Friday, December 4, 2015

Life inside an impossible Escher figure

When the M.C. Escher inspired puzzle video-game Monument Valley came out last year, I knew I had to check it out. The more so when it started getting rave reviews and winning awards. But with so many other things to hold my attention, I never managed to get round to it. The recent decision of the creators to make a version available for free prompted me to finally take a quick look. Not that it had been expensive. Rather, a tweet about the new free version happened to come when I had an hour or so of free time on my hands.

That free hour got immediately swallowed up, as did many more hours after that. I was hooked from the getgo.

Anyone who is intrigued by those impossible figures that Escher made famous, even those of you with little or no interest in puzzle (video-) games, will surely be captivated by Monument Valley, where the solution to many of the puzzles involves orienting the figure to create an illusion of a continuous object. For when the player views the object as continuous, characters in the game that traverse the figures can move along it. Impossible chasms that prevented a character’s progress suddenly disappear as you rotate the entire figure just the right way.

It’s not a learning game. I don’t see a player learning any new mathematics. But what it does is provide a rich, immersive experience of complex geometrical spaces from the inside. As a player, your task is to assist the princess on her quest, which involves finding her way through a fantasy world of Escher-like structures, the geometry of which you can sometimes change as you progress. By projecting yourself into the princess, you get a sense of what it would be like to live in such a world.

And a beautiful world it is. The creators, based in the UK and operating under the name UsTwo, have crafted a series of truly gorgeous fantasy worlds, which you encounter one after another. It is not so much a game as a collection of interactive pieces of art where you play with, and experience, geometric shapes.

In fact, it is the artistic creation that the developers bring to the work. The idea of taking Escher worlds and turning them into a puzzle game goes back to a 2007 video game called Echodrome, designed for the Sony Play Station 3 by the Japanese designer Jun Fujiki. By all accounts it was fiendishly difficult, and never broke out beyond a small group of hard-core puzzle aficionados.

Monument Valley shows the huge difference presentation can make. If you want to hold people’s attention, you often need to think carefully about the medium. The message on its own may not be enough. That holds in the math class or the math lecture hall as much as in a video game.

Regular readers of this MAA blog or my other blog will know that I have a long-standing interest in video games, particularly so as an educational medium, where I am professionally active as a player, a learning researcher, and an entrepreneur.

In fact, much of my career has involved looking for ways to use different media to make mathematics accessible to as many people as possible. I have authored many “popular mathematics books”, written for newspapers (MAA compilation of some of my articles here), worked on television programs (including A Mathematical Mystery Tour, BBC-tv 1984; Life by the Numbers, PBS 1998; and NUMB3RS, CBS, 2005-2008), and of course there is my regular Math Guy radio gig for NPR, which started in 1994. More recently, in 2012, I launched the first ever math MOOC on Coursera (the seventh session just ended). I even made a foray into using music, song, and dance, with the 2007 show Harmonius Equations. To me, video games are one more medium to carry mathematical content.

In fact, when it comes to K-12 mathematics, video games are in many ways the most effective medium we currently have to provide good math learning, as I tried to articulate in a book I wrote in 2011, and a presentation I gave at the big Teaching and Learning 2014 conference in Washington D.C. last year, a 20-min video summary of which is available here.

Until recently, there was relatively little research available to put any flesh onto educators’ beliefs/hopes/suspicions that video games could yield good math learning outcomes. That is starting to change. (Reports from two classroom studies, one of I was involved in, are due to be published in the International Journal of Serious Games this month. Preprints are available here and here.)

Certainly, the results obtained in those two papers raise more questions than they answer. (Moreover, pending further, and substantially larger, studies, the results themselves have to be viewed as tentative.) What we are seeing is that, for mathematics in the K-8 range, significant learning outcomes can be observed after a video-game intervention of as little as two hours play spread over a month or so. (Some measures show an increase of 20% over a comparison group.)

I’d seen reports earlier that made similar claims, and dismissed them as product- marketing masquerading as research. It was only when the first of the two particular studies I cited above came out in late 2014, carried out by Prof Jo Boaler’s research group at Stanford University’s Graduate School of Education, using my own math learning video game Wuzzit Trouble as the intervention, that I sat up and really took notice.

In fact, I did more than that. Together with a research colleague from Tampere University in Finland, Prof Kristian Kiili, who, like me, has founded a math-learning video game company, and who was spending the year at Stanford, we carried out our own study. (The second of the two papers I cited.) Kiili was developing a fractions learning game, Semideus, and wanted to see how well it could serve as an evaluation tool. So we repeated essentially the same study Boaler’s team had done, with Wuzzit Trouble as the intervention, but instead of a written pre- and post-test (which the Boaler team used), we used Semideus. The results were very similar to those obtained in the previous study. (With the added twist that this time we found transfer — in a game context — from the whole number arithmetic of Wuzzit Trouble to the fractional reasoning of Semideus.)

Something is going on, that’s for sure. But what? It did not take long to come up with a fairly long list of possible factors. Among the many things that a (well designed) math learning game can offer, all which are known to have a positive impact on learning, are:

  • Breaking the Symbol Barrier – human-friendly representation (not the traditional abstract symbols of math textbooks).
  • Focus on developing number sense and problem solving ability.
  • High level of engagement.
  • Instant feedback (both positive and negative).
  • Steady flow of dopamine – known to have positive impact on memory formation and consolidation.
  • Learning through failure – in a playful, safe environment.
  • “Failure” treated – and regarded – as “not yet succeeded”.
  • Constant sense of “I can do this on the next try.”
  • Lots of repetition – but at the demand of the student/player.
  • Student/player is in control.
  • Student/player has ownership.
  • Growth Mindset – good games encourage and develop this. (This is the important notion Carol Dweck is famous for.)
  • Fluid intelligence (Gf) – games require and develop this. (Loosely speaking, this is the ability to hold several pieces of information in the mind at the same time and reason fluidly with them.)

  • I have written about many (not all) of these factors in my series of video game learning articles in my blog profkeithdevlin. (See also the many writings and videos on games and learning by Prof James Paul Gee.)

    My current guess is that all of these factors, and likely others, are at play in those dramatic learning outcomes. The only way to find out for sure, of course, is to do more research. A lot more. Prof Kiili, now back in Finland, is already hard at work on that, as am I and some of my colleagues at Stanford. And we are by no means alone. The field is wide open. Stay tuned. (Even better, get involved.) Truly, it’s an exciting time to be involved in mathematics education.

    Meanwhile, I have to sign off. Monument Valley is calling.

Monday, November 2, 2015

Today is George Boole’s 200th Birthday

Today, November 2, 2015, marks the 200th anniversary of the birth of George Boole, one of the most influential mathematicians of all time – though it would be long after his death that his influence would manifest itself, when the growth of the modern digital age made significant aspects of our lives boolean. (To the degree that adjectival use of his name is no longer capitalized nor in need of italicization.)

Born in England, Boole spent the major part of his mathematical career as a professor at Queen’s College Cork, and the Irish mathematical community has been actively celebrating Boole’s life, work, and legacy throughout this year. Of particular note, is an Irish ballad, “The Mathematician - The Bould Georgie Boole”, specially written for the occasion and performed by the Arthur Céilí Band, which you can hear, with visual biographic accompaniment about Boole, on YouTube and Vimeo:

The lyric and a download link to the song are available at:

In a more academic vein, University College Cork has created a video biography available at:

And US-based Irish mathematician Colm Mulcahy has a celebratory article in Scientific American.

There is a lot more available on the Boole Bicentennial that digital search technology (part of Boole’s legacy) makes easy to find, so I’ll keep this post short and let you explore on your own.

Be sure to log on to Google today. The company logo for the day is an active demonstration of boolean algebra using colors.

Friday, October 2, 2015

Letter to a calculus student – The Sequel

Devlin’s Angle for July 2006 was titled Letter to a calculus student. In it, I tried to describe, as briefly but as effectively as I could, the deep beauty there is in calculus, a beauty that arises from the depth of human brilliance that it took for the human mind to find a way to tame the infinite, and bend it to our use, a beauty made the more so by the enormous impact calculus has had on life on Earth.

In my essay, I acknowledged that there was little chance any calculus student would be able to understand what I was trying to convey. I wrote:

“Those techniques [of calculus] are so different from anything you have previously encountered in mathematics, that it will take you every bit of effort and concentration simply to learn and follow the rules. Understanding those rules and knowing why they hold can come only later, if at all. Appreciation of the inner beauty of the subject comes later still. Again, if at all.

I fear, then, that at this stage in your career there is little chance that you will be able to truly see the beauty in the subject. Beauty - true, deep beauty, not superficial gloss - comes only with experience and familiarity. To see and appreciate true beauty in music we have to listen to a lot of music - even better we learn to play an instrument. To see the deep underlying beauty in art we must first look at a great many paintings, and ideally try our own hands at putting paint onto canvas. It is only by consuming a great deal of wine - over many years I should stress - that we acquire the taste to discern a great wine. And it is only after we have watched many hours of football or baseball, or any other sport, that we can truly appreciate the great artistry of its master practitioners. Reading descriptions about the beauty in the activities or creations of experts can never do more than hint at what the writer is trying to convey.

My hope then is not that you will read my words and say, "Yes, I get it. Boy this guy Devlin is right. Calculus is beautiful. Awesome!" What I do hope is that I can at least convince you that I (and my fellow mathematicians) can see the great beauty in our subject (including calculus). And maybe one day, many years from now, if you continue to study and use mathematics, you will remember reading these words, and at that stage you will nod your head knowingly and think, "Yes, now I can see what he was getting at. Now I too can see the beauty."

I then proceeded to describe, as articulately as I could, the beauty there is to be seen in calculus, or at least the beauty I see in it, taking the reader on a guided tour of the standard definition of the derivative, but from the perspective of how it takes advantage of what the human brain can do, while circumventing what it cannot.

I ended my essay by quoting poet William Blake’s Auguries of Innocence, saying:

That's what [the derivative limit formula] asks you to do: to hold infinity in the palm of your hand. To see an infinite (and hence unending) process as a single, completed thing. Did any work of art, any other piece of human creativity, ever demand more of the observer? And to such enormous consequence for Humankind? If ever any painting, novel, poem, or statue can be thought of as having a beauty that goes beneath the surface, then the definition of the derivative may justly claim to have more beauty by far.

As I noted above, I was really writing for my fellow mathematicians. I knew then, as I still acknowledge today, that what I had written was true: it is impossible to experience the beauty in many human creations until one has sufficient experience.

It was then, with great pleasure, that I received the following email a few weeks ago (on August 17), which I reproduce in its entirety, unedited, with the permission of the sender. I hope you enjoy it to. And, if you are a math instructor at a college or university, maybe print off this blog post and pin it somewhere on a corridor in the department as a little seed waiting to germinate.

* * * * * *
Western Washington University, in Bellingham, WA

Hi Dr. Devlin,

My name is Murray Pendergrass. I am a math student at Western Washington University, a small public liberal arts college in the Pacific Northwest where I am pursuing a BS in Mathematics.

Sometime around 2006 you authored a post on Devlin's Angle titled "Letter to a calculus student" and I suppose someone in the math department at my school enjoyed it because it has been tacked to a bulletin board on the math floor for quite sometime. I would have only been going into the 8th grade when it was originally posted, with absolutely no idea that I would ever become interested in mathematics. I did take a calculus course my junior year of high school, but I don't think I could even briefly explain what a derivative was by the time the course was over (time well spent, obviously).

I must have first seen your article either my sophomore or junior year of college, 2014 most likely. I would have either been in precalculus or calculus I (differential calculus), and still completely unaware that I would end up declaring a math major. At that time I would have still been a member of the business school. I was probably waiting outside a professor's office for office hours when the title caught my eye,

" 'Letter to a calculus student' … Hmm, maybe I should read this."

However being the impatient person that I am, I believe I started in and thought "ok this is boring, I'll check the next page and see if it gets better,

"Nope, second page is boring too. Oh well."

And I have to admit, it was not until last night that I actually read the whole thing through for the first time.

But not long after that first initial and brief encounter with the letter my passion for mathematics truly began to develop and I realized that you can actually major in math without being a child prodigy (yes I actually thought this for quite sometime). It would have been shortly after this time, less than a year ago, that I realized I wanted to major in math. Since changing majors, very few hours have been spent not working on math.

I was studying at school late last evening when I decided to take a break and cruise up and down the hallway when for the second time in my life I noticed the letter tacked to the bulletin board. I must walk past it every single day but it was not until last night that it caught my eye again and I thought "I've seen this before! Oh wow I should give it a shot now that I am passionate about math."

In the very first sentence you open with a quote by Bertrand Russell, someone I have taken great interest in over the last year since mathematical logic has become a particular interest of mine. I immediately knew this was going to be a whole different experience reading this letter, and I was right.

What provoked me to feel the need to write you this letter was that I feel I am a precise example of the reader you are mentioning when you say,

"And maybe one day, many years from now, if you continue to study and use mathematics, you will remember reading these words, and at that stage you will nod your head knowingly and think, 'Yes, now I can see what he was getting at. Now I too can see the beauty.' "

Just as you predicted, the first time I made an attempt to read the letter "there was little chance I could see the true beauty in math", a statement so true that I could not only fail to see the beauty in math but I could not even read a letter about someone else promising me that even though I couldn't see the beauty, it was there.

It was quite a shock to me to read the letter last night and realize what a strange coincidental experience it was to randomly come across it a year after diving head first into the world of mathematics. It felt like a testament to myself of the progress I have made in math over the last year, a type of progress that cannot be explained or noticed through grades or high marks but by reading and truly relating to a mathematicians admiration for the beauty in math.

Before college I lived a bit of a bumpy life, it was a long and interesting road getting to where I am now. I will spare you the details as this letter has already turned out to be longer than I expected but I can truly say that finding math has been the best thing that has ever happened to me. In a lot of ways it has set me free.  I am very grateful to have the opportunity to study math at a university, to study something I am passionate about, and to reflect on how my relationship with math has evolved. I also must note that I hope I don't sound naive! I know I have only been doing math for a little over a year, which might sound like child's play to a Doctor of Philosophy in Mathematics. I am ecstatic that I have reached the point where I can appreciate mathematical beauty and I am also confident that math will continue to fascinate me and reveal its beauty for many years to come. Like most things, math is a journey not a destination.

Overall, I just felt the need to write to you because I thought you might enjoy knowing that even 9 years after you wrote it there are still students thinking for the first time:

"Yes, now I can see what he was getting at. Now I too can see the beauty".

Thank you,

Murray Pendergrass

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!


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 mathematicians and mathematics as a profession, and he makes them primarily for the mathematics and mathematics education professions. So he strives to inform an audience who are interested in mathematics. This, of course, is very different from 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).

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.

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.

Sunday, February 1, 2015

The Greatest Math Teacher Ever?

Last month I wrote about the kind of mathematic learning experiences we need to design to prepare young people for life in the Twenty-First Century. I cited the hugely successful, pioneering educational work of the late Professor R L Moore of the University of Texas. This follow up article about Moore and his teaching method is a combination of two earlier Devlin’s Angle posts, from May 1999 and June 1999. Other than adding a short paragraph at the end leading to further information about Moore, the only changes to my original text are minor updates to adjust for the passage of time.

The set-up

Each year, the MAA recognizes great university teachers of mathematics. Anybody who has been involved in selecting a colleague for a teaching award will know that it is an extremely difficult task. There is no universally agreed upon, ability-based linear ordering of mathematics teachers, even in a single state in a given year. How much more difficult it might be then to choose a “greatest ever university teacher of mathematics.” But if you base your decision on which university teacher of mathematics has had (through his or her teaching) the greatest impact on the field of mathematics, then there does seem to be an obvious winner. Who is it?

Most of us who have been in mathematics for over thirty years probably know the answer. Let me give you some clues as to the individual who, I am suggesting, could justifiably be described as the American university mathematician who, through his (it’s a man) teaching, has had the greatest impact on the field of mathematics.

He died in 1974 at the age of 91.

He was cut in the classic mold of the larger-than-life American hero: strong, athletic, fit, strikingly good looking, and married to a beautiful wife (in the familiar Hollywood sense).

He was well able to handle himself in a fist fight, an expert shot with a pistol, a lover of fast cars, self-confident and self-reliant, and fiercely independent.

Opinionated and fiercely strong-willed, he was forever embroiled in controversy.

He was extremely polite; for example, he would always stand up when a lady entered the room.

He was a pioneer in one of the most important branches of mathematics in the twentieth century.

He was a elected to membership of the National Academy of Science, as were three of his students.

The method of teaching he developed is now named after him.

If you measure teaching quality in terms of the product - the successful students - our man has no competition for the title of the greatest ever math teacher. During his 64 year career as a professor of mathematics, he supervised fifty successful doctoral students. Of those fifty Ph.D.’s, three went on to become presidents of the AMS - a position our man himself held at one point - and three others vice-presidents, and five became presidents of the MAA. Many more pursued highly successful careers in mathematics, achieving influential positions in the AMS and the MAA, producing successful Ph.D. students of their own and helping shape the development of American mathematics as it rose to its present-day position of world dominance.

In the first half of the Twentieth Century, fully 25% of the time the president of the MAA was either a student or a grandstudent of this man.

Other students and grandstudents of our mystery mathematician served as secretary, treasurer, or executive director of one of the two mathematical organizations and were editors of leading mathematical journals.

After reaching the age of 70, the official retirement age for professors at his university, in 1952, he not only continued to teach at the university, he continued to teach a maximum lecture load of five courses a year - more than most full-time junior faculty took on. He did so despite receiving only a half-time salary, the maximum that state regulations allowed for someone past retirement age who for special reasons was kept on in a “part-time” capacity.

He maintained that punishing schedule for a further eighteen years, producing 24 of his 50 successful Ph.D. students during that period.

He only gave up in 1969, when, after a long and bitter battle, he was quite literally forced to retire.

In 1967, the American Mathematical Monthly published the results of a national survey giving the average number of publications of doctorates in mathematics who graduated between 1950 and 1959. The three highest figures were 6.3 publications per doctorate from Tulane University, 5.44 from Harvard, and 4.96 from the University of Chicago. During that same period, our mathematician’s students averaged 7.1. What makes this figure the more remarkable is that our man had reached the official retirement age just after to the start of the period in question.

Who was he?

The answer

His name is Robert Lee Moore. Born in Dallas in 1882, R. L. Moore (as he was generally known) stamped his imprint on American mathematics to an extent that only in his later years could be fully appreciated.

As I noted earlier, during 64 year career, the last 49 of them at the University of Texas, Moore supervised fifty successful doctoral students. Three of them went on to become presidents of the AMS (R. L. Wilder, G. T. Whyburn, R H Bing) -- a position Moore also held -- and three others vice-presidents (E. E. Moise, R. D. Anderson, M. E. Rudin), and five became presidents of the MAA (R. D. Anderson, E. Moise, G. S. Young, R H Bing, R. L. Wilder). Many more pursued highly successful careers in mathematics, achieving influential positions in the AMS and the MAA, producing successful Ph.D. students of their own -- mathematical grandchildren of Moore -- and helping shape the development of American mathematics as it rose to its present-day dominant position. That’s quite a record!

In 1931 Moore was elected to membership of the National Academy of Science. Three of his students were also so honored: G. T. Whyburn in 1951, R. L. Wilder in 1963, and R H Bing in 1965.

In 1973, his school, the University of Texas at Austin, named its new, two-winged, seventeen-story mathematics, physics, and astronomy building after him: Robert Lee Moore Hall. This honor came just over a year before his death. The Center for American History at the University of Texas has established an entire collection devoted to the writings of Moore and his students.

Discovery learning

Moore was one of the founders of modern point set topology (or general topology), and his research work alone puts him among the very best of American mathematicians. But his real fame lies in his achievements as a teacher, particularly of graduate mathematics. He developed a method of teaching that became widely known as “the Moore Method”. Its present-day derivative is often referred to as “Discovery Learning” or “Inquiry-Based Learning” (IBL).

One of the first things that would have struck you if you had walked into one of Moore’s graduate classes was that there were no textbooks. On each student’s desk you would see the student’s own notebook and nothing else! To be accepted into Moore’s class, you had to commit not only not to buy a textbook, but also not so much as glance at any book, article, or note that might be relevant to the course. The only material you could consult were the notes you yourself made, either in class or when working on your own. And Moore meant alone! The students in Moore’s classes were forbidden from talking about anything in the course to one another -- or to anybody else -- outside of class. Moore’s idea was that the students should discover most of the material in the course themselves. The teacher’s job was to guide the student through the discovery process in a modern-day, mathematical version of the Socratic dialogue. (Of course, the students did learn from one another during the class sessions. But then Moore guided the entire process. Moreover, each student was expected to have attempted to prove all of the results that others might present.)

Moore’s method uses the axiomatic method as an instructional device. Moore would give the students the axioms a few at a time and let them deduce consequences. A typical Moore class might begin like this. Moore would ask one student to step up to the board to prove a result stated in the previous class or to give a counterexample to some earlier conjecture -- and very occasionally to formulate a new axiom to meet a previously identified need. Moore would generally begin by asking the weakest student to make the first attempt -- or at least the student who had hitherto contributed least to the class. The other students would be charged with pointing out any errors in the first student’s presentation.

Very often, the first student would be unable to provide a satisfactory answer -- or even any answer at all, and so Moore might ask for volunteers or else call upon the next weakest, then the next, and so on. Sometimes, no one would be able to provide a satisfactory answer. If that were the case, Moore might provide a hint or a suggestion, but nothing that would form a constitutive part of the eventual answer. Then again, he might simply dismiss the group and tell them to go away and think some more about the problem.

Moore’s discovery method was not designed for -- and probably will not work in -- a mathematics course which should survey a broad area or cover a large body of facts. And it would obviously need modification in an area of mathematics where the student needs a substantial background knowledge in order to begin. But there are areas of mathematics where, in the hands of the right teacher -- and possibly the right students -- Moore’s procedure can work just fine. Moore’s own area of general topology is just such an area. You can find elements of the Moore method being used in mathematics classes at many institutions today, particularly in graduate courses and in classes for upper-level undergraduate mathematics majors, but few instructors ever take the process to the lengths that Moore did, and when they try, they rarely meet with the same degree of success.

Part of the secret to Moore’s success with his method lay in the close attention he paid to his students. Former Moore student William Mahavier (now deceased) addressed this point:

“Moore treated different students differently and his classes varied depending on the caliber of his students. . . . Moore helped his students a lot but did it in such a way that they did not feel that the help detracted from the satisfaction they received from having solved a problem. He was a master at saying the right thing to the right student at the right time. Most of us would not consider devoting the time that Moore did to his classes. This is probably why so many people claim to have tried the Moore method without success.”

Another famous (now deceased) mathematician who advocated -- and has successfully used -- (a modern version of) the Moore method was Paul Halmos. He wrote:

“Can one learn mathematics by reading it? I am inclined to say no. Reading has an edge over listening because reading is more active -- but not much. Reading with pencil and paper on the side is very much better -- it is a big step in the right direction. The very best way to read a book, however, with, to be sure, pencil and paper on the side, is to keep the pencil busy on the paper and throw the book away.”

Of course, as Halmos went on to admit, such an extreme approach would be a recipe for disaster in today’s over-populated lecture halls. The educational environment in which we find ourselves these days would not allow another R. L. Moore to operate, even if such a person were to exist. Besides the much greater student numbers, the tenure and promotion system adopted in many (most?) present-day colleges and universities encourages a gentle and entertaining presentation of mush, and often punishes harshly (by expulsion from the profession) the individual who seeks to challenge and provoke -- an observation that is made problematic by its potential for invocation as a defense for genuinely poor teaching. But as numerous mathematics instructors have demonstrated, when adapted to today’s classroom, the Moore method -- discovery learning -- has a lot to offer.

If you want to learn more about R. L. Moore and his teaching method, check out the web site:

But before you give the method a try, take heed of the advice from those who have used it: Plan well in advance and be prepared to really get to know your students. Halmos put it this way:

“If you are a teacher and a possible convert to the Moore method ... don’t think that you’ll do less work that way. It takes me a couple of months of hard work to prepare for a Moore course. ... I have to chop the material into bite-sized pieces, I have to arrange it so that it becomes accessible, and I must visualize the course as a whole -- what can I hope that they will have learned when it’s over? As the course goes along, I must keep preparing for each meeting: to stay on top of what goes on in class. I myself must be able to prove everything. In class I must stay on my toes every second. ... I am convinced that the Moore method is the best way to teach there is -- but if you try it, don’t be surprised if it takes a lot out of you.”

There is a caveat

When I wrote those two Devlin’s Angle posts back in 1999, I debated with myself whether to address a side to the Moore story that, particularly from a late Twentieth Century perspective, does not stand to his credit. The issue is race.

Moore’s racial attitude was nothing unusual for a white person who was born and lived most of his life in Texas in the late Nineteenth Century and the first three quarters of the Twentieth. When the Civil Rights Act was passed in 1964 (yes, that recently!), making racial discrimination illegal, Moore was already long past retiring age, and just five years short of actually vacating his university office. Moreover, no one who regularly reads a newspaper would believe that racial discrimination in America is a thing of the past. Moore’s racial views are still not unusual in Texas and elsewhere.

Were Moore not such a towering figure, his position on race (at least as demonstrated by his actions) would not merit attention. But like all great people, all aspects of his life become matters of scrutiny. Moore could have acted differently when it came to race, even back then, in Texas, but he did not. And from today’s perspective, that inevitably leaves an uncomfortable stain on his legacy.

In writing my two 1999 columns, I chose to focus on Moore the university teacher, in particular to raise awareness of discovery learning in mathematics. The focus of Devlin’s Angle is, after all, mathematics and mathematics teaching. I did not want to distract from that goal with what is clearly a side issue, particularly such an explosive one. Moore’s larger-than-life character was clearly a significant part of his success. His racism (or at least racist behavior) was not a part of that success story – if for no other reason than because he never accepted any Black students. So I did not raise the issue.

For the same reason, I have left this side of the Moore story to the end here. We can learn from Moore when it comes to designing good mathematics learning experiences, and even admire him as a highly gifted teacher, without condoning other aspects of his life, just as we can enjoy Wagner’s music without endorsing Nazism. I can however leave you with a pointer to an article posted online by Mathematics Professor Scott Williams on 5/28/99, about the same time my articles appeared (and possibly in response to the first of them). Like it or not, Williams’ post shines light on another side to the Moore story. We can learn things from great people in ways other than taking a class from them, and we can perhaps learn things they were not trying to teach us.


Paul R. Halmos (1975), The Problem of Learning to Teach: The Teaching of Problem Solving, American Mathematical Monthly, Volume 82, pp.466-470.
Paul R. Halmos (1985), I Want to Be a Mathematician, Chapter 12 (How to Teach), New York: Springer-Verlag, pp.253-265.
William S. Mahavier (1998), What is the Moore Method?, The Legacy of R. L. Moore Project, Austin, Texas: The University of Texas archives.

Thursday, January 1, 2015

Your Father’s Mathematics Teaching No Longer Works

Gender-challenged title courtesy of this famous 1988 Oldsmobile TV commercial:

The start of a New Year is traditionally a time when we resolve to make changes. Change is particularly imperative in US mathematics education, which is built on a (Nineteenth Century) pedagogic model that long since passed its expiry date.

In a nutshell, the school system we all grew up with was essentially developed in Nineteenth Century Britain to provided a global infrastructure to run the British Empire. In modern terms, the British Imperialists created an “Internet” and an “Internet of Things” using the best computational and manufacturing resources available at the time: people.

Controlled by an “Internet of human computers”: The British Empire in 1922.
Map from
While few of us in K-16 education today see it as merely a process to prepare young people for work, we inherited a system built to do just that.

Now we have an electronic, digital Internet, does it make sense to continue to use the old system?

What do Twenty-First Century citizens need from their education?

While not the only thing—not even close—equipping young people for work is still an important educational goal, both for the individual and for society as a whole. Accordingly, it makes sense for those of us in systemic education to be constantly aware of the skills that are actually required in the workplace. If those skills change, so should the education we provide.

A good place to start is by asking the leaders of the leading companies what they look for when hiring new employees. The table below shows us what skills the Fortune 500 companies were asking for in 1970, then again thirty years later in 1999.

Fortune 500 most valued skills, cited in
Linda Darling-Hammond et al, Criteria for High-Quality Assessment (2013)

It makes for dramatic reading. While the required skills ranked from 4 through 9 remained unaltered, the top three changed completely. The most important skill in 1970, writing, dropped to number 10, while skills two and three, computation and reading, respectively, dropped off the top ten list entirely.

The most important skill in the workplace at the start of the Twenty-First Century, according to those leading companies, is teamwork, which in a single generation had leapt up from number 10. The other two skills at the top, Problem Solving and Interpersonal Skills, were not even listed back in 1970.

Clearly, the world (of work) has changed, at least for those of us living in advanced societies. Unfortunately, for those of us in the United States, and many other parts of the world, our education system has failed to keep up.

In large part, this is because of the hard-to-avoid inertia that so often comes with national (or statewide) education systems. By and large, many politicians and bureaucrats are far less aware of rapidly changing workforce requirements than those in business, and politicians frequently pander to the often woefully uninformed beliefs of voters, who tend to resist change–especially change that will affect their children.

In the US, we see this dramatically illustrated by the widespread resistance to the Common Core State Standards. In the case of mathematics, just look at how closely the eight basic Mathematics Principles of the CCSS align to that Fortune 500 list of required Twenty-First Century skills:
  1. Make sense of problems and persevere in solving them.
  2. Reason abstractly and quantitatively.
  3. Construct viable arguments and critique the reasoning of others.
  4. Model with mathematics.
  5. Use appropriate tools strategically.
  6. Attend to precision.
  7. Look for and make use of structure.
  8. Look for and express regularity in repeated reasoning.
The fact is, any parent who opposes adoption of the CCSS is, in effect, saying, “I do not want my child prepared for life in the Twenty-First Century.” They really are. Not out of lack of concern for their children, to be sure. Quite the contrary. Rather, what leads them astray is that they are not truly aware of how the huge shifts that have taken place in society over the last thirty years have impacted educational needs.

Having lived through those changes, parents have (for the most part) been able to build on their own education and cope with new demands. “What worked for me will work for my children,” they say. (They say that even when it patently did not work for them!)

But the situation is very different for their children. They are being thrust straight into that new world. To prepare them for that, you need a very different kind of education: one based on understanding rather then procedural mastery, and on exploration rather than instruction.

One of the best summaries of this societal change, and the resulting need for educational shift, that I know is the 22 minutes TED Talk Build a School in the Cloud, given by the educational researcher Sugata Mitra, winner of the 2013 TED Prize.

In his talk, not only does Mitra explain why we need to make radical changes to education, he provides examples, backed by solid evidence, of how a “Fortune 500 oriented,” team-based, exploratory approach works. In the late 1990s and throughout the 2000s, Mitra conducted experiments in which he gave children in India access to computers. Without any instruction, they were able to teach themselves a variety of things, from English to DNA replication.

[See also Mitra’s earlier talk from 2010.]

Another good account of this need for educational change is provided by Sir Ken Robinson, also in a TED Talk (11 minutes).

These ideas are not new. Indeed, they are mainstream in educational research circles. They just have not permeated society at large.

For instance, Harvard physicist Eric Mazur has been teaching by Inquiry-Based Learning (IBL), to use one of several names for this general approach, for over twenty-five years, since he first noticed that instructional lectures simply do not work. He describes his approach, and the reasons for adopting it, in his 2009 talk Confessions of a Converted Lecturer (18 minutes, abridged version).

In mathematics, the IBL approach goes way back to the 1920s. I wrote about the best known proponent in two Devlin’s Angle posts back in 1999: May, June.

Moore’s ideas have been adapted and used successfully in present-day mathematics classrooms, as shown in the promotional video Creativity in Mathematics: Inquiry-Based Learning and the Moore Method (20 minutes).

Lest my account of R L Moore and that last video portrayal leaves you with the impression that IBL math is for bright college students, see also this WIRED magazine account of the success Mexican teacher Sergio Juárez Correa had when he took a Mitra inspired approach into a poor school in Matamoros, a city of half-a-million known more for its drug trade than for being at the forefront of Twenty-First Century mathematics education.

Remember, Bob Dylan sang this in 1964:


It’s long past time for the education system to catch up with the world outside the classroom. That should be our resolution for 2015.