Beyond “Beyond Explaining”

There’s a good reason why educators often talk about the need to move beyond explanations. People who don’t know much about teaching think all the action in teaching is about the clarity of the explanation. (That, and getting kids to listen to your ultra-clear explanation.)

There’s much more to the job than that, of course. Michael Fenton puts this nicely in a recent post:

In my first few years in the classroom, I held the notion that the best way to improve as a teacher was to hone my explaining skills. I figured that if I could explain things more clearly, then my students would learn more. […]

The best way to grow as a teacher is to develop my capacity to listen, to hear, to understand. […] This doesn’t mean that I’ll stop working on those other skills. But it does mean I have a new passion for learning about listening—really listening—to students and their thinking.

I think this focus on listening is wonderful, and Michael did say that he’s going to keep working on his other skills, which is a nuanced take. But what about the title of the post, “Beyond Explaining, Beyond Engaging”? Philosopher Eric Schwitzgebel has a helpful distinction between a piece’s headline view and its nuanced view:

Here’s what I think the typical reader — including the typical academic reader — recalls from their reading, two weeks later: one sentence. […] As an author, you are responsible for both the headline view and the nuanced view. Likewise, as a critic, I believe it’s fair to target the headline view as long as one also acknowledges the nuance beneath.

So let’s take on that headline: should we go “beyond explaining”? If we’re trying to improve our teaching, it could be that getting better at listening has a higher payoff than getting better at explaining. But my experience has been that there isn’t any strict hierarchy of payoffs in teaching. Teaching evolves in funny ways. Last year I taught an 8th Grade class that pushed on my classroom management. This year I’ve spent a good deal of time learning how to tutor students with learning disabilities. I’d hate to say that explaining is some sort of basic teaching skill, the sort of thing novices focus on but more experienced teachers don’t need. Teaching is weirder, more cyclical, more web-like than that.

Maybe, though, we should move beyond explaining because it’s easy — or because pretty much everybody knows how to do it well after their first two years on the job.

That might be true, for all I know. If I doubt it, it’s only because it’s only over the past year that I’ve really started to understand some of the things that make a good explanation hum and lead to great student thinking, instead of slack-eyed drooling from the back rows of the classroom.

Besides, a lot of what I’ve learned about explaining comes from outside math education writers and speakers. Which started me thinking that maybe this knowledge (if it’s even true) isn’t as well known to math teachers as it could be.

Here’s what I think I know about giving good explanations to kids:

1. Study Complete Mathematical Thoughts; Don’t “Roll Them Out”

This isn’t really an explanation, but it’s a complete example. Source: Algebra by Example

The first students I taught were subject to an especially painful type of instruction. I don’t know exactly how to describe it. Maybe an analogy would help. Imagine a magician (Ze Great Pershan-o) who is waaaay too detail-oriented: OK AND NOW CHECK IT OUT: I PUT MY HAND IN THE HAT! AND NEXT, I CLENCH MY FIST! HERE COMES THAT HAND SLOWLY COMING OUT OF THE HAT! ETC!


The phony enthusiasm was a problem. Another problem was that I was feeding the math one mini-idea at a time rather than presenting them the complete mathematical thought. I’ve come to think that when we do this — when we roll out the explanation, line by line — we lose a lot of kids.

Do you know that thing you do when you’re trying to understand something hard in a math textbook? How you put one finger at the top of the explanation, and then go line-by-line to make sure you understand each piece? But then you go back and try to make sense of the whole? That sort of self-explanation is where the learning can come from in an explanation, I think, and if we roll the explanation out, we’re making it harder for kids to look at how the pieces fit together.

So, when we’re ready to explain an idea to students, we ought to be offering them a complete mathematical thought. No need to dice it down to the atomic level, like Dumb Houdini or whatever.

Practically, this means that if I intend to “show the steps” in an explanation, I make a real effort to reveal them all at once. I project or photocopy artifacts like the ones that Algebra by Example create. If I’m working one-on-one, I’ll scribble a full example down on the page, rather than coaching a kid through that example step-by-step.

This idea, of course, isn’t my own. I came to understand this from reading about cognitive science, and especially cognitive load theory. There’s more to math than explaining, but there are ways of explaining things that preserve the math and others that make it harder for students to make meaning. Fully worked-out examples can help kids make meaning from our explanations, I think.

2. Use Arrows to Emphasize Process, Change and Action

This was an aha moment for me. I came to think people across education were saying similar things about how it’s easier for people to think about actions, rather than properties. You can hear this idea bouncing around research on how kids solve word problems, how our minds especially remember narratives, and what constitutes good writing.

Nobody has told me that these ideas are related. I’m a bit worried that I’m connecting totally unrelated people and ideas. Still, here’s the idea:

An interesting result of Cognitively Guided Instruction is that numerically equivalent word problems are often handled very differently by children. Consider these two problems:

Problem 1: Jill has 8 jellies, but a raccoon eats 5 of them. How many does she have left?

Problem 2: Jill has 8 jellies. A raccoon has 5 jellies. How many more jellies does Jill have?

Numerically equivalent problems — 8 minus 5 — but the first problem contains an action that is easy to represent. It would occur to a lot of kids that they could solve the first problem by dealing out 8 counters (or whatever) and then removing 5 off them from their pile. That’s directly modeling the action of Problem 1.

Problem 2 doesn’t contain an easy-to-notice action, so direct modeling would be less likely to occur to children. Carpenter and pals found that, in fact, kids didn’t use direct modeling strategies for Problem 2, and as a result Problem 2 was a bit trickier for kids to handle.

(This is 100% true, in my experience.)

Word problems with actions, essentially, contain stories that are easy for us to represent and understand. And stories are the sort of thing that our minds most easily grasp and symbolically represent. As Dan Willingham notes:

Research from the last 30 years shows that stories are indeed special. Stories are easy to comprehend and easy to remember, and that’s true not just because people pay close attention to stories; there is something inherent in the story format that makes them easy to understand and remember. Teachers can consider using the basic elements of story structure to organize lessons and introduce complicated material, even if they don’t plan to tell a story in class.

Incidentally, Tom Newkirk makes a similar observation about what constitutes good writing in his book Minds Made for Stories:


Later, Newkirk argues that part of what makes good non-fiction writers good is they find subjects and actions to metaphorically represent abstract structures. They turn “evolution is a process whereby genes are randomly mutated” into “mutagens are constantly attacking our genetic material, altering it in ways that have the power to change the direction of an entire species” or whatever. Action, action, action.

(I’m also pretty sure this connects to Anna Sfard’s work on the way we tend to turn mathematical processes into mathematical objects but I’m not sure I have all the pieces put together yet.)

Back to Planet Classroom: What does this mean for my teaching? Practically, a lot of annotated arrows.

Instead of an example of simplifying expressions that looks like this


…I try to recreate it with annotated arrows, to emphasize the actions involved:


Our minds privilege stories, which means that they privilege change over inaction. If our explanations can include more doing things to things, this can help kids see what we mean a bit more easily.

For a lot of fun use of arrows to represent the actions in otherwise static representations, check out David Wees’ new tumblr for mathematical annotations. A prime example:


Arrows — especially annotated arrows — can help transform examples (i.e. what correct work might look like) into explanations that help someone understand the examples.

3. Describe Mathematical Pictures, Though This is Harder Than You Might Expect

The What Works Clearinghouse (WWC) is this big federal initiative to try to sort through the evidence for various educational claims and give clear recommendations. The thing is, there is a ton of dissatisfaction with their standards for recommendations. Some people think their standards of evidence are weirdly strict. Others say they privilege large experimental or quasi-experimental studies over other forms of evidence.

Anyway, they have this report on helping students who struggle with math, and I like it. Their fifth recommendation is all about visuals:


Why are visuals important? At least partly because words don’t distract you from pictures — you can pay attention to both at once. (Unlike reading a slide and hearing it explained to you, where the words interfere with each other. You might find yourself doubly distracted in that situation.)

This relates to dual-coding theory, a theory from cognitive science that deserves to be better know in math education. Like worked-out examples or the privileged role of narrative, it’s a legitimately useful bit of cognitive science to know.

If you’re looking to teach a strategy, describing a (complete!) mathematical picture (with arrows!) can help.

“So to figure out 7 minus 2, I started with 2 and counted up until I got to 7 and that was 5, so I figured out that 7 – 2 is 5.”

The thing, though, is that it’s very easy to mess this up. A “mathematical picture” is not “a bunch of written numbers or words.” That’s not describing a picture with words — that’s just using spoken words to describe written words. I don’t think that helps as much, according to dual-coding. Words can distract you from words.

I’m not saying that board work in the above tweet is bad at all. My point is just that these equations are going to compete for attention with any spoken explanation in a way that (dual-coding says) a picture wouldn’t. (Though check out those annotated arrows!)

A problem: when I think about it, there are a lot of mathematical topics that I can’t think of a good picture for. And even for ones where I can (e.g. the connection between area and multiplication) those visual representations aren’t obviously connected to their numerical ones to kids. Those connections need to be carefully taught. Ideally, they’re built into a curriculum.

I mean, it’s obvious to me that you can carve up a rectangle into 4 quarters and this represents multiplying (x + 3)(x +7), but anyone who’s tried knows that this isn’t obvious to high school math students.

So while it’s great to aim for “describing a mathematical picture” as an ideal for explanation, we’re limited by the mathematical pictures that kids understand and that we know.

I love learning new pictures for mathematical ideas. I feel like this year I really realized the power of the visual representation of the Pythagorean Theorem to help my kids understand its meaning and use:

But there are a lot of topics where I don’t know good visuals to go with the numbers, equations or words. I’d love if we could find more of them.

What Beyond Explanations Shouldn’t Mean

I could be wrong, but I don’t see a lot of people writing or thinking about how to give good mathematical explanations. This is despite the fact that the vast majority of teachers I talk to say they give explanations often, even if they are a bit embarrassed by this. (They shouldn’t be, I think.)

And the vast majority of thinkers and writing about teaching would tell me that they aren’t anti-explanation, just against a mono-focus on explaining as the core of teaching.

So why doesn’t anyone write about giving good mathematical explanations? Three possibilities I can think of.

  1. Unlike me, pretty much already knows how to give good mathematical explanations.
  2. People don’t think that improving our explanations is worth the ink. It’s a low pay-off instructional improvement.
  3. Anti-explanation ideals make it trickier to talk about improving our explanations.

I’m pretty sure it’s not Possibility 1. I think Possibility 2 sounds good, and Possibility 3 is a solid maybe and is anyway related to 2.

Either way, now you know what I’ve recently figured out about explaining stuff to kids:

  • Represent complete mathematical thoughts
  • Use annotated arrows to emphasize action and change in those representations
  • If possible, describe mathematical pictures

So, what’s next? Just last week I tried out a new representation of solving trig problems with my geometry students. It flopped:

But then I made a little tweak, and it went better. Which got me thinking: we’ve got this whole internet thing. Why aren’t people sharing more of these images? Is it less fun for us to share pictures of our own work? Does it seem self-promoting in a way that sharing other things (e.g. activities) doesn’t?

I’m not sure. But I think that this work is valuable, and is worth sharing. Explanations are nitty gritty, but it’s important nitty gritty.

Book Review: Schoolteacher


Schoolteacher is a book that is referenced by pretty much every other scholarly book on education I’ve ever read. It’s a — the! — sociological study of k-12 teaching.

It’s also a book that I failed to read some three times before recently pushing myself through to the finish line.

It’s not that the book isn’t insightful. It is very insightful, the most revealing explanation of forces I’ve felt as a teacher that I’ve ever encountered. (More on that in a second.) It’s also not that the book is boring, or that it’s overly technical, and the dynamics that Dan Lortie describes aren’t even so complicated.

So I don’t really have a theory as to why I found it such a slog to get through my first three times.

What made the difference this last time was simple: I just skipped the boring parts. That took me to the second half of the book. That’s the part where Lortie starts including huge blocks of quotes from the teachers he interviewed, and these sections were much more fun to read:

Well, years ago I hit children. Of course I wouldn’t, you can’t do that now, but I have done that which I felt badly about. Recently I have taken work I didn’t like, I’ve just crumpled it up and threw it in the wastebasket. And afterward I have thought what a terrible thing to do because maybe that was the child’s best effort.

Yeah! Now we’re cooking.

So I read the second half that has all the interviews and then slogged back through that first half, though having a good deal of knowledge of Lortie’s argumentative strategy and goals.

Incidentally, reading this book for me felt eerily like reading Racism Without Racists, another sociology tract that I found impossibly boring until I skipped to the juicy quotes.

Sociologists, if you’re listening: don’t hide your quotes!


Lortie begins with the history of the teaching profession in America. Except he’s not actually interested in telling this story, he just wants to point out that a lot of things haven’t changed since the beginning of teaching in America.

So, teaching begins with a single teacher in a schoolhouse; teaching still happens with a single teacher in a closed room. In the beginning, it wasn’t that hard to become a teacher; it’s still not that hard. Schools used to be managed locally, by citizens; they still are. Things more-or-less stay the same in teaching.

This line of argument has been ruined by would-be-reformers who say ridiculous things about “21st century learning” or “the out-dated Prussian factory model of teaching.” But Lortie has a good point. How much has teaching changed?

Put it this way: if a 19th century teacher had a time machine and traveled to 2017, how long would it take them to get the hang of modern teaching? The biggest issue we’d have would probably be cutting-out the beatings and extreme (for us) emotional cruelties and shaming.

Actually, Lortie has something interesting to say about this too:

During the latter half of the nineteenth century and the early decades of the twentieth, laws and school custom changed; increasingly sharp limits were placed on the teacher’s use of physical punishment. There is a paradox in this transformation of values and practices: the teacher’s use of physical coercion was limited at about the same time compulsory education became the rule.

The presumption that students attended school voluntarily became void just when teachers were forced maintain their authority through persuasion and other leadership qualities. Discipline took on a different coloration under such conditions: teachers had to learn how to ‘motivate’ students regardless of whether they or their parents wished them to be in school.

Granting the premise that teaching has changed less than your typical profession (e.g. farming), why hasn’t it?

Lortie calls this conservatism, and he argues that resistance to change built in to the profession at pretty much every level. Students go to school and decide to become teachers. If they hated school, though, they don’t become teachers. These school-loving dorks partially choose teaching because of the service element, love of schools, and flexible hours. (“Teachers are sensitive to criticism about this.”) Is this a recipe for radical employees who are eager to rewrite the rules of school? No, Lortie argues, this is a recipe for a profession of people who basically like teaching the way they were taught.

I wasn’t so convinced by this. Sure, teachers go into teaching because they like the job as it is. But isn’t this true of literally every other chosen profession? If you go into medicine presumably you tend to be ok with the way medicine is presently delivered to patients. Maybe the professions that should experience the most radical employees are the ones that aren’t chosen, then?

The other plank of conservatism is the lack of strong teacher preparation. We sort of slide into teaching, mostly, even if we go through traditional teacher training. That’s because teacher training doesn’t fundamentally alter our prior ideas about teaching. Lortie calls it [THIS IS THE PHRASE THAT EVERYONE CITES PERPETUALLY FROM SCHOOLTEACHER] the “apprenticeship of observation.”

This might be a good point to mention that the book was published in 1975. Lortie keeps on mentioning that school unions are just starting to collectively bargain and the Women’s Liberation Movement is shaking things up. If you don’t believe something that Lortie says, you can always dismiss it on the grounds that it’s outdated.

The inherent conservatism of teaching would be much more worrisome to me if I had the perspective of a reformer. I don’t, though. I’m much more interested in the lived experiences of teachers. So it was the second half of the book that interested me more.


Before reading Schoolteacher I was familiar with the idea that teaching is a “special but shadowed” profession. (“Special but shadowed” is another catchy Lortie phrase.) To me, “special but shadowed” meant that in the eyes of outsiders teaching is both valuable and undesirable work. Lawyers have reputations for slime, traders for greed, but teachers are saintly in the eyes of others. But we’re also seen as refugees from more competitive jobs where we couldn’t cut it, and the money we make (good but not great) reflects this status.

(Interestingly, Lortie actually finds that a lot of his teachers are refugees from more competitive professions: “about a third of Five Towns [the district he interviewed] teachers reported that they had wanted to go into another line of work but were unable to do so because of external constraints.”)

It’s other people who see teaching as less-than. But after reading this book, what I came to think was that teachers see ourselves as doing special but shadowed work as well.

You can see this in the things teachers told Lortie in their interviews. Over and over again in these interviews, teachers reveal themselves to have work ideals that they aren’t able to live up to with any regularity.

Math teachers these days talk a lot about reaching every single student. Teachers in Lortie’s study shared these universalistic ideals too. To see how teacher expectations align with these ideals, Lortie asks a clever but indirect question — Please recall some occasion when you felt especially proud of something you achieved as a teacher. Please tell me about it.

If our expectations aligned with our ideals, a good day of teaching would be a day when the whole class understood something. But teachers, Lortie found, invariably mention successes with individual students rather than whole-group:

The most provocative difference between responses to the pride question and responses to other questions lies in the scope of outcomes claimed by the teacher. In speaking about their ideals, respondents emphasized reaching all students; some teachers, in fact, made such universality the focus of their answers. But the occasions associated with pride, in all but one instance, involved a single student or a small number of students. Pride, in short, is generated by ‘elitist’ outcomes, which are overtly rejected.

When Lortie asked teachers directly for their ultimate aims (I know it’s not easy to state clearly, but would you try to explain to me what you try most to achieve as a teacher? What are you really trying to do?), teachers talk about learning and achievement. But when he probed teacher expectations in indirect ways — Describe an outstanding teacher — teachers responded as if the best you could as a teacher is to nail the interpersonal element:

The elaborations elicited by direct questions concentrated on the ultimate outcomes of instruction, on learning changes in students. But when we ask teacher to describe outcomes achieved by outstanding colleagues, they emphasize results of a proximate and relational nature.

So on and so on. Name an ideal of the profession. How about “fulfilling each child’s potential?”

School systems often advertise their goals as including, for example, ‘the full realization of every child’s potential.’ It is clear that the aims of classroom teachers are less exalted.

I had always known that schools are placed in impossible positions by citizens and society. Schools are tasked with fixing democracy, creating citizens, Americanizing immigrants, training the workforce for everything and nothing, solving inequality, battling racism, being the ultimate solution to segregation and a million other problems that adults find intractable in larger society.

What I hadn’t considered was what this looks like for teachers. I had sort of imagined that teachers resist all of this, or at least that we understood what we can hope to achieve in our work.

What Lortie is saying is that teachers aren’t rejecting this rhetoric. We incorporate it in the form of our ideals and values. But we aren’t able to live up to them, because the form of the work makes living up to these values impossible.


You might wonder why teachers don’t just reject the crap rhetoric. Reach every child. Create a love for learning. Help children become curious people. If the lived reality of teaching is that — at best — it’s impossible to know if you’ve ever achieved any of these goals, shouldn’t the profession re-calibrate its expectations?

I think for Lortie the answer is about the way teachers are dependent on schools as institutions for our work. There are no independent classroom teachers, like doctors in private practice. Every teacher depends on being employed by a school, and every teacher has to suck it up because status-wise we’re controlled by administrators who are controlled by superintendents who are controlled by parents who are part of the same crazy democratic society that came up with all these unreasonable expectations in the first place. It’s a nice story, if one that I don’t know how to confirm.

In any event, though, here’s what we end up with: teachers are in professional tension. We have sky-high ideals and we also recognize that they’re rarely met.

My favorite chapter in the book describes this as “endemic uncertainty.” We never know if we’re doing a good job. The ideals are rarely fully achievable, so how do we know if we’re doing a good job? “Uncertainty is the lot of those who teach,” he says, and it’s a very good line.

How hard is it for teachers to figure out if we’re doing a good job? So hard that they had to change their questions about assessing outcomes because teachers kept on losing it and walking out of the interviews. Even with their revised question, prompting teachers to think about whether we’re achieving anything leads to some of the most depressing meditations possible on teaching:

I feel very inadequate and hopeless at times.

I do wonder, at the end of every single year, how much good have I done? And it’s hard to see.

You can go on for an eternity with nothing. They seem to be regressing.

This is all especially troubling for teachers because, for a lot of us, it’s the emotional rewards of teaching that mean the most to us.

A finding of Lortie’s is that teachers are focused on the present. The future and the past provide less consolation to us than they might be in other professions. We can’t look forward to that big raise, and we can’t try for lasting achievements. A lot of us won’t last a career in the classroom. As for the past, because of how easy it is to become a teacher and how uncertain we are of ever having achieved anything, we’re also not inclined to feel especially good about what we’ve already done. That puts a lot of pressure on the now. And right now, all we have is the kids in our room. When it’s going great we feel good. When it’s going badly we feel like crap. The past doesn’t matter, neither does the future. We’re present-oriented, and that means we’re chasing the rewards of the present.

But because of the endemic uncertainties of teaching, those rewards are rare.


This was the new, big idea I got while reading Lortie: teachers tend to be divided on ourselves. Our ideals and our expectations are majorly out of whack. And, if you believe Andy Hargreaves, this causes teachers to experience a tremendous amount of guilt.

This idea is one that I think helps me understand a bunch of things that lately I’ve been finding puzzling about teaching.

WARNING: This is just me making stuff up.

Teaching Puzzle #1: Teachers often talk like educational progressives, but teach like traditionalists. This is weird, when you think about it. Why talk about what “we discovered” in class if what actually happened was you asked a bunch of leading questions and one kid answered them?

This makes sense, though, because teacher ideals are different than teacher expectations. We should expect teachers to talk in the language of our ideals, especially when we’re around administrators, coaches or parents that represent those ideals.

Teaching Puzzle #2: Traditionalists about teaching make a big deal out of their differences with progressives, even though their classroom practices seem pretty indistinguishable from progressive practices.

But this makes sense too. Traditionalists reject the ideals of teaching and the rhetoric that results from those ideals. For example, you sometimes get people debating whether every child can do math at a high level is true or not. People who think that it’s really important to reject this are trying to free themselves of the oppressive, guilt-inducing ideals of the profession.

These seem like the two options for teachers who try to talk about teaching: reject the ideals or reject experience.


Another thing that people take from Lortie is his observation that, unlike other professions, teachers haven’t developed a big body of professional knowledge about how to teach. Teachers all pretty much agree that you learn to teach on the job, and that seems to be pretty much the best way to do it.

Everyone has an explanation for why this is. Lortie’s is complex. Teachers are focused on the present (presentism); they’re expecting to leave after a few years, or they’re expecting to take breaks from the job; they’re more-or-less happy with the way things are (conservatism); they don’t see other teachers teach very often (individualism); they don’t even know what success looks like.

Lortie doesn’t say this (or at least not strongly) but I think the ideals/expectation gap is a big factor. It’s the gap that produces that potent cocktail of guilt and doubt. Our belief in the ideals of teaching creates a huge difference between what we think success in teaching is supposed to look like and what we experience daily. What teachers really have is a set of tools we’ve developed for making painful compromises on our ideals.

When professors/teacher educators/consultants/coaches talk about teaching, it’s always about how to reach those ideals. (See: ambitious teaching.) So we get tips for improving our teaching, and even if they work teachers don’t believe that they work because we still haven’t met the ideals. We’re never succeeding, so we never know that we know anything.

Traditionalists say, great, time to throw out the ideals.

But who wants to throw out their ideals? Not the majority of teachers. The majority of teachers really do think that the goal of school should be to help every child discover a love of learning. (Hey, I believe that too.) So the traditionalist option is simply not available for the majority of teachers.

Traditionalists say, OK, let’s change the culture and make it an option. (They don’t say it like that, but that’s what I’d imagine some would say.)

The cost of producing technical knowledge: either your ideals or your reality.


There’s a missing choice in all this, I think. What if great teaching is about finding great ways to navigate the space between your ideals and reality?

Call it elevated nitty-gritty talk about teaching.

It’s not possible to reach every student in your class; this is an unreachable ideal. But it is an ideal — reaching more students is better. What’s are ways to move in that direction?

Everybody talks as if the goal of teaching is maximizing knowledge; it’s not, and teachers know this. We care about relationships in ways both selfish and not. We want kids to be happy in school. The ideals of schooling point towards the future — but teachers are focused on the present. This puts us in tension, but it’s a tension that we ought to have as we both want to prepare students for their futures while also helping them through school in humane and kind ways. This tension isn’t incidental to the work of teaching. It is the work of teaching. The point of teaching is balance. What are especially good ways to find this balance?

We want kids to learn stuff, but we also want them to have a love of learning. We don’t want learning to be painful. If the knowledge-maximizing teaching method turned out to be hooking up a small charge to a kid’s ears and lightly shocking them, I’d rather teach inefficiently, thank you very much. Teachers mostly don’t want learning to be painful. What are ways of teaching that balance the tension between difficulty and pleasure in learning?

(If you’re saying that you can have both, congratulations, you are speaking on behalf of the ideals of teaching.)

I think teachers could develop technical knowledge that looked like elevated nitty-gritty knowledge.

You have to assume, in the history of teaching, that pretty much everything has been tried. I’m sure there’s a good reason why this couldn’t possibly work.

Part of the reason why it might not work is because of the pressures on teachers to fully accept the ideals of schooling. What parent wants to hear that teachers consign some kids to failure (not what we mean) and what administrator wants to defend that sort of talk? And those who hope to influence teachers — reformers, teacher educators, consultants and researchers — fiercly defend the ideals of the profession at every conference and in every article. We might turn Lortie’s conservatism on its head here — these are precisely people who elected to leave teaching to promote the ideals.

But what about in teacher spaces online? Freed somewhat from the pressures of the ideals, might we find a little bit of space between the nitty gritty of the classroom and unrealistic dreams?


I have five books on my shelf that are all about why it’s so hard for reformers to change teaching*. They all draw heavily on Lortie, and it seems to me that this is what Schoolteacher is ultimately about.

* Tinkering Toward Utopia; Someone Has To Fail; From the Ivorty Tower to the Schoolhouse; Changing Teachers, Changing Times; Inside Teaching. I haven’t seriously read all of these.

I don’t know if I agree with everything in Schoolteacher, and I’m definitely sure I didn’t follow everything Lortie said. The book can get a bit listy at times, especially in those opening chapters.

Anyone who’s interested in changing teaching at any level ought to take a look at Schoolteacher. It has a lot of ideas, and a lot of pretty convincing quotes to support the latter chapters. I wasn’t always sure I could follow his arguments carefully enough to be sure they were true, but that didn’t bother me much. Lortie was trying to open up brand new space for others to explore, and based on my bookshelf and my read, it seems he succeeded.

How to Start Something In a #MTBoS

Why You Should Listen To Me

You shouldn’t.

OK Now Here’s The Post

Is it helpful or annoying to publicly muse on a community’s ability to get things done? Keep reading to find out!

The other day I was chatting on twitter, and out of the conversation came an idea to start a local math circle.

The other other day I was chatting on twitter, and we had the idea to start an organization that would help conference organizers find not-just-white-men to speak at their meetings.

What happens to these ideas? What do you need to do for them to actually happen?

The MTBoS has created and organized a lot of impressive things. Of these, Twitter Math Camp is objectively (objectively!) the most impressive. There are, of course, many others. I’m biased by my own involvement, but I think the Global Math Department is pretty impressive as well.

Every once in a while, these days, I hear about a new curricular website that somebody has put together. The single-purpose sites are collaborative, but not in the way that Twitter Math Camp is. Estimation180 or Open Middle or Visual Patterns or Math Mistakes all take ideas and materials from the community. There’s one or two people at the core, and then a community at the periphery.

To run an annual conference requires coordination of an entirely different sort. A significant number of people need to get their act together, together.

It seems to me that much of the ambitious coordination happening in the MTBoS right now is at the corporate level. A lot of the relational capital (so what if I made up a term) is being drawn into large organizational structures like Desmos, Illustrative Mathematics, NCTM. This is great — I’m a fan of each of those organizations.

My guess, though, is that this has slowed the pace of community organization at other levels. It takes a lot of people who trust each other and can get stuff done to make something happen. It seems to me that those people are getting busier and busier with their other highly-collaborative projects.

We’ve reached the part of the post where I speculate on what it takes to get a collaborative project off the ground in the MTBoS right now.

Step One: Decide that you actually care about this enough to make sure the ball doesn’t drop and the conversation doesn’t end. When conversation happens through asynchronous chat there’s always the chance that you’ll get ghosted. Every project that I’ve been part of lives or dies depending on whether there’s someone like this, someone who never says “sorry I’ve been crazy busy lately.” If this idea is going to happen, it might as well be you.

Step Two: Find one other person who you think is pretty close to your degree of commitment. Of course, you’ll never know, and people always are more committed at the start than they actually will turn out to be. This is normal, in my experience.

Step Three: Have a video chat or phone conversation ASAP. A phone conversation is best. The reason is because (a) having a phone conversation with a stranger is uncomfortable, and doing something mildly uncomfortable helps build trust and (b) you’re giving them your actual phone number and that (as ridiculous as it is to say) counts as an trust-building act in our near-dystopia too.

Step Four: I’m out of advice. Try to meet IRL if you can. Try to get more people involved, if you can. But the benefits of adding more people to your team, at first, are low. The main thing you need a collaborator for is to get past the part of your head that’s saying this idea is stupid and you’re an ego-maniac for thinking it was worth doing in the first place.

Two people is enough to start.

Step Later: Once things get going and you have a bunch of people involved, frequently test your group’s ability to keep working without you. Plan to step down, or to work on a side project for a year or two while someone else takes the lead. It’s sort of this sad thing where an organization takes a single high-energy person to get it going, but unless that person reduces their energy input the organization will never produce their own energy and become energy self-sufficient. (Yep, took that metaphor too far.)

I don’t know if this is true, of course, but it’s what I’ve seen so far.

It occurs to me that maybe I’m taking twitter too seriously again. True, I see a lot of ideas that never come to fruition. It’s true that the ephemeral nature of the Stream makes organization hard. But maybe that’s precisely what allows so many ideas to pop up. People would share far fewer ideas if they actually were intending to commit to them, and that’s fine.

But I think there’s room for more organization in the MTBoS than I currently see. And while I don’t think we should force anyone to do anything with their online experience — if people are happy with the way things are, that’s fine too — I suspect that people would take pleasure from forming the deeper relationships that come from doing something good together.

We have so much energy for talking about social justice, and yet there are no groups (that I know of) committed to promoting social justice in MTBoS. I think about this a lot. I don’t think it’s hypocrisy. I think we’ve communally lost the knack for organizing ourselves.

We aren’t aided by our tools. The technology we’re working with has been designed to get us to keep us using that technology. We need to fight that tendency if we want to form lasting groups online.

Back to writing a homework for tomorrow. Good night!

3rd Graders Tell You Whether or Not Fractions are Numbers

I’ve been thinking lately about what it means to be a number, for kids. My guess is that, for kids, something is a number if you can perform arithmetic operations with it. So, for example, when fractions are first introduced they aren’t seen as numbers, since you can’t count by them, add, subtract, multiply or divide them. Being a number involves living the life of a number, so to speak, and that life involves counting and arithmetic, mostly.

(Related: maybe measuring should be in that mix.)

If this idea is true, it would go counter to the conventional math education wisdom. When math education speakers or thinkers want to help kids realize that fractions are numbers, they usually recommend number line activities. The thinking seems to be that if we can show that fractions mingle with whole numbers, this will lend them legitimacy.

I think that kids find this unconvincing, though. I think being a number has more to do with whether the actions you can perform with numbers can also be performed with this new thing.

Or maybe it’s a 50/50 proposition? And both are important? Anyway, I wanted to look into this.

I asked my 3rd Graders, who have been studying fractions, whether they thought fractions were numbers. I asked them to write a few sentences explaining their thought. (I had to work to get those sentences. A bunch of them were very unhappy about writing.)

So, are fractions numbers?

Here are some of my favorite responses:

  • Yes, because to make a fraction you need numbers.
  • Fractions are a certain category of numbers because without numbers fractions would just be lines.
  • I do not think this is a number (1/2) when the 2 and the 1 are numbers, but not together, for 1 + 2 = 3 number, but this is not a number it is a half.
  • Yes a fraction is a number because a fraction is a number of pieces of something.
  • Fractions are numbers because I don’t know to explain it.
  • Fractions are numbers because you could add up to a number like 1/2 + 1/2 = 1 and a whole is a different way of saying 1.
  • NO a fraction is not a number a fraction is only part of a number.
  • Yes, because a fraction is a part of a whole number.
  • I think fractions are numbers because all math is numbers.
  • NO I think fractions are half of numbers. 1/2 is a fraction. 1 is a number.
  • A fraction: a problem of numbers in groups. Therefore not really a kind of number, but closely related.
  • Are fractions numbers?! Yes because 1/4 = 6 ’cause fractions are numbers but the bigger the number the less it is.

I have no idea what that last one means either.

So, how did my hypothesis hold up?

One kid used the fact that you can add fractions to make a whole number to decide that fractions are numbers. Though it wasn’t the mere fact of addition that helped this kid — it was the way that fractions could relate to whole numbers.

Reading these responses leads me to revise my hypothesis. Kids decide that fractions are numbers when fractions/whole numbers co-mingle in equations and problems.

The number line still, I think, is not necessarily relevant unless you count along the number line. Meaning, it’s not about the number line, it’s about the number line as a representation of counting that involves both whole numbers and fractions.

After asking my students to write their sentences, I led the class in some counting that I marked up on the number line: 0, 1/4, 2/4, 3/4, 1, 5/4, 6/4, 7/4, 2…

This, along with other arithmetic operations that involve both fractions and whole numbers, will help my kids increasingly see fractions as numbers.

UPDATE: I talked to my 3rd Graders about this again today. I just reminded them of the question, and then the room sort of exploded with arguments. Here are the arguments I heard.

When two parents have a kid that kid is still a person. So that’s like when you take a 1 and cut it up into thirds it’s still a number.

You can make numbers out of fractions like 1/2 + 1/2 so those have to be numbers too, because otherwise how could they make a number?

I like that these two arguments are sort of the flip sides of each other. I told my class that to me one of these sounded like “fractions are parents of numbers” and the other “fractions are children of numbers,” which a couple kids found hilarious.

One girl said that it really depended on the fraction. People talk about having half of something all the time, so half is a fraction. Other things — like 7/13 — that wouldn’t be a fraction because nobody talks about those. Later, I tried to put pressure on this position: what if I made a recipe that called for 7/13 of something? She said it really depended on what everybody did, not just me.

Her take reminds me of debates between prescriptivists and descriptivists about language.



[NBCT] Student and Professional Needs

So, you did it. You’ve gathered information about your kids from a lot of people. And then you used that knowledge to assess the class formatively and analyzed those results. Then you had kids self-assess and used a summative assessment to show that they actually learned something from this unit.

Congrats! You’re two-thirds of the way done with Component 4 of National Board Certification.

The final third has been hard for me to make sense of. It’s called “participation in learning communities,” and apparently proving that you participate in learning communities requires filling out a bunch of forms.

Screenshot 2017-03-14 at 2.27.04 PM.png

(Successfully pursuing NBCT may or may not show that you’re a better teacher, but it definitely does show that you’re willing to fill out a lot of forms.)

Cynicism aside, one thing that was driving me nuts was trying to figure out what the difference between a professional and a student need is for NBCT. Especially since you’re supposed to provide evidence that addressing your professional need impacted the students. Doesn’t that mean that every professional need is also a student need?

I’ve broken the code, though. The key is this passage:

Screenshot 2017-03-14 at 2.35.03 PM

This distinction aligns perfectly with the differing requirements of the professional and student need submissions. For the professional need you are supposed to describe something you needed to learn and show how you used colleagues/others to learn it. For the student need you don’t need to learn anything — you just need to recognize and identify something that would make a difference to kids in your school, and then you’re supposed to impact your colleagues/others.

Which is why you don’t necessarily need to provide evidence that the student need impacted your kids. This is teacher as advocate and leader, affecting your colleagues. When you’re a learner you need to be affected by your colleagues, and show an impact on your kids.

(There are parts of this process that I don’t enjoy, but I won’t pretend I don’t love the exegesis. Sue me.)

So, yay, I understand what I’m supposed to do. How can I do this? I’m usually pretty deferential around the office, and “teacher as advocate” doesn’t sit well with me. That said, why not share ideas with my colleagues? It would be good for me to do more of that, especially in, oh, the next month or so.

Here’s what I’m thinking.

Teacher as Advocate: Better Middle School Geometry Experiences

I’ve taught high school Geometry at my school for the last four years. It’s the course I’ve taught the most. And while kids do alright in my classes, I think our school could be better preparing kids for their high school geometry work.

First, they often come in to geometry without having thought much about angles as rotations, or as angles being greater than 180 degrees.

Second, they have inconsistent experiences with the Pythagorean Theorem.

Third, they have had inconsistent experiences with the relationship between shared angles and similarity.

In the next month I’ll try to share some of the things I know about middle school geometry with my colleagues. There are three things I’ll do to advocate for geometry in Grades 3-8 (which is what we cover):

  • Create and share resources for various geometric topics, and ask some of my colleagues to share them with their students and tell me what they find out about their geometric knowledge.
  • We have a shared curricular space in our department. I’ll make a page to share some geometric resources that are appropriate for various grade levels, and try to better organize some of the things our department is already doing.
  • I’ll share some of this work at one of our math department meetings.

Collecting evidence is always really tricky with these portfolios, and is most of the reason why I end up submitting at the last minute. (I find that things never really work out when I try to collect evidence after the fact. I need to know what I’m doing so I can collect evidence during the process.)

What could evidence be for this? It’s tricky, but I could collect student work from any assessments I make from my 4th, 8th and Geometry classes, and I could also try to collect the student work of any colleagues who try out my assessments. I’m also thinking that maybe, if I share a collected activity page via email, I could take snippets of any emails people write back to me about the resources I’ve shared.

This plan is a B+ plan.

Teacher as Learner: Learning Disabilities and Proofs

I’ve got a few kids in my geometry classes who have learning disabilities. I don’t know how to support them with understanding and creating complex proofs. These kids have attentional issues that are related to low working memory. This makes it hard for them to e.g. keep in mind the premise of an argument, or e.g. an earlier diagram after it’s been transformed into some new one.

I might as well focus on the next thing that I’m trying to teach — proofs of the Pythagorean Theorem. Is there any way that I can help my kids with learning disabilities make sense of these sorts of arguments?

I’ll ask the learning specialists in my school, but I’ll also ask smart people that I’m connected to online. As evidence of my learning I’ll excerpt our conversations. And as evidence of the student impact, well, hopefully I’ll help more of my kids learn these sorts of proofs.

And then if I can do that, then I just have to write the 12-page written commentary and I’m set.

And then, with the permission of an anonymous Pearson grader, I’ll be NBCT.

Sunday Poem – The Boatman

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(Source: Poetry Foundation)

Click through to the link and you can hear the poet reading it, which I recommend. I’m in the middle of understanding this. Here’s where I’m at:

  • The big, juicy lines here remind me of The Odyssey and other epic poetry. It seems likely to me that The Boatman is pointing us to Odysseus, and asking us to see the refugee as not just a victim but a person of ultimate bravery.
  • I don’t know anything about meter. Does the meter of this poem more specifically suggest epic works?
  • If that’s right, then the poem doesn’t go so far as to frame the refugee as entirely heroic, I think. The poem is clear that the refugee has been forced to the sea in the absence of any other options. What, then, does their epic heroism consist in?
  • (I have no clue. Maybe the poem suggests that real bravery consists in facing the horrors you’ve been dealt as opposed to seeking them out, from some love of adventure.)
  • The line-break that surprised me the most was “we fetched a child, not ours, from the sea, drifting face-//down in a life vest, its eyes taken by fish or the birds above us.”  Why break there? Why not keep “facedown” in the line?
  • I’m definitely going to read this again. What are your thoughts?

[NBCT] Self-Assessment

In the previous episode, I had given my students a formative assessment task and analyzed their responses by strategy.


I had no idea, though, how to do a meaningful self-assessment.

I liked what I ended up doing. Here is what I asked my students to work on today:


The figures are (again) taken from this Shell Center activity.

For as long as I’ve been thinking carefully about feedback and assessment, I’ve had a hard time getting excited about self-assessment. The whole point of assessment is that the assessor can direct your attention to things that you yourself have not seen. That makes self-assessment a pretty tough tool to use while you’re learning something.

If I had asked my kids “what could you have done better on this task” that would be lame. If you knew how to do something better, wouldn’t you? Unless you were lazy or tired or careless, and I’m interesting in teaching math. That other stuff is very rarely math.

My way out, though, is to reframe self-assessment as “assessing your own stuff against some other standard.”

I had noticed that some of my kids, on the initial task, weren’t finding the area using interesting structural features. Instead they were counting individual squares. That would hurt their ability to understand what the Pythagorean Theorem is saying (they wouldn’t be able to quickly check square areas) and would also hamstring their ability to understand proofs of the Pythagorean Theorem (since the proofs use those structural features).

Here is a self-assessment response from a kid who counted individual squares the first go around:


I would have described this student’s work on the initial task as strong, and her self-assessment was strong as well:


I thought this student could have dug a bit deeper.


I’d say that this activity worked at the level of “compare your approach to these two nice approaches that use nice geometric structure” for everybody, and then the rest of the activity worked well for like half the class.

So, I didn’t fall in love with self-assessment today. But I did figure out a way to do this part of my NBCT portfolio in a way that didn’t make me puke, and that I think helped kids learn something.

That counts as success, right?

Sunday Poem

Screenshot 2017-03-05 at 11.46.31 AM.png

Here’s my best read of this week’s poem: you have the frustrated artist, who finds himself completely capable of artfully representing reality but failing to apply that same art to imagined, fictional creations.

The poet contrasts painting with photography. Painting is what he aims for; the photographer’s lens merely captures snapshots. The snapshots are clearly attempts at assigning importance to the moments they capture, but frequently fail to do so effectively because they’re constrained by what actually happened (“paralyzed by fact”). The poet’s own work often feels just like photography, limited in this way.

But is this really a limitation? Vermeer is a painter who provides such realistic portrayals of reality that they seem almost photographed. And yet, they have a powerful effect. The artist — the photographer, the painter, the poet — does something wonderful, the poem concludes, by accurately representing actual moments. Our entire lives are composed of such short, delicate moments, and the photograph snapshot allows us to assign each of those unimportant moments some importance.

Thoughts and questions:

  • Why is the poem called “Epilogue”? Does it come at the end of some other work?
  • What does “all’s misalliance” mean?
  • Oh, wait, here’s a short essay about the poem.
  • From that essay: “the poem also recalls the classical recusatio (refusal), in which the speaker claims he is unable to write the kind of poem the occasion calls for.” That’s cool! I want to read more recusatios.
  • So is this poem imagined or recalled?
  • What does it mean to “tremble to caress the light”?
  • I like that line break — “I want to make // something imagined, not recalled.” But read the line break and it’s not just a felt inability to write imagined poems, but an inability to make anything at all. (As opposed to just recalling, I suppose?)


How Medicine is Like Education

Everyone’s favorite edu-game is comparing education to medicine. Do teachers garner the same respect as doctors? Should teachers be more like doctors? Is education like medicine?

I have no idea, and I sort of hate this game.*

In 2014 there were ~700,000 physicians and ~1,500,000 kindergarten and elementary teachers, with ~961,000 high school teachers. Which is just to say that there are a lot more teachers than doctors. (Source: BLS BLS BLS)

With the above as a caveat, a few things I recently read about medicine made me think of teaching. Not that they are definitive or representative of medicine, or anything. Just that they made me think. I have no idea how to put all the pieces together, but each made me think that in some little ways medicine might be a bit like teaching.

Doctors Don’t Have Time to Read Research and This Makes the Field Subject to Destructive Fads

I’ve been reading Dreamland (“The True Tale of America’s Opiate Epidemic”) and it’s consistently fascinating. Part of the story is about how the medical profession fell in love with opiate pain-killers after years of caution. How did this happen? Several trends converged, but an important part of the responsibility lies at the feet of the medical profession:

For many years, it was believed that pain protected against the development of addiction to opioid medications.


This false belief (med-myth?) had its source in some bad interpretations of research.


I read this page in Dreamland this afternoon, and the story felt familiar. Even more familiar was this tag to the story: “To actually look up Porter and Jick, to discover that it was a one-paragraph letter to the editor, and not a scientific study, requires going to a medical school library and digging up the actual issue, which took time most doctors didn’t have.”

Could the troubled relationship of research to practice be similar across many fields, and not just in teaching?

Medical School Doesn’t Prepare You for ‘Real Medicine’ and in ‘Real Medicine’ Often There is No Cure

House of God is a novel, a satire. Let’s start there: you shouldn’t take it literally.

It’s also from the 70s, and it is a very 70s book. Everyone’s constantly talking about Nixon. Everybody’s smoking. And everybody is having sex with everybody.

Seriously, when I told my doctor friend I was reading this book he took great pains to explain, You know it’s nothing like that anymore, for one there is way less S.E.X. (He spelled it, to save my toddler the embarrassment of hearing it I suppose.)

OK, so it’s somewhat dated and it’s satire. Still, it has passages like this:

Again, like the day before, most of what I’d learned at BMS about medicine either was irrelevant or wrong.

Sound familiar? Could it be that complaints about the relevance of professional training are more common in medicine than teachers usually think? (The question isn’t rhetorical. Could it be?)

The thing that really caught my attention, though, was this passage:

Talking about medicine, I told him with bitterness about my growing cynicism about what I could do, and he said “No, we don’t cure. I never bought that either. I went through the same cynicism…And yet, in spite of all our doubt, we can give something. Not cure, no. What sustains us is when we find a way to be compassionate, to love. And the most loving thing we do is to be with a patient, like you are being with me.

Which seems to me an entirely correct perspective on teaching as well.

Not quite sure how to do student self-assessment [NBCT C4]

As part of NBCT Component 4, you need to give a formative assessment. Last week, I gave this formative assessment to my high school geometry students:


Here was some of the work that my students came up with:


A wide variety of strategies and ideas, as well as levels of sophistication.

NBCT wants us to study and synthesize the results from the whole class.


This does not worry me. No problemo, there’s a lot to say here. I’m imagining doing something like this (or maybe this itself):


The student work analysis isn’t scaring me. The call for student self-assessment, on the other hand, gives me the jitters (the howling fantods):


The rubric idea? I hate that idea. And collecting a recording and making a transcript feels like a pain in the neck.

What other ways are there to have kids self-assess? Please, share your ideas. Here is what I’ve come up with so far:

  • Do what I always like to when asking kids to revisit their past work: do some whole-group activity that teaches them something related to the student work, and then ask them to improve their work. Usually I hand back a marked-up copy of their work — honestly, I think that’s an important part of revising — but I could just ask them to self-assess?
  • Maybe my whole-group activity could be teaching them different language for their strategies? And then they identify them in the student work?
  • Ooh, I sort of like this idea: what if I gave them a proof of the Pythagorean Theorem and asked them to compare their work to the proof diagram. Self-assess: how is your diagram similar or different from this one?


I dunno. Self-assessment isn’t something that’s an important part of my teaching at the moment, and I’m unsure whether it should be. Either way, I need to find a nice way to ask kids to self-assess for this portfolio.

Any ideas?