Math-Talk With My Daughter

I have eight-year-old twins—a boy, L, and a girl, M. One of my many hopes for them is that they grow up to enjoy math. They don’t have to ace their math classes, or study math in college. If they enjoy it, I will consider my work (in that department) done.

I think we’re on track so far, but the road has been a little bumpier with M. (I won’t go into too much detail here, but I have a theory—by no means original, I’m sure—as to why boys have historically been perceived as being “better at” math than girls.) I did have a wonderful exchange with her last night, though, and wanted to share that.

She has asked me many times if I would come into the bathroom while she’s in the shower so she and I can talk. I usually demur, because of both the weirdness/awkwardness and the likelihood that our conversation will distract her from actually washing herself. But last night she insisted that I comply because 1) she wanted to talk about math, and 2) while I’m on sabbatical she wanted to get at least a little one-on-one daddy-daughter time every day.

It’s like she knows exactly what buttons to push to get what she wants. (I don’t mean to imply that she was being insincere, but I do sometimes notice how easily I can be manipulated.)

Once she was in the shower, she called for me.

“So. What do you want to talk about?”
“Well, you’re the math teacher, so…”
“You want me to ask you a question?”
“Okay… Well, here’s one that I’ve asked you about in the past, but I don’t think we ever came up with a solid answer.”
“Is the sum of two even numbers always an even number?”
“… Yes.”
“Becaaause… 2+2 is 4, 4+4 is 8, 8+8 is 16, 16+16 is—”
“Okay… do the two numbers you’re adding have to be the same?”
“Oh. No. Because, like, 2+4 is 6. Which is even.”
“Okay. Well, so, it’s true in all those cases, but that doesn’t mean it’s true in all cases. Maybe it’s only true for numbers less than 100.”
“Okay, well, you come up with numbers where it doesn’t work.”

This was my first favorite moment. She put the onus on me to find a counterexample! I then taught her the word “counterexample,” and said, “I can’t think of a counterexample. But that doesn’t mean there isn’t one.”

After a little more thinking, she took a stab at an explanation. It started with, “Well, if two even numbers didn’t add up to an even number, then they would add up to an odd number. And odd numbers… are odd.”
“Okay… so what does that mean?”
“Well, ‘odd’ means ‘weird,’ so…” Even though I couldn’t see her face, I could tell she was smiling. She suggested that we table that question for the moment. So we did.

This was my second favorite moment. She was perfectly fine leaving the question unanswered. There was no, “Just TELL me!!!” or any kind of frustration. She just set it aside, perhaps to be picked up some other time.

“Do you want a shape question?”
“Yes! I love shapes! What kind of shape?”
“Well, what kind of shapes do you like?”
“Um… I like circles best.”
“Okay… How about this. Picture a square. Then picture a circle inside it that’s touching all four sides.”
I proceeded very slowly. “How much of the square’s area does the circle take up?” (We hadn’t talked much about “area” before; I was relying on her more “natural” understanding of the word.)
“… Are you counting the inside of the circle?”

Favorite moment #3. It makes perfect sense to me now that she would think of a circle as only consisting of the drawn boundary, but it took me a few seconds to catch on, since in my head, of course we were talking about the inside. It’s an area question! So that was a cool glimpse into her perception of shapes.

“… I… don’t even know how to answer a question like that!”
I assumed she meant a question that asks how much of one thing another thing takes up. “You could use percents.” (For whatever reason, percents are the part/whole concept they are currently most familiar with.)
“Okay… um… I think 60%.”
“How did you get that?”
“I think each of the little triangle thingies in the corners is 10%, so all together they’re 40%, so… 60%.”
“Ah. Interesting.”
“Actually… the corner things might be more like 5%.”
“Oh. Well, what would your answer be then?”
“… 80%.”
“Mmm. So you think the answer is somewhere between 60% and 80%?”
“Yeah. But… closer to 80%, I think. Like maybe 75%.”
“All right. So… how could we figure out how close we are to the right answer?”
Here she did a dramatic flourish with the curtain, and poked her head out. “It’s… not possible.”
“Yeah. Maybe we could invent a machine that takes a drawing you made, and… makes a copy… like it would be thick paper, like almost like cardboard, and it would put lights in each shape, and it would talk to you, and say, ‘This shape is 30%.’ Maybe we could email that to some engineers.”
“Yeah, maybe.”


“And then we could do… like, what about a square inside a circle?”
“… Oooh! That is an awesome question.”

Favorite moment #4. Asking “what-if” questions is something I try so hard to get my high school students—even the most “advanced” ones!—to do, and it’s like pulling teeth!

“So what do you thi—”
“I think it’d be the same percent.”
“Huh. Okay.”

At this point I checked in with her about her shower progress, and she confessed that she had not been doing any shampooing or soaping that whole time, so I had to leave.

As awkward—and probably untenable—as the shower context is, one thing I really like about it is that our conversations are necessarily entirely verbal. She can’t reach for pencil and paper (not that there’s anything wrong with that), and can’t even show me anything she’s doing on her fingers. That forces a kind of thinking she might not get to do too often.

Anyway, “Math Chats with M” might wind up being a category or tag on this blog. So far we both enjoy them, but I think we’ll have to be more intentional than ad-hoc about the venue.


Sabbatical Day 1

Okay, maybe it’s not technically Day 1. My colleagues are still on vacation. They report back for a Professional Day on Friday; that will be the first school obligation from which I will be absent. But my own kids are back in school today, so it’s my first opportunity to really take a crack at my Sabbatical To-Do List.

The list is loosely divided between “School/Math” and “Home” items. (Or, in other words, “Fun” and “Not-So-Fun” items.) I am going to focus on the “Home” items in these first few days, to see if I can get that list whittled down, but here are a few of the “School/Math” items that are on my radar:

About that last one. If you follow me on Twitter you know that I get to do some pretty cool stuff with my Advanced Topics students. I’ve always prided myself on just “going with the flow” in that course, and doing a lot of meandering, and deciding day-to-day what I will ask the students to do, both in and out of class. But I think that both I and they would benefit from more structure. Which is not exactly my strong suit. (By “structure” I mostly mean a consistent, transparent approach to things like assignments, assessments, and grades.) So I’m going to take some time to think about what kind of structure I could place on the course that would still allow it to maintain its free-wheeling nature.

I think some “backwards design” is in order. Step One will be answering questions like: How do I want my students to have changed by the end of the course? What skills and habits would I like them to have acquired? Only then does it make sense to think about what kind of structures would help me achieve those goals.

Okay, I’m off to the DPW. They owe me a $200 rebate check for the washer and dryer we bought. A rebate for which I initially applied on July 17th, and which has just been delayed, and delayed, and delayed. I don’t make scenes in public, but any more runaround, and I might decide to try it.

The Moment I Became a Mathematician

James Tanton recently published a piece on Medium entitled PERSONAL STORIES OF DISCOVERING MATHEMATICS. In it he shares his story, as well as those of two other mathematicians, of an experience from his youth that “hooked” him on mathematics. I have such a story as well, and I thought I’d share it here.

In elementary school, I don’t remember doing much more in math than basic arithmetic. And I was okay at it, I guess. I don’t know that I found it all that interesting, but I suppose it made sufficient sense to me.

What hooked me, though, was one day in fifth grade. My teacher, Mrs. Kennison, handed out a single piece of paper with a 10×10 grid of squares on it, with one question on top: “How many squares are there?”


It looked like this, except the question was hand-written. Because it was 1985.

My first response: “Psh. This is easy! Ten rows, ten columns, ten times ten is a hundred.”

Moments later: “Oh. Wait. Oh. Crap. There are… a LOT more than a hundred.” I was almost seeing random squares of different sizes light up all over the paper. The question was much harder than I originally thought, and counting all the squares felt like an impossible task.

Now. This is the moment. A fork in the road. I could have either declared the question “too hard,” given up, and moved on to something else, or stuck with it and tried to get an answer. What was it about fifth-grade me that made me decide to keep trying? Stubbornness? Curiosity? Competitiveness? I honestly don’t know. But it changed the course of my life.

I decided that it would be easier if I grouped the squares by size, and counted how many squares were in each group, then added them all up at the end. It would be a little time-consuming, but not too bad.

Okay, so how many 1×1’s? 100. Already did that. What about 2×2’s? I was careful to count “overlaps,” so I could see that there were essentially nine rows of nine 2×2’s, so 81. Then eight rows of eight 3×3’s: 64.

Waaaaaaaait a minute. THESE ARE JUST THE PERFECT SQUARES. If I just add up all the perfect squares between 1 (the entire square) and 100 (all the smallest ones), then THAT WILL BE MY ANSWER.

1+4+9+16+25+36+49+64+81+100 = 385. There are 385 squares.

Cue the dopamine.

It’s worth noting that this was excitement not about the actual calculation, but about what math empowered me to do. Namely, something that I at one time thought was impossible. And since that moment there have been hundreds if not thousands of times that math—or, perhaps more accurately, thinking mathematically—has allowed me to figure something out that had initially felt or appeared impossible.

Another great example is the classic question, “How many zeros are there at the end of 100! (one hundred factorial, or 100×99×98×97× … ×3×2×1)?” This number feels too huge to properly process, even with a handheld calculator to assist you. But the answer is findable without any technology whatsoever. And the route you take to get there reveals some interesting things about the whole numbers we know so well (or think we do).

The high school curriculum does not have very much room to make space for those moments. It makes me wonder how many life-changing opportunities we are missing.

A Risk I’m Taking

My first meetings of the school year are today. Students and their families will be in full force on Monday, and then the first day of classes is Tuesday, September 4th. Thus will begin my 19th year of teaching.

I will be teaching three sections (out of five) of Geometry (from CPM, a new curriculum for us), the one section of Precalculus Honors, and the one section of Advanced Topics (an elective, consisting of a handful of mostly seniors, with AP Calc as a co-requisite).

This year will be a little strange, as I go on sabbatical for the second half (starting December 20th). This has presented the department with some staffing challenges. We have someone taking over Geometry and Precalculus Honors from me, but no one to take Advanced Topics. At the end of last year, I was tasked with finding “something for them to do” in the second half that they would be able to work on mostly independently. As you can imagine, this was hard to do. I did not want them to “just watch online videos” to “learn” some new topic and then… do what? Present the material to the class? Teach it to their classmates? To what end? What student would be invested in this?

I wanted the ethos that I try to instill in them—what doing math is really all about—to remain even after I’ve left. No small order. Eventually I settled on this plan:

We will use two volumes from the Teacher Program Series published by the AMS (mostly written by the four Math-Yodas of EDC and PCMI: Bowen Kerins, Darryl Yong, Glenn H. Stevens, and Al Cuoco). These volumes are based on the problem sets that secondary teachers in the PCMI program work on every day. They are structured very carefully so as to allow the participants to do most of the discovering and connection-making themselves, through trying to solve the problems. Up through December, I will see them through Volume 3: Famous Functions in Number Theory. They will work on one problem set per week, and all the while I will talk with them about the spirit of the course and the text, and prepare them for the second half of the year, when they will work, mostly on their own, on Volume 4: Some Applications of Geometric Thinking.

It’s risky, and I still haven’t totally hammered out all the administrative nitty-gritty (i.e., what will assessments look like?), but I am feeling good about it. Definitely more so than if I had tried to get them to do something more traditional. I will keep you posted as to how it goes…

Freedom and Power

This is my contribution to Sam Shah’s Virtual Conference on Mathematical Flavors. I was honored, shocked, and thrilled that he asked me to be a keynote blogger, as this is something I’ve been thinking about, on and off, for years. The seed was planted with a comment on Dan Meyer’s blog, pointing out the difference between teaching math and teaching about math. One might reword Sam’s prompt as: We all teach our students math. But what are we teaching them—either explicitly or implicitly—about math?

I teach math because I can’t not teach it.

I have a passion for my subject that borders on obsession—which is both an asset and a liability as a math teacher. What makes it an asset is probably clear. (My highest marks on my student feedback surveys are always from the item, “My teacher is passionate about his/her subject area.”) The liability comes when I am prepping for a class, and casting about for interesting questions to ask. Invariably I will hit upon an idea or question that hooks me, and I am compelled to chase that rabbit down its hole, and then somehow it’s an hour or so later, and I am no closer to having something for my students to do.

Forget anti-drug: Math is my straight-up drug.

I started teaching because I wanted to do whatever I could to help students experience the same beauty and joy of doing mathematics that I do. I have learned a lot in the almost twenty years since, and I have also had time to reflect on what it is about mathematics that attracts me to it so strongly. The answer I keep coming back to is this: Math is freedom and power.

Or, put less tersely: Being able to think mathematically, meditate and ruminate on difficult questions, and persevere through challenges maximizes your freedom, empowering you to do things in your life you might not have thought possible.

(I daresay this is not unique to math. Blake Boles, in trying to define what “education” is in this piece, paraphrases John Taylor Gatto: “An education is the capacity to author your own life instead of merely accepting the one handed to you.” Love it.)

So how do I try to bring this to my classroom? How do I try to maximize my students’ freedom while still accomplishing my goals for them? How do I show them what learning mathematics enables them to accomplish? It is not easy. My students come to me with an already deeply-ingrained notion of what math is, so by the end of the school year, the best I can hope for is—as Sam says in his prompt for this conference—”moving the needle” a degree or two. But Lord knows I try.

The ideas I’m going to share here are by no means ground-breaking. They have been discussed and dissected in the #MTBoS since its inception, but for me they have been the keys to moving that needle over the last few years.

Take students’ ideas seriously.

I get it. We are given only so many minutes with our students each day, and we tend to have very specific ideas and plans for every one of those minutes. So when a student lobs something at us we are not expecting, it is tempting to treat it as an aberration to be dealt with in order to proceed with the plan, rather than an opportunity to explore and consider a new idea. Choosing to engage in the latter does three things. First, it shows the students that they are not mere receivers of information, they are generators of original ideas. Second, it shows them that I value, and am genuinely curious about, those ideas, which in turn helps them be more comfortable sharing them. Thirdly, it models more closely the true nature of doing mathematics: Playing around with new ideas, wondering things about them, and seeing where they lead us.

Doing this well requires not only an excellent pokerface (as Jenna Laib wrote about last week), but a pokervoice. When a student offers an answer to a question I’ve posed, nine times out of ten my response is simply, “What makes you say that?” in as judgment-neutral a tone as I can muster. Try as they might, they are unable to suss out my assessment of their contribution, and are forced to dissect it, carefully consider it, and justify it themselves.

Which leads us nicely into…

Give students authority to consider and evaluate both their own ideas and those proffered by others.

When I refuse to simply give students answers, and instead ask them to explain their reasoning, they learn very quickly that I am not the answer key. When this begins to dawn on them at the beginning of the year, there is usually a moment of panic, as when one is still learning to swim, and discovers that the water they’re in is a little deeper than they were anticipating. They complain. They get frustrated. But once they discover that the waters are in fact safe, and that they can in fact doggy-paddle, they begin doggy-paddling furiously.

My students are not used to being pushed to explain their reasoning. It is not a simple thing to learn—or to teach. It is not something that we can cover in a unit, have a test on, and move on. It is infused into all of our daily interactions in class. It’s the water we swim in. They practice their reasoning constantly without being told that that’s what they’re doing. And while I assess their reasoning formally fairly regularly¹, I am kind in my grading for most of the year. I’m playing a long game, and it’s usually not until the spring that some students will start to openly admit that, as difficult as explaining reasoning and proving a claim is, they actually kind of enjoy it.

Empower students to persevere through challenges.

I’ve found this to be the hardest. To abuse another metaphor, my students tend to pull the ripcord on their problem parachutes moments after they jump out of the plane. There are many factors causing them to do this, but most of them involve pressure of some sort. And the pressure that most negatively affects their perseverance potential is that of time. They have been taught—usually implicitly, via expectations of past teachers—that speed is important in doing mathematics. That being “good at math” means answering questions quickly. I try to change this just by slowing things down in general. Increasing my “wait time” before soliciting student thoughts or questions. Taking a few minutes, when a student wonders something aloud that I hadn’t considered, to go down that road a ways and see where it leads. Yes, sometimes this comes back to bite me toward the end of the year, when I have to worry about “covering” the requisite material. But in the end, it is completely worth it to me. My students are stressed out enough as it is; I want math class to be a place where they feel they can both relax and think deeply.

To help drive home the fact that speed is not king in mathematics, I regularly assign my students problems that are meant to be worked on over the course of a week. They can ask me for help with them, but my “help” typically comes in question form. “Well, what are your thoughts?” “What might you try?” “What else could you try?” “Forget what you’re being asked to find. What can you figure out, using the information you have?” “What do you need to know in order to answer the question?” By the end of the year, the hope is that I become so predictable in the questions I ask them when they’re “stuck” that they will eventually begin to ask them of themselves.

Even when I use these strategies well, it’s not always clear that they are having the desired effect. But every once in a while I get a clear indication that my messages are actually sinking in.

It was March or April of this year. We were two minutes shy of the opening bell ringing, and students were filing into my classroom for Geometry. Mostly freshmen, a handful of sophomores. There was some light conversation about which classes everyone had come from, and how they were feeling about entering this one. One freshman piped up with, “I really like this class, because I can say whatever I want.”

Now, this particular student had already developed a reputation (and not just in my class) for not only saying whatever she wanted, but saying it whenever she wanted. So I gave her a sidelong glance and said, “Could you… maybe… expand upon that a bit?”

She went on to say—though not in these exact words—that she felt comfortable sharing her ideas in my class, even when they were only partially formed, because she knew that they would at least be taken seriously and given careful consideration. It was the clearest indication that I’d had in a long time that the needle was, in fact, moving.

Postscript. Only after writing the bulk of this post did I realize that I came very close to simply writing the blog-post version of Dan Finkel‘s spectacular TEDx talk, Five Principles of Extraordinary Math Teaching. If you haven’t watched it, you must. Regularly.

¹ … and he lands the Triple Adverb!

Teacher Sedition

Welcome! My name is Matt Enlow, and this is my first post on my new blog.

First order of business: Explaining the name.

I teach math in the Upper School at Dana Hall, an independent all-girls school in Wellesley, MA.

Why do I teach math? In short, because I can’t NOT teach math.

If I did not teach math, I would still play with it in my spare time. I would still regularly baffle myself, and then try to un-baffle myself. I would still talk to people about the strange, beautiful things I had done, seen, and discovered. So I might as well get paid to do all of these things!

I teach math because I love it, and I want to do my best to get others to see in it what I see. Unfortunately, as odd as it may sound, sometimes I think I love math too much to teach it.

Imagine an artist, excited about the prospect of broadening young minds and helping them see the beautiful, transformative, sublime nature of artistic pursuits, being told that 90% of his curriculum must consist of designing corporate logos and pixel art. Or a musician who is made to teach her students how to write advertising jingles and design ringtones. The word “soul-crushing” comes to mind.

Because of the cultures and structures built around education in this country, I am made to take a subject that has inspired awe in every civilization in human history, chop it up into bland, easily-digestible pieces, and force them into my students’ mouths. I am also made to perpetuate practices that regularly convince many students that they don’t have what it takes to “do math,” permanently damaging their ability to learn and grow.

But I can’t. And I won’t.

Thus… Teacher Sedition.

This problem is not endemic to my school, by any means. I love my school, and have no plans to leave anytime soon. This problem is ubiquitous, and I would encounter it, in some form, wherever I taught. There are enough kindred spirits at Dana to sustain me. But I will do all that I can to work within my given constraints, and still show my students that they belong in math class regardless of perceived ability, that they can understand things that initially confuse them, and that the people who refer to math as “beautiful” might not be totally crazy.

My first post (well, okay, second) will be my contribution to Sam Shah’s Virtual Conference on Mathematical Flavors on August 15th. After that, this blog will feature:

  • cool math stuff I’ve been playing with
  • my attempts to change how I do things in my classes
  • my readings and other goings-on during my sabbatical (January-June 2019)
  • other musings

So, basically, my Twitter feed writ larger than 280 characters.

I know better than to promise any kind of posting frequency, but I’m going to set the bar relatively low, at one post per week. We’ll see how it goes.

Thanks for reading. I’m hoping that you get as much out of this endeavor as I do.