kitchen table math, the sequel: Devlin's Lament: the symbol barrier

Thursday, February 21, 2013

Devlin's Lament: the symbol barrier

(Cross-posted at Out In Left Field)

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.


Anne Dwyer said...

This statement:

"It simply is not the case that ordinary people cannot do everyday math."

is simply untrue. Obviously, the person who wrote this has nevver come in contact with 'ordinary people' in an ordinary setting.

If you ask some 'ordinary people'simple questions like how much tip do you leave or how much change should you get back or what is the interest rate on your mortgage and how much of your payment is interest, you will be met with either technology does this for me (I have a tip app on my phone) or shrugs. This is what I consider ordinary math. People may have basic math facts (or, in many cases, not) but they can't apply those skills. And isn't that what 'ordinary' math is all about?

If they can't do ordinary math without symbols, you can't expect them to be able to use the symbols since it adds a layer of abstraction to a skill that they don't possess.

Now my opinion may be biased because I come in contact with a lot of people who are taking basic math as adults. But there is a subset of the adult population who cannot do ordinary math. I often wonder how well they function in the real world.

momof4 said...

You are quite right; prior to the advent of the calculator and cash registers that totaled and added tax, ordinary people (who might not have finished HS) actually knew arithmetic. It wasn't even called "math"; it was basic arithmetic, fractions, decimals and percentages and the people in my small town actually used these skills every day and had mastered them by the end of 8th grade, if not earlier. They could calculate how much paint they needed for a room, how much lumber they needed to build a doghouse, calculate taxes and interest, or manipulate recipes for different numbers of people. You know; using math to solve real-world problems.

Allan Folz said...

RE: "ordinary people cannot do everyday math."

I understand Devlin to be saying all people are capable of everyday math with a modicum of proper training and skill development (ie. his software).

This is distinct from all people being able to be engineers and mathematicians (ie. play Rachmaninov).

That much I think we're all in agreement. The catch is how do you get there... hard, tedious "drill and skill" or fun, symbolic-free games and puzzles?

Well, drill and skill brought us two revolutions in productivity: first the industrial and then the technological.

The educational "experts" since Dewey(?) have had a good 30-40 years with their hands on the steering wheel and the results haven't exactly been an unquestioned resounding success.

cranberry said...

OK, I'm lost.

Letters and numbers are symbols. Words are symbols, too. Thus, there is a "symbolic barrier" to speech, as well. I know I think in words. Without symbols, I would think very differently.

Most people do not teach themselves to play the piano. The piano is a great first instrument, as you need only press a key to hear a note. However, a competent instructor will teach reading music alongside playing the piano. All of my children began with the piano, so I've seen this happen.

The piano student works through a curriculum of pieces, (set by the instructor) from easy to hard(er). The instructors I've seen have stressed the use of sheet music. Students are not encouraged to try to "wing it" while learning. They may be encouraged to create their own pieces.

A student who can play the piano will find it easier to learn other instruments, or to cold read sheet music. That doesn't mean there's a musical ability (these days) in trained musicians which can be separated from a fluid mastery of music symbols. Nor is not being able to read music a sign of a better musician.

My children's piano teacher claims that if students don't start lessons until after 7 year old (or so), they will lack the fluid ease with music those who start earlier may gain.

What is this math which does not rely upon symbols?

ChemProf said...

I'm curious, Cranberry, when does your teacher suggest starting? At least some of the piano teachers I've talked to said they wouldn't have a child start before 7, because they don't have the reading ability or the patience to practice yet, so it is interesting to run into another opinion. I know Suzuki starts earlier, around 3 or 4.

cranberry said...

ChemProf, the child must be able to read. There's also a size issue. One of my children began in kindergarten (early reader), the other two began in 1st or 2nd grade.

Public and private schools in this area begin band instruments in 4th grade. It's easier to start band instruments after a couple of years of keyboard instruction. Playing one note at a time is easier than playing multiple notes, although the young children won't have the lung capacity to play many notes in succession yet.

My older two play two or three instruments each and sing in various groups. They haven't been aiming at careers as professional musicians; music has been a source of joy and fun. As it is, it's a great advantage to be able to read music. They can both hear when a group or singer is "off"; being able to read music hasn't interfered with that skill.

Michael Weiss said...

It is really fascinating to me how much Devlin's argument relies on the metaphor "math = music", which allows him to distinguish between the performance of mathematics and its symbolic representation.

Change the metaphor, though, and see where you get. Is math more like music, or like poetry? Can you learn to be a poet / think poetically without learning to manipulate verbal symbols (words)? Or is math more like painting? Can you learn to be a painter / think like a painter without learning to manipulate visual symbols?

(It is worth noticing that Lockhart, who also uses the math = music analogy, does not reach the same conclusions as Devlin, in large part precisely because Lockhart uses multiple analogies, and also because Lockhart was himself trained as a research mathematician.)

Using symbols to represent abstract concepts is at the heart of "thinking mathematically". It is why we use the numeral "5" instead of a picture of five tally marks.

Anonymous said...

I know this is slightly off-topic, but some of my DD's swim teammates swam specifically for lung capacity improvement. They were mostly either singers or musicians. One female HS golfer improved her upper-body strength so much during her first fall-winter-spring swim team commitment that she was able to use the men's tee when golf season started. I don't know how widely known the swim effect is or how much training makes a significant difference. The over-13 training groups the girls were swimming with did at least 4-5k/day average, with an hour's running and/or medicine ball supplements. My DD was averaging at least 3k/day by age 10. FWIW

cranberry said...

Michael Weiss, aren't tally marks symbols too? Each mark = one item. Five marks = five items. Likewise, there's nothing inherent in the word "book" which links it to a bunch of paper bound together. When we use "book," we are avoiding the labor of defining the object anew each time, as symbols allow mathematicians to avoid the labor of defining a commonly known or agreed-upon concept anew each time.

K-12 math is a process of learning the language of math. However, we all use symbols all the time. I don't believe there's a "doing math" which can be separated from its symbolic language, the symbols in math allow greater levels of complexity. We know a fair amount about the mathematical systems of ancient peoples because we've found mathematical texts.

Thus, is it a symbolic barrier, or a complexity barrier? I think you'll run into the same compexity barrier in understanding history texts which assume readers possess the necessary background knowledge not to confuse Romans with Huguenots.

Michael Weiss said...

Michael Weiss, aren't tally marks symbols too?
Sure, of course. So too would a picture of five pipes be a symbol (because a picture of a pipe is not a pipe).

To be more explicit: Five solid objects (counters) is a symbol representing different objects. Five drawn tally marks is a symbol representing counters. A numeral is a symbol representing a specific quantity of things. The letter "n" is a symbol representing an arbitrary number. Eventually you are considering set set of all whole numbers, then sets that have "number-like" properties (groups, rings, fields), and so forth.

The point I am trying to make is that the motive force of mathematics points generally toward more abstract symbols. That is (arguably) what math is: the study of patterns and regularities (including "necessary consequences"), which must be expressed in symbols precisely because a pattern is more than its instances.

Allan Folz said...

It is really fascinating to me how much Devlin's argument relies on the metaphor "math = music", which allows him to distinguish between the performance of mathematics and its symbolic representation.

That's a good one. I'd totally missed that line of thought.

SteveH said...

"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."

Many people can believe something, but that doesn't make it true. The old nugget that tratidional math is all about rote learning is just plain wrong. Look at the textbooks. You can't be successful in math with rote memorization unless the teachers are incompetent. What happens with the best students who get to a STEM degree program in college? They usually aren't taking some other non-traditional path, so how, supposedly, did they make the transition from rote to non-rote? Obviously, something else is going on. But so many educators are bound and determined to blame traditional rote math because what they really want to do is to push their own bias of hands-on group learning in class. Devlin's arguments are based on the perception of a problem, not a real problem.

"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."

Devlin doesn't seem to make any distinctions about level; between those who appreciate and those who do. Even for those who do music or math, there are distinct levels. What might be a good approach for one level is not good enough for another.

You can start a child in piano from the top down using real music to develop skills or from the bottom-up using scales and Czerny or Hanon for finger exercises. It is highly unlikely that the top-down approach will ever end up at the same level as the scales and finger exercises bottom-up approach. One approach might be better for a student than the other, but you will never know that beforehand. The key point is that the top-down approach will never allow the student the opportunity to reach the same level as the bottom-up approach. There is a huge difference between the two approaches in terms of level of expectations. Top-down approach proponents assume that a top level can be achieved naturally with motivation and engagement, but that's just not true. Getting to a top level in math or music requires a long path of systematic skill development and a lot of non-natural pushing. If you don't develop the technical skills properly early on, then it is unlikely they will ever get done. I've always said that the comparison between top-down and bottom-up pedagogies is not meaningful because they target different end goals. A top-down approach can never reach the same levels as a bottom up approach. With a bottom-up skills approach, you can always drop to a lower level of expectations later on, but with a top-down approach, you can never make up the difference.

If you start with simple, fun, jazzy pieces when you are learning to play the piano and use those pieces in place of finger exercises, you are on a path that will never allow you to get to the top level in the music world. My son has had a steady diet of Hanon, Czerny, Bach inventions, Prelude & Fugues, and etudes over the years. He could have always lowered his skill and technical development (or dropped the piano), but one could never start with a fun approach and later on and hope to get to a top level.

SteveH said...

The two approaches are not equal. It's not just pedagogy. They offer different goals. You can always reduce expectations later on with a bottom-up skill based approach, but you can never go the other way with a top-down, engagement approach. Too often, educators select the top-down approach because it offers a better chance for relative statistical improvement for the average student. The only problem is that it condemns all students (without outside help) to a lower level even if they are the ones who could possibly reach a top level. In math, it's worse, because this lower level seems to be defined by doing reasonably well in a college algebra course.

"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"?"

Most schools separate students by 7th grade into the top path to calculus in high school and the math path to nowhere; struggling to master enough algebra and geometry to do well on the SAT, but never enough math to do well in a STEM degree program. Their math is a mountain top, not a base camp for higher level math in college. The problems happen in K-6.

Any real or imaningary symbol barrier is not the problem. The problem is what kind of math students had in K-6. It's a lack of seeing the linkage between skills and understanding. It's a fundamental misunderstanding of the difference between rote skills and the lack of flexibility and level of understanding. Students don't need a different (top-down) approach. They need a better skill development-based approach.

"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."

This is a trivial and self-serving analysis. It's the same as the idea that traditional math is all about rote learning. Devlin makes the usual assumption that there is some gentle, natural, not-a-put-off way to the same end result. That's not true, and it's very bad to start kids off on that path because there is no way back. In math, we KTM parents make up that difference while educators use our kids to think that their fuzzy ways can work.

SteveH said...

"At least some of the piano teachers I've talked to said they wouldn't have a child start before 7, because they don't have the reading ability or the patience to practice yet, so it is interesting to run into another opinion."

Some say that if you don't start piano or violin lessons by 5, it's too late. You have to train your body early or else it won't develop properly. I'm involved with a regional piano competition and we even get some Kindergarten kids playing in the lowest age bracket. My son started competing at age 7. He loved it. Who knew? We thought it was either going to work out really well or really badly.

We wanted to start our son on some instrument when he was young. We had no great plans for him. We just wanted to get him started early. When he was 5, I called the local violin teacher and after she heard how old he was, said that, well, she finds that girls are more mature than boys at that age. "Uh, OK. Thanks. Goodbye." We then started with a "piano 4 hands" class where a parent plays along with the child. He liked it and we soon followed up with individual private lessons.

I never saw an issue with starting too early. Maybe there would be a problem if parents force kids to do something they are not prepared for or that they don't like. Our son also started in baseball and soccer when he was 4 or 5. Nobody complained about how that was too young. BTW, that lasted only to about 5th grade.

In his first piano 4 hands class, they did have music, but that was not the central part of the lesson. When he started his individual private lessons (at 6), the teacher used the Bastien series of books. I saw no reason to not dive right in and start learning to read music. Private teachers can adapt to the needs of the individual child. I think that some parents are way too concerned about damaging their child somehow. I remember how some parents, who were all for skill development in sports, turned quite fuzzy and weird when it came to academics.

However, you have to be careful about the approach the teacher will take. If you want your child to not have doors closed early, you have to find a more traditional teacher; one who emphasizes basic skills. I know about some horror stories of teachers like that, but there are also the hidden horror stories of unfulfilled potential because teachers never taught the basics properly.

A top level in music is kind of like a top level in sports. It's very competitive. Our son is now in the process of working on his college audition repertoire, although he probably won't go to a conservatory or major in music. Students spend a year developing their 40 minutes of music for an audition the will last 15 minutes at most.

It's an interesting problem. How do you know beforehand how rigorous you should make an education? If you don't start out on a rigorous path, potential will be lost. But what if the child is not going to be at the top end in academics, music, or sports? Will you kill any future level of enjoyment?

I think of my son's baseball training that started at 5. Baseball is a really complex game, but the coaches all emphasized basic skills. All practices started with basic skills training like tossing duct tape potatoes. Then they had real games where everyone stayed at bat until they actually hit the ball. (They were soooo tedious!) They might be out at first, but they got to hit the ball. Actually, many could make it around the bases with an infield hit. The turning point seems to take place in about 5th grade, when the good or competitve kids branch off into the top leagues. The downside to this process is that there might not be much to offer those who were left behind. Then again, many high school sports have no cut policies, and most have music opportunities for all levels.

SteveH said...

Parents will never know about lost potential because it's hidden. Students will end up saying that "they are just not good in math". They will develop phobias and teachers will think that it's an IQ or attitude problem. If you come from a low SES family, they think that will explain it all.