The Math Myth that permeates “The Math Myth”

March 1 saw the publication of the book The Math Myth: And Other STEM Delusions, by Andrew Hacker. MAA members are likely to recognize the author’s name from an opinion piece he published in the New York Times in 2012, with the arresting headline "Is Algebra Necessary?"

Yes, I thought you’d remember it! It’s almost up there with John Lennon’s murder in terms of knowing where you were at the time you first heard of it. But just to be sure we are all on the same page, let me recap that, in that essay, Hacker, a retired college professor of political science who over the years had taught some non-majors math courses, laid out a case for dropping algebra as a required course in K-12 and college.

Before I dive into Hacker’s new book, you would be advised to refresh your memory of the case he presented in that article, since his book is essentially an extension of what he said then, expanded to cover the entire Common Core Mathematical Standards. Prior to writing this review, I wrote an article for the Huffington Post in which I summarized, with my commentary, his 2012 article, together with a recent interview he gave to the Chronicle of Higher Education.

In my article, I noted that Hacker has no idea what algebra really is. His focus is entirely on school algebra as it is very often taught, as a collection of rules for manipulating symbolic expressions. What his argument actually establishes, with sound arguments and good examples, are two conclusions I would agree with:

1. Algebra as typically taught in the school system is presented as a meaningless game with arbitrary rules that does more harm than good.
2. There are strong arguments for teaching algebra as it was originally developed and how professional mathematicians today view it.

I’ll leave you to read my HuffPost piece for more of the gory details. For the benefit of lay readers who may come to this site, I should though repeat here the brief summary I gave in that article of the difference between algebra (as mathematicians understand and practice it) and the rule-based-manipulation-of-symbolic-expressions that so often passes for algebra in our schools.

First codified by the Persian mathematician al-Khwarizmi in his book The Compendious Book on Calculation by Completion and Balancing (balancing = al-Jabr), written in Bhagdad around CE 820, algebra is a powerful method for solving numerical problems more efficiently than by arithmetic. It does so by introducing two new ways of handling numerical problems.

First, algebra provides methods for handling entire classes of numbers, rather than specific ones. (That’s where those x’s, y’s, and z’s come in, but that’s just an implementation detail introduced in France several centuries later.)

Second, it provides a way to find numerical answers not by computing, which is often very difficult, but by reasoning logically to hone-in on the answer, using whatever information is available. Thus, whereas in arithmetic you work forwards, starting with numbers and computing with them to arrive at an answer, in algebra you work backwards, starting by postulating an answer and reasoning logically to figure out what it is. True, this powerful application of human logical reasoning capacity frequently gets boiled down to mastering various symbolic procedures to “Solve for x,” but again that’s just a particular implementation. Numerical forensics would be a sexier, and more descriptive, term for the real thing.

The familiar symbolic expressions calculus usually taught in schools as “algebra” was a particular implementation of al-Khwarizmi’s ways of thinking introduced by the French mathematician François Viète in the 16th Century (700 years after algebra first began) to streamline paper-and-pencil problem solving. A more recent implementation of algebra is the computer spreadsheet.

Since his new book follows the same line of attack as his 2012 opinion piece, but with his sights widened from school algebra to the Common Core, instead of crafting another analytic essay, I will do what Hacker himself does, and list a number of examples to make my case. More precisely, I’ll select some of the 20 instances (in a book of just over 200 pages) where I found a claim that is either plain wrong, wildly misleading, or otherwise problematic, and ask where he went wrong. In marking 20 pages, it’s likely I missed some. There were so many wild and inaccurate claims, I frequently found myself skimming through.

First though, I should repeat what I said in my HuffPost article about his algebra piece. Just as his essay actually amounted to a strong argument in favor of teaching algebra to all students (albeit not the rule-based manipulations of formulas so often presented in place of algebra), so too his book includes a strong argument in favor of Common Core Math. In the same way that Hacker mischaracterized algebra in 2012, so too his portrayal of the CCSSM (Common Core State Standards for Mathematics) is totally at odds with the real thing—though not quite so far off if you turn your attention from the Standards themselves to some implementations of the CC.

One of the book’s flaws is that Mr Hacker seems to get carried away with the flow of his rhetoric, since for the most part his argument consists of the erection of a series of straw men which he then, in time-honored tradition, proceeds to attack.

“It’s a waste of time forcing kids to master azimuths and asymptotes,” he cries [not an exact quote] as early as page 2.

I had to look up the word azimuth, since in my entire career as a mathematician and mathematics educator, I had never come across it. According to Wikipedia, azimuth is a “concept used in navigation, astronomy, engineering, mapping, mining and artillery.” I ran a search for the word on the entire, 93-page CCSSM document and, as I expected, it did not turn up. Straw man.

Asymptotes are a different matter, of course, since a general sense of asymptotic behavior of functions is useful in many walks of life. The word is mentioned, but just once, in the CCSSM, in the section on Interpreting Functions (F-IF), where it says:

Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

That’s it. One mention, buried towards the end of the document, in the section that says the student should:

• Understand the concept of a function and use function notation
• Interpret functions that arise in applications in terms of the context
• Analyze functions using different representations

From the overall thrust of Hacker’s argument, I think it’s clear he believes this kind of knowledge is indeed important for everyone to have. But it’s also clear it is not a central pillar of the CC, to be used on page 2 to set the scene for what his book is about.

Unfortunately, this example is indeed a good characterization of his overall argument: to knock down straw men.

“We’re told that if our nation is to stay competitive, on a given morning all four million of our fifteen-year-olds will be studying azimuths and asymptotes,” he writes. (I am still on page 2, with over 200 more pages to go.) He provides no citation regarding who, exactly, is making this proclamation for the nation’s future. It’s not just disingenuously misleading, it’s about as far from reality as you could imagine, and not because of those azimuths. (See momentarily for the real story.)

He continues, “Then, to graduate from high school, they will face tests on radical notations and elliptical equations.”

To be sure, you will find mention of the word radical in the CCSSM, in the context of “Work with radicals and integer exponents” in the Section on Expressions and Equations (8.EE), which provides the helpful illustration,

“For example, estimate the population of the United States as 3 × 10⁸ and the population of the world as 7 × 10⁹, and determine that the world population is more than 20 times larger.”

Again, this is exactly the kind of thing Hacker says (towards the end of his book) students should be able to do! And it is entirely reasonable that they be asked to demonstrate that ability on a test.

“Elliptical equations” is another straw man.

The point is, what Hacker keeps attacking are straw men. The CCSS is just what its name implies, a set of standards. It is not a curriculum, nor does it specify anything remotely like a daily, or even weekly timetable. How and when teachers across the land cover the various standards is for them, or perhaps their school district, to decide. As far as the CCSS are concerned, teachers can operate fluidly, depending on how their class progresses. (And no one will even suggest that they mention azimuths, let alone force the class to master them.)

I would hazard a guess that Hacker has never looked at the CCSS document. Nor sat in on many math classes, as I have, and observed what actually goes on in today’s schools.

Caveat: I get to see classes to which, for one reason or another, I have been invited to visit. Likely they are some of the best, since their teachers invite me along so their students can talk for a while with someone who has devoted a career to mathematical research. I hear enough stories to be prepared to believe things are often a lot worse. Perhaps even as bad as Hacker says. But his book is purported to be about educational policy, not what you can actually find in good or bad classrooms.

Not only does Hacker give no indication he is familiar with the Common Core—the real one, not the azimuth-strewn, straw-man version he creates—he gives every indication he does not understand mathematics as it is practiced today. (He also does not know that pi is irrational, but I’ll come to that later.)

Certainly, the examples he selects to illustrate the irrelevancy (in today’s world) of some of the test problems students are asked to solve simply demonstrate that he is lacking the basic, every-day, number sense he is arguing for. Let me give just three examples.

On page 48, Hacker presents a question he took from an MCAT paper. It provides some technical data and asks what happens to the ratio of two inverse-square law forces between charges of given masses when the distance between them is halved. The context Hacker provides for this question is that medical professionals needs to be able to read and understand the mathematics used in technical papers. His claim is that this requirement does not extend to the physics of electrical and gravitational forces. In that, he is surely correct. But anyone with a grain of number sense will recognize at once that the setting is totally irrelevant. It’s a simple question about what happens to a ratio when the underlying scale is changed. The answer, of course, is nothing happens. It’s a ratio. The changes to the numerator and denominator cancel out. The ratio remains the same.

What this question is asking for is, Do you understand what a ratio is? Surely that is something that any medical professional who will have to read and understand journal articles would need to know. Hacker completely misses this simple observation, and presents the question as an example of baroque mathematical testing run amok.

On page 70, he presents a question from an admissions test for selective high schools. A player throws two dice and the same number comes up on both. The question asks the student to choose the probability that the two dice sum to 9 from the list 0, 1/6, 2/9, 1/2, 1/3. Hacker’s problem is that the student is supposed to answer this in 90 seconds. Now, I share Hacker’s disdain for time-limited questions, but in this case the answer can only be 0. It’s not a probability question at all, and no computation is required. It just requires you to recognize that you can never get a sum of 9 when two dice show the same number. As with the MCAT question, the question is simply asking, Do you understand numbers? In this case, do you recognize that the sum of two equal numbers can never be odd.

Finally, on page 101, Hacker presents a list of mathematics requirements high school students must meet in order to study at Harvard and similar universities. The list includes the names of various kinds of analytic functions. As usual, Hacker seems overwhelmed by the technical terms, or worries that the students will be, but all the list is asking for is that students can read graphs and charts and know what they represent in terms of growth and change. An essential skill, surely, for anyone in today’s information-rich world, not just students at elite universities.

You get the pattern surely? Hacker’s problem is he is unable to see through the surface gloss of a problem and recognize that in many cases it is just asking the student if she or he has a very basic grasp of number, quantity, and relationships. Yet these are precisely the kinds of abilities he argues elsewhere in the book are crucial in today’s world. He is, I suspect, a victim of the very kind of math teaching he rightly decries—one that concentrates on learning rules and mastering formal manipulations, with little attention to understanding.

This, surely, explains why he would write (page 96), “Reasoning mathematically may be a nice skill, but it is not relevant to most of life. We reason about many things: parenting, marriage, careers. Do we learn how to reason about these things by learning algebra?”

If he had asked instead if we learn such reasoning in a typical school algebra class, I would agree with his implied answer of “No.” But algebra arose by codifying the everyday reasoning people carried out—and still carry out today—about the numerical or quantity aspects of any human activity that involves them. (Trade, commerce, and civil engineering were the original applications.)

From that historical perspective, it is absolutely clear that learning algebra can help us master such reasoning. It helps by providing an opportunity to carry out that kind of reasoning free of the complexities a problem generally brings with it when it arises in a real world context.

The tragedy of The Math Myth is that Hacker is actually arguing for exactly the kind of life-relevant mathematics education that I and many of my colleagues have been arguing for all our careers. (Our late colleague Lynn Steen comes to mind.) Unfortunately, and I suspect because Hacker himself did not have the benefit of a good math education, his understanding of mathematics is so far off base, he does not recognize that the examples he holds up as illustrations of bad education only seem so to him, because he misunderstands them.

The real myth in The Math Myth is the portrayal of mathematics that forms the basis of his analysis. It’s the same myth you see propagated in Facebook posts from frustrated parents about Common Core math homework their children bring home from school.

In the interests of their overall cardiovascular health, I have to recommend that math educators do not read The Math Myth. But if you do, perhaps you should start with the final chapter, titled “Numeracy 101.” Here, at least, you will find things you are likely to agree with, as he lays out what he believes would be a good quantitative literacy course for college students.

But even there, where all seems warm and friendly and positive, you will be jolted by Hacker’s fundamental lack of knowledge of mathematics. He writes,

“Along with phenomena like earthquakes and cyclones, nature also has some numbers that control or explain how the world works. One of them is pi, whose 3.14159 goes on indefinitely, at least as far as we know.”

Yes, you read that last part correctly.

“Few people writing today … can make more sense of numbers” proclaims the Wall Street Journal on the cover of Hacker’s book. Well, if that’s the view of the newspaper that purports to have the expertise to cover the nation’s financial markets, it is only a matter of time before we have another financial meltdown.

Theorem: You are exceptional

“Everyone excels at something.” We hear it all the time, usually said to console someone who is miserable after underperforming at something. Parents, in particular, often fall back on it with their children. What few people realize, though, is that the statement can be mathematically verified. You need only consider a collection of 200 essentially independent human performance characteristics for 98% of people to measure as exceptional in at least one of them, where exceptional is defined as being in the top or bottom 1%. (The mathematics gives extremal values; if you want to effectively guarantee being in the top 1%, you need more characteristics. The phenomenon is asymptotic.)

This result is a consequence of a rather surprising, but little known, observation about high-order hypercubes: as the dimension increases, the proportion of points in the interior (i.e., not on the bounding shell) decreases without limit.

Here is how you can prove to your child, spouse, student, best friend, or whoever, that they—or you, as the circumstances may require—can or will excel at something.

Fig 1. The bell curve (normal distribution)

Everyone is familiar with the bell curve (normal distribution) showing the typical distribution of performance measures of a single characteristic across a sufficiently large population. This graph captures the fact that the scores for the majority of the population cluster around an “average”, middling value, with only a few individuals at either end (exceptionally poor or exceptionally good).

For the purposes of the multi-dimensional computation, we can start with a geometrically simpler model, namely the closed interval [0,100], as in Figure 2. We define the exceptional points to be those in the unit intervals at each end. In this model, for a single characteristic, only 2% of the population are exceptional. The remaining 98% are “normal."

Fig 2. A simple model of exceptionality in one characteristic

Now consider two characteristics, X and Y (assumed to be independent). The distribution then is represented by a 100x100 square with an inner 98x98 cube, as in Figure 2.

Fig 3. A simple model of exceptionality in two characteristics

An individual’s X measure is shown by the x-coordinate, their Y measure by the y-coordinate. The ordinary individuals are represented by points in the inner square; the exceptional individuals by points in the outer perimeter region.

The total number of points is 100x100. The number of normal points is 98x98. So the number of exceptional points is 10,000 – 9,604 = 396.

The proportion of exceptional points is thus 396/10,000 = 0.0396, i.e., 3.96%. Thus, more individuals are classified as exceptional when you consider two characteristics (3.96% as opposed to 2%).

Going to three characteristics, X, Y, and Z, the model will be a 100x100x100 cube with an inner 98x98x98 cube, as in Figure 4.

Fig 4. A simple model of exceptionality in three characteristics

The volume of the outer cube (representing the total population) is 1,000,000. The volume of the inner cube (representing the normal individuals) is 941,192. Thus the volume of the perimeter-region (representing the exceptional individuals) = 1,000,000 – 941,192 = 58,808. Hence, the proportion of exceptional individuals = 58,808/1,000,000 = 0.0588, i.e. 5.88%.

So far, everything seems fairly straightforward and reasonable. Going beyond three characteristics, the model is a hypercube of four or more dimensions, and we can no longer provide meaningful illustrations. But by now we have grown familiar with the idea that the model represents exceptional individuals by points in the outer 1% shell. To see what this entails, let’s jump to 10 characteristics, X₁,…,X₁₀. In that case, our model will represent the situation as a 100¹⁰ hypercube with an inscribed 98¹⁰ hypercube.

The volume of the outer hypercube (~ total population) = 100¹⁰. The volume of the inner hypercube (~ normal individuals) = 98¹⁰. Thus, the volume of the perimeter-region (~ exceptional individuals) = 100¹⁰ – 98¹⁰, and the proportion of exceptional individuals = (100¹⁰ – 98¹⁰)/100¹⁰. At this point, it’s time to bring in Wolfram Alpha to do the calculation. This gives the result that, with 10 characteristics, 18.29% of the population is exceptional.

With 100 characteristics, X₁,…,X₁₀₀, our model gives: Volume of hypercube (~ total population) = 100¹⁰⁰. Volume of inner hypercube (~ normal individuals) = 98¹⁰⁰. Volume of perimeter- region (~ exceptional individuals) = 100¹⁰⁰ – 98¹⁰⁰. Proportion of exceptional individuals = (100¹⁰⁰ – 98¹⁰⁰)/100¹⁰⁰. Calling on Wolfram Alpha again, we compute that with 100 characteristics, 86.74% of the population is exceptional.

With 200 characteristics, X₁,…,X₂₀₀, our model gives: Volume of hypercube (~ total population) = 100²⁰⁰. Volume of inner hypercube (~ normal individuals) = 98¹⁰⁰. Volume of perimeter-region (~ exceptional individuals) = 100²⁰⁰ – 98²⁰⁰. Proportion of exceptional individuals = (100²⁰⁰ – 98²⁰⁰)/100²⁰⁰. So with 200 characteristics, 98.24% of the population is exceptional. (Once again calling on the services of the normally unflappable Wolfram Alpha.)

And there’s our result.

Of course, we have been working with a model. As always, that entails making various assumptions and simplifications. If the final result surprises you, you have two choices. Either go back and modify your initial assumptions and generate another model. Or accept the result and modify the prejudices that led to your surprise.

In this case, we have to accept that in higher dimensions, almost all the material in an equal-sided, rectangular, solid (!) box is on the outer shell. The (solid) inside is almost empty.

When we consider more dimensions to a situation, the math can sometimes lead us to a counter-intuitive—but correct—conclusion we did not expect. Not everyone can accept that.

Yes, in this US Election Season, this is a story with a moral. 

Do your kids find learning math hard? There may be an app for that!

If you are like me, you probably sigh and switch off when you read an article with a title claiming kids’ math scores show significant improvement after using some great new app for a few minutes each day.

In which case, you may have paid little attention when a news article came out in Science magazine recently, reporting a new study showing that after just one year of parents using a bedtime-story-telling app called Bedtime Math with their young children, those kids were three months ahead of fellow students whose parents were using an app to provide non-mathematical stories. In fact, children of math-anxious parents showed even greater improvement, ending up six months ahead.

If you happened to see the article, you likely assumed it was essentially a piece of marketing, where a bogus “study” was carried out to produce the “results” the marketing folks wanted. After all, there are no magic bullets in the math ed world, right?

My reaction was very different. I immediately wanted to know more. The reason being, as I recounted in last month’s column, the same thing had happened a few months earlier with a math learning app I had created, a mobile game called Wuzzit Trouble. (Actually, I should have written “we” there. Paul McCartney may have sung that he had “got by with a little help from my friends,” but the reality was it took a lot of work by all four Beatles to make them a global phenomenon, and in the apps business it usually takes a whole team of highly talented people to create a great product. My team are listed here.) A Stanford classroom study led by Prof Jo Boaler had found significant math learning after just two hours play of Wuzzit Trouble spread over four weeks.

The Bedtime Math study was unlikely to have been a bogus marketing “study”, I felt, since it had been carried out at the University of Chicago, which is a great university. True, as the Science article noted, the study was funded by an entity called the Overdeck Family Foundation, whose chair, Laura Overdeck, a former astrophysics major at Princeton, established the nonprofit Bedtime Math Foundation, which created and supports the app. Some might read that and smell a rat – as some did when they first read of the Stanford Wuzzit Trouble study.

But to my mind such a reaction says more about the reader than the researchers. We are talking about an educational app made and distributed for free by a nonprofit organization. Why would anyone want to fake data? Really!

In fact, even if the app were for sale – say for a mind boggling $4.99 – the idea that seven researchers at a major university would fake a study about a small children’s app is simply not credible. As a number of news articles have made clear, the price for a university researcher faking a scientific study is dismissal from the university and the end of their career. When it does happen, the motivation is invariably massive career prestige and fame, or a huge follow-up research grant (or both). Neither of which are likely to result from an at best encouraging, small scale classroom study of Bedtime Math, Wuzzit Trouble, or any other kids’ app.

If a foundation or a company wanted to run a fake study for marketing purposes, they could simply do it themselves, or else farm it out to an unscrupulous, individual researcher. Such people are to be found, sometimes associated with universities. (Google “intelligent design” or “climate change denial” for examples.)

Certainly, James Stigler, a well known educational psychologist at UCLA, is not skeptical. Science quotes him as saying, "I think it's a fantastic study. But it is just the beginning."

Another respected scholar, Andee Rubin, a mathematician and computer scientist at the nonprofit education R&D company TERC in Cambridge, Mass, has a similar reaction. Science quotes him as observing, "I'm interested in teasing it apart and seeing what makes this effective."

Those are pretty much the same as the reaction I had to Stanford’s Wuzzit Trouble results, which prompted me to draw up the list of possible explanatory factors I published in my last column.

With both the full paper and a cover article available in Science, all I will do here is provide a brief summary of the cover article.

The Chicago team recruited 587 first-graders from 22 schools in the Chicago metropolitan area. The parents of each child were given a tablet computer with which to read to the child at bedtime. 420 families were told to use it to work through word problems related to counting, shapes, arithmetic, fractions, and probability using Bedtime Math. Another 167 families were instructed to use a reading app. With a standardized test, the researchers assessed all the subjects' mathematics performance at the beginning and end of the school year.

As you would expect, use of the reading app made little difference to the children's math performance. In contrast, children who used the math app two or more times per week outpaced peers whose family rarely used it, ending up three months ahead.

Perhaps most important, use of the app brought students whose parents said they were anxious about math up to par with those whose parents were at ease with the subject. Among children whose family rarely used the math app, those with math-phobic parents made only half as much progress as the children of parents comfortable with math.

The researchers make some suggestions as to what may be going on. My own best guess, based on several months reflections on the Stanford and (subsequent) Finnish studies of Wuzzit Trouble, are consistent with what they think. Namely:

We have created a system where learning is walled off from everyday life. Particularly in math. The “math classroom” operates according to its own rules. Even with a truly great teacher – and I have met many – there are many restrictions on what can be done. Not least because of an incessant rhythm of performance testing.

Go into most math classrooms and what you see will most likely bring to mind a room full of clerks in the pre-computer age when companies employed large numbers of numerically-able people to crunch their numbers. (Young people will have to rely on old photographs or depictions in movies.) Which was, of course, what the system was set up to provide.

The classroom certainly does not look remotely like a room full of professional mathematicians at work. The first words that might come to mind if you were to walk into such a room would be “fun”, “engagement”, “argument”, “passion”, “social interaction”.

Nor does it look like the human activity that hundreds of thousands of years of natural selection have inbred into us to maximize learning in the young: play. (Some wise person once said that “play is the work of the child.” I agree.)

And there is something else that evolution hard-wired into is: our love of stories. Effective political speeches are usually laden with stories of individual people, and for very good reason. Because they are powerful.

At which point, it’s probably a good idea to do what we math instructors tell our student to do: look for patterns. Well, what do we see when we look at professional mathematicians at work and kids in their ideal learning activity?

Fun, engagement, argument, passion, social interaction, play, stories.

Those are all, I would argue, essential ingredients for good learning. Yet you would have difficulty finding any in many math classrooms.

Indeed, society seems to have gotten into a mindset that these items are distractions that you have to eliminate to achieve good math learning. Even when good teachers do their best to inject some of those valuable features into their classes, they have to operate within a system that disapproves. And everyone knows that, most of all the kids.

No wonder then, that when you have a well designed, engaging app – a game, a puzzle, a family-supporting bedtime story provider, or whatever – you will likely get good results. Because apps, if properly designed, create their own environment.

Though I labored long and hard to create Wuzzit Trouble, and I am sure Laura Overdeck worked equally hard on Bedtime Math, all to good effect for sure, my strong suspicion is our apps work as well as they do in such a short time primarily because of what they are not. Namely they are not the typical school classroom approach to mathematics education.

How else can you explain dramatic results after a few hours engaging with an app, other than it unlocking what had hitherto been shackled by the chains of an Industrial Revolution conception of mathematics learning? Learning something that is genuinely new takes time and a lot of effort. Freeing something that is already there – if only in embryonic form – is much quicker. If so, then this means that our received wisdom that it takes a lot of hard work, repetitive practice, frustration, tears, and pathological levels of anxiety to achieve competency in mathematics may simply be very strong evidence that our approach sucks.

Life inside an impossible Escher figure

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Today is George Boole’s 200th Birthday

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

Born in England, Boole spent the major part of his mathematical career as a professor at Queen’s College Cork, and the Irish mathematical community has been actively celebrating Boole’s life, work, and legacy throughout this year. Of particular note, is an Irish ballad, “The Mathematician - The Bould Georgie Boole”, specially written for the occasion and performed by the Arthur Céilí Band, which you can hear, with visual biographic accompaniment about Boole, on YouTube and Vimeo:
https://www.youtube.com/watch?v=05IMBfkpn_M
https://vimeo.com/143768018.

The lyric and a download link to the song are available at:
https://arthurceiliband.bandcamp.com/releases.

In a more academic vein, University College Cork has created a video biography available at:
https://www.youtube.com/watch?v=y-eav8-EEY4.

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

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

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

Letter to a calculus student – The Sequel

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

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

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

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

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

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

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

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

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

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

* * * * * *

Western Washington University, in Bellingham, WA

Hi Dr. Devlin,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Thank you,

Murray Pendergrass

A Brilliant Young Mind: The IMO goes to the movies

Jo Yang as Zhang Mei and Asa Butterfield as Nathan Ellis in A Brilliant Young Mind. Credit: Samuel Goldwyn Films

Mainstream movies about mathematicians used to be a rarity, but are now fairly common. Good Will Hunting, Proof, A Beautiful Mind, Pi, Cube, The Bank, Travelling Salesman, The Imitation Game come immediately to mind. So too does Stand and Deliver if you include mathematics teaching.

The titles I just listed are such good movies, there is now a high bar to success in this growing genre. In particular, the movie has to have a good story, a strong cast, and it needs to get the math right – and moreover do so in a way that intrigues the audience but does not detract from the pace of the story. The new movie A Brilliant Young Mind, by British director Morgan Matthews, meets that standard.

Due to be released in the US on September 11, A Brilliant Young Mind, starring the hugely talented young British Actor Asa Butterfield (who starred as Ender in Ender’s Game, opposite Harrison Ford) was first screened in the UK last year, originally under the title “X + Y”, on which more later.

The film focuses on the International Mathematical Olympiad (IMO), the competition held annually around the world, where national teams of six pre-collegiate students compete for individual and team medals. The movie follows one particular British student as he goes through the grueling process of preparing for and taking the test to qualify for team pre-selection in the British National Mathematical Competition, going off to a training camp in Taiwan, where the final team of six is selected in a mock IMO competition, and then heading to Cambridge, England, for the international competition itself.

Both the mathematics and the mathematics competitions are handled well. (More later.) Mathematicians will not be disappointed on that score.

I have to admit that, on first viewing, I felt that the romantic thread between the Asa Butterfield character (Nathan Ellis) and the young female Chinese math whiz he meets at the training camp, played by Jo Yang, was a crude injection to create a movie with mainstream audience appeal. In particular, I thought the dramatic ending (you’ll have to watch the movie to find out if it is a happy or sad ending) was way over the top.

But then I watched the original BBC documentary that A Brilliant Young Mind director Matthews made back in 2006, on which he based the movie, and guess what? The story in A Brilliant Young Mind stays pretty close to real life! Right down to what at first viewing of the movie I thought were syrupy shots included purely for cinematic romantic effect. (Cue the rainbow in the background as the British and Chinese math whizzes travel by train through the British countryside. Taken right out of real life!)

So if my jaded-by-Hollywood mathematical colleagues find themselves, like me, lamenting to themselves, “Why do movie makers spoil the real story with all that romantic mush?” you should suppress that reaction at once. What the movie gives you is a dramatic (and dramatized) recreation of real life.

That, by the way, is why I like the movie’s original title in the UK: “X + Y”, a nicely succinct way to link the mathematics problem solving and the romantic engagement. Still the title A Brilliant Young Mind does convey the idea of a young version of the brilliant (but, like Butterworth’s Nathan Ellis, mentally troubled) John Nash in the movie A Beautiful Mind, and Matthews himself pulled on the same association with the title of his earlier documentary.

Despite taking his basic storyline straight out of real life, Matthews does (of course) take plenty of dramatic license in order to give us a watchable movie. He is, after all, telling a fictional story, albeit one based (unusually closely) on real life. Few in the audience will have much interest in the mathematics, or even math competitions (besides, perhaps, being surprised that such things exist), but everyone likes a good story about people. And that is what A Brilliant Young Mind delivers.

In particular, I suspect many of my fellow mathematicians will also balk at the portrayal of several of the British IMO team selectees as exhibiting various forms of autism. (In real life, good mathematicians of all ages run the full spectrum of human characteristics, with the vast majority of math whizzes being just like everyone else.) But that aspect too is what you will find in the documentary. (But see my postscript comments at the end.)

The IMO team members who the Butterfield character interacts with in the movie are also clearly based on real-life counterparts in the documentary. In particular, the student who has made his way onto the team by learning a lot of mathematical facts and procedures that he can regurgitate and apply at speed, but falters when it comes to having to apply original thinking. (Both the US and the UK math education systems encourage and reward that approach, which is why they do so poorly in the international PISA tests, which look for original thought. My Stanford colleague Jo Boaler has a new book on that misguided, and sad, state of affairs, Mathematical Mindsets, coming out in the Fall.)

The only complaint you could make about Matthews is the choices he made in selecting the footage he shot for the original documentary. But that is true for any documentary film. Matthews followed the UK 2006 IMO team through the entire competition process, and then told a story based on what he had captured.

Interestingly enough, another documentary film maker, George Csicery, followed the US IMO team at the same time. You can compare the two documentaries. Matthews’ BBC documentary is available online for free streaming. Csicery’s film Hard Problems, is available for purchase from the MAA ($19.95 to members).

Enough of all these words. We’re talking about a movie, after all. Time to watch some movie clips.

You can watch the official trailer for A Brilliant Young Mind here.

Mathematicians will particularly like the one short sequence where the movie shows a brilliant mathematical mind in action, solving a problem, which you can see in isolation in this officially sanctioned YouTube video. It is a superbly chosen (and acted) example. No fake numerical mumbo jumbo here. Genuine mathematical thinking in action. And good mathematical thinking to boot. Any math instructor would surely love to have a student produce such a solution for the class.

During the year the two documentaries were made, the IMO was held in Ljubljana, Slovenia (not Cambridge, England, as in the movie). You can see the actual problems the competitors faced here. (With sample solutions.)

Finally, for long lists of scenes in movies that feature a mathematician or a math problem, see here and here.

Enjoy the film!

EDITORIAL POSTSCRIPT

Both the movie and the BBC documentary raise some issues that concern me as a mathematician. The main danger of any documentary or movie is if viewers (and for a film like A Brilliant Young Mind, the audience may well include young kids showing an early interest in mathematics) get the impression that what they see is representative of the field. This of course, is true for pretty well any movie, be the topic crime detection, politics, business, law, the military, sports, or whatever. I think it is particularly worrying in mathematics, because most people have a very impoverished, and often completely erroneous perception of mathematics. Both of Matthews’ films trouble me on that count.

These thoughts are in no way a criticism of the movie I am reviewing. It is what it is. I think it is a good movie and I like it. (Though as I noted, in my case much of my positive evaluation comes from knowing that the things in the movie that initially turned me off as being unrealistic and contrived turned out to be true!) Rather, the issues I raise are general ones about the public perception of mathematics. In making both his documentary and the movie, Morgan Matthews set out to make good films. His goal was not to improve the public understanding of mathematics. That, on the other hand, is something I have spent a great deal of my career focusing on. Hence this postscript, separate from my review.

First, it has to be said that competition mathematics is in many respects a very different activity than the professional mathematics that most of us in the business pursue. For one thing, competition math requires speed, whereas many good mathematicians are slow thinkers. (I certainly am.)

There is also something very unusual about the kinds of problems that the IMO presents. Of necessity, they have to be solvable in at most an hour, and in many cases, the way to go about solving them is to be very conscious of that time limitation. They have to depend on seeing a particular insight or trick. Some people are naturally good at that kind of problem solving, but it can also be to some extent taught – which is what goes on at those IMO training camps. On the other hand, most mathematics problems that the pros grapple with are very different. In many cases, no one has any idea if there is a solution at all, or how long it may be.

Moreover, the connection between being good at competition math and having the aptitude to succeed in professional mathematics is not at all clear-cut. Certainly, some IMO medal winners have gone on to pursue mathematics at university level, but not all of them have gone on to lead a successful career in mathematics. (Some have.) And many of the best mathematicians in the world have never in their lives had any interest in competition math. Though the two domains do have abstract mathematics in common, they are in many ways very different activities.

So to a child or the parent of a child who shows aptitude toward mathematics, I would say this. If you fancy the idea of competition math, give it a try. If you do well, enjoy the experience. It will certainly show that you have some abilities that could help you succeed in a mathematical career. But if you find you do not enjoy it, or if you like it but do not do well, that in no way means you could not grow up to be a top rank mathematician.

Another unintended message that math competitions tend to convey is that you have to have a special talent for mathematics (a “math gene” if you like). This notion that mathematics is something for the “gifted and talented” is pervasive in many cultures, and it is total BS. The two most important factors in achieving success in mathematics are wanting to do math and growing up in a supportive, educationally rich, sociocultural environment. Not only is the world of mathematics replete with examples of world class mathematicians who will tell you flat out how many hours of effort it took them to get to that point, and how others helped them on their way, there is also a growing body of evidence from nueroscience studies to support the hypothesis that mathematics, like pretty well any other human endeavor, is 5% inspiration and 95% perspiration.

Society would do well to banish that term “gifted and talented” once and for all, and replace it with something more accurate. “Motivated and bloody hard working” is my nomination. (Individual mileages do, of course, vary.)

My final editorial remark is something I touched upon in my review. The movie, focuses on a small group of mathematics students, and one of them in particular, who exhibit various forms and degrees of autism. True, the same was true of the young students in the director’s earlier documentary, but that clearly reflects the particular perceptual lens the director brought to the project. That lens is made dramatically clear by the opening scenes in the documentary where we meet one of the competitors, Jos. Matthews set out to portray IMO participants as being unusual and different. And he found some.

Cleary, getting to represent their country in an international competition in of itself makes them different. But presenting them as intrinsically different is a definite editorial decision.

Contrast Matthews’ documentary with Csicery’s. In the latter, focusing on the US team at the same IMO, the director sets out to convey the very ordinariness of the participants, highlighting not what is different about them (they love math being the main thing) but how much they are just like any other kids of that age.

Csicery, as many readers of this column will know, makes documentary films about mathematicians and mathematics as a profession, and he makes them primarily for the mathematics and mathematics education professions. So he strives to inform an audience who are interested in mathematics. This, of course, is very different from Matthews, who sets out to make movies that intrigue viewers who do not necessarily have any interest in the topic, whatever it is. Csicery succeeds with his audience by being as accurate and representative as he can, while also managing to tell a story. Matthews has to tell a good human interest story that hangs on some strong characters, with everything else revealed during the film. They are doing different things.

I found Matthews’ documentary fascinating and highly engaging, and I’m really glad it inspired him to turn it into a movie. You should watch both.

But if what you want is to get a good overall sense of the world of competitive math, you should watch Csicery’s documentary. The two documentaries provide very different perceptions of the same IMO competition.