Yet Another Thing that Humans and Viruses Have in Common

When an epidemic rages through a population, at first it faces no immunity at all. The disease constantly encounters fresh meat. There is nothing that can stop it.

Eventually, the population gains some immunity, yada yada yada, the disease doesn’t spread as easily and the epidemic slows down. There is a point when the disease, on average, infects just one more person per infected person. You have heard this before. It is the “herd immunity threshold.”

What is more surprising is that even once the population has hit this immunity threshold, the epidemic continues to grow — for a time. Epidemics have a kind of momentum that pushes infections even past this threshold. If you know about this already, you probably learned about it the way I did: from reading experts discuss the COVID-19 pandemic.

I immediately liked the “momentum” but found myself having a difficult time thinking precisely about it. Then, while reading about mathematical ecology the other day, I learned something that helped it all snap into place for me: this happens with people too.

The notion of population momentum makes a lot more sense to me in a human case. Probably if I was a virus the epidemic case would be easier, but I am what I am. Wikipedia has a great exposition of it, including this handy chart:

In the first generation, the fertility rate is 4 and the 200 fertile people give birth to 400 children — some pretty robust population growth, given the age distribution of the population. Then, at time = 1, the fertility rate drops and parents have only two kids each, merely replacing the fertile population as the old population ages out (dies). Even though the fertility rate has dropped, there are still the result of the previous fertility boom at t = 0. Those 400 children are going to have two children each, and that’s going to help the population grow for a bit longer. Soon enough, though, the fertile population will just be replacing itself.

This phenomenon was first described by Nathan Keyfitz in 1971. He directs the idea to policymakers who are reluctant to offer contraception for fear that their countries will stop increasing in population. “In some countries hesitation in making contraception available is rationalized by the view that the country is not yet “full,” he writes. “Concern that total numbers will taper off prematurely is misplaced.” He goes on to explain how to calculate the total “ultimate” population once fertility reaches replacement levels.

It’s this exact same phenomenon that governs the growth of a virus, even after (say) a vaccine is introduced that brings the rate of infection down to 1. I find it interesting that some of the same population dynamics govern both humans and viruses. It suggests to me that a path towards better educating others about epidemic dynamics would be to start with human stuff.

Make a collection of worked examples for your classes

I’m filing away worked examples for my classes into a Google Slides presentation that I update daily. I’ve posted the presentation at the top of my Google Classroom page, the idea being this is a direct way for students to find examples. I don’t know if the kids will actually refer back to it, but I like collecting the examples now that my teaching leans so heavily on digital resources.

Here some of the examples I’m fond of:

We can’t agree on schools because they are compromise institutions

Coca Cola is a cola-producing, money-making enterprise. Hospitals are in the business of keeping people alive and healthy. Congress is in charge of making laws. Planned Parenthood supports reproductive health for women. Unions protect the interests of their members. And so on, and so on. All of this is relatively clear and straightforward.

Schools are different. Nobody can tell you exactly what it is that schools do.

(Update: my friend Chavi is a high-risk pregnancy doctor and author of a fantastic book that you should buy. She tells me, “You’ve got hospitals ALL WRONG. We are a TOTAL compromise. Social safety net, economically compromised model, plus restaurant and God knows what else. I like to point out that a really good hospital is also a really mediocre hotel.”)

Schools are in charge of teaching kids the things they need to get good jobs. That’s a matter of social justice. Or it’s about national security — you’ve got to keep up with the USSR Chinese.

Don’t forget about childcare, to support more people going into the workforce. That’s an important part of what schools do, their most basic function.

Actually — no! Schooling isn’t about jobs, national security or the economy. It’s about democracy. We need to create the next generation of informed citizens, prepared to vote and run for office and such. They need to know history, sure, but they also need to be schooled in democratic values. Kids need to learn that you can always improve by working hard. They also need to learn to respect their classmates, no matter if they come from a different place, worship differently, or have a different culture.

Really, though, school is about socialization. Kids need to be around other kids to be happy. They need to be part of a community where they get to make friends and find themselves. Schools give kids the chance to find themselves in passions like sports or music or art.

The point is that schools are all these things at the same time. Not everybody cares about all these things at the same time, but each purpose of schooling has its own cheering section in American society. We all care about schooling, but for different reasons.

David Labaree wrote a well-known essay describing this dynamic. In “Public Goods, Private Goods: The American Struggle Over Educational Goals” he lists three big categories for how Americans think about the purposes of schooling:

  • social efficiency (think, jobs and economics)
  • social mobility (think, meritocracy)
  • democratic equality (think, values of democracy)

This is the fundamental fact about schools: they are compromise institutions. They simply do not have a main purpose. They aren’t in the business of helping everybody learn the most that they can. Schools aren’t organized around giving everybody the same opportunities. They aren’t built to give meaningful experiences to children. They do all these things, but in a compromised way. Because they are compromise institutions.

A lot of “innovative” schools produce their innovations by going all-in on one of these goals while caring less about the others. You can certainly organize a school that produces learning gains — as long as you spend less time worrying about whether kids get more choices and chances to collaborate. And you can definitely organize a school that is completely untracked and gives everyone the same learning opportunities, but you’re going to have to give up a bit on some students’ individual achievements. You can get a Success Academy for a while, but eventually gravity will drag it down into the weird, tangled matrix of compromise that most schools have to grapple with.

I’m not saying that schools are compromise institutions and that this is a bad thing. Schools are weird. Public schooling isn’t that old — it’s a complex project that meets a lot of different people’s needs. It has evolved to do a reasonably good job keeping a majority of people happy, but that has involved a bit of layering and duct tape. Ah ok, the nation needs this? We’ll start teaching Biology. We’ll offer college classes. We’ll have music. We’ll do anti-drug talks.

There is nothing natural about schooling. When push comes to shove, and we have to rethink schooling with limited resources, don’t be surprised that we can’t easily come to agreement. Imagine you ran a local drug store and suddenly had to cut your inventory in half. Do you stop selling food? Keep the first-aid supplies but ditch the mouthwash? Your store has a lot of random stuff in it, because your store is a catchall for a lot of random needs that people have. You aren’t a grocer; you sell a lot of unrelated stuff. That’s what schools are like.

Many teachers have argued that in the autumn of 2020 schools should be entirely remote. After all, hybrid learning is going to be a pedagogical disaster, probably worse than remote learning. (It’s just remote learning but more confusing.) Why not go remote? That makes a lot of sense, if all you think about is learning.

A lot of professionals have argued that school should go back full-time, especially for the youngest students. These parents need childcare! How is the economy (and everything else) supposed to function when people are suddenly at home with their children. This is absolutely true. That this is the job of schools is a totally legitimate view.

Really, though, everybody should go back. (Safely, of course.) Kids need to be in a social environment, not miserable and locked up at home staring at a screen. It’s not good for kids. It’s not about childcare, it’s about thriving in a meaningful environment. (This is the view my own school has taken, I believe, and all grades are currently expected back in September.)

My point is not about who is right. Honestly, I don’t know! The point is that to understand why there is so much disagreement you need to understand that it’s not that people don’t value the economy, childcare, the social lives of children or anything else. It’s that schools are weird. They exist in a strange space and occupy a weird set of roles. They aren’t simply institutions; they are settings where a variety of institutions and forces play tug-of-war and compete.

That means that if you want schools to come back (or to go hybrid, or to go fully remote) you have to make a really complicated case that satisfies a lot of different desires. We don’t agree on what is the most important for children. We don’t agree on what schools are for.

This is why it’s difficult to figure out what to do with schools right now. We don’t have the resources to do all the weird things that schools used to do. (Smaller class sizes + staff with legit medical exemptions – painful state budgets – meaningful federal support = funding disaster.)

Schools are already weird compromise institutions. We should all expect it to take another weird compromise to make this next school year work.

Building Relationships in Online Math Camp

At some point last spring, with the ambulance sirens still constantly blaring but with my job more or less under control, it occurred to me how lucky we were that online teaching started late in the year. How much harder will it be at the start of the year, when we don’t have relationships with the kids yet?

This summer I’ve been teaching again at BEAM, an amazing organization that runs a network of math camps. The kids are all entering 7th Grade and are choosing to spend their summer learning math, programming, genetics, circuitry, statistics — my point is that camp is not quite the same gig as teaching during the school year. Still, the program is entirely online and I’d never met the students before. I now feel as if I have a bit more insight into the “how will it work with a new group of students” question.

In this post I’ll give the rundown as to what I’ve done in camp so far.

(I should note here that online teaching is a thing that many teachers do year in and year out. That’s not quite the same thing as camp or pandemic teaching, but it should keep up from saying things like “this whole project is impossible.” This parenthetical note is brought to you by StuckInTheMiddle, an online teacher currently writing a book about full-time online teaching for John Catt Edu, who has made this very point many times herself.)

OK: how do you build relationships with a new group of students while teaching online?

Here are my answers:

  • “How do I build relationships with students” is the wrong question.
  • What I’ve found much more pressing is whether the students trust and enjoy each other. Otherwise they can’t collaborate, don’t feel comfortable turning their cameras on, won’t ask questions, won’t discuss things in small groups, won’t do fun community-building stuff, and so on.
  • Looking back, the first few classes were designed to get kids participating and interacting with me. I don’t know if I regret the decision to start there, but it’s not enough.
  • Things got much better once I started thinking about how I could get kids talking to each other. Again, maybe that’s only possible after I did the earlier stuff, which I’ll detail in a second.
  • By the way, there are still a few students I’m worried that I’m not reaching at all. That happens at in-person camp too, but it’s hard to see what the problem is when we’re all online.

Here’s the rundown of how this went, day by day.

Day 1: I was worried about participation, but I had my plan. My plan was something we might call an “acclimation theory” of participation and trust. Students just have to get used to sharing their thinking and talking to me. I designed class to be all about participation.

I started with an algebraic number trick (via the late Don Steward). I asked everybody to type their result into the chat at the same time for a dramatic reveal.

This was fun and the chat was full of students being low-key blown away. We may be online, but fundamentally this is still a camp full of young math nerds. You can count on a lot of the math to just work.

After this tease (I always like to start with math right away) I described some course expectations.

Well, as the next few weeks would make clear, no dice on video on for every kid. I have basically given up on that and am focusing on measuring engagement in other ways.

On to the “mic check.” Kids were active on the chat, but this would be their chance to get used to sharing their voices on Zoom.

I’m conflicted about this mic check. On the one hand, it took a loooooong time. On the other hand, it did more or less guarantee that I had a conversation with every student, and they got to share some of the things they were into. And they got a chance to piggyback on each other’s enthusiasms — a lot of them love anime, video games, and YouTube. It’s also helpful to me, to hear everybody say their name once and to feel like I’m teaching children rather than usernames.

But it takes a LONG time.

After that, it was into math. I stuck to Zoom, figuring it would be a bit much to introduce a new tool on the first day. I presented a worked example and asked students to type explanations.

More Zoom math:

That was it for Day 1.

Day 2: With one of my groups, I did another mic check. But it just takes too long to ask everyone to unmute themselves and share their answer. It just really dragged on. So for the other group I made a video check-in, which was more successful. Again, I was operating under acclimation theory: just get used to it, and then you’ll feel comfortable. Here was my “turn on your video” prompt:

I should note that even as things are going well in my classes now, students don’t love having their video on. I am able to get most of the class to turn their video on for a minute or two to vote on something with their hands or thumbs, but that’s it. Probably not a battle worth waging, is my current thought.

Did another number trick, made another little example. This one tried to expose the structure of these tricks for the students:

Then I sent students to work on a Classkick assignment (link), our first.

Classkick is a very important tool in my online teaching, because it lets me see what students are working on as they are working on it. It’s absolutely critical to my teaching at the moment. It’s actually a pretty awesome tool in most ways.

I had a bit of dilemma though. It’s Day 2. Students still don’t know each other. If I send them to breakout rooms, it would just be entirely silent. Plus, that would be our first time using breakout rooms while they’re also trying to figure out a new tool. What do I do?

Well, I just assigned the Classkick problem set and kept everybody in the main room. Students would chat every once in a while, but overall it was very sleepy and slow. It felt OK, but not like we were really firing on all cylinders yet.

So much for Day 2.

Day 3: This was when I started to realize that something needed to change.

I ran Day 3 as I did Day 2. Worked examples that I asked students to discuss via chat:

Then, I assigned students to work on Classkick problems in Breakout rooms. And, as predicted, it was deathly silent.


This is where my “acclimation theory” started to run out of steam. Students were getting used to talking with their mics (because I was asking them to). They were chatting (some were talking to each other in chat but not much). They were working on problems. But why was the energy so … muted? It felt off.

That’s when I sat back and realized that it wasn’t about me: they weren’t acting like people stuck in a room full of strangers. That’s because, mostly, they were.

In my defense, the problem doesn’t really happen during my school life. By the time I meet students, they typically know each other much better than they know me. And though I’ve been teaching at camp for a while, the camp environment usually helps kids get to know each other real fast.

OK, so new theory: I need to be pushing students to talk to each other a lot more. Then they’ll feel more comfortable participating publicly.

Day 4: To be fair, I had fully intended to launch camp with something fun and game-like. I just couldn’t think of a way to tie it into my course, which is all about equations and solutions.

Now, though, I just needed a way to get kids talking to each other about something we’re all in on together.

Every teacher eventually gathers a few “break in case of emergency” activities. One of my favorites is the game Mastermind. There’s a totally fine online version that I like to play with my classes. It’s only related to my class indirectly, but I figured that was enough.

OK: it was a hit! And kids were talking to each other, because they wanted to get the code right at the end. I told them they could stay in my Zoom room for the after-class math time (it’s scheduled daily) and play Mastermind. Many did — it worked, as I should have known it would. The “break in case of emergency” stuff never lets you down.

With my new framework, I decided to start the Classkick problem (link) set right away after Mastermind. This time, I really thought about how to make breakout rooms work — I mean, how to get kids talking to each other in them.

Here was the first slide in the assignment:

What I did was:

  • There was something (the dumb tomato question) they had to talk about together.
  • Someone had to share their screen (to make it easier to talk about the same stuff)
  • I asked the counselors (who are teaching assistants) to more actively get kids talking about the math in their breakout rooms.

It went well! In the sense of, kids were talking and it felt more active and vibrant than class had felt up to that point.

Day 5: I reached back into the emergency kit and pulled out “Minesweeper.” Plus I imposed a coordinate grid on it so they could practice saying things like “there’s a bomb in (10, 6)” or whatever. Plus, it lets us talk about logical certainty in a way that mirrors “is this equation true for all values.” I used the Google Minesweeper game, which is nice and colorful.

I haven’t really given my students a chance to try a bunch of problems on their own. I figured today was a good day to take that leap. I assigned the Classkick problems (link) and asked my counselors to log-in to “teacher view” also, so we could keep an eye on what everyone was doing. (By the way, there is a co-teacher workaround with Classkick. Works like a charm.)

Here was the first slide:

It went well. Kids asked lots of questions — I told them to, and I was grateful that they did. Here was Slide 2:


Here are two other little things I had forgotten to mention:

  • To have some fun while students are trickling in, I’ve started awarding First, Second, Third place prizes to the funniest virtual backgrounds. (Then, I ask everyone to turn them back to normal so it doesn’t become a huge distraction in class.) I don’t think I would do this during school, because the virtual backgrounds are a huge pain in the ass, but for camp it’s been fun.
  • I start class with an extremely corny joke. I post the setup and ask kids to try to guess the punchline. “What do you call a factory that is only OK, not great, at making stuff?” I did do this during the school year, and it was kinda fun so I started doing it again during camp. It’s nice to have a little bit of inside humor.

(A “satisfactory.”)


I’m not sure that this will be as big a deal for my classes in the fall if we turn out to be fully online. Most students at my school have been in my school for a few years already. But I think I do have something like an answer to the relationship-building question:

  • Get kids learning and doing things and interacting with you, the teacher, yes.
  • Get kids interacting with each other and trusting each other. Participation isn’t just about getting used to a new environment, it’s about trusting and liking the other students in the room.
  • Some whole-group collaborative puzzles or games are good ways to start that. Getting kids talking to each other in smaller collaborative groups worked well for me to.
  • Then you can reinvest those relationships into more individual learning, and kids will be more willing to ask questions and share ideas.

That’s my thinking at the moment, anyway! There’s a lot more camp to go. And then the Fall of 2020. Then a vaccine.

Then, the vaccine fails. Then, Spring of 2021. Then, COVID-20 emerges. Then, the elephant flu. Then, COVID-FLU, a hybrid virus.

Then online teaching with kids at home and locked in your bedroom forever and ever and ever and ever.

Why my students like Deltamath

My survey question to 8th Graders was, if you liked Deltamath (a math practice site) as a distance learning tool, what did you like about Deltamath?

Here are the responses:

  • The very clear examples and how it is easy and not glitchy

  • There was always an example and there was a goal to work towards

I liked that if you didn’t understand something, it had examples to show you.

It felt totally normal like the homework we would usually have

Examples are clear easy to answer

It gave examples, helpful videos, and it was concrete. I also liked that it let you know if you were right or not.

I liked that it gave me a chance to put what I learned to use

I feel like the subjects are not super difficult but the problems are a good workout for ur brain

Also it tells you if you are right or wrong, so it helps you check answers

When it would show graphs and it would show you what you are trying to figure out.
  • Also that it explains what u do wrong after you get it wrong

It was fun and interactive, shifting to give you easier/harder questions if you were correct or incorrect.

By the way this last one is not true, Deltamath does not shift to giving you easier or harder questions. There is something psychologically fascinating about thinking that the software, which is essentially a random problem generator, is actually adapting to the student responses.

In any event, listen to what the students are saying: examples, examples, examples.

Update: Here are some more comments from students about Deltamath. (I lightly edited for legibility.)

  • Deltamath works well and is good for homework

I liked how Deltamath explained the problem if you didn’t understand it. Definitely my favorite out of all the tools we’ve used
  • It’s just easier than having to use paper and pen if i’m already on my computer in class

  • I like Deltamath because I am very familiar with and because it provides examples which are helpful for certain confusion topics.

  • It’s very organized which isn’t the case for a lot of other classes assignments

  • It’s 
also easier to work with other people

I’m kinda neutral on Deltamath, but I also feel like its a good way to learn

  • I liked Classkick more, I’m not sure why though
. I really don’t know why i think it may have just been the layout of the website or maybe just how you could do more types of assignments
  • I have no opinion
  • I really like getting the answer to the problem as soon as I solve them, like it’s really nice to have to get 5 questions right and learn what your doing wrong along the way instead of answering on work sheets and not being quite sure what your doing right until later
I like Deltamath because it is easy to use and it gives examples of question with step by step solving.

I asked students if they felt it was frustrating when they got something wrong. Here’s what they said to that.

Only if it takes away points
  • Deltamath can be frustrating when you get it wrong and there’s a penalty but otherwise it’s not that frustrating

Update to the update: here’s my other algebra class’s thoughts:

  • I love it, it is very organized
  • I liked the explanatory feature that it provided. Videos, examples etc…
  • I like Deltamath. Off the top of my head I can’t think of any part that is frustrating/not useful.
  • I think it was overall fine but bit less interactive than Desmos.
  • Delta math was my favorite because the questions were clear and although it wasn’t as visual as Desmos it still had some visual elements.
  • I like Deltamath a lot it’s really simple and easy to navigate
  • Desmos>>>>>>Deltamth because for me it helps to visualize it and Desmos does that
  • It was decent I mean the problems weren’t hard at all and I could complete them in the first 15 minutes of an asynchronous period. But Deltmath it self was an easy website to use.
  • I like it because it is easy to use and gives you clear feedback. And I feel that Classkick is hard to use and not great for online learning
  • I liked Deltamath because it was easy to understand what we were supposed to do and  when it was due.
  • Delta math was really good. The only thing that was a little frustrating was typing answers on keyboard. But otherwise it was very helpful, especially the help feature where it shows examples.

As a final question, I asked this class if they remembered times when they got frustrated and stopped working on the assignment. Do you remember which website you were using when you gave up on the assignment?

  • yes, on classkick
  • Yes, delta math, a few times actually
  • no not really
  • I think I got frustrated and stopped deltamath once
  • Yes a few times
  • Only a few times
  • Yes classkick
  • That never really happened to me
  • But I was very tired
  • mostly classkick
  • Yes. But I do not remember. It was not because of the site. More that I forgot about the assignment.
  • desmos
  • Sometimes I had trouble using classkick because it was  very technical and there were no specific inputs

Good auto-grading

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.

Screenshot 2020-04-02 at 9.00.06 PM

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

Screenshot 2020-04-02 at 9.00.19 PM

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:

Screenshot 2020-04-02 at 9.09.38 PM

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?

Screenshot 2020-04-02 at 9.09.55 PM

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

Screenshot 2020-04-02 at 9.14.36 PM

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

Screenshot 2020-04-02 at 9.15.40 PM

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:

Screenshot 2020-04-02 at 9.20.26 PM

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:

Screenshot 2020-04-02 at 9.23.41 PM

This is an individual student’s work:Screenshot 2020-04-02 at 9.22.36 PM

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:

Screenshot 2020-04-02 at 9.24.00 PM


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.