**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 …*

I like your term, the

symbol barrier. I hadn’t read of the research until recently (finally gotChildren Doing Mathematicsfrom the library and found it fascinating), but I’ve been aware of the symbol barrier for about 30 years, since I first started trying to teach math to preschoolers.I succeeded with my kids using the low-tech method of telling lots of little stories, and having them make up math stories of their own. Over and over, in informal settings like during the drive into town for soccer practice, for years and years. When the math symbols in a workbook were too abstract, I set the book aside and put the same problem into a story setting. I think kids naturally think in terms of characters, settings, plot, and analogies. It’s only after they’ve been bludgeoned by school math that they learn to fear story problems.

The combination of thinking about math in the context of story, of making up math stories, and of figuring out ways to come up with answers — all this made for a great foundation for the eventual introduction of symbols and school math. As you wrote in your book, “the skills come largely for free.”

I’ve tried to spread this idea to other homeschoolers (see, for example, this blog post), but so many people are primarily worried about how quickly their children can master the math facts. The idea of delaying symbolic manipulation in order to “just talk” doesn’t get much traction…

Hi Denise. Thanks for writing. Yes, we face a problem with parents who grew up in a world where it had some practical utility to have a collection of math facts at your fingertips, who do not realize that in today’s world the premium is on mathematical thinking, not only for the job market but also to be a productive citizen and live a rewarding life. Math facts are what we carry around in our pockets in a small digital device!

Actually, I think math facts are very important, and I’m a bit of a Luddite regarding calculators. I push my kids to do mental math whenever possible, which leads to them learning almost all of the math facts through repeated use in context. I think this develops the sort of flexible understanding of numbers, place value, and relationships that makes a good foundation for further study.

What I object to is a widespread emphasis on preschool and early-elementary children mastering abstract, symbolic manipulation by rote.

I think my problem with convincing homeschoolers to postpone math drill is similar to the problem you have convincing potential sponsors that your games teach mathematical thinking. It’s such a strong part of most people’s cultural understanding of mathematics that “real math” is school workbook math.

Agreed. My position is that math facts should no longer be the focus (or metric) of math ed. They are things you pick up through use. Just like we used to end up remembering a lot of phone numbers. (Pre-mobile era.) No one set out to memorize them, we just did so through frequent use.

“So why is it that many people believe mathematics is symbolic manipulation?”

I think I might know the answer to that one:

- People have a general tendency to mistake what they see (the artefact) with the real thing (all the circumstancial forces that make it work). I believe it is a natural result of ignorance.

Some examples:

- People mistake having a facebook-page with being present in social media, when it is really the interaction there that really matters. But it is the page that is most clearly visible.

- People mistake a plan document with planning. A document is just an artefact, and not necessarily worthwhile, as many business plans are a testament to. Planning (continuously thinking about future alternatives and contingencies), on the other hand, is immensely worthwhile. Alas, the document is what outside people actually see.

- People mistake badges, with rewards. They see players getting real exited about getting a badge, and conclude that the badge is what triggers it. In fact, it is overcoming a challenge that is rewarding, and the badge is just a symbolic representation of that. You can see this fallacy play out in tons and tons of websites that try to use game mechanics to motivate users/customers, by handing out worthless badges (i.e. badges handed out indiscriminately, after no challenge at all).

- People mistake mathematics with symbols, when it is actually a way of thinking. Since you can’t “see” a “way of thinking”.

I hope this was helpful to someone.

Nice article, btw.

Cheers,

Magne Matre Gåsland