Wednesday, June 14, 2017

Classroom Clickers Are Good; Except When They Are Not

Many math instructors use clickers in their larger lecture classes, and can cite numerous studies to show that they lead to more student attention and better learning. A recent research paper on clicker use devotes a page-long introductory section to a review of some of that literature. (Shapiro et al, Computers & Education 111 (2017), 44–59) But the paper—by clicker aficionadas, I should stress—is not all good news. In fact, its main new finding is that when clickers are used in what may be the most common way, they actually have a negative effect on student learning. This finding was sufficiently startling that EdSurge put out a feature article on the paper on May 25, which is how I learned of the result.

The most common (I believe) use of clickers is to provide students with frequent quiz questions to check that they are retaining important facts. (The early MOOCs, including my own, used simple, machine-graded quizzes embedded in the video lectures to achieve the same result.) And a lot of that research I just alluded to showed that the clickers achieve that goal.

So too does the latest study. All of which is fine and dandy if the main goal of the course is retention of facts. Where things get messy is when it comes to conceptual understanding of the material—a goal that almost all mathematicians agree is crucial.

In the new study, the researchers looked at two versions of a course (physics, not mathematics), one fact-focused, the other more conceptual and problem solving. In each course, they gave one group fact-based clicker questions and a second group clicker questions that concentrated on conceptual understanding in addition to retention of basic facts.

As the researchers expected, both kinds of questions resulted in improved performance in fact- based questions on a test administered at the end.

Neither kind of question led to improved performance in a problem-based test questions that required conceptual understanding.

The researchers expressed surprise that the students who were given the conceptual clicker questions did not show improvement in conceptual questions performance. But that was not the big surprise. That was, wait for it: students who were given only fact-based clicker questions actually performed worse on conceptual, problem solving questions.

To those of us who are by nature heavy on the conceptual understanding, not showing improvement as a result of enforced fact-retention comes as no big surprise. But a negative effect! That’s news.

By way of explanation, the researchers suggest that the fact-based clicker questions focus the student’s attention on retention of what are, of course, surface features, and do so to the detriment of acquiring the deeper understanding required to solve problems.

If this conclusion is correct—and is certainly seems eminently reasonable—the message is clear. Use clickers, but do so with questions that focus on conceptual understanding, not retention of basic facts.

The authors also recommend class discussions of the concepts being tested by the clicker questions, again something that comes natural to we concepts matter folks.

I would expect the new finding to have implications for game-based math learning, which regular readers will know is something I have been working on for some years now. The games I have been developing are entirely problem-solving challenges that require deep understanding, and university studies have shown they achieve the goal of better problem-solving skills. (See the December 4, 2015 Devlin’s Angle post.) The majority of math learning games, in contrast, focus on retention of basic facts. Based on the new clickers study, I would hypothesize that, even if a game were built on math concepts (many are not), unless the gameplay involves active, problem-solving engagement with those concepts, the result could be, not just no conceptual learning, but a drop in performance on a problem solving test.

Both clickers and video games set up a feedback cycle that can quickly become addictive. With both technologies, regular positive feedback leads to improvement in what the clicker- questions or game-challenges ask for. Potentially more pernicious, however, that positive feedback will result in the students thinking they are doing just fine overall—and hence have no need to wrestle more deeply with the material. And that sets them up for failure once they have to go beneath the surface fact they have retained. Thinking you are winning all the time seduces you to ease off, and as a result is the path to eventual failure. If you want success, the best diet is a series of challenges— that is to say, challenges in coming to grips with the essence of the material to be learned—where you experience some successes, some failures from which you can recover, and the occasional crash-and- burn to prevent over-confidence.

That’s not just the secret to learning math. It’s the secret to success in almost any walk of life.

Monday, May 22, 2017

The Math Gift Myth

My May post is more than a little late. The initial delay was caused by a mountain of other deadlines. When I did finally start to come up for air, there just did not seem to be any suitable math stories floating around to riff off, but I did not have enough time to dig around for one. That this has happened so rarely in the twenty years I have been writing Devlin’s Angle (and various other outlets going back to the early 1980s in the UK), that it speaks volumes against the claim you sometimes hear that nothing much happens in the world of mathematics. There is always stuff going on.

Be that as it may, when I woke up this morning and went online, two fascinating stories were waiting for me. What’s more, they are connected – at least, that’s how I saw them.

First, my Stanford colleague Professor Jo Boaler sent out a group email pointing to a New York Times article that quoted her, and which, she noted, she helped the author to write. Titled "No Such Thing as a Math Person," it summarizes the consensus among informed math educators that mathematical ability is a spectrum. Just like any other human ability. What is more, the basic math of the K-8 system is well within the capacity of the vast majority of people. Not easy to master, to be sure; but definitely within most people’s ability. It may be defensible to apply terms such as “gifted and talented” to higher mathematics (though I will come back to that momentarily), but basic math is almost entirely a matter of wanting to master it and being willing to put in the effort. People who say otherwise are either (1) education suppliers trying to sell products, (2) children who for whatever reason simply do not want to learn and find it reassuring to convince themselves they just don’t have the gift, or (3) mums and dads who want to use the term as a parental boast or an excuse.

Unfortunately, the belief that mathematical ability is a “gift” (that you either have or you don’t) is so well established it is hard to get rid of. Part of the problem is the way it is often taught, as a collection of rules and procedures, rather than a way of thinking (and a very simplistic one at that). Today, this is compounded by the rapid changes in society over the past few decades, that have revolutionized the way mathematics needs to be taught to prepare the new generation for life in today’s – and tomorrow’s – world. (See my January 1 article in The Huffington Post, "All The Mathematical Methods I Learned In My University Math Degree Became Obsolete In My Lifetime," and its follow up article (same date), "Number Sense: the most important mathematical concept in 21st Century K-12 education.")

With many parents, and not a few teachers, having convinced themselves of the “Math Gift Myth,” attempts over the past several decades to change that mindset have met with considerable resistance. If you have such a mindset, it is easy to see what happens in the educational world around you as confirming it. For instance, one teacher commented on The New York Times article:

“Excuse me? I'm a teacher and I refute your assertion. I have seen countless individuals who have problems with math – and some never get it. The same goes for English. But, unless you've spent years in the classroom, it takes years to fully accept that observation. The article's writer is a doctor, not a teacher; accomplishment in one field does not necessarily translate readily to another.”

Others were quick to push back against that comment, with one pointing out that her final remark surely argues in favor of everyone in the education world keeping up with the latest scientific research in learning. We are all liable to seek confirmation of our initial biases. And both teachers and parents are in powerful positions to pass on those biases to a new generation of math learners.

In her most recent book, Mathematical Mindsets: Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching, Prof Boaler lays out some of the considerable evidence against the Math Gift Myth, and provides pointers to how to overcome it in the classroom. The sellout audiences Boaler draws for her talks at teachers conferences around the world indicates the hunger there is to provide math learning that does not produce the math-averse, and even math-phobic, citizens we have grown accustomed to.

And so to that second story I came across. Hemant Mehta is a former National Board Certified high school math teacher in the suburbs of Chicago, where he taught for seven years, who is arguably best known for his blog The Friendly Atheist. His post on May 22 was titled "Years Later, the Mother Who 'Audited' an Evolution Exhibit Reflects on the Viral Response." Knowing Mehta’s work (for the record, I have also been interviewed by him on his education-related podcast), that title hooked me at first glance. I could not resist diving in.

As with The New York Times article I led off with, Mehta’s post is brief and to the point, so I won’t attempt to summarize it here. Like Mehta, as an experienced educator I know that it requires real effort, and courage, to take apart ones beliefs and assumptions, when faced with contrary evidence, and then to reason oneself to a new understanding. So I side with him in not in any way trying to diminish the individual who made the two videos he comments on. What we can do, is use her videos to observe how difficult it can be to make that leap from interpreting seemingly nonsensical and mutually contradictory evidence from within our (current!) belief system, to seeing it from a new viewpoint from which it all makes perfect sense – to rise above the trees to view the forest, if you will. The video lady cannot do that, and assumes no one else can either.

Finally, what about my claim that post K-12 mathematics may be beyond the reach of many individuals’ innate capacity for progression along that spectrum I referred to? Of course, it depends on what you mean by “many”. Leaving that aside, however, if someone, for whatever reason, develops a passionate interest in mathematics, how far can they go? I don’t know. Based on a sample size of one, me, we can go further than we think. I look at the achievement of mathematicians such as Andrew Wiles or Terrence Tao and experience the same degree of their being from a different species as the keen-amateur- cyclist-me feels when I see the likes of Tour de France winner Chris Froome or World Champion Peter Sagan climb mountains at twice the speed I can sustain.

Yet, on a number of occasions where I failed to solve a mathematics problem I had been working on for months and sometimes years, when someone else did solve it, my first reaction was, “Oh no, I was so close. If only I had tried just a tiny bit harder!” Not always, to be sure. Not infrequently, I was convinced I would never have found the solution. But I got within a hairsbreadth on enough occasions to realize that with more effort I could have done better than I did. (I have the same experience with cycling, but there I do not have a particular desire to aim for the top.)

In other words, all my experience in mathematics tells me I do not have an absolute ability limit. Nor, I am sure, do you. Mathematical proficiency is indeed a spectrum. We can all do better – if we want to. That, surely is the message we educators should be telling our students, be they in the K-8 classroom or the postgraduate seminar room.

Gifted and talented? Time to recognize that as an educational equivalent of the Flat Earth Belief. Sure, we are surrounded by seemingly overwhelming daily experience that the world is flat. But it isn’t. And once you accept that, guess what? From a new perspective, you start to see supporting evidence for the Earth being spherical.

Wednesday, April 5, 2017

Fibonacci and Golden Ratio Madness

The first reviews of my new book Finding Fibonacci have just come out, and I have started doing promotional activities to try to raise awareness. As I expected,  one of the first reviews I saw featured a picture of the Nautilus shell (no connection to Fibonacci or the Golden Ratio), and media interviewers have inevitably tried to direct the conversation towards the many fanciful—but for the most part totally bogus—claims about how the Golden Ratio (and hence the Fibonacci sequence) are related to human aesthetics, and can be found in a wide variety of real-world objects besides the Nautilus shell. [Note: the Fibonacci sequence absolutely is mathematically related to the Golden Ratio. That’s one of the few golden ratio claims that is valid! There is no evidence Fibonacci knew of the connection.]

For some reason, once a number has been given names like “Golden Ratio” and “Divine Ratio”, millions of otherwise sane, rational human beings seem willing to accept claims based on no evidence whatsoever, and cling to those beliefs in the face of a steady barrage of contrary evidence going back to 1992, when the University of Maine mathematician George Markovsky published a seventeen- page paper titled "Misconceptions about the Golden Ratio" in the MAA’s College Mathematics Journal, Vol. 23, No. 1 (Jan. 1992), pp. 2-19.

In 2003, mathematician, astronomer, and bestselling author Mario Livio weighed in with still more evidence in his excellent book The Golden Ratio: The Story ofPHI, the World's Most Astonishing Number.

I first entered the fray with a Devlin’s Angle post in June 2004 titled "Good Stories Pity They’re Not True" [the MAA archive is not currently accessible], and then again in May 2007 with "The Myth That Will Not Go Away" [ditto].

Those two posts gave rise to a number of articles in which I was quoted, one of the most recent being "The Golden Ration: Design’s Biggest Myth," by John Brownlee, which appeared in Fast Company Design on April 13, 2015.

In 2011, the Museum of Mathematics in New York City invited me to give a public lecture titled "Fibonacci and the Golden Ratio Exposed: Common Myths andFascinating Truths," the recording of which was at the time (and I think still is) the most commented-on MoMath lecture video on YouTube, largely due to the many Internet trolls the post attracted—an observation that I find very telling as to the kinds of people who hitch their belief system to one particular ratio that does not quite work out to be 1.6 (or any other rational number for that matter), and for which the majority of instances of those beliefs are supported by not one shred of evidence. (File along with UFOs, Flat Earth, Moon Landing Hoax, Climate Change Denial, and all the rest.)

Needless to say, having been at the golden ratio debunking game for many years now, I have learned to expect I’ll have to field questions about it. Even in a media interview about a book that, not only flatly refutes all the fanciful stuff, but lays out the history showing that the medieval mathematician known today as Fibonacci left no evidence he had the slightest interest in the sequence now named after him, nor had any idea it had several cute properties. Rather, he simply included among the hundreds of arithmetic problems in his seminal book Liber abbaci, published in 1202, an ancient one about a fictitious rabbit population, the solution of which is that sequence.

What I have always found intriguing is the question, how did this urban legend begin? It turns out to be a relatively recent phenomenon. The culprit is a German psychologist and author called Adolf Zeising. In 1855, he published a book titled: A New Theory of the proportions of the human body, developed from a basic morphological law which stayed hitherto unknown, and which permeates the whole nature and art, accompanied by a complete summary of the prevailing systems.

This book, which today would likely be classified as “New Age,” is where the claim first appears that the proportions of the human body are based on the Golden Ratio. For example, taking the height from a person's naval to their toes and dividing it by the person's total height yields the Golden Ratio. So, he claims, does dividing height of the face by its width.

From here Zeising leaped to make a connection between these human-centered proportions and ancient and Renaissance architecture. Not such an unreasonable jump, perhaps, but it was, and is pure speculation. After Zeising, the Golden Ratio Thing just took off.

Enough! I can’t bring myself to continue. I need a stiff drink.

For more on Zeising and the whole wretched story he initiated, see the article by writer Julia Calderone in business Insider, October 5, 2015, "The one formula that's supposed to 'prove beauty' is fundamentally wrong."

See also the blogpost on Zeising on the blog misfits’ architecture, which presents an array of some of the battiest claims about the Golden Ratio.

That’s it. I’m done.












Wednesday, March 8, 2017

Finding Fibonacci


Devlin makes a pilgrimage to Pisa to see the
statue of Leonardo Fibonacci in 2002.
In 1983, I did something that would turn out to have a significant influence on the direction my career would take. Frustrated by the lack of coverage of mathematics in the weekly science section of my newspaper of choice, The Guardian, I wrote a short article about mathematics and sent it to the science editor. A few days later, the editor phoned me to explain why he could not to publish it. “But,” he said, “I like your style. You seem to have a real knack for explaining difficult ideas in a way ordinary people can understand.” He encouraged me to try again, and my second attempt was published in the newspaper on May 12, 1983. Several more pieces also made it into print over the next few months, eliciting some appreciative letters to the editor. As a result, when The Guardian launched a weekly, personal computing page later that year, it included my new, twice-monthly column MicroMaths. The column ran without interruption until 1989, when my two-year visit to Stanford University in California turned into a permanent move to the US.

Before long, a major publisher contracted me to publish a collection of my MicroMaths articles, which I did, and following that Penguin asked me to write a more substantial book on mathematics for a general audience. That book, Mathematics: The NewGolden Age, was first published in 1987, the year I moved to America.

In addition to writing for a general audience, I began to give lectures to lay audiences, and started to make occasional appearances on radio and television. From 1991 to 1997, I edited MAA FOCUS, the monthly magazine of the Mathematical Association of America, and since January 1996 I have written this monthly Devlin’s Angle column. In 1994, I also became the NPR Math Guy, as I describe in my latest article in the Huffington Post.

Each new step I took into the world of “science outreach” brought me further pleasure, as more and more people came up to me after a talk or wrote or emailed me after reading an article I had written or hearing me on the radio. They would tell me they found my words inspiring, challenging, thought-provoking, or enjoyable. Parents, teachers, housewives, business people, and retired people would thank me for awakening in them an interest and a new appreciation of a subject they had long ago given up as being either dull and boring or else beyond their understanding. I came to realize that I was touching people’s lives, opening their eyes to the marvelous world of mathematics.

None of this was planned. I had become a “mathematics expositor” by accident. Only after I realized I had been born with a talent that others appreciated—and which by all appearances is fairly rare—did I start to work on developing and improving my “gift.”

In taking mathematical ideas developed by others and explaining them in a way that the layperson can understand, I was following in the footsteps of others who had also made efforts to organize and communicate mathematical ideas to people outside the discipline. Among that very tiny subgroup of mathematics communicators, the two who I regarded as the greatest and most influential mathematical expositors of all time are Euclid and Leonardo Fibonacci. Each wrote a mammoth book that influenced the way mathematics developed, and with it society as a whole.

Euclid’s classic work Elements presented ancient Greek geometry and number theory in such a well-organized and understandable way that even today some instructors use it as a textbook. It is not known if any of the results or proofs Euclid describes in the book are his, although it is reasonable to assume that some are, maybe even many. What makes Elements such a great and hugely influential work, however, is the way Euclid organized and presented the material. He made such a good job of it that his text has formed the basis of school geometry teaching ever since. Present day high school geometry texts still follow Elements fairly closely, and translations of the original remain in print.

With geometry being an obligatory part of the school mathematics curriculum until a few years ago, most people have been exposed to Euclid’s teaching during their childhood, and many recognize his name and that of his great book. In contrast, Leonardo of Pisa (aka Fibonnaci) and his book Liber abbaci are much less well known. Yet their impact on present-day life is far greater. Liber abbaci was the first comprehensive book on modern practical arithmetic in the western world. While few of us ever use geometry, people all over the world make daily use of the methods of arithmetic that Leonardo described in Liber abbaci.

In contrast to the widespread availability of the original Euclid’s Elements, the only version of Leonardo’s Liber abbaci we can read today is a second edition he completed in 1228, not his original 1202 text. Moreover, there is just one translation from the original Latin, in English, published as recently as 2002.

But for all its rarity, Liber abbaci is an impressive work. Although its great fame rests on its treatment of Hindu-Arabic arithmetic, it is a mathematically solid book that covers not just arithmetic, but the beginnings of algebra and some applied mathematics, all firmly based on the theoretical foundations of Euclid’s mathematics.

After completing the first edition of Liber abbaci, Leonardo wrote several other mathematics books, his writing making him something of a celebrity throughout Italy—on one occasion he was summonsed to an audience with the Emperor Frederick II. Yet very little was written about his life.

In 2001, I decided to embark on a quest to try to collect together what little was known about him and bring his story to a wider audience. My motivation? I saw in Leonardo someone who, like me, devoted a lot of time and effort trying to make the mathematics of the day accessible to the world at large. (Known today as “mathematical outreach,” very few mathematicians engage in that activity.) He was the giant whose footsteps I had been following.

I was not at all sure I could succeed. Over the years, I had built up a good reputation as an expositor of mathematics, but a book on Leonardo would be something new. I would have to become something of an archival scholar, trying to make sense of Thirteenth Century Latin manuscripts. I was definitely stepping outside my comfort zone.

The dearth of hard information about Leonardo in the historical record meant that a traditional biography was impossible—which is probably why no medieval historian had written one. To tell my story, I would have to rely heavily on the mathematical thread that connects today’s world to that of Leonardo—an approach unique to mathematics, made possible by the timeless nature of the discipline. Even so, it would be a stretch.

In the end, I got lucky. Very lucky. And not just once, but several times. As a result of all that good fortune, when my historical account The Man of Numbers: Fibonacci’s Arithmetic Revolution was published in 2011, I was able to compensate for the unavoidable paucity of information about Leonardo’s life with the first-ever account of the seminal discovery showing that my medieval role-model expositor had indeed played the pivotal role in creating the modern world that most historians had hypothesized.

With my Leonardo project such a new and unfamiliar genre, I decided from the start to keep a diary of my progress. Not just my findings, but also my experiences, the project's highs and lows, the false starts and disappointments, the tragedies and unexpected turns, the immense thrill of holding in my hands seminal manuscripts written in the thirteenth and fourteenth centuries, and one or two truly hilarious episodes. I also encountered, and made diary entries capturing my interactions with, a number of remarkable individuals who, each for their own reasons, had become fascinated by Fibonacci—the Yale professor who traced modern finance back to Fibonacci, the Italian historian who made the crucial archival discovery that brought together all the threads of Fibonacci's astonishing story, and the remarkable widow of the man who died shortly after completing the world’s first, and only, modern language translation of Liber abbaci, who went to heroic lengths to rescue his manuscript and see it safely into print.

After I had finished the Man of Numbers, I decided that one day I would take my diary and turn it into a book, telling the story of that small group of people (myself included) who had turned an interest in Leonardo into a passion, and worked long and hard to ensure that Leonardo Fibonacci of Pisa will forever be regarded as among the very greatest people to have ever lived. Just as The Man of Numbers was an account of the writing of Liber abbaci, so too Finding Fibonacci is an account of the writing of The Man of Numbers. [So it is a book about a book about a book. As Andrew Wiles once famously said, “I’ll stop there.”]

This post is adapted from the introduction of Keith Devlin’s new book Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World, published this month by Princeton University Press.

Friday, January 6, 2017

So THAT’s what it means? Visualizing the Riemann Hypothesis

Two years ago, there was a sudden, viral spike in online discussion of the Ramanujan identity

1 + 2 + 3 + 4 + 5 + . . . = –1/12

This identity had been lying around in the mathematical literature since the famous Indian mathematician Srinivasa Ramanujan included it in one of his books in the early Twentieth Century, a curiosity to be tossed out to undergraduate mathematics students in their first course on complex analysis (which was my first exposure to it), and apparently a result that physicists made actual (and reliable) use of.

The sudden explosion of interest was the result of a video posted online by Australian video journalist Brady Haran on his excellent Numberphile YouTube channel. In it, British mathematician and mathematical outreach activist James Grime moderates as his physicist countrymen Tony Padilla and Ed Copeland of the University of Nottingham explain their “physicists’ proof” of the identity.

In the video, Padilla and Copeland manipulate infinite series with the gay abandon physicists are wont to do (their intuitions about physics tends to keep them out of trouble), eventually coming up with the sum of the natural numbers on the left of the equality sign and –1/12 on the right.

Euler was good at doing that kind of thing too, so mathematicians are hesitant to trash it, rather noting that it “lacks rigor” and warning that it would be dangerous in the hands of a lesser mortal than Euler.

In any event, when it went live on January 9, 2014, the video and the result (which to most people was new) exploded into the mathematically-curious public consciousness, rapidly garnering hundreds of thousands of hits. (It is currently approaching 5 million in total.) By February 3, interest was high enough for The New York Times to run a substantial story about the “result”, taking advantage of the presence in town of Berkeley mathematician Ed Frenkel, who was there to promote his new book Love and Math, to fill in the details.

Before long, mathematicians whose careers depended on the powerful mathematical technique known as analytic continuation were weighing in, castigating the two Nottingham academics for misleading the public with their symbolic sleight-of- hand, and trying to set the record straight. One of the best of those corrective attempts was another Numberphile video, published on March 18, 2014, in which Frenkel give a superb summary of what is really going on.

A year after the initial flair-up, on January 11, 2015, Haran published a blogpost summarizing the entire episode, with hyperlinks to the main posts. It was quite a story.

[[ASIDE: The next few paragraphs may become a bit too much for casual readers, but my discussion culminates with a link to a really cool video, so keep going. Of course, you could just jump straight to the video, now you know it’s coming, but without some preparation, you will soon get lost in that as well! The video is my reason for writing this essay.]]

For readers unfamiliar with the mathematical background to what does, on the face of it, seem like a completely nonsensical result, which is the MAA audience I am aiming this essay at (principally, undergraduate readers and those not steeped in university-level math), it should be said that, as expressed, Ramanujan’s identity is nonsense. But not because of the -1/12 on the right of the equals sign. Rather, the issue lies in those three dots on the left. Not even a mathematician can add up infinitely many numbers.

What you can do is, under certain circumstances, assign a meaning to an expression such as

X1 + X2 + X3 + X4 + …

where the XN are numbers and the dots indicate that the pattern continues for ever. Such expressions are called infinite series.

For instance, undergraduate mathematics students (and many high school students) learn that, provided X is a real number whose absolute value is less than 1, the infinite series

1 + X + X+ X3 + X+ …

can be assigned the value 1/(1 – X). Yes, I meant to write “can be assigned”. Since the rules of real arithmetic do not extend to the vague notion of an “infinite sum”, this has to be defined. Since we are into the realm of definition here, in a sense you can define it to be whatever you want. But if you want the result to be meaningful and useful (useful in, say, engineering or physics, to say nothing of the rest of mathematics), you had better define it in a way that is consistent with that “rest of mathematics.” In this case, you have only one option for your definition. A simple mathematical argument (but not the one you can find all over the web that involves multiplying the terms in the series by X, shifting along, and subtracting—the rigorous argument is a bit more complicated than that, and a whole lot deeper conceptually) shows that the value has to be 1/(1 – X).

So now we have the identity

(*) 1 + X +X+ X3 + X+ … = 1/(1 – X)

which is valid (by definition) whenever X has absolute value less than 1. (That absolute value requirement comes in because of that “bit more complicated” aspect of the rigorous argument to derive the identity that I just mentioned.)

“What happens if you put in a value of X that does not have absolute value less than 1?” you might ask. Clearly, you cannot put X = 1, since then the right-hand side becomes 1/0, which is totally and absolutely forbidden (except when it isn’t, which happens a lot in physics). But apart from that one case, it is a fair question. For instance, if you put X = 2, the identity (*) becomes

1 + 2 + 4 + 8 + 16 + … = 1/(1 – 2) = 1/(–1) = –1

So you could, if you wanted, make the identity (*) the definition for what the infinite sum

1 + X + X+ X3 + X4 + …

means for any X other than X = 1. Your definition would be consistent with the value you get whenever you use the rigorous argument to compute the value of the infinite series for any X with absolute value less than 1, but would have the “benefit” of being defined for all values of X apart from one, let us call it a “pole”, at X = 1.

This is the idea of analytic continuation, the concept that lies behind Ramanujan’s identity. But to get that concept, you need to go from the real numbers to the complex numbers.

In particular, there is a fundamental theorem about differentiable functions (the accurate term in this context is analytic functions) of a single complex variable that says that if any such function has value zero everywhere on a nonempty disk in the complex plane, no matter how small the diameter of that disk, then the function is zero everywhere. In other words, there can be no smooth “hills” sitting in the middle of flat plains, or even one small flat clearing in the middle of a “hilly” landscape—the quotes are because we are beyond simple visualization here.

An immediate consequence of this theorem is that if you pull the same continuation stunt as I just did for the series of integer powers, where I extended the valid formula (*) for the sum when X is in the open unit interval to the entire real line apart from one pole at 1, but this time do it for analytic functions of a complex variable, then if you get an answer at all (i.e., a formula), it will be unique. (Well, no, the formula you get need not be unique, rather the function it describes will be.)

In other words, if you can find a formula that describes how to compute the values of a certain expression for a disk of complex numbers (the equivalent of an interval of the real line), and if you can find another formula that works for all complex numbers and agrees with your original formula on that disk, then your new formula tells you the right way to calculate your function for any complex number. All this subject to the requirement that the functions have to be analytic. Hence the term “analytic continuation.'

For a bit more detail on this, check out the Wikipedia explanation or the one on Wolfram Mathworld. If you find those explanations are beyond you right now, just remember that this is not magic and it is not a mystery. It is mathematics. The thing you need to bear in mind is that the complex numbers are very, very regular. Their two-dimensional structure ties everything down as far as analytic functions are concerned. This is why results about the integers such as Fermat’s Last Theorem are frequently solved using methods of Analytic Number Theory, which views the integers as just special kinds of complex numbers, and makes use of the techniques of complex analysis.

Now we are coming to that video. When I was a student, way, way back in the 1960s, my knowledge of analytic continuation followed the general path I just outlined. I was able to follow all the technical steps, and I convinced myself the results were true. But I never was able to visualize, in any remotely useful sense, what was going on.

In particular, when our class came to study the (famous) Riemann zeta function, which begins with the following definition for real numbers S bigger than 1:

(**) Zeta(S) = 1 + 1/2S + 1/3S + 1/4S + 1/5S + …

I had no reliable mental image to help me understand what was going on. For integers S greater than 1, I knew what the series meant, I knew that it summed (converged) to a finite answer, and I could follow the computation of some answers, such as Euler’s

Zeta(2) = π2/6

(You get another expression involving π for S = 4, namely π4/90.)

It turns out that the above definition (**) will give you an analytic function if you plug in any complex number for S for which the real part is bigger than 1. That means you have an analytic function that is rigorously defined everywhere on the complex plane to the right of the line x = 1.

By some deft manipulation of formulas, it’s possible to come up with an analytic continuation of the function defined above to one defined for all complex numbers except for a pole at S = 1. By that basic fact I mentioned above, that continuation is unique. Any value it gives you can be taken as the right answer.

In particular, if you plug in S = –1, you get

Zeta(–1) = –1/12

That equation is totally rigorous, meaningful, and accurate.

Now comes the tempting, but wrong, part that is not rigorous. If you plug in S = –1 in the original infinite series, you get

1 + 1/2-1 + 1/3-1 + 1/4-1 + 1/5-1 + …

which is just

1 + 2 + 3 + 4 + 5 + …

and it seems you have shown that

1 + 2 + 3 + 4 + 5 + . . . = –1/12

The point is, though, you can’t plug S = –1 into that infinite series formula (**). That formula is not valid (i.e., it has no meaning) unless S > 1.

So the only way to interpret Ramanujan’s identity is to say that there is a unique analytic function, Zeta(S), defined on the complex plane (apart from at the real number 1), which for all real numbers S greater than 1 has the same values as the infinite series (**), which for S = –1 gives the value Zeta(–1) = –1/12.

Or, to put it another way, more fanciful but less accurate, if the sum of all the natural numbers were to suddenly find it had a finite answer, that answer could only be –1/12.

As I said, when I learned all this stuff, I had no good mental images. But now, thanks to modern technology, and the creative talent of a young (recent) Stanford mathematics graduate called Grant Sanderson, I can finally see what for most of my career has been opaque. On December 9, he uploaded this video onto YouTube.



It is one of the most remarkable mathematics videos I have ever seen. Had it been available in the 1960s, my undergraduate experience in my complex analysis class would have been so much richer for it. Not easier, of that I am certain. But things that seemed so mysterious to me would have been far clearer. Not least, I would not have been so frustrated at being unable to understand how Riemann, based on hardly any numerical data, was able to formulate his famous hypothesis, finding a proof of which is agreed by most professional mathematicians to be the most important unsolved problem in the field.

When you see (in the video) what looks awfully like a gravitational field, pulling the zeros of the Zeta function towards the line x = 1/2, and you know that it is the only such gravitational field there is, and recognize its symmetry, you have to conclude that the universe could not tolerate anything other than all the zeros being on that line.

Having said that, it would, however, be really interesting if that turned out not to be the case. Nothing is certain in mathematics until we have a rigorous proof.

Meanwhile, do check out some of Grant’s other videos. There are some real gems!