# A typically wishy-washy take on discovery in math class

I think of myself as a non-discovery math sort of teacher, but every once in a while I find myself asking kids to discover stuff. I recently did this in my geometry classes, with a dot-paper area activity.

I really like dot paper. On dot paper you can make precise statements about area that typically generalize nicely to non-dot scenarios. The same basic relationships apply, it’s just easier to see them in a dot world.

My idea was to share this picture with students, and ask them to find the area of each shape. (This is the version that I marked up under the camera in class. None of the purple or blue ink was in the original.)

All of these shapes have the same height — 4 — but they otherwise differ. The first is a rectangle that has area 8. Next was a parallelogram: also area 8. Then a trapezoid. Many students came up with the idea of drawing a line to split it into a triangle and a rectangle — together they have area 8. This didn’t occur to every kid, though, so there was a good deal of neighbor-asking and chat to get the hang of those trapezoids.

Spoilers: every shape here has area 8! There’s something else they share too: the sum of their top and bottom bases is always 4. (The weirdest case is the triangle, that has a bottom of base of 4 and [arguably] a top base of length 0!)

So that was what I wanted my kids to come to notice and articulate. And I wanted it to be fun, and feel like they were discovering something new.

Not because I think that learning is more effective when kids discover something new, or that they’re working on their discovering skills or anything. Just because I think it’s fun for them to uncover patterns. It’s a cool part of math, and I’m trying hard to share more of the cool parts of math with my students, along with working on their skills and knowledge etc.

Here then is my take on discovery in math class:

• It can be fun to discover cool stuff about math.
• It takes longer for kids to understand something via discovering, and doesn’t really confer a learning advantage.
• But if the activity is relatively brief and I can spare the time, why not? I want my students to think math is fun and cool.

This is my unprincipled take. I like discovery for fun and color in class, and I don’t feel the need to aim for 100% efficiency in every second of my teaching.*

Whether you feel such a need — or need to feel such a need — probably varies a lot depending on your school, administration, students, etc. My guess, though, is that the vast majority of teachers don’t feel this need, and probably are correct in this feeling.

Lots of discovery activities are uninteresting to me. Though I absolutely love the practice exercises in Discovering Geometry, the discovery activities largely leave me cold. Here’s an example of the sort of thing I’ve fallen out of love with:

My issue with this activity isn’t that it’s discovery. It’s that it’s not such a cool pattern (most kids have seen it before taking a geometry class), so discovering it isn’t as surprising or fun. The activity takes a while — do it once, check, do it twice — and all that is probably to protect against the risk of incorrect measurement, which is another tedious aspect of this discovery activity. And, at the end of all this, what cool math have you uncovered? Relatively little — just a sum. You don’t see any new relationships or geometric structure that guarantees that the triangle will have such a sum.

Why bother with all this? I’ve decided that this sort of discovery activity isn’t much use for me. But that’s not a principled objection against discovery — it’s just that I don’t think this type of activity is worth it.

Speaking of “types of activity,” I think it’s fair to categorize this angle sum activity as discovering something easy to articulate. Check an example, check an example, check an example, woah it’s always the same. You don’t uncover the geometric relationship in this activity. I think that’s part of what makes it not much fun and sort of tedious.

I think discovering something hard to articulate (I’m not quite sure what to name this) tends to be more fun, more cool.

To illustrate this, here’s the conversation that went along with my “area 8” activity in class.

I began the conversation with a prompt to my students: what do all these shapes share? how do these shapes differ?

I called on James first.

James: They all have heights and bases of 4.

I said that I didn’t entirely get that, and asked if anyone also saw that. Robin came up to the board to point. She also subtly refined Jame’s claim.

Robin: A lot of them have a common theme that either the height is 4 or the base of 4.

Then Liam chimed in to make it precise and accurate:

Liam: The height is always 4. The bases are different though.

Luiz: Yeah, the bases are either 2 or 4…or no they vary. Sometimes it’s 1, 2, sometimes it’s none.

Then Sara chimes in. She started articulating a generalization — she was WORKING HARD to try to articulate some sort of generalization. Her first one wasn’t entirely accurate though. I loved how she put it as a question.

Sara: Wait, does that mean that any shape that has a height of 4 has an area of 8?

The class and I (and Sara) agreed: this just could not be true. (Luiz says: well you could have a base of 1000.) Jess tried to get clearer about what was special about all these shapes.

Jess: No this is just because they are all parallelograms and…wait the third one is sort of confusing. What is that?

[Insert a minute of discussion about which of these are trapezoids and which are parallelograms.]

After this, I decided that we wouldn’t be able to restart the conversation unless I summarized and took some notes. So I wrote some notes on the board.

This time, though, I decided to take a heavier hand to draw attention to something really cool. The kids hadn’t noticed it yet, and I wanted to make sure that they did.

I wrote the numbers that you saw in purple ink in my image. I wrote the top and bottom bases.

Sara: So couldn’t you say that it’s base 1 plus base 2, times the height and that’s the area. Like for shapes in general.

Very close! There are some gasps and agreements.

Samantha: So wait does that work for everything?

Sara: Yeah that’s what I’m asking.

Me: That’s a good question. I’m trying to find some dot paper.

Joe: Wait so does it?

That question just hangs there for a second. And here’s a choice I could make. I could act coy, refuse to answer, and insist that the thinking here come from the students.

But then you get this weird dynamic in class where kids never know if they’re getting a straight answer from the teacher or not. I don’t like that dynamic. I like it when kids ask questions about math, and I like that they can get a straight answer out of me. And would they spend more time thinking about this cool relationship if I answered that question, or if I refused to?

“The answer is yes, sort of.”

GASPS.

Sara: Does it have to have parallel lines? Does it have to be a trapezoid…wait does a parallelogram count as a trapezoid.

Good thinking, Sara!

While there’s thinking kind of just toppling out of Sara’s head, I’m searching for a blank piece of dot paper, because there’s something that I realize might help. I don’t want them to get too used to the area 8 case — that might lead to a false generalization, since Sara keeps on saying that it’s double the sum of the bases. (That’s true when the height is 4, but not when the height is something else.)

So I draw this:

We clocked in at about 10 minutes there. No question that this was not as effective as a worked example or something else more carefully designed for learning. But I wasn’t aiming for efficacy. I was aiming for those half-articulations, those gasps, that enthusiasm. And as long as I don’t come to worship those gasps and chase them exclusively, class will be a bit closer to being fun, cool.

# YouCubed is Sloppy About Research

I.

There’s a lot of sloppy talk of science that gets tossed around education. Every teacher knows this — or least, I hope they do — and I try not to get too worked up about it. I also try not to get worked up by people who wear backpacks on crowded subways. Not that it’s OK, but lots of people do it (the backpack thing) and picking any one person to bark at hardly seems like it would help.

The other thing is that people can be really passionate about sloppy science. This happens all the time, it’s nothing special about education. But passion makes it hard to talk critically about the research without it seeming like you’re attacking everything else that the person is passionate about.

I think a lot of the time it’s because we assume that the attack on the research isn’t really about the research, it’s about everything else it supports. It’s like, Why are you bothering to poke holes in [research that supports X]? You’d only do that if you were really against [X].

Which leads to an interesting question. Is it possible, at all, to avoid this trap? Is it possible to critique sloppy use of research without being heard as if you’re trashing a person, their organization, and everything they stand for?

II.

What if you ask a lot of rhetorical questions — does that help?

Anyway, let’s talk about YouCubed. I think there’s something that — if we slow down, turn off passion, turn on curiosity — we can agree is a serious mistake. Here’s a popular quote from their popular page, Mistakes Grow Your Brain:

When I have told teachers that mistakes cause your brain to spark and grow, they have said, “Surely this only happens if students correct their mistake and go on to solve the problem correctly.” But this is not the case. In fact, Moser’s study shows us that we don’t even have to be aware we have made a mistake for brain sparks to occur.

When teachers ask me how this can be possible, I tell them that the best thinking we have on this now is that the brain sparks and grows when we make a mistake, even if we are not aware of it, because it is a time of struggle; the brain is challenged and the challenge results in growth.

Indeed — this does sound really, really surprising! So, applied to math, it sounds like if you solve an equation incorrectly you learn from that experience (brain sparks?) in a way that you wouldn’t if you had solved it correctly. The mistake you made causes struggle — even if it’s not a struggle that you’re aware of. You might not feel as if you’re struggling, but your brain is on account of the mistake.

The citation of Moser is very clear, so, ok, let’s go upstream and check out Moser. Though a lot of research is behind paywalls, a lot of it isn’t and a quick Google Scholar search gives us a copy of the paper, here.

The study was an fMRI (EEG, see below. -MP) study. Here’s my understanding of the paper. The researchers wanted to know, on a neurological level, what’s different about how people with a growth mindset or a fixed mindset react to mistakes. So they gave everybody a test, to figure out if they had a fixed or a growth mindset. Then they hooked subjects up to fMRI EEG machines. While in the machines, participants worked on a task that yields a lot of mistakes. Researchers recorded their neural activity and then analyzed it, to see if there was a meaningful difference between the fixed and growth mindset groups, after making errors.

Spoilers: they found a difference.

Also: the whole paper is premised on people being aware of the errors that they made. This is not a subtle point, buried in analysis — the paper mentions this like nine times, including towards the end where they write, “one reason why a growth mind-set leads to an increased likelihood of learning from mistakes is enhanced on-line error awareness.”

This is so clearly different than what the YouCubed site says that I’m starting to doubt myself. The paper seems to be entirely about what happens when you realize you’ve made a mistake. Yet it’s cited as supporting the notion that you learn (“brain grows”) from mistakes, even if you aren’t aware of them.

Is there something that I’m missing here?

III.

This doesn’t seem to me like an isolated issue with YouCubed.

A while ago Yana Weinstein and I made a document together trying to collect errors in YouCubed materials, explain why, and suggest improvements. I don’t have much else to say about that, except that if you’re interested you might check it out here.

IV.

What strikes me about YouCubed is that the errors just seem so unnecessary. The message is a familiar one, and I’m OK with a lot of it: don’t obsess over speed, think about mindset, don’t be afraid of mistakes. But there’s this sloppy science that gets duct taped on to the message. What purpose does that serve?

There’s also the question of why so few people in the math education community talk about this. I mean, it’s not like we lack the critical capacity. Every so often I see people in math education whip out their skeptical tools to tear apart a piece of research. Why not with YouCubed?

I don’t want to be cynical, but I want to be truthful. The first reason, I think, is because the message of YouCubed (besides the science) is widely admired. A lot of teachers love it, and nobody wants to ruin a fun time.

But I don’t think we have to worry about that. We can talk about the science of YouCubed in a way that doesn’t entangle the rest of the YouCubed message.

The second reason is, I think, that YouCubed and Jo Boaler’s popularity makes it difficult for the most visible people in math education to seem critical — no one wants to turn on one of their own. Especially since Boaler has often been subject to unreasonable attacks in the past — nobody wants to be unfair, cruel or sexist to her.

I’m actually very sympathetic to that. But this is also why it’s important for people who aren’t part of the research or PD world to have platforms to discuss ideas. We don’t have the reputation or the connections to lose, and so we can take a closer look and ask, wait, does that really make sense?

Corrections:

Actual neuroscientist Daniel Ansari points out that I don’t know the difference between an EEG and an fMRI, which is true.

# NCTM Journals: Be Interesting, Not Useful

My department subscribes to Math Horizons, a journal “intended primarily for undergraduates interested in mathematics.” I really like it. I recently found an old issue around school, and was reminded how much fun it can be. Here are opening lines, pulled from articles in the issue:

• “The year 2014 is an especially good time to tell this tale of disguise, distance, disagreements, and diagonals.”
• “What made you decide to be a math major?”
• “Being in charge of a math club can be exhausting.”
• “Time to end it all, Ellen thought.”
• “More than 65 years ago, William Fitch Cheney Jr. conceived one of the greatest mathematical card tricks.”
• “I grew up around decks of cards.”

Following these openers, one can read interviews with mathematicians, longer pieces about the history of math, book reviews, mathematical exposition, and even fiction.

After rereading Math Horizons, I went searching around my apartment for an issue of an NCTM journal. I’ve subscribed to each of the three journals since first becoming a member, always hoping that the other journal would interest me more. I finally found the latest issue of smooshed in with a pile of other magazines.

Here are first sentences pulled from the September issue of TCM:

• “When you think of ‘modeling’ in the mathematics classroom, what comes to mind? With the inclusion of Model with mathematics as one of the Standards for Mathematical Practice (SMP), the Common Core (CCSSI 2010) puts forth a vision of modeling in the mathematics classroom that moves beyond using concrete materials or other visual representations to give meaning to mathematics.”
• “We recently conducted a randomized controlled trial that showed a significant impact of teachers’ lesson study, supported by mathematical resources, on both teachers’ and students’ understanding of fractions (Gersten et al. 2014; Lewis and Perry 2017).”

These are long. At risk of losing my own readers, I’ll include one last, even longer opening line:

• “I am always in pursuit of resources that will add to my knowledge as described in Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014), which posits how crucial it is for math educators to continue to “recognize that their own learning is never finished and continually seek to improve and enhance their mathematical knowledge for teaching, their knowledge of mathematical pedagogy, and their knowledge of students and learners of mathematics. (p. 99)”

In response to my frequent kvetching about the journals, an NCTM board member emailed me. He asked, “What would you like to see in the journal?”

Fair enough! I would like NCTM to publish interesting articles.

Nobody sets out to publish boring articles, of course. But I have reason to think that “is this interesting?” is not being asked nearly enough at the NCTM journals right now.

For instance: I recently completed a twenty-two question survey about the NCTM journals. Four of the questions asked me about what I found useful. What sort of articles do I find the most useful? The least? Which departments are useful or not to me?

To be fair, one question asked, “Would you be interested in reading articles about…? (check all that apply).” That makes a four-parts usefulness to one-part interestingness ratio, which sounds about right for what NCTM is currently putting out. Invert the ratio, and I don’t think the above quotes make the cut any longer.

The other thing about interest vs. usefulness is something Henri Picciotto calls “the seemingly obligatory genuflection at NCTM’s sacred texts, most recently Principles to Action.” He means the way so many of the pieces published include the line “…as demanded by the Standards for Mathematical Practice,” or “…as detailed in Principles to Actions.” And, in fact, all three editorial teams officially require articles to show consistency with Principles to Actions:

It’s simply hard to tell a good, interesting story about teaching while also projecting your adherence to a set of teaching standards. As a writer, you start losing options. One of the sturdiest formats I’ve found for writing about teaching is narrating learning. You develop some question, and then you take the reader along in your attempt to answer it. It is immeasurably harder to do this if in the very first sentence you announce that we already know how best to teach.

Each of those juicy opening lines from Math Horizons helps generate space to tell a story — about a trick, about a career, about a number. In turn, each of the NCTM openers eliminates space that might otherwise be occupied by a story.

An NCTM journal that aimed to be mostly interesting — four-parts to one, let’s say — could therefore change in these three ways:

1. Publish crisp, engaging writing that tries to capture attention.
2. Discourage writers from trying to adhere to standards; publish writing that disagrees with NCTM policy and teaching documents.
3. Seek articles from the range of reader interests: math, math history, classroom dilemmas, policy debates, interviews, and so on, and so on. Even research, but for heaven’s sake keep it interesting!

This won’t be an easy change to make. I know it will be difficult to find writers willing to veer from what NCTM has published in the past. A word of advice on the editorial process, then: don’t seek submissions, seek writers. Find people that you’d like to write, and then ask them to pitch ideas. When one strikes an editor’s eye as especially interesting, help the writer develop it. Ask for snippets, early thoughts, rough drafts, and help craft the pieces into something that you expect to capture reader interest.

And all of this is worth it, because courting interest is a matter of respect. A piece that doesn’t attempt to capture attention (like a textbook) projects the opposite message: you really ought to read this. And, after all, isn’t that the main message of NCTM to teachers? That you really ought to teach like this, because we have the standards, the experts, the research and the know-how to train and educate you. Sure, this may be a slog to read, but aren’t you a professional? And you’ll read what you need to for your professional development.

Of course, like the speaker who comes in with gimmicks and cheap jokes, writing can miss the mark the other way. Bad writing can suggest a lack of seriousness.

But when done well, engaging writing can project trust and respect to the reader. We know you’re busy and discerning, it says, and that you have the intelligence to decide how to think and what to think about. You and us both. But, how about this?

So, stop trying to be so useful, NCTM! Relax, and try to be interesting instead.

# My NCTM Benefits

I.

I’ve been getting those emails again — “30 Days Left of Your NCTM Benefits.”

If you talk to someone who works with NCTM, one of the first things they’ll tell you is the organization is committed to making big changes. Depending on the person, they’ll also tell you it’s because the organization is scared, staring straight into a crisis. Membership and income is dropping, and nothing they’ve tried has tempered this trend.

I’m committed to NCTM. But if the question is do you personally enjoy the benefits of membership?, my answer is “no.”

I mostly don’t enjoy the conferences, and it’s not easy for me to justify missing school for them. I mostly don’t enjoy the journals — I mean just that I don’t usually want to read them. And seeing as conference discounts and a journal are the major personal benefits that NCTM offers members, there’s just not much for me, personally speaking.

But what if the question isn’t about personal benefits, but of benefits to the field of math education? Volunteers often talk about the role of NCTM in making policy when explaining why they support NCTM: Sure, you can take or leave the personal benefits. But there is no other organization that has a voice for math education on the state or national level. When NCTM talks, people listen. Don’t you want to support that?

A lot of people are passionate on this point. In fact, I was once talking to a former employee of NCTM, and I suggested that I’d rather NCTM didn’t pay lobbyists to try to influence policy. We had been having a fairly radical talk: what if the organization eliminated conferences, changed the journals, restructured the volunteer board, etc. When I mentioned cutting lobbying, though, this person showed real emotion. This was unthinkable, she said. Every professional organization lobbies in Washington, and NCTM is a professional organization.

Maybe that’s right. All I can say is that I’ve never been asked by NCTM what my policy views are, so I don’t know how they can claim to represent them. A survey would be a nice start, but hardly enough. Maybe there should be debates at the national conference or something, or a right to vote on policy positions if you pay your dues. People sometimes worry that the policy work of NCTM is invisible to membership. I bet a lot more teachers would know about the policy arm if they could influence it.

It’s not hard to come up with a suggestions like these, and it’s not hard to come up with many others like it, that would increase the role of teachers in the organizations. Why haven’t these ideas been taken up yet? Why is NCTM structured the way it is?

II.

NCTM is an organization that wants math teachers to pay for the right to defer to experts. The experts are math education professors, consultants, coaches, administrators, and a few teachers on their way to becoming one of the above.

Here’s an excellent point that Henri Picciotto made to me: teachers are hardly the only members of NCTM. “The organization is not uniquely or even primarily a teachers’ organization,” he said. This sounds exactly right, and you can see it in everything that NCTM does.

Let’s get concrete. I love writing and reading, and I’ve thought about writing pieces for the NCTM teacher journals. I’ve been turned off, though, sinceI learned that my submission would be judged, in part, by its “consistency with the mathematics teaching practices as described in Principles to Actions.” (link)

In other words, you might not know what great teaching looks like, but NCTM does. If you have any great examples showing how to put NCTM’s vision into practice, by all means, submit an article. But if you’re not ready to sign on to the NCTM vision of teaching, well, best to take your writing elsewhere. And, hey, why shouldn’t you adhere to the NCTM vision? It simply describes elements of good teaching we all agree on. After all, it was written and packaged by the experts.

The “Principles to Actions” clause (and similar requirements across the journals and at conferences) is a shame. First, it narrows the bandwidth of ideas that we’re allowed to talk about in math education. Second, it makes for a duller reading experience; to me the journals have a sort of corporate feel to them. But the most significant thing of all, I think, is that it greatly reduces the creative work that teachers are encouraged to do, and teachers want to do creative work.

“Putting research into practice” can mean a lot of things, but most often it signifies that we all already agree on the best ways to teach. What’s left is to convince colleagues, boards, parents, kids, etc. And when you start with the sorts of vision-documents that NCTM has produced, you end up with very little left for teachers to do.

So across the board — for policy, journals, conferences, PD, publications, resources — NCTM’s pitch to teachers is: don’t you want us be your experts? And the question is, does that pitch still resonate with teachers if membership is dropping?

III.

NCTM seems to get that the internet changed things, but I think they’re wrong about why. It’s not just that there are free alternatives to NCTM publications, or that people now expect digital copies of stuff. The bigger problem for them is the web has allowed teachers to find alternatives to the institutional trust that NCTM currently seeks to trade on. Now, you can choose your experts.

So the current relationship — where teachers are asked to pay NCTM because they trust the experts — is no longer tenable for the organization. This leaves NCTM with really one alternative, which is to focus on what math teachers want, whatever that happens to be.

Is NCTM heading towards this? I don’t know. I do know that, for the first time in its history, NCTM now lists “Building Member Value” as a organizational goal. They are now institutionally committed to the following:

“NCTM fosters communities that engage members to improve the teaching and learning of mathematics.”

Which is great! I think that NCTM will do a better job earning members when it aims to serve the needs of those members, rather than asking us to pay for the right to be influenced. The direction of the organization needs to be reversed; math education professionals will need to trust teachers.

For the journals, this could mean publishing stuff that isn’t already 100%-certified nutritious. And it would mean, I think, that writers would have to start making the case for their vision of teaching without merely citing NCTM consensus documents for authority. But really, NCTM would have a mandate to publish whatever there are readers for in math education.

As far as the conferences go, I’m confused by the role that math plays in these math education conferences. The last NCTM conference I attended was Nashville, two years ago. I remember (and wrote about) being surprised why there wasn’t more learning and doing math for teachers at these conferences. What if it turned out that NCTM members wanted more chances to learn math with other teachers? What if we’ve heard enough about formative assessment?

But all of this is just fleshing out the details. NCTM won’t regain its membership by focusing on PD, making statements, or publishing new guidelines. It needs to stop trying to fix math education, and start serving its teachers.

Lately I’ve been struggling to finish a piece about growth mindset research, a topic that I can’t seem to leave alone. I always come back to it, for reasons that aren’t entirely clear to me.

Summer is ending, and teachers are putting their classrooms back together. A lot of classrooms — if social media is to be believed — have bulletin boards that look like this:

There were no bulletin boards like this at the high school I attended. I don’t remember if there were any bulletin boards at all — there must have been, but only for announcements and intramural schedules. No teacher would have dreamed of decorating their classrooms in this way. We wouldn’t have taken it seriously; it probably would have been destroyed the second the teacher turned their back to us.

****

I only had two math teachers in all of high school. Rabbi Weiss covered 9th, 11th and 12th Grade math for the honors track. He hated geometry, so he found someone else to cover that. Rabbi Weiss also taught me Talmud/Halacha in 10th Grade, so he was my teacher for all of high school.

Yeshivas, in my experience, are incredibly competitive places. (All-male yeshivas, I mean.) Who would make it to the top class? Who would be offered advanced placement in the Israeli schools? Who would win Torah Bowl? (Yep! Torah Bowl.)

Rabbi Weiss gave long, difficult exams in both math and Talmud. There were two competitions on every exam: who would finish first? who would finish last? Because Rabbi Weiss gave you as much time to finish these monster tests as you needed, and you could look up any sources you wanted (for Talmud — you were on your own for math). You could win for speed or you could win on endurance.

I won the speed competition on the first Talmud exam in 10th Grade, but that was the last time I won that. For every other test that year I was one of the last to finish. I’m not sure what changed.

(Come to think of it, grades were totally a part of our competition too. Getting a perfect score on one of those exams was another thing we fought for.)

There’s more to say about all of this: about Rabbi Weiss’ pedagogy, how badly I miss the summer Talmud classes that met in his basement, his sense of humor, and how even though all of us were highly competitive we were also best of friends, studying together and nudging each other along.

I have to say a bit more about Torah Bowl. I was made captain of a team, and how we made it to the championship. I wasn’t the fastest and never had scary-good memory for trivia, but I also drafted well and our crew was formidable. I could tell you about the legal question — from Bava Kamma if you care — we were asked in sudden death, to crown a winner. It was about damages: is such-and-such more like starting a fire, or digging a treacherous pit? And while I don’t remember the answer, I remember that I raised my hand and answered wrong, losing the contest.

****

I signed up for Multivariable Calculus in my first semester at college. I had just come from studying in a yeshiva in Israel for a year. I had a great time, but there was no question: I returned from Israel with a bad case of angst and melodrama. I was obsessed with questions of self-worth, all of which had been highlighted by the constant talk of “who’s a genius?” that permeated that world. This was the state of mind I was in when I started college.

Here’s what I wanted to do: I wanted to show up in tough classes and kick some ass, because otherwise what are you worth? You can only contribute that unique something if you have that unique something.

I wasn’t really prepared for Multivariable Calculus. Rabbi Weiss taught strictly through note-giving and homework-reviewing. It wasn’t terrible for us — along with a bunch of my classmates, I aced the AP exams — but it left me with relatively shallow reserves to draw on in my first college class.

Most importantly, though, I saw Multivariable Calculus as a referendum on me. I didn’t ask questions in class. It was only near the end of the semester that, sheepishly, I arrived at my teacher’s office for help. I profusely apologized for, like, a whole minute before my teacher (whose English wasn’t great) made it clear through intense eye-rolling that I was being ridiculous. Of course he was right — I was ridiculous.

In the end of the year, my classmates managed to get this guy a teaching award. I walked around campus rolling my eyes — haha, my turn now! — because they were giving an award to this guy? The guy who frequently stopped class to ask for English translations of mathematical terms? The guy who, I felt, had given me nothing, no life-vest, no rope, no help?

The big, big thing I was missing was that all the non-grumps in the class liked him precisely because he would ask questions. In doing so, he made everyone else feel as if they could ask too. That was the whole thing.

****

A few years ago I taught a 9th Grader who came with a warning: his teacher last year had been able to get nothing out of him. He shuts down, I was told, and this was absolutely confirmed by what I saw in class during the first few weeks.

At the start of my career, I would have diagnosed him with a struggle-allergy. He wasn’t willing to dig in; he was used to things being handed to him in math; he didn’t know that struggle is normal, a sure sign that learning is happening.

I don’t want to dismiss all of this, but I’ve found a different strategy more helpful. It’s simpler too — which is good, because I don’t do well with complex. I need simplicity in my teaching, as much as possible.

Here’s what I did for my 9th Grader: I told the entire class, “I want you all to ask me questions. Lots of questions. When you’re feeling stuck: ask me for help.”

And, then, when my 9th Grader didn’t ask me questions I walked over to him: “I really want you to ask me some questions if you’re stuck.”

When that didn’t work (“I’m doing fine Mr. P”) I went back to him and I said: “You’re going to start having an easier time with these problems when you start asking me some questions.”

And, finally, when he asked me a question, I answered it as best I could and said, “This was great — please keep asking questions.”

At risk of driving home the point a bit too strongly: I really, really wanted him to ask me questions.

When a student is working on their own — tinkering away, seemingly content — it might not be that they’ve embraced struggle. It might be that they’re embarrassed to ask for help. Kids sometimes end up thinking that you’re supposed to deal with problems on your own, and that in fact dealing with issues on your own is a sign of intelligence and academic worth. It’s certainly what I thought, sitting in the back of Rabbi Weiss’ class or in my professor’s office hours.

It’s the thing I look for, most of all, in evaluating how a student is doing. If they’re asking questions, they expect to learn. If they aren’t, it could very well be that they’ve given up, or are considering it.

I’m not great at classroom culture — kids like me OK, I think — but this is one thing I know that I do. It’s one thing, nothing complicated, but I beg kids to ask me questions. It’s how you grow.

****

On and off for the past seven years, I’ve been trying to learn more math. Not just to solve problems, but to learn a new discipline of math, or to relearn my college material in more depth.

Each summer I sign myself up for a new mathematical project; each year I fail. What I’ve realized, though, is that I can’t do this on my own. I need to ask for help. This summer has been my most exciting summer for learning math, and it’s entirely because I’ve realized that I just need help to learn new stuff.

(Shout out to Anna, Ben, Ben, David, Evelyn and anybody else who has helped me out with math over the past few months! Thank you.)

A lot of teachers — myself included — find it helpful at times to talk about the nature of mathematical work with students. So: mathematicians prove things; mathematicians struggle; mathematicians make mistakes; etc.

The thing is, though, that mathematicians do a lot of things. We get to pick and choose which aspects of mathematical culture we want to promote with kids. Mathematicians prove things, sure, but they also invent discriminatory algorithms. (Put that on a poster!) So we make choices.

It’s a choice to emphasize struggle, mistake-making and individual effort in our classes. What we’re trying to do is emphasize that one’s success in class is in one’s control. And that’s often true, but I don’t think that it mostly happens by trying harder on problems, which is what our growth mindset messages seem to emphasize.

Mathematicians struggle, it’s true, but mathematicians also ask for help. And when it comes to helping kids who have given up, I don’t find it helpful to emphasize the normality of struggle and frustration in math. (We’re all frustrated! might not be the most compelling sales-pitch on behalf of our subject to these students.)

****

Barry Mazur (another of my math teachers) helped prove Fermat’s Last Theorem:

KEN RIBET: I saw Barry Mazur on the campus, and I said, “Let’s go for a cup of coffee.” And we sat down for cappuccinos at this cafe, and I looked at Barry and I said, “You know, I’m trying to generalize what I’ve done so that we can prove the full strength of Serre’s epsilon conjecture.” And Barry looked at me and said, “But you’ve done it already. All you have to do is add on some extra gamma zero of m structure and run through your argument, and it still works, and that gives everything you need.” And this had never occurred to me, as simple as it sounds. I looked at Barry, I looked at my cappucino, I looked back at Barry, and I said, “My God. You’re absolutely right.”

Mathematicians ask questions. Sometimes these questions are fun and playful, but other times the questions are more straightforward: can you help me understand this?

I wish that a teacher had told me — no, begged me — to ask questions that weren’t aimed at impressing anybody. Maybe then I could have been better equipped for math in college, and I wouldn’t have had to run away from it after that first taste. There’s so much more that I could have learned during those years if I had been more comfortable seeking clarity from those who had it.

So, put it on a poster: When you feel stuck, ask for help.

# A Quick One, On Politics and Teaching

I’m watching Grace’s talk (which you should watch too) and thinking about her question:

Is teaching necessarily political?

This is a question that I find tremendously tricky — though I sometimes feel alone in finding it so, and I often do a terrible job explaining my trouble. I’ll try again here.

In watching Grace’s talk, I see a difference between two ways of arguing for viewing teaching through a political lens:

2. You must adopt a political lens because teaching is political, and you have to open your eyes up to reality.

The second metaphor is behind talk of being “woke.” Right? It’s saying things just are a certain way. You need to see teaching as political just as you must see the world as round. Wake up!

This reminds me of a favorite passage from Maimonides’ treatise on sin and recovery:

“Ye that sleep, bestir yourselves from your sleep, and ye slumbering, emerge from your slumber, examine your conduct, turn in repentance, and remember your Creator!”

To see teaching as non-political is to slumber; to realize that it’s not is to open your eyes.

For whatever reason, though, this language feels wrong to me. It’s the first way of putting things that I’m much more comfortable with. Not that teaching is necessarily political, but that we can choose to see it as such, and that we should because it’s the right thing to do.

(I feel nervous sharing these rough thoughts. Some might accuse me of getting caught up in language, but what can I say? The question is one of language, and I’m caught up in it.)

In a comment on one of Grace’s incisive posts, I tried to draw an analogy between teaching as necessarily political and teaching as necessarily spiritual to try to make sense of this all. I’ll quote it here, but definitely go and read Grace’s post in its entirety:

Is teaching spiritual? Well, to someone who sees the world through spiritual lenses it certainly is! Every interaction — each moment — is stuffed with spiritual potential. Our sense for the spiritual is, arguably, tied up with the experiences of kindness, connection, understanding. We’re also capable of casual cruelty, and that mundane disregard for other people is the opposite of what it means to be spiritually engaged in a moment. In short, each moment in teaching is potentially spiritual, so let’s go out and say it: teaching is spiritual work. (Even when you fail to sense it, or treat the moment as mundane.)

At the same time, the classroom is not a religious center and there is a great deal of spiritual activity that would be inappropriate in a classroom context. In that sense, teaching is not spiritual, i.e. there is not widespread agreement among parents, students, educators and other stakeholders that there ought to be spiritual activity in the classroom. (Certainly not that there ought to be any particular sort of spiritual activity present.)

So is teaching inherently spiritual? It depends what you mean.

(a) A spiritual person (I guess I am) could say, yes, absolutely. Teaching is, or it can be, spiritual work. (And the absence of spiritual meaning is taking a sort of spiritual stand, too.)
(b) On the the other hand, spirituality is not an agreed upon purpose of schools or schooling. So you can bring spirituality to the fore of your classroom, but there are risks involved. (Like losing your job, or offending someone who has a strong opposition to spirituality or your particular spiritual message.)

We might also ask, SHOULD schools be more spiritual?

All of this feels as if it’s closely parallel to what we talk about when we talk about whether teaching is political.

The way of thinking about this that I find most natural is that teaching is not necessarily political, though it’s possible to see all of teaching through a political lens, and I really think that you should.

Why see teaching through a political lens, if it’s not necessarily political? Because it’s the right thing to do for your students. It’ll sensitize you to a host of issues that — whether or not they help increase test scores or get kids into college — will make your classroom a more humane place for your students. People need to be loved and understood; your students are people. A political perspective helps.

But I admit to being entirely unsure of this, and confused as to whether there is really any real difference here. Is there anything important at stake between these two ways of arguing for seeing teaching as political? Are these just two ways of saying the same thing, or two fundamentally different perspectives on politics and teaching?

I don’t know, and I don’t know if I’ve articulated where I’m at in a way that can convince you that I’m not trying to stir up shit or to cause trouble, and I also don’t know if I’ve convinced anyone that this is coming from a place of really sincere concern for doing right by my students. I don’t know why this question feels important and elusive to me, but it does.

And now go watch Grace’s talk! It really is great.

# 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.