Some feedback makes kids give up, stop thinking, or feel bad. In my view, this is almost always feedback that doesn’t help learning. And this is why so much auto-grading is bad feedback — it doesn’t help learning. (In general, motivation is tangled up with success in ways it’s difficult to separate.)

Auto-graded work sometimes makes kids feel bad. This is when the auto-grading doesn’t lead to learning, or makes it seem like learning will be impossible. It’s not exactly mysterious why this is. Most of the time when a computer tells you that something is wrong, that’s it. So you’re wrong. What are you supposed to do with that information, as a learner? If you knew how to do it right, you’d have done it right.

What would the ideal learner do when they get the “wrong answer” info? In some cases, they’d take a close look at their steps and try to suss out the error, essentially discovering the correct way to solve the problem on their own. But in a lot of cases, kids get a question wrong because they don’t know how to do it, or they fundamentally misunderstand the problem. An ideal learner in that case would seek out the information they’re missing, from a text, a video, a friend or a teacher.

Auto-grading in my experience works best when it makes those ideal behaviors easier. I sometimes play around on the Art of Problem Solving’s Alcumus site, just for fun. It automatically tells me if I’ve said the right answer or not (though it gives me two chances and it lets me give up if I want). Then, there’s always a worked-out solution provided. It’s right there, waiting for me to read it. And then, it gives me a chance to rate the quality of the explanation (which I find empowering in some cases).

The first incorrect notice gives me a chance to discover my own mistake and learn something from it. The second incorrect notice gives me a chance to study an example. And then I have a chance to practice similar problems (because the computer will continue to provide them). It feels very oriented towards growth. I can’t solve every problem on that site right now, but I’m confident that with enough time I could.

Deltamath does this nicely as well, though of course not every student reads every explanation or watches every video. It works best in a classroom, where students can ask each other or me if they get something wrong — again, it’s using auto-grading in a context that makes it even easier to act as an ideal student would.

I’d also like to suggest that there isn’t a meaningful difference between auto-grading and a lot of the “insta” feedback that kids get in current Desmos activities. If a kid understands what a graph means, then they understand that their answer didn’t produce the correct graph. If they don’t understand why, you’ll see those same giving up activities that auto-grading can produce — or they’ll do guess and check with the graph until they get a correct answer, which in some cases is not a bad idea — get the answer, and then try to figure out why it’s correct. In either event, Desmos currently employs a great deal of de facto auto-grading in their activities.

One way Desmos could help is by making it easy for teachers to connect students to learning. You might make it easy for teachers to attach examples or explanations to a wrong answer. You might make it easy for students to ask the teacher a question via a textbox if they get an answer wrong and they can’t figure out the problem. You might enable teachers to include a brief explanation with the wrong answer, and then let kids rate the quality of that explanation. (Really, check out Alcumus.)

There are smart ways to do auto-grading, I think. The smartest way, though, is to make sure it’s happening in the context of a lot of interaction between students and a teacher.

(Cross-posted as a comment to Dan’s post.)

## Dan Anderson Made Me Do It

Dan started it here, describing a day in the life of a math teacher these days.

Well, here I am, typing this while my kids are playing with a pair of broken laptops at my feet. Apparently my wife and I have been on our laptops enough that it’s inspired our children to play along.

I was hoping to do some work now, but my daughter Shuli (aged 2) refused to nap. Right now she grabbed Yosef’s face (he’s 5) and pulled on his cheeks as hard as she could. I am watching this disaster unfold as I type. Yosef just kicked her in the face. Shuli went on his back and slapped him.

Now, Shuli is climbing on me, and I’m typing while elbowing her, to keep her off the keyboard. This is more-or-less how it is.

“Which bomputer is the dirty one?” she asks. “Yosef is not giving me a bomputer?”

I just heard my toast ding. While I typed this, Yosef grabbed her first and she yelped. Now they’re slapping each other again. Should I get my toast?

I teach soon. I taught a live 4th Grade class this morning; this afternoon it’s my geometry students. The other three classes had assignments that (in principle) they’re working on.

Finding time to put the assignments together is, obviously, a challenge. Yosef is now holding up a pillow and hitting Shuli with it while she runs into it, like a blocking drill. More slapping, more yelling. “Get off me, you stupid jet pant,” he says.

“Is Shuli taking a nap?” he asks. “She really needs one.” How about we all take a nap, kid?

Since I can’t put together any of the assignments during the day, I’ve moved to the night. Last night I put on Netflix (I can’t even remember what) and set up all of today’s assignments. Oh, yeah, it was a Jim Gaffigan stand-up thing.

Here was one of his jokes: “If you want to know what it’s like to have four kids, imagine that you’re drowning…and then someone hands you a baby.”

“At the count of three, go to where Mommy is standing,” the boy says to the girl.

“Boo!” they shout at my wife.

After my 4th Grade class I went to the grocery store today. We’re in NYC where apparently you have to fight tooth and nail with your neighbors to get a delivery slot. We live close to the grocery store, so after class I took out the granny cart and went to the store to replenish our fridge. I managed to spend a couple hundred dollars, which makes me glad that my wife and I still have our teaching jobs. This would all be infinitely harder if we were out of work.

When I came back, my wife was in a meeting and the kids were wrapping up their first burst of TV time. (Sid the Science Kid and Daniel Tiger are the current offerings, but it changes weekly.) I checked in on the Google Classroom for my 3rd Graders. Engagement in that class had been low, so I’ve started posting half of a joke and asking students to guess the punchline.

“Stop running,” my wife and I yell to Yosef at the same time. I look up from the computer I’m typing this at. Our downstairs neighbors aren’t thrilled with the nose from our apartment, and I can’t blame them. I don’t like it either. When this is all over, we’re going to take our kids to a specialist who can teach them how to walk without stomping.

It’s raining today, is the trouble. (“STOP RUNNING,” this time I yell it.) We live in a seven-floor building that is part of a complex of four, and there is a nice big yard behind it. Usually I take the kids out there where they can run, jump, search for broken glass and used bottles, kick a ball around. I’ve taken to giving them splash time in muddy puddles on rainy days, but they get covered in mud and get too cold to be out after just twenty minutes; it’s not worth it.

I’ve spent the last few weeks saying that this whole thing was unsustainable, but it’s been six weeks now and that doesn’t feel quite right. This is sustainable, we can keep doing it. It’s just a grind, and a lot of days it sucks.

## 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.

## After three days of distance learning, here’s what I now understand

I feel like we all know the issues with online learning. If not, we all will soon enough. No need to make this post all about the problems, though it would be strange not to mention them at all. Here are the main ones:

• A social environment is more motivating than being stuck in your bedroom alone
• You can explain stuff, and people can ask questions, but it’s really hard to know what’s going on for kids who aren’t asking questions
• I am also taking care of my two young biological children
• You can’t tell what’s on kids’ pages as they are writing
• My own two children are absolutely going to destroy our two-bedroom apartment
• It is very difficult to show anything in to the teacher while asking a question
• The two-year old is hitting the five-year old in the face over and over with her Elmo figurine

So, what do you do? Justin Reich had been very generous in talking through some ideas with me, and he helped give me a way to think about it. “Organize around individual check-ins,” he wrote. “When I talk to the best full time virtual school teachers, they say they spend a considerable portion of time following up with individual kids.”

Designing materials for a lesson like this is not easy. If you start with individual check-ins, there’s a nasty chain of deduction that pretty quickly leads you away from normal classroom practice:

• You can only check-in and tutor individual students unless most students have something productive to do
• You’re going to need a few days to individually check-in with all the kids you need to
• That means that everybody needs something that they can do pretty-much independently for several days

It’s not really easy to design materials for that.

Well, I should say that the difficulty really depends on the group. I have a high-flying 8th Grade class that is seeming relatively easy to design for. I’ve used ASSISTments (buggy, but good) to assign practice problems from Illustrative Math, with Desmos extension problems. That’s been smooth.

But, yeah, all my other classes have been harder.

The way I’m seeing it right now, you’re looking for materials that have three qualities:

1. The assignment lasts a couple days. This reduces planning time, and also gives me more of a window for checking in and helping my students with the assignment.
2. The materials include clear examples and support. This is how kids are going to do the assignment more independently. Fawn just wrote a very nice post about making worksheets that start with models and then fade the support, teaching kids what correct answers look like.
3. There are challenges at the end that are optional. Because some kids will finish things in a day and want more math, and they deserve it. If they want it. If they want to go outside and take a walk while maintaining nice physical distancing conduct, they should do that too.

The hardest, hardest thing so far has been finding time to put materials together while taking care of my own two biological children. My wife is also teaching her middle school students from home, and every evening we’re scraping together a schedule that lets us both do our jobs. I’m really so grateful to be getting a paycheck at all right now, I can’t really complain. But it’s not easy for me to put nice-looking things together.

So, here it is, a sloppy thing that I’ll share. It’s the resources that I gave my high school geometry students as their assignment this week, and it’s been going OK.

***

The document is here.

We were studying similarity before school closed. I had gotten as far as practicing setting up and solving proportions.

Starts with an example:

• I talked it through with the whole-group on Zoom, which I might as well not have done. I had to work it through one-on-one with the kids who I knew would need it. Still, it was worth the five minutes to reassure some kids that they understood this so that I could focus on the individuals. I’d say the whole-group is worth it, but not worth a huge investment of time.
• A lot of my kids don’t have printers so they’re reading my worksheets on their phones, so I’ve been trying to write in big, clear fonts. Not sure if it’s helping, but it’s the rationale behind those huge letters.

After the example comes practice. I send kids into breakout rooms and tell them they can leave the computer and work alone if they want, but they can also collaborate. All I ask is that they remain more or less around so that I can call them back if I need to, and so far that’s been OK.

Then another example:

Then, ideally, would have been some indirect measurement practice except it’s BEEN ONE HELL OF A WEEK and I put what I could find into the worksheet even though it’s not a perfect match:

Then, the extension problems:

And also:

***

I did something similar for my 4th Grade class, and it went better than Day 1 did.

Example:

Practice (this time with more fading of the supports):

Extension challenge:

And it went better, in the sense that I was able to have some mathematical conversations with people and figure out what they understood, what they didn’t understand, help people, that’s the whole point of this job, right? The key, when you strip everything else away, is figuring out what people understand and helping them understand it?

But you’re basically flying blind in this distance teaching because you can’t see what anybody is doing at a given moment. Maybe there’s some digital tool that can help give you a better picture of what kids understand and can do, but without that the main thing is the individual conversation. This is a structure that tries to plan around that.

Oh, and remember how we used to share resources online and but then we stopped because blogging died? Can we bring blogs back, if just for a few months?

***

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.

## Will schools close for coronavirus?

First, listen to the experts. Here is who I have found helpful to follow:

I’ve been trying to figure out what is going on and what the response actually will be. This has been an attempt to just make sense of what various experts are saying, and think about the implications. I know nothing about this beyond what I’ve learned in the last few weeks, like the rest of us.

Here is what I think is going on:

It’s tempting to ask, “how lethal is this virus?” It’s also tempting to summarize that lethality in a number. But the mortality rate of a virus is not some fixed fact about the virus. The reason why the usual flu isn’t so very lethal is because it’s seasonal, there are vaccines, we have experience treating it, and so on.

So this new virus has a mortality rate, and for vulnerable populations it does seem to be many times worse than the usual flu. But mortality is not simply about biology. And that’s where the real potential for crisis begins.

Because this virus is new to humanity and spreads so rapidly, dangerous cases can pile up very quickly. If hospitals are overwhelmed with cases and don’t have enough nurses, doctors, beds, masks, and so on, the virus suddenly becomes much deadlier. If you have a choice between everybody getting sick over the course of a month or spreading that out over many months — you would definitely want that second option. All things being equal, the mortality rate will be lower.

Not all things are equal, though. Stretching out the timing — dragging this whole thing out as long as humanly possible — also buys time for scientists and medical professionals to figure out how to treat the sick. It makes sure that doctors themselves aren’t all sick at once, and available to work. It gives the government time to fund mask manufacturers so supplies aren’t immediately depleted. And, of course, it gives researchers time to work on a vaccine.

When it comes to pandemics, slower is better.

A lot of what has been happening so far feels like an attempt to contain the disease. We’re used to a world where people are quarantined to keep the disease from spreading. For example, the WHO has been speaking about how it is important not to give up on containment of the virus as a goal:

“As long as you have these discrete outbreaks … there is the opportunity to control them — to get on top of these and contain them and prevent a lot of disease and ultimately death,” says Dr. Bruce Aylward, a senior adviser to the director general of the World Health Organization. “That’s the big message we saw in China — and one of the big surprises.”

Other experts, though, have been puzzled by why the WHO is talking like this. From their point of view, the opportunity for containment has long passed:

As best I can tell, containment was never really a serious possibility. It’s just that even a failed containment strategy gives a government a head start on slowing things down, which is the real public health goal.

What will that mean, in the months ahead?

It’s hard to know exactly, because we’re constantly learning more about the virus, the disease, and potential public responses to it. It was initially unclear whether kids were even getting sick and spreading the virus, since their symptoms have been so mild. My sense is that evidence is starting to come together that children do get infected with the disease. If their symptoms are more mild, does that mean that they infect others at lower rates? We don’t know this yet, and governments need to know this.

What I’m getting the sense of, though, is that schools won’t be closed in the initial stages of local epidemics — for now.

There are a good number of school closings, but they seem to be focused on wealthier schools or school districts that are more responsive to the needs and fears of parents. They also include colleges and universities where online learning is easier to put together.

Public schools, though, are a big deal to close. And even experts who are raising alarm bells are reminding people that shuttering schools isn’t necessarily the right response. There are confusing network effects of closing a school. If you close a school where kids are getting their meals, will that send more kids to an overburdened hospital? Will nurses have to stay home to take care of their children? Will kids gather outside of school anyway? (This was apparently something that happened during H1N1 closures.)

That said, school closures, even big public closures, are clearly happening across the world. Japan, Italy, Saudi Arabia, they’ve all closed all of their schools.

Based on what’s happening in Seattle, it seems to me that this is the escalation of responses we can expect:

1. Early isolation – When the virus is new to a place, a school (or anyplace else) will shut down the moment someone gets tested positive. There’s still a chance at some kind of containment of the virus early on. This is when public health officials want to isolate and chase the cases.
2. Individual caution – When the disease is plausibly in your neighborhood, it seems that public health officials then urge individuals to start changing their habits. Be on guard, they say. You may have this disease — don’t mess around, isolate yourself is you are feeling sick. Wash your hands seriously. Cut down on contact. Reasonably avoid crowds. On the margins, be more careful. And this has an impact.
3. Easy shut downs – As cases pile up, places that can begin to shut down. Private schools or districts that aren’t responsible for providing a lot of social services begin to shut, when cases are a degree or two away. People are encouraged to work from home as much as possible. Big events are cancelled or postponed.
4. Tougher shut downs – If things continue, then bans on gatherings above a certain size can be imposed.

And so on, and so on. And if a community starts approaching the capacity to handle cases, and there’s a risk of running out of healthcare resources…that’s when the serious moves are pulled out. And, as far as I can tell, that’s when large schools districts with lots of poverty will begin to close, for several weeks.

My sense is that various countries have been able to avoid these tougher shut downs, for now, by aggressively pursuing 1-3. Part of the work of 1-3 is extremely widespread testing for the virus, and the US has gotten a slow start on this despite adequate warning. (It doesn’t seem like this is entirely the fault of the White House, but it seems like they didn’t help things along at all. Here’s a piece about what we know about the testing rollout.) That’s somewhat scary, but my understanding is that testing is ramping up in the US. Hopefully we’ll be able to move smoothly on that front, from now on, but who knows.

All this, though, is just the first wave of the virus. Who knows, at this point, whether we’ll get any reduction in cases when the weather gets warmer. Even if it does, though, we’ll have to deal with all of this again in the fall. We can expect waves, though future waves will (hopefully) not be as severe. Anyone who already fell ill will have some immunity (as researchers have recently confirmed). And who knows, maybe we’ll know more about treating it. Hopefully we’ll have a better sense of what measures are effective by then at slowing its spread.

But we should expect a few waves, I think, over the next year or two. That’s my impression.

## Want to get emails from me?

I’m a big fan of blogging, but I read a fair number of email newsletters. Here’s why I like email, as a reader:

• It’s easy to read text in email.
• It gets delivered to me.
• It feels more private than the very public spaces of social media (and blogging, sort of).

Here’s why I find the idea of email attractive, as a writer:

• I don’t have to pay attention to some sharing or popularity metric.
• It’s easy for people to respond and have a conversation, if they’d like to.
• People either sign up for emails or don’t. I don’t have to use Twitter to share it.*

Like a lot of people, I feel ambivalent about twitter and the demise of blogs. I like spending time on twitter, but I want to carve out some non-twitter space for myself. Really, I’m looking for balance. I can imagine a version of my online life that would look something like

1. Spend a couple hours on Twitter a week
2. Updating my websites with more permanent essays, projects, resources
3. Sending out emails to people who think that hearing from me every few weeks sounds like a blast.

Basically, I’m looking to fight back and capture a bit more quiet in my life. Thinking hasn’t felt easy lately. The news, yeah, but also my parenting and teaching, both of which are just SO LOUD. It saps up my energy, sometimes.

Anyway, let’s call this as an experiment. If at least ten people say they’d like to get emails from me (I’m not calling it a newsletter) then I’ll give it a shot. You can sign up here or give me your email address some other way.

## 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.