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?
Polynomial factoring — as those of us steeped in high school algebra know — is the art of “unmultiplying” an algebraic expression. One of these tricks for unmultiplying an expression is the difference of squares identity. My favorite uses of it involve arithmetic:
In school math, however, the difference of squares is typically used in the context of algebraic factoring exercises:
And children are often asked to commit to memory the general form of this rule:
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. How could factoring be useless if it played such a large role in 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 has been the best of the bunch for my needs.
“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. Could Mesopotamian scribes show us how to teach factoring? 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 under Nazi rule. (True, to the Nazis he made a point of his “full-blooded Arianness”. In correspondence, though, he was disposed against the regime. He’s clearly guilty of cowardice and self-interest, but it’s hard to know quite how harshly to judge the past.)
He wrote the first comprehensive textbook on modern algebra, and later 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. Sometimes he reported finding modern theorems lurking in the work of the ancients. These included various identities that today we would teach as factoring, including the difference of squares.
Our knowledge of Mesopotamian mathematics comes from clay tablets found in Iraq. Some of the tablets (like Plimpton 322) contain calculation tables, while others are collections of word problems with solutions. Intriguingly, we think most of these documents are pedagogical artifacts, either used for instruction or practice. (Some of them have errors!)
Here’s a “real-world” problem from a clay tablet called MS 5112:
“The field and 2 ‘equal-sides’ heaped [added together] give 120. The field and the same side are what?”
This is equivalent to the modern-day equation . Van der Waerden’s claim about the difference of squares formula — that the Mesopotamians knew and used it — largely depends on how they solved problems such as those found on MS 5112.
Modern algebra students learn how to use the difference of squares to solve equations, but not for equations like Modern students would only use the difference of squares when the equation is explicitly presented as a difference of squares, e.g. or . These ancient sources are using the difference of squares transformation as their go-to move for solving quadratic equations.
When presented with a problem such as , the Mesopotamians would typically transform the expression into a difference of two squares.
Pictorially, the right chunk of this rectangle — the — is cut in half down the middle…
…and pasted at the bottom of the left chunk, creating a difference-of-squares arrangement:
This was the fundamental step in their solution of a quadratic equation.
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, and we’ve done so by seeing as a difference of squares.
To me, this is a surprising connection. I’ve known about this method for solving equations for years, but have never seen it through the lens of the difference of squares identity. Factoring may seem frivolous, but van der Waerden argues that it was a central part of how Mesopotamians did mathematics.
All the above — the “real world” word problem and its solution — comes to us in the language of geometry: fields, squares, lengths, areas. Van der Waerden, of course, noted this:
From the very beginning, algebra has always been closely connected with geometry. In Babylonian problem texts, the unknown quantities are very often called “length” and “width”, and their product “area”. The product of a number by itself is called “square”, the number itself “side” (of the square).
For van der Waerden, this is all besides the point; it’s just a geometric sheen over an algebraic essence:
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.
He also writes that “in ancient civilizations geometry and algebra cannot well be separated,” but that is because algebra was being performed in a thoroughly geometric context. Modern students may use symbols and ancient ones used shapes, but all are doing algebra.
These days, most historians of math do not agree with this picture — they see the Mesopotamian work as essentially geometric, not algebraic. True, it was algorithmic — there was a definite procedure that was repeatedly used — but what the Mesopotamians passed on were methods for manipulating areas and lengths, not numbers.
The current perspective is the result of historians taking a fuller view of the ancient world than that taken by the earlier generation of researchers. Current historians know a lot about the Mesopotamians: about their geography, culture, society, economy, etc. The first generation of historians of Mesopotamian mathematics, in contrast, were mainly mathematicians-turned-historians who had narrower interests — people like good-old Nazi-tolerating van der Waerden.
Mathematicians tend to see math as a set of truths universally held and recognized. (Carl Gauss may or may not have suggested communicating with aliens by etching an enormous Pythagorean Theorem diagram into the Siberian tundra, but they don’t tell stories like that about chemists.) It’s only natural that when mathematicians turned to the past (another alien world) they would see algebraic continuity, not difference.
Current historians see the difference, though. Through a better understanding of Mesopotamian language they have arrived at translations that attempt to better represent the mathematics as it was, not as it is. What an early mathematician-historian translated as “coefficient” is now translated as “projection,” a subtle change with important implications: “When expressed in these very concrete terms, Old Babylonian algebra becomes not arithmetical but geometrical and metric: concerned not with abstract numbers but with measured lines, areas, and volumes,” Eleanor Robson writes.
It’s exciting to look at the past and seek insight into modern teaching dilemmas. But, if their mathematics was fundamentally different from our’s, is this project even possible?
There is another instance of factoring the difference of squares appearing in discussions of ancient mathematics. It involves a connection between the Pythagorean Theorem and the difference of squares. Here too, the connection was made by an earlier generation of scholars and has more recently been challenged by contemporary historians.
Like van der Waerden, Otto Neugebauer also began his career as a mathematician in Germany. When the Nazis asked him to sign a loyalty oath, though, he refused and was suspended from work. He continued on in Germany until 1939, when the Nazis took over his mathematical journal and he made his way to the United States.
Neugebauer is especially known for his work with Mesopotamian clay tablets. More than any other scholar, he was responsible for uncovering mathematics in these ancient records.
Plimpton 322 is a clay tablet containing a carefully organized table of numbers:
At first, nobody thought Plimpton 322 was special. But Otto Neugebauer took another look at the table and announced that this was actually a mathematical treasure: a Babylonian record of Pythagorean triples (i.e. whole numbers that could be sides of a right triangle, like 3/4/5 or 5/12/13).
How did these ancient mathematicians produce this table? This is where, for Neugebauer, factoring the difference of squares comes in.
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:
It’s not obvious that both and both must be square numbers, but they do. 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 .
So, there you have it. Pick two numbers, swap them in for and and you get yourself values for and (you can get too) and you have an triple. Tada: the ancient Mesopotamian method for finding Pythagorean triples!
Once again, though, this historical connection has been questioned. Eleanor Robson wrote a fantastic article challenging Neugebauer’s 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.
Part of the problem, again, is that Neugebauer’s idea is intensely algebraic, whereas in ancient Iraq the mathematics was chiefly geometric. Part of the problem is also that Neugebauer didn’t know what these sorts of tables were typically used for in Mesopotamia, so he misunderstood their cultural use.
Whether or not it reflects history, the mathematics here is solid. The Pythagorean Theorem is connected to factoring a difference of squares, just as the factoring connects to solving .
The historical question is whether this mathematics would have been meaningful to the ancients. The pedagogical question is whether it could be meaningful to our students.
So: can the studying the past help us better teach factoring?
It’s tempting to cull specific ideas from this history. The connection of factoring the difference of squares to solving quadratics and the Pythagorean Theorem are still knocking around my head. I don’t know if there’s a way to bring these connections to my students, and I also don’t know if they’d enjoy them as much as I do. I don’t know yet — I’m going to have to think on this for a while more.
I’m wondering, though, if there’s maybe a more general lesson about teaching algebra to take from all this.
The mistake of the early mathematician-historians was to see too much of algebra in the cut-and-paste geometry of the Mesopotamians. 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 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, makes me a little bit 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 this history of algebra could be that we should worry very, very worried about whether these more comprehensible contexts are really aids for understanding the difficult things.
Each context is its own little world, and the sense that we can make of it is not easily bridged to some other area. In particular, there is nothing simple about moving from geometry to algebra.