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

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

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

**STREAM**

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

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

Here is the PROLOGUE of The Math Gene 3/

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

I have written about many (not all) of the factors listed above in a series of video-game learning articles in this blog. (Starts here.) Taking a broader perspective than mathematics, the many writings and video interviews on games and learning by James Paul Gee have much to say that can help us understand how those factors and others can effect learning.]

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

**Selected sources
**Berkowitz, Schaeffer, Maloney, Peterson, Gregor, Levine, Beilock, 2015. Math at home adds up to achievement in school, Science 09 Oct 2015: Vol. 350, Issue 6257, 196-198.

Kiili, Devlin, Perttula, Tuomi, 2015. Using video games to combine learning and assessment in mathematics education,

Pope & Mangram, 2015. Wuzzit Trouble: The Influence of a Digital Math Game on Student Number Sense,

Popovic, 2014. “Learning basic Algebra by playing 1.5h”. Center for Game Science, Uni of Washington.

Riconscente, 2013. Results From a Controlled Study of the iPad Fractions Game Motion Math,

Wendt & Rice, 2013. “Evaluation of ST Math in the Los Angeles Unified School District”, report by WestEd.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

*where this account continues…*

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

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

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

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

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

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

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

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

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

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

MATH: Ditto.

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

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

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

BIKE: I ended my previous post by promising to

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

It took me four attempts to complete that initial climb!And therein lies one of the biggest secrets of being able to solve a difficult math problem.”

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

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

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

What are we to make of this strange process?

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

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

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

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

Which leads me to this.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Here is our brief basking point.

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

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

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

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

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

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

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

*It took me four attempts to complete that initial climb!*

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

*To be continued …*

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

[ADDED LATER: We renamed the company BrainQuake (brainquake.com) ]

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

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

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

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

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

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

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

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

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

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

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

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

For more details about InnerTube Games and *Wuzzit Trouble*, visit our website: http://innertubegames.net.

If ever there were an article that repeatedly utilized faulty logic, this was it. Evidently written from an advocacy viewpoint, the authors obviously got carried away, allowing their advocacy position to stretch and twist logic well beyond breaking point.

I don’t normally comment on math ed advocacy articles, since people tend to be so firmly entrenched in their position that no amount of evidence and reasoning will prompt them to reflect. But this particular article was so far off the mark, and the logic so abused, I could not resist picking up my teacher’s red pen and going through it paragraph by paragraph, annotating as I did.

(Actually I generally use a green pen, since red is known to have negative consequences on student motivation. In this case, the two authors are accomplished academics, well able to handle the back and forth of scholarly debate, where we attack one another’s ideas but not the people, so ink color is not an issue. Besides, for this medium I worked at a keyboard, using boldface to signify my comments to their normal-typeface article.)

So, original article in regular type, my commentary in bold. Here we go. (It’s long, as was the original article, much of which I have to quote in order to critique it.)

* * * * * * * * *

There is a great progressive tradition in American thought that urges us not to look for the aims of education beyond education itself. Teaching and learning should not be conceived as merely instrumental affairs; the goal of education is rather to awaken individuals’ capacities for independent thought. Or, in the words of the great progressivist John Dewey, the goal of education “is to enable individuals to continue their education.”

This vision of the educational enterprise is a noble one. It doesn’t follow, however, that it is always clear how to make use of its insights. If we are to apply progressive ideals appropriately to a given discipline, we need to equip ourselves with a good understanding of what thinking in that discipline is like. This is often a surprisingly difficult task. For a vivid illustration of the challenges, we can turn to raging debates about K-12 mathematics education that get referred to as the “math wars” and that seem particularly pertinent now that most of the United States is making a transition to Common Core State Standards in mathematics.

At stake in the math wars is the value of a “reform” strategy for teaching math that, over the past 25 years, has taken American schools by storm. Today the emphasis of most math instruction is on — to use the new lingo — numerical reasoning.

**No. Numerical reasoning is just one aspect of math instruction. Analytic reasoning, logical reasoning, relational reasoning, and conceptual understanding are just as important and equally stressed. The basic components of K-12 education were elaborated at length by a blue ribbon panel of experts assembled by the National Academies of Science in a 2001 National Academies Press volume titled Adding It Up: Helping Children Learn Mathematics. **

This is in contrast with a more traditional focus on understanding and mastery of the most efficient mathematical algorithms.

**What is meant by “efficient” here? For many centuries, it was a crucial ability to be able to carry out numerical computations in the head or by paper-and-pencil. The “standard algorithms” were developed in India in the first centuries of the Current Era, and further honed by traders and engineers in the Iraq-Persia region, in order to make mental paper-and-pencil calculation most efficient. (The medium then was either a smooth patch of sand, a sandbox, a parchment, or some form of tablet.) **

**Those standard algorithms sacrificed ease of understanding in favor of computational efficiency, and that made sense at the time. But in today’s world, we have cheap and readily accessible machines to do arithmetical calculations, so we can turn the educational focus on understanding the place-value system that lies beneath those algorithms, and develop the deep understanding of number and computation required in the modern world, and prepare the ground for learning algebra.**

A mathematical algorithm is a procedure for performing a computation. At the heart of the discipline of mathematics is a set of the most efficient — and most elegant and powerful — algorithms for specific operations.

**“At the heart of the discipline”! Totally untrue. This reduces mathematics to computational arithmetic. The standard arithmetical algorithms were developed by, and for, traders, to facilitate commercial activity. Those algorithms were never at the heart of mathematics, not even when they were developed. Anyone who says this, exhibits so little knowledge of what mathematics is, they should not purport to be sufficiently expert to pontificate on mathematics education.**

The most efficient algorithm for addition, for instance, involves stacking numbers to be added with their place values aligned, successively adding single digits beginning with the ones place column, and “carrying” any extra place values leftward.

**Actually, these are only the most (computationally) efficient algorithms if the computation is done using paper-and-pencil. For mental calculation, left-right algorithms are far more efficient. But this is a red herring, since the focus in education should be learning and understanding, and there are algorithms that are far more efficient in achieving those goals.**

What is striking about reform math is that the standard algorithms are either de-emphasized to students or withheld from them entirely.

**De-emphasized, yes, for the reason I alluded to above. The need for a strong focus on those particular algorithms evaporated with the dawn of the computer age. No good teacher would withhold mention or discussion of the standard algorithms, not least because they have huge historical significance. That last remark of the authors is simply not true (though I dare say you could find the occasional teacher who acted is such a way).**

In one widely used and very representative math program — TERC Investigations — second grade students are repeatedly given specific addition problems and asked to explore a variety of procedures for arriving at a solution. The standard algorithm is absent from the procedures they are offered.

**Note that this is what is done in the second grade, when students are just starting out on their mathematical learning.**

Students in this program don’t encounter the standard algorithm until fourth grade, and even then they are not asked to regard it as a privileged method.

**So much for the authors’ earlier assertion about the standard algorithms being withheld! As to those algorithms not being treated as privileged methods, that is as it should be in today’s world. They were (rightly) privileged for many centuries when calculation had to be done by hand. But those days are gone.**

The battle over math education is often conceived as a referendum on progressive ideals, with those on the reform side as the clear winners. This is reflected, for instance, in the terms that reformists employ in defending their preferred programs. The staunchest supporters of reform math are math teachers and faculty at schools of education.

**Just stop and think about this for a moment. Where would you expect to find people who know most about mathematics education? Dare I say, math teachers and faculty at schools of education? By way of analogy, consider this statement. “The staunchest supporters of the need for cleanliness and the use of improved medical procedures are the doctors and nurses who work in hospitals.” You don’t say! If you get sick, you consult a medical professional – someone who has spent years studying the subject and has demonstrated their knowledge and ability. Why not proceed in the same fashion when it comes to education?**

While some of these individuals maintain that the standard algorithms are simply too hard for many students, most take the following, more plausible tack. They insist that the point of math classes should be to get children to reason independently, and in their own styles, about numbers and numerical concepts. The standard algorithms should be avoided because, reformists claim, mastering them is a merely mechanical exercise that threatens individual growth.

**This is such a blatant misrepresentation, I am suspicious of the authors’ motives in writing this. The standard approach in current beginning mathematics education is to begin by providing opportunities for children to reason independently (a hugely valuable ability in today’s world!), and then introduce algorithms, starting with algorithms designed for educational efficiency, and then moving on to algorithms optimized for hand-calculation efficiency. (It is arguable that it would make sense in today’s world to spend some time also looking at the algorithms used by computers, since they are not the standard paper-and-pencil algorithms, and comparison of different algorithms can help students gain deep understanding of number and computation. That could perhaps come later in the educational journey.)**

**The authors end the paragraph by repeating once more their false claim that “the standard algorithms should be avoided”. The “threatening growth” comment would have substance if mechanical mastery of the standard algorithms were the students’ only exposure to computational methods. But they are not.**

The idea is that competence with algorithms can be substituted for by the use of calculators, and reformists often call for training students in the use of calculators as early as first or second grade.

**No they do not. I do not know a single teacher who advocates calculator use in the second grade. I can’t say with certainty that you won’t find a self-proclaimed “reformist” who has made such a call, but it definitely is not “often”.**

Reform math has some serious detractors. It comes under fierce attack from college teachers of mathematics, for instance, who argue that it fails to prepare students for studies in STEM (science, technology, engineering and math) fields.

**You can find (a few) college professors who say evolution is false, but they are not in the mainstream. College professors enjoy enormous freedom in what and how they teach, so you can find all kinds of examples. But as someone whose career is almost exclusively spent in academia, who travels extensively and meets other academics across the US and around the world, I have yet to meet anyone who argues strongly that reform math fails to prepare students for studies in STEM. What I do hear a lot is complaints about a lot (not all) of K-12 education failing to prepare students adequately. My sense is the problem is quality of teaching as much if not more than curriculum, though the two are not necessarily independent.**

These professors maintain that college-level work requires ready and effortless competence with the standard algorithms and that the student who needs to ponder fractions — or is dependent on a calculator — is simply not prepared for college math.

**The first part of this statement is totally false. (Unless the phrase “these professors” refers to a couple of professors the authors happen to know.) Familiarity with the standard algorithms plays no role in college STEM. To say mastery of those particular algorithms is crucial to STEM is like saying using a Mac better prepares you for STEM than using a PC. Having a good sense of, and facility with, number, including fractions, is absolutely vital, and the algorithms currently taught in K-12 were developed to maximize that outcome. **

**Having been teaching university level mathematics around the world for 45 years, I know that in the days when the standard algorithms were the main focus, the results in terms of college-preparedness were terrible. Except for a few students (that few very likely including the article’s authors), the classical teaching methods simply did not work. If they had worked, you would not find so many adults who say they cannot do math! That failure of the old method is what led to the introduction of alternative approaches using algorithms optimized for learning.**

They express outrage and bafflement that so much American math education policy is set by people with no special knowledge of the discipline.

**Really? I mean, really? Other than a few outliers, I have not heard a deluge of outrage. Reform math is a result of an extensive collaboration between math teachers, mathematics education faculty, and mathematicians, including that blue-ribbon committee of the National Academy of Sciences I mentioned earlier, which published that huge volume on the basic of mathematics education almost fifteen years ago. Hardly “people with no special knowledge of the discipline.”**

Even if we accept the validity of their position, it is possible to hear it in an anti-progressivist register. Math professors may sound as though they are simply advancing a claim about how, for college math, students need a mechanical skill that, while important for advanced calculations, has nothing to do with thinking for oneself.

**I doubt they would sound that way. In any case, I am not sure what point the authors are trying to make here.**

It is easy to see why the mantle of progressivism is often taken to belong to advocates of reform math. But it doesn’t follow that this take on the math wars is correct. We could make a powerful case for putting the progressivist shoe on the other foot if we could show that reformists are wrong to deny that algorithm-based calculation involves an important kind of thinking.

What seems to speak for denying this? To begin with, it is true that algorithm-based math is not creative reasoning. Yet the same is true of many disciplines that have good claims to be taught in our schools. Children need to master bodies of fact, and not merely reason independently, in, for instance, biology and history. Does it follow that in offering these subjects schools are stunting their students’ growth and preventing them from thinking for themselves? There are admittedly reform movements in education that call for de-emphasizing the factual content of subjects like biology and history and instead stressing special kinds of reasoning. But it’s not clear that these trends are defensible. They only seem laudable if we assume that facts don’t contribute to a person’s grasp of the logical space in which reason operates.

**I still don’t know for sure what the authors are trying to say here. To my knowledge, no teacher has ever said facts are not important. I can only assume that authors are erecting a huge straw man, but it’s so ludicrous it does not deserve more than this brief dismissal.**

The American philosopher Wilfrid Sellars was challenging this assumption when he spoke of “material inferences.” Sellars was interested in inferences that we can only recognize as valid if we possess certain bits of factual knowledge. Consider, for instance, the following stretch of reasoning: “It is raining; if I go outside, I’ll get wet.” It seems reasonable to say not only that this is a valid inference but also that its validity is apparent only to those of us who know that rain gets a person wet. If we make room for such material inferences, we will be inclined to reject the view that individuals can reason well without any substantial knowledge of, say, the natural world and human affairs. We will also be inclined to regard the specifically factual content of subjects such as biology and history as integral to a progressive education.

**More of the same. The horse is long dead. It was never born, for heavens sake. Stop flogging it.**

These remarks might seem to underestimate the strength of the reformist argument against “preparatory” or traditional math. The reformist’s case rests on an understanding of the capacities valued by mathematicians as merely mechanical skills that require no true thought.

**The second sentence here is the exact opposite of actuality. The reformist case rests on knowing that mathematicians value mechanical skills that are based on sound understanding and can be utilized in a creative, thoughtful, reflective way.**

**[I am going to skip the authors’ next few paragraphs as theoretical cognitive philosophy. You can the entire article in its original posting.]**

It is important to teach [the standard algorithms] because, as we already noted, they are also the most elegant and powerful methods for specific operations. This means that they are our best representations of connections among mathematical concepts. Math instruction that does not teach both that these algorithms work and why they do is denying students insight into the very discipline it is supposed to be about.

**The first sentence is okay if you delete the qualifier “the most elegant”. The second sentence displays total ignorance of mathematics. (A very odd thing, since the second listed author is an accomplished research mathematician.) The third sentence needs a bit of analysis.**

**The standard algorithms are a very good historical hack, improved over many generations, that enabled people to do complex arithmetic calculations with paper-and-pencil (or its early equivalent). As long as students learn at least one algorithm for each basic arithmetical operation that gives them an understanding of number and the number system, they will gain insight into number and arithmetic. But number and arithmetic are not what the discipline [mathematics] “is supposed to be about.” Again, the authors’ words indicate that have not a clue what mathematics is about. (Once more puzzling, given the second author’s credentials.) As it happens, there are algorithms that are better suited than the standard ones for gaining insight into number and arithmetic, and those are the ones currently used in “reform mathematics education.” Teaching other methods, including the standard algorithms, can increase that important insight, but the justification for including the classical algorithms is largely historical.**

(Reformists sometimes try to claim as their own the idea that good math instruction shows students why, and not just that, algorithms work. This is an excellent pedagogical precept, but it is not the invention of fans of reform math. Although every decade has its bad textbooks, anyone who takes the time to look at a range of math books from the 1960s, 70s or 80s will see that it is a myth that traditional math programs routinely overlooked the importance of thoughtful pedagogy and taught by rote.)

**Nonsense. No one makes such a claim. There have always been good teachers providing good education. There have always been, unfortunately, poor teachers providing poor education. This parenthetical paragraph is another straw man.**

As long as algorithm use is understood as a merely mechanical affair, it seems obvious that reformists are the true progressivists. But if we reject this understanding, and reflect on the centrality to mathematical thought of the standard algorithms, things look very different. Now it seems clear that champions of reform math are wrong to invoke progressive ideals on behalf of de-emphasizing these algorithms. By the same token, it seems clear that champions of preparatory math have good claims to be faithful to those ideals.

**I cannot imagine a paragraph that is more the exact opposite of actuality.**

There is a moral here for progressive education that reaches beyond the case of math. Even if we sympathize with progressivists in wanting schools to foster independence of mind, we shouldn’t assume that it is obvious how best to do this. Original thought ranges over many different domains, and it imposes divergent demands as it does so. Just as there is good reason to believe that in biology and history such thought requires significant factual knowledge, there is good reason to believe that in mathematics it requires understanding of and facility with the standard algorithms.

**The authors were doing fine until they wrote the second part of this last sentence. The standard algorithms offer no privileged insight into arithmetic, let alone the far broader discipline of mathematics. Their value today is primarily historical. For the period after our ancestors developed symbolic writing up to the invention of the modern computer, the standard algorithms were of great value. They no longer offer anything of unique pedagogic value other than variety. Other algorithms are better suited to learning.**

Indeed there is also good reason to believe that when we examine further areas of discourse we will come across yet further complexities. The upshot is that it would be naïve to assume that we can somehow promote original thinking in specific areas simply by calling for subject-related creative reasoning. If we are to be good progressivists, we cannot be shy about calling for rigorous discipline and training.

**An apple pie paragraph that everyone will agree with.**

The preceding reflections do more than just speak for re-evaluating the progressive credentials of traditional, algorithm-involving math. They also position us to make sense of the idea, which is as old as Plato, that mathematics is an exalted form of intellectual exercise. However perplexing this idea appears against the backdrop of the sort of mechanical picture favored by reformists, it seems entirely plausible once we recognize that mathematics demands a distinctive kind of thought.

**Actually, what the preceding reflection indicates is that the authors haven’t any real understanding of mathematics or mathematics learning. (Second author puzzlement again.) They certainly give the impression they have no idea what reform mathematics is about, since the reformists’ position is as far removed from a “mechanical picture” as can be imagined.**

Whether you view mathematics as a collection of procedures or a way of thinking (see my last post), math is something you *do*. Or cannot *do*, as the case may be.

When I meet people for the first time and tell them my profession, they frequently reply, “I never could do math.” What they never say is, “I don’t know math.” Everyone, whether mathematically able or not, realizes that math is not stuff that you know, but an *activity *you* do*.

Of course, sleeping, sitting on the beach daydreaming, and watching TV are also activities, but they are passive activities. I am using the word “activity” in its stronger sense. That stronger sense certainly includes mental activity. As a simple rule of thumb, you know something is an activity in my sense if doing it makes you tired. By that metric, math is one of the most strenuous activities I know — and I’m one of those people who spend their weekends cycling over mountain passes for seven or more hours at a stretch.

What is the most efficient way to learn how to *do* something? We all know the answer, and we did so long before Nike turned it into a commercial slogan: *Just do it! *

If you want to learn to ride a bike, drive a car, ski, play tennis, play golf, play chess, play the piano, and so forth, you don’t start out by attending a lecture or reading a book. Those can be useful supplements when you have reached a sufficient level of proficiency and want to get better. But learning from a lecture or a book require interpretation and assimilation of incoming *information* (a static commodity), and that in turn requires sufficient prior understanding. No, what you do is start to do it.

Very likely you don’t start out doing it unaided. You seek guidance, from a parent, relative, friend, instructor, professional coach, or whatever. And in the course of helping you learn, that person may well give you instructions and advice. But they do so in the course of you performing the activity you are trying to learn, when what they say makes sense and has immediate, recognizable value.

With everyone, it seems, in agreement that mathematics is an activity, and given our collective experience that mastering an activity is best achieved through *doing* it, we have to ask ourselves how mathematics education has come to be dominated by the math textbook?

Though there is an argument to be made about the self-interest of textbook publishers, the fact is that mathematics instruction has been delivered through textbooks since the subject began. Archimedes’ *Method*, Euclid’s *Elements*, al Khwarizmi’s *Al-Jabr*, Leonardo of Pisa’s *Liber abbaci*, and on throughout mathematical history, the symbol-heavy, written text has been the primary vehicle for storing and disseminating mathematical knowledge.

Why? Because putting words and symbols on a flat surface was the only technology available for the task!

But video games — or rather, video game technologies — provide us with an alternative. The digital framework in which a typical video game is embedded is dynamic and interactive, and can provide the experience of moving around in a 3D world. In other words, video game technologies provide platforms or environments suited (by design) for *action*. Which makes them ideal for representing and doing mathematics (an activity).

The task facing the designer of a video game to provide good mathematics learning experiences is to represent the mathematics using the natural affordances of the medium. This means putting aside the familiar symbolic representation. My own experience, having been doing this for over five years now, and working with others doing the same thing, is that it is initially very difficult. People have been using symbolic representations or one form or another for several millennia and that has conditioned how we think of mathematics. But it is worth making the effort, because the potential payoff is massive: it will circumvent the Symbol Barrier, which I discussed in the third post in this series.

In addition, by representing the mathematics in a medium-native fashion, we will minimize, and in some cases eliminate, the degree to which “doing the math” detracts from the game mechanics. For some students — the ones with a natural affinity to mathematical thinking — this is not a big deal, since they will gain satisfaction from solving the mathematical problem, but for many students, advancement in the game will be the main driver.

I should stress that what I am advocating is not watering down mathematical thinking to a “video game version” of mathematical thinking. At a conceptual level, it is the same thinking; only the representation is changing. Once the student has mastered mathematical thinking presented in “video-game language”, a teacher could use that experience as a foundation on which to base instruction in the symbolic representation of the same concepts and thinking.

That last step is an important one, in part because mastery of symbolic mathematics is what is required to perform well in standardized math tests, and regardless of your views on the educational value of such instruments, they are currently a fact of life for our students. But there are two other reasons why it is important to transition the students to symbolic mathematics. First, mastering multiple representations greatly assists good conceptual learning, and the abstract symbolic representation, by virtue of its abstractness, is particularly powerful in that regard. Second, the symbolic representations make it much easier to apply mathematical thinking to a wide variety of new problems in novel domains.

I’ll pursue these ideas further in subsequent postings. In the meantime, let me leave you with three examples of video games that present mathematics in a medium-native fashion: Motion Math, Number Bonds, and Jiji. Notice that in each case the mathematical concept involved is represented in a medium-native, and *dynamic* fashion. The player interacts *directly* with the concept, not indirectly via a symbolic representation, in the same way that a person playing a piano interacts directly with the music, not indirectly via a symbolic musical score.

To my mind, this is one of the most significant, and potentially disruptive benefits of using video games in mathematics education: they offer the possibility of direct manipulation of mathematical concepts, thereby circumventing the symbol barrier. Achieving this direct connection to the concepts is not easy. Those three games may look simple. Indeed, to the player, they are simple, and that is the point! But I know for a fact that all three took some very smart folks a lot of time and effort to produce. That’s usually the case with any tool that looks simple and works naturally. Designing simplicity is hard.

]]>The vast majority of video games that claim to teach mathematics do not actually do that. Rather, what they do is provide a means for students to practice what they have already been taught. For the most part, the focus is on basic computational skills.

A good example is the first-person shooter Timez Attack. Mastery of the multiplication bonds (times tables in parent-speak) is an extremely useful thing to achieve, and the sooner the better. All it requires is sufficient repetition, and I know of no better way to achieve that than with an entertaining video game.

Such games are the low hanging fruit for the math ed video game designer, and like most low hanging fruit, it has pretty well all been picked, leaving game designers coming into the math ed space having to look elsewhere for a useful application of their talents. The good news is, since repetitive practice of basic computational skills is a tiny part of learning mathematics — albeit an important part, in my view (some educators disagree) — most of the fruit in the math ed orchard is still waiting to be picked. The bad news is, that fruit is a lot higher up, and thus more difficult to reach.

The difficulty hits you as soon as you decide to go for more than mastery (ideally to fluency) of already taught basic computational skills. Are you going to approach mathematics as a collection of procedures or as a way of thinking? These are not completely separate classifications; indeed, the latter is in many ways a broader conception than the former. But they do tend to cash out in very different forms of pedagogy. (Spoiler: instruction versus guided-discovery.)

This distinction is to a great extent relatively recent. Until the nineteenth century, mathematicians viewed the discipline as a collection of procedures for solving various kinds of problems. Originally, the problems studied arose in the world. Then, in due course, the focus widened to include more abstract problems arising within mathematics itself. Proficiency in math meant being able to carry out calculations or manipulate symbolic expressions to solve problems.

By and large, high school mathematics is still very much based on that earlier tradition, so few people outside the professional mathematical community are aware that in the middle of the 19th century, a revolution took place.

Working in the revolution’s epicenter, the small university town of Göttingen in Germany, the mathematicians Lejeune Dirichlet, Richard Dedekind, and Bernhard Riemann pioneered a new, broader conception of mathematics, where the *primary* focus was not performing a calculation or computing an answer, but formulating and understanding abstract concepts and relationships. This was a shift in emphasis from *doing* to *understanding*.

For the Göttingen revolutionaries, mathematics was about “Thinking in concepts” (*Denken in Begriffen*). Mathematical objects were no longer thought of as given primarily by formulas, but rather as carriers of conceptual properties. Proving was no longer a matter of transforming terms in accordance with rules, but a process of logical deduction from concepts.

Of course, during the course of this conceptual thinking, mathematicians still made use of procedures. What changed was the primary emphasis. The reason for the change? An increase in complexity, in science, technology, business, society, and, derivatively, within mathematics itself. In a simple world, a few well-practiced procedures can generally get you by. But when things get more complex, you need understanding in order to select from a variety of different procedures, to fix old procedures that no longer work, and to develop new ones.

I give this somewhat lengthy detour through recent mathematical history not because it has a direct bearing on how we teach K-12 mathematics. The one attempt to modify K-12 education to take account of the 19th century shift in mathematics as practiced by the professional mathematicians, the “New Math” movement of the 1960s, was so badly bungled that even a professional mathematician, Tom Lehrer, satirized it. (It was also hardly “new math” at the time, being already a century old.) Rather, I am stressing the distinction between math-as-procedures and math-as-thinking because it is now extremely relevant to the way we educate our next generation of citizens. The complexity of 21st Century life is such that ordinary citizens now need to upgrade their mathematical knowledge and abilities the same way the professional mathematicians did in the mid nineteenth century. The changes in society, and in particular technology and the way we do business, that were made possible by the newer, richer, and more powerful mathematics that developed in the 20th Century, now affect us all in the 21st.

I discussed the growing importance of “mathematical thinking” in a “Devlin’s Angle” column for the MAA back in 2010, and summarized those arguments in a more recent article in the *Huffington Post*. My purpose here is not to argue for any one approach to the design of video games to help students learn math. Heavens, the medium is so new, and there are so few games of any real educational merit, there is scope for a wide variety of approaches. Give me *any* video game that plays well and helps students learn math and I’ll applaud, whatever the pedagogy.

What distresses me is that the medium offers so much promise for good mathematics learning, it is a waste of time, effort, and money to focus on the lowest level — repetitive practice of the basic, procedural, computational skills. We’ve done that. Let’s move on.

Step 1 for the math ed video game designer today is, to recap, deciding whether to develop a game to help students master mathematics procedures or to develop powerful mathematical thinking capacities. As readers of my book will already know, I favor the latter, in large part because mastery of mathematical thinking capacity carries mastery of procedures along with it, just as the person who sets out to build a house will have to develop skills in bricklaying, carpentry, plumbing, and so on, along the way. But as I said a moment ago, the challenge we face in K-12 mathematics education is so great, and video games offer such potential, hitherto largely untapped, I’ll settle for any approach that works.

It’s your call which view of mathematics you take, pre-1850 or post. Both have strong track records. But you do have to make that call, as it will affect every design choice you make from then on. Engineers who set out to build a bicycle and then act as if they are building a car tend not to succeed, even though both are transportation devices. I’ll try to make this blog series helpful whichever way you make the call.

]]>In designing a video game to help students learn mathematics, it’s important not to over-estimate the capabilities of the medium. Breathless articles about the imminent arrival of HAL-like artificial intelligence notwithstanding, the day is not yet here (and in my view won’t be for a long time, if indeed at all) when we can take the human teacher out of the loop. If you really want to develop video games that contribute in a significant way to mathematics education, you should view them as supplementary educational materials or tools to be used by teachers.

A major problem with video games, or more generally any mechanized educational delivery system, is that the system has no way of knowing what the player, or student, is learning. That a player who moves up a level in a video game has learned **something** is clear. Video games are all about learning. But all you can reliably conclude from a player’s leveling up is that she or he has leveled up. It could have been happenstance.

Okay, but what if the player keeps leveling up? Surely that is not just chance? Possibly not; indeed, given good game design, probably not. The player must have learned something. But what? It might be precisely what that game activity was designed to teach. But it could be something quite different.

This is not a problem unique to video games, or to educational technology in general. It’s a fundamental problem about teaching and learning.

Take a look at the following video from the well-known educational consultant Marilyn Burns:

If you are like me the first time I saw this video, when you heard Cena’s answer in the class you concluded that she understood place value representation. She certainly gave the right answer. Moreover, **to those of us who do understand place-value**, her verbally articulated reasoning indicated she had conceptual understanding. But she had nothing of the kind, as the subsequent interview made clear.

And therein lies the problem. The human brain is a remarkable pattern-recognizing device. It will even discern a pattern – usually many patterns – in a random display of dots on a screen, where by definition there is no pattern. But is it the pattern the brain recognizes “right” pattern? Cena clearly recognized a pattern, and it yielded the “right answer.” But was it place-value? Perhaps some aspect, but we have no way of knowing

Of course, video games are highly interactive and ongoing. Surely, with video game learning a false understanding will eventually become apparent. Eventually the player will demonstrate that something has gone wrong. Right?

Unfortunately not, as was discovered in 1973, albeit not in the context of a video game but something with similar features. In what rapidly became one of the most famous and heavily studied papers in the mathematics education research literature, Stanley Erlwanger exposed the crippling limitations of what at the time was thought to be a major step forward in mathematics education: Individually Prescribed Instruction (IPI).

Though not a video game, nor indeed delivered by any technology beyond printed sheets of paper, IPI was very similar to an educational video game, in that it presented students with a series of mathematical problems that were selected and delivered at a rate thought to be ideally suited to the individual student, leading the student forward in the same way good level design does in a video game. Without a doubt, if IPI has problems, so too will video games. And as Erlwanger’s paper “Benny’s Conception of Rules and answers in IPI Mathematics” showed, IPI had problems. Big problems. How big? It didn’t work.

The subject of Erlwanger’s study was a twelve-year-old boy called Benny, chosen because he was doing particularly well on the program, moving rapidly from level to level, scoring highly at each stage. As Erlwanger states in his paper, Benny’s teacher, who was administering the program for Benny, felt sure that his pupil could not have progressed so far without having a good understanding of previous work.

Erlwanger’s research methodology was essentially the same as the approach Marilyn Burns used. He interviewed Benny to see what the boy understood. And when he did, a large can of worms spilled out. Though he got high scores on all the question sheets, Benny had almost no understanding of any mathematics, and a totally warped view of what mathematics is, to boot.

Being bright, Benny had quickly worked out a strategy for tacking the IPI question sheets. His strategy was based in part on pattern recognition, and in part on developing a theory about how the game was constructed – yes, he viewed it as a game! And he did what any smart kid would do, he figured out how to game the game.

What the designers of the IPI program had intended was that gaming the game required mastering the mathematics. Unfortunately, there is no way to prevent people, particularly smart ones, from coming up with alternative systems.

In Benny’s case, this involved developing a complete set of rules for adding, subtracting, multiplying and dividing fractions. Though his rules were symbolic manipulation procedures that made no sense mathematically, they enabled him to move through the sheets faster than everyone else in his cohort group, scoring 80% or better at each stage.

Whenever his rules yielded wrong answers, he simply adapted them to fit the new information he had acquired.

When asked by Erwanger, Benny was able to provide consistent, coherent explanations of his methods and why they worked. He was also very confident in his performance, and would stick to his explanations and would not alter his answers when pressured.

I won’t spend time here going through the details. You can read it all in Erlwanger’s paper, which is available here. Anyone who is about to embark on designing a video game for mathematics education should read that paper thoroughly. You need to know what you are up against. (The same dangers arise with gamification, and for the same reason.)

What I will do is say briefly what the fundamental issue is. The designer of the video game (just like the developers of the IPI worksheets) starts with an understanding of the mathematics to be learned, and creates a system to deliver it. The player, or student, does not yet know that mathematics, so they approach the system as what they see: a video game in our case or a series of quizzes in Benny’s. In both cases, the rewards come not from mastery of the underlying mathematics, but from successful completion of the challenge *qua challenge*. Indeed, with many educational video games, that’s the whole point: turn mathematics learning into a game!

In Benny’s case, not only was he successful in “playing the game”, in the process he developed an entire conception of mathematics as consisting of pointless questions that have a range of possible correct answers, one of which the test maker (in our case, read game developer) had decided, according to some secret but arbitrary set of rules, to declare as the “correct” one. Benny saw his task as to figure out the arbitrary rules the test-maker was using.

Only when you understand the nature of mathematics does Benny’s strategy seem crazy. Without such understanding, his approach is perfectly sensible. He does not know about math, but he already knows a lot about people and about playing games of different kinds. And when this particular game keeps telling him he is doing well, and making progress, he has no reason to change his basic assumptions.

Anyone who sets out to develop a math ed video game needs to have a strategy to avoid falling into the Benny Trap. Personally, I know of no way to do that with any hope of success other than conducting Marilyn Burns type player interviews throughout the development cycle. Fortunately, game developers are already used to doing lots of player testing. Mostly, they are checking for playability and engagement. With an educational video game, they need to augment those tests with interviews to see what is being learned.

That should at least ensure that the game will stand a chance of achieving the educational goal you want. The next issue to address is the circumstances under which the game will be played, and in particular the role of the (human) teacher.

*To be continued …*