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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.

From al-Khwārizmī to Steve Jobs

Published January 5, 2012 algebra 2 Comments
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The sixth and last in a series. See the November 20 entry, “What is algebra?” for the first, the December 13 entry “When did algebra begin?” for the second, the December 19 entry “The  golden age of Arabic mathematics” for the third, the December 25 entry “al-Khwārizmī” for the fourth, and the December 30 entry “What is algebra good for?” for the fifth.

History tends to focus on key individuals, when in fact most advances are the cumulative affect of the contributions of many. In the case of algebra, claims that al-Khwārizmī invented algebra are not sustainable. As I have explained in previous articles in this short series, the chain leading to algebra goes back at least to the ancient Babylonians, and to modern eyes Diophantus’s book Arithmetica was clearly a book on algebra. Nevertheless, al-Khwārizmī does deserve the credit for establishing algebra as a major collection of intellectual tools.

He can’t be credited with establishing it as a branch of mathematics, howver, since the mathematicians of the Arabic period did not view the methods they developed as anything other than a set of very valuable practical tools. (Likewise, Diophantus viewed his work as a sophisticated form of arithmetic, as the title of his famous work suggests.) Viewing algebra as a discipline in its own right came later.

Al-Khwārizmī’s greatness is in the same category as Euclid nine centures earlier, as Leonardo of Pisa four hundred years later, or as Steve Jobs in our own time: their impact on society and thence the course of history. None of these four were the original inventors or discoverers of the seminal developments we associate with their names. Their greatness was not one of original discovery – though both Euclid and al-Khwārizmī may well have contributed some of the methods they described in their seminal books and we know from his other works besides Liber abbaci that Leonardo was a first-rate, original mathematician. Rather, all four had the highly unusual ability to take a collection of powerful new ideas and package and present them to society in a manner that made them acceptable to – indeed eagerly sought-after by – a wide range of people. In our present-day society we tend to focus on the priority of discovery and invention, as epitomised by the status we award Nobel Laureates, but initial discovery would be of little value were others not able to take the new knowledge and use it to change society.

Each of Euclid, al-Khwārizmī, and Leonardo (and Steve Jobs) were followed by many others who carried the torch forwards, and they too deserve credit.

Among the hundreds of Arabic mathematicians who helped to develop and spread algebraic knowledge after al-Khwārizmī, several stand out as worthy of special mention. I’ll list a few.

Abū Kāmil The Egyptian born Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad ibn Shujā (c. 850 – c. 930) was the first major Arabic algebraist after al-Khwārizmī. By all accounts he was a prolific author. There are references to works with the titles Book of fortune, Book of the key to fortune, Book of the adequate, Book on omens, Book of the kernel, Book of the two errors, and Book on augmentation and diminution. None of these have survived. Works that did survive include the Book on algebra, the Book of rare things in the art of calculation, Inheritance by means of algebra, and the Book on surveying and geometry.

The Book on algebra (Kitāb fi al-jabr wa al-muqābala) is arguably Abū Kāmil’s most influential work. It expanded on al-Khwārizmī’s Algebra. Whereas the latter was aimed at the general public, Abū Kāmil wrote more for other mathematicians, assuming familiarity with Euclid’s Elements. He extended the range of polynomials studied beyond al-Khwārizmī to include 8th powers.

Al-Karajī  A century after Abū Kāmil did his work, around 1000 C.E., another major advance in algebra was made by the Persian mathematician and engineer Abū Bakr ibn Muḥammad ibn al Ḥusayn al-Karajī, who lived from c. 953 to c. 1029. His three major works were Al-Badi’ fi’l-hisab (Wonderful on calculation), Al-Fakhri fi’l-jabr wa’l-muqabala (Book of al-Fakhri on the Art of Algebra), and Al-Kafi fi’l-hisab (Sufficient on calculation).

Al-Fakhri  is regarded as one of the key works on the path that led to the final separation of algebra from geometry as a discipline in its own right. Al-Karajī gave a systematic treatment of reducible higher-degree equations. He studied the algebra of exponents, and was the first to state explicitly that the sequence x, x2, x3, … could be extended indefinitely, and likewise the reciprocals 1/x, 1/x2, 1/x3, …

Omar Khayyám Shortly after al-Karajī died, another famous Arab scholar came onto the scene: Omar Khayyám. Although in the West he is better known today as a poet, he was a first rate mathematician.

Al-Khayyám, more fully Ghiyath al-Din Abu’l-Fath Umar ibn Ibrahim al-Nisaburi al-Khayyámi, was born on 18 May, 1048 in Nishapur, Persia (now Iran), and died there on 4 December, 1131. As a young man he studied philosophy, and went on to be an outstanding mathematician and philosopher. By the time he was 25, he had written several books, covering arithmetic, geometry, algebra, and music. His major work in algebra was an analysis of polynomial equations titled Treatise on the Proofs of Algebra Problems.

Al-Khayyám approached mathematics primarily as a geometer, firmly rooted in the Greek tradition. Whereas Abū Kāmil and al-Karajī presented algebra as a method for numerical problem-solving, al-Khayyám viewed it as a tool for theoretical geometers.

al-Samawʾal Several further advances in algebra were made around the mid-twelfth century by a teenager (yes, that’s right, a teenager) called Ibn Yaḥyā al-Maghribī al-Samawʾal, who was born around 1130 in Baghdad. His parents were Jewish, his father a literature scholar and Rabbi from Morocco, his mother from Basra, in Iraq.

Although his initial interest as a child was to become a doctor, al-Samawʾal proved to be a child prodigy in mathematics, and the study of medicine was soon relegated to second place (but not abandoned). He began to study the Hindu methods of calculation when he was thirteen or so. Rapidly finding himself ahead of his teachers, he continued on his own, reading the works of Abū Kāmil, al-Karajī, and others. By the time he was eighteen years old he had read almost all the available mathematical literature. He wrote his most famous treatise, al-Bahir fi’l-jabr (The brilliant in algebra), when he was just nineteen years old.

Mathematicians before al-Samaw’al had begun to develop what contemporary historians have called the “arithmetization of algebra”. Al-Samaw’al was perhaps the first to give this development a precise description, writing that it involved “operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known.” This can be regarded as a significant step toward the development of modern algebra.

In all, al-Samaw’al is reported to have written 85 books or articles, though most have not survived. He died in Maragha, Iran, around 1180.

Further advances in algebra were made in the Maghreb in the twelfth to fifteenth century, by a highly organized teacher-student network linked to mosque and madrasah teaching. The Maghrebs used abbreviations for both unknowns and their powers and for operations; another innovation in the chain that culminated in the development of modern symbolic algebra in Europe in the 16th century.

* * *

For the next episode in the development and growth of algebra, when the ideas found their way to Europe, see my recent book The Man of Numbers: Fibonacci’s Arithmetic Revolution. (And for a comparison between Fibonacci’s role and that of Steve Jobs, see the companion e-book Leonardo and Steve.)

Acknowledgement
I am greatful to Professor Jeffrey Oaks of the University of Indianapolis for his assistance in the prepartion of the essays in this series. In particular, he supplied me with preprints of his forhtcoming articles for Springer Verlag’s upcoming Encyclopedia of Sciences and Religions (2012): “Mathematics and Islam”, “Arithmetic and Islam”, “Algebra and Islam”, and “Geometry and Islam”, which I drew on heavily. He also commented in detail on a more substantial work from which these essays were abridged.

What is algebra good for?

Published December 30, 2011 algebra 2 Comments
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The fifth in a series. See the November 20 entry, “What is algebra?” for the first, the December 13 entry “When did algebra begin?” for the second, the December 19 entry “The  golden age of Arabic mathematics” for the third, and the December 25 entry “al-Khwārizmī” for the fourth.

Modern algebra is generally acknowledged to have begun with the appearance around 830 CE of al-Khwārizmī’s book al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqābala. What better source can there be to find the answer to that perennial student question, “What is this stuff good for?” In the introduction to his seminal work, al-Khwārizmī stated that its purpose was to explain:

… what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.

It doesn’t get much more practical and useful than that! Either in 830 or today! Throughout history, the nations that led the world in mathematics led the world in commerce, industry, and science. In the 9th century, Baghdad was the commercial, industrial, and scientific center of world. In the 13th century, the leadership role crossed the Mediterranean to Italy, then over the ensuing centuries continued gradually westwards through Europe, crossing the Atlantic to the East coast of the US in the middle third of the 20th century, arriving in California in the 1980s, and likely to cross the Pacific (back) to China within the next couple of decades.

Al-Khwārizmī’s strong emphasis on practical applications typified Arabic texts of the time, every bit as much as the intense focus on applications of mathematics and science you find in today’s Silicon Valley.

The book was divided into three parts. The first part was devoted to algebra, giving the rules together with 39 worked problems, all abstract. Then came a short section on the rule of three and mensuration. Two mensuration problems dealing with surveying were solved with algebra. Finally, al-Khwārizmī presented a long section on inheritance problems solved by algebra.

The term al-jabr (“restoration” or “completion”) in al-Khwārizmī’s title refers to a procedure whose modern counterpart is eliminating negative terms from a (linear or quadratic) equation by adding an appropriate qantity to both sides of the equation. For example, using one of al-Khwārizmī’s own examples (but expressed using modern symbolic notation), al-jabr  transforms

 x2 = 40x – 4x2

into

5x2 = 40x.

The other key term in the title, al-muqābala (“confrontation”) refers to the process of eliminating identical quantities from the two sides of the equation. For example, (again in modern notation) one application of al-muqābala simplifies

50 + 3x + x2 = 29 + 10x

to

21 + 3x + x2 = 10x

and a second application simplifies that to

21 + x2 = 7x.

Procedurally (but not conceptually) these are the methods we use today to simplify and solve equations. Hence, a meaningful, modern English translation for Hisâb al-Jabr wa’l-Muqābala would be, simply, “Calculation with Algebra.”

The symbolic notation is not the only difference between medieval algebra and its present-day counterpart. The medieval mathematicians did not acknowledge negative numbers. For instance, they viewed “ten and a thing” (10 + x) as a composite expression (it entails two types of number: “simple numbers” and “roots”), but they did not see “ten less a thing” (10 – x) as composite. Rather, they thought of it as a single quantity, a “diminished” 10, or a 10 with a “defect” of x. The 10 retained its identity, even though x had been taken away from it. When an x was added to both sides of an equation, the diminished 10, (10 – x), was restored to its rightful value. Hence the terminology.

The first degree unknown, our x, was usually called shay’ (“thing”), but occasionally jidhr (“origin” or “base”, also “root” of a tree, giving rise to our present-day expression “root of an equation”). The second power, our x2, was called māl (a sum of money/property/ wealth). Units were generally counted in dirhams, a denomination of silver coin, occasionally simply “in number”.  For example, al-Khwārizmī’s (rhetorical) equation “a hundred ten and two māls less twenty-two things equals fifty-four dirhams” corresponds to our symbolic equation 110 + 2x2 – 22x = 54.

Arabic authors typically explained the methods of algebra in two stages. First they provided an explanation of the names of the powers, described six simplified forms of equations and their solutions, and gave rules for operating on polynomials and roots. They then followed this introduction by a collection of solved problems which illustrated the methods.

Their solutions followed a standard template:

Stage 1: an unknown quantity was named (usually referred to as a “thing”), and an equation was set up.

Stage 2: the equation was simplifed to one of six canonical types.

Stage 3: the appropriate procedure was applied to arrive at the answer.

Because they allowed only positive coefficients, they had to consider six equation types, rather than the single template ax2 + bx + c = 0   we use today:

(1) māls equals roots (in modern terms, ax2 = bx),

(2) māls equals numbers (ax2 = c),

(3) roots equals numbers (bx = c),

(4) māls and roots equals numbers (ax2 + bx = c),

(5) māls  and numbers equals roots (ax2 + c = bx),

(6) māls equals roots and numbers (ax2 = bx + c).

We see how al-Khwārizmī  used the two simplification steps in Stage 2, al-jabr wa’l-muqābala, (“restoration and confrontation”) in his solution to a quadratic equation, which he described in these words:

 If some one say: “You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times.” Computation: You say, ten less thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts.

The American scholar Jeffrey Oaks has translated this (fairly literally) as follows, adding headings to assist the reader:

Enunciation
If [someone] said, ten:  you divided it into two parts.  You multiplied one of the parts by itself, which is the same as eighty-one times the other.

Setting up and simplifying the equation
The rule for this is that you say ten less a thing by itself is a hundred and a mal less twenty things [which] equal eighty-one things. Restore the hundred and a mal by the twenty things and add them to the eighty-one [things].  This yields:  a hundred and a mal equal a hundred roots and a root.

Solving the simplified equation
So halve the roots, which yields fifty and a half, and multiply it by itself, which yields two thousand five hundred fifth and a fourth. Subtract from it the hundred, leaving two thousand four hundred fifty and a fourth.  Take its [square] root, which is forty-nine and a half. Subtract it from half the roots, which is fifty and a half.  There remains one, which is one of the two parts.

Using modern notation, and substituting the letter x for “thing”, al-Khwārizmī was solving the equation

(10 − x)2 = 81x

which can be written in the equivalent form

x2 + 100 = 101x

Al-Khwārizmī did not state the equation

(10 − x)2 = 81x

Rather, he set up the equation

100 + x2 – 20x = 81x.

Nothing like the equation (10 − x)2 = 81x was ever stated in medieval algebra; the left side of such an expression entails what was then an unrealized operation. Medieval algebraists worked out all operations before stating equations, so al-Khwārizmī did not begin with (10 − x)2 = 81x, as we would, rather he first worked out the multiplication.

Having demonstrated methods for solving linear and quadratic equations, al-Khwārizmī proceeded to examine how to manipulate algebraic expressions. For example he showed how to multiply out specific numerical instances

(a + bx) (c + dx)

expressing everything in words, not symbols.

He ended the first section of the book by presenting the solutions to 39 problems.

In the following section, al-Khwārizmī presented solutions to some mensuration problems, including rules for finding the area of figures such as the circle and for finding the volume of solids such as the sphere, cone, and pyramid.

The final part of the book dealt with the complicated Islamic rules for inheritance, which involved the solution of linear equations.

* * *

COMING UP: In the final article in this series I’ll summarize some of the amazing developments in algebra that were made in the Arabic period subsequent to al-Khwārizmī.

al-Khwārizmī

Published December 25, 2011 algebra 2 Comments
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The fourth in a series. See the November 20 entry, “What is algebra?” for the first, the December 13 entry “When did algebra begin?” for the second, and the December 19 entry “The  golden age of Arabic mathematics” for the third.

Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī (c.780 – c.850 CE) was one of the most significant figures in the development of modern algebra. Yet we know virtually nothing about his life.

There is even some confusion in the literature as to his full name. Most present-day sources give it as Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī, which can be translated as “Father of ʿAbdallāh, Mohammed, son of Moses, native of the town of al-Khwārizmī”. References to Abū Jaʿfar Muḥammad ibn Mūsā al-Khwārizmī are erroneous in this context; that was a different person

Al-Khwārizmī wrote several books, two of which had a huge impact on the growth of mathematics, one focused on arithmetic, the other on algebra. He aimed both at a much wider audience than just his fellow scholars. As with Euclid and his Elements, it is not clear whether al-Khwārizmī himself developed some of the methods he desribed in his books, in addition to gathering together the work of others, though a later author, Abū Kāmil, suggested that his famous predecessor did develop some of the methods he presented in his books.

The first of al-Khwārizmī’s  two most significant books, written around 825, described Hindu-Arabic arithmetic. Its original title is not known, and it may not have had one. No original Arabic manuscripts exist, and the work survives only through a Latin translation, which was most likely made in the 12th century by Adelard of Bath. The original Latin translation did not have a title either, but the Italian bibliophile Baldassare Boncompagni gave it one when he published a printed edition in the 19th century: Algoritmi de numero Indorum (“al-Khwārizmī on the Hindu Art of Reckoning”). The Latinized version of al-Khwārizmī’s name in this title (Algoritmi) gave rise to our modern word “algorithm” for a set of rules specifying a calculation. In English, the work is sometimes referenced as On the Calculation with Hindu Numerals, but it is most commonly referred to simply as “al-Khwārizmī’s Arithmetic.”

Al-Khwārizmī’s second pivotal book, completed around 830, was al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqābala. The phrase al-jabr wa’l-muqābalah translates literally as “restoration and confrontation,” or more loosely as “reducing (or solving) an equation.” The title of the book translates literally as “The Abridged Book on Calculation by Restoration and Confrontation”, but a more colloquial rendering would thus be “The Abridged Book on Algebra”. It is an early treatise on what we now call “algebra,” that name coming from the term al-jabr in the title. Scholars today usually refer to this book simply as “Al-Khwārizmī’s Algebra.” There are seven Arabic manuscripts known, not all complete. One complete Arabic copy is kept at Oxford and a Latin translation is kept in Cambridge. Two copies are in Afghanistan.

In Algebra, al-Khwārizmī described (but did not himself develop) a systematic approach to solving linear and quadratic equations, providing a comprehensive account of solving polynomial equations up to the second degree.

The Algebra was translated into Latin by Robert of Chester in 1145, by Gherardo of Cremona around 1170, and by Guglielmo de Lunis around 1250.  In 1831, Frederic Rosen published an English language translation. In his preface, Rosen wrote:

ABU ABDALLAH MOHAMMED BEN MUSA, of Khowarezm, who it appears, from his preface, wrote this Treatise at the command of the Caliph AL MAMUN, was for a long time considered as the original inventor of Algebra.        …   …   …      From the manner in which our author [al-Khwārizmī], in his preface, speaks of the task he had undertaken, we cannot infer that he claimed to be the inventor. He says that the Caliph AL MAMUN encouraged him to write a popular work on Algebra: an expression which would seem to imply that other treatises were then already extant.

In fact, algebra (as al-Khwārizmī described it in his book) was being transmitted orally and being used by people in their jobs before he or anyone else started to write it down. Several authors wrote books on algebra during the ninth century besides al-Khwārizmī, all having the virtually identical title  Kitāb al-ğabr wa-l-muqābala. Among them were Abū Hanīfa al-Dīnawarī, Abū Kāmil Shujā ibn Aslam, Abū Muḥammad al-ʿAdlī, Abū Yūsuf al-Miṣṣīṣī, ‘Abd al-Hamīd ibn Turk, Sind ibn ʿAlī, Sahl ibn Bišr, and Šarafaddīn al-Tūsī.

In addition to his two books on mathematics, al-Khwārizmī wrote a revised and completed version of Ptolemy’s Geography, consisting of a general introduction followed by a list of 2,402 coordinates of cities and other geographical features. Titled Kitāb ṣūrat al-Arḍ (“Book on the appearance of the Earth” or “The image of the Earth”), he finished it in 833. There is only one surviving Arabic copy, which is kept at the Strasbourg University Library. A Latin translation is kept at the Biblioteca Nacional de España in Madrid.

* * *

COMING UP NEXT: Al-Khwārizmī’s answer to that perennial student question, “What is algebra good for?” Plus a look at the contents of his seminal book, including an explanation of what exactly was being “restored” in the process for which al-Khwārizmī’s Arabic term was al-jabr.

* * *

Al-Khwārizmī on National Public Radio: I talked about al-Khwārizmī and the birth of algebra with host Scott Simon in my occasional “Math Guy” slot on NPR’s Weekend Edition on December 24.

The golden age of Arabic mathematics

Published December 19, 2011 algebra 6 Comments
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The third in a series. See the November 20 entry, “What is algebra?” for the first and the December 13 entry “When did algebra begin?” for the second.

On 14 September 786, Harun al-Rashid became the fifth Caliph of the Abbasid dynasty. From his court in the capital city of Baghdad, Harun ruled over the vast Islamic empire, stretching from the Mediterranean to India. He brought culture into his court and encouraged the widespread pursuit of learning.

Al-Rashid had two sons, the elder al-Amin, the younger al-Mamun. Harun died in 809 and there was an armed conflict between the brothers. Al-Mamun won the armed struggle and al-Amin was defeated and killed in 813. Following this, al-Mamun became Caliph and ruled the empire from Baghdad.

Al-Mamun continued the patronage of learning started by his father. With his encouragement, scholars of the time set about collecting and writing down in books all available practical knowledge, much of which had hitherto been transmitted only orally, including mathematics and folk astronomy. They translated into Arabic works of Greek and Indian science.

Many of the works collected and created may have been housed in a library called the House of Wisdom, though there is no evidence to support the commonly repeated claims that (1) it was massive, (2) it was founded by al-Mamun, or (3) translations were carried out there.

The tradition of learning, writing, and translation begun by al-Rashid and al-Mamun continued for the next quarter century, making the Islamic civilization the center of world knowledge. The aristocracy and other wealthy groups within Muslim society supported the appropriation of all practical and scientific knowledge they could acquire. They employed scholars to translate into Arabic works by Indian, Sasanian, and especially Greek authors, and mathematicians recorded on paper all that was known of arithmetic, algebra, and mensuration, which had hitherto been communicated orally by traders. In addition to the mathematical sciences (arithmetic, geometry, optics, mathematical astronomy, etc.), they also translated texts on geography, astrology, philosophy, medicine, agriculture, alchemy, and even falconry.

Greek works formed the bulk of the material translated. In addition, the more scientifically oriented mathematicians adopted the Greek tradition of definitions, axioms, and propositions with rigorous proof, and astronomers embraced the Greek idea of geometric models of planetary motion. Within this framework, Indian techniques were incorporated into this new Arabic/Islamic mathematics.

In addition to the translations, scholars wrote commentaries and criticisms of the ancient mathematics and made their own original contributions. For example, in the 9th century, Thābit ibn Qurra (d. 901) translated several works of Archimedes, wrote commentaries on Euclid’s Elements and Ptolemy’s Almagest, critiqued Euclid’s definition for the composition of ratios of numbers, and derived and proved new formulas for volumes of solids of revolution.

When the sources of Greek and other foreign texts was finally exhausted, scholars continued to produce new results in all branches of mathematics. For instance, in the 11th century, Ibn al-Haytham made major contributions to optics and geometry, and at the start of the 12th century, al-Khāyyamī wrote his book on algebra.

Over a thousand mathematical manuscripts from the period have survived, about half of them dating before the 15th  century.

Al-Khwārizmī, who may have studied and worked in the House of Wisdom, was one of the earliest contibutors to this vast undertaking, and arguably had the most impact of all the mathematicians involved. But his books – he wrote one on Hindu arithmetic in addition to the one on algebra – should be viewed as part of this larger movement.

At the time, algebra was viewed primarily as a practical, numerical problem solving technique, not the autonomous branch of mathematics it became later. Indeed, the greatest contribution of Arabic mathematical work to society was its development as a set of practical tools.

Three systems of practical calculation were taught and practiced in the medieval Islamic world: finger reckoning, Hindu arithmetic, and the base 60 system of the astronomers. Merchants preferred finger-reckoning, which worked for numbers up to 10,000. Finger reckoning was used to solve problems by various methods, such as double false position and algebra. Al-Khwārizmī is known to have written a work, now lost, called Book of Adding and Subtracting, in the early 9th century, which was probably devoted to the use of finger reckoning. (If so, it was probably the earliest written text on the subject.)

The Arabic mathematicians referred to the numerals 1, 2, 3, etc., as Hindī numerals, because they acquired the system from India. These numerals were already in use in the Middle East by the 7th century CE. The earliest known Arabic text describing the system is al-Khwārizmī’s Book on Hindī Reckoning, written in the early 9th  century, which survives only in Latin translation. The original algorithms for calculating in this system were devised for use on a dust board, where erasing is easy. In the middle of the 10th century, al-Uqlīdisī introduced new algorithms for use with pen and paper. The Arabic mathematicians introduced the concept of decimal fractions, wihich al-Uqlīdisī described for the first time.

Unlike Diophantus, most of the Arabic authors, including al-Khwārizmī, wrote their algebra almost entirely in words. For example, where we would write down the symbolic equation  x + 1 = 2, they might write “The thing plus one equals two” (and very occasionally “The thing plus 1 equals 2”). This is generally known as the rhetorical form, and remained in common use right up to the 16th  century. This is, however, a notational distinction, not one of content. Commentators who refer to “rhetorical algebra” as being a form of algebra distinct from “literal algebra” are in error. For, although the Arabic authors wrote their books rhetorically, with no notation even for numbers, they did not solve problems rhetorically. Throughout most of Arabic algebra, problems were worked out on some ephemeral surface, by writing the coefficients and numbers in Hindu form. For example, they would write

1  2  1

to mean x2 + 2x + 1. Later, Arabic scholars in the Maghreb developed a truly algebraic notation, with symbols for the words representing the powers of the unknown, but even they they would resort to rhetorical text to communicate the result of a calculation.

Symbolic algebra, where full symbolism is used, is generally credited in the first instance to the French mathematician François Viète (1540 –1603), followed by René Descartes (1596 – 1650), though traces can be discerned in the writings of some Arabic mathematicians as early as the 13th century.

* * *

In my next two articles in this short series, I’ll say a bit about  al-Khwārizmī and take a look at the contents of his seminal book on algebra. In particular, I’ll give his answer to that perennial student question, “What is algebra good for?”

I use terms like “Arabic mathematics” in the standard historical fashion to refer to the mathematics done where and when the primary language for scholastic texts was Arabic. Mathematics, like all of science, belongs to the world.

When did algebra begin?

Published December 13, 2011 algebra 11 Comments
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The second in a series. See the November 20 entry, “What is algebra?” for the first.

Two key features of algebra as we understand the word today are:

1. Reasoning about numbers by recognizing patterns across numbers;

2. Solving a problem by introducing a term for an unknown and then, starting with what is known, reasoning to determine its value.

We first see the emergence of both features of algebra in the mathematics of ancient Babylonia, around 2,000 BCE.

Several hundred of the many thousands of Babylonian’s cuneiform-inscribed clay tablets that have been found are devoted to mathematics. They show that those ancient mathematicians had systematic procedures for solving geometric problems involving the determination of lengths and areas of figures. Today, we would solve those kinds of problems using linear and quadratic equations and indeterminate systems of linear equations. Their methods amounted to a form of geometric algebra that could be applied to solve problems beyond overtly geometric examples such as calculating the perimeters or areas of various plane figures or the volumes of solid objects: arithmetic problems arising in trade and commerce, for example, and other financial transactions such as inheritance. In addition, the Babylonians considered problems that seemed to have had no practical application, pursuing them purely for recreation. Although they described their procedures in terms of specific lengths and areas, they did so in a way that made it clear they applied in general, and in that sense they were starting to think algebraically, by recognizing patterns across quantities.

Moreover, some of their writings show the second characteristic feature of algebra, namely introducing an unknown and then reasoning to find its value. In their case, however, the unknown was not numeric but geometric – an unknown line on which they performed geometrical operations to get the answer.

In reasoning with unknown quantities, the Babylonians went further than other early civilizations with a mathematical tradition, such as the Egyptians, the Chinese, and the early Greeks, all of the first millennium BCE. Our knowledge of the mathematics of those peoples comes from works such as the Rhind papyrus, The Nine Chapters of the Mathematical Art, and Euclid’s Elements, respectively. The approach described in those documents was, like that of the Babylonians, fundamentally geometric and exhibited reasoning about patterns of quantities, but we do not find the introduction of an unknown followed by an argument to determine its value.

It is with the work of the Greek mathematician Diophantus  (ca. 210–290 CE) that we first find clearly recognizable algebra, where the unknowns represent numbers whose values are to be determined. Around 250 CE, Diophantus, who lived in Alexandria in Egypt, wrote a multi-volume work, Arithmetica, which its title notwithstanding was an algebra book. Its author used letters (literals) to denote the unknowns and to express equations, but that is a purely notational distinction. He also was one of the first mathematicians to use negative numbers in calculations. He showed how to solve equations by using two techniques called restoration and confrontation. In modern terms, these correspond more or less (but not precisely) to (1) adding a quantity to both sides of an equation to eliminate a negative term on one side, and (2) eliminating like terms from both sides. He used these techniques to solve polynomial equations involving powers up to 6.

Almost four hundred years later, the Indian mathematician Brahmagupta (598–668 CE) likewise displayed recognizable algebra, in his book Brahmasphutasiddhanta, where he described the first complete arithmetic solution (including zero and negative solutions) to quadratic equations.

Following Diophantus and Brahmagupta, the next major step in the development of algebra – and it was huge – took place in the period generally referred to as “Arabic mathematics” or “Muslim mathematics”, a significant outpouring of mathematical activity stretching from the 8th century to the end of the 16th. Indeed, the word algebra itself comes from the Arabic word al-jabr, which occurs in the title of a highly influential book by the Persian mathematician al-Khwārizmī, completed around 830: al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqābala. The phrase al-jabr wa’l-muqābalah translates literally as “restoration and confrontation,” but more loosely means “solving an equation.”

That period will be the focus of my next article on algebra.

What is algebra?

Published November 20, 2011 algebra 30 Comments
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We hear a lot about the importance that all children master algebra before they graduate from high school. But what exactly is algebra, and is it really as important as everyone claims? And why do so many people find it hard to learn?

Answering these questions turns out to be a lot easier than, well, answering a typical school algebra question, yet surprisingly, few people can give good answers.

First of all, algebra is not “arithmetic with letters.” At the most fundamental level, arithmetic and algebra are two different forms of thinking about numerical issues. (I should stress that in this article I’m focusing on school arithmetic and school algebra. Professional mathematicians use both terms to mean something far more general.)

Let’s start with arithmetic. This is essentially the use of the four numerical operations addition, subtraction, multiplication, and division to calculate numerical values of various things. It is the oldest part of mathematics, having its origins in Sumeria (primarily today’s Iraq) around 10,000 years ago. Sumerian society reached a stage of sophistication that led to the introduction of money as a means to measure an individual’s wealth and mediate the exchange of goods and services. The monetary tokens eventually gave way to abstract markings on clay tablets, which we recognize today as the first numerals (symbols for numbers). Over time, those symbols acquired an abstract meaning of their own: numbers. In other words, numbers first arose as money, and arithmetic as a means to use money in trade.

It should be noticed that counting predates numbers and arithmetic by many thousands of years. Humans started to count things (most likely family members, animals, seasons, possessions, etc.) at least 35,000 years ago, as evidenced by the discovery of bones with tally marks on them, which anthropologists conclude were notched to provide what we would today call a numerical record. But those early humans did not have numbers, nor is there any evidence of any kind of arithmetic. The tally markers themselves were the record; the marks referred directly to things in the world, not to abstract numbers.

Something else to note is that arithmetic does not have to be done by the manipulation of symbols, the way we are taught today. The modern approach was developed over many centuries, starting in India in the early half of the First Millennium, adopted by the Arabic speaking traders in the second half of the Millennium, and then transported to Europe in the 13th Century. (Hence its present-day name “Hindu-Arabic arithmetic.”) Prior to the adoption of symbol-based, Hindu-Arabic arithmetic, traders performed their calculations using a sophisticated system of finger counting or a counting board (a board with lines ruled on it on which small pebbles were moved around). Arithmetic instruction books described how to calculate using words, right up to the 15th Century, when symbol manipulation began to take over.

Many people find arithmetic hard to learn, but most of us succeed, or at least pass the tests, provided we put in enough practice. What makes it possible to learn arithmetic is that the basic building blocks of the subject, numbers, arise naturally in the world around us, when we count things, measure things, buy things, make things, use the telephone, go to the bank, check the baseball scores, etc. Numbers may be abstract — you never saw, felt, heard, or smelled the number 3 — but they are tied closely to all the concrete things in the world we live in.

With algebra, however, you are one more step removed from the everyday world. Those x’s and y’s that you have to learn to deal with in algebra denote numbers, but usually numbers in general, not particular numbers. And the human brain is not naturally suited to think at that level of abstraction. Doing so requires quite a lot of effort and training.

The important thing to realize is that doing algebra is a way of thinking and that it is a way of thinking that is different from arithmetical thinking. Those formulas and equations, involving all those x’s and y’s, are merely a way to represent that thinking on paper. They no more are algebra than a page of musical notation is music. It is possible to do algebra without symbols, just as you can play and instrument without being ably to read music. In fact, traders and other people who needed it used algebra for 3,000 years before the symbolic form was introduced in the 16th Century. (That earlier way of doing algebra is nowadays referred to as “rhetorical algebra,” to distinguish it from the symbolic approach common today.)

There are several ways to come to an understanding of the difference between arithmetic and (school) algebra.

  • First, algebra involves thinking logically rather than numerically.
  • In arithmetic you reason (calculate) with numbers; in algebra you reason (logically) about numbers.
  • Arithmetic involves quantitative reasoning with numbers; algebra involves qualitative reasoning about numbers.
  • In arithmetic, you calculate a number by working with the numbers you are given; in algebra, you introduce a term for an unknown number and reason logically to determine its value.

The above distinctions should make it clear that algebra is not doing arithmetic with one or more letters denoting numbers, known or unknown.

For example, putting numerical values for a, b, c  in the familiar formula

in order to find the numerical solutions to the quadratic equation

 is not algebra, it is arithmetic.

In contrast, deriving that formula in the first place is algebra. So too is solving a quadratic equation not by the formula but by the standard method of “completing the square” and factoring.

When students start to learn algebra, they inevitably try to solve problems by arithmetical thinking. That’s a natural thing to do, given all the effort they have put into mastering arithmetic, and at first, when the algebra problems they meet are particularly simple (that’s the teacher’s classification as “simple”), this approach works.

In fact, the stronger a student is at arithmetic, the further they can progress in algebra using arithmetical thinking. For example, many students can solve the quadratic equation  x2 = 2x + 15  using basic arithmetic, using no algebra at all.

Paradoxically, or so it may seem, however, those better students may find it harder to learn algebra. Because to do algebra, for all but the most basic examples, you have to stop thinking arithmetically and learn to think algebraically.

Is mastery of algebra (i.e., algebraic thinking) worth the effort? You bet — though you’d be hard pressed to reach that conclusion based on what you will find in most school algebra textbooks. In today’s world, most of us really do need to master algebraic thinking. In particular, you need to use algebraic thinking if you want to write a macro to calculate the cells in a spreadsheet like Microsoft Excel. This one example alone makes it clear why algebra, and not arithmetic, should now be the main goal of school mathematics instruction. With a spreadsheet, you don’t need to do the arithmetic; the computer does it, generally much faster and with greater accuracy than any human can. What you, the person, have to do is create that spreadsheet in the first place.  The computer can’t do that for you.

It doesn’t matter whether the spreadsheet is for calculating scores in a sporting competition, keeping track of your finances, running a business or a club, or figuring out the best way to equip your character in World of Warcraft, you need to think algebraically to set it up to do what you want. That means thinking about or across numbers in general, rather than in terms of (specific) numbers.

Of course, the need for algebra does not make it any easier to learn — though I think that spreadsheets can provide today’s students with more meaningful and fulfilling applications than problems about trains leaving stations or garden hoses filling swimming pools, that my generation had to endure. But in a world where our very national livelihood depends on staying ahead of the technology curve, it is crucial that we equip our students with the kind of thinking skills today’s world requires. Being able to use computers is one of those skills. And being able to use a computer to do arithmetic requires algebraic thinking.

In future postings I’ll describe the growth of algebra through the ages.

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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