# Making Long Problems Shorter

Some math questions have long complicated answers. This is a problem for teachers, because students get lost in the middle of complex stuff. They end up not learning and feeling bad about it. This is not good.

This past year I taught calculus for the first time. There’s no way around it: some of those solutions take a lot of steps. Moreover, my class was a sort of intro to calc for people who aren’t especially enthusiastic about algebra. This was a class where kids apologized for being bad at math on the first day of school.

What could I, an award-winning* worked example enthusiast, do to help these students?

* Correction 6/15/2021: I have not won any awards.

One approach, one I don’t favor, is to break a complex problem up into many mini-problems. That way, students only have to think about a smaller part at any one moment. This is called “scaffolding.” It’s the approach Active Calculus takes for some of their problems:

Look, that’s still a lot of words and questions. And you can’t forget part (c) once you solve it — you haven’t learned anything until you take all five parts and make sense of them together. I do not like this, it is bad.

What I’ve started to think is that good teaching sometimes involves turning long ideas into a series of shorter ones. But in a way so that after each relatively short task you have learned something complete, gotten a full meal. The pieces add up to something greater, but each part is a meaningfully whole thought. You can breathe in between each piece without the entire structure falling apart.

For longer calculus problems this year, here was my routine for doing that.

First, I would warm up students to the lesson with any skills they’d need for the day’s learning. I was just hoping to help them notice and remember anything they’d need for what was coming next. For this related rates lesson, I knew that the chain rule was going to be important, so we started with that.

Then, I would slowly develop the problem, making sure everyone understood the question. “If we’re inflating a spherical balloon, what happens to the volume as it inflates? What happens to the radius?”

“Imagine we inflate the balloon with a constant flow of air, maybe using a pump or something? Will the volume change constantly? What about the radius?”

If I want to make sure students get it, I’ll ask them to take a guess as to what the answer will be. “Will the radius grow at a constant rate? Will it grow faster at first and then slow down? Or slower, then fast? What’s your guess?” (There is a lovely animation somebody made to go along with this problem. I projected it while collecting guesses.)

When it was time to dive into the solution, I would first give a “headline news” version of it. I’d project it on the board, and then talk through it.

“Here’s the plan of attack for this problem. We can write a function that connects the radius to the volume. But we’re going to turn that into an equation that connects the rate of change of volume to the rate of change of the radius. How? The derivative! We’ll treat volume and the radius as if they are not constants but functions of time, and we’ll use the chain rule to differentiate in terms of time. Then we’ll sub in all the given info from the problem — which rate of change do we already know? that’s right! — and solve for dr/dt.”

This approach is cribbed directly from Richard Catrambone’s research on “subgoal learning.”

Then I start revealing steps in the solution, filling in the empty spaces. What function do we have that connects the volume of a sphere to the radius? How do we differentiate? etc.

I would give students time to study this, time to explain it to a partner, and time to answer some self-explanation questions about it. I had previously explained in the abstract why we’d need to use the chain rule in the second step — I’d ask students to articulate that principle on their own. And I would ask students what if the diameter were 16 inches instead of 12 — what would the answer be then?

Their job now is to understand the details of this particular solution and connect it to the generalized outline I had already presented.

And once we’re comfortable with this solution, we’ll develop another question and ask students to solve just a bit of it on their own. In doing this we’re giving them something short to think about in the context of a much larger problem. These are “completion problems,” a type of task identified and studied by van Merrienboer (for example, here).

And it keeps going:

And so on:

Anyway, I think a lot of what I do as a teacher is I make shorter cycles of learning compared to a lot of other people. I’ll do inquiry, but for a few minutes. I’ll do worked examples, but I structure it so that it’s made of several self-contained tasks that (mostly) stand on their own. Is that just what we in the biz call “scaffolding”? I guess, but it’s not “scaffolding” in the sense of giving kids lots of help to climb up a very tall structure. More like just building a bunch of smaller structures that are just as valuable as the big tall one.

I don’t know what to call it, but it seemed to work well for some of my highly anxious, algebraphobic calculus students.

# Making practice work with distance learning

My story with distance learning, so far, looks like this:

• You need a plan, and a conversation with Justin Reich helped me form one. Create assignments that kids can work on more-or-less independently, and then try to check in with as many as you can.
• I needed materials for that plan, so I put together assignments with examples, scaffolding and extension challenges. I wrote a post summarizing what I understood after three days of this plan, which was not much.
• After a week spent thinking about the kind of work kids could do in our distance learning set-up, it became clear that I was at risk of coming in contact with precious little mathematical thinking. And when I was getting kids to hand in work, it was often after they had completed an assignment, i.e. kind of late to help them with it.
• During my second week of distance learning, I focused on the workflow. A big goal was to figure out if I could get kids to upload photos of their work, something I deemed essential.
• After that week, I wrote a second post about how I was making sure kids were able to send their work to me. I mentioned three paths that were working OK — typing in the chatbox of Zoom (with all messages privately sent to me), typing in a google doc, or uploading images into a google doc.

And, now, at the end of Week 3, I need to largely retract that second post.

The main reason was because I was largely dismissive of tech tools in those first two weeks. I mean, everything we’re using is a tech tool. But I meant the apps, the endless stream of tech tools that people have been recommending over the past few weeks. Off the top of my head, those include: CueThink, FlipGrid, OneNote, Microsoft Whiteboard, GoFormative, Equatio, and so many more.

I ignored these tools at first for two reasons:

• Who’s got the time?
• I don’t want kids to have to learn a new tool.

But there are two tools that stand out, and those are Desmos and Classkick. Go on over to Rachel’s blog and read her on-target comparison of the pros and cons of each tool. She also has examples of materials she has adapted for Classkick, and she’s a great designer of Desmos custom activities.

My main purpose in writing this post is to apologize for the earlier mistake. At the end of this third week of distance teaching, I want to summarize what my classes currently look like.

***

Students log on to Google Classroom. Twice a week they have live classes on Zoom, and I post the meeting link there.

The other two days, I have a day-labeled assignment waiting for them.

Whole-group interactive lessons on Zoom are probably the smoothest part of this. I use slides, and I’ve become pretty adept at annotating them using Zoom’s tools. I pepper students with questions that they then respond to in the chat. When I set the chat to private, students get a direct channel for sharing their thinking with me. This is a wonderful picture into who is participating, what people are thinking. Teaching is basically a conversation, and the chat makes sure we’re able to have it.

Then, we move into practice. That’s when I have started to lean extremely heavily on Desmos and Classkick.

These tools are simple for students to use because, as Rachel notes in her post, kids can just click a link and go to the activity. They are simple for me to use, because I can take my existing resources and post them online.

Twice this week, I took activities I wanted my 4th Grade students to work on and brought them into Desmos activities. Nothing fancy. First, a decimals worksheet:

As students were working on these practice problems, I was able to watch what they were doing and find ways to have conversations with them. Desmos lets you test a new tool for typing little feedback comments (though kids frequently don’t see them). Most important is the big-picture view of where kids are, something that roughly stands in for those moments when you’re looking around at your students and just watching and figuring out what is going on, can they do this thing?

I next took one of my favorite puzzle pages from the Beast Academy books and ported it into a different Desmos activity:

The Desmos teacher dashboard is, once again, extremely helpful.

Because I have knee-jerk skepticism about tech tools, I was initially dismissive of Classkick. But once I saw it in action, I realized that, in the distance learning context, it is very similar to Desmos. It isn’t built for math (so no math type) but it is built for letting teachers import worksheets, have kids work on them through the computer, and enable kids to ask and receive help on specific problems.

This time I was going even more basic: I just wanted to post a review worksheet for students to work on independently this afternoon. I took a page out of my new favorite collection of worksheets and quickly turned it into a Classkick assignment. It looked like this:

This afternoon, while students were working, I was able to monitor their thinking as it came in. (I was “on call” for questions in Classkick, where kids can raise their hands and request help through a chat box. The chat is great — it feels like AOL Instant Messenger.)

Here was a sample of my view of things:

This is an individual student’s work:

I can leave comments through that chat function, or I can leave notes on the slide itself. A student raised her hand to ask for help, so I came in and left and note and an unfinished diagram on her slide:

***

And that’s it, basically. It avoids the awkward need for students to take pictures of their work with their web camera. I’m still open to students turning in their work that way, but I’m not currently encouraging it. These tools seem to do the trick better.

So, in sum, that’s where I’m at. I’m currently using these tech tools reluctantly but enthusiastically. We’re living in a world where you necessarily have to use a tech tool for your teaching. All I’ve done is realize that a web camera and Google Docs are often clumsier for math practice than these other tools.

So, in short, my teaching this week used chat to make whole-group lessons interactive. Then, for practice or assignments, I used Desmos or Classkick, both of which make student thinking more visible. Which enables me to then make informed decisions about how to respond.

None of which is nearly working as well as teaching in an actual classroom would. But it’s much better than when kids were working on their own, invisible to me, for the longest time. So this is a step forward, and where my teaching is at right now.

# What are the fastest, simplest ways to get student work while using Zoom and Google Classroom?

In case you missed it, here is the current situation:

• NYC, where I live, is at the center of the coronavirus epidemic in the United States, and our governor is saying we are 14-21 days from the apex of the epidemic.
• But, we’re still teaching!
• I teach math to young children (3rd and 4th) as well as to older students (8th and 9th).
• My wife is also a teacher.
• My two young children are what some people would call “loud.”
• And what if one of them slips off the bed while they’re jumping on it? And we have to go to the hospital?

This is my second week doing distance learning. I previously wrote about what I had figured out in the first few days of this wild experiment, and my school now has each class doing two live sessions and two asynchronous sessions every week. The live sessions are on Zoom; I post materials on Google Classroom.

By the end of last week it was clear that my next, big problem was to figure out how to make more of my students’ work visible. To me, I mean.

Before going forward: I am not interested in your favorite app. At least, not yet. There are dozens of apps that aim to help students and teachers communicate over the computer. I’m sure some of them are useful, but I just can’t deal with a new tech tool right now.

Here is what I have figured out.

****

## Before Class

It took me a few days to realize that it’s much easier to organize Google Classroom around classwork instead of announcements. Here is how I’ve started to organize my Google Classroom pages:

I’m organizing things this way mostly so that students can clearly see what they are supposed to do on each day. The instructions are clear: go to Classwork, to the opening assignment, start the next if you are ready, etc.

The most important thing, though, is that each learning activity becomes its own “assignment.” During week 1 I was creating large documents that students were working on over multiple days. This was good in one sense, because I had to post only one thing. But it became very difficult to monitor the progress of kids through the assignment at all. And then it became tricky to modify the plan in the middle of the week by adding on other bits of classwork.

This is a compromise position now, and I think it’s working. It’s certainly easier for me to figure out what I’ve assigned. This also seems to be the correct way to share solutions and notes, rather than trying to organize them by topic.

## During Class

There are basically three good ways to collect student thinking during class.

1. The chat box in Zoom.
2. Creating a separate doc for every student, and asking them to type in the doc.
3. Taking a picture of their work and posting it.

Imagine these three ways of collecting work plotted on a grid. One axis keeps track of how easy it is to share your idea with the teacher. Chat is easiest, moving to a doc and typing is still pretty easy, but it’s pretty tough to take a picture of your work and post it to your teacher.

The other axis is about how easy it is to express yourself using a tool. Drawing on a piece of paper is the easiest, most natural approach. You can type at length in a doc, and a chat allows you only to type quick responses.

Here is how I’ve been using these three methods for communicating with kids over the course of the lesson.

## Opening Problem

I saw a common mistake in my students’ work yesterday, so this morning I wanted to talk about similar triangles couched between a pair of parallels.

This was my “opening” assignment. But how should I collect this work?

Students need to be looking at the image while responding (or else having to hold a lot of stuff in their heads while shifting back and forth between screens). It’s also not hard to type responses to these questions. And I wanted to ask a series of questions, and that might be hard to type in a chat.

I went for “type your response in a google doc” and I think it went OK. Here is what I saw from the students.

My 4th Graders can do this, my students with shaky tech can more or less do this. Monitoring them is a little bit slow, but it’s OK. As I monitor, I start saying things over Zoom about the work. “Hey Anna, got your work. I left you a little comment.” “Jay, good stuff; here’s what you should try for the last one.” You can also leave little private comments on the side that kids can see pretty quickly. Classroom is good at this.

## During a Live Lesson

This is where Zoom chat is great. You don’t want kids going back and forth between Zoom and some doc during the live, most interactive part of the lesson. I’ll quote my last post:

There is a way to sort of do whole-group instruction that opens up a tiny window into how students are thinking. It uses the chat box.

You can change the chat settings so that all chat messages are sent privately to the host of the meeting. At first, I did this because I wanted to cut down on random chat chatter. But then I realized it’s a private way that students can respond to questions.

Here is a routine I’m finding useful during whole-group discussions:

1. Ask a question. I state it, and if possible I also write the question in the chat box. That way, everyone receives the question even if they missed it when I said it.

2. Then, everyone types in their answer to the question. As the answers come in, I acknowledge receipt. “Thanks, Emma.” “Got it, Jake.” I’ll comment on wrong answers without calling individual students out. “Careful, if you’re writing (x – 2)(x -3) that’s factored form.” I can also address individual students: “Tommy, what you wrote is fine.”

3. Then, I’ll share the answer. I’ll take questions, rinse and repeat for however much whole-group time I’ve decided on for the lesson. Attention spans for this vary by class and by age, of course. More than 15 minutes is probably pushing it, I think.

## Classwork or Asynchronous Assignments

If you can get away with using some online practice tool, that’s great. In general these tools are less useful in this situation than I would have hoped. The problems do not ramp up in difficulty; they’re only useful for practicing something that students are already reasonably proficient at. I use Deltamath, but it doesn’t work for my 4th Graders. I’m looking at IXL for the little kids, but it looks like it’ll be very tricky to coach kids through signing up and getting used to the tool — I’m holding off on that for now. ASSISTments is too buggy for me to use right now, and I find it difficult to navigate their reports on what the students did.

If you can get away with using a doc and asking kids to respond in it, by all means. But this is incredibly limiting. To see why, let me take you back to my Sunday night. My 4th Graders were supposed to work independently on an assignment on Monday afternoon. I decided to do review; I found a useful set of fractions worksheets from the Math in Focus books. I really wanted to use them:

But! How are kids going to fill this out in a doc? Do I need to recreate this from scratch? That would take so much work.

Same with an assignment for my 8th Graders — I wanted them to try sketching a parabola. Same with my geometry students — I wanted to see their work setting up proportions. Same with my 4th Graders today — I want to study decimals, and wanted them to shade in 0.7 of a whole square.

This leads me to my final and most important thing to share, which is how to coach kids through sharing pictures of their work through Google Classroom.

## How to Coach Kids Through Sharing Pictures

My first obstacle was getting a “student view” of my classes. Google does not have this feature, and my school only allows us to add people with our school accounts to Google Classroom. And they weren’t thrilled by my suggestion that they give me a second email account under the name “fake_pershan.”

So under my personal gmail account I made a class called “Pretend Math” and registered a new personal gmail account (“fakepershan”) and created an assignment for myself.

Then I shared my screen and walked students through the steps.

That solves the problem for most of my students. It’s not so hard to take a picture with a phone and then upload it to the assignment. I walked my older students through these steps.

BUT: my youngest students don’t have phones. And I can’t get their parents in front of the screen to coach them through this.

SO: my brilliant colleague pointed out that there is a simpler way to do this in docs, with a webcam.

In 4th Grade today I created an assignment focused on teaching photo sharing.

I shared my “Pretend Math” screen and showed them how to use the tool. Then, I showed them how to open the assignment, and then every kid went to this doc. (I created a separate doc for each student, which Google Classroom lets you do.)

The big question is whether their webcam pictures would be good enough for me to learn anything from. The answer so far seems to be “sort of.”

To test whether my 4th Graders could actually pull this off with some real math, I sent them to another brief assignment, this time connected to content our class was learning.

This is what I got:

So, not perfect. And some of them were worse than not perfect.

But this is a crucial option that I needed for working with my younger students. And in each of my classes of older students, I have kids whose phones either are not working or they don’t have smart phones. This is a crucial method that kids can do on their own to turn in photos of their own work.

But, of course, this is sort of annoying. I tried to use it for my opening problem in a class or two and that turned out to be a bit of a mistake, as it took far too long for students to submit their work. I would have been better off using one of the other methods.

## Feedback

Once students have handed something in to Classroom, things are actually pretty good. Google lets you comment on the work itself via highlighting and commenting, but I’ve found it more useful to give a quick written comment that appears under the assignment itself.

I should say that Desmos has released a tool for adding written comments to students in their activities. There are two reasons why I’m not ready to use this for my classes:

• My resources aren’t already loaded into Desmos.
• Desmos is at some other site, rather than my one-stop shop Google Classroom.

But if I had more time and if my students were doing a lot of Desmos, I could see that as being useful. It seems roughly equivalent to what Classroom lets me do with assignments, but it makes it easier for me to assign Desmos “worksheets” for students to do on their own. I’ll need to do that for a lot of graphing activities down the line, probably.

***

Phew! OK, that’s all I’ve got. Good luck, out there, and let me know what you’ve figured out.

# Crossword Puzzles from Beast Academy

I do love the Beast Academy books. My 3rd Graders are working on multiplication, and the Beast Academy books have these crossword puzzles. The kids love them:

(Why do the kids love them? Oh, I don’t know. If interest = “this is new” times “I can do this” then I guess this has enough going on that it feels new. And the puzzle is self-checking, which probably validates that “I can do it” feeling. That’s all I’ve got.)

Every puzzle can both be solved and studied. I’ve made it a habit to encourage my students to ask questions about the puzzles we solve, and I usually do this by sharing a question or two that I have.

My question was, can you make this puzzle using only multiplication?

And my kids’ questions were:

• Can you make one of these crosswords only using subtraction?
• Are the blanks really necessary?
• Could the puzzle be smaller? Larger? Could it be 5 x 4? 8 x 2?
• Could it be shaped like a path?

The next class, I gave students some blank crosswords and asked them to see if they could fill in the blanks in a way that worked.

Which was interesting. But the kids wanted more, so I sat down to make more crosswords in the original style. The original puzzles always include mostly multiplication, and then one addition and one subtraction equation, always on the right and bottom sides.

So I set out to make a few puzzles in this style. I started filling in the boxes, and got stuck. Then I tried again — still not working. I started to get that familiar good/bad feeling that happens with math. It’s the feeling of “oh this is harder than I thought” but also “there might be something here!”

Over lunch, I interrupted two of my colleagues and recruited them into the problem. (I was happy to return the favor after they’ve done the same to me so many times.) We filled out the crossword with variables.

Using these variables, the puzzle is only possible if $ac+bd =ab-cd$. My colleague pointed out that you might factor this a bit and then solve for d:

$d = \frac{a(b - c)}{(b + c)}$

A few things about that equation:

• It means that the whole puzzle is determined by just three of those variables.
• d is a whole number, so (b + c) needs to go into a(b – c).

This is not a ton to work on, but suppose that the sum of b and c is chosen to be a prime number. It clearly won’t go into (b – c). So that means a will have to be a multiple of b + c.

That seems to work!

This leaves me with a bunch of questions, though. Does this characterize all the possible crossword puzzles? I feel like this finds one specific way of getting a crossword that works here, but is it really the only way? Also, I haven’t really thought about whether I could use any multiple of b + c. I think I can, just because it’s worked whenever I’ve tried it so far, but it would be better to understand why.

There’s a math textbook that I like that makes the case that there is significant mathematics that has been developed by teachers, just for the sake of having nice examples to give to students. I always like when that sort of thing happens, a nice mathematical surprise that appears sometimes when you remember to look for it.

# I flipped around this Illustrative Math lesson — it went pretty well!

Every year, this lesson has given my students trouble:

Here were the problems I had with the lesson:

• My students didn’t have strategies for making sense of why the faster bug would go on the bottom. Kids would quickly misidentify which line was which bug, and we’d have to back up and talk about that for a while before doing anything else.
• So they didn’t really get a chance to engage with the math. And I wasn’t sure what the math exactly was, beyond this little tricky graph that puts time on the y-axis.
• Also the formatting was tricky, because the ladybug/ant race happens on one page and the graph on the other. You don’t necessarily need cognitive science to tell you that swapping between two very separated images makes learning hard, but it does.

Each of the Illustrative Mathematics lessons has a Summary at the end of the lesson. It’s good, but meant as a reference — it’s not really designed for classroom use.

So here was my redesign idea:

• Turn the summary into sample student work i.e. a worked example
• Pair that with analysis prompts and a follow-up question i.e. an example-problem pair
• Redesign the actual materials so that the graph and the race are next to each other

Here’s what I did:

We used the original “warm up” from the materials. Then shifted into the example-based materials I created. Then the redesigned activity from the lesson itself.

It went well! Here’s how I know that it went well: most kids got to the extension questions, and the students were able to focus their on those.

That seems to me the basic tradeoff. If you leave ideas a bit more implicit, then kids will spend more time uncovering them. That can be good, mathematical thinking, of course. The other choice is to make things more explicit at the outset. Then, maybe you have a better shot of diving into what would otherwise be “advanced” “challenge” problems.

I usually make that second choice, and part of why is because I think good mathematical thinking can happen with the example-based materials I shared. After that warm-up (where I asked kids to notice as many details about the graphs as possible and didn’t really push the “wonder” question) I covered up the “student solution” and showed my students the “problem” I had created. Then, I uncovered the student work. There was a pause — followed by “oh!” and “ah that makes sense.” There’s a little mathematical thrill you get every time you figure something out — a few kids got that when I revealed the work.

Then I asked kids to talk about it with partner, and then to solve a similar problem with neighbors. I listened in on conversations and was able to figure out if kids were understanding the example or not. A few times I inserted myself into conversations to help. And then I led a discussion about the example where kids shared the following ideas:

• That the ratio between the heights of points on each line that are directly atop each other stays equal.
• That you could also compare points that are at the same horizontal.
• That if the axes were swapped the top line would represent the faster racer, because then the top point covers more distance in the same time.

And the kids most eager to share were not the ones who usually solve problems with the most confidence (and therefore least likely to share if all of this had come through problem solving).

And then we did the activity that had given my kids trouble each year of the past, and they were able to be struggle productively i.e. they had the “compare points on the same vertical/horizontal” strategy. And they got to extension problems.

I’m going to keep looking for chances to do make this same trade.

# What we’re debating when we debate “misconceptions”

Is ‘misconceptions’ a bad word? I’ve had the conversation about misconceptions a number of times, most recently when I wrote this post. Here is a bit from the conclusion:

We see misconceptions in children because it really is true that there’s stuff that they don’t yet know. Noticing this doesn’t have to be an act of violence — in fact, I don’t think that it usually is. Usually it’s like me playing with my son and noticing there’s stuff he doesn’t yet know how to do, even as my mind is blown because oh my god my son is into puzzles! When did our baby turn into a kid?

Is it good pedagogy to ask people who don’t already see their pedagogy as abusive to forswear from using words that they use all the time? Isn’t this exactly the sort of “intellectual violence” that we’re being urged to refrain from? Shouldn’t we start with the way people actually see the world, rather than asking them to use language that is not their own?

That excerpt did not convince anybody at all, but my goal here isn’t to convince. Really all I want to do is bring up something I learned about the constraints of this argument.

There are a couple people I’ve met who have flirted with the idea of cutting out all evaluative language from discussions of teaching, but it’s largely an unsustainable position. You can’t cut out value from teaching, and the thought that you can is a bad mistake. Even if you don’t talk of “misconception” you’re still in need of language to describe thinking that isn’t yet what it could be. Maybe there are no misconceptions, but there is thinking that is e.g. inflexible, procedural, memorized, additive-but-not-yet-multiplicative, trick-reliant, stage one, whatever it is you want to say.

Plus, the math education community very clearly want to be able to understand problematic language and ideas for what they are. We want to be able to call ideas or patterns of thought racist, sexist, colonialist, etc. That’s very different than the “all thinking is just thinking” position.

And so the discussion is only ever about what is particularly harmful (or not) about the term “misconception” and its popular usage. Though people frequently talk about the issues with evaluative language in general when discussing misconceptions, that argument just confuses things. We need to be able to talk about thinking in terms of what it could, even should ideally be.

So there are really just two questions that are relevant for this discussion. Is the term “misconception” particularly harmful, compared to other evaluative language? And even if the term is intrinsically fine, is it used in particularly harmful ways?

I’ve shared my answers, but I’d make the case that those are the right questions.

# YouCubed, Reviewed

This exponents activity is neither original nor at all an interesting version of the idea. It’s no better than what most teachers would make on their own, if they wanted to teach exponent rules inductively.

Better versions of this are readily available in practically any textbook, but Illustrative Math has a totally free and online unit on exponents that does this activity better. It’s less tedious and repetitive and it asks questions to push students towards generalizations, rather than asking kids to churn out rows and notice the structure at the very end (“discovery”).

Yes, it’s at a Grade 8 level, but this lesson is pretty much there too. And if you can wait a few months, you’ll have the high school version available too.

# “Equity” is dead, long live equity

By the time organizations — even organizations whose work I really like — start using the language of equity to advertise their work, it’s a sign that we’ve overtaxed the latest bit of edu lingo. “Equity” is at that point in the edu fad life cycle; it’s beginning to mean just about anything.

I don’t know if there’s anything to do about this. I think this is less about education and more about the corporate world — business lingo isn’t much better than edu lingo. People want to signal that they get it, without getting too bogged down in what exactly “getting it” entails.

The thing I try to remind myself is to be specific and to use familiar, boring words whenever possible. In place of stuffing meaning into abstract terms, I try to put it into sentences. And instead of “equity” I try to talk about the particulars: unsafe classrooms, hot schools, bad water, inexperienced teachers, and so on. This is my personal resistance to the educational world’s endless desire for catchy language, as I think it’s really all we’ve got.

# Some questions about the problem of teachers leaving the classroom

Is it actually a problem for kids? Would schools be more effective places if more teachers on the margins of leaving were to stay in the classroom? How do we know? Is there a correlation between ambitious and teaching skill? What is the correlation?

Do master teacher programs improve learning for a district?

How much of the stress in education about people leaving the classroom could be explained by how uniquely meaningful working with children is? After all, going into management involves a change at work across professions. (Sales managers don’t go on sales calls; you leave the regular police work to get a desk job; you still do rounds occasionally but mostly you don’t see patients, etc.) How much of the problem is that there is a huge emotional gap between teaching and higher-paying work that keeps teachers in the classroom, marginally?

Would people be more effective at their administrative jobs if they were partly in the classroom? Would they be more influential?

William Carlos Williams was a doctor by day, poet by night. No one suggests that there should be more doctor/poet jobs. How do we decide what sorts of jobs their ought to be?

I find this so confusing. What questions do you have? Comments are open.

# Some of my assumptions for communicating about teaching

These all might be wrong, but I think some of them are worth exposing. Maybe you’ll help me see how I’m wrong?

1. When have something I want to say about teaching or learning, there is a temptation to coin a new word that identifies a new concept. I try to avoid this temptation.

Suppose, for example, that I get up at a conference and say “math should be sticky.” There are some risks. First, there’s the risk someone will spend a lot of time puzzling over what I mean by “sticky,” remember the phrase, and have no idea what I meant by it in context. (This happens often — people remember memorable tags but struggle to articulate what they mean.) Probably then I’ll start hearing people say that I believe that you should teach in such-and-such a way because it’s “sticky” when that’s not what I meant. There’s also a risk that my word will have connotations that I didn’t expect. (Oh, you think “sticky” is gross and bad? Oops.)

So as a rule — a writing rule, a speaking rule — I try very hard to only use words that I think everybody pretty much uses in the same way.

This is not easy, because (I associate this thought with Ilana Horn) the meanings people assign to seemingly clear words like “discover” in teaching varies a great deal. I might say “worksheet” and you might imagine “evil packet that kids work on in silence and struggle” and I imagine “a bunch of problems on a page that hit the sweet spot for kids, who are asking questions and talking together about math.”

So it’s not easy, but I do try. It helps to keep an eye out for words (like “worksheet”) that could be misunderstood, and to replace those with context and sentences that make it clearer what’s happening and what I’m imagining.

2. I try to avoid advocating for practices unconditionally. What I mean is that I never say “we should do this in class more!” without suggesting when it might be useful to do that in class. I’m thinking about this right now with worked examples. I think example-based learning is great and cool and fun, but I would never give a talk (I think) calling for greater use of examples in teaching. Instead, though, I would give a talk describing situations that especially call for worked-examples and teaching people how examples can be useful in that context. (Here are two: “examples as feedback” and “examples as models for really complex thinking.”)

Likewise, I try never to talk in general about teaching, or about teaching math in general. I try to stay conditional.

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These two things, I think, make communication about teaching easier. As a consequence, I think it ensures that nobody thinks that I mean something I don’t mean, and nobody thinks that I have solutions to many of their teaching problems, or a message that would revolutionize math teaching.

And, as a further result of that, what I have to say is less broadly meaningful, polarizing and also less popular. That’s the tradeoff, I think. Clarity for popularity.

Addendum: I have nobody in particular in mind with this post, but it was inspired by a lot of the tweets I saw from the NCTM conference. I’ll say that the “unconditional” thing was inspired by advocacy for a lot of the thinking prompts that don’t call for precise answers — numberless word problems, goal-free problems, estimation problems, notice/wonder, etc.

These are all incredibly useful, but (I think) far more useful when a topic is new to a student. So I think the general direction is that these more open prompts are great ways in, but you sort of want to call for more and more precision in your prompts as the learning progresses.

I was once talking to a friend who felt burned by Estimation180. Why, I asked. Well, she was trying to use it every day to improve her students’ number sense, but it hadn’t worked. She was disillusioned.

I’m not disillusioned. I know that Estimation180 tasks are useful in some situations and less useful in others. I have some thoughts about where and when they’re useful in my teaching. I try to stick to talking about that when I’m talking about teaching and estimation.