It was Twenty Years Ago Today

The title of the famous Beatles song does not exactly apply to Devlin’s Angle. The online column (now run on a blog platform, but unlike most blogs, still subject to an editor’s guiding hand) is in its twentieth year, but it actually launched on January 1, 1996.

In last month’s column, I looked back at the very first post. It was a fascinating exercise to try to put myself back in the mindset of how the world looked back then, which was about the time when the World Wide Web was just starting to find its way onto university campuses, but had not yet penetrated the everyday lives of most of the world’s population.

That period of intense technological and societal change – looking back, it is clear it was just beginning, in the first half of the 1990s being more evolutionary rather than the revolutionary that was soon to follow – and the strong sensation of change both underway and pending, is reflected in some of the topics I chose to write about each month in that first year. Here is a list of those first twelve posts, with hyperlinks.

• "Good Times," January, 1996
• "Base Considerations," February, 1996
• "Deep Blue," March, 1996
• "Are Mathematicians Turning Soft?," April, 1996
• "The Five Percent Solution," May, 1996
• "Laws of Thought," June, 1996
• "Tversky's Legacy Revisited," July, 1996
• "Of Men, Mathematics, and Myths," August, 1996
• "Dear New Student," September, 1996
• "Wanted: A New Mix," October, 1996
• "Spreading the Word," November, 1996
• "Moment of Truth," December, 1996

Along with essays you might find in a mathematics magazine for students (February, June, July, August, November, December), there are reflections on where mathematics and its role in the world might be heading in the next few years.

January’s post, about the growth of computer viruses in the digital domain, was clearly in that Brave New World vein, as I noted last month, and in February I focused on another aspect of the rapid growth of the digital world, with a look at the ongoing debate about the future of Artificial Intelligence. Though that field has undoubtedly made many advances in the ensuing two decades, the core argument I summarized there seems as valid today as it did then. Digital devices still do not “think” in anything like a human fashion (though these days it can sometimes be harder to tell the difference).

The posts for April, May, and October looked at different aspects of the “Where is mathematics heading?” question. Of course, I was not claiming then, nor am I suggesting now, that the core of pure mathematics is going to change. (Though the growth of Experimental Mathematics in the New Millennium was a new direction, one I addressed in a Devlin’s Angle post in March 2009.) Rather, I was taking a much broader view of mathematics, stepping outside the mathematics department of colleges and universities and looking at the way mathematics is used in the world.

The October post, in particular, turned out to be highly prophetic for my own career. Shortly after the terrorist attack on the World Trade Center on September 11, 2001, I was contacted by a large defense contractor, asking if I would join a large team they were putting together to bid for a Defense Department contract to find ways to improve intelligence analysis. I accepted the offer, and worked on that project for the next several years. (From my perspective, that project and the work that followed did not end uniformly well, as I lamented in an AMS Notices opinion piece in 2014.) When that project ended, I did similar work for a large contractor to the US Navy and another project for the US Army. In all three projects, I was living in the kind of world I portrayed in that October, 1996 column.

In fact, my professional life as a mathematician for the entire life of Devlin’s Angle has been in that world – a way of using mathematics I started to refer to as “mathematical thinking.” In a Devlin’s Angle post in 2012, I tried to articulate what I mean by that term. (The term is used by others, sometimes with different meanings, though I see strong overlaps and general agreements among them all.) That same year, I launched the world’s first mathematical MOOC on the newly established online course platform Coursera, with the title “Introduction to Mathematical Thinking”, and published a book with the same title.

With the world as it is today, in particular the pervasive (though largely hidden) role played by mathematics and mathematical ideas in almost every aspect of our lives, I would hazard a guess that there are far more people using “mathematical thinking” than there are people doing mathematics in the traditional sense.

If so, that would make the professions of mathematician and mathematics educator two of the most secure careers in the world. For there is one thing in particular you need in order to engage in (effective) mathematical thinking about a real world problem: an adequate knowledge of, and conceptual understanding of, mathematics. In fact, that need was ever so, but it often tended to be overlooked in the pre-digital eras, when doing mathematics meant engaging in a lot of paper-and- pencil, symbolic computations, which meant that the bulk of mathematics instruction focused on computation, with wide ranging knowledge and conceptual understanding often getting short shrift.

But those days are gone. Today, we carry around in our pockets devices that give us instant access to pretty well all of the world’s mathematical information and computational procedures we might need to use. (Check out Wolfram Alpha.) But the thinking still has to be done where it always has: in our heads.

Against Answer Getting

"Correct answers are essential... but they're part of the process, they're not the product. The product is the math the kids walk away with in their heads." —Phil Daro

If you have not already watched Phil Daro's 17-minute video Against Answer Getting, you should do so right away. (I'll keep this post short to give you enough time to watch it in its entirety.)

Daro, a longtime mathematics educator and leading figure in the national mathematics education community, is currently director of the San Francisco field site of SERP, the Strategic Education Research Partnership. He was one of the mathematics educators who played a leading role in the formulation of the mathematics Common Core State Standards. (You know, one of those knowledgeable experts the StopCommonCore brigade keep claiming were not involved in CCSS development.)

The video is full of powerful insights that the mathematics education community has accumulated over many years of research. My opening quote sums up the focus of the video. Here is another one I like:

"Mathematics does not break down into lesson-sized pieces." —Phil Daro

This particular quote resonates with me. I adopted the same principle in the design of my MOOC Introduction to Mathematical Thinking, currently about halfway through its fifth run.

Daro's focus, both in the video and in his work in general, is K-12 mathematics education. But it is very relevant to those of us in college-level mathematics education. When students come to college with a perception that mathematics is about "answer getting," we face the very uphill task of ridding them of that misleading mindset.

True, for hundreds of years, getting answers was a key component of learning and doing mathematics. But these days, if we want answers in mathematics, we generally use one of a number of digital technologies. The job of today's mathematician (or typical user of mathematics) is problem solving. The part that requires a human mind is when the problem has a novel aspect. It was precisely to put the focus on the thinking part that I named my MOOC the way I did.

The principle requirement for being able to solve a novel problem is conceptual understanding. That is why the issues Daro raises in that video are so central to the mathematics education of the citizens of tomorrow.

The outdated mindset about the purpose of mathematics that many students bring with them when they transition from school to college is not the only problem many have to overcome. A parallel issue manifests itself when they start to learn about mathematical proofs (if they follow the mathematics path).

My MOOC students are currently right in the middle of that part of the course (proofs), and many are having a very hard time coming to understand what role proofs play and what (therefore) constitutes a good proof.

The dominant perception is that proofs are what mathematicians produce in order to determine mathematical truth. That, of course, is true (at least in an idealistic sense that guides mathematical progress), but as with arithmetic answer getting, it is only part of the story. And in terms of actual mathematical practice, a very small part of the story.

As with answer getting in K-12 math, achieving a logically correct proof is a binary target (right or wrong), which make both very easy to evaluate for correctness and assign a numerical grade. (Ka-ching!)

But let's pause and ask ourselves how proofs work in practice. If you want to know if Fermat's Last Theorem is true, you consult a reliable source. Today, any moderately knowledgeable mathematician will tell you the answer: "Yes." Now you know.

But what if you want to know why it is true. That's when you need to look at a proof.

In terms of mathematical practice, proofs are about understanding. They are communicative devices we construct to convince ourselves and to convince others.

In my MOOC, because I cannot assume the students have access to individualized, expert feedback on their work, I do not ask them to construct proofs. But I do present them with a range of purported proofs, some correct, others not, and ask them to evaluate them. The evaluation is in terms both of logical correctness and communicative effectiveness.

To do this, I ask them to look at each purported proof in terms of five different factors: one logical correctness, the others focusing on communicative issues. Though the five factors are not independent variables, I ask them to treat them as such when evaluating a proof.

This is the part of the course where those students who have had some exposure to proofs in the K-12 system tend to do worse than those who are new to proofs. They are simply not able to approach a proof other than in the "answer getting" mode of "Is it logically correct?"

This shows up dramatically with extremal cases. When I present them with a carefully constructed argument that is logically correct but provides no explanation, they will give it high marks across the board. But faced with an argument that is superbly articulated but has a logical flaw, they are psychologically unable to evaluate the structure of the argument. "It's wrong," they keep saying. End of story (for them).

Of course, extremal examples are atypical, and often difficult to wrap our minds around. That's what makes them so valuable as learning devices. It's when the classroom rubber hits the road and we find ourselves using mathematical thinking in our lives or careers that it becomes important to have good communication skills.

Pick up a more advanced level mathematics book or research article and the chances are high that the arguments presented will contain errors. (Actually, the book does not have to be advanced. Euclid's Elements is littered with "proofs" that are not logically sound.) But if the arguments are well laid out, with adequate explanations, a suitably skilled reader can fix them as they go along—possibly with help from someone else. (That's definitely the case with Elements, though it took two thousand years before David Hilbert noticed that Euclid's own arguments left a lot of work to be done to make them genuine "proofs.")

It's the same in software engineering. Any useful program will have bugs. But if the code is well structured, and adequately annotated, someone else can dive in and fix it whenever a flaw manifests. A good computer programmer is not someone who writes error-free, working code; it is someone who writes working code that can easily be fixed or modified.

I'll leave it as an exercise for the reader to identify the analogous issue in the natural sciences.

If those of us in the education business want to do the best we can to prepare our students for life in the 21st century, we need to recognize that in an era when technologies provide instant answers (facts), the one ability they will need above anything else is (creative, reflective) thinking. 

Most Math Problems Do Not Have a Unique Right Answer

One of the most widely held misconceptions about mathematics is that a math problem has a unique correct answer.

(Some of those who hold that view also think that there is just one correct way to get that answer. A far smaller group, to be sure, but still a worryingly large number. Still, my focus here is on the first false belief.)

Having earned my living as a mathematician for over 40 years, I can assure you that the belief is false. In addition to my university research, I have done mathematical work for the U. S. Intelligence Community, the U.S. Army, private defense contractors, and a number of for-profit companies. In not one of those projects was I paid to find "the right answer." No one thought for one moment that there could be such a thing.

So what is the origin of those false beliefs? It's hardly a mystery. People form that misconception because of their experience at school. In school mathematics, students are only exposed to problems that (a) are well defined, (b) have a unique correct answer, and (c) whose answer can be obtained with a few lines of calculation.

But the only career in which a high school graduate can expect to continue to work on such problems is academic research in pure mathematics—and even then (and again speaking from many years of personal experience), cleanly specified problems that have (obtainable) "right answers" are not as common as you might think.

Since the vast majority of students who go through school math classes do not end up as university research mathematicians, whereas many do find themselves in careers that require some mathematical ability, it's reasonable to ask why their entire school mathematics education focuses exclusively on one tiny fraction of all possible mathematics problems.

The answer can be found by looking at the history of mathematics. Starting with the invention of numbers around 10,000 years ago, people developed mathematical methods to solve problems they faced in the world: arithmetic and algebra to use in trade and engineering, geometry and trigonometry for building and navigation, calculus for scientific research, and so forth.

While some of that mathematics was required only by specialists (e.g. calculus), arithmetic and parts of algebra in particular were essential for everyday living. As a consequence, mathematicians wrote books from which ordinary people could learn how to calculate. From the very earliest textbooks (Babylonian tablets, Indian manuscripts, etc.), two kinds of problems were presented: algorithm ("recipes") problems that showed the steps to be carried out to do a particular kind of computation, presented without any context, and word problems, designed to help people learn how to apply a particular algorithm to solve a real world problem. Ancient and medieval textbooks had many hundreds of such problems, so that a trader (say) could find a problem almost identical in form to the one he (and back then use of mathematics was primarily a male activity) actually wanted to solve in his business. If he were lucky, all he would have to do is substitute his own numbers for those in the book's worked word problem. In other cases, the book might not provide an exact match, but by working through five or six problems that were close in form, the individual could learn how to solve his real problem.

For the majority of people, that was enough. Life simply did not require anything more. The problems they faced in their everyday activities for which mathematics was needed were simple and routine. The mathematical word problems that today seem so unrealistic were by and large remarkably similar to the problems ordinary citizens faced every day.

"When do I need to leave home in order to catch that train?" There wasn't an app to tell you the answer; you had to calculate it yourself. That word problem about trains leaving stations in your math class showed you how.

Arithmetic, in particular, was an essential, basic life skill that remained so until the development of devices that automated the process in the 1960s. I am a member of the last generation for whom the question "What do I need arithmetic for?" simply did not arise. (We asked it about other parts of mathematics.)

But that computer technology that eliminated the need for people to be good calculators led to a world in which there is a huge demand for higher order mathematical skills, starting with algebra. I wrote about this change in this column back in 1998, in a piece titled "Forget 'Back to Basics.' It's Time for 'Forward to (the New) Basics.'" Looking back at what I wrote then, I am amazed at just how much things have changed in the intervening 16 years. In September of that year, Google was founded, and the Web became a dominant force in our lives and our work.

Today, we have instant access to vast amounts of information and to unlimited computing power. Both are now utilities, much like water and electricity. And that has led to a revolution in the mathematics ordinary citizens need in order to lead a fulfilling, productive life. In a world where procedural (i.e., algorithmic) mathematics is available at the push of a button, the need has shifted to what I and others have been calling mathematical thinking.

I wrote about this in my September 2012 Devlin's Angle. Broadly speaking, mathematical thinking is a way of approaching problems that is based on classical mathematics, but takes account of the fact that computation (both numeric and symbolic) can be readily done by machines.

In practical terms, what this means is that people can now focus all their attention on real-world problems in the form they are encountered. Knowing how to solve an equation is no longer a valuable human ability; what matters now is formulating the equation to solve that problem in the first place, and then taking the result of the machine solution to the equation and making use of it.

In the 1960s, we got used to the fact that the arithmetic part of solving a mathematical problem could be done by machines. Now we are in a world where almost all the procedural mathematics can be done by machines.

Of course, this does not mean we should stop teaching procedural mathematics to the next generation, any more than the introduction of pocket calculators meant we should stop teaching arithmetic. But in both cases, the reason for teaching changes, and with it the way we should teach it. The purpose shifts from mastering procedures—something that was necessary only when there were no machines to do that part—to understanding the concepts sufficiently well to make good use of those machines.

Though this change in emphasis has been underway for some years now, it did not garner much attention in the United States until the rollout of the Common Core State Standards, which are very much geared towards the mathematical thinking needs of the 21st century. The degree to which many parents were shortsighted by the shift was made clear when some of them took to social media to complain about the kinds of homework questions their children were being asked to do. While some of those questions were truly, truly awful, others garnering a lot of critical SM comments were actually extremely good.

What was particularly ironic was that many parents, faced with being unable to assist their child with elementary grade arithmetic homework, did not draw the obvious conclusion: "Gee, if I cannot understand something as basic as integer arithmetic—however it is done—there must have been something really lacking in my own education." Instead, they jumped to the totally off-the-wall conclusion that the current educational system must be wrong.

That's like waking up in the morning to find your car won't start and saying, "Oh dear, the laws of physics don't work." The smart person says, "I need to replace the battery."

I'll tell you something. I was taught math the "old-fashioned way" too, and some of those student arithmetic worksheets were new to me when I first saw them. But regardless of any views I might have as to how it is best taught in today's world, it didn't take a lot of effort to figure out what those kids were doing on those worksheets posted on Facebook. It was just whole number arithmetic for heavens sake! Anyone who understands the basic ideas of whole number arithmetic can figure it out.

It was not my training as a professional mathematician that helped me here. It was the simple fact that I understand whole number arithmetic, something that goes back to my early childhood, when I did not even know there was such a thing as a professional mathematician, let alone aspire to be one. Unfortunately, many Americans were never taught to understand arithmetic, they were just trained to execute procedures. It's not their kids who are being short-changed. They—the parents—were!

Breezing into this fray is University of Wisconsin mathematics professor Jordan Ellenberg, with his new book How Not To Be Wrong. I knew I would find a kindred spirit when I read the book's subtitle: “The Power of Mathematical Thinking.” With a Stanford MOOC and an associated textbook both called Introduction to Mathematical Thinking, how could I not?

Ellenberg's title is superb. In one fell swoop, it casts aside that old misconception that mathematics provides "right answers," replacing it with the far more accurate description that it is a great way to stop you being wrong. For, like me, he focuses not on the internal activities of pure mathematics, rather on how mathematics is used in today's real world.

To be sure, also like me, Ellenberg has devoted a lot of his career to working in pure mathematics, so he loves searching for those "right answers," and he enjoys the subject in its own terms. We both know that there are eternal truths within mathematics (a better term would be "tautologies") and have experienced the thrill of going after them. But we both realize that what we do as pure mathematicians is a very specialist pursuit. The society that supports us when we do that does so largely because of the payoff in terms of the benefits that emerge when mathematical thinking is applied to real world problems.

Ellenberg's book is chock full of examples of those benefits, from many walks of life, presented with a delightfully light touch. He grabs the reader's attention with his very first example, taken from the Second World War. The U. S. military chiefs wanted to reduce the number of warplanes that were being shot down. The obvious solution was to add more armor to protect them. But armor adds weight, which limits the distances that can be flown and the duration of the mission, as well as increasing the production cost. So the question was, where is the most effective place to put that extra protection?

To answer this question, the chiefs brought in a team of mathematicians to analyze the evidence and determine what parts of the aircraft were most likely to be hit. They examined the fuselages of all the damaged planes that had flown back after being hit to see where the most damage was. It turned out that the engines had an average of 1.11 bullet holes per square foot, the fuel system had 1.55, the fuselages 1.73, and the rest of the plane 1.8.

So where was the optimal place to add extra armor? According to the data, the fuselages took a lot of hits, while engines suffered the least damage. So an obvious suggestion was to add armor to the fuselages. But that was not what the mathematicians suggested. Their solution was to add the armor to the engines, the part that had fewer hits when the planes got back.

And they were right. I'll leave you to figure out why that is the best solution. It's a great example of mathematical thinking. After you have convinced yourself why adding armor to the engines was the best strategy, you should buy a copy of Ellenberg's book and gain some understanding of just what mathematical thinking is, and why it is a crucial ability in today's world.

(My own book on mathematical thinking is more of a "how to" guide, as is my MOOC. Another, excellent book on mathematical thinking, that is somewhere between Ellenberg's and mine, is Burger and Starbird's The 5 Elements of Effective Thinking.)

Finally, and to some extent switching gears (and definitely switching media), I want to draw your attention to a new video game, DragonBox Elements, by the Norwegian-based educational technology company WeWantToKnow. The company made a splash with its first game, DragonBox (Algebra) a couple of years ago.

Unlike my own work in educational videogames, through my company BrainQuake, which is very strongly focused on real-world mathematical thinking, the DragonBox folks are seeking to enhance and strengthen school mathematics.

When I first played the new Elements game, I was initially confused, since I approached it with a Geometer's Sketchpad expectation. But Elements is not a geometry construction/exploration tool. The focus is on the importance of providing justification for steps in a proof. Knowing why something is true. And that is not only a key feature of GOFM (“Good Old Fashioned Math”), as was taught for two thousand years, it's one of the aspects of mathematics that is characteristic of mathematical thinking (as used in the real world). Euclid, the author of the first Elements (the book), would surely have approved.

The modern world has not made GOFM redundant. What has changed, and drastically, is the way GOFM fits in with the rest of human activities. Unless you are going to make a career for yourself in pure mathematics research, GOFM today is simply an amazingly powerful tool for acquiring one of the most important cognitive capacities in the 21st century: mathematical thinking.

In today's world, most of the important problems are complex and multi-faceted. There are few right answers. As Ellenberg demonstrates, mathematical thinking can help you choose better answers—and avoid being wrong.

The Power of Dots

On June 29, the New York Times ran a story about the Common Core Mathematics Standards. If ever you wanted proof of the dismal mathematics education most Americans have been provided, you will find it in the story’s “human interest lede,” which described one mother’s response to seeing her daughter’s homework. By taking the daughter out of school to teach her herself “the old fashioned way” she herself had been subjected to, this well-meaning parent was ensuring that, as had clearly been the case for the mother, the daughter too would not be exposed to real mathematical thinking—the kind that in today’s world is a key to the most attractive jobs. Instead she would be subjected to the same, dreary, rote-skills-drills inflicted on previous generations—a process designed to train people for routine work in the pre-computer era, but so hopelessly inadequate for the 21st century that parents are un-equipped to figure out for themselves the simple (albeit unfamiliar) math homework their children are assigned.

Surely, if mathematics education should achieve one thing, it is develop the ability to figure things out for yourself. We’re not talking the Riemann Hypothesis here; the focus is basic school arithmetic, for heaven’s sake.

To continue with the Times article, arrays of dots seemed to figure large in this parent’s dislike of the Common Core. She felt it was pointless to spend time drawing and staring at arrays of dots.

True, it would be possible—and I am sure it happens—to generate tedious, and largely pointless, “busywork” exercises involving drawing arrays of dots. But the image of a Common Core math worksheet the Times chose to illustrate its story showed a very sensible, and deep use of dot diagrams, to understand structure in arithmetic. Much like the (extremely deep) dot array at the top of this article, which I’ll come to in a moment.

To the girl’s parent, mathematics is about numbers, but that’s just a surface feature. It’s really about structure. And throughout the ages, mathematicians have used the most simple symbols possible to bring out and understand that structure: namely, dots and lines.

The Times’ parent, so dismissive of time spent drawing and reflecting on dot diagrams, would, I am sure, think it a waste of time to devote any effort trying to make sense of the dot diagram I used to open this post. She would, I have no doubt, find it incomprehensible that an individual with a freshly-minted Ph.D. in mathematics would spend many months—at taxpayers’ expense—staring day-after-day at either that one diagram, or seemingly minor variations he would start each day by sketching out on a sheet of paper in front of him.

Well, I am that mathematician. That diagram helped me understand the framework that would be required to specify an infinite mathematical object of the third order of infinitude (aleph-2) by means of a family of infinite mathematical objects of the first order of infinitude (aleph-0). The top line of dots represents an increasing tower of objects that come together to form the desired aleph-2 object, and each of the lower lines of dots represent shorter towers of aleph-0 objects. In the 1970s, a number of us used those dot diagrams to solve mathematical problems that just a few years earlier had seemed impossible.

That particular kind of dot diagram was invented by a close senior colleague (and mentor) of mine, Professor Ronald Jensen, who called it a “morass.” He chose the name wisely, since the structure represented by those dots was extremely complex and intricate.

In contrast, the simple, rectangular array implicitly referred to in the New York Times article is used to help learners understand the much simpler (but still deep, and far more important to society) structure of numbers and the basic operations of arithmetic, as was well explained in a subsequent blog post by mathematics education specialist Christopher Danielson. The fact is, dot diagrams are powerful, for learners and world experts alike.

The problem facing parents (and many teachers) today, is that the present student generation is the one that, for the first time in history, is having to learn the mathematics the professionals use—what I and many other pros have started to call “mathematical thinking” in order to distinguish it from the procedural skills so important in past times.

The reason for that is that in the world today’s students will graduate into, computation is as plentiful as water or electricity. The smartphone we carry around with us is much faster, and more accurate, in carrying out mathematical procedures than any human.

In a single generation, society’s need for mathematical mastery has gone from procedural computation, to being able to make effective and reliable use of an effectively unlimited amount of automated computation. To put it bluntly, mastery of computational skills is no longer a marketable asset. The ability to make good use of computational power is where it’s at in math today.

For almost all the three thousand years of mathematical development, the focus in mathematics was calculation (numerical, symbolic, or geometric). Learning mathematics meant learning how to perform those calculations, which boiled down to achieving mastery of various procedures. Mastery of any one procedure could be achieved by rote learning—doing many examples, all essentially the same—leaving the only truly creative mental task that of recognition of which procedure to apply to solve which problem.

Numerical and symbolic calculation (arithmetic and algebra) are so simple and routine that we can program computers to do it for us. That is possible because calculation is essentially trivial. Perceiving and understanding structure, on the other hand, is something that (at least at the present time) requires human insight. It is not trivial and it is difficult. Dot diagrams can help us come to terms with that difficulty.

When movie director Gus Van Sant was faced with introducing the lead character, Will Hunting (played by Matt Damon) in the hit 1997 film Good Will Hunting, establishing in one shot that the hero was an uneducated (actually, self-educated) mathematical genius, the first encounter we had with Will showed him drawing a dot diagram on a blackboard in an MIT corridor.

You can be sure that when an experienced movie director like Gus Van Sant selects an establishing shot for the lead character, he does so with considerable care, on the advice of an expert. By showing Will writing a network of dots on a blackboard, Van Sant was right on the button in terms of portraying the kind of thing that professional mathematicians do all the time.

The one bit of license Van Sant took was that the diagram we saw Matt Damon writing was not the solution to a problem that had taken an MIT math professor two years to solve. (Unless MIT math professors are a lot less smart than we are led to believe!) It was a real solution to a real math problem, all right. I am pretty sure it was chosen because it fitted nicely on one blackboard and looked good on the screen. It absolutely conveyed the kind of (dotty) activity that mathematicians do all the time—the kind of (dotty) thing I did in my early post-Ph.D. years when I was working with Prof Jensen’s morasses.

But it’s actually a problem that anyone who has learned how to think mathematically should be able to solve in at most a few hours. Numberphile has an excellent video explaining the problem.

So, New York Times story parent, I hope you reconsider your decision to take your daughter out of school to teach her the way you were taught. The kind of mathematics you were taught was indeed required in times past. But not any more. The world has changed dramatically as far as mathematics is concerned. As with many other aspects of our lives, we have built machines to handle the more routine, procedural stuff, thereby putting a premium on the one thing where humans vastly outperform computers: creative thinking.

Those dot diagrams are all about creative thinking. A computer can understand numbers, and process millions of them faster than a human can write just one. But it cannot make sense of those dot diagrams. Because it does not know what any particular array of dots means! And it has no way to figure it out. (Unless a human tells it.)

Next month I’ll look further into the distinction between old-style procedural mathematics and the 21st-century need for mathematical thinking. In particular, I’ll look at an excellent recent book, Jordan Ellenberg’s How Not to be Wrong.

The book’s title is significant, since it recognizes that the vast majority of real-world mathematical problems do not have a unique right answer, and that the real power of mathematical thinking is making sure you are not wrong. (The book’s subtitle is “The power of mathematical thinking.”)

I’ll also look at a new mathematics video game that also focuses on mathematical thinking, this time, school-room Euclidean geometry. It’s called DragonBox Elements.

You might want to check out both.

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 in Devlin’s more wide ranging blog profkeithdevlin.org.

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

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

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?

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

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.

The short, narrow descent

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.

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.

(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 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

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 to 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, almost 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 ...

See "How Mountain Biking Can Provide the Key to the Eureka Moment"

23 and Me. Play it again, Sam

It’s one of the most famous lines from one of the most famous movies of all time, Casablanca. Except it’s not what Ilsa, played by Ingrid Bergman, actually said, which was “Play it once, Sam, for old times' sake . . . [NO RESPONSE] . . . Play it, Sam. Play 'As Time Goes By.'”

This month’s column is in response to the emails I receive from time to time asking for a reference to articles I have written for the MAA since I began on that mathemaliterary journey back in 1991. (Yes, I just made that word up. Google returns nothing. But it soon will.)

I first started writing monthly articles for the MAA back in September 1991 when I took over as editor of the Association’s monthly print magazine FOCUS. When I stepped down as FOCUS editor in January 1996, the MAA launched its website, and along with it Devlin’s Angle.

During that time, in addition to moving from print to online, the MAA website went through two overhauls, leaving the archives spread over three volumes:

January 1996 – December 2003

http://www.maa.org/external_archive/devlin/devlin_archives.html

January 2004 – July 2011

http://www.maa.org/external_archive/devlin/devangle.html

August 2011 – present

http://devlinsangle.blogspot.com

Throughout those 23 years, I’ve wandered far and wide across the mathematical and mathematics education landscape. But three ongoing themes emerged. None of them was planned. In each case, I simply wrote something that generated interest – and for one theme considerable controversy – and as a result I kept coming back to it.

I continue to receive emails asking about articles I wrote on the first two of those three themes, and the third is still very active. So I am devoting this month’s column to providing an index to those three themes.

I’ll start with the most controversial: what is multiplication? This began innocently enough, with a throw-away final remark to a piece I wrote back in 2007. I little knew the firestorm I was about to unleash.

What is Multiplication?

September 2007, What is conceptual understanding?

http://www.maa.org/external_archive/devlin/devlin_09_07.html

June 2008, It Ain't No Repeated Addition

http://www.maa.org/external_archive/devlin/devlin_06_08.html

July-August 2008, It's Still Not Repeated Addition

http://www.maa.org/external_archive/devlin/devlin_0708_08.html

September 2008, Multiplication and Those Pesky British Spellings

http://www.maa.org/external_archive/devlin/devlin_09_08.html

December 2008, How Do We Learn Math?

http://www.maa.org/external_archive/devlin/devlin_12_08.html

January 2009, Should Children Learn Math by Starting with Counting?

http://www.maa.org/external_archive/devlin/devlin_01_09.html

January 2010, Repeated Addition - One More Spin

http://www.maa.org/external_archive/devlin/devlin_01_10.html

January 2011, What Exactly is Multiplication?

http://www.maa.org/external_archive/devlin/devlin_01_11.html

November 2011, How multiplication is really defined in Peano arithmetic

http://devlinsangle.blogspot.com/2011_11_01_archive.html

Mathematical Thinking

I first started making the distinction between mathematics and mathematical thinking in the early 1990s, when an extended foray into mathematical linguistics and then sociolinguistics led to an interest in mathematical cognition that continues to this day.

April 1996, Are Mathematicians Turning Soft?

http://www.maa.org/external_archive/devlin/devlinangle_april.html

October 1996, Wanted: A New Mix

http://www.maa.org/external_archive/devlin/october.html

September 1999, What Can Mathematics Do For The Businessperson?

http://www.maa.org/external_archive/devlin/devlin_9_99.html

January 2008, American Mathematics in a Flat World

http://www.maa.org/external_archive/devlin/devlin_01_08.html

February 2008, Mathematics for the President and Congress

http://www.maa.org/external_archive/devlin/devlin_02_08.html

October 2009, Soft Mathematics

http://www.maa.org/external_archive/devlin/devlin_10_09.html

July 2010, Wanted: Innovative Mathematical Thinking

http://www.maa.org/external_archive/devlin/devlin_07_10.html

September 2012, What is mathematical thinking?

http://devlinsangle.blogspot.com/2012/08/what-is-mathematical-thinking.html

MOOCS

No introduction necessary. MOOCs are constantly in the news. Though I was one of the early pioneers in developing the Stanford MOOCs that generated all the media interest in 2012, and I believe the first person to offer a mathematics MOOC (Introduction to Mathematical Thinking), the idea goes back to a course given at Athabasca University in Canada, back in 2008.

May 2012, Math MOOC – Coming this fall. Let’s Teach the World

http://devlinsangle.blogspot.com/2012_05_01_archive.html

November 2012, MOOC Lessons

http://devlinsangle.blogspot.com/2012_11_01_archive.html

December 2012, The Darwinization of Higher Education

http://devlinsangle.blogspot.com/2012/12/the-darwinization-of-higher-education.html

January 2013, R.I.P. Mathematics? Maybe.

http://devlinsangle.blogspot.com/2013_01_01_archive.html

February 2013, The Problem with Instructional Videos

http://devlinsangle.blogspot.com/2013_02_01_archive.html

March 2013, Can we make constructive use of machine-graded, multiple-choice questions in university mathematics education?

http://devlinsangle.blogspot.com/2013_03_01_archive.html

September 2013, Two Startups in One Week

http://devlinsangle.blogspot.com/2013_09_01_archive.html

More about MOOCs

In addition to the MOOC articles listed above, I have also written articles about the topic in my own blog MOOCtalk.org and for the Huffington Post. Here are the references:

MOOCTALK

An irregular series of posts starting on May 5, 2012

http://mooctalk.org

HUFFINGTON POST

December 2013, MOQR, Anyone? Learning by Evaluating

http://devlinsangle.blogspot.com/2013_12_01_archive.html

March 2, 2013, MOOCs and the Myths of Dropout Rates and Certification

http://www.huffingtonpost.com/dr-keith-devlin/moocs-and-the-myths-of-dr_b_2785808.html

March 27, 2013, Can Massive Open Online Courses Make Up for an Outdated K-12 Education System?

http://www.huffingtonpost.com/dr-keith-devlin/massive-open-online-courses_b_2946591.html

August 19, 2013, MOOC Mania Meets the Sober Reality of Education

http://www.huffingtonpost.com/dr-keith-devlin/mooc_b_3741625.html

November 18, 2013, Why MOOCs May Still Be Silicon Valley's Next Grand Challenge

http://www.huffingtonpost.com/dr-keith-devlin/why-moocs-remain-silicon-_b_4289739.html

MOQR, Anyone? Learning by Evaluating

Many colleges and universities have a mathematics or quantitative reasoning requirement that ensures that no student graduates without completing at least one sufficiently mathematical course.

Recognizing that taking a regular first-year mathematics course—designed for students majoring in mathematics, science, or engineering—to satisfy a QR requirement is not educationally optimal (and sometimes a distraction for the instructor and the TAs who have to deal with students who are neither motivated nor well prepared for the full rigors and pace of a mathematics course), many institutions offer special QR courses.

I’ve always enjoyed giving such courses, since they offer the freedom to cover a wide swathe of mathematics—often new or topical parts of mathematics. Admittedly they do so at a much more shallow depth than in other courses, but a depth that was always a challenge for most students who signed up.

Having been one of the pioneers of so-called “transition courses” for incoming mathematics majors back in the 1970s, and having given such courses many times in the intervening years, I never doubted that a lot of the material was well suited to the student in search of meeting a QR requirement. The problem with classifying a transition course as a QR option is that the goal of preparing an incoming student for the rigors of college algebra and real analysis is at odds with the intent of a QR requirement. So I never did that.

Enter MOOCs. A lot of the stuff that is written about these relatively new entrants to the higher education landscape is unsubstantiated hype and breathless (if not fearful) speculation. The plain fact is that right now no one really knows what MOOCs will end up looking like, what part or parts of the population they will eventually serve, or exactly how and where they will fit in with the rest of higher education. Like most others I know who are experimenting with this new medium, I am treating it very much as just that: an experiment.

The first version of my MOOC Introduction to Mathematical Thinking, offered in the fall of 2012, was essentially the first three-quarters of my regular transition course, modified to make initial entry much easier, delivered as a MOOC. Since then, as I have experimented with different aspects of online education, I have been slowly modifying it to function as a QR-course, since improved quantitative reasoning is surely a natural (and laudable) goal for online courses with global reach—that “free education for the world” goal is still the main MOOC-motivator for me.

I am certainly not viewing my MOOC as an online course to satisfy a college QR requirement. That may happen, but, as I noted above, no one has any real idea what role(s) MOOCs will end up fulfilling. Remember, in just twelve months, the Stanford MOOC startup Udacity, which initiated all the media hype, went from “teach the entire world for free” to “offer corporate training for a fee.” (For my (upbeat) commentary on this rapid progression, see my article in the Huffington Post.)

Rather, I am taking advantage of the fact that free, no-credential MOOCs currently provide a superb vehicle to experiment with ideas both for classroom teaching and for online education. Those of us at the teaching end not only learn what the medium can offer, we also discover ways to improve our classroom teaching; while those who register as students get a totally free learning opportunity. (Roughly three-quarters of them already have a college degree, but MOOC enrollees also include thousands of first-time higher education students from parts of the world that offer limited or no higher education opportunities.)

The biggest challenge facing anyone who wants to offer a MOOC in higher mathematics is how to handle the fact that many of the students will never receive expert feedback on their work. This is particularly acute when it comes to learning how to prove things. That’s already a difficult challenge in a regular class, as made clear in this great blog post by “mathbabe” Cathy O’Neil. In a MOOC, my current view is it would be unethical to try. The last thing the world needs are (more) people who think they know what a proof is, but have never put that knowledge to the test.

But when you think about it, the idea behind QR is not that people become mathematicians who can prove things, rather that they have a base level of quantitative literacy that is necessary to live a fulfilled, rewarding life and be a productive member of society. Being able to prove something mathematically is a specialist skill. The important general ability in today’s world is to have a good understanding of the nature of the various kinds of arguments, the special nature of mathematical argument and its role among them, and an ability to judge the soundness and limitations of any particular argument.

In the case of mathematical argument, acquiring that “consumer’s understanding” surely involves having some experience in trying to construct very simple mathematical arguments, but far more what is required is being able to evaluate mathematical arguments.

And that can be handled in a MOOC. Just present students with various mathematical arguments, some correct, others not, and machine-check if, and how well, they can determine their validity.

Well, that leading modifier “just” in that last sentence was perhaps too cavalier. There clearly is more to it than that. As always, the devil is in the details. But once you make the shift from viewing the course (or the proofs part of the course) as being about constructing proofs to being about understanding and evaluating proofs, then what previously seemed hopeless suddenly becomes rife with possibilities.

I started to make this shift with the last session of my MOOC this fall, and though there were significant teething troubles, I saw enough to be encouraged to try it again—with modifications—to an even greater extent next year.

Of course, many QR courses focus on appreciation of mathematics, spiced up with enough “doing math” content to make the course defensibly eligible for QR fulfillment. What I think is far less common—and certainly new to me—is using the evaluation of proofs as a major learning vehicle.

What makes this possible is that the Coursera platform on which my MOOC runs has developed a peer review module to support peer grading of student papers and exams.

The first times I offered my MOOC, I used peer evaluation to grade a Final Exam. Though the process worked tolerably well for grading student mathematics exams—a lot better than I initially feared—to my eyes it still fell well short of providing the meaningful grade and expert feedback a professional mathematician would give. On the other hand, the benefit to the students that came from seeing, and trying to evaluate, the proof attempts of other students, and to provide feedback, was significant—both in terms of their gaining much deeper insight into the concepts and issues involved, and in bolstering their confidence.

When the course runs again in a few week's time, the Final Exam will be gone, replaced by a new course culmination activity I am calling Test Flight.

How will it go? I have no idea. That’s what makes it so interesting. Based on my previous experiments, I think the main challenges will be largely those of implementation. In particular, years of educational high-stakes testing robs many students of the one ingredient essential to real learning: being willing to take risks and to fail. As young children we have it. Schools typically drive it out of us. Those of us lucky enough to end up at graduate school reacquire it—we have to.

I believe MOOCs, which offer community interaction through the semi-anonymity of the Internet, offer real potential to provide others with a similar opportunity to re-learn the power of failure. Test Flight will show if this belief is sufficiently grounded, or a hopelessly idealistic dream! (Test flights do sometimes crash and burn.)

The more people learn to view failure as an essential constituent of good learning, the better life will become for all. As a world society, we need to relearn that innate childhood willingness to try and to fail. A society that does not celebrate the many individual and local failures that are an inevitable consequence of trying something new, is one destined to fail globally in the long term.

For those interested, I’ll be describing Test Flight, and reporting on my progress (including the inevitable failures), in my blog MOOCtalk.org as the experiment continues. (The next session starts on February 3.)