Tuesday, June 7, 2016

Infinity and Intuition

On May 30, Gary Antonick’s always interesting Numberplay section in the New York Times featured a contribution by Berkeley mathematician Ed Frenkel on the difficulties the human mind can encounter when trying to come to grips with infinity. If you have not yet read it, you should.

Infinity offers many results that are at first counter-intuitive. A classic example is Hilbert's Hotel, which has infinitely many rooms, each one labeled by a natural number printed on the door: Room 1, Room 2, Room 3, etc., all the way through the natural numbers. One night, a traveler arrives at the front desk only to be told be the clerk that the hotel is full. "But don't worry, sir," says the clerk, "I just took a mathematics course at my local college, and so I know how to find you a room. Just give me a minute to make some phone calls." And a short while later, the traveler has his room for the night. What the clerk did was ask every guest to move to the room with the room number the next integer. Thus, the occupant of Room 1 moved into Room 2, the occupant of Room 2 into Room 3, etc. Everyone moved room, no one was ejected from the hotel, and Room 1 became vacant for the newly arrived guest.

This example is well known, and I expect all regular readers of MAA Online will be familiar with it. But I expect many of you will not know what happens when you step up one level of infinity. No sooner have you started to get the hang of the countable infinity (cardinality aleph-0), and you encounter the first uncountable infinity (cardinality aleph-1) and you find there are more surprises in store.

One result that surprised me when I first came across it concerns trees. Not the kind the grow in the forest, but the mathematical kind, although there are obvious similarities, reflected in the terminology mathematicians use when studying mathematical trees.

A tree is a partially ordered set (T,<) such that for every member x of T, the set {y in T : y < x} of elements below x in the tree is well ordered. This means that the tree has a preferred direction of growth (often represented as upwards in diagrams), and branching occurs only in the upward direction. It is generally assumed that a tree has a unique minimum element, called the root. (If you encounter a tree without such a root, you can simply add one, without altering the structure of the remainder of the tree.)

Since each element of a tree lies at the top of a unique well ordered set of predecessors, it has a well defined height in the tree - the ordinal number of the set of predecessors. For each ordinal number k, we can denote by T_k the set of all elements of the tree of height k. T_k is called the k'th level of T. T_0 consists of the root of the tree, T_1 is the set of all immediate successors of the root, etc.

Thus, the lower part of a tree might look something like this:



(It could be different. There is no restriction on how many elements there are on each level, or how many successors each member has.)

A classical result of set theory, sometimes called König's Lemma, says that if T is an infinite tree, and if each level T_n, for n a natural number, is finite, then T has an infinite branch, i.e., an infinite linearly ordered subset.

It's easy to prove this result. You define a branch {x_n : n a natural number} by recursion. To start, you take x_0 to be the root of the tree. Since the tree is infinite, but T_1 is finite, there is at least one member of T_1 that has infinitely many elements above it. Let x_1 be one such element of T_1. Since x_1 has infinitely many elements above it and yet only finitely many successors on T_2, there is at least one successor of x_1 on T_2 that has infinitely many elements above it. Let x_2 be such an element of T_2. Now define x_3 in T_3 analogously so it has infinitely many elements of the tree above it, and so on. This simple process clearly defines an infinite branch {x_n : n a natural number}.

Having seen why König's Lemma holds, it's tempting to argue by analogy that if you have an uncountable tree T (i.e., a tree whose cardinality is at least aleph-1) and if every level T_k, for k a countable ordinal, is countable, then T has an uncountable branch, i.e., a linearly ordered subset that meets level T_k for every countable ordinal k.

But it turns out that this cannot be proved. It is possible to construct an uncountable tree, all of whose levels T_k, for k a countable ordinal, are countable, for which there is no uncountable branch. Such trees are called Aronszajn trees, after the Russian mathematician who first constructed one.

Here is how to construct an Aronszajn tree. The members of the tree are strictly increasing (finite and countably transfinite), bounded sequences of rational numbers. The tree ordering is sequence extension. It is immediate that such a tree could not have an uncountable branch, since its limit (more precisely, its set-theoretic union) would be an uncountable strictly increasing sequence of rationals, contrary to the fact that the rationals form a countable set.

You build the tree by recursion on the levels. T_0 consists of the empty sequence. After T_k has been constructed, you get T_(k+1) by taking each sequence s in T_k and adding in every possible extension of s to a strictly increasing (k+1)-sequence of rationals. That is, for each s in T_k and for each rational number q greater than or equal to the supremum of s, you put into T_(k+1) the result of appending q to s. Being the countable union of countably many sets, T_(k+1) will itself be countable, as required.

In the case of regular recursion on the natural numbers, that would be all there is to the definition, but with a recursion that goes all the way up through the countable ordinals, you also have to handle limit ordinals - ordinals that are not an immediate successor of any smaller ordinal.

To facilitate the definition of the limit levels of the tree, you construct the tree so as to satisfy the following property, which I'll call the Aronszajn property: for every pair of levels T_k and T_m, where k < m, and for every s in T_k and every rational number q that exceeds the supremum of s, there is a sequence t in T_m which extends s and whose supremum is less than q.

The definition of T_(k+1) from T_k that I just gave clearly preserves this property, since I threw in EVERY possible sequence extension of every member of T_k.

Suppose now that m is a limit ordinal and we have defined T_k for every k < m. Given any member s of some level T_k for k < m, and any rational number q greater than the supremum of s, we define, by integer recursion, a path (s_i : i a natural number) through the portion of the tree already constructed, such that its limit (as a rational sequence) has supremum q.

You first pick some strictly increasing sequence of rationals (q_i : i a natural number) such that q_0 exceeds the supremum of s and whose limit is q.

You also pick some strictly increasing sequence (m_i : i a natural number) of ordinals less than m that has limit m and such that s lies below level m_0 in the tree.

You can then use the Aronszajn property to construct the sequence (s_i : i a natural number) so that s_i is on level m_i and the supremum of s_i is less than q_i.

Construct one such path (s_i : i a natural number) for every such pair s, q, and let T_k consist of the limit (as a sequence of rationals) of every sequence so constructed. Notice that T_k so defined is countable.

It is clear that this definition preserves the Aronszajn property, and hence the construction may be continued.

And that's it.

NOTE: The above article first appeared in Devlin’s Angle in January 2006. Seeing Frenkel’s Numberplay article prompted me to revive it and give it another airing.

Wednesday, May 4, 2016

Algebraic Roots – Part 2

What does it mean to “do algebra”? In Part 1, published here last month, I described how algebra (from the Arabic al-Jabr) began in 9th Century Baghdad as a way to approach arithmetical problems in a systematic way that scales. It was a way of thinking, using logical reasoning rather than (strictly speaking, in addition to) arithmetical calculation, and the first textbook on the subject explained how to solve problems that way using ordinary language, not symbolic expressions. Symbolic algebra was introduced later, in 16th Century France.

Just as the formal algorithms of Hindu-Arabic arithmetic make it possible to do arithmetic in a purely procedural, rule-following way (without the need for any thought), so too symbolic algebra made it possible to solve algebraic problems by manipulating symbolic expressions using formal rules, again without the need for any thought.

Over the ensuing centuries, schools focused more and more exclusively on the formal, procedural rules of arithmetic and symbolic algebra, driven in part by the needs of industry and commerce to have large numbers of people who could carry out computations for them, and in part for the convenience of the school system.

Today, however, we have digital devices that carry out arithmetical and algebraic procedural calculations for us, faster and with greater accuracy, shifting society’s needs back to arithmetical and algebraic thinking. This is why you see the frequent use of those terms in educational circles these days, along with number sense. (All three terms are so common that definitions of each are easily found on the Web by searching on the name, as is also the case for the more general term mathematical thinking.)

As more (and hopefully better) technological aids are developed, the nature of the activity involved in solving an arithmetical or algebraic problem changes, both for learning and for application. The fluent and effective use of arithmetical calculators, graphing calculators (such as Desmos), spreadsheets, computer algebra systems (such as Mathematica or Maple), and Wolfram Alpha, are now marketable skills and important educational goals. Each of these tools, and others, provides a different representation of numbers, numerical problems, and algebraic problems.

One consequence of this shift that seemed to take an entire generation of parents off guard, is that mastery of the “traditional algorithms” for solving arithmetic and algebraic problems, which were developed to optimize human computations and at the same time create an audit trail, and which used to be the staple of school mathematics instruction, became a much less important educational goal. Instead, it is evidently far more valuable for today’s students to spend their time working with algorithms optimized to develop good arithmetical and algebraic thinking skills, that will (among other things) support fluent and effective use of the new technologies.

I said “evidently” above, since to those of us in the education business, it was just that. With hindsight, however, it seems clear that in rolling out the Common Core State Standards, those in charge should have put much more effort into providing that important background context that was evident to them but, clearly, not evident to many people not working in mathematics education.

I was not involved in the CCSS initiative, by the way, but I doubt I would have done any better. I still find it hard to wrap my mind round the fact that the “evident” (to me) need to modify mathematics education to today’s world is actually not at all evident to many of my fellow citizens—even though we all live and work in the same digital world. I guess it is a matter of the educational perspective those of us in the math ed business bring to the issues.

But even those of us in the education business can sometimes overlook just how much, and how fast, things have changed. The most recent example comes from a highly respected learning research center, LearnLab in Pittsburgh (formerly called the Pittsburgh Science of Learning Center), funded by the National Science Foundation.

The tweet shown below caught my eye a few weeks ago.



The tweet got my attention because I am familiar with DragonBox, and include it in the (very small) category of math learning apps I usually recommend. (I also know the creator, and have given occasional voluntary feedback on their development work, but I have no other connection to the company.)

“Ineffective”? “#dragonboxfail”? Those are the words used in the tweet. But neither can possibly be true. DragonBox provides an alternative representation for linear equations in one unknown. Anyone who completes the game (for want of a better term) has demonstrated mastery of algebraic thinking for single variable linear problems. Period. (There is a separate issue of the representation that I will come to later.)

Indeed, since the mechanics in DragonBox are essentially isomorphic to the rules of classical symbolic algebra (as taught in schools for the last four hundred years), completing the game demonstrates mastery of those mechanics too. From a logical perspective then, the tweet made no sense. All very odd for an official tweet from a respected, federally-funded research institute. Suspecting what must be going on, I looked further.

The tweet was in response to a review of DragonBox, published by EdSurge. I recognized the name of the reviewer, Brady Fukumoto, a former game developer I had meet a few times. It was a well analyzed review. Overall, I agreed with everything Brady said. In particular, he spent some time comparing “doing algebra in the DragonBox representation” to “doing algebra using the traditional symbolic equations representation”, pointing out how much richer is the latter—but noting too that the former can result in higher levels of student engagement. Hardly the “promote” of a product that LearnLab accused him of. Indeed, Brady correctly summarized, and referenced (with a link) the Carnegie Mellon University study the LearnLab tweet implicitly referred to.

I recommend you read Brady’s review. It gets at many aspects of the “what does it mean to do algebra?” issue. As does playing DragonBox itself, which toward the end gradually replaces its initial “game representation” with the standard symbolic equation representation on a touch screen (a process often referred to as deconcretization).

Unlike the tweet, the CMU paper was careful in stating its conclusion. The authors say, and Brady quotes, that they found DragonBox to be “ineffective in helping students acquire skills in solving algebra equations, as measured by a typical test of equation solving.” (The emphasis is mine.)

Now we are at the root of that odd tweet. (One should not make too much of a tweet, of course. Twitter is an instant medium. But, rightly or wrongly, tweets in the name of an organization or a public figure are generally viewed as PR, presenting an authoritative, public stance.) The folks at LearnLab, their knowledge of educational technology notwithstanding, are assuming a perspective in which one particular representation of algebra is privileged; namely, the traditional symbolic one. (Which is the representation they adopt in developing their own algebra instruction app, an Intelligent Tutoring System called Lynnette.) But as I pointed out last month, that representation became the dominant one entirely by virtue of what was at that time the best available distribution technology: the printing press.

With newer technologies, in particular the tablet computer (“printed paper on steroids”), other representations are possible, some better suited to learning, others to applications. To be sure, there are learning benefits to be gained from mastering symbolic algebra, perhaps even from doing so using paper-and-pencil, as Brady points out in his review. But at this stage in the representational technology development, we should adopt a perspective of all bets being off when it comes to how to best represent algebra in different contexts. I think it highly unlikely that we will ever again view algebra as something you learn or do exclusively by using a pen to pour symbols onto a page.

Indeed, with his background in video game design, Brady ends his review by rating DragonBox according to three metrics:

Fun Factor – A: I collected all 1,366 stars available in DragonBox 1 and 2 and had a great time.

Academic Value – B: I worry that many will underestimate the effort needed to transfer DragonBox skills to practical algebra proficiency.

Educational Value – A+: Anytime a kid leaves a game with thoughts like, “algebra is fun!” or “hey, I’m really good at math!” that is a huge win.

The LearnLab researchers are locked into the second perspective: what he calls Academic Value. (So too is Brady, to some extent, with his use of the phrase “practical algebra proficiency” to mean “symbolic algebra proficiency.”)

Make no mistake about it, transfer from mastery in an interactive engagement on a tablet to paper-and-pencil math is not automatic, as both Brady and the CMU researchers observe. To modify the old horse aphorism, DragonBox takes its players right to the water’s edge and dips their feet in, but still the players have difficulty drinking. (My best guess is that, for most learners it takes a good teacher to facilitate transfer.)

I note in passing that initially I had difficulty playing DragonBox. My problem was, classical, symbolic algebra is a second language to me that I have been fluent in since childhood and use every day. I found it difficult mastering the corresponding actions in DragonBox. Transfer is difficult in both directions.

At the present moment in time, those of us in education (or learning research) should absolutely not assume any one representation is privileged. Particularly so when it comes to learning. In that respect, Brady is right to note that DragonBox’s success in terms of his third metric (essentially, attitude and engagement) is indeed “a huge win.”

In the world in which our students will live their lives, arithmetic, algebra, and many other parts of mathematics, should be learned, and will surely be applied, in multimedia environments. All the evidence available today suggests that mastery of the traditional symbolic representation will be a crucial ingredient in becoming proficient at arithmetic and algebra. But the more effective practitioners are likely to operate with the aid of various technological tools. Indeed, for some future practitioners, mastery of the traditional symbolic representation (which is, remember, just a user interface to a certain kind of thinking) may turn out to be primarily just a key step in the cognitive process of achieving conceptual understanding, not used directly in applications, which may all be by way of mathematical reasoning tools.

Exactly when, in the initial learning process, it is best to introduce the classical symbolic representation is as yet unclear. What the evidence of countless generations of students-turned-parents makes abundantly clear, however, is that teaching only the classical symbolic approach is a miserable failure. That much is affirmed every time a parent posts on social media that they are unable to understand a Common Core math question that requires nothing more than understanding the place-value representation of integers. (Which is true of most of the ones I have seen posted.)

There is some evidence (see for example Jo Boaler’s new book) that a more productive approach is to use learning technologies to develop and sustain student engagement and develop a growth mindset, and provide learning environments for safe, productive failure, with the goal of developing number sense and general forms of creative problem solving (mathematical thinking), bringing in symbolic representations and specific techniques as and when required.

**Full declaration: I should note that my own work in this area, some of it through my startup company BrainQuake, adopts this philosophy. The significant learning gains obtained with our first app were in number sense and creative problem solving for a novel, complex performance task. Acquisition of traditional “basic skills” with our app comes about (intentionally, by design) as a valuable by-product. The improvement we see in the basic skills category is much more modest, and may well be better achieved by a tool such as LearnLab’s ITS. In a world where we have multiple representations, it is wise to make effective use of them all, according to context. It is not a case of an interface “fail”; to say that (with or without a hashtag) is to remain locked in past thinking. Easy to do, even for experts. Rather, in an era when algebra is being forced to return to its roots of being a way of thinking to help us solve practical problems, using all available representations in unison can provide us with a major win.

Monday, April 4, 2016

Algebraic roots – Part 1

Fig. 1: A problem from the first ever algebra textbook.
The first ever algebra text book was written in Baghdad around 830CE, by the Persian mathematician Muhammad ibn Musa al-Khwarizmi, our modern word “algebra” coming from the Arabic term al-Jabr, a technique for balancing an equation, described in the book. If you were a student – or a teacher – back then, the problem shown above (Figure 1) is the kind of thing you would be faced with in your math class. It is a direct translation from the original Arabic of a problem in al-Khwarizmi’s book.

Most modern readers, on seeing this for the first time and being told it is an algebra problem, are surprised that there are no symbols. Yet it is clearly not an “algebra word problem” in the usual sense. It’s just about numbers. It is, in fact, a quadratic equation problem. Figure 2 below is the same problem as we would present it in an algebra textbook today.

Fig. 2: Al-Khwarizmi's quadratic equation in modern notation.

Symbolic algebra, as we understand it today, was not introduced until the Sixteenth Century, when the French mathematician François Viète took what until then had been a discipline presented in prose, and turned it into the symbolic process we are familiar with today.

This is not to say that mathematicians back in Ninth Century Persia did not use symbolic expressions in their work. They surely did. The issue is how they presented it in textbook form. In the days when books were handwritten and duplicated by hand-copying, the author of a mathematics book was faced with a problem that other writers did not have to worry about: faithful copying. Copying of manuscripts was largely done by monks in monasteries. While masters of the written word – they did, after all, “live by a book” — few monks mastered mathematics, and hence could not be relied upon to create an accurate copy of anything other than prose. Aware of this issue, authors of mathematics books wrote everything in prose.

With the introduction of the printing press in the Fifteenth Century, however, everything changed. Indeed, one of the first printed books published after Gutenberg printed his famous edition of the Bible was an Italian book on practical arithmetic. True, to handle a symbolic textbook, you have first to master the linguistic rules for reading, writing, and manipulating symbolic expressions, but once you do, algebra becomes a whole lot easier to do, as a line-by-line comparison of Figures 1 and 2 makes abundantly clear. (Actually, it’s lines-by-line!)

Notice, however, that the two presentations of the quadratic problem specify the same problem, and both solutions are, from a logical deduction point of view, the same. To some extent, the al-Khwarizmi’s prose version describes what goes on in your head when you solve the problem. At least — and this is where I am going with this — it does if you solve the problem by thinking about it.

With the symbolic presentation, it is possible to reduce the solution of an algebra problem to the mindless (literally), algorithmically-specified manipulations of symbols. Ever since the invention of the printing press, generations of students quickly discovered that you can pass an algebra test by mastering a collection of symbolic-manipulation rules. No understanding necessary. Moreover, when taught this way, the teacher’s job became immeasurably easier. It is easier to teach rules to be followed than to develop thinking skills, and it is easy to evaluate a student’s work if the goal is simply to check that it accords with the rules and arrives at the correct answer. (Indeed, teachers soon realized that the quickest way to grade a student’s work was to first see if the answer is correct, and only if it is not look at the symbolic working.)

While the student in Ninth Century Baghdad solved (linear and quadratic) equations by performing essentially the same steps as a student would today, with the problem presented in words, and the solution written out (presumably) in words, it can’t be carried out in a mindless fashion. The human mind can learn to follow rules for manipulating symbols, without knowing what they mean, but words are so much an integral part of human thinking that we cannot use them without their having meaning (albeit possibly a meaning other than the one intended by the author of an algebra book).

There is, then, a potential loss in taking algebra from a prose presentation to a symbolic one: namely, the student can lose the appreciation that algebra is a powerful way of thinking with countless uses in the everyday world. Instead of algebra being a codification of human logical thinking that emerges from within, it becomes a set of externally imposed, and often arbitrary-seeming rules to be mastered by repetitive practice. The natural, relevant, and empowering becomes the artificial, pointless, and tedious. (Those of us who like symbolic algebra see beyond the rules.)

“When will I ever use algebra?” today’s student justifiably asks. In terms of rule-based, symbol manipulation, the answer is, for most people (not all – and this is educationally significant), “Never.” But in terms of algebra, that codified way of thinking that has evolved and developed considerably since al-Khwarizmi’s day, the answer is, “All the time.” (Whenever you use a spreadsheet, for example.)

In the introduction to his algebra book, al-Khwarizmi declared that he was presenting

“... what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.”

This was cutting edge stuff back then. It doesn’t get much more practical than that!

As al-Khwarizmi explains, he was asked to write his book by the Caliph, who recognized the importance — for trade and engineering in particular (both of which were crucial to the regional society at the time) — of making those new methods of calculation widely available. The Caliph’s reasoning was as sound and significant then as it would be today. When a society reaches a state of development where trade and commercial and financial activity go beyond two people engaging in one-off transactions, it needs a more efficient tool than basic arithmetic. What is required is arithmetic-at-scale. When you boil it down to its essence, that is what algebra is. Al-Khwarizmi’s book codifies and formalizes the numerical reasoning that people use in their daily personal and professional lives in a fashion that enables them to operate at scale.

In the years since the printing press made it possible to produce algebra textbooks that used symbolic representations, the focus in the algebra class has gradually shifted from being about sophisticated reasoning about numbers to an often mindless game of symbol manipulation. For several centuries that could be justified on the grounds that the only effective way for society to be able to handle the arithmetic-at-scale required to advance was to train lots of people to carry out the necessary calculations. And for that, the most efficient way is to use rule-based, symbolic manipulation. The people carrying out those calculations no more had to understand what they were doing than the electronic calculator on your iPhone has to understand what it is doing. All that matters it that it – the human symbolic-algebraist or the calculator app — gets the right answer.

But now that those of us in more advanced societies (and in a great many less advanced societies, for that matter) do have ready access to those powerful calculating devices, devices that in addition to performing numerical calculations can also solve algebraic problems (arithmetic-at-scale, aka the electronic spreadsheet), the once-important societal need for many human symbolic calculators has gone away. What is required today is that people can make effective use of those new tools. That has shifted the emphasis back from symbolic-rule-mastery to the kind of formalized, rigorous thinking about quantitative matters that, thanks to al-Khwarizmi, we call algebra. Only now, we are back to the realm, not of symbol manipulation, but codified, logical, rigorous thinking about issues in our lives and in the world we inhabit.

To be sure, symbolic algebra is not going away. It is way too powerful to ignore. But whereas it used to be possible to provide a rationale for teaching algebra as pure, rule-based symbolic manipulation (albeit a societal rationale that views people as fodder for industry), it makes no sense to teach it that way today.

Which is why the Common Core now directs the focus not on the symbolic rules that dominated math instruction for centuries past, but on sophisticated mathematical thinking skills that develop and require a deeper understanding of numbers. This is why there is now so much talk of “number sense” and why Mary and Johnny are coming home from school with homework questions that their parents often find strange and occasionally incomprehensible.

In other words, algebra has returned to its roots. (Pun intended.)

END OF PART 1
In Part 2 of this commentary, to be published here next month, I will look at how those same digital technologies that have rendered obsolete much of what used to constitute K-12 algebra education, have provided new ways to teach the subject that are ideally suited to the way we use — and will increasingly use even more — algebra. After all, if the printing press turned algebra from prose to symbolic expressions, what will algebra look like now that the digital computer, and in particular the tablet device, has largely replaced the printing press?

NOTE: I realize that there is little in this month’s post that is new to MAA members. But as I know from emails and comments I receive, Devlin’s Angle posts find their way to a wide variety of readers, occasionally onto the desks of governors, education administrators, and others who play a role in the nation’s education system. With so much media attention currently being given to a mathematics education proposal being made by an individual having little knowledge of mathematics or current mathematics education (see last month’s column), I thought it timely to bring us back to an appreciation of algebra (i.e., algebraic thinking) that was apparent to a Ninth Century Caliph in Baghdad, and which is even more relevant to our lives today than it was back then.

I cannot avoid ending by observing that 2016 will surely go down as the year when the US media devoted more media space and time to individuals pontificating on topics they knew almost nothing about, than they did to experts, of which the United States has large numbers with global reputations. I think many editors would benefit from a (good) course in algebraic thinking.

Tuesday, March 1, 2016

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.

Thursday, February 11, 2016

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. 


Friday, January 1, 2016

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.

Friday, December 4, 2015

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.