COVID-19 and God’s List: assessing personal risk from the novel coronavirus

How dangerous is the novel coronavirus, and how does that risk compare with other risks we face in our lives? A bit of number sense and a few simple calculations allow us to visualize, and hence compare, the risks.

Imagine God has a book with the names of all people living in America today. Against each name, he has entered the cause of their death. (He’s God, so he has the death entries now.) He won’t let you see your entry. You do, however, have a chance to look at the book. Not close enough to read the individual entries, but close enough to discern the separate entries as such. You ask him to re-order the list grouped by the listed causes of death. (It’s a digital book.)

The biggest group is labeled “heart disease”. Roughly 1 in 6 of the names on the list are in that group. As a decimal, approximately 0.167. [That’s a figure quoted regularly in the medical profession. For example, this article. The exact values are not important to get an overall picture of the risks.]

This year, God had to add a new category: COVID-19. Right now, you are worried about it. You should be, and should do everything you can to avoid it. But in the overall scheme of things, how big a risk is it? Let’s run a few numbers.

Well, let’s suppose the infection reaches 60% of the population. That is the point where the growth ceases to be exponential (because there are not enough non-immune hosts left to fuel such rapid growth). The penetration could go higher, but at a much slower rate, which means medical services are not overwhelmed and more sick people can be saved.

The best current estimate we have for the mortality rate is 1%. (Congressional testimony of Dr. Anthony Fauci, director of the U.S. National Institute of Allergy and Infectious Diseases.)

So, in a population of 327M, the proportion who will be infected and subsequently die is 327 x 0.6 x 0.01M, which works out to 1.96M. As a proportion of the entire population, that is 0.006.  Six in every thousand, or 0.6%. Putting it another way, 994 of every 1,000 people in the United States today will survive.

That sounds encouraging, and it is, but it’s a lot higher than the risks associated with most of the actions we take in daily life, such as getting on a plane, driving across town, or crossing the road. Moreover, in particularly severe cases, the illness can be very unpleasant, even if you survive. All in all, it’s definitely something you should try to avoid.

But how does the risk of COVID-19 death stack up against the other things you may die of? More specifically, where does COVID-19 come on God’s List? The answer is, pretty low down the (reordered) list. The group of names that have that label in the “cause of death” column is (roughly) just 1/28th the size of the heart disease group.  [.006/.167 = 1/28 approx.]

But be cautious. COVID-19 presents a different kind of risk than heart disease. The latter is something that creeps up over your lifetime, and, if you get medically examined regularly and you are found to have it, you usually have plenty of time to modify your activities and habits to slow it down or even stop it developing further.

With COVID-19, however, it is a risk that (we all hope) you will only face in 2020 (because by 2021 we hope to have an effective vaccine), and the length of time you have to change your behavior to save yourself is measured in a few weeks (if you are following the news from a reliable source), or maybe just days. On the other hand, the behavioral changes required are very simple, albeit involving a major disruption of your daily life. Thoroughly wash your hands regularly, apply hand-sanitizer after coming into contact with any person, pet, or object, avoid touching your face, and keep at least six feet from any other person.

Nevertheless, in terms of what God has entered against your name as the cause of your death, it’s highly unlikely to be COVID-19. It’s 28 times more likely to be “Heart failure”.

In other words, what the numbers tell us is that you are extremely unlikely to die from COVID-19, whatever you do. But, assuming you’d rather live than die (and would rather avoid a terrible illness even if you survive it), then if there were actions you could take to make your odds for not dying from this virus, you should take them.

As we just saw, the actions required are a nuisance, but you only have to do them until the virus simply goes away (not likely) or we find an effective vaccine (highly likely within a year, maybe a bit more). So COVID-19 presents a numerically small danger made acute by being entirely within a single year or a year-and-a-half. Avoidance is simple, and is required for a relatively short time, but it’s a nuisance.

In the case of heart disease, the same conclusions follow, but the time-scale is very different. In this case, it’s one of a small collection of illnesses (around ten) that you are highly likely to die from, but the threat lasts your entire life. Avoidance measures are also simple (principally diet, exercise, and not smoking), but need to be sustained.

The take-home message, then, is that during this year (and maybe into next), COVID-19 is a threat you should not treat lightly. But in terms of God’s List, this is most unlikely to be what he has entered against your name. (“Heart disease” is far more likely.)

So, given how heavily the overall odds are stacked in your favor, the incentive to take COVID-19 avoidance measures is huge, since they are almost certain to be effective. Besides, with a virus epidemic, avoidance measures any one of us takes benefit all of society, including our families, friends, and loved ones.

If you do come through the COVID-19 epidemic, then, on January 1, 2021, with the novel coronavirus threat behind you (or at least soon to be so), you will be in a position to make a New Year Resolution to make sure your diet and exercise habits can fend off heart disease. That’s a deadly threat that continues throughout your life. And the odds that this killer is the one that gets you are a lot higher than for that virus you just dealt with. Why get through one threat and then squander your chances to avoid a much greater one?

NOTE: This article has a good description of the various ways COVID-19 can spread and how we can prevent them.

The math behind Apollo 11


Buzz Aldrin on the lunar surface, July 20, 1969, photographed by Neil Armstrong

It’s been over three years since I last posted to this blog. Not that I stopped writing. In addition to my long running monthly Devlin’s Angle for the Mathematical Association of America, early this year I launched the Stanford Mathematics Outreach Project (, which has its own blog. The former is aimed, fairly broadly, at high school on college level mathematics instructors and, to some extent, their students. The latter focuses on K-12 mathematics teachers, and was designed to complement, and collaborate with, Stanford University Mathematics Education Professor Jo Boaler’s youcubed project.

Having retired from full-time employment at Stanford at the end of last year, however, I find I have time to reflect on other mathematics related issues that don’t entirely fit on either of those blogs. Time then, to revive this outlet. To be sure, it was frequently a toss-up whether to post on or Devlin’s Angle, and the same has already occurred with the SUMOP blog. What tipped the scales in favor of reviving this blog was the upcoming Fiftieth Anniversary of the Apollo 11 moon landing on July 20, 1969.

On June 23, CNN broadcast a new documentary of the Apollo mission, using video footage never previously seen. You can watch the whole thing (just under two hours), for a small charge, on YouTube or Amazon Prime. For reasons that will momentarily become clear, I had to watch it at the first opportunity. So I joined the many others (mostly around my age, I suspect) to view the first screening. Soon after the movie started, I decided to live-tweet my reactions. The remainder of this post is a transcription of that twitter stream, with the inevitable twitter-typos cleaned up. [For the record, the original twitter stream can be found here.]


Watching the CNN documentary about Apollo 11. My context. At the time, in July 1969 I was a 22 year-old, English speaking, white male with a new math degree. Apollo was an incredible achievement for the group I was a de facto member of 1/

I was fluent in all of NASA’s languages: English, mathematics, white male machismo. By 2001, when I wrote my book The Math Gene, my understanding of math had increased to appreciate its role in the creation and management of human teams  2/

Here is the PROLOGUE of The Math Gene 3/

In fact, as I have said often, I became a mathematician because I wanted to be part of the space era. See for example my recent video interview 4/

Had I been born just a few years earlier, to American parents rather than English, I would have done all I could to be one of those math and engineering geeks at Johnson Space Flight Center whose names were at the time as familiar to me as were those of my family 5/

Watching that video footage today, now as a holder of a US Passport and with several years of service as a mathematician to three organizations in the US Department of Defense, my reactions are complex. 6/

The thrill is still there. I would have loved to have been part of it, not merely an avid follower from the other side of the Atlantic. After I came to Stanford in 1987, with NASA Ames Research Center just down the road, I met a few of those who were involved. 7/

Moreover, some of my research found applications at NASA in the later missions of the International Space Station. But, though they involved way more complex engineering, they lacked the sheer drama of Apollo 11, … /8

… a mission that was conducted entirely by following a carefully worked out mathematical plan, in an environment where any one of millions of possible small miscalculations would bring the mission to a tragic end 9/

But here in 2019, I watched the video with other emotions. One was the degree to which Apollo 11 was, at least in its public presentation, almost exclusively a white males operation. I saw just one woman and one black male in one of the control rooms. 10/

Of course, as everyone knows thanks in large part to the movie Hidden Figures, NASA’s reality was far from that, with African American women mathematicians in particular playing a crucial role from the very early missions on 11/

That American society has moved on since the 1960s (at least in part) was brought home vividly by the dramatic contrast between the all-white-males video footage and the commercial breaks, with their carefully selected representatives of modern US society 12/

But America has also moved backwards. Instead of taking inspiration from those in the Apollo era, and seeking to do more great things, but this time in a far more inclusive way, we have an entire major party whose goal is to MAGA by going back to the 60s social divisions 13/

To do this, they try to kill anything that depends on more than 1960s scientific knowledge, and insist we do things the way they were back then.  But that is not what Apollo was about!!! 14/

Apollo was about taking a bold step into the unknown. To join together and move forward into the future – “to boldly go …”, as the best known fiction of space exploration famously proclaims 15/

It doesn’t have to be space. It could be anything we choose. To those of us who felt “in the club” (i.e., white , male, math-geeky, and fluent in English), Apollo was a thrilling life experience. How about creating another one, but this time one that includes all Americans? 16/

In education? Health care? Curing famine? Addressing climate change? There’s no shortage of big goals. Indeed, those four tower above a Moon landing in terms of importance to society. All it takes is the will. Apollo happened only because President Kennedy expressed the will 17/END

The beginnings of Education Science?

[This is a greatly expanded version of the essay published in Edge as my response to the Annual Edge Question for 2016.]

If you google the term “education science”, the search engine will return almost 2 billion hits. Yet even those of us who might be regarded as “in the field” will admit that there is no “there” there—nothing that could legitimately be called a science.

The education field is much like medicine was in the Nineteenth Century, a human practice, guided by intuition, experience, and occasionally inspiration. It took the development of modern biology and biochemistry in the early part of the Twentieth Century to provide the solid underpinnings of today’s science of medicine.

To be sure, since the second half of Twentieth Century, a great deal of work has been done on pedagogy, and much has been learned—though depressingly little has found its way into the classroom. But even the most ardent participant in that work would admit that the field could not be called a science, alongside, say, chemistry, physics, or medical science.

But that may be about to change. To me—a mathematician who became interested in mathematics education in the second half of my career—it seems that we may at last be seeing the emergence of a genuine science of learning. Given the huge significance of education in human society, if that is the case, then this will represent a major advance for Humanity.

At the risk—nay certainty—of raising the ire of many researchers, I should start out by observing that I am not basing my assessment on the rapid growth in popularity of educational neuroscience. You know, the kind of study where a subject is slid into an fMRI machine and asked to solve math puzzles. Those studies are valuable, and are undoubtedly science, but at the present stage, at best they provide some very tentative clues about how people learn, and little specific in terms of how to help people learn. (A good analogy would be trying to diagnose an engine fault in a car by moving a thermometer over the hood.)

Yes, I follow educational neuroscience research (mostly at Internet distance), and am often in admiration of the ingenuity of the researchers, but I don’t see it as even close to providing a solid basis for education the way, say, the modern theory of genetics advanced medical practice. One day? Maybe. But not yet.

Rather, I am encouraged in thinking we are seeing the emergence of a science of learning by the possibilities Internet technology brings to the familiar, experimental cognitive science approach.

The problem that has traditionally beset learning research has been its huge dependence on the individual teacher, which makes it near impossible to run the kinds of large scale, control group, intervention studies that are par-for-the-course in medicine. (No, I am not about to argue that computers will replace teachers! On the contrary, I am firmly of the opinion that teaching is inherently and inescapably a human–human endeavor.)

The problem raised by that inescapable centrality of the human teacher is that classroom studies invariably end up as studies of the teacher as much as of the students.

In fact, it is even worse. What those studies frequently measure is as much, if not more, the effect of the home environment of the students than what goes on in the classroom.

For instance, news articles often cite the large number of successful people who as children attended a Montessori school, a figure hugely disproportionate to the relatively small number of such schools. Now, it may well be the case (I think it is) that the Montessori educational principles are good, but it’s also the case that such schools are magnets for passionate, dedicated teachers and the pupils that attend them do so because they have parents who go out of their way to enroll their offspring in such a school, and already raise their children in a learning-rich home environment.

Internet technology offers an opportunity to carry out medical-research-like, large scale control group studies of classroom learning that can significantly mitigate the “teacher effect” and “home effect”, allowing useful studies of different educational techniques to be carried out. Provided you collect the right data, Big Data techniques can detect patterns that cut across the wide range of teacher–teacher and family–family variation, allowing useful educational conclusions to be drawn.

An important factor is that a sufficiently significant part of the actual learning is done in a digital environment, where every action can be captured.

This is not easily achieved. The vast majority of educational software products operate around the edges of learning: providing the learner with information; asking questions and capturing their answers (in a machine-actionable, multiple-choice format); and handling course logistics with a learning management system.

What is missing is any insight into what is actually going on in the student’s mind—something that can be very different from what the evidence shows, as was dramatically illustrated for mathematics learning several decades ago by a study now famously referred to as “Benny’s Rules”, where a child who had aced a whole progressive battery of programmed learning cycles was found (by a lengthy, human–human working session) to have constructed an elaborate internal, rule-based “mathematics” that enabled him to pass all the tests with flying honors, but was extremely brittle and bore no relation to actual mathematics.

But real-time, interactive software allows for much more than we have seen flooding out of tech hotbeds such as Silicon Valley.

To date, the more effective uses of interactive technology from the viewpoint of running large-scale, comparative learning studies, have been by way of learning video games—so-called game-based learning. But it remains an open question how significant is the game element in terms of learning outcomes. My bet, based on a few small studies, is on it being an important factor, but this is a question that can and will be answered by solid, scientific studies.

And there have been such studies. But it would be easy to have missed them.

With large numbers of technology companies, (as well as book publishers and occasionally media moguls) vying for education customers—a trend driven almost exclusively by expertise looking for new markets (hammers seeking new nails)—and making often wildly extravagant claims in the process, the small number of studies in the past few years that have shown some remarkable results have not garnered much media attention.

That is not entirely the fault of the news media. With results still largely tentative, and for the most part not fully explained, those involved in those studies (I am one of them) have been reluctant to go out on a limb with bold statements.

That reluctance is heightened by the depressing reality that the vast majority of so-called “learning games” are of a very poor quality and offer little or no real learning, relying instead on bold marketing claims. The last thing serious researchers want to do is find their claims dismissed as yet more vacuous hype.

To cut to the chase, in the case of elementary through middle school mathematics learning (which is the research I am familiar with), what has been discovered, by a number of teams, is that digital learning interventions of as little as ten minutes a day, for three to five days a week, over a course of as little as one month, can result in significant learning gains when measured by a standardized test—with improvements of as much as 16% in some key thinking skills. (I list some sources at the end.)

That may sound like an educational magic pill. It almost certainly is not. I believe it’s an early sign that we know even less about learning than we thought we did.

For one thing, part of what is going on is that many earlier studies measured the wrong things—knowledge rather than thinking ability. The learning gains found in the studies I am referring to are not in the area of knowledge acquired or algorithmic procedures mastered, rather in high-level problem solving ability. (So the standardized tests used cannot be multiple-choice; they require human grading or game-based assessment techniques.)

What is exciting about these findings, is that in today’s information- and computation- rich environment, those very human problem-solving skills are the ones now at a premium.

Like any good science, and in particular any new science, this work has generated far more research questions than it has answered.

Indeed, it is too early to say it has answered any questions. Most of us embarked on the studies with the more modest ambitions of developing new learning tools, having no expectation of finding such dramatic outcomes.

In the case of math learning, among the many factors that may be at play, all of which a (well designed) math learning game can offer, and all which are known to have a positive impact on learning, are:

  • Use of a 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 the factors listed above in a series of video-game learning articles in this blog. (Starts here.) Taking a broader perspective than mathematics, the many writings and video interviews on games and learning by James Paul Gee have much to say that can help us understand how those factors and others can effect learning.]

So, as of now we what we have are a scientifically sound method to conduct experiments at scale, some very suggestive early results, and a resulting long and growing list of research questions, all of which are testable. The sure looks to me like the beginnings of a genuine science of learning.

Selected sources
Berkowitz, Schaeffer, Maloney, Peterson, Gregor, Levine, Beilock, 2015. Math at home adds up to achievement in school, Science  09 Oct 2015: Vol. 350, Issue 6257, 196-198.
Kiili, Devlin, Perttula, Tuomi, 2015. Using video games to combine learning and assessment in mathematics education, International Journal of Serious Games, October 2015, 37-55.
Pope & Mangram, 2015. Wuzzit Trouble:  The Influence of a Digital Math Game on Student Number Sense, International Journal of Serious Games, October 2015, 5-22.
Popovic, 2014. “Learning basic Algebra by playing 1.5h”. Center for Game Science, Uni of Washington.
Riconscente, 2013. Results From a Controlled Study of the iPad Fractions Game Motion Math,  Games and Culture 8(4), 186-214.
Wendt & Rice, 2013. “Evaluation of ST Math in the Los Angeles Unified School District”, report by WestEd.

What is a proof, really?

What is a mathematical proof? Way back when I was a college freshman, I could give you a precise answer: A proof of a statement S is a finite sequence of assertions S(1), S(2), … S(n) such that S(n) = S and each S(i) is either an axiom or else follows from one or more of the preceding statements S(1), …,S(i-1) by a direct application of a valid rule of inference.

But I was so much older then, I’m younger than that now.

After a lifetime in professional mathematics, during which I have read a lot of proofs, created some of my own, assisted others in creating theirs, and reviewed a fair number for research journals, the one thing I am sure of is that the definition of proof you will find in a book on mathematical logic or see on the board in a college level introductory pure mathematics class doesn’t come close to the reality.

For sure, I have never in my life seen a proof that truly fits the standard definition. Nor has anyone else.

The usual maneuver by which mathematicians leverage that formal notion to capture the arguments they, and all their colleagues, regard as proofs is to say a proof is a finite sequence of assertions that could be filled in to become one of those formal structures.

It’s not a bad approach if the goal is to give someone a general idea of what a proof is. The trouble is, no one has ever carried out that filling-in process. It’s purely hypothetical. How then can anyone know that the purported proof in front of them really is a proof?

I wrote about this dilemma in my MAA “Devlin’s Angle” column way back in 1996, in an article titled Moment of Truth.

I picked up the theme again in 2003 with my Devlin’s Angle piece When is a Proof?

These days I have a very pragmatic perspective on what a proof is, based on the way people use them in the day-to-day world of mathematics:

Proofs are stories that convince suitably qualified others that a certain statement is true.

If I present you with a proof, and you have the appropriate background knowledge and ability, you can – usually with some time and effort – as a result of reading my story, become convinced that what I claim is true.

But if you take that as your working definition of proof, you have to acknowledge it is fundamentally about communication, not truth. In particular, whether an argument classifies as a proof depends as much on the intended reader as on its creator.

Of course, in order to function in that way, the “story” has to be pretty heavily constrained.

Moreover, the creators and the consumers of those stories have to be familiar with the genre. That part takes time to acquire.

On the other hand, once a person becomes familiar with both the genre and the particular mathematical focus, reading and understanding those stories becomes natural and fluent.

The system works – as any professional mathematician will affirm. It’s how mathematics advances.

To an outsider, however, the whole thing is usually incomprehensible.

Today, many proofs stretch over several pages, not infrequently hundreds of pages. A key feature that allows such proofs to function effectively in the mathematical community is that many steps are left out.

In some cases this is because the step has already been established, either by the same author in a previous piece of work, or by someone else. In such cases, the author simply refers the reader to that source.

In other cases, the author judges that the intended reader should be capable of supplying the missing steps on the fly. The author may provide a hint to help the reader provide the missing steps, but not always.

There is, then, a huge element of audience design in constructing effective proofs. A proof designed for an undergraduate mathematics class is in general very different from one constructed to present at a research seminar.

To the beginner, trying to make the transition from high school mathematics to university level, coming to terms with real proofs is not only difficult, it can be traumatic, with a once comforting illusion of crisp, clean certainty rapidly giving way to a panicked feeling of sinking into shifting quicksand.

At this point, it can be of some comfort to learn that Euclid screwed up big-time when he penned his famous geometry proofs in Elements. Yes, those iconic proofs may seem logically sound, and indeed for two thousand years were held up as models of logically sound reasoning. But as David Hilbert observed in the late Nineteenth Century, Euclid’s arguments are riddled with logical holes.

To give just one example, he often tells you to construct a point by intersecting an arc of a circle with a straight line. But how do you know there is an intersection? Sure, when you draw the arc and the line on a sheet of paper, the arc may cross over the line. But do they actually intersect? That is, do they have an actual (dimensionless) point in common?

That is not only not obvious, it takes a lot of work to answer. (The answer is, it depends on the underlying number system. But it requires some deep machinery not developed until the Nineteenth Century.)

Of course, high school teachers rarely, if ever, tell their students that the geometry proofs they are presented as models are at best sketches of how proofs can be constructed. As a result, those students typically enter university with a totally false impression of what a proof is. In particular, they believe proofs are fundamentally and exclusively about truth, and that they are either right or wrong.

In reality, proofs are about truth, but not fundamentally, and definitely not exclusively. The key property of a proof is not that it is logically correct (it almost certainly is not, but more pertinent, how could you ever be sure it is?), rather that it is expressed in a manner that enables a suitably qualified reader to fill in any holes they notice, to check any steps they have any doubt about, and to correct any errors they find (as they surely will if they dig deep enough).

It’s very much like software engineering, where the most important thing about a program is not that it is bug free (it almost certainly is not), rather that – in addition to working – it is structured and annotated so that someone else can come along later and either fix bugs or else modify the code to do something else.

Ridding high school graduates of the “proofs are about logical correctness” misconception is generally a difficult (for both instructor and student) and painful (for the student) process. Just what it entails has been a focus of a study I have been making in my MOOC Introduction to Mathematical Thinking, currently being offered for the fifth time. I describe my most recent observations in a new post on my other blog,,

where this account continues…

How mountain biking can provide the key to the Eureka moment

Because this blogpost covers both mountain biking and proving theorems, it is being simultaneously published by the Mathematical Association of America in my Devlin’s Angle series. 

In my post last month, I described my efforts to ride a particularly difficult stretch of a local mountain bike trail in the hills just west of Palo Alto. As promised, I will now draw a number of conclusions for solving difficult mathematical problems.

Most of them will be familiar to anyone who has read George Polya’s classic book How to Solve It. But my main conclusion may come as a surprise unless you have watched movies such as Top Gun or Field of Dreams, or if you follow professional sports at the Olympic level.

Here goes, step-by-step, or rather pedal-stroke-by-pedal-stroke. (I am assuming you have recently read my last post.)

BIKE: Though bikers with extremely strong leg muscles can make the Alpine Road ByPass Trail ascent by brute force, I can’t. So my first step, spread over several rides, was to break the main problem – get up an insanely steep, root strewn, loose-dirt climb – into smaller, simpler problems, and solve those one at a time.

MATH: Breaking a large problem into a series of smaller ones is a technique all mathematicians learn early in their careers. Those subproblems may still be hard and require considerable effort and several attempts, but in many cases you find you can make progress on at least some of them. The trick is to make each subproblem sufficiently small that it requires just one idea or one technique to solve it.

In particular, when you break the overall problem down sufficiently, you usually find that each smaller subproblem resembles another problem you, or someone else, has already solved.

When you have managed to solve the subproblems, you are left with the task of assembling all those subproblem solutions into a single whole. This is frequently not easy, and in many cases turns out to be a much harder challenge in its own right than any of the subproblem solutions, perhaps requiring modification to the subproblems or to the method you used to solve them.

BIKE: Sometimes there are several different lines you can follow to overcome a particular obstacle, starting and ending at the same positions but requiring different combinations of skills, strengths, and agility. (See my description last month of how I managed to negotiate the steepest section and avoid being thrown off course – or off the bike – by that troublesome tree-root nipple.)

MATH: Each subproblem takes you from a particular starting point to a particular end-point, but there may be several different approaches to accomplish that subtask. In many cases, other mathematicians have solved similar problems and you can copy their approach.

BIKE: Sometimes, the approach you adopt to get you past one obstacle leaves you unable to negotiate the next, and you have to find a different way to handle the first one.

MATH: Ditto.

BIKE: Eventually, perhaps after many attempts, you figure out how to negotiate each individual segment of the climb. Getting to this stage is, I think, a bit harder in mountain biking than in math. With a math problem, you usually can work on each subproblem one at a time, in any order. In mountain biking, because of the need to maintain forward (i.e., upward) momentum, you have to build your overall solution up in a cumulative fashion – vertically!

But the distinction is not as great as might first appear. In both cases, the step from having solved each individual subproblem in isolation to finding a solution for the overall problem, is a mysterious one that perhaps cannot be appreciated by someone who has not experienced it. This is where things get interesting.

Having had the experience of solving difficult (for me) problems in both mathematics and mountain biking, I see considerable similarities between the two. In both cases, the subconscious mind plays a major role – which is, I presume, why they seem mysterious. This is where this two-part blogpost is heading.

BIKE: I ended my previous post by promising to

“look at the role of the subconscious in being able to put together a series of mastered steps in order to solve a big problem. For a very curious thing happened after I took the photos to illustrate this post. I walked back down to collect my bike from … where I’d left it, and rode up to continue my ride.

It took me four attempts to complete that initial climb!

And therein lies one of the biggest secrets of being able to solve a difficult math problem.”

BOTH: How does the human mind make a breakthrough? How are we able to do something that we have not only never done before, but failed many times in attempts to do so? And why does the breakthrough always seem to occur when we are not consciously trying to solve the problem?

The first thing to note is that we never experience the process of making that breakthrough. Rather, what we experience, i.e., what we are conscious of, is having just made the breakthrough!

The sensation we have is a combined one of both elation and surprise. Followed almost immediately by a feeling that it wasn’t so difficult after all!

What are we to make of this strange process?

Clearly, I cannot provide a definitive, concrete answer to that question. No one can. It’s a mystery. But it is possible to make a number of relevant observations, together with some reasonable, informed speculations. (What follows is a continuation of sorts of the thread I developed in my 2000 book The Math Gene.)

The first observation is that the human brain is a result of millions of years of survival-driven, natural selection. That made it supremely efficient at (rapidly) solving problems that threaten survival. Most of that survival activity is handled by a small, walnut-shaped area of the brain called the amygdala, working in close conjunction with the body’s nervous system and motor control system.

In contrast to the speed at which our amydala operates, the much more recently developed neo-cortex that supports our conscious thought, our speech, and our “rational reasoning,” functions at what is comparatively glacial speed, following well developed channels of mental activity – channels that can be built up by repetitive training.

Because we have conscious access to our neo-cortical thought processes, we tend to regard them as “logical,” often dismissing the actions of the amygdala as providing (“mere,” “animal-like”) “instinctive reactions.” But that misses the point that, because that “instinctive reaction organ” has evolved to ensure its owner’s survival in a highly complex and ever changing environment, it does in fact operate in an extremely logical fashion, honed by generations of natural selection pressure to be in synch with its owner’s environment.

Which leads me to this.

Do you want to identify that part of the brain that makes major scientific (and mountain biking) breakthroughs?

I nominate the amygdala ­– the “reptilean brain” as it is sometimes called to reflect its evolutionary origin.

I should acknowledge that I am not the first person to make this suggestion. Well, for mathematical breakthroughs, maybe I am. But in sports and the creative arts, it has long been recognized that the key to truly great performance is to essentially shut down the neo-cortex and let the subconscious activities of the amygdala take over.

Taking this as a working hypothesis for mathematical (or mountain biking) problem solving, we can readily see why those moments of great breakthrough come only after a long period of preparation, where we keep working away – in conscious fashion – at trying to solve the problem or perform the action, seemingly without making any progress.

We can see too why, when the breakthrough (or the great performance) comes, it does so instantly and surprisingly, when we are not actively trying to achieve the goal, leaving our conscious selves as mere after-the-fact observers of the outcome.

For what that long period of struggle does is build a cognitive environment in which our reptilean brain – living inside and being connected to all of that deliberate, conscious activity the whole time – can make the key connections required to put everything together. In other words, investing all of that time and effort in that initial struggle raises the internal, cognitive stakes to a level where the amygdala can do its stuff.

Okay, I’ve been playing fast and loose with the metaphors and the  anthropomorphization here. We’re really talking about biological systems, simply operating the way natural selection equipped them. But my goal is not to put together a scientific analysis, rather to try to figure out how to improve our ability to solve novel problems. My primary aim is not to be “right” (though knowledge and insight are always nice to have), but to be able to improve performance.

Let’s return to that tricky stretch of the ByPass section on the Alpine Road trail. What am I consciously focusing on when I make a successful ascent?

BIKE: If you have read my earlier account, you will know that the difficult section comes in three parts. What I do is this. As I approach each segment, I consciously think about, and fix my eyes on, the end-point of that segment – where I will be after I have negotiated the difficulties on the way. And I keep my eyes and attention focused on that goal-point until I reach it. For the whole of the maneuver, I have no conscious awareness of the actual ground I am cycling over, or of my bike. It’s total focus on where I want to end up, and nothing else.

So who – or what – is controlling the bike? The mathematical control problem involved in getting a person-on-a-bike up a steep, irregular, dirt trail is far greater than that required to auto-fly a jet fighter. The calculations and the speed with which they would have to be performed are orders of magnitude beyond the capability of the relatively slow neuronal firings in the neocortex. There is only one organ we know of that could perform this task. And that’s the amygdala, working in conjunction with the nervous system and the body’s motor control mechanism in a super-fast constant feedback loop. All the neo-cortex and its conscious thought has to do is avoid getting in the way!

These days, in the case of Alpine Road, now I have “solved” the problem, the only things my conscious neo-cortex has to do on each occasion are switching my focus from the goal of one segment to the goal of the next. If anything interferes with my attention at one of those key transition moments, my climb is over – and I stop or fall.

What used to be the hard parts are now “done for me” by unconscious  circuits in my brain.

MATH: In my case at least, what I just wrote about mountain biking accords perfectly with my experiences in making (personal) mathematical problem-solving breakthroughs.

It is by stepping back from trying to solve the problem by putting together everything I know and have learned in my attempts, and instead simply focusing on the problem itself – what it is I am trying to show – that I suddenly find that I have the solution.

It’s not that I arrive at the solution when I am not thinking about the problem. Some mathematicians have expressed their breakthrough moments that way, but I strongly suspect that is not totally true. When a mathematician has been trying to solve a problem for some months or years, that problem is always with them. It becomes part of their existence. There is not a single waking moment when that problem is not “on their mind.”

What they mean, I believe, and what I am sure is the case for me, is that the breakthrough comes when the problem is not the focus of our thoughts. We really are thinking about something else, often some mundane detail of life, or enjoying a marvelous view. (Google “Stephen Smale beaches of Rio” for a famous example.)

This thesis does, of course, explain why the process of walking up the ByPass Trail and taking photographs of all the tricky points made it impossible for me to complete the climb. True, I did succeed at the fourth attempt. But I am sure that was not because the first three were “practice.” Heavens, I’d long ago mastered the maneuvers required. It was because it took three failed attempts before I managed to erase the effects of focusing on the details to capture those images.

The same is true, I suggest, for solving a difficult mathematical problem. All of those techniques Polya describes in his book, some of which I list above, are essential to prepare the way for solving the problem. But the solution will come only when you forget about all those details, and just focus on the prize.

This may seem a wild suggestion, but in some respects it may not be entirely new. There is much in common between what I described above and the highly successful teaching method of R.L. Moore. For sure you have to do a fair amount of translation from his language to mine, but Moore used to demand that his students did not clutter their minds by learning stuff, rather took each problem as it came and then try to solve it by pure reasoning, not giving up until they found the solution.

In terms of training future mathematicians, what these considerations imply, of course, is that there is mileage to be had from adopting some of the techniques used by coaches and instructors to produce great performances in sports, in the arts, in the military, and in chess.

Sweating the small stuff will make you good. But if you want to be great, you have to go beyond that – you have to forget the small stuff and keep your eye on the prize.

And if you are successful, be sure to give full credit for that Fields Medal or that AMS Prize where it is rightly due: dedicate it to your amygdala. It will deserve it.

Want to learn how to prove a theorem? Go for a mountain bike ride

Because this blogpost covers both mountain biking and proving theorems, it is being simultaneously published by the Mathematical Association of America in my Devlin’s Angle series.

Mountain biking is big in the San Francisco Bay Area, where I live. (In its present day form, using specially built bicycles with suspension, the sport/pastime was invented a few miles north in Marin County in the late 1970s.) Though there are hundreds of trails in the open space preserves that spread over the hills to the west of Stanford, there are just a handful of access trails that allow you to start and finish your ride in Palo Alto. Of those, by far the most popular is Alpine Road.

My mountain biking buddies and I ascend Alpine Road roughly once a week in the mountain biking season (which in California is usually around nine or ten months long). In this post, I’ll describe my own long struggle, stretching over many months, to master one particularly difficult stretch of the climb, where many riders get off and walk their bikes.

[SPOILER: If your interest in mathematics is not matched by an obsession with bike riding, bear with me. My entire account is actually about how to set about solving a difficult math problem, particularly proving a theorem. I’ll draw the two threads together in a subsequent post, since it will take me into consideration of how the brain works when it does mathematics. For now, I’ll leave the drawing of those conclusions as an exercise for the reader! So when you read mountain biking, think math.]

Alpine Road used to take cars all the way from Palo Alto to Skyline Boulevard at the summit of the Coastal Range, but the upper part fell into disrepair in the late 1960s, and the two-and-a-half-mile stretch from just west of Portola Valley to where it meets the paved Page Mill Road just short of  Skyline  is now a dirt trail, much frequented by hikers and mountain bikers.

Alpine Road. The trail is washed out just round the bend

Alpine Road. The trail is washed out just round the bend

A few years ago, a storm washed out a short section of the trail about half a mile up, and the local authority constructed a bypass trail. About a quarter of a mile long, it is steep, narrow, twisted, and a constant staircase of tree roots protruding from the dirt floor. A brutal climb going up and a thrilling (beginners might say terrifying) descent on the way back. Mountain bike heaven.

There is one particularly tricky section right at the start. This is where you can develop the key abilities you need to be able to prove mathematical theorems.

So you have a choice. Read Polya’s classic book, or get a mountain bike and find your own version of the Alpine Road ByPass Trail. (Better still: do both!)

My mountain bike at the start of the bypass trail

My mountain bike at the start of the bypass trail

When I first encountered Alpine Road Dirt a few years ago, it took me many rides before I managed to get up the first short, steep section of the ByPass Trail.

What lies around that sharp left-hand turn?

What lies around that sharp left-hand turn?

It starts innocently enough – because you cannot see what awaits just around that sharp left-hand turn.

The short, narrow descent

The short, narrow descent

After you have made the turn, you are greeted with a short narrow downhill. You will need it to gain as much momentum as you can for what follows.

I’ve seen bikers with extremely strong leg muscles who can plod their way up the wall that comes next, but I can’t do it that way. I learned how to get up it by using my problem-solving/theorem-proving skills.

The first thing was to break the main problem – get up the insanely steep, root strewn, loose-dirt climb – into smaller, simpler problems, and solve those one at a time. Classic Polya.

But it’s Polya with a twist – and by “twist” I am not referring to the sharp triple-S bend in the climb. The twist in this case is that the penalty for failure is physical, not emotional as in mathematics. I fell off my bike a lot. The climb is insanely steep. So steep that unless you bend really low, with your chin almost touching your handlebar, your front wheel will lift off the ground. That gives rise to an unpleasant feeling of panic  that is perhaps not unlike the one that many students encounter when faced with having to prove a theorem for the first time.

Steep. If you are not careful, your front wheel will lift off the ground.

Steep. If you are not careful, your front wheel will lift off the ground.

The photo above shows the first difficult stretch. Though this first sub-problem is steep, there is a fairly clear line to follow to the right that misses those roots, though at the very least its steepness will slow you down, and on many occasions will result in an ungainly, rapid dismount. And losing momentum is the last thing you want, since the really hard part is further up ahead, near the top in the picture.

Also, do you see that rain- and tire-worn groove that curves round to the right just over half way up – just beyond that big root coming in from the left? It is actually deeper and narrower than it looks in the photo, so unless you stay right in the middle of the groove you will be thrown off line, and your ascent will be over. (Click on the photo to enlarge it and you should be able to make out what I mean about the groove. Staying in the groove can be tricky at times.)

Still, despite difficulties in the execution, eventually, with repeated practice, I got to the point of  being able to negotiate this initial stretch and still have some forward momentum. I could get up on muscle memory. What was once a series of challenging problems, each dependent on the previous ones, was now a single mastered skill.

[Remember, I don’t have super-strong leg muscles. I am primarily a road bike rider. I can ride for six hours at a 16-18 mph pace, covering up to 100 miles or more. But to climb a steep hill I have to get off the saddle and stand on the pedals, using my body weight, not leg power. Unfortunately, if you take your weight off the saddle on a mountain bike on a steep dirt climb, your rear wheel will start to spin and you come to a stop – which on a steep hill means jump off quick or fall. So I have to use a problem solving approach.]

Once I’d mastered the first sub-problem, I could address the next. This one was much harder. See that area at the top of the photo above where the trail curves right and then left? Here is what it looks like up close.

The crux of the climb/problem. Now it is really steep.

The crux of the climb/problem. Now it is really steep.

(Again, click on the photo to get a good look. This is the mountain bike equivalent of being asked to solve a complex math problem with many variables.)

Though the tire tracks might suggest following a line to the left, I suspect they are left by riders coming down. Coming out of that narrow, right-curving groove I pointed out earlier, it would take an extremely strong rider to follow the left-hand line. No one I know does it that way. An average rider (which I am) has to follow a zig-zag line that cuts down the slope a bit.

Like most riders I have seen – and for a while I did watch my more experienced buddies negotiate this slope to get some clues – I start this part of the climb by aiming my bike between the two roots, over at the right-hand side of the trail. (Bottom right of picture.)

The next question is, do you go left of that little tree root nipple, sticking up all on its own, or do you skirt it to the right? (If you enlarge the photo you will see that you most definitely do not want either wheel to hit it.)

The wear-marks in the dirt show that many riders make a sharp left after passing between those two roots at the start, and steer left of the nobbly root protrusion. That’s very tempting, as the slope is notably less (initially). I tried that at first, but with infrequent  success. Most often, my left-bearing momentum carried me into that obstacle course of tree roots over to the left, and though I sometimes managed to recover and swing  out to skirt to the left of that really big root, more often than not I was not able to swing back right and avoid running into that tree!

The underlying problem with that line was that thin looking root at the base of the tree. Even with the above photo blown up to full size, you can’t really tell how tricky an obstacle it presents at that stage in the climb. Here is a closer view.

The obstacle course of tree roots that awaits the rider who bears left

The obstacle course of tree roots that awaits the rider who bears left

If you enlarge this photo, you can probably appreciate how that final, thin root can be a problem if you are out of strength and momentum. Though the slope eases considerably at that point, I – like many riders I have seen – was on many occasions simply unable make it either over the root or circumventing it on one side – though  all three options would clearly be possible with fresh legs. And on the few occasions I did make it, I felt I just got lucky – I had not mastered it. I had got the right answer, but I had not really solved the problem. So close, so often. But, as in mathematics, close is not good enough.

After realizing I did not have the leg strength to master the left-of-the-nipple path, I switched to taking the right-hand line. Though the slope was considerable steeper (that is very clear from the blown-up photo), the tire-worn dirt showed that many riders chose that option.

Several failed attempts and one or two lucky successes convinced me that the trick was to steer to the right of the nipple and then bear left around it, but keep as close to it as possible without the rear wheel hitting it, and then head for the gap between the tree roots over at the right.

After that, a fairly clear left-bearing line on very gently sloping terrain takes you round to the right to what appears to be a crest. (It turns out to be an inflection point rather than a maximum, but let’s bask for a while in the success we have had so far.)

Here is our brief basking point.


The inflection point. One more detail to resolve.

As we oh-so-briefly catch our breath and “coast” round the final, right-hand bend and see the summit ahead, we come – very suddenly – to one final obstacle.


The summit of the climb

At the root of the problem (sorry!) is the fact that the right-hand turn is actually sharper than the previous photo indicates, close to a switchback. Moreover, the slope kicks up as you enter the turn. So you might not be able to gain sufficient momentum to carry you over one or both of those tree roots on the left that you find your bike heading towards. And in my case, I found I often did not have any muscle strength left to carry me over them by brute force.

What worked for me is making an even tighter turn that takes me to the right of the roots, with my right shoulder narrowly missing that protruding tree trunk. A fine-tuned approach that replaces one problem (power up and get over those roots) with another one initially more difficult (slow down and make the tight turn even tighter).

And there we are. That final little root poking up near the summit is easily skirted. The problem is solved.

To be sure, the rest of the ByPass Trail still presents several other difficult challenges, a number of which took me several attempts before I achieved mastery. Taken as a whole, the entire ByPass is a hard climb, and many riders walk the entire quarter mile. But nothing is as difficult as that initial stretch. I was able to ride the rest long before I solved the problem of the first 100 feet. Which made it all the sweeter when I finally did really crack that wall.

Now I (usually) breeze up it, wondering why I found it so difficult for so long.

Usually? In my next post, I’ll use this story to talk about strategies for solving difficult mathematical problems. In particular, I’ll look at the role of the subconscious in being able to put together a series of mastered steps in order to solve a big problem. For a very curious thing happened after I took the photos to illustrate this post. I walked back down to collect my bike from the ByPass sign where I’d left it, and rode up to continue my ride.

It took me four attempts to complete that initial climb!

And therein lies one of the biggest secrets of being able to solve a difficult math problem.

To be continued …

The Wuzzits – Free at Last

In which the word free has several meanings.

As regular readers of this blog will know, I’ve been looking at, thinking about, reflecting on, writing about, and playing video games for many years. I’ve also been working on creating my own video games, working with a small group of highly talented individuals at a company I co-founded a couple of years ago, InnerTube Games, to create high quality casual games that embody mathematical concepts and procedures in a fundamental way.

[ADDED LATER: We renamed the company BrainQuake ( ]

Earlier this year, in an article in American Scientist magazine, I said a little bit about the simple (though to some surprising) metaphor for learning mathematics that guides our design, and provided a screen-shot first glimpse of our pending initial release: Wuzzit Trouble. I also discussed a few other video games that adopted a similar approach to the design of games designed to develop mathematical thinking ability – rather than the  rote practice of basic skills that the vast majority of “math ed video games” focus on.

A screen shot from Wuzzit Trouble

A screen shot from Wuzzit Trouble

Last week, a bit later than the release date published American Scientist, we were finally able to release our game, Wuzzit Trouble.  In our game, the aim is to use increasingly sophisticated analytic thinking to help the cute little Wuzzit characters break free from the traps they have got caught in.

When the game broke free from Apple’s clutches, a free download was all that was required for players from ages 8 to 80 to get to work freeing the Wuzzits. That’s three uses of “free”. A fourth was our approach broke free of the familiar tight binding between mathematical thinking and the manipulation of symbols on a page. (See my February 2012 post on this blog.)

One of our greatest worries was that many people think that mathematical thinking is the manipulation of symbols on a page according to specific rules. (My Stanford colleague, Professor Jo Boaler, has studied this phenomenon. See for example, my account of her work in an article for the Mathematical Association of America.) For anyone with that view, our game would not appear to offer anything particularly new or different. That would mean they would fail to grasp the power of our design metaphor, as I had described in my American Scientist article and in a short video (3 min) we released at the same time as the game.

That will likely be a problem we continue to face. It will, I fear, mean that some people we would like to reach will dismiss our game. (On one remarkable occasion, an anonymous reviewer of a funding application we submitted to cover the development costs of our game, after playing an early prototype, declared that there was not enough mathematical content. All I can say to anyone who thinks that is, give the game a try and see how far you get – see later for the fine print that accompanies that challenge.)

Fortunately, the first review of our game, published in Forbes on the day Apple released it in the App Store,  was written by an educational technology writer who understood fully what we are doing. That initial review set the tone for many follow-up articles. We were off to a good start.

We were also greatly helped by Apple’s decision to feature our game, which appeared front and center on the App Store website for educational apps.

Apple features Wuzzit Trouble on its release

Apple features Wuzzit Trouble on its release

Other websites that track and report on the apps world followed suit, and before long we found ourselves in the Top Fifty of new educational apps. People seemed to “get it.”

Presenting a mathematics video game that does not have equations, formulas, or other symbolic mathematics all over the screen is just one way we are different from the vast majority of math learning games. Another is that we built the game to allow players of different ages and mathematical abilities to be able to enjoy the game.

As I describe at the end of another short video, if all you want to do is free all the Wuzzits, all you need is basic whole number arithmetic, which means the game can provide a young child lots of practice with basic number work.

But if someone older wants to get lots of stars and bonus points as well, much more effort is required. (Just check out the solution to one of the puzzles I describe on that video.) This is what we mean when we say Wuzzit Trouble provides a challenge to any player between the ages 8 and 80.

But this is already way too much text. Writing about video games is like writing movie reviews. Both are designed to be experienced, not read about. Just download the game and try it for yourself. And if you are so inclined, take me up on The Math Guy Challenge.

For more details about InnerTube Games and Wuzzit Trouble, visit our website:


I’m Dr. Keith Devlin, an emeritus mathematician at Stanford University, an author, the Math Guy on NPR’s Weekend Edition, and an avid cyclist. (The header photo is me halfway up Mt. Baldy in Southern California.)

New book 2017

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