Wednesday, August 8, 2018

How a Fields Medal led to a mathematical roller-coaster journey

By Keith Devlin

You can follow me on Twitter @profkeithdevlin

First, congratulations to Caucher Birkar, Alessio Figalli, Peter Scholze, and Akshay Venkatesh on being awarded the Fields Medal, an award that for regular “Devlin’s Angle” readers needs neither introduction nor description. (If it does, use Google.)

With Fields Medals awarded only (at most) once every four years to mathematicians who produce truly exceptional mathematics before they turn forty, few of us who enter the field come close to getting one. (Indeed, in some ways they are more akin to Olympic Gold Medals than the Nobel Prizes with which they are usually compared. Few club athletes will get one of those either.)  On the other hand, many of us earn our doctorates, or build our careers, by understanding a new approach or mastering a new technique that led to a Fields Medal. Just as Bill Gates copied the groundbreaking Macintosh interface to create Windows, so too it can pay off handsomely, and often quickly, for a young mathematician to “reverse-engineer” a Fields-Medal-winning new result and try to use it to solve a different – though often related – problem.

In fact, sometimes, the medal-winning breakthrough has such broad applicability that is initiates an entire new subfield of mathematics. That was exactly how I began my mathematical career almost a half a century ago. An interest in computing, initiated by a high-school summer internship writing software for British Petroleum (using the very first digital computer delivered to the city I grew up in), stayed with me throughout my undergraduate years, culminating with me interviewing for a job at IBM on graduation. But I was put off by the strong corporate culture and the lack of intellectual freedom I feared would come with joining Big Blue. Instead, I decided to go for a PhD in the general area of computing. Unfortunately, this was before Computer Science was a recognized discipline, and though it was possible to pursue graduate research related to computing, mathematically speaking there was not much of a “there” there back then.

The one mathematically-intriguing little “there” was a relatively new subject called Automata Theory that I had come across references to. Moreover, one of the pioneers of that field, John Shepherdson, was a professor of mathematics at the University of Bristol, just over a hundred miles from London, where I had just graduated. As chair of department, Shepherdson had built up a strong research team of experts in different branches of Mathematical Logic, the subfield of mathematics that provided the mathematical tools for Automata Theory. And so it was then that I applied to do a PhD at Bristol. That was in the fall of 1968.

Once I arrived in Bristol, everything changed. Among the mathematics graduate-student community at Bristol University, all the buzz – and there was a lot of it – was about an emerging new field called Axiomatic Set Theory. Actually, the field itself was not new. But as a result of a Fields Medal winning new result, it had recently blossomed into an exciting new area of research.

Not long after Georg Cantor’s introduction of abstract Set Theory in the late 19th century, Bertrand Russell came up with his famous paradox, concerning the set of all sets that are not members of themselves. To escape from the paradox – more accurately, to rescue the appealing, natural notion of using abstract sets as the basic building block out of which to construct all mathematical objects – Ernst Zermelo formulated a seemingly simple set of axioms to legislate the formation of sets. With an important addition from Abraham Fraenkel in 1925, that axiom system seemed to provide an adequate basis for the construction of all the objects of mathematics, while avoiding Russell’s Paradox. Zermelo-Fraenkel Set Theory, as it became known, rapidly came to be regarded as the "Grand Unified Theory" of mathematics, the basic system on which everything else is built. It was generally referred to as ZFC, the “C” denoting the Axiom of Choice, a basic principle Zermelo included but which was sufficiently controversial that its use was often acknowledged explicitly.

While the ZFC axiom system was indeed sufficient to ground all of mathematics, there were a small number of seemingly-simple questions about sets that no one could answer using just those axioms. The most notorious by far went back to Cantor himself: Cantor’s Continuum Problem asks how many real numbers there are? Of course, one answer is that there are an infinite number of such. But the ZFC axioms allow the construction of an entire system of infinite numbers of increasing size, together with an arithmetic, that can be used to provide a “count” of any set whatsoever. The smallest such infinite number, aleph-0, is the number of natural numbers. After aleph-0, the next infinite number is aleph-1. Then aleph-2, and so on. (It’s actually a lot more complicated than that, but let’s leave that to one side for now.)

Cantor showed that the real continuum, the set of all real numbers, has an infinite size strictly larger than aleph-0, so it must be at least aleph-1. But which aleph exactly was it? It’s tempting (on the grounds of pure laziness) to assume it’s aleph-1, an assumption known as the Continuum Hypothesis (CH). But there is no known evidence to support such an assumption.

In 1940, Kurt Goedel contructed a set-theoretic model of ZFC in which CH is true, thereby demonstrating that CH could never be proved false. But that does not imply that it is true. Maybe it was possible to construct another model in which CH was false. If so, then CH would be completely undecidable, based on the ZFC axioms. This would mean that the ZFC axioms are not sufficient to answer all reasonable questions about sets.

In 1963, Paul Cohen, a young mathematician at Stanford University, found such a model. Using an ingenious new method for constructing models of set theory that he called forcing, Cohen was able to create a model of ZFC in which CH is false. That result earned him the Fields Medal in 1966.

By 1968, when I went to the University of Bristol to commence my doctoral work, Cohen’s new method of forcing had been shown to have wide applicability, making it possible to prove that a number of long-standing, unanswered mathematical questions were in fact undecidable in the ZFC system. This opened up an exciting new pathway to getting a PhD. Learn how to use the method of forcing and then start applying it to unsolved mathematical problems, of which there was no shortage. Large numbers of beginning graduate students did just that, and by the time I joined a group of them, a few months after arriving at Bristol, the field was red hot. My interest in computation did not go away, but it would be over two decades before I would pick it up again. At 21 years of age, with a newly minted bachelors degree in mathematics under my belt, I had a mathematical research career to build, and axiomatic set theory was by far the most exciting field to do it in. I jumped onto the roller coaster and joined in the fun.

Working in my newly chosen field was just like working in any other branch of mathematics. Each day, you woke up and attempted to prove various mathematical statements using logically rigorous reasoning. To an observer looking over your shoulder, doing that involved scribbling formulas on paper and manipulating them in an attempt to construct a proof, just like any other branch of mathematics. The “rules of the game” were exactly the same as in any other branch of mathematics as well. The only difference was the nature of the answers you obtained – on the rare occasion when you did so. (Mathematics research is 95% failure. Actually, the failure rate may be higher than that; we have a far worse batting average than any professional baseball hitter.) In what those of us in this new field called “classical mathematics,” the goal was to prove statements about mathematical objects were true or false. In the new mathematics of undecidability proofs, the goal was to prove that statements about mathematical objects were undecidable (in the ZFC system). In both cases, the result was a rigorous mathematical statement (a theorem) justified by a rigorous mathematical argument (a proof).

From the perspective of mathematics as a whole, this meant that, thanks to Cohen, mathematicians had a new way to answer a mathematical question. Classically, there had been just two possibilities: true and false. If you can do neither, you had failed to find an answer.  Now, there was a third possibility: (provably) undecidable. What had previously been failure could now become success. Absence of a definite answer could be replaced by getting a definitive answer. Lack of knowledge could be replaced by knowledge.

The two decades following Cohen saw a whole range of unsolved mathematical problems proved undecidable, as a whole army of us jumped into the fray. Some results were easy. Success came quickly to those smart enough or lucky enough (or both) to find an unsolved problem that yielded relatively easily to the forcing technique. Others took much longer to resolve, and a few resisted all attempts (and have done so to this day). But by the start of the 1980s, the probability of success had dropped to that in other areas of mathematics. From then on, for most young mathematicians, getting a PhD by solving an undecidability problem meant finding some relatively minor variant of a result someone else had already obtained. The party was over.

Looking back, I realize that I was simply very lucky to be starting my mathematical career when a productive new subfield was just starting up. By going to the University of Bristol to do a PhD in Automata Theory, I found myself in the right place at the right time to jump ship and have the time of my life. When the field started to settle down and slowdown in the 1980s, I started to lose interest. Not in mathematics, just in that particular area as my main research focus. My main interest shifted elsewhere as my attention was caught by some new mathematics being developed at Stanford (as with forcing, Stanford turned out to be the place that generated the new ideas that caught my attention). That new mathematics was closely intertwined with my earlier high school interest in computing. But that’s another story.

Thursday, July 5, 2018

By Keith Devlin

You can follow me on Twitter @profkeithdevlin

21st Century Math: The Movie

All my Devlin’s Angle posts this year so far have studied the dramatic shift that took place over the past twenty-five years in the way professional mathematicians “do the math” in order to solve real-world problems. There have been parallel changes in the way pure mathematicians work as well, but those changes are somewhat less visible, and not as dramatic. In any case, I have been focusing on mathematics in the wild.

Those changes in how math is done have put pressure on global education systems to catch up. In previous posts, I addressed these changing educational needs, but overall, there has been a considerable lag. In the United States, many of the better, selective, private schools have adjusted, but little has changed in the math classrooms of most state-funded schools. There are a number of reasons for that lack of action, some educationally valid, others resulting from Americans’ proclivity to treat mathematics education as a political football. But that is another story.

The fact is, however, the mathematical world has changed significantly, it is not going to change back, and sooner or later the educational system must catch up. Hopefully sooner, given that today’s students will enter a world and a workforce where no one does calculations anymore – where by “calculation” I mean performing any form of algorithmic procedure.

In May, I participated in Maths for the 21st Century, a global mathematics education summit in Geneva, Switzerland, organized to discuss the new way mathematics is being done and how best to prepare students to live and work in such a world. Both the United States Department of Education and the OECD’s (Organization for Economic Cooperation and Development’s) PISA educational testing organization were represented at the summit.

The half hour talk I gave at the summit is in many ways a summary (absent all the details) of my series of posts for the MAA. So I thought I would wrap up the series, at least for now, by pointing you to the video of my presentation. The main summit page, linked above, also provides a link to a lightly abridged PDF version of the deck I used to accompany my talk.

My experience in giving public talks on this topic over the past several years has been that it evokes two very different reactions. Engineers and scientists in the audience, for the most part, nod along in agreement with everything I say. I am, after all, just describing the way they have been working for twenty years. In contrast, teachers, or at least a great many of them, often show surprise, confusion, and not infrequently hostility. Many parents react similarly.

Why is that? Well, to repeat an arguably over-used quotation from the great Paul Newman movie Cool Hand Luke, “What we have here is failure to communicate.” After my talks, I am often left feeling like the Paul Newman character, Luke, in that clip. However, for the analogy to work, Luke has to represent not me but the entire mathematics community.

Teachers are taken aback to be told that calculation is less relevant in today’s world. I believe this is because no one in the mathematics business – that is, the business of using mathematics to solve real-world problems – has taken the trouble to inform teachers that the entire game has changed, and in what ways. It’s time we bring better communication to this issue. My series of Devlin’s Angle posts this year is one of my latest attempts to do just that. The Geneva video I am directing you to is another.

Wednesday, June 6, 2018

Cycling can be such a drag – and math can tell you exactly how much

By Keith Devlin

You can follow me on Twitter @profkeithdevlin

Last month, my two greatest passions collided: mathematics and cycling. On my way back from a short biking trip to the Californian Central Coast (gorgeous in the late spring, when the grass is still green and the wildflowers are in full bloom), I stopped off in Morgan Hill (home of the American Institute of Mathematics) to watch part of Stage 4 of the seven stage, Amgen Tour of California. Wandering around the start area, where the teams warm up on stationary trainers and the big vendors show off their wares, I noticed a couple of guys in one corner, promoting a curious looking bike.

Actually, it wasn’t the bike itself that looked unusual, it was the strange, overly large fenders over the two wheels. Had the bike been in a regular bike shop, I would have thought they were splash guards for image-conscious, athletic-leaning commuters to use on rainy days. But the exhibits at a professional cycle racing event are aimed at hard-core cyclists, whose passion revolves around razor-thin saddles, aerodynamic bike design, and low bicycle weight. Whatever those fenders were for, they were not for keeping a rider dry. It had to be about performance. But modern, racing-bikes are designed to be as light and thin as possible – the wheels have tires just a tad over 20mm wide – so those bulky-looking fenders seemed completely out of place. I could not resist asking for an explanation.

The conversation soon got very mathy, and I quickly sensed an opportunity for a Devlin’s Angle post showing the power of thinking mathematically about everyday activities—be those activities work- or leisure- related. It would be a natural continuation of my recent series of posts (starting in January) on how mathematics education needs to change to prepare people for life in the 21st century.

The strange looking fenders were designed by Garth Magee, a former aerospace engineer from Southern California, who like me is a keen cyclist. I had already gotten some clue as to what the fenders were for from the company name on a poster: Null Winds Technology. The fenders must have something to do with reducing drag. (If so, then we need to start calling them “fairings”.) That would also explain Magee’s presence at the Time Trial stage of the Tour of California. Modern professional time-trialing is all about aerodynamic bicycle design, with all the major international bicycle manufacturers spending small fortunes on computer-aided designs and hours of testing in wind tunnels. Magee’s main purpose in Morgan Hill was likely to raise interest from some of the other manufacturers present at the event, I surmised. In addition, professional time-trialing is heavily regulated to ensure competitive cycling’s “purity” and “fairness”, and any kind of wheel fairings are banned in competition, so Magee’s product was surely not aimed at professional racers. He was likely seeking to sell to amateur riders, like myself.

[Disclaimer: Magee told me he had formed his company in 2012 to turn his discovery/invention into a product people can use. I have no involvement with the company, I don’t own any of their products, and I am not actively promoting their products. My focus here is on the really cool application of mathematical thinking.]

From past research into what was known about the mathematics of cycling, I knew it is complicated, uses advanced techniques, and has yet to fully explain the physics of cycling. You can get a sense of just how advanced and complicated it is by perusing the description on Wikipedia.

As any experienced cyclist will tell you, at anything beyond very low speeds, the greatest resistance to forward motion is due to the air. That’s why professional cycle racers stay close together in tight packs (“pelotons”) or long lines (“pacelines”), where the few rides at the front sacrifice their chance of winning in order to shield their teammates from the wind. In still air, the headwind is caused entirely by the cyclist’s own forward motion (at speeds up to 30 mph on the flat). If there is a headwind, the resistance is greater. Moreover, it increases with the cube of the rider’s speed relative to the air. That’s why riding a bike into a strong head wind is so hard. The Wikipedia article gives you the key formula, which I reproduce below.

Another nice summary of the relevant math you can access online can be found here.

What the math tells you is that wind drag is a huge problem that every cyclist encounters when trying to go faster. At a speed of approximately 7 mph, overcoming air resistance takes about half of your effort (with ground friction on the tires and mechanical resistance in the drive train accounting for the other half). As you go faster, that cubic growth starts to flex its muscles, which means you have to increasingly flex your muscles in order to overcome air resistance that demands a larger and larger proportion of your total effort. At around 15 mph, approximately 70% of your effort is being used to overcome air resistance; at 20 mph it takes roughly 85% of your effort. At the top end, a typical average speed for a flat stage in the Tour de France is about 29 mph. At that speed, over 90% of the effort needed to maintain this speed is used to overcome air resistance.

Needless to say, the bicycle manufacturing industry has put in a lot of effort over the years to try to minimize the effect of headwind drag. The results of those efforts, explained for cyclists rather than mathematicians, are nicely summarized in two articles that you will find online at: and

With so much research put into the problem of headwind drag, you would think the industry had done as much as could be done. But as Magee showed, there were still more efficiencies to be obtained. His observation is an excellent illustration of the power of mathematical thinking.

The mathematics I’ve summarized so far treats the bicycle as a single item, not an assembly of components. (An instance of the mathematician’s standard approach, as encapsulated in the quip “Consider a spherical cow not subject to frictional forces.”) Magee focused on the effect of headwind on the wheels.

To be sure, he was not the first to do that. Most of the racers at the Morgan Hill time-trial rode bicycles with solid rear wheels. Though considerably heavier than the more common spoked wheels, a solid wheel creates far less rotational drag than does a regular wheel, where the spokes create turbulence as they cut through the air. (The only reason the pros don’t use a solid wheel at the front as well is because it would make the bicycle highly unstable, with a slight crosswind likely to send the bike and the cyclist out of control. In indoor races on banked tracks, you do see bikes with two solid wheels.)

And now, as that last paragraph should indicate (note the word turbulence), we are deep into aerodynamics. The field Magee worked in for many years. What Magee did, that no one had previously done (at least to the point of taking out a patent on a design), was observe that, with wind resistance increasing with the cube of the speed through the air, the resistance increases rapidly as you go up the wheel from the axle to the top of the wheel (where the top of the tire is moving at twice the speed of the bicycle), and as you go down from the axle to the ground it decreases to 0 relative to the ground. What would happen, Magee asked, if a shield were to keep the wind from hitting the fast-moving top portion of the wheel? Sure, it would add some weight, but with that cubic function to contend with, it seemed likely the drag reduction could be significant.

In particular, what portion of the wheel should be protected to optimize any gain due to overall reduced wind resistance on the wheel? This is where a bit of good-old-fashioned math comes into play. The chart below is the key to Magee’s fairing design.

According to the math, there would be little or no benefit when riding in still air (or with a tailwind). But in a headwind, the difference should be noticeable to the rider. The stronger the headwind, the greater the benefit. To test his invention out, he turned to his friend Robert Keating, a former bike racer and triathlete who teaches triathlon and works at a local Triathlon shop in Los Angeles. (I met Keating at the Morgan Hill event, where he and Magee were jointly demoing the new device.) A number of road tests showed that the idea worked as the math said it should. You can see a video of one recent test here. This, folks, is the power of mathematics in action.

Just as with the UPS routing problem I had the students at Nueva School look at in January (see my previous posts for January 2018 onwards), Magee’s problem was all about optimization. Not unique right answers; rather better performance.

Yes, there was a fair amount of sophisticated calculation to be done. But the key was to approach the problem in the right way, so that mathematical power could do its work.

Note, too, that all the background mathematics needed to solve the design problem can be found on the internet. Indeed, in writing this article, I simply used Google to locate suitable sources. (Remember, Google was the very first modern tool on that chart of modern tools to do mathematics I presented in my February post.)

As I keep saying, in today’s world, in using mathematical thinking to help solve a problem, you (usually) don’t need to re-invent the wheel. Those days are largely gone. Today, you mostly need to understand mathematics in a fundamental, conceptual way so you can make an existing wheel do the work for you.

In Magee’s case, of course, that was true both literally as well as in my original metaphorical sense.

All in all, it’s a superb example of 21st century mathematical problem solving.

Finally, I note that I was originally motivated to reverse-engineer the UPS routing algorithm because of the strange movements (and non-movements) of a bicycle I was shipping from California to Princeton. So might it be that math is particularly well suited to problems involving bicycles. Not at all. Bicycles figure in both examples because they interest me. After all, I was only at the Tour of California Time Trial because I am passionate about cycling, and it was my mathematical bent that prompted me to approach the Null Winds display and ask for an explanation. (Remember also my 2014 Devlin’s Angle post on mountain biking and proving theorems.)

The real lesson is that mathematical thinking can be applied to almost anything, particularly if the question is “Can we make it better (in some way)?” You are not interested in biking? Fine. Think about something that really does interest you. The chances are high—very high—that mathematics could be used to make it better in some way.

Once you have decided what to optimize, use the wide range of tools that are now freely available to start to find out how you might do it. Go as far as you can, then seek help from a mathematician. Your passion, experience, and domain knowledge coupled with the mathematician’s experience at using math to solve problems can make a powerful team.

Of course, to proceed this way, as I discussed in my earlier posts this year, you do need to understand what mathematics (really) is and how it can be used. But that is really all you need. Yes, it’s a big “all”. But it’s THE “all”. That’s why it needs to be the focus of mathematics education in the 21st century.

LABELS: mathematical thinking, problem solving, aerodynamics, wind resistance, bicycle mathematics.

Wednesday, May 2, 2018

Calculation was the price we used to have to pay to do mathematics

By Keith Devlin

You can follow me on Twitter @profkeithdevlin

Ever since mathematics got properly underway around 3,000 years ago, there was only one way to achieve access to the field. You had to spend many years developing a fairly extensive calculation skillset. In the first instance, to pass the graduation and entrance examinations to gain initial access to the field. Then, once accepted into the world of mathematics, calculation of one kind or another was what all mathematicians spent the bulk of their mathematical time doing. Arguably, for most of mathematics history, the subject really was, to a large extent, primarily about calculation of one form or another. Newton, Leibniz, Bernoulli (any of them), Fermat, Euler, Riemann, Gauss, and the other greats of times past, were all superb masters of calculation. (We should also include Boole, since his famous Boolean algebra is also a calculation system.)

But whereas most laypersons seem to think that calculation is all there is to mathematics, surely none of the greats did. Calculation was an important tool (more accurately, a set of tools) you needed to do mathematics, they must have realized, but the essence of mathematics is much more, a plateau of knowledge that transcends all the calculation techniques.

In the 19th Century, that somewhat tacit understanding became explicit. The increasing complexity of the problems mathematicians tackled led to a series of results that defied the human intuition. (Several of them were referred to as “paradoxes”.) This led to an intense period of mathematical introspection, where the primary focus was not performing a calculation or computing an answer, but formulating and understanding abstract concepts and relationships. In other words, a shift in emphasis from doing to understanding. What had previously been implicit, became full-on explicit.

Mathematical objects were no longer thought of as given primarily by formulas, but rather as carriers of conceptual properties. Proving something was no longer a matter of transforming terms in accordance with rules, but a process of logical deduction from concepts. Mathematics was reconceptualized as “thinking in concepts” (Denken in Begriffen).

This was, in every sense, a mathematical revolution, with the primary revolutionaries being
leading mathematicians such as Lejeune Dirichlet, Richard Dedekind, Augustin-Louis Cauchy, Karl Weierstrass, and Bernhard Riemann.

To give just one instance of the shift, prior to the nineteenth century, mathematicians were used to the fact that a formula such as  y = x2 + 3x – 5  specifies a function that produces a new number y from any given number x. Then the revolutionary Dirichlet came along and said, forget the formula and concentrate on what the function does in terms of input-output behavior. A function, according to Dirichlet, is any rule that produces new numbers from old. The rule does not have to be specified by an algebraic formula. In fact, there's no reason to restrict your attention to numbers. A function can be any rule that takes objects of one kind and produces new objects from them.

This definition legitimized functions such as the one defined on real numbers by the rule:
If x is rational, set f(x) = 0; if x is irrational, set f(x) = 1.

Of course, you cannot draw a graph of such a monster. Instead, mathematicians began to study the properties of abstract functions, specified not by some formula but by their behavior. For example, you can investigate questions such as, is the function one-one, injective, surjective, continuous, differentiable, etc.?

For any mathematician alive today, mathematics is a subject that studies formally-defined concepts, with a focus on the establishment of truth (based on accepted axioms), with various forms of calculation (numerical, algebraic, set-theoretic, logical, etc.) being tools developed and used in the pursuit of those goals. That’s the only kind of mathematics we have known.

Except, that is, when we were at school. By and large, the 19th Century revolution in mathematics did not permeate the world’s school systems, which remained firmly in the “mathematics is about calculation” mindset. The one attempt to bring the school system into the modern age (in the US, the UK, and a few other countries), was the 1960s “New Math”. Though well-intentioned, its rollout was disastrous, in large part because very few teachers understood what it was about – and hence could not teach it well. The confusion caused to parents (other than mathematician parents) was nicely encapsulated by the satirical songwriter and singer Tom Lehrer (who taught mathematics at Harvard, and did understand New Math), in his hilarious, and pointedly accurate, song New Math.

As a result of the initial chaos, the initiative was quickly dropped, and school math remained largely unchanged while real-world uses of mathematics kept steadily changing, leaving the schools increasingly separated from the way people did math in their jobs. Eventually, the separation blew up into a full-fledged divorce. That occurred in the late 1980s. The divorce was finalized on June 23, 1988. That was the date when Steve Wolfram released his mammoth software package Mathematica. Within a few short years of that release, if not on the release-date, Mathematica (and a similar package released a few months later in Canada, Maple) could answer pretty well any school or university math exam question with at least a grade B+, and very often an A.

The days when calculation (of pretty well any kind, not just numerical) was the price humans had to pay to do mathematics were over.

Given that thirty years have passed since that initial epochal moment, and most of the world has still not woken up to the fact that the entire mathematical world has changed dramatically and forever, let me repeat the core of that statement in caps.


To be sure, after that symbolic 1988 date, it took a few years for the change to percolate through the system, gain momentum, and eventually reach critical mass. Three further developments were also hugely significant: the birth of the World Wide Web in 1989 and the browser in 1993, and the launch of Wolfram Alpha in 2009. (Others might want to add other factors. I’m being selective here.)

Talking about being selective, I’ve mentioned Wolfram products twice now. Though I was a member of Wolfram’s Mathematica Advisory Board in the first few years, I have no stake in or involvement with the company. While both Mathematica and Alpha were indeed major players in changing the way mathematics is done – particularly in applied settings – I am citing those particular products largely as icons, using two specific products to represent a range of new digital tools that were being developed around the world at that time. While Wolfram’s systems were ones I myself made early use of in my work, other mathematicians were also active in that digital mathematical revolution, using different systems. Still, Mathematica was the system that caught the public attention.

Since the turn of the new Millennium, I doubt if anyone making professional use of mathematics in their job, or indeed any adult using mathematics in their everyday lives, has taken out paper-and-pencil and followed a classical algorithm to add, subtract, multiply or divide numbers in an array of real-life size, or perform complex algebraic reasoning to solve systems of equations, or solve problems using calculus, or any other established mathematical procedure. Not only would it now be a waste of valuable human time and energy doing something a cheap machine can do in far less time with no possibility of error, but many of the problems that people encounter in their careers and lives have simply too much data for the human mind to handle. Those same digital tools that have made the execution of mathematical procedures unnecessary have also come to dominate and drive our world, so many of the problems that require mathematics in their solution are now simply beyond human capacity. That’s why Amazon Web Services has become such a behemoth for data storage and processing.

But that does not mean humans no longer need to have some mathematical skills. On the contrary, they are as crucial as ever – unless you are willing to be totally reliant on others, but personally, I have never felt comfortable doing that with things that are part of my life every day. What has changed are the specific mathematical skills required today. There are plenty of things computers cannot do or do poorly. Genuinely creative thinking and analogical reasoning are two obvious ones – though with today’s massive cloud computing resources we can use systems that provide an approximation often adequate for the purpose, and on occasion can be better than humans.

Mostly, however, where you need humans is going from a real-world challenge situation to formulating one or mathematical tasks that can help you make progress. Sometimes, progress means solving a real-world problem in the sense of getting a specific answer (say, a number), but much more commonly it’s about finding a better method, where “better” can mean faster, cheaper, safer, or whatever other criterion is important, and where the change may involve developing a new method or improving an existing one.

This way of using mathematics was the focus of that mini-course I gave at a California school (Nueva School) in January of this year, that I wrote about in the February, March, and April posts to this column.

Though several mathematicians and mathematics education scholars expressed agreement with what I wrote, my articles brought some critiques from teachers and parents. The critiques all made reference to my asides about the Common Core State Standards in the first two of the posts. Since “Devlin’s Angle” no longer seems to be a target for the CCSS social media trolls (likely because the yield of issues to react to relative to the length and substance of most of my posts makes it less rewarding to them), I made some efforts to find out what exactly it is about the CCSSs that they objected to. As far as I could ascertain, the issue was inevitably (and predictably) to do with particular implementations of the Standards in specific curricula or (and this seems to be the most common occurrence) claims that a particular homework exercise was a “Common Core exercise”, which of course it cannot be since the CCSS are, as the name indicates, purely a set of standards to attain, not in any way a curriculum or curriculum content.

More generally, in fact, pretty well all critiques of the CCSS are due to a complete misunderstanding of what they are, why they are, and what they say. The issue was nicely dealt with in this 2014 article in the Hechinger Report.

My reason for bringing the Common Core into my series of posts was to point out that the standards were developed precisely to help guide school districts, schools, and teachers in the tricky task of updating K-12 mathematics education to adequately prepare tomorrow’s citizens for life and work in a world where calculation is no longer a central pillar of mathematics.

Having said that, I should point out that the above statement in no way implies that we should drop the teaching of basic arithmetic and algebra from the school system. As I discussed in some length in the third of my Nueva-inspired articles, the change that is required in K-12 math education is not so much in the mathematical topics but the reason they are now being taught and, in consequence, the way they should be taught.

Teaching for execution is no longer the primary driver, since no one using mathematics in the real world does that anymore. What is now of crucial importance is teaching for understanding. Digital systems outperform humans to an insane degree when it comes to execution. But they don’t understand; people have to supply that.

I leave you with an image I pulled from one of those Common Core social media rants some time ago. (I no longer remember the exact source.)

Typical social media posts about Common Core mathematics.
I have three comments about the post on the left. First, the mathematics in the bottom left is not some fancy new algorithm, it is what a child wrote down in reasoning (sensibly) about a particular arithmetic problem. Second, if you are unable to follow what the child is doing, you would have trouble making effective use of mathematics in today’s world. It’s pretty basic. (Your kid just did it, right?) Third, if you are a parent and you don’t see why it is important that today’s school students acquire those math reasoning skills, please don’t communicate your skepticism to your children. Doing so would be a great disservice, to your child, to your child’s math teacher, and to society. The mathematical world has changed significantly. That occurred over twenty years ago. It is not going to change back. Sit back, relax, be encouraging, and let the kids take over. They do just fine with it.

REFERENCE: During the period when the computer revolutionized how mathematics is done, I edited the American Mathematical Society’s “Computers and Mathematics” section of their monthly notices publication, sent to all members. I wrote about the column and that period in general in a paper that I submitted to the Proceedings of the Jon Borwein Commemorative Conference, held in 2017. Borwein, who died tragically young in 2016, was a leading pioneer in bringing digital technologies into mathematics. You can access a preprint of the paper HERE.

Wednesday, April 4, 2018

How today’s pros solve math problems: Part 3 (The Nueva School course)

By Keith Devlin

You can follow me on Twitter @profkeithdevlin

NOTE: This article is the final installment of a four-episode mini-series posted here starting in mid-January. In writing it, I have assumed my readers have read those three earlier pieces.

At the end of last month’s post, I left readers with a (seemingly) simple arithmetic problem. I prefaced the problem with the following two instructions:

1. Solve it as quickly as you can, in your head if possible. Let your mind jump to the answer.

2. Then, and only then, reflect on your answer, and how you got it.

The goal here, I said, is not to get the right answer, though a great many of you will. Rather, the issue is how do our minds work, and how can we make our thinking more effective in a world where machines execute all the mathematical procedures for us?

Here is the problem.

PROBLEM: A bat and a ball cost $1.10. The bat costs $1 more than the ball. How much does the ball cost on its own? (There is no special pricing deal.)

What answer did you get? And what did you learn from the subsequent reflection?

Before I continue, I should note that the use of this problem (which you can find in many puzzle books and on countless websites) in the context of trying to maximize the human mind’s innate abilities in order to become good 21st Century mathematical thinkers, is due to Gary Antonick, with whom I co-taught a Stanford Continuing Studies adult education course last fall. It was in that course that I gave the second iteration of the UPS problem I subsequently based my Nueva School course on. The discussion of the bat-and-ball problem that follows is the one Antonick presented in our course.

Now to the problem itself. The most common answer people give instantly to this problem is that the ball costs 10¢. It’s wrong (and many realize that is the case soon after their mind has jumped to that wrong number). What leads many astray is that the problem is carefully worded to run afoul of what under normal circumstances is an excellent strategy. (So if you got it wrong, you probably did so because you are a good thinker with some well-developed problem-solving strategies— problem-solving heuristics is the official term, and I’ll get to those momentarily. So take heart. You are well placed to do just fine in 21st Century mathematical thinking. You simply need to develop your heuristics to the next level.)

Here is, most likely, what your mind did to get to that 10¢ answer. As you read through the problem statement and came to that key phrase “cost more,” your mind said, “I will need to subtract.” You then took note of the data: those two figures $1.10 and $1. So, without hesitation, you subtracted $1 from $1.10 (the smaller from the larger, since you knew the answer has to be positive), getting 10¢.
Notice you did not really perform any calculation. The numbers are particularly simple ones. Almost certainly, you retrieved from memory the fact that if you take a dollar from a dollar-ten, you are left with 10¢. You might even have visualized those amounts of money in your hand.

Notice too that you understood the mathematical concepts involved. Indeed, that was why the wording of the problem led you astray!

What you did is apply a heuristic you have acquired over many financial transactions and most likely a substantial number of arithmetic quiz questions in elementary school. In fact, the timed tests in schools actively encourage such a “pattern recognition” approach. For the simple reason that it is fast and usually works!

We can, therefore, formulate a hypothesis as to why you “solved’ the problem the way you did. You had developed a heuristic (identify the arithmetic operation involved and then plug in the data) that is (a) fast, (b) requires no effort, and (c) usually works. This approach is a smart one in that it uses something the human brain is remarkably good at—pattern recognition—and avoids something our minds find difficult and requiring effort to master (namely, arithmetic calculation).

Of course, primed by the context in which I presented this particular problem, you probably expected there to be a catch. So, after letting your mind jump to the 10¢ answer, you likely took a second stab at it (or, if you were anxious about “getting a wrong answer,” made this your first solution) by applying an algorithm you had learned at school. Namely, you reasoned as follows:

Let x = cost of bat and y = cost of the ball. Then, we can translate the problem into symbolic
form as x + y = 1.10 ,   x = y + 1

Eliminate x from the two equations by algebra, to give
1.10 – y = y + 1

Transform this by algebra to give
0.10 = 2y

Thus, dividing both sides by 2, you conclude that
y = 5¢.

And this time, you get the correct answer.

You may, in fact, have been able to carry out this procedure in your head. When I was at school, I could do algebraic manipulations far more complicated than this in my head, at speed. But, truth be told, since I started outsourcing arithmetic to machines many decades ago, I have lost that skill, and now have to write down the equations and solve them on paper. (This is a confirmation, if any were needed, that arithmetic calculations do not come naturally to the human brain. Over the years, as my mental arithmetic skills have declined, my pattern recognition abilities have not diminished, but on the contrary have dramatically improved, as I learned—automatically, through exposure—to recognize ever more fine-grained distinctions.)

Whether or not you can do the calculation in your head, it is of course entirely formulaic and routine. Unlike the first method I looked at (a heuristic that is fast and usually right), this method is an algorithmic procedure, it is slow (much slower than the first method, even when the algebraic reasoning is carried out in your head), but it always works. It is also an approach that can be executed by a machine. True, for such a simple example, it’s quicker to do it by hand on the back of an envelope, but as a general rule, it makes no sense to waste the time of a human brain following an algorithmic procedure, not least because, even with simple examples it is familiarly easy to make a small error that leads to an incorrect answer.

But there is another way to solve the problem. It’s the way I addressed it, and, according to Antonick, who has given it to many professional mathematicians and asked them to vocalize their solutions, the way many math pros solve it. Like the first method we looked at, it is a heuristic, hence instinctive and fast, but unlike the first heuristic method, it always works.

This third method requires looking beyond the words, and beyond the symbols in the case of a problem presented symbolically, to the quantities represented. Though I (and likely other mathematicians) don’t visualize it quite this way (in my case it is more of a vague sense-of-size), the following image captures what we do.

As we read the problem, we form a mental sense of the two quantities, the cost of the ball-on-its-own and the cost of the bat-plus-ball, together with the stated relation between them, namely that the latter is $1 more than the former. From that mental image, where we see the $1.10 total consists of three pieces, one of which has size $1 and the other two of which are equal, we simply “read off” the fact that the ball costs 5¢. No calculation, no algorithm. Pure pattern recognition.

This solution is an example of Number Sense, the critical 21st Century arithmetic skill I wrote about in the January 1, 2017 Huffington Post companion piece to the article I published on the same day as my article about all my math skills becoming obsolete, which I referred to in my last post here on Devlin’s Angle.

It is, I suggest, hard to imagine how a computer system could solve the problem that way. (Of course, you could write a program so it can perform that particular pattern recognition, but the essence of number sense is that you can apply it to many numerical problems you come across.)

Those three ways to solve the bat-and-ball problem I just outlined are examples of what the famous Australian (pure) mathematician Terrence Tao has called (in his blog), respectively, pre-rigorous thinking, rigorous thinking, and post-rigorous thinking. You can also listen to him explain these three categories in a short video in the Numberphile series.

Post-rigorous heuristic thinking is how today’s math pros use mathematics to solve real-world problems. In fact, as Tao makes clear, post-rigorous thinking is what the pros use most of the time to solve abstract problems in pure math. The formal, symbolic, rigorous stuff comes primarily at the end, to check that the solution is logically correct, or at various intermediate points to make those checks along the way.

In the case of solving real-world problems, the pros almost always turn to technology to handle any algebraic deductions. In contrast, though pure mathematicians sometimes do use those technology products as well, they often find it much quicker, and perhaps more fruitful in terms of gaining key insights, to do the algebraic work by hand.

So, one of the big question facing math teachers today is, how do we best teach students to be good post-rigorous mathematical thinkers?

In the days when the only way to acquire the ability to use mathematics to solve real-world problems involved mastering a wide range of algorithmic procedures, becoming a mathematical problem solver frequently resulted in becoming a post-rigorous thinker automatically.

But with the range of tools available to us today, there is a good reason to assume that, with the right kinds of educational experiences, we can significantly shorten (though almost certainly not eliminate) the learning path from pre-rigorous, through rigorous thinking, to post-rigorous mathematical thinking. The goal is for learners to acquire enough effective heuristics.

To a considerable extent, those heuristics are not about “doing math” as such. Rather, they are focused on making efficient and effective use of the many sources of information available to us today. But before you throw away your university-level textbooks, you need to be aware that the intermediate step of mastering some degree of rigorous thinking is likely to be essential. Post-rigorous thinking is almost certainly something that emerges from repeated practice at rigorous thinking. Any increased efficiency in the education process will undoubtedly come from teaching the formal methods in a manner optimized for understanding, as opposed to optimized for attaining procedural efficiency, as it was in the days when we had to do everything by hand. See Daniel Willingham’s excellent book Why Don’t Students Like School? for a good, classroom-oriented look at what it takes to achieve mastery in a discipline.

Now to that UPS routing problem that was the focus of my Nueva School course. [You will find it discussed here.] Here are some of the hints and suggestions about solving the problem I made to the students in the three courses where I used it. Whether they followed my advice was entirely up to them. The purpose of the course was not to solve the problem unaided—even an entire semester would not be enough time for that with students who had never approached a problem the way the pros do. Rather, it was to give them an experience of the method.

First, they had to work in teams of three to five. I let them select the teams, but said it would be good if at least one person on each team felt they were “good at math.”

Then, start out by using Google to find out what you can about the problem domain, and any attempts made by others to solve it.

Whenever you come across a reference to a concept, an approach, or a method that you suspect might be relevant, use general Web resources like Wikipedia to get an initial understanding of what they are and what they can do.

Follow any leads your search brings up to solutions of problems that look similar. Note what methods were used to solve them.

If you come across references to others who have worked on the problem, or a similar one, send them a brief email. You may not get a reply, but occasionally you will, and it could be invaluable. (I receive such emails all the time. Mostly I do not have time to respond, but occasionally one lands in my inbox when I have a spare moment, and I happen to know something that might help, so I shoot back a brief reply, often just a reference to a particular source.)

When you get to a point where you need to perform a specific calculation, perhaps because you have found a solution to a very similar problem someone else has obtained and published, but your data is different, use Wolfram Alpha. It is structured so you can use pattern recognition (of formulas) to identify the appropriate technique and then edit the example provided to be the one you want to solve.

Reinforce your use of Wolfram Alpha by using YouTube to find suitable videos that provide you with quick tutorials on the technique.

The resources I just mentioned are all listed on that chart of “Important Mathematical Technology Tools” I published with the first two articles in this series.

As it turns out, with the UPS routing problem, the sequence of steps I have outlined so far quickly leads to identification of a small number of possible solution techniques for which there are many very accessible YouTube videos, and in fact, for this problem there is no need to go much further into my list of tools, if at all.

You should, though, check out the various other resources on my list, to see what they offer. Each new problem has to be approached afresh, in its own terms. Twitter is on my list because it is my list, and I have sufficiently many math-expert Twitter followers that a quick tweet can often yield just the information I need, saving me having to send out a slew of emails to people I think might be able to help. LinkedIn is also idiosyncratic to me, since I have a good network of mathematics and technology professionals I can contact. But the other resources are pretty generic.

Ideally, everything goes much more smoothly if you can avail yourself the services of a math consultant to assist you in negotiating the various resources. (I was that consultant to the teams in the three courses I gave.)

Interestingly, in the final meeting of my Princeton class (which was the fist time I used the UPS problem in a course), after having the student teams present their solutions, I gave the solution I had obtained, at the end of which two individuals came up to me to say they hoped I had not minded their sitting in on the class. (It was an experimental course, and there had been strangers sitting in for one or two sessions throughout the semester, so I had not paid them any attention.) They were, they said, postdocs working with Professor X, who was a math consultant for UPS and had worked on the algorithm the class and I had been trying to reverse engineer. Hence their curiosity-driven attendance on the last day! Unbeknownst to me, my final lecture had been my oral exam!

“How did I do?”, I asked. “You got it pretty well right,” they replied.

Which was nice for me, but it would not have mattered if I had followed a different track. What was important from an educational standpoint was the process.

Something else I suggested to the class was to come up with a solution—any solution—as soon as you can. “Don’t worry if it is optimal or even right,” I said. “Just check it by computation, perhaps in the form of a spreadsheet simulation. Once you have some solution that you can check (in the case of my UPS problem, check against the shipping data I supplied, or any other UPS data you can find on the Web), you can iterate to find a better one. It might turn out that your first solution, or your first three or four, won’t even get you to first base, but in the process of formulating and checking those initial attempts, you will inevitably gain insight into the problem you are trying to solve. Remember, computation is cheap, fast, and essentially limitless.”

If you are not familiar with this way of solving math problems, it may not seem like an approach that will work. But it does. It is, in fact, how all of today’s pros do it!

If you have not already done so, now is a good time to check out the dictionary definition of the word heuristic! Here is Wikipedia’s (at the time of writing):

“A heuristic technique (from the ancient Greek for “find” or “discover”), often called simply a heuristic, is an approach to any problem-solving, learning, or discovery that employs a practical method not guaranteed to be optimal or perfect, but sufficient for the immediate goals. Where finding an optimal solution is impossible or impractical, heuristic methods can be used to speed up the process of finding a satisfactory solution. Heuristics can be mental shortcuts that ease the cognitive load of making a decision. Examples of this method include using a rule of thumb, an educated guess, an intuitive judgment, guesstimate, stereotyping, profiling, or common sense.”

Without an expert consultant, the heuristics approach to solving real-world problems can work, but it definitely goes a lot faster, and with a far great likelihood of success, if you have a math expert you can call on. Not to “do any math.” Computer systems handle those parts. Rather to help you negotiate the vast array of resources at your disposal and select the most promising one(s) to try next. For that is what using mathematics to solve a real-world problem really boils down to these days: managing resources.

And managing resources is something humans are innately good at. Natural selection always favors those creatures which are best able to manage the available resources. We are here as present-day humans because as a species we are good at doing that. What is new in the case of mathematical problem solving is that pieces of mathematics (formulas, equations, procedures, algorithms, techniques) are now among the “intellectual Lego pieces” (freely accessible on the Web) we can use as we assemble a solution.

As the students in my three courses could, in principle, attest, you don’t need vast expertise in mathematics to work this way. You just need to be a good thinker able to work in a small team. I say, “in principle,” since I think it highly likely most of not all the students felt they did not do much at all by way of using math to solve a problem. But that, I would say, is because they have a conception of “using math to solve a problem” rooted in the Nineteenth Century, if not the Fourth Century BCE. From my perspective, they absolutely were able to do what I just said they did.

Of course, they were not as good at it as I am. I’ve been at this game a lot longer, and, make no mistake about it, experience counts. (I think it is close to being the only thing that counts.)

What does not count, at least to any extent even remotely approaching the prohibitive degree it used to, is the ability to “do the math.” You just need to be able to select (hopefully, with help from someone with experience) the right pieces from the available online mathematical smorgasbord, and stitch them together in an appropriate way.

This kind of problem-solving doesn’t feel like math (as we all came to love or hate), that’s for sure. In fact, it doesn’t even feel like work. Once they got into the swing of it, even the students who declared they were not good at math or did not enjoy it, found they were having a good time, working in teams in a creative, explorative way. For the fact is, properly approached, humans enjoy problem-solving. (That’s another consequence of natural selection— problem-solving, particularly group problem solving, is one of our species’ key survival advantages.)

In fact, another way to look at the recent revolution in how we “use math” to solve real-world problems, is that it has brought “using math” into the mainstream of human group activities we naturally find enjoyable. At heart, mathematical thinking is little more than formalized common sense. It always has been. Which means it is something we can all do. (In my 2000 book The Math Gene, I presented an evolutionary explanation for the human brain’s acquisition of the ability to do mathematics, which implied that mathematical capacity is in the human gene pool, and hence available for all of us to “switch on.”) What caused many people problems over the centuries was that, before we had technologies that could handle the formal symbol-manipulation stuff, the only way to employ our innate capacity for mathematical thinking was to train the brain to do those manipulations. But manipulating algebraic symbols with logical precision is most definitely not something our brains evolved to do. (Our early ancestors’ lives on the savannas did not present much by way of a need for algebra.) So we find it very hard. Only with great effort over several years can we train our brains to do such work. And even then, we are error-prone.

Incidentally, practically everything I have said in this article applies to the way 21st Century coders work. In coding as in mathematics, the days are long gone when it was all about writing thousands of lines of instructions. The modern-day mathematician’s Web resource MathOverflow (on my chart of useful math tools) was modeled on, and named after, the coding world’s StackOverflow. Both groups of professionals use heuristics. In today’s world, highly regarded math problem solvers and good coders have simply acquired a richer and more effective set of heuristics than the ones who are less highly ranked. And for the most part, developing heuristics is a result of reflective experience, not some innate talent.

And there you have it. The primary goal in 21st Century mathematics-education-for-all is the development of a good repertoire of heuristics.

I’ll leave you with a graphical summary of Tao’s categorization of the three kinds of mathematical thinking we can bring to problem-solving. I introduced this categorization above to provide a perspective on the three phases each one of us has to go through to become proficient mathematical (real-world) problem solvers. But it also provides an excellent summary of three historical stages of mathematical thinking as it has evolved over the past ten thousand years or so, from the invention of numbers in Sumeria, where the mathematical thinking of the time was accessible to all, through three millennia of formal mathematics development, where many people were never able to make effective use of it, and now into the third phase, where, because of technology, mathematical thinking can once again be accessible to all.

To be sure, we do not know the degree to which people have to master rigorous thinking to become good post-rigorous thinkers. As I already noted, I don’t for a second imagine that stage can be by-passed. (See the Willingham book I cited.) But, given today’s technological toolkit, including search, social media, online resources like Wolfram Alpha and Khan Academy, and a wide array of online courses, it is absolutely possible to master most of the rigorous thinking you need “on the job,” in the course of working on meaningful, and hence motivational and rewarding, real-world problems.

This is not to say there is no further need for teachers. Far from it. Very few people are able to become good mathematical thinkers on their own. Newtons and Ramanujans, who achieved great things with just a few books, are extremely rare. The vast majority of us need the guidance and feedback of a good teacher.

What the inevitable transition to 21st Century math learning requires is that mathematics teachers operate very differently than in the past. The days where you need a live person to deliver information are largely over. Today, teaching is much more a matter of being a coach and mentor. To be sure, you can occasionally find such teaching on the Internet, but it works only if you can be one-on-one with that teacher. I expect there will be change, but I don’t expect an economy of scale. If I had to make a guess, I would predict that in due course you will find your (specialist) math teacher by going online to a website, where you will be paired with a practicing 21st Century math professional who spends part of each day coaching and mentoring students.

LABELS: mathematical thinking, problem-solving, rigorous thinking, pre-rigorous thinking, post-rigorous thinking, Terrence Tao, social media in mathematics

Friday, March 9, 2018

How Today’s Pros Solve Math Problems: Part 2

By Keith Devlin

You can follow me on Twitter @profkeithdevlin

CHANGE OF PLAN: When I wrote last month’s post, I said I would conclude the description of my Nueva School Course this time. But when I sat down to write up that concluding piece, I realized it would require not one but two further posts. The course itself was the third iteration of an experiment I had tried out on a university  class of non-science majors and an Adult Education class. This series of articles is my first attempt to try to describe it and articulate the thinking behind it. As is often the case, when you try to describe something new (at least it was new to me), you realize how much background experience and unrecognized tacit knowledge you have drawn upon. In this post, I’ll try to capture those contextual issues. Next month I’ll get back to the course itself.

We all know that mathematics is not always easy. It requires practice, discipline and patience,  as do many other things in life.  And if learning math is not easy, it follows that teaching math is not easy either. But it can help both learner and teacher if they know what the end result is supposed to be.

In my experience, many learners and teachers don’t know that. In both cases, the reason they don’t know it is that no one has bothered to tell them. There is a general but unstated assumption that everyone knows why the teaching and learning of mathematics is obligatory in every education system in the world.  But do they really?

There are two (very different) reasons for teaching and learning mathematics.

One reason is that it is a way of thinking that our species has developed over several thousand years, that provides wonderful exercise for the mind, and yields both challenging intellectual pleasure and rewarding aesthetic beauty to many who can find their way sufficiently far into it. In that respect, it is like music, drama, painting,  philosophy, natural sciences, and many other intellectual human activities. This is a perfectly valid reason to provide everyone with an opportunity to sample it, and make it possible for those who like what they see to pursue it as far as they desire. What it is not, is a valid reason for making learning math obligatory throughout elementary, middle, and high school education.

The argument behind math’s obligatory status in education is that it is useful; more precisely, it is useful in the practical, everyday world. This is the view of mathematics I am adopting in the short series of “Devlin’s Angle” essays of which this is the third. (There will be one more next month. See episode 1 here and episode 2 here.)

Indeed, mathematics is useful in the everyday practical world. In fact, we live in an age where mathematics is more relevant to our lives than at any previous time in human history. 

It is, then, perfectly valid to say that we force each generation of school students to learn math because it is a useful skill in today’s world. True, there are plenty of people who do just fine without having that skill, but they can do so only because there are enough other people around who do have it.

But let’s take that argument a step further. How do you teach mathematics so that it prepares young people to use it in the world? Clearly, you start by looking at the way people currently use math in the world, and figure out how best to get the next generation to that point. (Accepting that by the time those students finish school, the world’s demands may have moved on a bit, so those new graduates may have a bit of catch up and adjustment to make.)

If the way the professionals use math in the world changes, then the way we teach it should change as well.  Don’t you think? That’s certainly what has happened in the past.

For instance, in the ninth century, the Arabic-Persian speaking traders around Baghdad developed a new, and in many instances more efficient, way to do arithmetic calculations at scale, by using logical reasoning rather than arithmetic. Their new system, which quickly became known as al-jabr after one of the techniques they developed to solve equations, soon found its way into their math teaching.

When Hindu-Arabic arithmetic was introduced into Europe in the thirteenth century, the school systems fairly quickly adopted it into their arithmetic teaching as well. (It took a few decades, but knowledge moved no faster than the pace of a packhorse back then. I tell the story of that particular mathematics-led revolution in my 2011 book The Man of Numbers.)

The development of modern methods of accounting and the introduction of financial systems such as banks and insurance companies, which started in Italy around the same time, also led to new techniques being incorporated into the mathematical education of the next generation.

Later, when the sixteenth century French mathematician François Viète introduced symbolic algebra, it too became part of the educational canon.

In each case, those advances in mathematics were introduced to make mathematics more easy to use and to increase its application. There was never any question of “What is this good for?” People eagerly grabbed hold of each new development and made everyday use of it as soon as it became available.

The rise of modern science (starting with Galileo in the seventeenth century) and later the Industrial Revolution in the nineteenth century, led to still more impetus to develop new mathematical concepts and techniques, though some of those developments were geared more toward particular groups of professionals. (Calculus, for example.)

To make it possible for an average student or worker to make use of each new mathematical concept or technique, sets of formal calculating rules (algorithmic procedures) were developed and refined. Once mastered, these made it possible to make use of the new mathematics to handle—in a practical way—the tasks and problems of the everyday world for which those concepts and techniques had been developed to deal with in the first place.

As a result of all those advances, by the time the Baby Boomers came onto the educational scene in the 1950s, the curriculum of mathematical algorithms that were genuinely important in everyday life was fairly large. It was no longer possible for a student to understand all the underlying mathematical concepts and techniques behind the algorithms and procedures they had to learn. The best that they could do was master, by repetitive practice, the algorithmic procedures as quickly as possible and move on. [A few of us had difficulty doing that. We wanted to understand what was going on. By and large, we frustrated our teachers, who seemed to think we were simply troublesome slow learners. Some of us eventually learned to “play the mindless algorithm game” in class to pass the test, but kept struggling on our own to understand what was going on, setting us on a path to becoming mathematics professors in the 1970s.]

It was while that Boomer generation was going through the school system that mathematics underwent the first step of a seismic shift that within a half of a century would completely revolutionize the way mathematics was done. Not the pure mathematics practiced by a few specialists as an art—though that too would be impacted by the revolution to some extent. Rather, it was mathematics-as-used-in-the-world that would be radically transformed.

The first step of that revolution was the introduction of the electronic desktop calculator in 1961. Although, mechanical desktop calculators had been available since the turn of the Twentieth Century, by and large their use was restricted to specialists—often called “computers” in businesses. [I actually had a summer-job with British Petroleum as such a specialist in my last three years at high school, and it was in my final year in that job that the office I worked in acquired its first electronic desktop calculator and the British Petroleum plant bought its first digital computer, both of which I learned to use.] But with the increasing availability of electronic calculators, and in particular the introduction of pocket-sized versions in the early 1970s, their use in the workplace rapidly became ubiquitous. Mathematics underwent a major change. Humans no longer needed to do arithmetic calculations themselves, and professionals using arithmetic in their work no longer did.

It was not too many years later that, one by one, electronic systems were developed that could execute more and more mathematical procedures and techniques, until, by the late 1980s, there were systems that could handle all the mathematical procedures that constituted the bulk of not only the school mathematics curriculum, but the entire undergraduate math curriculum as well. The final nail in the coffin of humans needing to execute mathematical procedures was the release of the mathematics system Mathematica in 1988, followed soon after by the release of Maple.

In the scientific, industrial, engineering, and commercial worlds, each new tool was adopted as soon as it became available, and since the early 1990s, professionals using mathematical techniques to carry out real-world tasks and solve real-world problems have done so using tools like Mathematica, Maple, and a host of others that have been developed.

Simultaneously, colleges and universities quickly incorporated the use of those new tools into their teaching. And while the cost of the more extensive tools put their use beyond most schools, the graphing calculator too was quickly brought into the upper grades of the K-12 system, after its introduction in 1990.

Yet, while the pros in the various workplaces changed over to the new human-machine-symbiotic way of doing math with little hesitation, most educators, exhibiting very wise instincts, proceeded with far more caution. The first wave of humans to adopt the new, machine-aided approach had all learned mathematics in an age when you had to do everything yourself. Back then, “computers” were people. For them, it was easy and safe to switch to executing a few keystrokes to make a computer run a procedure they had carried out by hand many times themselves. But how does a young person growing up in this new, digital-tools-world learn how to use those new tools safely and effectively?

To some extent, the answer is (and was) obvious. You teach not for smooth, proficient, accurate execution of procedures, but for broad, general understanding of the underlying mathematics. The downplay of execution and increased emphasis on understanding are crucial. Computers outperform us to ridiculous degrees (of speed, accuracy, size of dataset,  and information storage and retrieval) when it comes to execution of an algorithm. But they do not understand mathematics. They do not understand the problem you are working on. They do not understand the world. They don't understand anything. 

People, on the other hand, can understand, and have a genetically inherited desire to do so.

But just how do you go about teaching for the kind of understanding and mastery that is required for students to transition into worlds and workplaces dominated by a wide array of new mathematical tools, where they will encounter work practices that involve very little by way of hand execution of algorithms?

We know so little about how people learn (though we do know a whole lot more than we did just a few decades ago), that most of us with a stake in the education business are rightly concerned about making any change that would effectively be a massive experiment on an entire generation. So we can, and should, expect small steps, particularly in systemic education.

In the U.S., the mathematicians who developed the mathematical guidelines for the Common Core State Standards made a good first attempt at such a small step. True, it quickly ran into difficulties when it came to implementing the guidelines in a large and complex public educational system that is answerable to the public. But that is surely a temporary hiccup. Most of the problems at launch came from a lack of effective ways to assess the new kind of learning. Those problems can be and are being fixed. Which is just as well. For, although it’s possible to argue for tinkering with specific details of the Common Core State Standards guidelines, in terms of setting out a broad set of educational goals to aim for, there is no viable alternative first step. The pre-1970s educational approach is no longer an option.

In the meantime, individual teachers at some schools (particularly, but not exclusively, private schools) have been trying different approaches, in some cases sharing their experiences on the MTBOS (Math Twitter Blog-O-Sphere), making use of another technological tool (social media) now widely available. [For a quick overview of one global initiative to support and promote such innovations, the OECD’s Innovative Pedagogies for Powerful Learning project (IPPL), see this recent article from the Brookings Institution.]

The mini-course I gave at Nueva School in the San Francisco Bay Area last January, which I talked about in the first of this short series of essays, is one such experiment in teaching mathematics in a way that best prepares the next generation for the world they will live and work in after graduation. I tested it first with a class of non-science majors in Princeton in the fall of 2015 and then again with an Adult Education class at Stanford in the fall of 2017. The Nueva School class was its third outing.

With the above backstory now established, next month I will describe that course and talk about how today’s pros “do the math”. (Again, let me stress, I am not talking here about “pure math”, the academic discipline carried out by professional mathematicians in universities and a few think tanks. My focus here is on using math in the everyday world.)

In the meantime, I’ll leave you with a simple arithmetic problem that I will discuss in detail next time.

It comes with two instructions:

  1. Solve it as quickly as you can, in your head if possible. Let your mind jump to the answer.
  2. Then, and only then, reflect on your answer, and how you got it.
The goal here is not to get the right answer, though a great many of you will. Rather, the issue is how do our minds work, and how can we make our thinking more effective in a world where machines execute all the mathematical procedures for us?

Ready for the problem? Here it is. 

PROBLEM: A bat and a ball cost $1.10. The bat costs $1 more than the ball. How much does the ball cost on its own? (There is no special pricing deal.)