Wednesday, February 20, 2013

Devlin's Lament: the "symbol barrier"

In an article in the most recent issue of American Scientist entitled "The Music of Math Games," Keith Devlin (head of the Human-Sciences and Technologies Advanced Research Institute at Stanford University and NPR's "math guy") says that learning math should be like learning to play the piano. In doing so, he recalls (but does not credit) Paul Lockhart's Lament ("A piano student's lament: how music lessons cheat us out of our second most fascinating and imaginative art form"), which I blogged about here.

Though Devlin is no literary virtuoso, not all of what he writes here is mushy metaphor. He begins with a discussion of educational software, and here his points are clear and consistent with my own experience. Most "math games" and "math education" software programs I've seen don't make mathematics an organic part of the games or activities. Instead, math problems--mostly arithmetic problems of the "mere calculation" variety--are shoe-horned into non-mathematical situations. Here they serve simply as tasks you must complete before moving through the current non-mathematical activity or on to the next non-mathematical activity.

As Devlin writes:

To build an engaging game that also supports good mathematics learning requires... understanding, 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 platform on which the game will be played.
The same is true of language learning. Most linguistic software taps only superficial aspects of language, and, as I know from personal experience, it takes great effort to build a program that does more than that.

Where I begin to part ways with Mr. Devlin is in his discussion of traditional math and what he thinks is an excessive emphasis on symbols:
Many people have come to believe mathematics is the memorization of, and mastery at using, various formulas and symbolic procedures to solve encapsulated and essentially artificial problems. Such people typically have that impression of math because they have never been shown anything else...
By and large, the public identifies doing math with writing symbols, often obscure symbols. Why do they make that automatic identification? A large part of the explanation is that much of the time they 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.
This approach to math, Devlin suggests, is at odds with the resolutions of a "blue-ribbon panel of experts" serving on the National Research Council’s Mathematics Learning Study Committee ("Adding it Up: Helping Children Learn Mathematics," National Academies Press, 2001). In Devlin's words: these resolutions hold that math proficiency consists of:
the aggregate of mathematical knowledge, skills, developed abilities, habits of mind and attitudes that are essential ingredients for life in the 21st century. They break this aggregate down to what they describe as “five tightly interwoven” threads. The first is conceptual understanding, the comprehension of mathematical concepts, operations and relations. The second is procedural fluency, defined as skill in carrying out arithmetical procedures accurately, efficiently, flexibly and appropriately. Third is strategic competence, or the ability to formulate, represent and solve mathematical problems arising in real-world situations. Fourth is adaptive reasoning—the capacity for logical thought, reflection, explanation and justification. Finally there’s 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.
Ah, "21st century skills," "habits of mind," "conceptual understanding," "real-world situations," "explanation," "disposition"...--all this makes me wonder about the ratio of mathematicians to math eduation "experts" on this blue-ribbon panel. (It should be noted that Devlin himself is not, strictly speaking, a mathematician; he holds a Ph.D. in logic from the University of Bristol, and, while affiliated with Stanford, is not a member of the Stanford math department.)

Standing in the way of these lofty goals is what Devlin calls the "symbol barrier":
For the entire history of organized mathematics instruction, where we had no alternative to using static, symbolic expressions on flat surfaces 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.
To the rescue comes... Devlin's math education software program:
With video games, we can circumvent the barrier. Because video games are dynamic, interactive and controlled by the user yet designed by the developer, they are the perfect medium for representing everyday mathematics, allowing direct access to the mathematics (bypassing the symbols) in the same direct way that a piano provides direct access to the music.
Devlin's notion that a well-designed math video game can help students meet the National Academy's goals for math education rests on two assumptions. One is that students can achieve a sufficient level of mastery in mathematics without symbols. The other is that playing such video games is to math what playing the piano is to music.

To address the first claim, Devlin elaborates the analogy to music:
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 too is mathematics created and carried out (and by many of us enjoyed) in the mind. At its heart, mathematics is a mental activity—a way of thinking—one that over several millennia of human history has proved to be highly beneficial to life and society.
But there's an important difference between math and music--and a reason why no one confuses music with a musical score. Music has a privileged place in subjective experience. Along with sensations like color, taste, and smell, it produces in us a characteristic, irreduceable, qualitative impression--an instance of what philosophers call "qualia." Just as there's no way to capture the subjective impression of "redness" with a graph of its electromagnetic frequency, or of "chocolate" with a 3-D model of its molecular structure, so, too, with the subjective feeling of a tonic-dominant-submediant-mediant-subdominant-tonic-subdominant-dominant chord progression. Embedded in what makes music what it is to us is the qualia of its chords and melodies.

Like most other, more abstract concepts ("heliocentric," "temporary"), mathematic concepts don't generally evoke this qualia sensation. What makes math beautiful are things like eloquence, patterns, and power. Unlike a Bach fugue translated homomorphically into, say, a collage of shapes, mathematical concepts can be be translated into different representational systems without losing their essence and beauty.

Devlin argues that while we might write down symbols in the course of doing real-life math, it is primarily a "thinking process," and that "at its heart, mathematics is a mental activity—a way of thinking." I agree. Indeed, math is much more appropriately compared with thoughts than with music. But this makes math symbols the mathematical equivalent of linguistic symbols. While thoughts, like math, can be expressed in a number of different symbol systems, you need some sort of symbol system in order to represent your own thoughts and to understand the thoughts of others.

This is especially true of abstract thoughts--and of abstract math. As Devlin himself admits, "the advanced mathematics used by scientists and engineers is intrinsically symbolic. "What isn't intrinsically symbolic, Devlin claims, is "everyday mathematics":
The kind of math important to ordinary people in their lives... is not, and it can be done in your head. 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.)
OK, but what does this mean for education? Are we going to decide before the end of middle school which students are going to become scientists, engineers, and mathematicians, and only help those students scale the "symbol barrier"? For a barrier it certainly is, as Devlin himself notes: "people can become highly skilled at doing mental math and yet be hopeless at its symbolic representations."

But Devlin is too busy appreciating the (well-studied) math skills of Brazilian street vendors, who do complex arithmetic calculations in their heads with 98% accuracy, and supposedly without the help of symbols (even mental ones?), to realize the educational implications of the fact that "when faced with what are (from a mathematical perspective) the very same problems, but presented in the traditional symbols, their performance drops to a mere 35 to 40 percent accuracy." No, not everyone is going to become an engineer. But not all non-engineers are going to become Brazilian street vendors.

It's ironic how deeply Devlin appreciates the difficulty that "ordinary people" have with the symbol barrier without appreciating what this says about their educational needs:
It simply is 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. Only someone who has mastery of symbolic mathematics can recognize the problems encountered in the two contexts as being “the same.”
Instead of seeing this as a reason for exposing children to mathematical symbols early and often, Devlin sees this as reason to create computer games that somehow teach math non-symbolically.

He calls this "adaptive technology," a term that should raise red flags. In a recent blog post, I wrote about how assistive technology often becomes yet another excuse not to teach basic skills. Kids with dyslexia struggle mightily with the symbol system of written language; should they instead learn everything through text-to-speech and speech-to-text devices, and never learn how to read and write?

Devlin makes a few other strained comparisons to the piano:
The piano metaphor can be pursued further. There’s a widespread belief that you first have to master the basic skills to progress in mathematics. That’s total nonsense. It’s like saying you have to master musical notation and the performance of musical scales before you can start to try to play an instrument—a surefire way to put someone off music if ever there was one.
No it's not; it's like saying you have to master simple scales and exercises before you move on to Rachmaninoff.
The one difference between music and math is that whereas a single piano can be used to play almost any tune, a video game designed to play, say, addition of fractions, probably won’t be able to play multiplication of fractions. This means that the task facing the game designer is not to design one instrument but an entire orchestra.
Can one create a video game that functions "as an instrument on which a person can 'play' mathematics?"
Can this be done? Yes. I know this fact to be true because I spent almost five years working with talented and experienced game developers on a stealth project at a large video game company, trying to build such an orchestra.
What does Devlin's software do? The last two paragraphs of this article function as an extended but not very informative infomercial. Here's the most informative excerpt:
Available in early March, Wuzzit Trouble is a game where players must free the Wuzzits from the traps they’ve inadvertently wandered into inside a castle. Players must use puzzle-solving skills to gather keys that open the gearlike combination locks on the cages, while avoiding hazards.
Puzzle solving? As I argue in my last post on math games, existing games already offer some version of this, and it isn't math. This, indeed, is one of the other problems with so-called math education software.

Devlin suggests his software is different:
Unlike the majority of other casual games, it is built on top of sound mathematical principles, which means that anyone who plays it will be learning and practicing good mathematical thinking—much like a person playing a musical instrument for pleasure will at the same time learn about music.

Wuzzit Trouble might look and play like a simple arithmetic game, and indeed that is the point. But looks can be deceiving. The puzzles carry star ratings, and I have yet to achieve the maximum number of stars on some of the puzzles! (I never mastered Rachmaninov on the piano either.) The game is not designed to teach. The intention is to provide an “instrument” that, in addition to being fun to play, not only provides implicit learning but may also be used as a basis for formal learning in a scholastic setting.
If you say so. But I wonder how much it will cost schools (and society) to find out whether this latest incarnation of "math education" software helps prepare students to become mathematicians, scientists, engineers--or Brazilian street vendors.


Allan Folz said...

Wuzzit Trouble is a game where players must free the Wuzzits from the traps they’ve inadvertently wandered into inside a castle.

Hmmm, that sounds vaguely familiar. Where have I heard that before... "in a bizarre laboratory accident, Dr. Brain's mind got mixed up with his lab rat's... Unlock the secrets hidden deep within his eccentric mind as you puzzle over the twists and turns packed into over a thousand wild, wacky brain-busting adventures!"

Castle == Eccentric Mind
Mathematical Principles == Puzzles

(For the record everything from Angry Birds to Sim City has mathematical principles at their core. Usually they're called rules. Or 'rule engines' if you want to flash your nerd badge.)

Of course they could do a lot worse than re-implement Sierra's Dr. Brain. Actually, it would be pretty easy to do a lot worse. Especially if they took 5 yrs to do it. ;-) I'm friends with a guy that was the lead developer on Dr. Brain. It was written in 9 months and got a #1 rating by Consumer Reports. Sierra bungled the production and quietly starved it of marketing leading to an inglorious death in the marketplace, but among those lucky enough to get a copy they loved it. Interestingly, you can see the game being played on You Tube.

But I don't want to sound like an overly critical jerk. I hope they have something solid. It's just that a lot of people have tried before. It's a very hard problem. I think people would get further if they'd try learning from history rather than acting like nobody's ever tried any of this stuff before.

Auntie Ann said...
This comment has been removed by the author.
Auntie Ann said...

ED Hirsch ( :

"Why should vocabulary size be related to achieved intelligence and real-world competence? Though the intricate details of cognitive abilities are under constant study and refinement, it’s possible to give a rough answer. The space where we solve our problems is called “working memory.” For everyone, even geniuses, it’s a small space that can hold only a few items in suspension for only a few seconds. If one doesn’t make the right connections within that space, one has to start over again. Hence, one method for coping and problem solving is to reduce the number of items that one has to make sense of at any moment. The psychologist George A. Miller called that process “chunking.” Telephone numbers and Social Security numbers are good examples. The number (212) 374-5278, written in three chunks, is a lot easier to cope with than 2123745278.

"Words are fantastically effective chunking devices. Suppose you put a single item into your working memory—say, “Pasteur.” So long as you hold in your long-term memory a lot of associations with that name, you don’t need to dredge them up and try to cram them into your working memory. The name serves as a brief proxy for whatever aspects will turn out to be needed to cope with your problem. The more readily available such proxies are for you, the better you will be at dealing with various problems. Extend this example to whole spheres of knowledge and experience, and you’ll realize that a large vocabulary is a powerful coping device that enhances one’s general cognitive ability."

Symbols are the vocabulary of math. That's why, to a mathematician, something like this can be extremely beautiful:

e^(i*pi) + 1 = 0

Looking at that, they see the concepts of 0, of identity and equality, of imaginary numbers, of sine and cosines, and of natural logarithms. All that and more in 7 little symbols.

Anonymous said...

What a sad commentary. As someone who both does math and plays music, I must say that Devlin appears to have sold his soul to the devil (who is a pointy-haired mushy woo-woo 'education expert').

No, Devlin, you really do have to master scales to be a good musician. If you do not master scales, you will never be a very good pianist, or violinist, only a hack. This is why my violin teacher says, and this is what every real musician I know says, and they are experts, not Devlin.

And likewise, you really do have to master symbols, and arithmetic, to be good at math.

What fascinates me most about this foolish argument is how across the board it is - how it is applied to every single educational discipline with abandon. With reading: you don't have to master phonics, try whole language instead. With language: you don't have to master grammar, develop your (ten-year-old) voice instead. With math: who needs arithmetic. And now with music, don't bother doing scales, just express your feelings banging on the piano. The inmates have truly taken over the asylum.

This is all a recipe for becoming an incompetent, illiterate, innumerate, self-centered boor. And it's laid out by a cabal of "experts" who are so mysteriously taken with the woo-woo that they seem to be conspiring to pull the ladder of mastery up after them.

I speak four languages fluently. I spent time mastering the grammar in each one. Yes, I lived abroad too, but the truth is that most returned peace corps volunteers do not master the language - all they remember years later is street slang - because they didn't combine study with practice. I did the work.

I aced the GRE math section (750) - because I can do arithmetic (yes, with symbols) in my head. I am learning to play the violin at 45, along with my 8 year old son - because we do scales for a half hour every day, before we work on my new piece. There is no mystery, no special secret. It's just work

I was at the bakery the other day, and bought a sourdough boule and a sfilatino. The clerk punched it in and was surprised to see the $7.25 waiting in my hand. She was amazed by the mastery of math I showed at being able to add $5.50 and $1.75 correctly.

Ignorance and innumeracy is nothing new, judging by her age. The problem is that now the "blue ribbon panels" are giving in, and deciding to join rather than beating them.

New rule: nobody whose degree is in education (the softest, most cheat-infested major at any university), and has not achieved mastery of a real subject (say, math, engineering, or foreign language), can be considered an expert in education, and the majority participation of such people on a "blue ribbon panel" disqualifies their findings.

BTW, some of the few math games I think might convey some of the spirit of numeracy rather than just throwing intermittent sums in the middle of goofy games, are Factor Samurai and Dragon Box.

Barry Garelick said...

Speaking of music and math, Devlin strikes me as the Liberace of math.