Archive for the 'mathematics education' Category

The beginnings of Education Science?

Published January 18, 2016 education research , education science , educational technology , mathematics education , video game design 1 Comment

[This is a greatly expanded version of the essay published in Edge as my response to the Annual Edge Question for 2016.]

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, International Journal of Serious Games, October 2015, 37-55.
Pope & Mangram, 2015. Wuzzit Trouble:  The Influence of a Digital Math Game on Student Number Sense, International Journal of Serious Games, October 2015, 5-22.
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,  Games and Culture 8(4), 186-214.
Wendt & Rice, 2013. “Evaluation of ST Math in the Los Angeles Unified School District”, report by WestEd.

The Wuzzits – Free at Last

Published September 3, 2013 mathematical thinking , mathematics education , video game design 2 Comments
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In which the word free has several meanings.

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.

A screen shot from Wuzzit Trouble

A screen shot from Wuzzit Trouble

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.

Apple features Wuzzit Trouble on its release

Apple features Wuzzit Trouble on its release

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.

Faulty logic in the new Math Wars skirmish

Published June 19, 2013 mathematics education 62 Comments

Mathematicians love puzzling about self-referential statements such as “This sentence is false” (ask yourself if it is true or false and see what happens), and analyzing them has led to significant insights in the way mathematics is used to model real world phenomena. But I suspect the authors of a recent New York Times opinion piece did not see the ironical self-reference in their title The Faulty Logic of the ‘Math Wars’, published on June 16.

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.

How to design video games that support good math learning: Level 6

Published April 6, 2012 mathematical thinking , mathematics , mathematics education , video game design 25 Comments
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Show me the action!
Part 6 of a series 

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.

How to design video games that support good math learning: Level 5

Published March 30, 2012 mathematics education , video game design 7 Comments
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Procedures or thinking?
Part 5 of a series 

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.

How to design video games that support good math learning: Level 4

Published March 1, 2012 mathematics education , video game design 11 Comments
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“Benny’s Rules” Still Rule
Part 4 of a series 

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 …

How to design video games that support good math learning: Level 3

Published February 22, 2012 mathematics education , video game design 6 Comments
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The Symbol Barrier
Part 3 of a series

In my view, the most significant single benefit that video games offer to mathematics education is their capacity to overcome the biggest obstacle to practical mastery of middle school math: the symbol barrier. Yet to date, practically none of the now hundreds of math ed video games available have even begun to address it. In part, I suspect, because the developers of those games were probably not aware of the issue.

Chances are you have never heard of the symbol barrier either. Certainly not by that name, I agree. That term is mine, and I started using it only recently (when I realized that video games provided the key to overcome it). But the problem itself has been familiar to mathematics learning specialists for twenty years, and it created a considerable stir when it was first observed. The first main chapter of my recent book on mathematics education video games, after the opening chapter that sets the scene, is devoted to a fairly lengthy discussion of the issue.

To understand the symbol barrier, and appreciate how pervasive it is, you have to question the role symbolic expressions play in mathematics.

When a TV or movie director wants the audience to know that a particular character is a mathematician, somewhere in that character’s first scene you will see her or him writing symbols – on a piece of paper, on a blackboard, or, quite likely, on a window or a bathroom mirror. (Real mathematicians never do that, but it looks cool on the screen.) This character-establishing device is so effective because, as the director knows very well, people universally identify doing math with writing symbols, often obscure symbols.

Why do we make that automatic identification? Part of the explanation is that much of the time we spent in the school mathematics classroom was devoted to the development of correct symbolic manipulation skills, and symbol-filled books are the standard way to store and distribute mathematical knowledge. So we have gotten used to the fact that mathematics is presented to us by way of symbolic expressions.

But just how essential are those symbols? After all, until the invention of various kinds of recording devices, symbolic musical notation was the only way to store and distribute music, yet no one ever confuses music with a musical score.

Just as music is created and enjoyed within the mind, so to is mathematics created and pursued (and by many of us enjoyed) in the mind. At its heart, mathematics is a mental activity – a way of thinking. Not a natural way of thinking, to be sure; rather one that requires training to learn and concentration to achieve. But a way of thinking that over several millennia of human history has proved to be highly beneficial to life and society.

In both music and mathematics, the symbols are merely static representations on a flat surface of dynamic mental processes. Just as the trained musician can look at a musical score and hear the music come alive in her or his head, so too the trained mathematician can look at a page of symbolic mathematics and have that mathematics come alive in the mind.

So why is it that many people believe mathematics is symbolic manipulation? And if the answer is that it results from our classroom experiences, why is mathematics taught that way? I can answer that second question. We teach mathematics symbolically because, for many centuries, symbolic representation has been the most effective way to record mathematics and pass on mathematical knowledge to others.

Still, given the comparison with music, can’t we somehow manage to break free of that historical legacy?

Well, things are not quite so simple. Like all analogies, the comparison of mathematics with music, while helpful, only takes you so far. Although mathematical thinking is a mental activity, for the most part the human brain can do it only when supported by symbolic representations. In short, the symbolic representation seems far more crucial to doing mathematics than is musical notation for performing music. (We are all aware of successful musicians who cannot read or write a musical score.) In fact, much of mathematics – including all advanced mathematics – deals with symbolically defined, abstract entities. Without the symbols, there would be no entities to reason about.

The one exception, where the brain does not require the aid of symbolic representations (and where the comparison with music holds well) is what for several years now I have been calling “everyday mathematics.” This is the collection of mathematical concepts, operations, and procedures that are an essential part of everyday life skills for today’s world – the mathematical equivalent of the ability to read and write. (In contrast to the mathematics required for science, engineering, economics, advanced finance, and many parts of business, where fluency with symbolic expressions is essential.)

Roughly speaking, everyday mathematics comprises counting, arithmetic, proportional reasoning, numerical estimation, elementary geometry and trigonometry, elementary algebra, basic probability and statistics, logical thinking, algorithm use, problem formation (modeling), problem solving, and sound calculator use. (Yes, even elementary algebra belongs in that list. The symbols are not essential. For much of its roughly fifteen-hundred-year history, algebra was not written down symbolically, rather was recorded, described, and taught using ordinary language, with terms like “the unknown” where today we would write an “x”.)

True, people sometimes scribble symbols when they do everyday math in a real-life context. But for the most part, what they write down are the facts needed to start with, perhaps the intermediate results along the way and, if they get far enough, the final answer at the end. But the doing math part is primarily a thinking process – something that takes place primarily in your head. Even when people are asked to “show all their work,” the collection of symbolic expressions that they write down is not necessarily the same as the process that goes on in their heads when they do math correctly. In fact, people can become highly skilled at doing mental math and yet be hopeless at its symbolic representations.

It is with everyday mathematics that the symbol barrier emerges.

In the early 1990s, three researchers, Terezinha Nunes (then at the University of London, England, now at Oxford University), Analucia Dias Schliemann, and David William Carraher (both of the Federal University of Pernambuco in Recife, Brazil) embarked on an anthropological study in the street markets of Recife. With concealed tape recorders, they posed as ordinary market shoppers, seeking out stalls being staffed by young children between 8 and 14 years of age. At each stall, they presented the young stallholder with a transaction designed to test a particular arithmetical skill. The purpose of the research was to compare traditional instruction (which all the young market traders had received in school since the age of six) with learned practices in context. In many cases, they made purchases that presented the children with problems of considerable complexity.

What they found was that the children got the correct answer 98% of the time. “Obviously, these were not ordinary children,” you might imagine, but you’d be wrong. There was more to the study. Posing as shoppers and recording the transactions was only the first part. About a week after they had “tested” the children at their stalls, the three researchers went back to the subjects and asked each of them to take a pencil-and-paper test that included exactly the same arithmetic problems that had been presented to them in the context of purchases the week before, but expressed in the familiar classroom form, using symbols.

The investigators were careful to give this second test in as non-threatening a way as possible. It was administered in a one-on-one setting, either at the original location or in the subject’s home, and the questions were presented in written form and verbally. The subjects were provided with paper and pencil, and were asked to write their answer and whatever working they wished to put down. They were also asked to speak their reasoning aloud as they went along.

Although the children’s arithmetic had been close to flawless when they were at their market stalls – just over 98% correct despite doing the calculations in their heads and despite all of the potentially distracting noise and bustle of the street market – when presented with the same problems in the form of a straightforward symbolic arithmetic test, their average score plummeted to a staggeringly low 37%.

The children were absolute number wizards when they were at their market stalls, but virtual dunces when presented with the same arithmetic problems presented in a typical school format. The researchers were so impressed ­– and intrigued – by the children’s market stall performances that they gave it a special name: they called it street mathematics.

As you might imagine, when the three scholars published their findings (in the book Street Mathematics and School Mathematics, Cambridge University Press, Cambridge, UK, 1993), it created a considerable stir. Many other teams of researchers around the world carried out similar investigations, with target groups of adults as well as children, and obtained comparable results. When ordinary people are faced with doing everyday math regularly as part of their everyday lives, they rapidly achieve a high level of proficiency (typically hitting that 98% mark). Yet their performance drops to the 35 to 40% range when presented with the same problems in symbolic form.

It is simply not the case that ordinary people cannot do everyday math. Rather, they cannot do symbolic everyday math. In fact, for most people, it’s not accurate to say that the problems they are presented in paper-and-pencil format are “the same as” the ones they solve fluently in a real life setting. When you read the transcripts of the ways they solve the problems in the two settings, you realize that they are doing completely different things. (I present some of those transcripts in my book.) Only someone who has mastery of symbolic mathematics can recognize the problems encountered in the two contexts as being “the same.”

That, my friend, is the symbol barrier. It’s huge and it is pervasive. For the entire history of organized mathematics instruction, where we had no alternative to using static, symbolic expressions on flat surfaces in order to store and distribute mathematical knowledge, that barrier has prevented millions of people from becoming proficient in a cognitive skill-set of evident major importance in today’s world, on a par with the ability to read and write.

With video games, we can circumvent the barrier.

To be continued …

How to design video games that support good math learning: Level 2.

Published January 31, 2012 mathematics education , video game design 11 Comments
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The educational goal
Part 2 of a series.

Anyone setting out to design a video game to help students learn mathematics should start out by reading – several times, from cover to cover – the current “bible” on K-12 mathematics education. It is called Adding it Up: Helping Children Learn Mathematics, and was published by the National Academies Press in 2001. The result of several years work by the National Research Council’s Mathematics Learning Study Committee, a blue-ribbon panel of experts assembled to carry out that crucial millennial task, this invaluable volume sets out to codify the mathematical knowledge and skills that are thought to be important in today’s society. As such, it provides the best single source currently available for guidelines on good mathematics instruction.

The report’s authors use the phrase mathematical proficiency to refer to the aggregate of mathematical knowledge, skills, developed abilities, habits of mind, and attitudes that are essential ingredients for life in the twenty-first century. They then break this aggregate down to what they describe as “five tightly interwoven” threads:

Conceptual understanding – the comprehension of mathematical concepts, operations, and relations

Procedural fluency – skill in carrying out arithmetical procedures accurately, efficiently, flexibly, and appropriately

Strategic competence – the ability to formulate, represent, and solve mathematical problems arising in real-world situations

Adaptive reasoning – the capacity for logical thought, reflection, explanation, and justification

Productive disposition – a habitual inclination to see mathematics as sensible, useful, and worthwhile, combined with a confidence in one’s own ability to master the material.

The authors stress that it is important not to view these five goals as a checklist to be dealt with one by one. Rather, they are different aspects of what should be an integrated whole. On page 116 of the report, they say [emphasis in the original, image reproduced with permission]:

The most important observation we make here, one stressed throughout this report, is that the five strands are interwoven and interdependent in the development of proficiency in mathematics. Mathematical proficiency is not a one-dimensional trait, and it cannot be achieved by focusing on just one or two of these strands. … [W]e argue that helping children acquire mathematical proficiency calls for instructional programs that address all its strands. As they go from pre-kindergarten to eighth grade, all students should become increasingly proficient in mathematics.

In my book, I describe in some detail how to incorporate these educational goals (actually, to be faithful to the NRC Committee’s recommendation, I should say “educational goal”, in the singular) into good game design for a video game that seeks to help children learn mathematics. In this post, I’ll simply distill from that discussion eight important things to avoid. Try using this list to evaluate any math ed video game on the market. Very few – and I mean VERY few – pass through this filter.

  • AVOID: Confusing mathematics (a way of thinking) with its (symbolic) representation on a static, flat surface. (cf. music and music notation.)
  • AVOID: Presenting the mathematical activities as separate from the game action and game mechanics.
  • AVOID: Relegating the mathematics to a secondary activity when it should be the main focus.
  • AVOID: Reinforcing the perception that math is an obstacle that gets in the way of doing more enjoyable things.
  • AVOID: Reinforcing the perception that math is an arbitrary hurdle to be overcome, or circumvented, in order to progress .
  • AVOID: Encouraging the student to try to answer quickly, without reflection.
  • AVOID: Reinforcing the belief that math is just a large bag of isolated facts and tricks.
  • AVOID: Reinforcing the perception that math is so intrinsically uninteresting it has to be sugar coated.

I’ll be referring to Adding It Up a lot in this series. I shall also discuss many things TO DO when designing a good video game that support good learning, not just what to avoid. As you might (and for sure should) realize, with two challenging goals, good game and good learning, designing a successful math ed video game is difficult. Very difficult. If you do not have an experienced and knowledgable mathematics education specialist on your team, you are not going to succeed. Period.

Game programmers who think that because they were good at basic math (and they have to have been to become successful programmers) they can design a video game that will provide good learning are deluding themselves.

It’s easy to underestimate the depth of expertise of professionals in areas other than our own. Let me stress this point from the perspective of a hypothetical math educator who knows how to program in html5 and decides to create the next Angry Birds.

S/he might well think, “I can write code that produces screen action like that.” Indeed s/he probably can; it’s not hard. But as any experienced game developer will attest, the coding is the easiest part. The huge success of Angry Birds is not an accident. It is a result of brilliant design on many levels. (See this article for an initial, eye-opening summary of some of what went in to making that success.) The expertise it took to be able to create that game was acquired over many years. The Helsinki, Finland based Rovio game studio built ten other games, picking up a ton of increased expertise and insights along the way, before they reached the design heights of Angry Birds.

To build a successful game, you have to understand, at a deep level, what constitutes a game, how and why people play games, what keeps them engaged, and how they interact with the different platforms on which the game will be played. That is a lot of deep knowledge. On its own, being able to code is not enough.

To build a game that supports good mathematics learning, requires a whole lot more.  You have to understand, at a deep level, what mathematics is, how and why people learn and do mathematics, how to get and keep them engaged in their learning, and how to represent the mathematics on the different platforms on which the game will be played. That too is a lot of deep knowledge. On its own, being “good at math”, or at least the relevant math, is not enough.

If you are a game developer who happens to have both kinds of expertise, then go ahead and build a game on your own. But I have yet to meet such a person. For the rest of us, the answer is clear. You need a team, and that team must have all the expertise you will require to do a good job. If that team does not include, in particular, an experienced, knowledgable, math education specialist, then you are not a good engineer. You are an amateur.

How to design video games that support good math learning: Level 1.

Published January 29, 2012 mathematics education , video game design 7 Comments
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Cut scene

Most designers of video games to support mathematics learning make a number of fundamental mistakes. That does not mean they don’t create a game that gets played, and it may be that the game helps some children develop better mathematical ability, or at least memorize or master some basic skills. But minimal results like that are like justifying the creation of the piano because it supports the playing of a two-fingered rendering of Twinkle, twinkle, little star.

The fact is (and I know this is a fact because I have seen hard evidence that was obtained as part of a four-year project I took part in to develop and test elements of video games to support middle-school mathematics learning), the medium offers huge potential to completely revolutionize mathematics education at the K-8 levels, and to enhance it at the high school and university levels. But to date, that potential has at best been sniffed at.

Last year, I published a book outlining some of the factors that need to be taken into consideration when deciding whether to incorporate video games into schooling, including purchases made by parents to assist their children. Though I did include some tips for game developers on ways to incorporate mathematical learning activities into a video game, my main audience was the mathematics teaching community and my primary focus was (therefore) on the pedagogic issues. In this series of articles, my focus will be different. I’ll lay out some of the principles that I believe game developers should follow to create video games that support good learning. In particular, I’ll describe how video games can overcome the single biggest obstacle to mastery of mathematical thinking ability.

But that’s all for the future. Many video games begins with a “cut scene” or “cut sequence” that explains the background a player needs to know in order to play. Likewise for the series of articles I’m starting here. How did we get to be where we are now? In particular, how did I, an older university professor with graying hair, get to be both a gamer and a video game advocate? [The remainder of this introductory discussion is abridged from my 2011 book. I’ll pull material from that source throughout this series of articles, but much of what I say will be new.]

Since video games first began to appear, many educators have expressed the opinion that they offer huge potential for education. The most obvious feature of video games driving this conclusion is the degree to which games engage their players. Any parent who has watched a child spend hours deeply engrossed in a video game, often repeating a particular action many times to perfect it, will at some time have thought, “Gee, I wish my child would put just one tenth of the same time and effort into their math homework.” That sentiment was certainly what first got me thinking about educational uses of video games 25 years ago. “Why not make the challenges the player faces in the game mathematical ones?” I wondered at the time. In fact, I did more than wonder; I did something about it, although on a small scale.

This was in the mid 1980s when I was living in the United Kingdom. Soon after small personal computers appeared, so too did video games to play on them. My two (then very young) daughters particularly liked a fast-action game called Wizard’s Lair, which they played on our first personal computer, an inexpensive machine called the Sinclair Spectrum. Wizard’s Lair offered a two-dimensional, bird’s eye view of a subterranean world. My daughters spent hours playing that game. (A new version of the game, using three-dimensional graphics, was developed and released recently by IGN Entertainment.)

Meanwhile, the brand new computer in their elementary school was sitting largely unused. About the same time that Clive Sinclair (in due course, Sir Clive) introduced the Spectrum computer, the British government mandated that every elementary school in the nation be supplied with a computer. (That’s right, one per school—the early 1980s were the Stone Age of personal computers.) But when my daughters’ school received delivery of its shiny new machine, there was no software to run on it. Apparently no one thought of that. Mindful of what I had observed my daughters doing at home, I wrote a simple mathematics education game, in which the player had to use the basic ideas of coordinate geometry in order to discover buried treasure on an island, viewed in two dimensions from directly above. While it did not offer the excitement of Wizard’s Lair, the children at the school seemed to enjoy playing it. Perhaps offering a prelude of things to come, the game even made a financial profit. I wrote it for free, and the school PTA sold copies to other local schools. I think we sold five or six copies in total, each one on an audiocassette tape. I did not give up my day job.

After that one brief foray into the game development world, computer games remained for me a parental observation activity until 2003, by which time I was living in the United States and working at Stanford University. The previous year, Stanford communication Professor Byron Reeves and I started a new interdisciplinary research program at Stanford called Media X. Media X carries out research in collaboration with industries, mostly large high tech companies, and one early focus of Media X research was the growing interest in using video-game technology in education. In the fall of 2003, Media X organized a two-day workshop on the Stanford campus called Gaming-2-Learn, to which we invited roughly equal numbers of leading commercial game developers and education specialists from universities interested in developing educational games.

The main take-home message from the Gaming-2-Learn conference was that it took a lot of money and effort to design and build a good video game, and no one knew for sure it would be successful until it was finished and released. Both the cost and the uncertainties increase significantly when you try to incorporate mathematics learning into the game in a meaningful way. That perhaps explains why the vast majority of math-ed games that come out look like forced marriages of video games with traditional instruction of basic skills. It’s hard enough just getting a game out, without trying to integrate mathematical thinking into the gameplay.

Still, I was hooked on the promise, if not the then actuality. Following the Gaming-2-Learn event, I decided to take advantage of my location in the heart of Silicon Valley, and started to have conversations with, and in due course collaborate with, professional video game designers. That gave me insights into the organizational and engineering complexities involved in commercial video game production, leading eventually to my partnering with some colleagues from the commercial video game industry to form a video game company last year. More on that as the series develops.

Meanwhile, for all those skeptics out there who believe video games have little to offer education, I’ll leave you with a challenge. Find one – that’s right, just one – video game that is not about learning.

ABOUT ME

I’m Dr. Keith Devlin, an emeritus mathematician at Stanford University, an author, and was for many years “the Math Guy” on NPR’s Weekend Edition. Off duty, I’m an avid cyclist. (The header photo is me halfway up Mt. Baldy in Southern California.)

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