Writing is allowed to be hard

What makes this post weird, for me, is it started with having something to say. Lately, this is not how I write. Here is the origin story of my last several posts:

And so on. Now, I don’t want to be facetious. It’s not like I start these projects without any thought about what I’m going to say. Usually it’s sort of a nascent take. It’s often extremely tentative: maybe I’ll end up saying…

The point isn’t that I go in to a piece of writing without anything in mind. The point is that all these recent posts have required active development. Through a combination of research, drafting and editing, I figure out what the post is about well after I decide to write it.

I mention all of this because I’ve been talking to people recently about why they stopped (or never started) blogging. Before you misunderstand my purpose, there’s nothing wrong with not blogging. Seriously: do whatever you want. I never want to be the guy to criticize someone for not doing something. As long as nobody’s getting hurt, don’t-do to your heart’s content.

Here’s the thing. A lot of people were telling me that they don’t blog because they don’t have ideas, or because they’ve already said what they want to say, or they don’t have the time, and so on and so on. These are all entirely legitimate reasons not to write — along with the very best reason, which is “I don’t feel like it.”

I worry, though, that in the online math teacher community (mtbos) the dominant, default view about writing is that it’s supposed to be easy. The expectation in our community is that writing about teaching is most appropriate as either an organic expression of your views or as a casual, nearly-personal record of your professional practice.

Now: this isn’t such a big deal! There is no crisis in the mathtwitterblogosphere — the community is growing, and pretty much everyone is having a fun, meaningful time. I certainly don’t see myself as a dork Cassandra.

(OK fine, just a bit.)

Here’s what I think might’ve happened. Blogging was a fantastic medium on which to build a math education community. The community’s initial growth was enabled by a particularly flexible type of writing — relatively quick posts that shared a brief, relatively unsexy thing about teaching. This wasn’t the only way to blog, but it was a fantastic, accessible genre for teachers who were new to the community. It was easy to dive in, and a lot of generous engagement resulted while knowledge and resources accumulated.

Along with this success, the community developed a series of (totally reasonable and beneficial!) norms around accessibility. Blogging doesn’t need to be anything fancy, and you don’t even need to worry about a reader — write for yourself, and if other people find it helpful? Hey, that’s a bonus.

People are justifiably sensitive about this point so let me say it again: I am not critiquing this view on blogging, or even its prominence in the blog-o’-land. It’s a message that maximizes accessibility, and that is probably the most important value of our community.

I think that now might be an especially good time to remind people that there’s another way to write in this community, which is to slowly, painstakingly, dutifully carve out posts. And — thinking entirely personally here — it’s just so, so much fun to write like that. You should try it! Taking writing seriously is a hoot.

Let’s get the costs out of the way: I spend a ton of my free time reading and writing. Call it whatever you want — hobby, avocation, craft — but it’s time-consuming. It’s also sometimes unnatural, in the sense that I have to search for something to say, and I need to figure out how to say it. (I still fire off a quick sharing post from time to time, but I’m drifting away from it.) And, because I work hard on this stuff, I sometimes get frustrated when my work is ignored or when I see myself as having failed.

So much for costs. The benefits: seriously, it’s a blast. I learn so much more from crafting a piece than from a post like this one, where I’m sort of just yapping. And, if the past is any indication, I’ll probably be a bit disappointed with the response to this post. Some folks will like it, others won’t, and that’ll be that. My longer, more complex pieces, though, have generated incredibly meaningful responses. I’m blown away by the comments people have left on these posts, and my email correspondence has been rich as well. And that’s all I’ve ever really wanted from this blogging thing — to get to write and to have it mean something real to my peers.

(It’d be nice to have writing in legit publications so my parents could have something to talk about, but that would just be a cherry on top of my current situation.)

What I’ve found, after a lot of stumbling and searching, is that an especially fruitful genre for me is review. Some of the most fun I’ve had writing (generating the most exciting responses) has come when I read a difficult book or article as best I can and try to make sense of it in writing.

I would love to read more complex, critical writing about reading, especially from teachers: won’t someone humor me?

Another type of post that I’m finding especially fun is the research/practice post. I find it a tricky balance. You need to tell two stories at once, taking care to weave them together without sublimating experience to research or dismissing serious findings. This type of piece also gives me that awesome feeling I had when I started blogging and people were still sharing the unsexy things — the feeling that, potentially, any classroom moment could be transformed into a post and thereby be significant beyond the moment itself.

This is another type of post that, while I suppose anyone in education could write it, is especially interesting to me coming from people in classrooms.

There’s a third type of post that I’ve been trying to figure out how to handle. I really want to get better at writing straight math. I want to learn how to apply what I know about teaching to the sort of content that I’m interested in learning about. And I’m also interested in using writing as an engine and discipline for learning new mathematics. My experience with the history of algebra essay was totally energizing; I’m ready for more.

But I’m also eager to read more writing about mathematics from the people who know the most about helping other people make sense of it. It’s a type of writing that is particularly apt for teachers to do, and yet I don’t see much of it.

These three areas — the review essay, the research/practice post, straight math — are some of my favorite types of writing to read, and I am especially interested in reading them from teacher-writers. My purpose here isn’t to nay-say what anyone else is doing. I just want to share how much fun, how rewarding it’s been to explore these areas in my own writing, and to try to entice someone else to start down a similar path.

These kinds of writing will always be hard and time-consuming. But so is making incredible math videos or putting together a presentation. I think there’s a community of writers out there in mtbos interested in playing around with writing, but I don’t think it’s come together quite yet. And maybe there are some people that are looking for a way in on blogging, but haven’t figured out how to make it click yet.

My message, then, is that writing is allowed to be easy, but it doesn’t need to be. Writing can be an effortful process that ends, but doesn’t start, with having something to say. It can involve research, months of planning, asking friends for editing and revising, revising, revising. And, when everything clicks, this sort of writing yields rewards different in kind to the rewards for the more common modes of blogging.

Blogging can be very, very hard but so much more fun.

What I’ve Learned About Practicing Multiplication Facts

I.

During this past school year, I started practicing math facts in a new way with my 3rd and 4th Graders. The name I came up with for the routine was “Forwards and Backwards Practice.”

Like all my classroom ideas, it was lazy and simple. I handed a piece of blank paper to each kid. I told everyone that we’d be doing an activity in two rounds, that they should write “Round 1” at the top of their papers. Then I wrote the “forwards” and “backwards” problems on the board.

The “forwards” problems were pretty familiar to my kids. Solve the equation; put a number in the blank to make the equation true:

4 x ___ = 28, 8 x 4 = ___, ____ x 7 = 42

The “backwards” questions were more open-ended. On the board, I simply wrote three numbers:

21; 42; 81

I explained that for these I wanted the kids to write as many multiplications as they could remember that equaled each number. Accurate “backwards” answers for 21 would be 3 x 7, 1 x 21, etc.

As kids were wrapping these questions up, I called attention back to the board. If there was a common mistake, this is when I mentioned it. I shared accurate answers to each question, emphasizing what I wanted to emphasize.

Then, I erased the board. I told kids that there would be a second round of questions in a minute that were very closely related. Take a minute, I said, and study the multiplication we just reviewed. Try to remember as much of these as you can. When a minute is up, you’ll flip your page over to the blank side for Round 2.

Here’s what I did, basically: I swapped the forwards and the backwards questions. The backwards questions were now forwards equations, and the forwards were now put in backwards form.

That means that the corrections and practice that the kids got in Round 1 are relevant for Round 2. If a kid is just starting to practice 7 x 3, then they get a chance to study it and try to remember it for a problem that is coming right up, moments after they study.

That’s why I like this routine. It packs a pretty virtuous cycle into a fairly quick package:

• Get some explicit instruction in response to your work
• Study
• Try to remember it

One thing I like about this routine is that it solves a problem I was having with other whole-group practice, which was some kids were finishing my practice much before others. I didn’t want to end the activity, but the quick finishers needed something to do. Backwards practice is something that sort of “naturally” differentiates. It’s end-goal is vague; kids interpret it according to their understanding of multiplication, so they each student tends to find appropriate math to work on, and my speed-demons don’t force me to call a quick end to the activity.

Depending on the group and their confidence, knowledge, etc., I might vary how closely the questions in Round 2 resemble the questions in Round 1. If kids are really at the beginning of their learning of multiplication, the Round 2 questions might very closely resemble the ones in Round 1. Or we might keep them all “forwards” practice, just knock out different numbers in the equation. Or change the direction (i.e. from 3 x 7 to 7 x 3).

II.

I also use flashcards with my 3rd and 4th Graders.

When I first introduced them, I was very, very nervous. I tend to worry about the most anxious kids in my class, and I had two nervous wrecks in my 3rd Grade group. (One was receiving medical care for his stress.) How would they react to all this? They were already shutting down when I gave out worksheets. Flash cards would only be worse.

So, I introduce flashcards. I get each kid a little plastic decks and I get a ton of colorful index cards. (It turns out that you need both of these things to make this work, because otherwise kids lose their cards or mix them up with other decks. I tried to pull this off with envelopes and white flash cards last year and it was a total disaster.)

We slowly start filling out cards with multiplication (and addition) problems. I ask kids to practice, and I explain what practicing means, and I tell them what good practice looks like. (“None of this stuff where you’re shouting out answers while someone else is thinking. We don’t want to take away someone else’s chance to think.”) And then I give them a good chunk of time to start practicing.

Things looked good in class, but you never know for sure, so I asked kids at the end of class to write a bit about how they liked practicing math with their decks. I’m very interested in what one of my high-anxiety kids thinks, so I grab him at the end. What did you write, O? What were you thinking?

What he tells me is really interesting. He says that he really prefer the cards because they only show one problem at a time. When he sees a page with a ton of problems on it he gets overwhelmed, distracted, stressed out. But cards are significantly less stress for him.

The year goes on. There are a few groups that are getting a bit competitive when they practice, which I come to think is fine as long as I keep an eye on it. I do maintenance on their practice: be nice; you can write another problem as a “starter problem”; make sure everyone you’re practicing with has a chance to answer; you can do this by yourself; throw out a few cards that are too easy. I ask questions: are the cards too easy? are they too hard?

Are you enjoying this practice? I ask that often, because I’m sort of surprised by how much they’re enjoying themselves. But they are, really.

Flashcards are just great for practice. The answer is right there — if you get it wrong you get correction and a nudge in the right direction. (Math facts is the sort of thing that it really does help to get quick corrections on.)

There are other benefits too. Like O said, you only see one problem at a time. You can go fast, you can go slow. You can turn the cards over and do “backwards practice.” You can take the deck home and practice by yourself. You can quickly take it out if you finish an activity quickly — it can go on the menu.

One challenge I’ve had with flashcards is that some kids persist in using really inefficient strategies when practicing with their decks. This is because they are basically choosing how long to spend on each card in their decks. This is attenuated somewhat by kids practicing together but it’s something that I had to keep an eye on while they were practicing.

III.

When I wanted a bit more control over which fact families my students practiced, I used dice games:

It’s another dumb, easy thing. The only problem here is that there are no corrections when kids are practicing. I had my students write down their results for this sort of practice, but I often couldn’t catch mistakes quickly enough to be useful for their practice.

IV.

I want to help my students commit as many multiplication facts as they can to memory. I don’t want to feteshize math fact automaticity — some kids do OK without this knowledge — but it’s really useful knowledge for learning more math. Why wouldn’t I try to help my kids commit their math facts to memory?

What’s the best way to do this? Well, you need a theory as to how kids come to commit facts to memory. As I’ve written about before, my perspective is you learn what you practice. If you want to remember facts, you have to practice remembering them. And if you don’t practice remembering them — if you only ever practicing skip-counting to derive them — you’ll probably never come to memorize them.

This helps me navigate the world of multiplication practice, where controversy abounds.

Take, for example, speed practice. Daniel Willingham and Daniel Ansari recently wrote a post defending speed practice. I left a comment arguing that we needed to know why speed can help kids in their practice before we defend it:

In one study I read (about fluency software) I learned that students with learning disability did not improve their addition fluency through untimed practice. Why? Because during untimed practice, the students simply DERIVED the facts rather than trying to RECALL them. In other words, you’d see a lot of kids in front of screen counting out 3 + 9 with their fingers instead of trying to recall them from memory. The kids were already pretty good at using this strategy, and the untimed practice allowed them to keep doing what they were good at.

I see this in my own students too. It’s not so much that timed practice is helpful for learning directly, as much as it creates a context in which kids practice the things you’d like them to practice.

A solution is timed practice with immediate fact instruction. (You got 3 + 9 wrong? OK, 3 + 9 = 12. Try again.)

[…]
The worst case scenario is that teachers give kids a full worksheet of problems, and kids can’t directly recall ANY of them. Instead, kids work on using strategies to derive the facts. The teacher says to solve as many as you can, but the students can only correctly answer that many questions using direct recall — with strategies, there’s not enough time. Time pressure (along with the long list of problems) generates anxiety, which makes it harder still to answer problems correctly. None of this produces fact fluency.

Based on talking to colleagues and other math educators, this worst case scenario is in fact prevalent in US classrooms. These “Mad Minute” activities could be used appropriately, but they are instead often given to novices who are not prepared to draw on their mostly memorized facts for the activity. And, I think, this probably does generate feelings of helplessness and anxiety.

As a result of all this, when I think about fact practice I end up asking myself this question all the time: Will the kids be practicing derivation or recall?

And here’s a fundamental follow-up: Kids can’t practice recall unless they are being prompted with the correct answers during the practice.

I really don’t like Mad Minute activities because they don’t prompt you with corrections or instruction in the fact during the activity. So you can’t really learn anything from the activity unless you’re “almost there.” Maybe it helps you practice pulling out the fact from memory, but it can’t help you learn that fact with automaticity without some sort of prompting during practice.

That’s why I like splitting up practice into two rounds, as I do during “forwards/backwards” practice. I get to give prompting/instruction in between the rounds, and then kids get a chance to practice with it during Round 2.

It’s also why I like practice with flashcards, especially if kids are reminded to try to figure out the answer as quickly as they can. (They basically do this anyway.) While I worried that this would be stressful for my kids, I’ve actually found the opposite. Flashcards, the way I use them in class, tend to be less stressful than other conventional practice activities (like long problem sets).

The absence of prompting/corrections is a downside of my dice practice, though it’s attenuated somewhat by the way the problems will reappear as kids cycle through the different boxes and repeat factors. Still, it’s a form of practice that probably would be better at helping kids have a chance to practice strategies rather than remembering.

I think it’s important to be thoughtful here. Math facts aren’t the be-all of school math, but they do make a difference for kids’ future learning.

The fundamental disagreement I have with a lot of people in math education is that I don’t think that practice using a strategy helps kids commit facts to memory. (Though I do believe that having efficient strategies does help kids commit facts to memory. Both knowing efficient strategies and recall practice are important for developing automaticity. I have citations for this. See also the Willingham/Ansari piece.)

And my fundamental displeasure about the debate is how rarely it gets into the classroom details. So, you’ve got a position on how multiplication should be taught? Does it fit on a slide? Do people take pictures of it with their phones during conferences? Tweet it, retweet it, like it?

That’s great, seriously, but let’s talk the nitty gritty. What are your activities? What does your class look like? What is it that you do?

When measures of steepness disagree

I.

My students know a lot more about skiing than I do. I grew up in Skokie, IL — an exceptionally flat place, we went sledding down at a pile of garbage called ‘Mount Trashmore’ in Evanston — but a lot of my students go on vacations to resorts and stuff in the winter.

Once or twice, a Jewish youth group took me to Wisconsin to ski. Wisconsin sort of has hills. A midwestern ski resort is the sort of place where you can choose whether to slide down a hill on skis or an inflatable tube. It is also home to the tamest “Black Diamond” slopes in the country — colder but otherwise not much different than the slides my son plays on at the park.

Anyway, that’s what I know about skiing. Glad to get that off my chest.

II.

Towards the beginning of my trigonometry unit — after studying the tangent ratio for several days — I showed this picture to my geometry class. In whole-group, I asked my students to notice as much as they could, and after that I asked the class to try to figure out what all the numbers represented:

When I teach trigonometry, one of my first goals is to help kids see that angles and the tangent ratio both are measures of steepness. Trigonometry is the art of moving between these two different measures. With a trig table or a calculator you can take an angle and look up its associated ratios, and you can look things up the other way (ratios to angles) too. This is true for all the trig functions, and my students encounter it first in the context of the height-to-width ratio.

If you’re trying to describe the steepness of a ski slope — again, not a major concern growing up in Skokie — you could talk about the height:width ratio, or you could talk about the angle of inclination. That chart above rates the difficulty of ski slopes in terms of the angle, but it just as well could have done it in terms of ratios. (I asked my students to draw slopes with heights and widths in each zone.)

The American with Disabilities Act describes the appropriate steepness of a ramp in terms of both measures:

ADA Ramp Specifications Require a 1:12 ramp slope ratio which equals 4.8 degrees slope or one foot of wheelchair ramp for each inch of rise. For instance, a 30 inch rise requires a 30 foot handicap wheelchair ramp.

Every ramp, hill, slide or mountain has a steepness. To bring that physical concept into the realm of mathematics, we have to measure it. But there are many ways to measure steepness, and often we want to be able to move between them. That’s a big part of what trigonometry is.

III.

Before really launching into the trig unit, I task kids with a series of “Which Is Steeper?” problems.

Along with everything else, these problems also really help kids use the height:width ratio as a measure of steepness.

What I’m looking for is for kids to fluently use three little micro-skills:

• when two ramps are the same height (or the same width), the ramp with less width (or more height) will be steeper
• when heights are different, scale one ramp until its dimensions match the other’s, and then directly compare the other dimension
• in general, compare the steepness of two ramps by dividing the height by the width and comparing the ratios

The way I see it, these micro-skills are important background knowledge to support the procedures for finding missing sides of triangles using trig — especially if you come into this work without a lot of comfort with ratios and setting up and solving equations like $\frac{5}{x} = \frac{17}{19}$.

(I do a lot to help kids with ratios, but I don’t usually focus on setting up and solving the equations. Maybe I should.)

IV.

Once I think my kids are getting comfortable using the height:width ratios to find missing sides of right triangles, I show them the physical trig table. There is so much for kids to learn from the trig table — I think it’s a shame when students move straight to looking up values on the calculator.

The most amazing thing about the trig table — at least it’s my favorite thing, and kids often get excited by it — is what happens as we approach 90 degrees. The sine and cosine functions change a bit, of course, but the tangent values just explode:

Kids often are surprised by this, but it makes a lot of sense. Adding another degree of steepness always makes the height:width ratio larger, but not always by the same amount. If your ski slope is very, very flat, then going up by a degree doesn’t increase the ratio very much. If your slope is a double black diamond, though, upping the steepness by a degree leads to a radical change in the ratio, a huge increase.

I always try to use this as an opportunity to introduce some important language to my students: the relationship between steepness ratios and angles is non-linear; a small change in the angle doesn’t always have the same impact on the ratio’s size.

When I think of multiple ways of measuring things, I usually think of pairs of measures that stand in a linear relationship. The nurses measure my newborn daughter’s weight in terms of grams and pounds. When you lose a pound of weight you’re losing 453.92 grams — always. It doesn’t matter how much or how little you weigh. A pound and 453.92 grams are simply interchangeable.

But a lot of pairs of things in the world vary in non-linear ways. In a sense, an additional year of investment is worth more in the future than it’s worth now; a falling ball drops faster as time goes on. I don’t know how many opportunities there are to study this in terms of measurements, but it seems a fruitful arena for chipping away at the assumption that everything is linear.

V.

And, now, we get to the question that has been bugging me for the last few months: How much steeper is an 89 degree ramp than an 88 degree one? A lot or a little?

Remember: whether with ski-slopes or with ramps, there are two ways to measure steepness. You can measure it in terms of the angle or in terms of the ratio.

From the point of view of angles, the 89 degree ramp is just as different from the 88 degree ramp as a 21 degree ramp is from a 20. Which is to say, a bit steeper.

But look at the ratios! Maybe we should think in terms of height:width, in which case the 89 degree ramp is much steeper than the 88 degree ramp, especially compared to what happens when you add a degree of steepness lower down the trig chart.

One way out of this conundrum would be to assert that one of the measures of steepness is the actual, true measure of steepness. But any choice seems arbitrary. Both angles and ratios seem perfectly fine. Why choose one over the other?

(Maybe we’d try to further plant things on a human foundation; how much more effort would it take to climb up each of these ramps? Let’s run experiments that measure physical exertion; maybe we could use physics to model this. Steepness would just then be an expression of human exertion. This is a weird idea.)

Another way out could be to deny that there is any single thing that we’re measuring at all. Maybe steepness isn’t one single thing — it has an angle dimension and a ratio dimension. But what does that mean?

I really have no idea what to think. As we near 90 degrees it seems that the two measures of steepness disagree on how much of a difference a small change makes. Which means that we’re measuring the same quantity (steepness) with tools that are fundamentally incompatible.

What does it mean for two measures to be incompatible? What other measures are like this?

In trying to sort this all out — and I hope it’s clear that I’m awfully confused — I’ve been also thinking about something Freddie deBoer wrote about educational testing:

Incidentally, it’s a very common feature of various types of educational, intelligence, and language testing that scores become less meaningful as the move towards the extremes. That is, a 10 point difference on a well-validated IQ test means a lot when it comes to the difference between a 95 and a 105, but it means much less when it comes to a difference between 25 and 35 or 165 and 175. Why? In part because outliers are rare, by their nature, which means we have less data to validate that range of our scale.

Could that help us think about what’s going on with steepness? Clearly there is no such validation problem when it comes to the steepness of right triangles — we can always draw more! — but maybe there is something analogous going on. We might say: it just doesn’t mean very much to get precise about how steep a very steep ski slope is. Numbers break down, our measures of steepness fall apart, and all we can say about very steep things is just the tautological thing — they’re pretty damn steep.

That is, there just is no way to precisely talk about the steepness of a very steep ramp, as the measures disagree.

But that seems weird too, and I’m lead to the conclusion that I don’t understand this very well at all.

High School Algebra in Ancient Mesopotamia

I.

On an online forum for discussing math, a user named Mr. Javascript  (his bio: “If you’ve ever gone to the doctor, purchased insurance, or used a credit card, my code may have been executed.”) took a swing at polynomial factoring:

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:

$25 - 4 \rightarrow (5 + 2)(5 - 2)$

$100 - 1 \rightarrow (10 + 1)(10 - 1)$

$400-9 \rightarrow (20 + 3)(20 - 3)$

In school math, however, the difference of squares is typically used in the context of algebraic factoring exercises:

$x^2 - 9 \leftrightarrow (x + 3)(x - 3)$

$a^2 x^2 - 9b^2 \leftrightarrow (ax + 3b)(ax - 3b)$

$\frac{a^2 x^2}{100} - \frac{9b^2}{121} \leftrightarrow (\frac{ax}{10} + \frac{3b}{11})(\frac{ax}{10} - \frac{3b}{11})$

And children are often asked to commit to memory the general form of this rule:

$a^2 - b^2 = (a + b)(a - b)$

It’s these algebraic factoring exercises that frustrate people like Mr. and Mrs. Javascript.

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 CivilizationsA 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?

II.

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 $x^2 + 2x = 120$. 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 $x^2 + 2x = 120.$ Modern students would only use the difference of squares when the equation is explicitly presented as a difference of squares, e.g. $x^2 - 9 = 0$ or $100 - 4x^2 = 0$. 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 $x^2 + bx = c$, the Mesopotamians would typically transform the $x^2 + bx$ expression into a difference of two squares.

Pictorially, the right chunk of this rectangle — the $bx$ — 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 $\frac{b^2}{4}+c$; the side length is $\sqrt{\frac{b^2}{4}+c}$; the missing length, $x$ is  $\sqrt{\frac{b^2}{4}+c} - \frac{b}{2}$. We have just come very, very close to deriving the quadratic formula, and we’ve done so by seeing $x(b + x)$ 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.

III.

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?

IV.

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:

$A^2 + B^2 = C^2$

But it’s equally true that the Pythagorean Theorem is saying something about a difference of squares:

$A^2 = C^2 - B^2$

Which means that you could just as well put it like this:

$A^2 = (C + B)(C - B)$

It’s not obvious that both $(C + B)$ and $(C-B)$ both must be square numbers, but they do. Call the first square number $s^2$ and the second $t^2$. Which means that the following two equations are true:

$C + B = s^2$

$C - B = t^2$

Add those two equations together, and you get a new one.

$2C = s^2 + t^2$

Subtract them, and you get an equation for $b$.

$2B = s^2 - t^2$

So, there you have it. Pick two numbers, swap them in for $s$ and $t$ and you get yourself values for $b$ and $c$ (you can get $a$ too) and you have an $A^2 + B^2 = C^2$ 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 $x^2 + bx = c$.

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.

V.

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.