The Perfect Medium
There has to be a rational explanation for the huge disparity between the high performance level that ordinary people can achieve when they have to use “street mathematics” in everyday settings, and the terrible results many exhibit when faced with equivalent tasks in paper-and-pencil, symbolic form. Why do so many people claim “I just can’t do math,” when all the evidence indicates that they can do it just fine when they need to-provided it is not the school variety of math? Come to that, why is math so often presented using symbols, both in schools and elsewhere? The Trouble with SymbolsIf you ask someone to describe or to draw a picture of someone doing mathematics, almost certainly the image you will get is of someone writing symbols-on a piece of paper, a blackboard, or, if they are taking their cue from a number of portrayals of mathematicians in the movies or on television in recent years, on a window or a bathroom mirror. (Real mathematicians never do that, but it looks cool on the screen.) We identify doing math with writing symbols, often obscure symbols. Why? Because for the two and a half thousand years since the ancient Greeks initiated modern mathematics around 500 BCE, that was the only effective way to record mathematics and pass on mathematical knowledge to others.Notice that I didn’t say that was how people do math. If you think about it, doing math is something that takes place primarily in your head. What you write down when you are doing math are the facts you need to start with, perhaps the intermediate results along the way and, if you get far enough, the final answer
at the end. But the doing math part is primarily a thinking process. And as the studies carried out in Recife and elsewhere demonstrate conclusively, even when people are asked to “show all their work,” the symbolic stuff that they write down is not necessarily the same as what 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.So why do we still bother teaching symbolic math? There are some excellent reasons. One is that it is essential for modern life that there are enough people around who are really good at symbolic math. Science and engineering depend heavily on symbolic math. A second is that it is a key gateway to more advanced mathematics and a lot of other subjects as well. Still another is that the process of trying to master school algebra has proven scholastic benefits across the educa-tional spectrum, regardless of how well the student does on the school algebra exam. But how often do ordinary people find themselves faced with having to solve a paper-and-pencil, symbolic math problem? Outside of school and college, and leaving aside those people who solve math problems for recreation-and there are a lot more of those people than is popularly assumed-modern life sim-ply doesn’t require such ability.Except, of course, when it comes to passing tests. Though there is a lot wrong with the way the United States designs mathematics tests, society has to have measures of performance. At the moment, my focus is on designing video games to develop effective mathematical thinking, but video games can also play a role in improving test scores even in the United States, and I’ll come to that in Chapter 12.What modern life does require-unless you are content to live the life of a mollusk, not knowing or understanding what is going on around you or being vulnerable to being ripped off and misled at every opportunity-is a good working facility with numbers and quantity. If you think that a 10% price increase this week followed by a 10% price reduction next week will bring the price back to where it started, then sooner or later you are likely to be ripped off. (The final price will be 99% of the original.) If you think that a 50% chance of rain Saturday and a 50% chance of rain Sunday means a 100% chance of rain over the weekend, you are likely to be in for a surprise. (The correct answer is that the chance of rain over the weekend is 75%.) If you think that a newspaper headline saying that the rate of inflation is falling means things will become less expensive, you are in for a shock when prices continue to rise. (A decrease in the inflation rate means that prices are not going up as fast as before; but they can still be going up.) You can’t read a newspaper or watch the TV news without being inundated with statistics, and that includes the sports reports.But think about it for a minute. We’re not talking here about doing math symbolically, school fashion. What modern life requires-and it requires it
in spades-is the “in-the-head” stuff. The stuff that, as study after study has demonstrated, and which you may well have experienced yourself, almost anyone can pick up and get good at (or at least as good as they need to be).So why do we try to teach people how to develop this useful, everyday ability by forcing them to learn and practice all those symbolic rules-a method that for most people demonstrably fails? Even when you come across someone who is able to ace all the symbolic math tests, and there seems to be at least one such person in every school math class, they are often unable to apply that ability in a real-world setting. Teaching everyday math by way of symbolic mathematics denies mastery of the former-which the Recife study and others like it show is within everyone’s grasp-to all those who fail to master the latter, which a great many do. So why do we teach numerical mastery the way we do? Because for 2000 years, it has been the only way available at a system-wide level. If all teenagers could be put in a series of engaging, real-life situations that mattered to them, then like the Recife children, these students would eventually start to develop good, real-life number skills. But that approach simply is not practical.Note that I am not saying we should abandon trying to teach algebra and focus entirely on mastering arithmetic in real contexts. We should teach both, for reasons I’ve mentioned. But acquisition of everyday number skills should not depend upon mastery of symbolic mathematics. On the contrary, mastery of number skills should precede the attempt to learn symbolic mathematics. To be sure, sufficient mastery of symbolic mathematics is a crucial part of achieving mastery of working with numbers. But the general ideas represented by the symbols need to be grounded in sufficient real-life, concrete experiences. Though all of this is well known, the current approach largely tends to approach these skills in the reverse order. How did we arrive at this situation? Math for AllUntil the beginning of the thirteenth century, mathematics was a specialist occupation. The European manufacturers and traders would employ people to do their mathematics for them. Those people would learn their skills by working alongside an expert as an apprentice, not unlike the young children in the marketplace in Recife. Much of what they did involved either finger arithmetic, a remarkable system that worked for numbers up to 10,000, or the use of mechanical aids such as counting boards-flat surfaces ruled with lines for place-value, on which the user moved “counters” or pebbles-and the various kinds of abacus. In 1202, a young Italian mathematician named Leonardo of Pisa (who a later historian would dub “Fibonacci”) completed a book called Liber abbaci (spelled with two b’s, which makes the title translate as the “Book of Calculating”)
in which he described how to do calculations symbolically. It was one of the first teach-yourself arithmetic textbooks oriented to practical mathematics usage in the western world, and the one having by far the greatest impact on the spread of symbolic calculation in Europe. The methods Leonardo described had been developed in India several hundred years earlier and brought by Muslim traders to North Africa. Leonardo encountered them when, as a teenager, he joined his father who had been posted there as a Pisan customs official. The methods Leonardo recorded are essentially the same ones taught in school math classrooms ever since. You can see for yourself: an English language translation of Leonardo’s book was finally published 800 years later in 2002.1With the new approach to teaching math brought about by the completion of Liber abbaci (and literally hundreds of similar works that followed), it was no longer necessary to learn mathematics one-on-one as an apprentice. You could obtain a copy of the manuscript (and later a printed book) and teach yourself. Or a teacher could use the text to show a classroom full of students how to do it. What had until then been a specialist profession mastered by only a few suddenly became accessible to all.But this democratization of mathematics came at a price. When most people solve a mathematical problem, they do so as a step in achieving something else-to manufacture something, to exchange currency, or to buy or sell a product. To use the methods described by Leonardo to solve a real-world problem, you first have to convert the problem into symbolic form, then apply the rules to solve the symbol problem, then translate the symbolic answer back into real-world terms. To help people master this part of the task, the infamous mathematical “word problems” (also known as “story problems”) were developed. (Liber abbaci is full of them, and they can be found in much earlier mathematical works, such as ancient Babylonian tablets and Egyptian papyri!) But as generations of students have discovered, it takes a huge amount of effort to master word problems, and many never get the hang of them. The human brain simply is not well suited to do that kind of thing. (I explain why in my book The Math Gene.)Still, the approach Leonardo described was the only game in town. And so, for 800 years since the appearance of Liber abbaci, that is how arithmetic has been taught. Geometry and algebra have been taught using textbooks for even longer-Euclid’s famous multi-volume geometry and algebra textbook Elements, written around 350 BCE, looks much the same as any present day geometry or algebra textbook. When you have only one method to teach an important and useful skill-in the case of interest to us, real-world arithmetic-to a large number of people, then that method is what you use, and everyone has to struggle through the best they can. 1 Laurence Sigler, Fibonacci’s Liber Abaci, Springer-Verlag, New York, 2002.