Michael Disagrees With Tweets

In what may or may not begin a new series on this blog, I will now (politely and lovingly, I hope) disagree with a tweet.

On the internet, nobody knows if you can manage a classroom or not. Maybe twitter can solve this. Currently, you get a blue “verified account” check next to your name if you did something cool to deserve it, like being rich or popular. Maybe we could have something like that in education. (I’m a verified red apple educator!)

Until then, there’s no way to tell online who can or can’t run a classroom.

I suppose it’s true that someone who has never run a classroom probably can’t, and these people shouldn’t try to tell you about managing behavior. But take Tom. I don’t know Tom. I have no idea what sort of a teacher he was when he was in the classroom. How would Tom’s standard apply to Tom? How can I know if Tom can run a classroom or not?

This is always how it is with teaching. We don’t have access to each other’s classrooms, so we can only rely on each other’s descriptions of teaching. That’s true for everybody, teachers and non-teachers alike.

This matters a lot more to ex-teachers than to teachers, I think. The relationship between teachers and non-teachers is complicated. You might think that teachers are just suspicious of non-teachers, and that’s true, but we also care the most about what some non-teachers say. Someone on twitter once pointed out to me that classroom teachers are generally suspicious of non-teachers but very trusting of a few chosen non-teacher experts who have credibility. This struck me as totally true.

As a consequence of all this, some non-teachers find it helpful to try to hold on to the status of in-the-know teacher even though they have left the classroom.

To which I say, it’s not worth it. Don’t bother. The kindness that teachers offer other teachers isn’t because of a presumption that this other teacher gets it, or that they have useful information to offer that non-teachers don’t. Rather, I’d say, it’s just that: kindness. I would posit that it’s not that teachers are more trusting of others in the classroom, just that we try to be nice to each other, because the job is hard and knowledge is tentative and we all know how little status we each have. Once you leave the classroom your status has just bumped up in the education world, and that extra-kindness can no longer protect you from the skepticism of other teachers.

Which is fine, because you can still influence teachers in the one way you ever could: by describing what it is that you think will work.


Mental Math Gone Wrong?

Maybe this was a good idea, maybe not.

I was trying to figure out how to start class. My 8th Graders have been studying the Pythagorean Theorem. I knew I wanted to start with some mental math* but wasn’t sure how to start.

This desire to often begin class with some mental math is, at this point, sort of an instinct. On the one hand you need instincts when you’re planning class, because otherwise everything takes forever as you get sucked into a recursive vortex of decision-making. But is it a good instinct? I don’t know how to think about that.

The way I teach the Pythagorean Theorem, being able to mentally chunk a tilted square into triangles and squares (rather than trying to count each square or triangle) is an important part of the skill. It helps kids quickly see the area of squares, freeing up their attention to focus on the relationship between the squares built on the sides of triangles.

Yesterday, we explicitly talked about the Pythagorean Theorem in terms of the area of squares built on a right triangle’s sides. The plan for class was for kids to get better at using it in all sorts of different problems.

So, I decided to build a string of squares built on the hypotenuses of right triangles, and ask kids to find the square-areas in sequence, building up to a generalization. We start: What’s this square’s area? Put a thumb up (please don’t wave a hand in someone else’s face) when you’ve decided. What is the area? How do you know? OK here’s your next tilted square, etc.

image (6).jpg
Sloppy picture so you know it’s the real deal

Here’s where my teaching got sort of mushy. The really important skill isn’t finding the area of tilted squares. What kids really are going to want to know, later on, is the Pythagorean relationship between right triangle sides and areas.

So here’s the question: did this string of problems draw attention to the important math?

Turns out, it didn’t. Kids made the generalization in the last step (as far as I could tell from eavesdropping on their conversations) entirely on the basis of the earlier examples. And those areas were found by chunking up the area. In other words, this was arithmetic-generalization. They didn’t use the Pythagorean relationship.

What were my options, when I realized this? I was happy that kids were able to mentally dissect these tilted-squares, but was a bit disappointed that they didn’t start noticing Pythagorus here. I lost a chance to help them try out using that relationship. Since the rest of the class was designed to help them practice this theorem, it became important for me to prompt their memory of it at the start.

What can you do, right? Impossible to predict kids perfectly. Except that I could have prompted the Pythagorean relationship after the first example didn’t go the way I expected it to. I could have said — after I made sure that students were not going to — that this tilted square’s area could be found using Pythagorus, and then I’m sure I would have gotten more kids to play out this relationship in their minds for the rest of the string.

That’s not what happened, though, so I weakly finished the string with my own personal observation that, hey, we could’ve used PT here. The kids shrugged. OK. I pulled out a quick problem that did prompt kids to use the Pythagorean Theorem, but by then I’m not sure I had everybody on board. We finish, and kids are getting jittery. We’ve used up* whatever whole-group learning time we were going to get at the start of class, so I started problem-solving time.

That’s definitely how I see things right now, at least. Again, I don’t know if this instinct is a good one.

Class went OK after that. But I’m still trying to figure out whether I did this right. Should I have designed the initial string differently? Should I have reacted differently?

(And, the sort of meta-question I have is what exactly it would mean for me — or any teacher — to know how to do this better. Where does that knowledge come from? Can it be shared?)

Two Ideas

Everybody should do more mental math — mental algebra, geometry, calculus, topology, whatever — at pretty much every level of math.

Whenever you’re tempted to write comments on a student’s paper, just pick out one common issue from the class’ work and start class responding to that, somehow.



Stephen King taught high school English for two years:

I wasn’t having much success with my own writing, either. Horror, science fiction, and crime stories in the men’s magazines were being replaced by increasingly graphic tales of sex. That was part of the trouble, but not all of it. The bigger deal was that, for the first time in my life, writing was hard. The problem was the teaching. I liked me coworkers and loved the kids — even the Beavis and Butt-Head types in Living with English could be interesting — but by most Friday afternoons I felt as if I’d spent the week with jumper cables clamped to my brain. If I ever came close to despairing about my future as a writer, it was then.

“Jumper cables clamped to my brain.” I totally believe and experience this. It’s an obvious fact of my life…but is it true? Why would it be?

My work isn’t as intellectually involved as e.g. being a grad student, researcher, journalist, etc. Teachers don’t have to regularly learn new facts or disciplines. We don’t make our living as readers, writers or thinkers.

We don’t even work especially long hours. Yes, yes, endless grading. But even taking grading into account, it’s unclear to me how many extra hours we actually put in. I know, personally, that I tend to way over-estimate my out-of-work hours. I tend to count all sorts of quasi-work into the bucket, like all that time that I’m thinking about looking at student work but instead I’m writing a blog post on a Sunday night.

Grading and planning are like (to get back to King) little evil vampire children that rap on the window while we’re catching a break after a long day. Let us in, they say, you need to.

(I just finished reading Salem’s Lot which stars Matt Burke, veteran teacher, which is how I ended up down this road.)

The Bureau of Labor Services surveyed teachers and instead of asking how many hours they worked in a week, asked them how many they worked yesterday. (This includes out of school work.) The stunning results: responses amount to just under a forty-hour work week. We even, on average in this survey, work less on the weekends than a comparison groups that contains health care professionals, business and financial operations professionals, architects and engineers, community and social services professionals, managers.

I’m inclined to believe the more modest hour-estimates of the BLS, as they fit what I see in myself and colleagues at the different places I’ve taught. (I’d also say that, in the places I’ve taught, there are outlier teachers who just go nuts with work. If the BLS stuff doesn’t fit your picture, you might be such a teacher.)

But I’m also inclined to think that King’s brain wouldn’t be depleted if he were a journalist or a researcher or a bond-trader who worked till ten every night.

(Speaking of work hours and exhaustion: I read King’s Under the Dome while working fifteen-hour work days as a delivery truck driver in my summer after graduation. I’ve never been as desperate for a book as I was while working that job. That book took over my life while I was working — my wife [then girlfriend] still teases me about it. The job involved driving around campus, picking up and dropping off recycled furniture, which sort of wore me out. I was physically exhausted but mentally starving and I’d collapse in bed with that book for some of the most satisfying hours of reading I’ve experienced in my life.)

What could there be about teaching that makes it mentally exhausting? Or is this just standard working-adult exhaustion?

I can’t think of anything, which makes me wonder if Steve and I are making this up.

One thing I know about teaching, though, is that you rarely know if you’ve taught well. And that fits with what I know about writing — that it’s lonely work, done in a quiet space over long-periods of time. Unlike teaching, there are certain stone-cold ways of knowing that you’ve done good work — the acclaim of readers — but the lead-up to that moment (if it ever arrives) is the ultimate marshmallow test.

Maybe this is it: teaching exhausts us in a way that kills our willingness to write.

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.