The wife and I are sitting here on a Saturday night doing some algebra homework. We are factoring polynomials and we both had the same thought at the same time: when are we going to use this?
For some reason, factoring makes people angry. I’ve never entirely understood why. Is it useless? Mostly, but it’s unclear to me it’s especially useless. It’s probably a bit more useless than knowing how to ride a unicycle, a bit less than the punchline of the first Peanuts comic (“How I hate him!”), which means that it fits nicely within the range of useless stuff I learned and enjoyed as a teen.
Part of the problem is that factoring is too much of one thing, not enough of another. It’s typically introduced to students as a method for solving polynomial equations. But it’s never the only method taught. If you hate or fear algebraic manipulation, are you going to solve an equation by factoring? Not if you can graph it. And if algebraic manipulation is your speed, why bring a spoon to a knife fight? The quadratic formula or completing the square could be your go-to.
So, nobody’s students likes factoring. (Sit down, Honors Algebra.) It seems frivolous and useless. Which is why I was a bit surprised to see it coming up again and again while reading about ancient mathematics.
I’ve been on a bit of a math history kick lately. I started with The Beginnings and Evolution of Algebra, a book I found while scanning the shelves at school for some summer reading. Beginnings and Evolution seems to heavily rely on van der Waerden’s dry but important Geometry and Algebra in Ancient Civilizations. A search for an up-to-date, well-written version of all this led me to Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century, which is the best of the bunch for my needs…
…and then things started spiraling out of control. Katz’s material on Mesopotamia sent me to Eleanor Robson, and this gem of a piece by her is fantastic. Read Robson for sentences like “the conceptual transition from arc to angle was slow and halting” and “Sherlock Holmes, if he ever made it to Babylon, would have been over 100 miles away from the action.”
“Using the history of algebra, teachers of the subject can increase students’ overall understanding of the material.” This is from Katz and Parshall, at the start of Taming the Unknown. Are we to take teaching tips from Mesopotamian scribes? What exactly can a modern teacher glean from mathematical history?
Not many people have five words in their name, but most people aren’t Bartel Leendert van der Waerden. Though a student of Emmy Noether (who was Jewish) he managed to hold on to his university position in Germany during the Nazi rule. (On one hand, when talking to the Nazis he made a point of his “full-blooded Arianness”. On the other, he seemed disposed against the regime in fairly conventional ways. In sum, he’s guilty of cowardice and self-interest. Aren’t we all?)
He wrote the first comprehensive textbook on modern algebra, and later his interests turned to the history of mathematics. In both Scientific Awakening and Geometry and Algebra in Ancient Civilizations, he put ancient sources in conversation with a modern mathematical perspective. Often, he discovered modern theorems lurking in the work of the ancients. These included various identities that today we would teach as factoring:
This third identity — the difference of squares — is studied by high school math students all over the world. Personally, I love the way this identity can be used as a mental math shortcut, allowing for seemingly heroic acts of paperless computation. So, for example, if you transform into a difference of squares, you can conclude that the product must be 9,999:
And, for extra fun, you can sometimes play the game backwards, starting with a difference of squares and transforming it into a product. From brilliant.org, here is a particularly clever use of this transformation:
In school math, however, the identity is typically used as an exercise in of itself:
And so on, and so on, until the font of the terms becomes far too small to read.
Van der Waerden is very clear; he believes the ancient Mesopotamians knew of this (and other) factoring identity:
For the longest time, I pored over van der Waerden’s books in search of a source that could justify his claim. I found none, and grew frustrated. Where is the proof that Babylonians knew and used the difference of squares identity? And why doesn’t van der Waerden bother to carefully show this?
Now, though, my best read is he thought that it was just blindingly obvious that they used the difference of squares. This is because the Mesopotamian method for solving quadratic equations (“completing the square”) just is an instance of the difference of squares transformation.
When presented with a problem such as , the Mesopotamians would typically transform this expression into a difference of squares.
Pictorially, this rectangle of area ““…
…is cut and pasted into a difference-of-squares arrangement:
This was the fundamental step in their solution of a quadratic equation. If the initial diagram had area c (as it does, since ) then this difference of two squares does too.
And then things get rolling: the area of the full square is ; the side length is ; the missing length, is . We have just come very, very close to deriving the quadratic formula.
This, I think, is why van der Waerden doesn’t bother to carefully source his claim. The difference of squares identity is in constant use by Mesopotamian mathematicians, as they go about their business of solving area problems.
Shocking, right? A factoring identity was crucially useful.
This is lovely, but is it algebra?
Van der Waerden certainly thought so. As in other ancient sources, Mesopotamian math was couched entirely in the language of shapes — squares, lengths, sides, rectangles, area. There was no symbolic language, no equations or expressions. Just words and numbers.
Nevertheless, he thought that all of this was essentially a geometric sheen over an algebraic core. After speculating on how Mesopotamian mathematicians might have geometrically derived the difference of squares identity, he cautions his reader:
But we must guard against being lead astray by the geometric terminology. The thought processes of the Babylonians were chiefly algebraic. It is true that they illustrated unknown numbers by means of lines and areas, but they always remained numbers.
If this is so, then perhaps ancient mathematicians would use the differences of squares identity just as we could, for quick mental computation in special cases, or for factoring expressions.
In time, though, van der Waerden’s view would become extremely controversial. (Well, at least as heated as these things get in history of math.)
All of this is new to me, so I’m worried that I’m going to get this wrong, but here’s what I’m gathering. The first major breakthroughs in the history of ancient math came from modern mathematicians — people like good-old Nazi-tolerating van der Waerden. Their contributions can’t be minimized — they put ancient mathematics in terms moderns can understand.
More recently, the study of ancient mathematical texts has been taken over by historians trained as historians. These scholars understand ancient societies in a way that the mathematicians-turned-historians could not. These more recent scholars believe that mathematicians like van der Waerden erred by seeing too much of the past in modern mathematical terms.
The current consensus seems to be that it’s an error to see the Mesopotamians as engaging in algebra. But what should we call it? It was clearly algorithmic — they used the same procedure for solving quadratic situations again and again. It was also, clearly, geometric. The problems were couched in geometric language, and it seems that rather than conceiving of their solutions as numbers they conceived of them as lengths and areas that were constructed.
The earlier, debated term for describing this style of math was “geometrical algebra,” but that’s fallen out of favor. In Taming the Unknown the authors seem to prefer “cut and paste” to describe this mathematical style — I like that! (I also sort of like quasi-algebraic geometry, not despite but because it’s a mouthful. Fun to say!)
It’s not that ancient Mesopotamians weren’t “abstract thinkers” or something. The scribes who produced these quadratic equations weren’t solving practical problems. From Taming: “The problems in geometrical algebra that show up on the clay tablets are artificial, in the sense that they do not reflect actual problems that the surveyors (or the scribes) needed to solve.”
In fact, no one can quite figure out why they were solving them in the first place. Taming summarizes two interesting proposed contexts for the math:
Various scholars have speculated about these origins, however. Jens Hoyrup, for example, believes that they were originally developed as riddles to demonstrate the professional competence of surveyors and scribes, while Eleanor Robson holds that the development of techniques for accurately calculating lines and areas reflected a concern with justice, part of the ideology of the Mesopotamian states around 2000 BCE.
Either way, modern scholars have caused trouble for the idea that what our high school students study is actually ancient. While the Mesopotamians knew how to cut-and-paste a rectangle into the difference of two squares, that didn’t mean they also knew our factoring identities. And while our students might study factoring identities, that surely doesn’t mean that they would see their work in what is found on clay tablets in Iraq.
We tend to think of mathematics as universal, Eleanor Robson points out. Studying the history of math shows us that this is not so: for example, the Mesopotamians didn’t use radii to find the area of circles. (Area was entirely a function of circumference.) They had no concept of angle measure. And their math was largely concerned with geometric algorithms, rather than numerical ones.
Each year, in my high school geometry classes, we study a bit of cut-and-paste math. We do this while studying area formulas. (I first learned this approach from the CME Geometry text.)
Can you cut a triangle, and rearrange its pieces into a rectangle? How about a trapezoid?
Remember: you can’t throw pieces away, and you can’t add anything either! (Somehow I always forget to say this and kids end up giving the triangle a haircut and declaring victory.)
Most years, a student ends up wondering whether it’s possible to cut and rearrange the pieces of a rectangle to make a square. My first instinct has always been, sure, it must be possible, but I’ve never known how to.
(Come to think of it, is it really always possible? Suppose the rectangle is 1 ft by 19 ft. We would have to find a way to cut and rearrange the pieces of the original into a square whose sides are each . That feels impossible…)
While reading ancient Greek mathematics, I learned that Euclid provided an algorithm for constructing a square with area equal to that of a rectangle.
(Come to think of it, if you can construct the square, you must be able to cut and paste the original rectangle into it. Right? Right?? I’m confused.)
This isn’t quite how Euclid did it, but it’s closely related. Start with a rectangle of a certain height (H) and width (W). It’s area would be .
Suppose that we were able to make a right triangle that looked like this:
Then, the Pythagorean Theorem would tell us, this diagram follows:
Do you see it? Do you see the difference of squares in the bottom left square? Here’s the rest of the argument.
Which means: that square is four times bigger than our original rectangle. A fourth of it, though, is a square of area equal to what we started with.
Tada! (Well, sort of “tada.” I skipped all the construction details. You can read more about this here.)
The way Euclid did it doesn’t exactly involve the difference of squares identity, but I think there’s something important about this version of the proof. The difference of squares identity is deeply connected to the Mesopotamian cut-and-paste method. It’s also deeply connected to the Pythagorean Theorem, though this rarely comes up in school math.
We typically introduce the Pythagorean Theorem as a sum of squares relationship:
But it’s equally true that the Pythagorean Theorem is saying something about a difference of squares:
Which means that you could just as well put it like this:
If you want to go down a number theory road, this line of reasoning can lead to a method for generating an unending collection of integers that make the Pythagorean Theorem true. (See APPENDIX below.)
The older generation of scholars — the mathematicians-turned-historians — thought that they had recognized this Pythagoras/Factoring one-two punch in a Mesopotamian clay tablet called “Plimpton 322.” They thought that the tablet represented a list of Pythagorean triples, and they inferred that the difference of squares method was used to generate them.
Here’s what Plimpton 322 looks like, by the way:
Eleanor Robson wrote a fantastic article challenging this view. She argues on both mathematical and contextual basis that this table can’t represent Pythagorean triples. For her, this is just another example of mathematicians not understanding Mesopotamia on its own mathematical and social terms.
But even if this idea is not quite as ancient as initially supposed, there is no doubt that Plato, Euclid and Diophantus used functions like this one to generate Pythagorean triples. And the connection between factoring the difference of squares and the Pythagorean Theorem is mathematically true, no matter the historical record.
A curriculum is a funny thing. In high school, students study area formulas, the Pythagorean Theorem and factoring a difference of squares. And yet, it would seem to never do a teacher any good to get into all of the above, except as an extension or something. What was a natural part of an ancient curriculum has no place in our own.
So: can the studying the past help us better teach factoring?
At one level, I think the answer is “no.” The needs of Mesopotamian scribes and Greek phiosopher-mathematicians are too different from the needs of our students. We live in different worlds, and that applies to mathematics too.
At a less-practical level — the level of understanding — I think this historical excursion helped me better understand a few important points about teaching factoring to students.
Solve by factoring:
These sort of problems, though, basically never showed up in the ancient history, because they almost never studied these trinomial equations. Because these situations always had an area interpretation, it didn’t make sense to make an expression equal to zero. Instead, the equations always involved two terms. Factoring didn’t involve generic trinomials. It involved the special, factoring formulas.
I really have no taste for curriculum advocacy — I teach what I’m expected to teach — but I do think it could make sense to separate the difference of squares from other factoring cases. Perhaps, I’d teach difference of squares (and the binomial expansion, i.e. first.
The other thought I have is mushier, and more general. The mistake of the early mathematician-historians was to see too much of algebra in the cut-and-paste geometry of the Mesopotamians and Greeks. What they failed to understand was the extent to which this ancient math was fully geometrified. It was fully and thoroughly geometry, all the way down.
It seems weird, then. Why didn’t the Mesopotamians and Greeks make the leap to algebra? And why don’t our students make these same connections?
In the history of education there have been people who have made very strong claims about the similarity of children’s development to the historical development of cultures. This is wrong — and often racist and colonialist, as it assumes that other cultures are further behind in an inevitable path towards the present.
But historians of mathematics have a more nuanced view of Mesopotamia now. It’s not that ancient cultures knew — or failed to know — algebra, as much as they had their own sort of algorithmic geometry. It made sense to them, and it needs to be understand in its own context and time.
All of this, though, might make us a little more pessimistic about the usefulness of geometry for helping students learn algebraic concepts. The geometry of cut-and-paste really is different from the algebra of factoring. It’s only when you understand both that you can look back and see the connections between them, as van der Waerden did.
When faced with a tough topic, math teachers often like to change the context — add a story, move to pictures, put things in geometric terms. A lesson from the history of math is to be very, very worried about whether these more comprehensible contexts are really aids for understanding the difficult things. There is nothing simple about moving from geometry to algebra.
APPENDIX: ON PYTHAGOREAN TRIPLES AND THE DIFFERENCE OF SQUARES
Where were we? Ah, right:
It’s not obvious that both and have to be square numbers, but they do (I think), and let’s call the first square number and the second . Which means that the following two equations are true:
Add those two equations together, and you get a new one.
Subtract them, and you get an equation for .
We had already said that .
So, there you have it. Pick two numbers, swap them in for and and you get yourself a new Pythagorean triple!
Thanks again, difference of squares.