Tuesday, September 1, 2015

A Brilliant Young Mind: The IMO goes to the movies

Jo Yang as Zhang Mei and Asa Butterfield as Nathan Ellis in A Brilliant Young Mind. Credit: Samuel Goldwyn Films

Mainstream movies about mathematicians used to be a rarity, but are now fairly common. Good Will Hunting, Proof, A Beautiful Mind, Pi, Cube, The Bank, Travelling Salesman, The Imitation Game come immediately to mind. So too does Stand and Deliver if you include mathematics teaching.

The titles I just listed are such good movies, there is now a high bar to success in this growing genre. In particular, the movie has to have a good story, a strong cast, and it needs to get the math right – and moreover do so in a way that intrigues the audience but does not detract from the pace of the story. The new movie A Brilliant Young Mind, by British director Morgan Matthews, meets that standard.

Due to be released in the US on September 11, A Brilliant Young Mind, starring the hugely talented young British Actor Asa Butterfield (who starred as Ender in Ender’s Game, opposite Harrison Ford) was first screened in the UK last year, originally under the title “X + Y”, on which more later.

The film focuses on the International Mathematical Olympiad (IMO), the competition held annually around the world, where national teams of six pre-collegiate students compete for individual and team medals. The movie follows one particular British student as he goes through the grueling process of preparing for and taking the test to qualify for team pre-selection in the British National Mathematical Competition, going off to a training camp in Taiwan, where the final team of six is selected in a mock IMO competition, and then heading to Cambridge, England, for the international competition itself.

Both the mathematics and the mathematics competitions are handled well. (More later.) Mathematicians will not be disappointed on that score.

I have to admit that, on first viewing, I felt that the romantic thread between the Asa Butterfield character (Nathan Ellis) and the young female Chinese math whiz he meets at the training camp, played by Jo Yang, was a crude injection to create a movie with mainstream audience appeal. In particular, I thought the dramatic ending (you’ll have to watch the movie to find out if it is a happy or sad ending) was way over the top.

But then I watched the original BBC documentary that A Brilliant Young Mind director Matthews made back in 2006, on which he based the movie, and guess what? The story in A Brilliant Young Mind stays pretty close to real life! Right down to what at first viewing of the movie I thought were syrupy shots included purely for cinematic romantic effect. (Cue the rainbow in the background as the British and Chinese math whizzes travel by train through the British countryside. Taken right out of real life!)

So if my jaded-by-Hollywood mathematical colleagues find themselves, like me, lamenting to themselves, “Why do movie makers spoil the real story with all that romantic mush?” you should suppress that reaction at once. What the movie gives you is a dramatic (and dramatized) recreation of real life.

That, by the way, is why I like the movie’s original title in the UK: “X + Y”, a nicely succinct way to link the mathematics problem solving and the romantic engagement. Still the title A Brilliant Young Mind does convey the idea of a young version of the brilliant (but, like Butterworth’s Nathan Ellis, mentally troubled) John Nash in the movie A Beautiful Mind, and Matthews himself pulled on the same association with the title of his earlier documentary.

Despite taking his basic storyline straight out of real life, Matthews does (of course) take plenty of dramatic license in order to give us a watchable movie. He is, after all, telling a fictional story, albeit one based (unusually closely) on real life. Few in the audience will have much interest in the mathematics, or even math competitions (besides, perhaps, being surprised that such things exist), but everyone likes a good story about people. And that is what A Brilliant Young Mind delivers.

In particular, I suspect many of my fellow mathematicians will also balk at the portrayal of several of the British IMO team selectees as exhibiting various forms of autism. (In real life, good mathematicians of all ages run the full spectrum of human characteristics, with the vast majority of math whizzes being just like everyone else.) But that aspect too is what you will find in the documentary. (But see my postscript comments at the end.)

The IMO team members who the Butterfield character interacts with in the movie are also clearly based on real-life counterparts in the documentary. In particular, the student who has made his way onto the team by learning a lot of mathematical facts and procedures that he can regurgitate and apply at speed, but falters when it comes to having to apply original thinking. (Both the US and the UK math education systems encourage and reward that approach, which is why they do so poorly in the international PISA tests, which look for original thought. My Stanford colleague Jo Boaler has a new book on that misguided, and sad, state of affairs, Mathematical Mindsets, coming out in the Fall.)

The only complaint you could make about Matthews is the choices he made in selecting the footage he shot for the original documentary. But that is true for any documentary film. Matthews followed the UK 2006 IMO team through the entire competition process, and then told a story based on what he had captured.

Interestingly enough, another documentary film maker, George Csicery, followed the US IMO team at the same time. You can compare the two documentaries. Matthews’ BBC documentary is available online for free streaming. Csicery’s film Hard Problems, is available for purchase from the MAA ($19.95 to members).

Enough of all these words. We’re talking about a movie, after all. Time to watch some movie clips.

You can watch the official trailer for A Brilliant Young Mind here.

Mathematicians will particularly like the one short sequence where the movie shows a brilliant mathematical mind in action, solving a problem, which you can see in isolation in this officially sanctioned YouTube video. It is a superbly chosen (and acted) example. No fake numerical mumbo jumbo here. Genuine mathematical thinking in action. And good mathematical thinking to boot. Any math instructor would surely love to have a student produce such a solution for the class.

During the year the two documentaries were made, the IMO was held in Ljubljana, Slovenia (not Cambridge, England, as in the movie). You can see the actual problems the competitors faced here. (With sample solutions.)

Finally, for long lists of scenes in movies that feature a mathematician or a math problem, see here and here.

Enjoy the film!

EDITORIAL POSTSCRIPT

Both the movie and the BBC documentary raise some issues that concern me as a mathematician. The main danger of any documentary or movie is if viewers (and for a film like A Brilliant Young Mind, the audience may well include young kids showing an early interest in mathematics) get the impression that what they see is representative of the field. This of course, is true for pretty well any movie, be the topic crime detection, politics, business, law, the military, sports, or whatever. I think it is particularly worrying in mathematics, because most people have a very impoverished, and often completely erroneous perception of mathematics. Both of Matthews’ films trouble me on that count.

These thoughts are in no way a criticism of the movie I am reviewing. It is what it is. I think it is a good movie and I like it. (Though as I noted, in my case much of my positive evaluation comes from knowing that the things in the movie that initially turned me off as being unrealistic and contrived turned out to be true!) Rather, the issues I raise are general ones about the public perception of mathematics. In making both his documentary and the movie, Morgan Matthews set out to make good films. His goal was not to improve the public understanding of mathematics. That, on the other hand, is something I have spent a great deal of my career focusing on. Hence this postscript, separate from my review.

First, it has to be said that competition mathematics is in many respects a very different activity than the professional mathematics that most of us in the business pursue. For one thing, competition math requires speed, whereas many good mathematicians are slow thinkers. (I certainly am.)

There is also something very unusual about the kinds of problems that the IMO presents. Of necessity, they have to be solvable in at most an hour, and in many cases, the way to go about solving them is to be very conscious of that time limitation. They have to depend on seeing a particular insight or trick. Some people are naturally good at that kind of problem solving, but it can also be to some extent taught – which is what goes on at those IMO training camps. On the other hand, most mathematics problems that the pros grapple with are very different. In many cases, no one has any idea if there is a solution at all, or how long it may be.

Moreover, the connection between being good at competition math and having the aptitude to succeed in professional mathematics is not at all clear-cut. Certainly, some IMO medal winners have gone on to pursue mathematics at university level, but not all of them have gone on to lead a successful career in mathematics. (Some have.) And many of the best mathematicians in the world have never in their lives had any interest in competition math. Though the two domains do have abstract mathematics in common, they are in many ways very different activities.

So to a child or the parent of a child who shows aptitude toward mathematics, I would say this. If you fancy the idea of competition math, give it a try. If you do well, enjoy the experience. It will certainly show that you have some abilities that could help you succeed in a mathematical career. But if you find you do not enjoy it, or if you like it but do not do well, that in no way means you could not grow up to be a top rank mathematician.

Another unintended message that math competitions tend to convey is that you have to have a special talent for mathematics (a “math gene” if you like). This notion that mathematics is something for the “gifted and talented” is pervasive in many cultures, and it is total BS. The two most important factors in achieving success in mathematics are wanting to do math and growing up in a supportive, educationally rich, sociocultural environment. Not only is the world of mathematics replete with examples of world class mathematicians who will tell you flat out how many hours of effort it took them to get to that point, and how others helped them on their way, there is also a growing body of evidence from nueroscience studies to support the hypothesis that mathematics, like pretty well any other human endeavor, is 5% inspiration and 95% perspiration.

Society would do well to banish that term “gifted and talented” once and for all, and replace it with something more accurate. “Motivated and bloody hard working” is my nomination. (Individual mileages do, of course, vary.)

My final editorial remark is something I touched upon in my review. The movie, focuses on a small group of mathematics students, and one of them in particular, who exhibit various forms and degrees of autism. True, the same was true of the young students in the director’s earlier documentary, but that clearly reflects the particular perceptual lens the director brought to the project. That lens is made dramatically clear by the opening scenes in the documentary where we meet one of the competitors, Jos. Matthews set out to portray IMO participants as being unusual and different. And he found some.

Cleary, getting to represent their country in an international competition in of itself makes them different. But presenting them as intrinsically different is a definite editorial decision.

Contrast Matthews’ documentary with Csicery’s. In the latter, focusing on the US team at the same IMO, the director sets out to convey the very ordinariness of the participants, highlighting not what is different about them (they love math being the main thing) but how much they are just like any other kids of that age.

Csicery, as many readers of this column will know, makes documentary films about mathematics and mathematics as a profession, and he makes them primarily for the mathematics and mathematics education professions. So he strives to inform an audience who are interested in mathematics. This, of course, is very different from Matthews, who sets out to make movies that intrigue viewers who do not necessarily have any interest in the topic, whatever it is. Csicery succeeds with his audience by being as accurate and representative as he can, while also managing to tell a story. Matthews has to tell a good human interest story that hangs on some strong characters, with everything else revealed during the film. They are doing different things.

I found Matthews’ documentary fascinating and highly engaging, and I’m really glad it inspired him to turn it into a movie. You should watch both.

But if what you want is to get a good overall sense of the world of competitive math, you should watch Csicery’s documentary. The two documentaries provide very different perceptions of the same IMO competition.

Saturday, August 1, 2015

Hard fun – video games creep into the math classroom

This month’s musings were inspired by the appearance of Greg Toppo’s excellent new book The Game Believes in You: How Digital Play Can Make Our Kids Smarter. In it, Toppo, who is USA Today's national K-12 education writer, does an excellent job of not only surveying the current scene in educational video games, he also exhibits a deep understanding of, and appreciation for, the educational potential of well designed video games. I have gone on record as saying it will likely turn out to be the most influential book on the role of video games in education since James Paul Gee’s 2003 classic What Video Games Have to Teach Us About Learning and Literacy.

Like it or loath it, video games are slowly finding their way into the nation’s math classes, as teachers and parents increasingly see video games as a valuable educational resource. For instance, according to a recently published survey designed by the Joan Ganz Cooney Center, 55% of teachers report having their students play video games at least once a week, with 47% of teachers saying low-performing students benefited most from the use of games. (Games and Learning, 2015)

Well-designed educational video games offer meaningful learning experiences based on principles of situated learning, exploration, immediate feedback, and collaboration. The power of experiential learning in engaging contexts that have meaning for learners has been demonstrated in several studies (e.g. Lave, 1988; Nunes et al, 1993, Shute & Ventura 2013).

But when it comes to education, not all games are equal. Of the many mathematics education video games (or gamified apps) available today (Apple’s App Store lists over 20,000), the majority focus on traditional drill to develop mastery of basic skills, particularly automatic recall of fundamental facts such as the multiplication tables. They require repetition under time pressure. Such games make no attempt to teach mathematics, to explore mathematical concepts, or to help students learn how to use mathematical thinking to solve real world problems. Their purpose is purely to make repetitive drill more palatable to students.

The proliferation of such games is in large part a consequence of the mathematics education many Americans have experienced: teacher and textbook instruction emphasizing isolated facts, procedures, memorization, and speed.

So widespread is this educational model in the US, that many American parents, teachers, and game developers think that this is the nature of mathematics, a perception that can result in underdeveloped mathematical proficiency. (See, for example, Boaler 2002; Boaler 2008; or Fosnot & Dolk 2001.)

While command of basic computation skills was a valuable asset to previous generations, in an era where fast, accurate computation is cheaply and readily available (in our pockets and briefcases, and on our desks), the crucial ability has shifted to what is often called mathematical proficiency: the ability to solve a novel problem that requires creative, multi-step reasoning, making appropriate use of computational technology as and when required.

The National Research Council’s recognized this significant change in the nation’s mathematical needs in its seminal 2001 recommendations for the future of US K-12 mathematics education, which laid out the case for the promotion of mathematical problem solving ability, built on number sense. Number sense involves being mathematically proficient with numbers and computations. It moves beyond the basics to developing a deep understanding about properties of numbers, and thinking flexibly about operations with numbers.

The last few years have seen the emergence of a tiny handful of video games designed to meet the educational requirements laid out by the National Research Council. Games such as Mind Research Institute’s K-5 focused Jiji games, Motion Math, DragonBox, Refraction, Slice Fractions, and my own Wuzzit Trouble. These games represent mathematics in a fashion native to the game medium (not the “symbolic” representations developed for the static page). They present the player with conceptually deep, complex problem solving tasks wrapped up in a game mechanic.

As such, these games leverage the representational power of personal computers and touch-screen devices to provide students with a means to interact with mathematical concepts in a direct fashion, not mediated through a symbolic representation, thereby facilitating exploration and learning through interactive problem solving.

In this context, it is worth reminding ourselves that the dominance of the printed symbol in the systemic mathematics education world is itself a product of the then-available technology, namely the invention of printing press in the 15th Century. Modern devices allow us to greatly expand on the symbolic interface, which for many people is a known barrier to mathematics learning (Nunes et al 1993, Devlin 2011).

References
Boaler, Experiencing school mathematics: Traditional and reform approaches to teaching and their impact on student learning, Revised and Expanded Edition. Mahwah, N.J. : L. Erlbaum, 2002.

Boaler, “Promoting ‘relational equity’ and high mathematics achievement through an innovative mixed‐ability approach,” British Educational Research Journal, vol. 34, no. 2, pp. 167–194, Apr. 2008.

Fosnot & Dolk, Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction. Portsmouth, NH: Heinemann, 2001.

Devlin, Mathematics education for a new era: video games as a medium for learning. CRC Press, 2011.

Games and Learning report, 2015. http://www.gamesandlearning.org/2014/06/09/teachers-on-using-games-in-class/#

Lave, 1988. Cognition in Practice: Mind, Mathematics and Culture in Everyday Life (Learning in Doing), Cambridge University Press.

National Research Council, Adding It Up: Helping Children Learn Mathematics. Washington, DC: National Academies Press: National Academy Press, 2001, pp. 1–462.

Nunes, Carraher, & Schliemann, 1993. Street Mathematics and School Mathematics, Cambridge University Press.

Pope, Boaler, & Milgram 2015. “Wuzzit Trouble: The Influence of a Digital Math Game on Student Number Sense”, submitted to International Journal of Serious Games.

Shute & Ventura, 2013. Stealth Assessment: Measuring and Supporting Learning in Video Games, MIT Press.

Wednesday, July 8, 2015

Is Math Important?

This month’s column is short on words, because I want to give you time to watch a great video (1 hr 18 min in length) from the recent Aspen Ideas Festival. It’s a panel discussion (actually, two discussions, back-to-back) hosted by New York Times journalist David Leonhardt. The topic is the question that I have chosen as the title for this post: Is math important? What makes this particularly worth watching is the selection of speakers and the views they express.

From the mathematical world there are Steven Strogatz of Cornell University and Jordan Ellenberg of the University of Wisconsin, and from mathematics education research there is Jo Boaler of Stanford University. They are joined by David Coleman, President of the College Board, education writer Elizabeth Green, author of the recent book Building a Better Teacher, Pamela Fox, a computer scientist working with Khan Academy, and financier Steve Rattner.

The conversation is lively and informative, and moves along at a brisk, engaging pace, with each speaker given time to provide in-depth answers (a refreshing antidote to the idiotic “received wisdom” that today’s viewers are not capable of watching a video longer than two-and-a-half minutes, a Big Data statistic that almost certainly says more about the abysmal engagement quality of most videos than about audience attention span).

That’s it from me. Here is the video.

Tuesday, June 2, 2015

PIACC – PISA for grown-ups

Greetings from 37,000 feet. As I write these words, I am on my way from San Francisco, California, to Boston, Massachusetts, to participate in a two-day workshop at Harvard, sponsored by the OECD (Organisation for Economic Co- operation and Development), to look at what should go into the math tests that will be administered to children around the world for PISA 2021.

PISA, the Programme for International Student Assessment, gets such extensive press coverage each time one of its reports is published, that it really needs no introduction. Americans have grown used to the depressing fact that US school children invariably perform dismally, ranked near the bottom of the international league tables, with countries like Japan and Finland jostling around at the top.

But chances are you have not heard of PIACC – the Programme for the International Assessment of Adult Competencies. The OECD introduced this new program a few years ago to investigate the nation-based adult skillsets that are most significant to national prosperity in a modern society: literacy, numeracy, and problem solving in a technology-rich environment (PS-TRE).

Whereas the PISA surveys focus on specific age-groups of school students, PIAAC studied adults across the entire age range 16 to 65.

The first report based on the PIAAC study was published in fall 2013: OECD Skills Outlook 2013: First Results from the Survey of Adult Skills.

A subsequent OECD report focused on PIACC data for US adults. The report’s title, Time for the U.S. to Reskill, gives the depressing-for-Americans headline that warns you of its contents. The skill levels of American adults compared to those of 21 other participating OECD countries were found to be dismal right across the board. The authors summarized US performance as “weak on literacy, very poor on numeracy,” and slightly below average on PS-TRE.

“Broadly speaking, the weakness affects the entire skills distribution, so that the US has proportionately more people with weak skills than some other countries and fewer people with strong skills,” the report concluded.

I have not read either OECD report. As happened when I never was able to watch the movie Schindler’s List, it is one of those things I feel I ought to read but cannot face the depression it would inevitably lead to. Rather, for airplane reading on my flight from Stanford to Harvard, I took with me a recently released (January 2015) report from the Princeton, NJ-based Educational Testing Service (ETS), titled AMERICA’S SKILLS CHALLENGE: Millennials and the Future.

The ETS report disaggregates the PIAAC US data for millennials—the generation born after 1980, who were 16–34 years of age at the time of the assessment.

The millennial generation has attained more years of schooling than any previous cohort in American history. Moreover, America spends more per student on primary through tertiary education than any other OECD nation. Surely then, this report would not depress me? I would find things to celebrate.

Did I? Read on.

This month’s column is distilled from the notes I made as I read through the ETS report. (These are summarizing notes. I did not bother to quote exactly, or even to use quotation marks when lifting a passage straight from the report. The originals of all the reports cited here are all freely available on the Web, so please go to the source documents to see what was originally written.)

A central message emerging from the ETS report is that, despite all the costly and extensive education, US millennials on average demonstrate relatively weak skills in literacy, numeracy, and problem solving in technology-rich environments, compared to their international peers. Sigh.

And this is not just true for millennials overall, it also holds for our best performing and most educated young adults, for those who are native born, and for those from the highest socioeconomic background. Moreover, the report’s findings indicate a decrease in literacy and numeracy skills for US adults when compared with results from previous adult surveys.

Some of the data highlights:
  • In literacy, US millennials scored lower than 15 of the 22 participating countries.
  • In numeracy, US millennials ranked last.
  • In PS-TRE, US millennials also ranked last.
  • The youngest segment of the US millennial cohort (16- to 24-year-olds), who will be in the labor force for the next 50 years, ranked last in numeracy and among the bottom countries in PS-TRE.
Even worse for those of us in higher education, this dismal picture holds for those with higher education:
  • US millennials with a four-year bachelor’s degree scored third from bottom in numeracy.
  • US millennials with a master’s or research degree were fourth from bottom.
All very depressing. I fear that this state of affairs will continue all the time US education continues to be treated as a political football, with our nation’s children and their teachers treated as pawns while various groups fight political battles, and make decisions, based not on learning research (of which there is now a copious amount, much of it generated in US universities) but on uninformed beliefs and political ideology. [You were surely waiting for me to throw in my two cents worth of opinion. There it is.]

To finish on a high note, we Americans famously like winners. So let’s raise a glass to the nations that came out on top in the rankings (in order, top first):

Literacy: Japan, Finland, Netherlands

Numeracy: Japan, Finland, Belgium

PS-TRE: Japan, Finland, Australia

In their own way, each of these countries seems to be doing education better than we are.

Yet here’s the fascinating thing. I’ve spent time in all of those countries. They each have a lot to offer, and I like them all. I also was born and grew up in the UK, moving to the US as an adult in 1987. I am a lifelong educator. But for all its faults (and its education system is just one of a legion of things America does poorly) I’d rather live where I do now, in the USA, with Italy in second place. But that’s another story. A complicated story. (If you think California is a separate nation, and in many ways it is, then my preference statement needs further parsing.) Doing well on global tests of educational attainment is just one factor that we can use to measure quality of life.

Thursday, May 7, 2015

Time to re-read (or read) What’s Math Got To Do With It?

Back in June 2010, I wrote a post to this blog in which I summarized a new book on K-12 mathematics teaching by a former Stanford colleague of mine, Prof Jo Boaler. At the time, though I had met Jo a few times, I did not really know her; rather I was just one of many mathematical educators who simply admired her work, some of which she described in the book What's Math Got To Do With It?, parts of which were the primary focus of my post.

Not long after my post appeared, Jo returned to Stanford from the UK, and over time we got to know each other better. When I formed my mathematics educational technology company BrainQuake in 2012, I asked her to be a founding member of its Board of Academic Advisors, all of whom are listed here. When she was putting the final touches to the new edition of her book, just published, she asked me to write a cover-quote, which I was pleased to do.

I say all of this by way of disclosure.* For my primary aim in writing this month’s column is to persuade you to read (or re-read) my earlier post, and ideally Jo’s book. The research findings she describes in the book highlight the lasting damage done to generations of K-12 students (and possibly consequent damage to the US economy when that generation of students enters the workforce) by continuing adherence to a classroom mathematics pedagogy that portrays math as a rule-based process of answer-getting, rather than a creative enterprise of understanding and problem solving.

The woefully ill-informed “debate” about the benefits of the US Common Core State Mathematics Standards that has been fostered in between the appearances of the two editions of Boaler’s book, make her message even more important than it was when the first edition came out in 2009. While CCSS opponents espouse opinions, Boaler presents evidence – lots of it – that supports the approach to K-12 mathematics learning the CCSS promotes.

If you want to see more of Prof Boaler’s efforts to improve K-12 mathematics education, see her teachers’ resource site YouCubed, or sign up for her online course How to Learn Math: for Teachers and Parents, which starts on June 16.

Also, check out her latest post in The Hechinger Report where she presents some recent data about the problems caused by a lot of old-style rule-memorization math instruction.

* NOTE: Prof Boaler’s Stanford research team also recently completed a small pilot study of BrainQuake’s mathematics learning (free-) app Wuzzit Trouble, first reported by education technology journalist Jordan Shapiro in an April 27, 2015 article in Forbes Magazine. (Prof Boaler is an academic advisor to BrainQuake but does so as a volunteer, and has no financial stake in the company.)

Wednesday, April 1, 2015

The Importance of Mathematics Courses in Computer Science Education

The confluence of two events recently reminded me of an article I wrote back in 2003 about the role of mathematics courses in university computer science education. [Why universities require computer science students to take math, Communications of the Association for Computing Machinery, Vol 46, No 9, Sept 2003, pp.36-39.]

The first event was a request for me to be an advisor on a research project to develop K-12 computer science programs. The second was a forum discussion in my Mathematical Thinking MOOC, currently in the middle of its sixth session.

My MOOC attracts a lot of mid-career computer professionals, who bring a different perspective to some of the issues the course considers. The forum thread in question focused on what is meant by a statement of the form “Let x be such that P(x).“ In mathematics, use of this statement requires that there exists an object satisfying P. If the existence is not known, you should express the statement counterfactually, as “Let x be an object such that P(x), assuming such an object exists.”

Some of the computer scientists, however, instinctively interpreted the statement “Let x be such that P(x)” as a variable declaration. This led them to give an “incorrect” answer to a question that asked then to identify exactly where the logic of a particular mathematical argument broke down. The logic failed with the selection of an object x that was not known to exist. In contrast, those computer scientists felt that things went wrong when the argument subsequently tried to do something with that x. That, they observed in the discussion, was where the program would fail.

It was a good discussion, that highlighted the distinction between the currently accepted view of mathematics as primarily about properties and relations, and the pre-nineteenth century view that it was at heart procedural. As such, it served as a reminder of the value of mathematics courses in computer science education, and vice versa.

The remainder of this post is what I wrote in the CACM back in 2003 (very lightly edited). I still agree with what I wrote then. (That is by no means always the case when I look at things I wrote more than a decade earlier.) I suspect that now, as then, some will not agree with me. (I actually received some ferociously angry responses to my piece.) Here goes.

Some years ago, I gave a lecture to the Computer Science Department at the University of Leeds in England. Knowing my background in mathematics — in particular, mathematical logic — the audience expected that my talk would be fairly mathematical, and on that particular occasion they were right. As I glanced at the announcement of my talk posted outside the lecture room, I noticed that someone had added some rather telling graffiti. In front of the familiar header “Abstract” above the description of my talk, the individual had scrawled the word “Very”.

It was a cute addition. But it struck me then, and does still, many years later, that it spoke volumes about the way many CS students view the subject. To the graffiti writer, operating systems, computer programs, and databases were (I assumed) not abstract, they were real. Mathematical objects, in contrast, so the graffiti-writer likely believed — and I have talked to many students who feel this way — are truly abstract, and reasoning about them is an abstract mental pursuit. Which goes to show just how good we humans are (perhaps also how effective university professors are) at convincing ourselves (and our students) that certain abstractions are somehow real.

For the truth is, of course, that computer science is entirely about abstractions. The devices we call computers don’t, in of themselves, compute. As electrical devices, if they can be said to do anything, it’s physics. It is only by virtue of the way we design those electrical circuits that, when the current flows, obeying the laws of physics, we human observers can pretend that they are doing reasoning (following the laws of logic), performing a numerical calculation (following the laws of arithmetic), or searching for information. True, it’s a highly effective pretence. But just because it’s useful does not make it any less a pretence.

Once you realize that computing is all about constructing, manipulating, and reasoning about abstractions, it becomes clear that an important prerequisite for writing (good) computer programs is an ability to handle abstractions in a precise manner. Now that, as it happens, is something that we humans have been doing successfully for over three thousand years. We call it mathematics.

This suggests that learning and doing mathematics might play an important role in educating future computer professionals. But if so, then what mathematics? From an educational point of view, in order to develop the ability to reason about formal abstractions, it is largely irrelevant exactly what abstractions are used. Our minds, which evolved over tens of thousands of years to reason (largely imprecisely) about the physical world, and more recently the social world, find it extremely difficult accepting formal abstractions. But once we have learned how to reason precisely about one set of abstractions, it takes relatively little extra effort to reason about any other.

But surely, you might say, even if I’m right, when it comes to training computer scientists, it makes sense to design educational courses around the abstractions the computer scientists will actually use when they graduate and go out to work in the technology field. Maybe so (in fact no, but I’ll leave that argument to another time), but who can say what the dominant programming paradigms and languages will be four years into the future? Computing is a rapidly shifting sand. Mathematics, in contrast, has a long history. It is stable and well tested.

Sure, there is a good argument to be made for computer science students to study discrete mathematics rather than calculus. But, while agreeing with that viewpoint, I believe it is often overplayed. Here’s why I think this.

A common view of education is that it is about acquiring knowledge — learning facts. After all, for the most part that is how we measure the effectiveness of education: by testing the students’ knowledge. But that’s simply not right. It might be the aim of certain courses, but it’s definitely not the purpose of education. The real goal of education is to improve minds — to enable them to acquire abilities and skills to do things they could not do previously. As William Butler Yeats put it, “Education is not about filling a bucket; it’s lighting a fire.” Books and USB memory sticks store many more facts than people do — they are excellent buckets — but that doesn’t make them smart. Being smart is about doing, not knowing.

Numerous studies have shown that if you test university students just a few months after they have completed a course, they will have forgotten most of the facts they had learned, even if they passed the final exam with flying colors. But that doesn’t mean the course wasn’t a success. The human brain adapts to intellectual challenges by forging and strengthening new neural pathways, and those new pathways remain long after the “facts” used to develop them have faded away. The facts fade, but the abilities remain.

If you want to prepare people to design, build, and reason about formal abstractions, including computer software, the best approach surely is to look for the most challenging mental exercises that force the brain to master abstract entities — entities that are purely abstract, and which cause the brain the maximum difficulty to handle. And where do you find this excellent mental training ground? In mathematics.

Software engineers may well never apply any of the specific theorems or techniques they were forced to learn as students (though some surely will, given the way mathematics connects into most walks of life in one way or another). But that doesn’t mean that those math courses were not important. On the contrary. The main benefit of learning and doing mathematics, I would argue, is not the specific content; rather it’s the fact that it develops the ability to reason precisely and analytically about formally defined abstract structures.

Monday, March 9, 2015

Pi Day, Cyclical Motion, and a Great Video Explanation of Multiplication


March 14 is Pi Day, the day in the year when we celebrate the world’s most famous mathematical constant.

Back in 1988, on March 14, a physicist called Larry Shaw organized the first Pi Day celebration at the Exploratorium in San Francisco, where he worked. It was meant to be just a one-off, fun event to get kids interested in math. Children were invited in to march around one of the Exploratorium’s circular rooms and end up eating fruit pies. But the idea took off, and ever since, March 14 has been Pi Day. Not just at the Exploratorium, but with celebratory events organized all across the United States, and in other parts of the world.

In case you haven’t twigged it, we celebrate Pi Day on March 14 because, in American date format, that day is 3.14, which is pi to two decimal places.

This year is a particularly special, once-in-a-century Pi Day, since the American format date this year is 3.14.15, pi to four decimal places. If you want more pi-accuracy, drink a toast to pi at time 9:26:53 (AM or PM), to get the first nine places 3.141592653.

That degree of accuracy, by the way, is more than enough for practical purposes. If you use that value to calculate the circumference of the Earth, the answer will be accurate to within 1/4 inch.

Though we have known since the 18th Century that pi is irrational (indeed, transcendental, thereby demonstrating that you cannot square a circle), calculating approximate values of pi has a long history. In ancient times, Babylonians, Egyptians, Greeks, Indians, and Chinese mathematicians calculated the first three or four places, and found fraction approximations like 22/7 and 355/113.

In the 16th century, a German who presumably had a lot of time on his hands spent most of his life computing pi to 36 places, and a 19th century American went all the way to 707 places, but he mad a mistake after 527 places, so the last part of his answer was wrong.

In more recent times, computers have been used to compute pi to well over a trillion places, in part for sport, but also to test the accuracy of high speed supercomputers.

Of course, PI Day isn’t really just about pi, it’s an excuse to celebrate all of mathematics, and in particular stimulate interest in mathematics among children and young adults. You will find Pi Day events in schools and colleges, at science museums, and other venues. Teachers, instructors, and students organize all kinds of math-related events and competitions. The value of pi simply sets the date.

With this year’s special Century edition, some large organizations are putting on celebratory events, among them the Museum of Mathematics in New York City (details of the event here), the Computer History Museum just south of San Francisco (details here), and the NASA Space Center in Houston (see here). And at the big Teaching and Learning Conference in Washington D.C. this week, I’m hosting a Pi Celebration at 8:00AM on Saturday morning.

There are many other celebrations. Check to see what is going on in your area. If there is a large science or technology organization nearby, they may well be putting on a Pi Day event.

The media have been getting in on the act too. NPR will air one of my short Math Guy conversations with Weekend Edition host Scott Simon this Saturday morning, and today’s New York Times ran a substantial article about pi by their regular Numberplay contributor Garry Antonick.

Antonick led off with a short pi-related problem I provided him with, and in honor of the Pi Day of the Century, in place of the traditional photo of me at a blackboard, he picked an action shot of me cresting a mountain on a bicycle (pi motion if ever there were) in a Century (100 mile) ride back in 2013.

He could not resist bringing in the famous Euler Identity, linking the five most significant constants of mathematics, pi, e, i, 0, and 1. This has always been my favorite mathematical identity, and Antonick quotes from a magazine article I wrote about it a few years ago.

But truth be told, it is not my favorite pi fact. For the simple reason, it’s not really about pi, rather it is about multiplication and exponentiation. Pi gets in because both operations involve the number.

My favorite pi fact, ever since I first came across it as a teenager (one of several eye-opening moments that motivated me to become a mathematician), is Leibniz’s series (sometimes called Gregory’s series), which dates from the 17th century. You write down an endless addition sum that starts out 1/1, minus 1/3, plus 1/5, minus 1/7, plus 1/9, etc. All the reciprocals of the odd numbers, with alternating signs.

Since this sum goes on for ever, you can’t actually add it up term by term, but you can use mathematical techniques to determine the answer a different way. And that answer is pi/4.

What does pi have to do with adding the reciprocals of the odd numbers? As with Euler’s Identity, Leibniz’s series provides a glimpse of the deep structure of numbers and arithmetic that lies just beneath the surface.

Talking of which, I caused a huge stir a few years ago when I ran a series of Devlin’s Angle posts trying to rid people (in particular, math teachers) of their false (and educationally dangerous) belief that multiplication is repeated addition (and exponentiation is repeated multiplication).

The initial series ran in June, July-August, and September 2008. When the barrage of facts I referenced in the third of those posts failed to stem the flood of disbelieving reactions of readers, I ran a lengthy post in January 2011 trying to convey the truly deep (and powerful) structure of multiplication.

Still to little avail. Put repeated addition in the same bin as evolution by natural selection, climate change, and the Golden Ratio. For many people, no amount of facts can overturn a long held and cherished belief. It’s a common human trait – fortunately not a universal one, else we’d still be living in caves and mud huts. (A politician who says “I am not a scientist” is effectively saying “I don’t understand the difference between building my mansion and a mud hut.”)

Unfortunately, as a wordsmith, I did not, and do not, possess the skill to provide a really good explanation of multiplication. I had to resort to spinning a multi-faceted story based on scaling. Someone who does have what it takes to tell the story properly, using video, is Stanford mathematics and computer science senior undergraduate Grant Sanderson. His recent video on Euler’s Identity is the best explanation of addition and multiplication I have ever seen. Period. Antonick embeds it in his New York Times piece. It deserves widespread circulation.

The video actually goes on to discuss the exponential function, and then the Euler Identity, but I suspect many viewers will get lost at that point. The exponential function is pretty sophisticated. Much more so than addition and multiplication. In contrast, all it takes to understand those two staples of modern numerical life is to get beyond the ultimately misleading concepts many of us form in the first few years of our lives. Do that, and Sanderson’s video provides the rest.

As is so often said, a picture can be worth a thousand words. Sanderson demonstrates that a motion picture can be worth a hundred thousand.

NOTE: I did try song a few years ago, collaborating with a Santa Cruz choral group called Zambra. The result can be found here. There’s lots of pi stuff in those compositions. But it’s primarily musical interpretation of mathematics, not explanation. (For instance, check out our rendering of Leibniz’s series.)

Finally, I often get asked why we use the Greek letter pi to denote the ratio of the circumference of a circle to its perimeter of a circle to its diameter.. This convention goes back to the 18th Century.