just a reference to a particular source.)<br />
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When you get to a point where you need to perform a specific calculation, perhaps because you have found a solution to a very similar problem someone else has obtained and published, but your data is different, use Wolfram Alpha. It is structured so you can use pattern recognition (of formulas) to identify the appropriate technique and then edit the example provided to be the one you want to solve.<br />
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Reinforce your use of Wolfram Alpha by using YouTube to find suitable videos that provide you with quick tutorials on the technique.<br />
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The resources I just mentioned are all listed on that chart of “Important Mathematical Technology Tools” I published with the first two articles in this series.<br />
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As it turns out, with the UPS routing problem, the sequence of steps I have outlined so far quickly leads to identification of a small number of possible solution techniques for which there are many very accessible YouTube videos, and in fact, for this problem there is no need to go much further into my list of tools, if at all.<br />
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You should, though, check out the various other resources on my list, to see what they offer. Each new problem has to be approached afresh, in its own terms. Twitter is on my list because it is my list, and I have sufficiently many math-expert Twitter followers that a quick tweet can often yield just the information I need, saving me having to send out a slew of emails to people I think might be able to help. LinkedIn is also idiosyncratic to me, since I have a good network of mathematics and technology professionals I can contact. But the other resources are pretty generic.<br />
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Ideally, everything goes much more smoothly if you can avail yourself the services of a math consultant to assist you in negotiating the various resources. (I was that consultant to the teams in the three courses I gave.)<br />
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Interestingly, in the final meeting of my Princeton class (which was the fist time I used the UPS problem in a course), after having the student teams present their solutions, I gave the solution I had obtained, at the end of which two individuals came up to me to say they hoped I had not minded their sitting in on the class. (It was an experimental course, and there had been strangers sitting in for one or two sessions throughout the semester, so I had not paid them any attention.) They were, they said, postdocs working with Professor X, who was a math consultant for UPS and had worked on the algorithm the class and I had been trying to reverse engineer. Hence their curiosity-driven attendance on the last day! Unbeknownst to me, my final lecture had been my oral exam!<br />
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“How did I do?”, I asked. “You got it pretty well right,” they replied.<br />
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Which was nice for me, but it would not have mattered if I had followed a different track. What was important from an educational standpoint was the process.<br />
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Something else I suggested to the class was to come up with a solution—<b><i>any</i></b> solution—as soon as you can. “Don’t worry if it is optimal or even right,” I said. “Just check it by computation, perhaps in the form of a spreadsheet simulation. Once you have some solution that you can check (in the case of my UPS problem, check against the shipping data I supplied, or any other UPS data you can find on the Web), you can iterate to find a better one. It might turn out that your first solution, or your first three or four, won’t even get you to first base, but in the process of formulating and checking those initial attempts, you will inevitably gain insight into the problem you are trying to solve. Remember, computation is cheap, fast, and essentially limitless.”<br />
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If you are not familiar with this way of solving math problems, it may not seem like an approach that will work. But it does. It is, in fact, how all of today’s pros do it!<br />
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If you have not already done so, now is a good time to check out the dictionary definition of the word heuristic! Here is Wikipedia’s (at the time of writing):<br />
<span style="background-color: yellow;"><br /></span> “A heuristic technique (from the ancient Greek for “find” or “discover”), often called simply a heuristic, is an approach to any problem-solving, learning, or discovery that employs a practical method not guaranteed to be optimal or perfect, but sufficient for the immediate goals. Where finding an optimal solution is impossible or impractical, heuristic methods can be used to speed up the process of finding a satisfactory solution. Heuristics can be mental shortcuts that ease the cognitive load of making a decision. Examples of this method include using a rule of thumb, an educated guess, an intuitive judgment, guesstimate, stereotyping, profiling, or common sense.”<br />
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Without an expert consultant, the heuristics approach to solving real-world problems can work, but it definitely goes a lot faster, and with a far great likelihood of success, if you have a math expert you can call on. Not to “do any math.” Computer systems handle those parts. Rather to help you negotiate the vast array of resources at your disposal and select the most promising one(s) to try next. For that is what using mathematics to solve a real-world problem really boils down to these days: managing resources.<br />
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And managing resources is something humans are innately good at. Natural selection always favors those creatures which are best able to manage the available resources. We are here as present-day humans because as a species we are good at doing that. What is new in the case of mathematical problem solving is that pieces of mathematics (formulas, equations, procedures, algorithms, techniques) are now among the “intellectual Lego pieces” (freely accessible on the Web) we can use as we assemble a solution.<br />
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As the students in my three courses could, in principle, attest, you don’t need vast expertise in mathematics to work this way. You just need to be a good thinker able to work in a small team. I say, “in principle,” since I think it highly likely most of not all the students felt they did not do much at all by way of using math to solve a problem. But that, I would say, is because they have a conception of “using math to solve a problem” rooted in the Nineteenth Century, if not the Fourth Century BCE. From my perspective, they absolutely were able to do what I just said they did.<br />
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Of course, they were not as good at it as I am. I’ve been at this game a lot longer, and, make no mistake about it, experience counts. (I think it is close to being the only thing that counts.)<br />
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What does not count, at least to any extent even remotely approaching the prohibitive degree it used to, is the ability to “do the math.” You just need to be able to select (hopefully, with help from someone with experience) the right pieces from the available online mathematical smorgasbord, and stitch them together in an appropriate way.<br />
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This kind of problem-solving doesn’t feel like math (as we all came to love or hate), that’s for sure. In fact, it doesn’t even feel like work. Once they got into the swing of it, even the students who declared they were not good at math or did not enjoy it, found they were having a good time, working in teams in a creative, explorative way. For the fact is, properly approached, humans enjoy problem-solving. (That’s another consequence of natural selection— problem-solving, particularly group problem solving, is one of our species’ key survival advantages.)<br />
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In fact, another way to look at the recent revolution in how we “use math” to solve real-world problems, is that it has brought “using math” into the mainstream of human group activities we naturally find enjoyable. At heart, mathematical thinking is little more than formalized common sense. It always has been. Which means it is something we can all do. (In my 2000 book<span style="color: blue;"> </span><span style="color: blue;"><a href="https://www.amazon.com/Math-Gene-Mathematical-Thinking-Evolved/dp/0465016197" target="_blank">The Math Gene</a></span>, I presented an evolutionary explanation for the human brain’s acquisition of the ability to do mathematics, which implied that mathematical capacity is in the h<span style="background-color: transparent; color: blue; font-family: "times new roman"; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;"><a href="https://www.amazon.com/Math-Gene-Mathematical-Thinking-Evolved/dp/0465016197" target="_blank"></a></span>uman gene pool, and hence available for all of us to “switch on.”) What caused many people problems over the centuries was that, before we had technologies that could handle the formal symbol-manipulation stuff, the only way to employ our innate capacity for mathematical thinking was to train the brain to do those manipulations. But manipulating algebraic symbols with logical precision is most definitely not something our brains evolved to do. (Our early ancestors’ lives on the savannas did not present much by way of a need for algebra.) So we find it very hard. Only with great effort over several years can we train our brains to do such work. And even then, we are error-prone.<br />
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Incidentally, practically everything I have said in this article applies to the way 21st Century coders work. In coding as in mathematics, the days are long gone when it was all about writing thousands of lines of instructions. The modern-day mathematician’s Web resource MathOverflow (on my chart of useful math tools) was modeled on, and named after, the coding world’s StackOverflow. Both groups of professionals use heuristics. In today’s world, highly regarded math problem solvers and good coders have simply acquired a richer and more effective set of heuristics than the ones who are less highly ranked. And for the most part, developing heuristics is a result of reflective experience, not some innate talent.<br />
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And there you have it. The primary goal in 21st Century mathematics-education-for-all is the development of a good repertoire of heuristics.<br />
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I’ll leave you with a graphical summary of Tao’s categorization of the three kinds of mathematical thinking we can bring to problem-solving. I introduced this categorization above to provide a perspective on the three phases each one of us has to go through to become proficient mathematical (real-world) problem solvers. But it also provides an excellent summary of three historical stages of mathematical thinking as it has evolved over the past ten thousand years or so, from the invention of numbers in Sumeria, where the mathematical thinking of the time was accessible to all, through three millennia of formal mathematics development, where many people were never able to make effective use of it, and now into the third phase, where, because of technology, mathematical thinking can once again be accessible to all.<br />
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To be sure, we do not know the degree to which people have to master rigorous thinking to become good post-rigorous thinkers. As I already noted, I don’t for a second imagine that stage can be by-passed. (See the Willingham book I cited.) But, given today’s technological toolkit, including search, social media, online resources like Wolfram Alpha and Khan Academy, and a wide array of online courses, it is absolutely possible to master most of the rigorous thinking you need “on the job,” in the course of working on meaningful, and hence motivational and rewarding, real-world problems.<br />
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This is not to say there is no further need for teachers. Far from it. Very few people are able to become good mathematical thinkers on their own. Newtons and Ramanujans, who achieved great things with just a few books, are extremely rare. The vast majority of us need the guidance and feedback of a good teacher.<br />
<br />
What the inevitable transition to 21st Century math learning requires is that mathematics teachers operate very differently than in the past. The days where you need a live person to deliver information are largely over. Today, teaching is much more a matter of being a coach and mentor. To be sure, you can occasionally find such teaching on the Internet, but it works only if you can be one-on-one with that teacher. I expect there will be change, but I don’t expect an economy of scale. If I had to make a guess, I would predict that in due course you will find your (specialist) math teacher by going online to a Math-Teacher-Match.com website, where you will be paired with a practicing 21st Century math professional who spends part of each day coaching and mentoring students.<br />
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LABELS: mathematical thinking, problem-solving, rigorous thinking, pre-rigorous thinking, post-rigorous thinking, Terrence Tao, social media in mathematicsMathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-55504178834703941652018-03-09T12:19:00.001-05:002018-03-12T08:48:13.966-04:00How Today’s Pros Solve Math Problems: Part 2By Keith
Devlin<br />
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<span style="color: black; font-family: "" "calibri" "" , serif;">You can
follow me on Twitter @profkeithdevlin</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span><br />
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<span style="color: black; font-family: "" "calibri" "" , serif;">CHANGE OF
PLAN:<i style="color: black;"> When I wrote last month’s post, I said I would conclude the
description of my <a href="http://devlinsangle.blogspot.com/2018/01/deja-vu-all-over-again.html" target="_blank">Nueva School Course</a></i></span><i><span style="color: black; font-family: "" "calibri" "" , serif;"> this time. But when I sat down to write up that
concluding piece, I realized it would require not one but two further posts.
The course itself was the third iteration of an experiment I had tried out on a
university class of non-science majors and an Adult Education class. This series
of articles is my first attempt to try to describe it and articulate the
thinking behind it. As is often the case, when you try to describe something
new (at least it was new to me), you realize how much background experience and
unrecognized tacit knowledge you have drawn upon. In this post, I’ll try to
capture those contextual issues. Next month I’ll get back to the course itself.
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<span style="color: black; font-family: "" "calibri" "" , serif;">We all
know that mathematics is not always easy. It requires practice,
discipline and patience, as do many other things in life. And if learning math is not
easy, it follows that teaching math is not easy either. But it
can help both learner and teacher if they know what the end result is supposed
to be. </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">In my
experience, many learners and teachers don’t know that. In both cases, the
reason they don’t know it is that no one has bothered to tell them. There is a
general but unstated assumption that everyone knows why the teaching and
learning of mathematics is obligatory in every education system in the world.
But do they really?</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">There are
two (very different) reasons for teaching and learning mathematics. </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">One
reason is that it is a way of thinking that our species has developed over
several thousand years, that provides wonderful exercise for the mind, and
yields both challenging intellectual pleasure and rewarding aesthetic beauty to
many who can find their way sufficiently far into it. In that respect, it is like music, drama,
painting, philosophy, natural sciences, and many other intellectual human
activities. This is a perfectly valid reason to provide everyone with an
opportunity to sample it, and make it possible for those who like what they see
to pursue it as far as they desire. What it is not, is a valid reason for
making learning math obligatory throughout elementary, middle, and high school
education.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">The
argument behind math’s obligatory status in education is that it is useful;
more precisely, it is useful in the practical, everyday world. This is the view
of mathematics I am adopting in the short series of “Devlin’s Angle” essays of
which this is the third. (There will be one more next month. See episode 1 <a href="http://devlinsangle.blogspot.com/2018/01/deja-vu-all-over-again.html" target="_blank">here</a> </span><span style="color: black; font-family: "" "calibri" "" , serif;">and
episode 2 <a href="http://devlinsangle.blogspot.com/2018/02/how-todays-pros-solve-math-problems.html" target="_blank">here</a>.</span><span style="color: black; font-family: "" "calibri" "" , serif;">) </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">Indeed, mathematics <b>is </b>useful in the everyday practical world. In fact, we live in an age where mathematics is more relevant to our lives than at any previous time in human history. </span><br />
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<span style="color: black; font-family: "" "calibri" "" , serif;">It is, then, perfectly valid to say that we force each generation of school students to
learn math because it is a useful skill in today’s world. True, there are
plenty of people who do just fine without having that skill, but they can do so
only because there are enough other people around who do have it.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">But let’s
take that argument a step further. How do you teach mathematics so that it prepares
young people to use it in the world? Clearly, you start by looking at the way
people currently use math in the world, and figure out how best to get the next
generation to that point. (Accepting that by the time those students finish
school, the world’s demands may have moved on a bit, so those new graduates may
have a bit of catch up and adjustment to make.)</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">If the
way the professionals use math in the world changes, then the way we teach it
should change as well. Don’t you think?</span><span style="color: black; font-family: "times new roman" , serif;"> </span>That’s
certainly what has happened in the past.</div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">For
instance, in the ninth century, the Arabic-Persian speaking traders around
Baghdad developed a new, and in many instances more efficient, way to do
arithmetic calculations at scale, by using logical reasoning rather than
arithmetic. Their new system, which quickly became known as <i>al-jabr</i>
after one of the techniques they developed to solve equations, soon found its
way into their math teaching. </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">When
Hindu-Arabic arithmetic was introduced into Europe in the thirteenth century,
the school systems fairly quickly adopted it into their arithmetic teaching as
well. (It took a few decades, but knowledge moved no faster than the pace of a
packhorse back then. I tell the story of that particular mathematics-led
revolution in my 2011 book <a href="https://www.amazon.com/Man-Numbers-Fibonaccis-Arithmetic-Revolution/dp/0802778127/ref=sr_1_1?ie=UTF8&qid=1306990867&sr=8-1" target="_blank">The Man of Numbers</a></span><span style="color: black; font-family: "" "calibri" "" , serif;">.)</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">The
development of modern methods of accounting and the introduction of financial
systems such as banks and insurance companies, which started in Italy around
the same time, also led to new techniques being incorporated into the
mathematical education of the next generation.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">Later,
when the sixteenth century French mathematician <span style="background: white;">François
Viète introduced symbolic algebra, it too became part of the educational canon.
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<span style="background: white; color: black; font-family: "" "calibri" "" , serif;">In each case, those advances in mathematics were introduced
to make mathematics more easy to use and to increase its application. There was
never any question of “What is this good for?” People eagerly grabbed hold of
each new development and made everyday use of it as soon as it became
available.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="background: white; color: black; font-family: "" "calibri" "" , serif;">The rise of modern science (starting with Galileo in the seventeenth century) and later
the Industrial Revolution in the nineteenth century, led to still more impetus
to develop new mathematical concepts and techniques, though some of those
developments were geared more toward particular groups of professionals.
(Calculus, for example.)</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">To make
it possible for an average student or worker to make use of each new
mathematical concept or technique, sets of formal calculating rules (<i>algorithmic
procedures</i>) were developed and refined. Once mastered, these made it
possible to make use of the new mathematics to handle—in a practical way—the
tasks and problems of the everyday world for which those concepts and
techniques had been developed to deal with in the first place. </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">As a
result of all those advances, by the time the Baby Boomers came onto the
educational scene in the 1950s, the curriculum of mathematical algorithms that
were genuinely important in everyday life was fairly large. It was no longer
possible for a student to understand all the underlying mathematical concepts
and techniques behind the algorithms and procedures they had to learn. The best
that they could do was master, by repetitive practice, the algorithmic
procedures as quickly as possible and move on. [A few of us had difficulty
doing that. We wanted to understand what was going on. By and large, we
frustrated our teachers, who seemed to think we were simply troublesome slow
learners. Some of us eventually learned to “play the mindless algorithm game” in
class to pass the test, but kept struggling on our own to understand what was
going on, setting us on a path to becoming mathematics professors in the
1970s.] </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">It was
while that Boomer generation was going through the school system that
mathematics underwent the first step of a seismic shift that within a half of a
century would completely revolutionize the way mathematics was done. Not the
pure mathematics practiced by a few specialists as an art—though that too would
be impacted by the revolution to some extent. Rather, it was mathematics-as-used-in-the-world that would be radically transformed.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">The first
step of that revolution was the introduction of the electronic desktop
calculator in 1961. Although, mechanical desktop calculators had been available
since the turn of the Twentieth Century, by and large their use was restricted
to specialists—often called “computers” in businesses. [I actually had a
summer-job with British Petroleum as such a specialist in my last three years
at high school, and it was in my final year in that job that the office I
worked in acquired its first electronic desktop calculator and the British
Petroleum plant bought its first digital computer, both of which I learned to
use.] But with the increasing availability of electronic calculators, and in
particular the introduction of pocket-sized versions in the early 1970s, their
use in the workplace rapidly became ubiquitous. Mathematics underwent a major
change. Humans no longer needed to do arithmetic calculations themselves, and
professionals using arithmetic in their work no longer did.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">It was
not too many years later that, one by one, electronic systems were developed
that could execute more and more mathematical procedures and techniques, until,
by the late 1980s, there were systems that could handle all the mathematical
procedures that constituted the bulk of not only the school mathematics
curriculum, but the entire undergraduate math curriculum as well. The final
nail in the coffin of humans needing to execute mathematical procedures was the
release of the mathematics system <i>Mathematica</i> in 1988, followed soon after by the
release of <i>Maple</i>.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">In the
scientific, industrial, engineering, and commercial worlds, each new tool was
adopted as soon as it became available, and since the early 1990s,
professionals using mathematical techniques to carry out real-world tasks and
solve real-world problems have done so using tools like <i>Mathematica</i>, <i>Maple</i>,
and a host of others that have been developed. </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">Simultaneously,
colleges and universities quickly incorporated the use of those new tools into
their teaching. And while the cost of the more extensive tools put their use beyond most schools, the graphing calculator too was quickly
brought into the upper grades of the K-12 system, after its introduction in
1990.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">Yet,
while the pros in the various workplaces changed over to the new <i>human-machine-symbiotic</i>
way of doing math with little hesitation, most educators, exhibiting very wise
instincts, proceeded with far more caution. The first wave of humans to adopt
the new, machine-aided approach had all learned mathematics in an age when you
had to do everything yourself. Back then, “computers” were people. For them, it
was easy and safe to switch to executing a few keystrokes to make a computer
run a procedure they had carried out by hand many times themselves. But how
does a young person growing up in this new, digital-tools-world learn how to
use those new tools safely and effectively?</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">To some
extent, the answer is (and was) obvious. You teach not for smooth, proficient,
accurate execution of procedures, but for broad, general understanding of the
underlying mathematics. The downplay of execution and increased emphasis on
understanding are crucial. Computers outperform us to ridiculous degrees (of
speed, accuracy, size of dataset, and information storage and retrieval)
when it comes to execution of an algorithm. But they do not understand
mathematics. They do not understand the problem you are working on. They do not
understand the world. They don't understand anything. </span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">People,
on the other hand, can understand, and have a genetically inherited desire to
do so.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">But just
how <b><i>do</i></b> you go about teaching for the kind of understanding and
mastery that is required for students to transition into worlds and workplaces
dominated by a wide array of new mathematical tools, where they will encounter
work practices that involve very little by way of hand execution of algorithms?
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<span style="color: black; font-family: "" "calibri" "" , serif;">We know
so little about how people learn (though we do know a whole lot more than we
did just a few decades ago), that most of us with a stake in the education
business are rightly concerned about making any change that would effectively
be a massive experiment on an entire generation. So we can, and should, expect
small steps, particularly in systemic education.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">In the
U.S., the mathematicians who developed the mathematical guidelines for the
Common Core State Standards made a good first attempt at such a small step.
True, it quickly ran into difficulties when it came to implementing the
guidelines in a large and complex public educational system that is answerable
to the public. But that is surely a temporary hiccup. Most of the problems at
launch came from a lack of effective ways to assess the new kind of learning.
Those problems can be and are being fixed. Which is just as well. For, although
it’s possible to argue for tinkering with specific details of the Common Core
State Standards guidelines, in terms of setting out a broad set of educational
goals to aim for, there is no viable alternative first step. The pre-1970s
educational approach is no longer an option.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" "calibri" "" , serif;">In the
meantime, individual teachers at some schools (particularly, but not
exclusively, private schools) have been trying different approaches, in some
cases sharing their experiences on the <a href="https://mtbos.org/" target="_blank">MTBOS</a></span><span style="color: black; font-family: "" "calibri" "" , serif;"> (Math
Twitter Blog-O-Sphere), making use of another technological tool (social media)
now widely available. [For a quick overview of one global initiative to support
and promote such innovations, the OECD’s <i>Innovative Pedagogies for Powerful
Learning</i> project (IPPL), <a href="https://www.brookings.edu/blog/education-plus-development/2018/02/08/innovation-in-everyday-teaching-no-more-waiting-for-superman/" target="_blank">see this recent article</a> </span><span style="color: black; font-family: "" "calibri" "" , serif;">from the Brookings Institution.] </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="color: black; font-family: "" "calibri" "" , serif;">The
mini-course I gave at Nueva School in the San Francisco Bay Area last January,
which I talked about in the first of this short series of essays, is one such experiment in teaching
mathematics in a way that best prepares the next generation for the world they
will live and work in after graduation. I tested it first with a class of
non-science majors in Princeton in the fall of 2015 and then again with an
Adult Education class at Stanford in the fall of 2017. The Nueva School class
was its third outing.</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="color: black; font-family: "" "calibri" "" , serif;">With the
above backstory now established, next month I will describe that course</span><span style="color: black; font-family: "times new roman" , serif;"> </span>and talk about how today’s pros “do the math”.
(Again, let me stress, I am not talking here about “pure math”, the academic
discipline carried out by professional mathematicians in universities and a few
think tanks. My focus here is on using math in the everyday world.)</div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="color: black; font-family: "" "calibri" "" , serif;">In the
meantime, I’ll leave you with a simple arithmetic problem that I will discuss
in detail next time. </span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="color: black; font-family: "" "calibri" "" , serif;">It comes
with two instructions:</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
<ol start="1" style="margin-top: 0in;" type="1">
<li class="MsoNormal" style="line-height: normal; margin-bottom: 0in; vertical-align: baseline;"><span style="font-family: "" "calibri" "" , serif;">Solve
it as quickly as you can, in your head if possible. Let your mind <b><i>jump</i></b>
to the answer.</span></li>
<li>Then,
<b><i>and only then</i></b>, reflect on your answer, and <b><i>how you got
it</i></b>.</li>
</ol>
<ol start="1" style="margin-top: 0in;" type="1">
</ol>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="color: black; font-family: "" "calibri" "" , serif;">The goal
here is not to get the right answer, though a great many of you will. Rather,
the issue is how do our minds work, and how can we make our thinking more
effective in a world where machines execute all the mathematical procedures for
us?</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="color: black; font-family: "" "calibri" "" , serif;"><br /></span>
<span style="color: black; font-family: "" "calibri" "" , serif;">Ready for the problem? Here it is. </span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="color: black; font-family: "" "calibri" "" , serif;">PROBLEM:
A bat and a ball cost $1.10. The bat costs $1 more than the ball. How much does
the ball cost on its own? (There is no special pricing deal.)</span><span style="color: black; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;">
</div>
<br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-328022757845701212018-02-07T08:00:00.000-05:002018-02-08T11:57:32.295-05:00How today’s pros solve math problems: Part 1Last month, I wrote about my recent experience teaching a three-day mini-course in the <a href="http://www.nuevaschool.org/" target="_blank">Nueva School</a> January electives “Intersession” program. What I left out was a description of the course itself. I ended with the below diagram as a teaser. I said that, when reading in the usual left-right-down reading order, these were the technology tools that I typically turn to when I start to work on solving a new problem.<br />
<br />
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A number of mathematicians commented on social media that their list would be almost identical to mine. That did not surprise me. My chart simply captures the way today’s pros approach new problems. A number of math teachers expressed puzzlement. That too did not surprise me. The current mathematics curriculum is still rooted in a conception of “doing math” that developed to meet society’s needs in the 19th Century.<br />
<br />
Actually, I should point out that the diagram above is not exactly the one I published last month. I have added an icon for a spreadsheet. A mathematician in Austria emailed me to say I should have included it. The two of us had corresponded in the past about the use of spreadsheets in mathematics, both in problem solving and in teaching, and we were (and are) very much on the same page as to their usefulness in a wide variety of circumstances. My excuse for overlooking it the first time round was that it was only the second technological tool I brought into my mathematics arsenal, so far back in my career that I had long ago stopped thinking of it as something new. (The first piece of “new tech” I adopted was the electronic calculator, and that too did not appear in my chart.) I suspect that almost all math teachers, and indeed, pretty well all of society, make frequent use of calculators and spreadsheets, not only in their professional activities but in their social and personal lives as well. Still, the spreadsheet is such a powerful, ubiquitous mathematics tool, I should have included it, and now I have. (Its use definitely figured in the guidance I gave to the Nueva School class.) I have placed it in the position in my list that, on reflection, I find I turn to in order of frequency.
<br />
<br />
Some of the responses I received from teachers indicated that I need to clarify that, by “solving a mathematical problem”, I mean using mathematics to solve a real-world problem. The problem we worked on at Nueva School was one UPS worked on not long ago: “What is the most efficient way to route packages from place to place?” More on that later. A simpler example in the same vein is when we ask ourselves “Which kind, model, and hardware configuration of mobile phone best meets my needs within my current budget?”—an example where, for most of us, the item’s cost is high enough for us to weigh the (many) options fairly carefully.
<br />
<br />
This is clearly not the same as “solving a math problem” in a typical math textbook. For example, “What are the roots of the equation <i>x</i><sup>2</sup> + 3<i>x</i> – 5 = 0?” Those kinds of questions are, of course, designed to provide practice in using various specific, sharply focused, mathematics techniques, procedures, formulas, or algorithms.
<br />
<br />
Those techniques, procedures, etc. are the basic building blocks for using mathematics to solve problems in real life, but they don’t really present much of a <i>problem</i>, in the sense the word is used outside the math class. Indeed, the reason it can be valuable to master those basic techniques, etc. is that being able to use them fluidly means they <i>won’t</i> be a problem (in the sense of an <i>obstacle</i>) that gets in the way of solving what really is a mathematical <i>problem</i> (e.g., which phone to buy). That, of course, is why we call them basic skills. But having mastery of a range of <i>basic</i> skills does not make a person a good problem solver any more than being a master bricklayer makes someone an architect or a construction engineer.
<br />
<br />
My focus then, is on using math to solve real-world problems. That’s where things are very different from the days when I first learned mathematics. Back in the 1950s and 60s, when I went through the school system, we spent a huge amount of time mastering algorithms and techniques for performing a variety of different kinds of numerical and symbolic calculations, geometric reasoning, and equation solving. We had to. In order to solve any real-world problem, we had to be able to crank the algorithmic and procedural handles.
<br />
<br />
Not any more. These days, that smartphone in your pocket has access to cost-free cloud resources that can execute any mathematical procedure you need. What’s more, it will do it with a dataset of any real-world size, with total accuracy to whatever degree you demand, and in the majority of cases in a fraction of a second.
<br />
<br />
To put it another way, all those algorithms, techniques, and procedures I spent years mastering, all the way through to earning my bachelors degree in mathematics, became obsolete within my lifetime, an observation I wrote about <a href="https://www.huffingtonpost.com/entry/all-the-mathematical-methods-i-learned-in-my-university_us_58693ef9e4b014e7c72ee248" target="_blank">in an article</a> in the <i>Huffington Post</i> in January of last year.
<br />
<br />
So, does that mean all that effort was wasted? Not at all. Discounting the fact that in my case, I was able to make good use of those skills and knowledge for several decades before the march of technology rendered them obsolete, the one thing that I gained as a result of all that procedural learning that is as valuable today as it was back then, was the ability to think mathematically. I wrote about one aspect of that “mathematical thinking” mental ability, number sense, in a simultaneously published <a href="https://www.huffingtonpost.com/entry/number-sense-the-most-important-mathematical-concept_us_58695887e4b068764965c2e0" target="_blank">follow-up piece</a> to that<i> Huffington Post</i> article.
<br />
<br />
In today’s world, all the algorithmic, computational, algebraic, geometric, logical, and procedural skills that used to take ten years of effort to master can now be bought for $699. At least, that amount (the price of an iPhone 8, which I chose for illustration) is all it costs to give you access to all those skills. Making effective use of that vast powerhouse of factual knowledge and procedural capacity requires considerable ability. Anyone who mastered mathematics the way I did acquired that ability as an automatic by-product of mastering the basic skills. But what does it take to acquire it in an age when all those new tools are widely available?
<br />
<br />
The answer, <i>of course</i> (though not everyone involved in the mathematics education system thinks it is obvious, or even true), is that the educational focus has to shift from procedural mastery to <i>understanding</i>. Which is precisely the observation that guided the Common Core initiative in the United States. Yes, I know that the current leadership of the US Department of Education <a href="https://www.edsurge.com/news/2018-01-16-betsy-devos-touts-personalized-learning-slams-common-core-and-reform-efforts" target="_blank">believes that the Common Core is a bad idea</a>, but that is an administration that also believes the future of energy lies in fossil fuels, not renewables, and the highly qualified, career-professional contacts I have in the Department of Education have a very different view.
<br />
<br />
How do you acquire that high-level skill set? The answer is, the same way people always did: through lots of practice.
<br />
<br />
But be careful how you interpret that observation. What need to be practiced are <i>the kinds of activities that you would use as a professional</i>—or at least a <i>competent user</i> of mathematics—in the circumstances of the day. In my school days, that meant we had to practice with highly constrained, “toy” problems. But with today’s technologies, we can practice on <i>real-world problems</i> using <i>real-world</i> data.
<br />
<br />
Almost inevitably, when you do that, you find you frequently need to drop down to suitably chosen “toy problem” variants of your task in order to understand how a particular online tool (say) works and what it can (and cannot) do. But today, the purpose of, say, inverting a few 2x2 or 3x3 matrices is not (as it was in my day) so you can become fluent at doing so, and certainly not because you will actually invert by hand that 100x100 matrix that has just reared its ugly head in your real-world problem. No, you just need to get a good understanding of what it means to invert a matrix, why you might do so, and what problems can arise.<br />
<br />
And you know what? That’s rarely a problem. Once you have identified a mathematical technique you need to understand, the chances are high you will find not one but a dozen or more YouTube videos that explain it to you.
<br />
<br />
These new tools certainly don’t solve the problem for you. [Well, sometimes they may do, but in that case it wasn’t a problem that required the time of a mathematician. Better to move on and put your efforts into a problem that cannot be solved by an app in the Cloud!] All that these fancy new tools have done is change the level at which we humans operate.
<br />
<br />
At heart, that shift is no different from the level-shift introduced in the 9th Century when traders in and around Baghdad developed techniques for doing routine arithmetic calculations at scale, by performing operations not on specific numbers but on classes of numbers. One of the techniques they developed was called, <i>al-jabr</i>, a term that ended up giving the name we use today to refer to that new kind of calculation procedure: algebra.
<br />
<br />
Throughout mathematics’ history, mathematicians have calculated and reasoned logically with the basic building blocks of the time. Today’s procedures (that have to be executed) turn into tomorrow’s basic entities (<i>on which you operate</i>). A classic example is differential calculus, where functions are no longer viewed as rules that you execute to yield new numbers from old numbers, but higher-level objects <i>on which you operate</i> to produce new functions from old functions.
<br />
<br />
So (finally), what exactly did we do in that Nueva School mini-course to illustrate the way today’s pros use math to solve a problem? The problem, remember, was this: Reverse engineer the core algorithm than UPS uses to route packages from origin to destination?
<br />
<br />
To start the class off—they worked in small teams of three or four—I provided a small amount of information to get them started:<br />
<ol>
<li>Tracking information for a fairly large, heavy case, including a partially dismantled bicycle, I had shipped from Petaluma, California to Fair Haven, New Jersey, in 2015. See image below.</li>
<li>I told them I sent the case by “three day select.”</li>
<li>I reported that my package went by plane from Louisville, Kentucky, to the UPS facility in Newark, where it was immediately loaded onto a truck, and was delivered to the intended Fair Haven destination with just a few hours to spare within the three-day period guaranteed.</li>
</ol>
That information, I told the class, was enough to figure out how the routing algorithm worked. [This itself is useful information that I did not have when I first solved the problem, but they had to figure it out by the end of the course, so I was happy to give them additional information.] In solving this problem, they could elicit my help as their “math consultant,” to call on with specific questions when required. But they had to carry out the key steps.
<br />
<br />
They could, of course, use the various tools in my “modern math tools” chart, and any others they could find. (Since the UPS routing algorithm is an extremely valuable trade secret, they would not find that online, of course.)
<br />
<br />
Next month, I’ll tell you how they got on. In the meantime, you might like to see how far you can get with it. Happy problem solving! Happy mathematical thinking!
<br />
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<a href="https://4.bp.blogspot.com/-SX-pwAkG6FY/WnN-tTM1kyI/AAAAAAAALBo/HmqKtPNEQwscQb0J7XwSCT2LG4YTs3SfACLcBGAs/s1600/Bike_data.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="768" data-original-width="1024" height="480" src="https://4.bp.blogspot.com/-SX-pwAkG6FY/WnN-tTM1kyI/AAAAAAAALBo/HmqKtPNEQwscQb0J7XwSCT2LG4YTs3SfACLcBGAs/s640/Bike_data.jpg" width="640" /></a></div>
<i>Part 2 will appear next month.</i><br />
<br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-11164661543675779552018-01-23T10:31:00.002-05:002018-01-23T11:13:35.475-05:00Déjà vu, all over againI gave a short course at a local high school recently. Three days in a row, two hours a day, to
fifteen students. To my mind, it was a huge success. By the end of the course, the students had
successfully reverse-engineered UPS’s core routing/scheduling algorithm. In fact, they spent the
last half hour brainstorming how UPS might improve their efficiency. (My guess is the company
had long ago implemented, or at least considered, the ideas the kids came up with, but that
simply serves to illustrate how far they had come in just six hours of class-time.)
<br />
<br />
To be sure, it was not an average class in an average high school. <a href="http://www.nuevaschool.org/" target="_blank">Nueva School</a>, located in the
northern reaches of Silicon Valley, is private and expensive (tuition runs at $36,750 for an 8th gader), and caters to students who have already shown themselves to be high achievers.
Many Silicon Valley tech luminaries send their children there, and some serve on the board.
They have an excellent faculty. Moreover, the fifteen students in my class had elected to be
there, as part of their rich, January, electives learning experience called “Intersession”.
<br />
<br />
I was familiar with the school, having been invited to speak at their annual education
conference on a couple of occasions, but this was the first time I had taught a class.
<br />
<br />
Surprisingly, the experience reminded me of my own high school education, back in the UK in
the early 1960s. My high school was a state run, selective school in the working class city of
Hull, a major industrial city and large ocean fishing and shipping port. Socially and financially, it
was about as far away as you could get from Nueva School on the San Francisco Peninsula, and
my fellow students came from very different backgrounds than the students at Nueva.
<br />
<br />
What made my education so good was a highly unusual set of historical circumstances. Back
then, Hull was a fiercely socialist city that, along with the rest of the UK, was clawing its way out
of the ravages of the Second World War. For a few short years, the crippling English class
system broke down, and an entire generation of baby boomers entered the school system
determined to make better lives for themselves—and everyone else. (“Me first” came a
generation later.)
<br />
<br />
We had teachers who had either just returned from fighting the war (the men on the
battlefields, the women in the factories or in military support jobs), or were young men and
women just starting out on their teaching careers, having received their own school education
while the nation was at war. There was a newly established, free National Health Service, an
emerging new broadcasting technology (television) run by a public entity, a rapidly growing
communications systems (a publicly funded telephone service), and free education, including
government-paid- for university education for the 3 percent or so able to pass the challenging
entrance exams.
<br />
<br />
We were the generation that the nation was dependent on to rebuild, making our way through
the education system in a social and political environment where the class divisions that had
been a part of British life for centuries had been (temporarily, it turned out) cast aside by the need to fight a common enemy across the English Channel. The result was that, starting in the middle of the 1960s, a “British Explosion” of creative scientific, engineering, and artistic talent
burst forth onto the world. Within our individual chosen domains, we all felt we could do
anything we set our minds to. And a great many of us did just that. About half my high school
class became highly successful people. That from a financially impoverished, working class
background.
<br />
<br />
It was short lived, lasting but a single generation. I was simply lucky to be part of it.
<br />
<br />
What brought it all back to me was finding myself in a very similar educational environment in
my three days at Nueva School. The circumstances could hardly be more different, of course.
But talking and working with those students, I sensed the same thirst to learn, the same drive
to succeed (in terms they set for themselves), and the same readiness to keep trying I had
experienced two generations earlier. It felt comfortingly—and encouragingly—familiar.
<br />
<br />
But I digress. In fact, I’ve done more than digress. I’ve wandered far from my intended path. Or
have I? The point I want to get across is that when it comes to learning, success is about 5
percent talent, 35 percent the teachers and students around you, and 60 percent desire and
commitment. (I just made up those figures, but they represent more or less how I see the
landscape, having been an education professional for half a century.)
<br />
<br />
It turns out that, in today’s world, given those ingredients, in roughly those proportions, it is
possible for a small group of people, in the space of just a few days, to make significant
progress in solving a major problem of massive societal importance. (If you can figure out how
UPS performs its magic, you can do the same thing with many other large organizations,
Walmart, Amazon, United Airlines, and so on.)
<br />
<br />
How can it be possible to take a small group of students, still in high school, and make solid
progress on a major mathematical problem like that? It would not have been possible in my
school days. The answer is, in today’s world, everyone has access to the same rich toolset the
professionals use. Moreover, most of those tools—or at least, enough of them—are free to
anyone with access to a smartphone or a personal computer. You just have to know how to
make effective use of them.
<br />
<br />
Next month, I will describe how my Nueva class went about the UPS project. (I had done it once
before, with a non-science majors undergraduate class at Princeton University. Doing it with
high school students confirmed my belief that a group with less academic background could
achieve the same result, in the process providing me with some major-league ammunition to
back up my oft-repeated—and oft-ignored or disputed—claim that K-12 mathematics
education is in need of a major (and I mean MAJOR) makeover. (After the invention of the
automobile, it made more sense to teach people how to drive than how to look after a horse. I
feel the math ed argument should end with that razor-sharp analogy, but it rarely does.)
<br />
<br />
As I say, that discussion is for next month. But let me leave you with a teaser. Actually, two
teasers. One is my January 1, 2017 opinion piece in the <i>Huffington Post</i>, "<a href="https://www.huffingtonpost.com/entry/all-the-mathematical-methods-i-learned-in-my-university_us_58693ef9e4b014e7c72ee248" target="_blank">All The MathematicalMethods I Learned In My University Math Degree Became Obsolete In My Lifetime</a>." The
other teaser is the diagram I will end with. It summarizes some of the most useful tools that
a professional mathematician today uses when starting to work on a new problem. (Note: I’m
talking about using math to solve real-world problems here. Pure mathematics is very different,
although all the tools I will mention <i>can</i> be of use to a pure mathematician.)
<br />
<br />
This is my set of “most useful tools,” I should note, and reading the diagram left-to- right, top to
bottom, the tools I list are roughly in the order I have used them in working on various projects
over the past fifteen years. Other mathematicians might produce different collections and
different orders. But they won’t be <i><b>that</b></i> much different, and I’ll bet they all begin with the same
first tool.
<br />
<br />
If you find this diagram in any way surprising, you likely have not worked in today’s world of
mathematical problem solving. If you find it surprising <b><i>and</i></b> are in mathematics education, I
respectfully point out that this is the mathematical toolset that your students will need to
master in order to make use of math in the world they will inhabit after graduation. You may or
may not like that. If you don’t like it, then that is unfortunate. Mathematical problem solving is
simply done differently today. It just is.
<br />
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<br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-46466326448319262292017-12-14T15:37:00.001-05:002017-12-14T15:37:14.152-05:00Clash of representations<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-ZhAuLxqHGCk/Wi7xCCFyiiI/AAAAAAAAK_c/oWSNle3TL5IZbcWyEsU8cmZ_n25JO2N4gCLcBGAs/s1600/Original_tweet.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1326" data-original-width="960" height="400" src="https://2.bp.blogspot.com/-ZhAuLxqHGCk/Wi7xCCFyiiI/AAAAAAAAK_c/oWSNle3TL5IZbcWyEsU8cmZ_n25JO2N4gCLcBGAs/s400/Original_tweet.jpg" width="289" /></a></div>
<br />
The pie chart in the above tweet jumped out of the page when it appeared in my twitter feed
on September 14. My initial shock at seeing the figure 1% attached to a region of the pie chart
that was evidently almost 25% of the total area of the disk did not last long, of course, since the
accompanying text made it clear what the diagram was intended to convey. The 1% label
referred to the section of the population being discussed, whereas the pie-chart indicated the
share of taxes paid by that group. Indeed, the image was an animated GIF; when I clicked on it,
the region labeled “1%” shrank, culminating with the chart on the right in the image shown
below:<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-n1mBg8RGILw/Wi7yQdbx0AI/AAAAAAAAK_0/WzRnrTYoETYWdxVcKRW2mOmXsCRNAcGswCLcBGAs/s1600/Pie_GIF.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="474" data-original-width="851" height="221" src="https://3.bp.blogspot.com/-n1mBg8RGILw/Wi7yQdbx0AI/AAAAAAAAK_0/WzRnrTYoETYWdxVcKRW2mOmXsCRNAcGswCLcBGAs/s400/Pie_GIF.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
But here’s the thing. Even after I had figured out what the chart was intended to convey, I still
found it confusing. I wondered if a lay-reader, someone who is not a professional
mathematician, would manage to parse out the intended meaning. It was not long before I
found out. The image below shows one of the tweets that appeared in response less than an
hour later:<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-g7-5kLuiETo/Wi7xl0FPZ2I/AAAAAAAAK_o/haf-Gqyva8MP-61rhIEnwiUHsDMPjEoWwCLcBGAs/s1600/Reply.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="169" data-original-width="925" height="71" src="https://2.bp.blogspot.com/-g7-5kLuiETo/Wi7xl0FPZ2I/AAAAAAAAK_o/haf-Gqyva8MP-61rhIEnwiUHsDMPjEoWwCLcBGAs/s400/Reply.jpg" width="400" /></a></div>
As I had suspected, a common reaction was to dismiss the chart as yet another example of a
bad data visualization created by an innumerate graphics designer. Indeed, that had been my
initial reaction. But this particular example is more interesting. Yes, it is a bad graphic, for the
simple reason that it does not convey the intended message. But not because of the
illustrator’s innumeracy. In fact, numerically, it appears to be as accurate as you can get with a
pie chart. The before and after charts do seem to have regions whose areas correspond to the
actual data on the tax-payer population.
<br />
<br />
This example was too good to pass up as an educational tool: asking a class to discuss what the
chart is intended to show, could lead to a number of good insights into how mathematics can
help us understand the world, while at the same time having the potential to mislead. I was
tempted to write about it in my October post, but wondered if I should delay a couple of
months to avoid an example that was at the heart of a current, somewhat acrimonious party-political debate. As it turned out, the September 30 death of the game-show host Monty Hall
resolved the issue for me—I had to write about that—and then November presented another
“must do” story (the use of mathematics in election jerrymandering). So this month, with the
background political, tax votes now a matter of historical record, I have my first real
opportunity to run this story.
<br />
<br />
The two-month delay brought home to me just how problematic this particular graphic is. Even
knowing in advance what the issue is, I still found I had to concentrate to “see” the chart as
conveying the message intended. That “1%” label continued to clash with the relative area of
the labeled region.
<br />
<br />
It’s a bit like those college psychology-class graphics that show two digits in different font sizes,
and ask you to point to the digit that represents the bigger integer. If the font sizes clash with
the sizes of the integers, you take measurably longer to identify the correct one, as shown
below:<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-yx8NR9Vkn7A/Wi7x4Gw6ioI/AAAAAAAAK_s/YsP1uhGBpuslAZ2iCfYGR5xqwVwxoNn2ACLcBGAs/s1600/Stroop.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="393" data-original-width="759" height="206" src="https://1.bp.blogspot.com/-yx8NR9Vkn7A/Wi7x4Gw6ioI/AAAAAAAAK_s/YsP1uhGBpuslAZ2iCfYGR5xqwVwxoNn2ACLcBGAs/s400/Stroop.jpg" width="400" /></a></div>
For me, the really big take-home lesson from the tax-proposal graphic is the power of two
different mathematical representations of proportions: pie charts and numerical percentages.
Each, on its own, is instant. In the case of the pie chart, the representation draws on the innate
human cognitive ability to judge relative areas in simple, highly symmetrical figures like circular
disks or rectangles. With percentages, there is some initial learning required—you have to
understand percentages—but once you have done that, you know instantly what is meant by
figures such as “5%” or “75%."
<br />
<br />
But how do you get that understanding of the meaning of numerical percentages? For most of
us (I suspect all of us), it comes from being presented (as children) with area examples like pie
charts and subdivided rectangles. This sets us up to be confused, bigly, by examples where
those two representations are used in the same graphic but with the percentage representing
something other than the area of the segment (or what that area is intended to represent).
<br />
<br />
The message then, from this particular example—or at least the message I got from it—is that
powerful graphics are like any powerful tool, their power for good depends on using them
wisely; if used incorrectly, they can confuse and mislead. And make no mistake about it,
numbers are incredibly powerful tools. Their invention <b><i>alone</i></b> is by far the greatest
mathematical invention in human history. That’s why in every nation in the world, math is the
only mandated school subject apart from the native language.
<br />
<br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-15294729129469802322017-11-16T16:40:00.000-05:002017-12-11T15:52:16.423-05:00Mathematics and the Supreme CourtAmerican courts have never appeared to be very receptive to mathematical arguments,
in large part, some (including me) have assumed, because many judges do not feel
confident evaluating mathematical reasoning and, in the case of jury trials, no doubt
because they worry that unscrupulous, math-savvy lawyers could use formulas and
statistics to fool jury members. There certainly have been some egregious examples of
this, particularly when bogus probability arguments have been presented. Indeed, one
classic misuse of conditional probability is now known as the “<a href="https://en.wikipedia.org/wiki/Prosecutor%27s_fallacy" target="_blank">prosecutor’s fallacy</a>."<br />
<br />
Another example where the courts have trouble with probability is in cases involving
DNA profiling, particularly Cold Hit cases, where a DNA profile match is the only hard
evidence against a suspect. I myself have been asked to provide expert testimony in
some such cases, and I wrote about the issue in this column in <a href="https://www.maa.org/external_archive/devlin/devlin_09_06.html" target="_blank">September</a> and <a href="https://www.maa.org/external_archive/devlin/devlin_10_06.html" target="_blank">October</a>
of 2006.
<br />
<br />
In both kinds of case, the courts have good reason to proceed with caution. The
prosecutor’s fallacy is an easy one to fall into, and with Cold Hit DNA identification there
is a real conflict between frequentist and Bayesian probability calculations. In neither
case, however, should the courts try to avoid the issue. When evidence is presented,
the court needs to have as accurate an assessment as possible as to its reliability or
veracity. That frequently has to be in the form of a probability estimate.
<br />
<br />
Now the courts are facing another mathematical conundrum. And this time, the case
has landed before the US Supreme Court. It is a case that reaches down to the very
foundation of our democratic system: How we conduct our elections. Not how we use
vote counts to determine winners, although that is also mathematically contentious, as I
wrote about in this column in <a href="https://www.maa.org/external_archive/devlin/devlin_11_00.html" target="_blank">November</a> of 2000, just before the Bush v Gore Presidential
Election outcome ended up before the Supreme Court. Rather, the issue before the
Court this time is how states are divided up into electoral districts for state elections.
<br />
<br />
How a state carves up voters into state electoral districts can have a huge impact on the
outcome. In six states, Alaska, Arizona, California, Idaho, Montana, and Washington,
the apportioning is done by independent redistricting commissions. This is generally
regarded—at least by those who have studied the issue—as the least problematic
approach. In seven other states, Arkansas, Colorado, Hawaii, Missouri, New Jersey,
Ohio, and Pennsylvania, politician commissions draw state legislative district maps. In
the remaining 37 states, the state legislatures themselves are responsible for state
legislative redistricting. And that is where the current problem arises.
<br />
<br />
There is, of course, a powerful temptation for the party in power to redraw the electoral
district maps to favor their candidates in the next election. And indeed, in the states
where the legislatures draw the maps, both major political parties have engaged in that
practice. One of the first times this occurred was in 1812, when Massachusetts
governor Elbridge Gerry redrew district boundaries to help his party in an upcoming
senate election. A journalist at the <i>Boston Gazette</i> observed that one of the contrived districts in Gerry’s new map looked like a giant salamander, and gave such partisan redistricting a name, combining <i>Gerry</i> and <i>mander</i> to create the new word <i>gerrymander</i>.
Though Gerry lost his job over his sleight-of- hand, his redistricting did enable his party
to take over the state senate. And the name stuck.
<br />
<br />
Illegality of partisan gerrymandering is generally taken to stem from the 14th
Amendment, since it deprives the smaller party of the equal protection of the laws, but it
has also been argued to be, in addition, a 1st Amendment issue—namely an
apportionment that has the purpose and effect of burdening a group of voters’
representational rights.
<br />
<br />
In 1986, the Supreme Court issued a ruling that partisan gerrymandering, if extreme
enough, is unconstitutional, but it has yet to throw out a single redistricting map. In large
part, the Supreme Court’s inclination to stay out of the redistricting issue is based on a
recognition that both parties do it, and over time, any injustices cancel out, as least
numerically. Historically, this was, generally speaking, true. Attempts to gerrymander
have tended to favor both parties to roughly the same extent. But in 2012, things took a
dramatic turn with a re-districting process carried out in Wisconsin.
<br />
<br />
That year, the recently elected Republican state legislature released a re-districting map
generated using a sophisticated mathematical algorithm running on a powerful
computer. And that map was in an altogether new category. It effectively guaranteed
Republican majorities for the foreseeable future. The Democrat opposition cried foul, a
Federal District Court agreed with them, and a few months ago the case found its way
to the Supreme Court.
<br />
<br />
That the Republicans come across as the bad actors in this particular case is likely just
an accident of timing; they happened to come to power at the very time when political
parties were becoming aware of what could be done with sophisticated algorithms. If
history is any guide, either one of the two main parties would have tried to exploit the
latest technology sooner or later. In any event, with mathematics at the heart of the new
gerrymandering technique, the only way to counter it may be with the aid of equally
sophisticated math.
<br />
<br />
The most common technique used to gerrymander a district is called “packing and
cracking." In packing, you cram as many of the opposing party’s voters as possible into
a small number of “their” districts where they will win with many more votes than
necessary. In cracking, you spread opposing party’s voters across as many of “your”
districts as possible so there are not enough votes in any one of those districts to ever
win there.
<br />
<br />
A form of packing and cracking arises naturally when better-educated liberal-leaning
voters move into in cities and form a majority, leaving those in rural areas outnumbered
by less-educated, more conservative-leaning voters. (This is thought to be one of the
factors that has led to the increasing polarization in American politics.) Solving that
problem is, of course, a political one for society as a whole, though mathematics can be
of assistance by helping to provide good statistical data. Not so with partisan
gerrymandering, where mathematics has now created a problem that had not arisen
before, for which mathematics may of necessity be part of the solution.
<br />
<br />
When Republicans won control of Wisconsin in 2010, they used a sophisticated
computer algorithm to draw a redistricting map that on the surface appeared fair—no
salamander-shaped districts—but in fact was guaranteed to yield a Republican majority
even if voter preferences shifted significantly. Under the new map, in the 2012 election,
Republican candidates won 48 percent of the vote, but 60 of the state’s 99 legislative
seats. The Democrats’ 51 percent that year translated into only 39 seats. Two years
later, when the Republicans won the same share of the vote, they ended up with 63
seats—a 24-seat differential.
<br />
<br />
Recognizing what they saw as a misuse of mathematics to undermine the basic
principles of American democracy, a number of mathematicians around the country
were motivated to look for ways to rectify the situation. There are really two issues to be
addressed. One is to draw fair maps—a kind of “positive gerrymandering.” The other is
to provide reliable evidence to show that a particular map has been intentionally drawn
to favor one party over another, if such occurs, and moreover to do so in a way that the
courts can understand and accept. Neither issue is easy to solve, and without
mathematics, both are almost certainly impossible.
<br />
<br />
For the first issue, a 2016 Supreme Court ruling gave a hint about what kind of fairness
measure it might look kindly on: one that captures the notion of “partisan symmetry,”
where each party has an equal opportunity to convert its votes into seats. The
Wisconsin case now presents the Supreme Court with the second issue.
<br />
<br />
When, last year, a Federal District Court in Wisconsin threw out the new districting map,
they cited both the 1st and 14th Amendments. It was beyond doubt, the court held, that
the new maps were “designed to make it more difficult for Democrats, compared to
Republicans, to translate their votes into seats.” The court rejected the Republican
lawmakers’ claim that the discrepancy between vote share and legislative seats was
due simply to political geography. The Republicans had argued that Democratic voters
are concentrated in urban areas, so their votes have an impact on fewer races, while
Republicans are spread out across the state. But, while that is true, geography alone
does not explain why the Wisconsin maps are so skewed.
<br />
<br />
So, how do you tell if a district is gerrymandered? One way, that has been around for
some time, is to look at the geographical profile. The <i>gerrymandering score</i>, G, is
defined by:<br />
<i>G</i> = <i>gP/A</i>, where<br />
g: the district’s boundary length, minus natural boundaries (like coastlines and rivers)<br />
P: the district’s total perimeter<br />
A: the district’s area<br />
The higher the score, the wilder is the apportionment as a geographic region, and
hence the more likely to have been gerrymandered.
<br />
<br />
That approach is sufficiently simple and sensible to be acceptable to both society and
the courts, but unfortunately does not achieve the desired aim of fairness. And, more to
the point in the Wisconsin case, use of sophisticated computer algorithms can draw
maps that have a low gerrymandering score and yet are wildly partisan.
<br />
<br />
The Wisconsin Republicans’ algorithm searched through thousands of possible maps
looking for one that would <b><i>look reasonable</i></b> according to existing criteria, but would
favor Republicans <i><b>no matter what the election day voting profile might look like</b></i>. As
such, it would be a statistical <i><b>outlier</b></i>. To find evidence to counter that kind of approach,
you have to look at the results the districting produces when different voting profiles are
fed into it.
<br />
<br />
One promising way to identify gerrymandering is with a simple mathematical formula
suggested in 2015, called the “<a href="https://www.brennancenter.org/sites/default/files/legal-work/How_the_Efficiency_Gap_Standard_Works.pdf" target="_blank">efficiency gap</a>." It was the use of this measure that
caused, at least in part, the Wisconsin map to be struck down by the court. It is a simple
idea—and as I noted, simplicity is an important criterion, if it is to stand a chance of
being accepted by society and the courts.
<br />
<br />
You can think of a single elector’s vote as being “wasted” if it is cast in a district where
their candidate loses or it is cast in a district where their candidate would have won
there anyway. The efficiency gap measures those “wasted” votes. For each district, you
total up the number of votes the winning candidate receives in excess of what it would
have taken to elect them in that district, and you total up the number of votes the losing
candidate receives. Those are the two parties’ “wasted votes” for that district.
<br />
<br />
You then calculate the difference between those “wasted-vote” totals for each of the two
parties, and divide the answer by the total number of votes in the state. This yields a
single percentage figure: the <b><i>efficiency gap</i></b>. If that works out to be greater than 7%,
the systems developers suggest, the districting is unfair.
<br />
<br />
By way of an example, let’s see what the efficiency gap will tell us about the last
Congressional election. In particular, consider Maryland’s 6 th Congressional district,
which was won by the Democrats. It requires 159K votes to win. In the last election,
there were 186K Democrat votes, so 186K – 159K = 26K Democrat votes were
“wasted,” and 133K Republican votes, all of which were “wasted.”
<br />
<br />
In Maryland as a whole, there were 510K Democrat votes “wasted” and 789K
Republican votes “wasted.” So, statewide, there was a net “waste” of 789K – 510K =
279K Republican votes.
<br />
<br />
There were 2,598M votes cast in total. So the efficiency gap is 279K/2598K = 10.7% in
favor of the Democrats.
<br />
<br />
I should note, however, that the gerrymandering problem is currently viewed as far more
of a concern in state elections than in congressional races. Last year, two social scientists published <a href="http://www-personal.umich.edu/~jowei/gerrymandering.pdf" target="_blank">the results</a> they obtained using computer simulations to measure
the extent of intentional gerrymandering in congressional district maps across most of
the 50 states. They found that on the national level, it mostly canceled out between the
parties. So banning only intentional gerrymandering would likely have little effect on the
partisan balance of the U.S. House of Representatives. The efficiency gap did,
however, play a significant role in the Wisconsin court’s decision.
<br />
<br />
Another approach, developed by a team at <a href="https://services.math.duke.edu/projects/gerrymandering/" target="_blank">Duke University</a>, takes aim at the main idea
behind the Wisconsin redistricting algorithm—searching through many thousands of
possible maps looking for ones that met various goals set by the creators, any one of
which would, of necessity, be a <b><i>statistical outlier</i></b>. To identify a map that has been
obtained in this way, you subject it to many thousands of random tweaks. If the map is
indeed an outlier, the vast majority of tweaks will yield a fairly unremarkable map. So,
you compare the actual map with all those thousands of seemingly almost identical, and
apparently reasonable, variations you have generated from it. If the actual map
produces significantly different election results from all the others, when presented with
a range of different statewide voting profiles, you can conclude that it is indeed an
“outlier” — a map that could only have been chosen to deliberately subvert the
democratic process.
<br />
<br />
And this is where we—and the Supreme Court—are now. We have a problem for our
democracy created using mathematics. Mathematicians looking for mathematical ways
to solve it, and there are already two candidate “partisan gerrymandering test” in the
arena. Historically, the Supreme Court has proven resistant to allowing math into the
courtroom. But this time, it looks like they may have no choice. At least as long as state
legislatures continue to draw the districting maps. Maybe the very threat of having to
deal with mathematical formulas and algorithms will persuade the Supreme Court to
recommend that Congress legislates to enforce all states to use independent
commissions to draw the districting maps. Legislation under pain of math. We will know
soon enough.
<br />
<br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-34989335412024262412017-10-11T17:45:00.002-04:002017-10-12T09:37:29.227-04:00Monty Hall may now rest in peace, but his problem will continue to frustrate<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-uMMlSwZYqjw/Wd6Q2LQyGXI/AAAAAAAAK7w/bNey7uxYHNQh1ducy7jasljp7VEP915PwCLcBGAs/s1600/MontyHallDoors.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="312" data-original-width="516" height="241" src="https://1.bp.blogspot.com/-uMMlSwZYqjw/Wd6Q2LQyGXI/AAAAAAAAK7w/bNey7uxYHNQh1ducy7jasljp7VEP915PwCLcBGAs/s400/MontyHallDoors.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Monty Hall with a contestant in <i>Let's Make a Deal</i>.</td></tr>
</tbody></table>
The news that American TV personality Monty Hall died recently (<a href="https://www.nytimes.com/2017/09/30/obituaries/monty-hall-dead-lets-make-a-deal.html?hp&amp;action=click&amp;pgtype=Homepage&amp;clickSource=story-heading&amp;module=second-column-region&amp;region=top-news&amp;WT.nav=top-news&amp;_r=0" target="_blank">The New York Times, September 30</a>) caused two groups of people to sit up and take note. One group, by far the larger, was
American fans of television game shows in the 1960s and 70s, who tuned in each week to his
show “Let’s Make a Deal.” The other group include lovers of mathematics the world over, most of
whom, I assume, have never seen the show.
<br />
<br />
I, and by definition all readers of this column, are in that second category. As it happens, I have
seen a key snippet of one episode of the show, which a television documentary film producer
procured to use in a mathematics program we were making about probability theory. Our
interest, of course, was not the game show itself, but the famous — indeed infamous —
“Monty Hall Problem” it let loose on an unsuspecting world.
<br />
<br />
To recap, at a certain point in the show, Monty would offer one of the audience participants
the opportunity to select one of three doors on the stage. Behind one, he told them, was a
valuable prize, such as a car, behind each of the other two was a booby prize, say a goat. The
contestant chose one door. Sometimes, that was the end of the exchange, and Monty would
open the door to reveal what the contestant had won. But on other occasions, after the
contestant had chosen a door, Monty would open one of the two unselected doors to reveal a
booby prize, and then give them the opportunity to switch their selection. (Monty could always
do this since he knew exactly which door the prize was hidden behind.)
<br />
<br />
So, for example, if the contestant first selects Door 2, Monty might open Door 1 to reveal a
goat, and then ask if the contestant wanted to switch their choice from Door 2 to Door 3. The
mathematical question here is, does it make any difference if the contestant switches their
selection from Door 2 to Door 3? The answer, which on first meeting this puzzler surprises
many people, is that the contestant doubles their chance of winning by switching. The
probability goes up from an original 1/3 of Door 2 being the right guess, to 2/3 that the prize is
behind Door 3.
<br />
<br />
I have discussed this problem in <i>Devlin’s Angle</i> on at least two occasions, the most recent being
<a href="https://www.maa.org/external_archive/devlin/devlin_12_05.html" target="_blank">December 2005</a>, and have presented it in a number of articles elsewhere, including national
newspapers. That on each occasion I have been deluged with mail saying my solution is
obviously false was never a surprise; since the problem is famous precisely because it presents
the unwary with a trap. That, after all, is why I, and other mathematics expositors, use it! What
continues to amaze me is how unreasonably resistant many people are to stepping back and
trying to figure out where they went wrong in asserting that switching doors cannot possibly
make any difference. For such reflection is the very essence of learning.
<br />
<br />
Wrapping your mind around the initially startling information that switching the doors doubles
the probability of winning is akin to our ancestors coming to terms with the facts that the Earth
is not flat or that the Sun does not move around the Earth. In all cases, we have to examine
how it can be that what our eyes or experience seem to tell us is misleading. Only then can we
accept the rock-solid evidence that science or mathematics provides.
<br />
<br />
Some initial resistance is good, to be sure. We should always be skeptical. But for us
and society to continue to advance, we have to be prepared to let go of our original belief when
the evidence to the contrary becomes overwhelming.
<br />
<br />
The Monty Hall problem is unusual (though by no means unique) in being simple to state and
initially surprising, yet once you have understood where your initial error lies, the simple
correct answer is blindingly obvious, and you will never again fall into the same trap you did on the first encounter. Many issues in life are much less clear-cut.
<br />
<br />
BTW, if you have never encountered the problem before, I will tell you it is not a trick question.
It is entirely a mathematical puzzle, and the correct mathematics is simple and straightforward.
You just have to pay careful attention to the information you are actually given, and not remain
locked in the mindset of what you initially <i><b>think</b></i> it says. Along the way, you may realize you
have misunderstood the notion of probability. (Some people maintain that probabilities cannot
change, a false understanding that most likely results from first encountering the notion in
terms of the empirical study of rolling dice and selecting colored beans from jars.) So reflection
on the Monty Hall Problem can provide a valuable lesson in coming to understand the hugely
important concept of mathematical probability.
<br />
<br />
As it happens, Hall’s death comes at a time when, for those of us in the United States, the
system of evidence-based, rational inquiry which made the nation a scientific, technological,
and financial superpower is coming under dangerous assault, with significant resources being
put into a sustained attempt to deny that there are such things as scientific facts. For scientific
facts provide a great leveler, favoring no one person or one particular group, and are thus to
some, a threat.
<br />
<br />
The late Carl Sagan warned of this danger back in 1995, in his book <i><a href="https://www.amazon.com/Demon-Haunted-World-Science-Candle-Paperback/dp/B00EQBY4TW/ref=sr_1_2?ie=UTF8&qid=1506948921&sr=8-2&keywords=the+demon+haunted+world" target="_blank">The Demon-Haunted World:Science as a Candle in the Dark</a></i>, writing:<br />
<blockquote>
“I have a foreboding of an America in my children’s or my grandchildren’s time — when
the United States is a service and information economy; when nearly all the key
manufacturing industries have slipped away to other countries; when awesome
technological powers are in the hands of a very few, and no one representing the public
interest can even grasp the issues; when the people have lost the ability to set their
own agendas or knowledgeably question those in authority; when, clutching our
crystals and nervously consulting our horoscopes, our critical faculties in decline,
unable to distinguish between what feels good and what’s true, we slide, almost
without noticing, back into superstition and darkness. ...”</blockquote>
Good scientists, such as Sagan, are not just skilled at understanding what is, they can
sometimes extrapolate rationally to make uncannily accurate predictions of what the future
might bring. It is chilling, but now a possibility that cannot be ignored, that a decade from now,
I could be imprisoned for writing the above words. Today, the probability that will happen is
surely extremely low, albeit nonzero. But that probability could change. As mathematicians, we
have a clear responsibility to do all we can to ensure that Sagan’s words do not describe the
world in which our children and grandchildren live.
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<br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-25439525252888978332017-09-20T14:02:00.000-04:002017-09-20T14:22:37.383-04:00The Legacy of Jonathan Borwein<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-eabNsYY8UW4/WcKs4cjposI/AAAAAAAAK6A/wXD7oNry-8MJx2A2RprinA6kiFXemNzIwCEwYBhgL/s1600/keith%2Bdevlin.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="891" data-original-width="1600" height="222" src="https://4.bp.blogspot.com/-eabNsYY8UW4/WcKs4cjposI/AAAAAAAAK6A/wXD7oNry-8MJx2A2RprinA6kiFXemNzIwCEwYBhgL/s400/keith%2Bdevlin.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Keith Devlin and Jonathan Borwein talk to host Robert Krulwick on stage at the World Science Festival in 2011.</td></tr>
</tbody></table>
<br />
At the end of this week I fly to Australia to speak and participate in the <a href="https://carma.newcastle.edu.au/meetings/jbcc/">Jonathan Borwein
Commemorative Conference</a> in Newcastle, NSW, Borwein’s home from 2009 onwards, when he
moved to the Southern hemisphere after spending most of his career at various Canadian
universities. Born in Scotland in 1951, Jonathan passed away in August last year, leaving behind
an extensive collection of mathematical results and writings, as well as a long list of service
activities to the mathematical community. [For a quick overview, read the brief <a href="http://experimentalmath.info/blog/2016/08/jonathan-borwein-dies-at-65/">obituary</a>
written by his long-time research collaborator David Bailey in their joint blog Math Drudge. For
more details, check out his <a href="https://en.wikipedia.org/wiki/Jonathan_Borwein">Wikipedia entry</a>.]
<br />
<br />
Jonathan’s (I cannot call him by anything but the name I always used for him) career path and
mine crossed on a number of occasions, with both of us being highly active in mathematical
outreach activities and both of us taking an early interest in the use of computers in
mathematics. Over the years we became good friends, though we worked together on a project
only once, co-authoring an expository book on experimental mathematics, titled <i><a href="https://www.amazon.com/Computer-Crucible-Introduction-Experimental-Mathematics/dp/1568813430/ref=asap_bc?ie=UTF8" target="_blank">The Computer as Crucible</a></i>, published in 2008.
<br />
<br />
Most mathematicians, myself included, would credit Jonathan as the father of experimental
mathematics as a recognized discipline. In the first chapter of our joint book, we defined
experimental mathematics as “the use of a computer to run computations—sometimes no
more than trial-and- error tests—to look for patterns, to identify particular numbers and
sequences, to gather evidence in support of specific mathematical assertions that may
themselves arise by computational means, including search.”
<br />
<br />
The goal of such work was to gather information and gain insight that would eventually give rise
to the formulation and rigorous proof of a theorem. Or rather, I should say, that was Jonathan’s
goal. He saw the computer, and computer-based technologies, as providing new tools to
formulate and prove mathematical results. And since he gets to define what “experimental
mathematics” is, that is definitive. But that is where are two interests diverged significantly.
<br />
<br />
In my case, the rapidly growing ubiquity of ever more powerful and faster computers led to an
interest in what I initially called “soft mathematics” (see my 1998 book<i> <a href="https://www.amazon.com/Goodbye-Descartes-Logic-Search-Cosmology/dp/0471251860/ref=asap_bc?ie=UTF8" target="_blank">Goodbye Descartes</a></i>) and
subsequently referred to as “mathematical thinking,” which I explored in a number of articles
and books. The idea of mathematical thinking is to use a mathematical approach, and often
mathematical notations, to gather information and gain insight about a task in a domain that
enables improved performance. [A seminal, and to my mind validating, example of that way of
working was thrust my way shortly after September 11, 2001, when I was asked to join a team
tasked with improving defense intelligence analysis.]
<br />
<br />
Note that the same phrase “gather information and gain insight” occurs in both the definition
of experimental mathematics and that of mathematical thinking. In both cases, the process is
designed to lead to a specific outcome. What differs is the nature of that outcome. (See my
2001 book <i><a href="https://www.amazon.com/Infosense-Turning-Information-Into-Knowledge/dp/0716741644/ref=asap_bc?ie=UTF8" target="_blank">InfoSense</a></i>, to get the general idea of how mathematical thinking works, though I
wrote that book before my Department of Defense work, and before I adopted the term “mathematical thinking.”)
<br />
<br />
It was our two very different perspectives on the deliberative blending of mathematics and
computers that made our book <i>The Computer as Crucible</i> such a fascinating project for the two of us.
<br />
<br />
But that book was not the first time our research interests brought us together. In 1998, the
American Mathematical Society introduced a new section of its ten-issues- a-year <i>Notices</i>, sent
out to all members, called “Computers and Mathematics,” the purpose of which was both
informational and advocacy.
<br />
<br />
Though computers were originally invented by mathematicians to perform various numerical
calculations, professional mathematicians were, by and large, much slower at making use of
computers in their work and their teaching than scientists and engineers. The one exception
was the development of a number of software systems for the preparation of mathematical
manuscripts, which mathematicians took to like ducks to water.
<br />
<br />
In the case of research, mathematicians’ lack of interest in computers was perfectly
understandable—computers offered little, if any, benefit. (Jonathan was one of a very small
number of exceptions, and his approach was initially highly controversial, and occasionally
derided.) But the writing was on the wall—or rather on the computer screen—when it came to
university teaching. Computers were clearly going to have a major impact in mathematics
education.
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The “Computers and Mathematics” section of the AMS <i>Notices</i> was intended to be a change
agent. It was originally edited by the Stanford mathematician Jon Barwise, who took care of it
from the first issue in the May/June 1988 Notices, to February 1991, and then by me until we
retired the section in December 1994. It is significant that 1988 was the year Stephen Wolfram
released his mathematical software package Mathematica. And in 1992, the first issue of the
new research journal <i>Experimental Mathematics</i> was published.
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<br />
Over its six-and- a-half years run, the column published 59 feature articles, 19 editorial essays,
and 115 reviews of mathematical software packages — 31 features 11 editorials, and 41
reviews under Barwise, 28 features, 8 editorials, and 74 reviews under me. [The <i>Notices</i>
website has a <a href="http://www.ams.org/notices/199502/devlinsixyear.pdf" target="_blank">complete index</a>.] One of the feature articles published under my watch was
“Some Observations of Computer Aided Analysis,” by Jonathan Borwein and his brother Peter,
which appeared in October 1992. Editing that article was my first real introduction to
something called “experimental mathematics.” For the majority of mathematicians, reading it
was their introduction.
<br />
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From then on, it was clear to both of us that our view of “doing mathematics” had one feature
in common: we both believed that for some problems it could be productive to engage in
mathematical work that involved significant interaction with a computer. Neither of us was by
any means the first to recognize that. We may, however, have been among the first to conceive
of such activity as constituting a discipline in its own right, and each to erect a shingle to
advertise what we were doing. In Jonathan’s case, he was advancing mathematical knowledge;
for me it was about utilizing mathematical thinking to improve how we handle messy, real-world problems. In both cases, we were engaging in mental work that could not have been
done before powerful, networked computers became available.
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<br />
It’s hard to adjust to Jonathan no longer being among us. But his legacy will long outlast us all. I
am looking forward to re-living much of that legacy in Australia in a few days time.
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<br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-19532749839105122632017-08-07T14:44:00.000-04:002017-09-07T16:41:07.060-04:00What are universities for and how do they work?Exactly 30 years ago, I and my family arrived in the U.S. from the U.K. to take up a one-year visiting position in the mathematics department at Stanford University. (We landed on July 28, 1987.) That one year was subsequently extended to two, and in the end we never returned to the U.K. A very attractive offer of a newly endowed chair in mathematics at Colby College in Maine provided the pull. But equally significant was a push from the U.K.
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<br />
The late 1980s were a bad time for universities in Britain, as Prime Minister Margaret Thatcher launched a full-scale assault on higher education, motivated in part by a false understanding of what universities do, and in part by personal vindictiveness stemming from her being criticized by academics for her poor performance as Minister for Education some years earlier. My own university, Lancaster, where I had been a regular faculty member since 1977, had been a source of some of the most vocal criticisms of the then Minister Thatcher, and accordingly was dealt a particularly heavy funding hit when Prime Minister Thatcher started to wield her axe. A newly appointed vice chancellor (president), with a reputation for tough leadership as a dean, was hired from the United States to steer the university through the troubled waters ahead.
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One of the first decisions the new vice chancellor made was to cut the mathematics department faculty by roughly 50%, from around 28 to 14. (I forget the actual numbers.) The problem he faced in achieving that goal was that in the British system at the time, once a new Lecturer (= Assistant Professor) had passed a three-year probationary period, they had tenure for life. The only way to achieve a 50% cut in faculty was to force out anyone who could be “persuaded” to go. That boiled down to putting pressure on those whose reputation was sufficiently good for them to secure a position elsewhere. (So, a strategy of “prune from the top,” arguably more productive in the garden than a university.)
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In my case, the new vice chancellor made it clear to me soon after his arrival that my prospects of career advancement at Lancaster were low, and I could expect ever increasing teaching loads that would hamper my research, and lack of financial support to attend conferences. As a research mathematician early in my career, with my work going well and my reputation starting to grow, that prospect was ominous. Though I was not sure whether he would ever actually follow through with his threat, it seemed prudent to start thinking in terms of a move, possibly one that involved leaving the U.K.
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Then, just as all of this was going on, out of the blue I got the invitation from Stanford. (I had started working on a project that aligned well with a group at Stanford who had just set up a new research center to work on the same issues. As a result, I had gotten to know some of them, mostly by way of an experimental new way to communicate called “e-mail,” which universities were just starting to use.)
<br />
<br />
In my meeting with the vice chancellor to request permission to accept the offer and discuss the arrangements, I was told in no uncertain terms that I would be wise not to return after my year in California came to an end. The writing was on the wall. Lancaster wanted me gone. In addition, other departmental colleagues were also looking at opportunities elsewhere, so even if I were to return to Lancaster after my year at Stanford, it might well be to a department that had lost several of its more productive mathematicians. (It would have been. The vice chancellor achieved his 50% departmental reduction in little more than two years.)
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Yes, these events were all so long ago, in a different country. So why am I bringing the story up now? The answer, is that, as is frequently observed, history can provide cautionary lessons for what may happen in the future.
<br />
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Those of us in mathematics are deeply aware of the hugely significant role the subject plays in the modern world, and have seen with every generation of students how learning mathematics can open so many career doors. We also know sufficient mathematics to appreciate the enormous impact on society that new mathematical discoveries can have—albeit in many cases years or decades later. To us, it is inconceivable that a university—an institution having the sole purpose of advancing and passing on new knowledge for the good of society—would ever make a conscious decision to cut down (especially from the top), or eliminate, a mathematics department.
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But to people outside the universities, things can look different. Indeed, as I discovered during my time as an academic dean (in the U.S.), the need for mathematics departments engaged in research is often not recognized by faculty in other departments. Everyone recognizes the need for each new generation of students to be given some basic mathematics instruction, of course. But mathematics research? That’s a much harder sell. In fact, it is an extremely hard sell. Eliminating the research mathematicians in a department and viewing it as having a solely instructional role can seem like an attractive way to achieve financial savings. But it can come at a considerable cost to the overall academic/educational environment. Not least because of the message conveyed to the students.
<br />
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As things are, students typically graduate from high school thinking of mathematics as a toolbox of formulas and procedures for solving certain kinds of problems. But at university level, they should come to understand it as a particular way of thinking. To that end, they should be exposed to an environment where tasks can be approached on their own terms, with mathematicians being one of any number of groups of experts who can bring a particular way of thinking that may, or may not, be effective.
<br />
<br />
The educational importance of having an active mathematics research group in a university is particularly important in today’s world. As I noted in <a href="http://www.huffingtonpost.com/entry/all-the-mathematical-methods-i-learned-in-my-university_us_58693ef9e4b014e7c72ee248" target="_blank">an article</a> in <i>The Huffington Post</i> in January, pretty well all the formulas and procedures that for many centuries have constituted the heart of a university mathematics degree have now been automated and are freely available on sites such as <a href="https://www.wolframalpha.com/examples/Math.html" target="_blank">Wolfram Alpha</a>. Applying an implemented, standard mathematical procedure to solve, say, a differential equation, is now in the same category as using a calculator to add up a column of numbers. Just enter the data correctly and the machine will do the rest.
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In particular, a physicist or an engineer (say) at a university can, for the most part, carry out their work without the need for specialist mathematical input. (That was always largely the case. It is even more so today.) But one of the functions of a university is to provide a community of experts who are able to make progress when the available canned procedures do not quite fit the task at hand. The advance of technology does not eliminate the need for creative, human expertise. It simply shifts the locus of where such expertise is required. Part of a university education is being part of a community where that reliance on human expertise is part of the daily activities; a community where all the domain specialists are experts in their domains, and able to go beyond the routine.
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It is easy to think of education as taking place in a classroom. But that’s just not what goes on. What you find in classrooms is instruction, maybe involving some limited discussion. Education and learning occur primarily by way of interpersonal interaction in a community. That’s why we have universities, and why students, and often their parents, pay to attend them. It’s why “online universities” and MOOCs have not replaced universities, and to my mind never will. The richer and more varied the community, the better the education.
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Lest I have given the impression that my focus is on topline research universities, stocked with award winning academic superstars, let me end by observing that nothing I have said refers to level of achievement. Rather it is all about the attitude of mind and working practices of the faculty. As long as the mathematics faculty love mathematics, and enjoy doing it, and are able to bring their knowledge to bear on a new task or problem, they contribute something of real value to the environment in which the students learn. It’s a human thing.
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A university that decides to downgrade a particular discipline to do little more than provide basic instruction is diminishing its students educational experience, and is no longer a bona fide university. (It may well, of course, continue to provide a valuable service. The university, my focus in this essay, is just one form of educational institution among many.)
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<br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-80369957071276771682017-07-13T12:18:00.000-04:002017-07-13T12:51:06.793-04:00The Power of Simple RepresentationsThe great mathematician Karl Freidrich Gauss is frequently quoted as saying “What we need are
notions, not notations.” [In “About the proof of Wilson's theorem,” <i>Disquisitiones
Arithmeticae</i> (1801), Article 76.]
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<br />
While most mathematicians would agree that Gauss was correct in pointing out that concepts,
not symbol manipulation, are at the heart of mathematics, his words do have to be properly
interpreted. While a <i>notation</i> does not matter, a <i>representation</i> can make a huge difference. The
distinction is that developing or selecting a representation for a particular mathematical concept
(or <i>notion</i>) involves deciding which features of the concept to capture.
<br />
<br />
For example, the form of the ten digits 0, 1, … , 9 does not matter (as long as they are readily
distinguishable), but the usefulness of the Hindu-Arabic number system is that it embodies base-
10 place-value representation of whole numbers. Moreover, it does so in a way that makes both
learning and using Hindu-Arabic arithmetic efficient.
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<br />
Likewise, the choice of 10 as the base is optimal for a species that has highly manipulable hands
with ten digits. Although the base-10 arithmetic eventually became the standard, other systems
were used in different societies, but they too evolved from the use of the hands and sometimes
the feet for counting: base-12 (where finger-counting used the three segments of each of the four
fingers) and base-20 where both fingers and toes were used. Base-12 arithmetic and base-20
arithmetic both remained in regular use in the monetary system in the UK when I was a child
growing up there, with 12 pennies giving one shilling and 20 shillings one pound. And several
languages continue to carry reminders of earlier use of both bases — English uses phrases such
as “three score and ten” to mean 70 (= 3x20 + 10) and French articulates 85 as “quatre-vingt cinq
(4x20 + 5).
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Another number system we continue to use today is base-60, used in measuring time (seconds
and minutes) and in circular measurement (degrees in a circle). Presumably the use of 60 as a
base came from combining the finger and toes bases 10, 12, and 20, allowing for all three to be
used as most convenient.
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These different base-number representation systems all capture features that make them useful to
humans. Analogously, digital computers are designed to use binary arithmetic (base 2), because
that aligns naturally with the two states of an electronic gate (open or closed, on or off).
<br />
<br />
In contrast, the <i>shapes</i> of the Hindu-Arabic numerals is an example of a superfluous feature of
the representation. The fact that it is possible to draw the numerals in a fashion whereby each
digit has the corresponding number of angles, like this<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-YVDUy7CSxgo/WWeRZ91JdiI/AAAAAAAAK2c/FLqZuX8IVZogvX3K5mJN1OT1d89oYXWUwCLcBGAs/s1600/H-A_numerals.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="210" data-original-width="1558" height="43" src="https://3.bp.blogspot.com/-YVDUy7CSxgo/WWeRZ91JdiI/AAAAAAAAK2c/FLqZuX8IVZogvX3K5mJN1OT1d89oYXWUwCLcBGAs/s320/H-A_numerals.png" width="320" /></a></div>
may be a historical echo of the evolution of the symbols, but whether or not that is the case (and
frankly I find it fanciful), it is of no significance in terms of their use—the <i>form</i> of the numerals
is very much in Gauss’s “unimportant notations” bucket.
<br />
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On the other hand, the huge difference a <i>representation system</i> can make in mathematics is
indicated by the revolutionary change in human life that was brought about by the switch from
Roman numerals and abacus-board calculation to Hindu-Arabic arithmetic in Thirteenth Century
Europe, as I described in my 2011 book <i><a href="https://www.amazon.com/Man-Numbers-Fibonaccis-Arithmetic-Revolution/dp/0802778127/ref=sr_1_1?ie=UTF8&qid=1306990867&sr=8-1" target="_blank">The Man of Numbers</a></i>.
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Of course, there is a sense in which representations do not matter to mathematics. There is a
legitimate way to understand Gauss’s remark as a complete dismissal of how we represent
mathematics on a page. The notations we use provide mental gateways to the abstract notions of
mathematics that live in our minds. The notions themselves transcend any notations we use to
denote them. That may, in fact, have been how Gauss intended his reply to be taken, given the
circumstances.
<br />
<br />
But when we shift our attention from mathematics as a body of eternal, abstract structure
occupying a Platonic realm, to an activity carried out by people, then it is clear that notations
(i.e., a representation system) are important. In the early days of Category Theory, some
mathematicians dismissed it as “abstract nonsense” or “mere diagram chasing”, but as most of us
discovered when we made a serious attempt to get into the subject, “tracing the arrows” in a
commutative diagram can be a powerful way to approach and understand a complex structure.
[Google “the snake lemma”. Even better, watch actress Jill Clayburgh <a href="https://www.youtube.com/watch?v=etbcKWEKnvg" target="_blank">explain it</a> to a graduate
math class in an early scene from the 1980s movie <i>It’s My Turn</i>.]
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A well-developed mathematical diagram can also be particularly powerful in trying to
understand complex real-world phenomena. In fact, I would argue that the use of mathematical
representations as a tool for highlighting hidden abstract structure to help us understand and
operate in our world is one of mathematics most significant roles in society, a use that tends to
get overlooked, given our present day focus on mathematics as a tool for “getting answers.”
Getting an answer is frequently the end of a process of thought; gaining new insight and
understanding is the start of a new mental journey.
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<br />
A particularly well known example of such use are the <a href="https://en.wikipedia.org/wiki/Feynman_diagram" target="_blank">Feynmann Diagrams</a>, simple
visualizations to help physicists understand the complex behavior of subatomic particles,
introduced by the American physicist Richard Feynmann in 1948.
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-mOGUeuoP5MA/WWeR0IIEQbI/AAAAAAAAK2g/pzXCt-qIDssO22H9ykXN7mqbf5v3vN_aQCLcBGAs/s1600/Feynmann_Diagram.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="366" data-original-width="574" height="204" src="https://4.bp.blogspot.com/-mOGUeuoP5MA/WWeR0IIEQbI/AAAAAAAAK2g/pzXCt-qIDssO22H9ykXN7mqbf5v3vN_aQCLcBGAs/s320/Feynmann_Diagram.png" width="320" /></a></div>
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A more recent example that has proved useful in linguistics, philosophy, and the social sciences
is the “completion diagram” developed by the American mathematician Jon Barwise in
collaboration with his philosopher collaborator John Perry in the early 1980s, initially to
understand information flow.
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<a href="https://1.bp.blogspot.com/-fEXr7O3UZjc/WWeR6upRpsI/AAAAAAAAK2k/oxW8sjjYlYw-C2FCa4s8GHSnpfNjJfJNgCLcBGAs/s1600/CompletionDiagram.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="558" data-original-width="892" height="250" src="https://1.bp.blogspot.com/-fEXr7O3UZjc/WWeR6upRpsI/AAAAAAAAK2k/oxW8sjjYlYw-C2FCa4s8GHSnpfNjJfJNgCLcBGAs/s400/CompletionDiagram.png" width="400" /></a></div>
<br />
A discussion of one use of this diagram can be found in a survey article I wrote in the volume
<i>Handbook of the History of Logic</i>, Volume 7, edited by Dov Gabbay and John Woods (Elsevier,
2008, pp.601-664), a <a href="https://web.stanford.edu/~kdevlin/Papers/HHL_SituationTheory.pdf" target="_blank">manuscript version</a> of which can be found on my Stanford homepage. That
particular application is essentially the original one for which the diagram was introduced, but
the diagram itself turned out be to be applicable in many domains, including improving
workplace productivity, intelligence analysis, battlefield command, and mathematics education.
(I worked on some of those applications myself; some <a href="https://web.stanford.edu/~kdevlin/papers.html" target="_blank">links to publications</a> are on my
homepage.)
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<br />
To be particularly effective, a representation needs to be simple and easy to master. In the case of
a representational diagram, like the Commutative Diagrams of Category Theory, the Feynmann
Diagram in physics, and the Completion Diagram in social science and information systems
development, the representation itself is frequently so simple that it is easy for domain experts to
dismiss them as little more than decoration. (For instance, the main critics of Category Theory in
its early days were world famous algebraists.) But the mental clarity such diagrams can bring to
a complex domain can be highly significant, both for the expert and the learner.
<br />
<br />
In the case of the Completion Diagram, I was a member of the team at Stanford that led the
efforts to develop an understanding of information that could be fruitful in the development of
information technologies. We had many long discussions about the most effective way to view
the domain. That simple looking diagram emerged from a number of attempts (over a great many
months) as being the most effective.
<br />
<br />
Given that personal involvement, you would have thought I would be careful not to dismiss a
novel representation I thought was too simple and obvious to be important. But no. When you
understand something deeply, and have done so for many years, you easily forget how hard it
can be for a beginning learner. That’s why, when the MAA’s own James Tanton told me about
his “Exploding Dots” idea some months ago, my initial reaction was “That sounds cute," but I did
not stop and reflect on what it might mean for early (and not so early) mathematics education.
<br />
<br />
To me, and I assume to any professional mathematician, it sounds like the method simply adds a
visual element on paper (or a board) to the mental image of abstract number concepts we already
have in our minds. In fact, that is exactly what it does. But that’s the point! “Exploding Dots”
does nothing for the expert. But for the learner, it can be huge. It does nothing for the expert
because it represents on a page what the expert has in their mind. <i>But that is why it can be so
effective in assisting a learner arrive at that level of understanding!</i> All it took to convince me
was to watch Tanton’s <a href="https://vimeo.com/204368634" target="_blank">lecture video</a> on Vimeo. Like Tanton, and I suspect almost all other
mathematicians, it took me <i>many years of struggle</i> to go beyond the formal symbol manipulation
of the classical algorithms of arithmetic (developed to enable people to carry our calculations
efficiently and accurately in the days before we had machines to do it for us) until I had created
the mental representation that the exploding dots process capture so brilliantly. Many learners
subjected to the classical teaching approach never reach that level of understanding; for them,
basic arithmetic remains forever a collection of incomprehensible symbolic incantations.
<br />
<br />
Yes, I was right in my original assumption that there is nothing new in exploding dots. But I was
also wrong in concluding that there was nothing new. There is no contradiction here.
Mathematically, there is nothing new; it’s stuff that goes back to the first centuries of the First
Millennium—the underlying idea for place-value arithmetic. Educationally, however, it’s a big
deal. A very big deal. Educationally explosive, in fact. Check it out!
<br />
<br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-13132498731378487762017-06-14T16:53:00.000-04:002017-06-14T16:53:23.620-04:00Classroom 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 />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-63231293777438662402017-05-22T15:19:00.000-04:002017-06-01T11:34:58.430-04:00The 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.
Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-1143250936552300722017-04-05T13:24:00.003-04:002017-04-05T13:25:09.912-04:00Fibonacci 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 />
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<br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-47310623271147221752017-03-08T11:48:00.001-05:002017-03-08T11:51:20.847-05:00Finding 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>
Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-2516188730140164076.post-82339844828493692762017-02-08T09:01:00.000-05:002017-02-08T09:01:01.742-05:00Hans 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;">
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Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com1tag:blogger.com,1999:blog-2516188730140164076.post-42407722580530248592017-01-06T00:02:00.000-05:002017-01-09T15:56:49.437-05:00So 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 />
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<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!
Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0