## 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:

## 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?

• “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.

## 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.

## 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.

## Equations and Equivalence in 3rd Grade

So I was stupidly mouthing off online to some incredibly serious researchers about equivalence and the equals sign and how it’s not that hard of a topic to teach when — OOPS! — my actual teaching got in the way.

I had done the right thing. In my 3rd Grade class I wanted to introduce “?” as a symbol for an unknown so I put up some equations on the board:

15 = ? x 5

3 + ? = 10

10 + 3 = 11 + ?

And I was neither shocked, nor did I blink, when a kid told me that the last equation didn’t make any sense. Ah, I thought, time to nip this in the bud.

I listened to the child and said I understood, but that I would like to share how it does make sense. I asked whether anyone knew what the equals sign meant, and one kid says “makes” and the next said “the same as.” Wonderful, I said, because that last equation is just saying the left side equals the same as the right side. So what number would make them the same? 2? Fantastic, let’s move on.

Then, the next day, I put a problem on the board:

5+ 10 = ___ + 5

And you know what comes next, right? Consensus around the room is that the blank is 15. “But didn’t we say yesterday that the equals sign means ‘the same as’?” I asked. A kid raised her hand and explained that it did mean that, but the answer should still be 15. Here’s how she wanted us to read the equation, as a run on:

(5 + 10 =  15 ) + 5

Two things were now clear to me. First, that my pride in having clearly and decisively taken care of this issue was misguided. I needed to do more and dig into this more deeply.

The second thing is that isn’t this interesting? You can have an entirely correct understanding of the equals sign and still make the same “classic” mistakes interpreting an actual equation.

I think this helps clear up some things that I was muddling in my head. When people talk about the need for kids to have a strong understanding of equivalence they really are talking about quite a few different things. Here are the two that came up above:

• The particular meaning of the equals sign (and this is supposed to entail that an equation can be written left-to-right or right-to-left, i.e. it’s symmetric)
• The conventional ways of writing equations (e.g. no run ons, can include multiple operations and terms on each side)

But then this is just the beginning, because frequently people talk about a bunch of other things when talking about ‘equivalence.’ Here are just a few:

• You can do the same operations to each side (famously useful for solving equations)
• You can manipulate like terms on one side of an equation to create a true equation (10 + 5 can be turned into 9 + 6 can be turned into 8 + 7; 8 x 7 can be turned into 4 x 14; 3(x + 4) can be turned into 3x + 12, etc.)

When a kid can’t solve 5 + 10 = ___ + 9 correctly or easily using “relational understanding,” this is frequently blamed on a kid’s understanding of the equals sign, equivalence or the particular ways of relating 5 + 10 to __ + 9. But now I’m seeing clearly that these are separate things, and some tend to be easier for kids than others.

So, this brings us to the follow-up lesson with my 3rd Graders.

I started as I usually do in this situation, by avoiding the equals sign. I find that a double arrow serves this purpose well, so I put up an arrow relationship on the board:

2 x 6 <–> 8 + 4

I pointed out that 2 x 6 makes 12 and so does 8 + 4. Could the kids come up with other things like this, I asked?

They did. I didn’t grab a picture, but I was grateful that all sorts of things came up. Kids were mixing operations nicely, like 12 – 2 <–> 5 + 5, in general it felt like this was not hard, kids knew exactly what I meant and could generate lots of ideas.

My next move was to pause and introduce the equals sign into this conversation. Would anyone mind if I replaced that double arrow with an equals sign? This is just what the equal sign means, anyway. No problem, that went fine also.

Kids were even introducing great examples like 1 x 2 = 2 x 1, or 12 = 12. Wonderful.

Then, I introduced the task of the day, in the style of Open Middle (R) (TM) (C):

Yeah, I quickly handwrote it with a sharpie. It was that sort of day.

I carefully explained the constraints. 10 – 2 + 7 + 1 was a true equation, but wouldn’t work for this puzzle. Neither would 15 – 5 = 6 + 4. And then I gave the kids time to search for solutions, as many as they could find.

Bla bla, most kids were successful, others had trouble getting started but everyone eventually had some success. Here are some pictures of students who make me look good:

Here is a picture of a student who struggled, but eventually found a solution:

Here is a picture of the student from the class I was most concerned with. You can see the marks along his page as he tries to handle things like 12 – 9 as he tries subtracting different numbers from 12. I think there might have been some multiplying happening on the right side, not sure why. Anyway:

The thing is that just the day before, this last student had almost broken down in frustration over his inability to make sense of these “unconventional” equations. So this makes me look kind of great — I did it! I taught him equivalence, in roughly a day. Tada.

But I don’t think that this is what’s going on. The notion of two different things being equal, that was not hard for him. In fact I don’t think that notion is difficult for very many students at all — kids know that different additions equal 10. And it was not especially difficult for this kid to merge that notion of equivalence with the equals sign. Like, no, he did not think that this was what the equals sign meant, but whatever, that was just on the basis of what he had thought before. It’s just a convention. I told him the equals sign meant something else, OK, sure. Not so bad either.

The part that was very difficult for this student, however, was subtracting stuff from 12.

Now this is what I think people are talking about when they talk about “relational understanding.” It’s true — I really wish this student knew that 10 + 2 <–> 9 + 3, and so when he saw 12 he could associate that with 10 + 2 and therefore quickly move to 9 + 3 and realize that 12 – 3 = 9. I mean, that’s what a lot of my 3rd Graders do, in not so many words. That is very useful.

So to wrap things up here are some questions and some provisional answers:

Q: Is it hard to teach or learn the concept of equivalence.

A: No.

Q: Is it hard to teach the equals sign and its meaning?

A: It’s harder, but this is all conventional. If you introduce a new symbol like “<–>” I don’t think kids trip up as much. They sometimes have to unlearn what they’ve inferred from prior experiences that were too limited (i.e. always putting the result on the right side). So you’re not doing kids any favors by doing that, it’s good to put the equations in a lot of different forms, pretty much as soon as kids see equations from the first time in K or 1st Grade. I mean why not?

Q: If kids don’t learn how equations conventionally work will that trip them up later in algebra?

A: Yes. But all of my kids find adding and subtracting itself to be more difficult than understanding these conventions. My sense is that you don’t need years to get used to how equations work. You need, like, an hour or two to introduce it.

Q: Does this stuff need to be taught early? Is algebra too late to learn how equations work?

A: I think kids should learn it early, but it’s not too late AT ALL if they don’t.

I have taught algebra classes in 8th and 9th Grade where students have been confused about how equations work. My memories are that this was annoying because I realized too late what was going on and had to backtrack. But based on teaching this to younger kids, I can’t imagine that it’s too late to teach it to older students.

I guess it could be possible that over the years it gets harder to shake students out of their more limited understanding of equations because they reinforce their theory about equations and the equals symbol. I don’t know.

I see no reason not to teach this early, but I think it’s important to keep in mind that in middle school we tell kids that sometimes subtracting a number makes it bigger and that negative exponents exist. Kids can learn new things in later years too.

Q: So what makes it so hard for young kids to handle equations like 5 + 10 = 6 + __?

A: It’s definitely true that kids who don’t understand how to read this sort of equation will be unable to engage at all. But the relational thinking itself is the hardest part to teach and learn, it seems to me.

Here is a thought experiment. What if you had a school or curriculum that only used equal signs and equations in the boring, limited way of “5 + 10 = ?” and “6 x ? = 12” throughout school, but at the same time taught relational thinking using <–> and other terminology in a deep and effective way? And then in 8th Grade they have a few lessons teaching the “new” way of making sense of the equals sign? Would that be a big deal? I don’t know, I don’t think so.

Q: There is evidence that suggests learning various of the above things helps kids succeed more in later algebra. Your thoughts?

A: I don’t know! It seems to me that if something makes a difference for later algebra, it has to be either the concept of equivalence, the conventions of equations, or relational thinking.

I think the concept of equivalence is something every kid knows. The conventions of equations aren’t that hard to learn, I think, but they really only do make sense if you connect equations to the concept of equivalence. The concept of equivalence explains why equations have certain conventions. So I get why those two go together. But could that be enough to help students with later algebra experiences? Maybe. Is it because algebra teachers aren’t teaching the conventions of equations in their classes? Would there still be an advantage from early equation experience if algebra teachers taught it?

In the end, it doesn’t matter much because young kids can learn it and so why bother not teaching it to them? Can’t hurt, only costs you an hour or two.

But the big other thing is relational thinking. Now there is no reason I think why relational thinking has to take place in the context of equations. You COULD use other symbols like double arrows or whatever. But math already has this symbol for equivalence, so you might as well teach relational thinking about addition/subtraction/multiplication/division in the context of equations. And that’s some really tricky, really important mathematics to learn. A kid being able to understand that 2 x 14 is equal to 4 x 7 is important stuff.

It’s important for so many reasons, for practically every reason that arithmetic is the foundation of algebra. I can’t list them now — but it goes beyond equations, is my point. Relational thinking (e.g. how various additions relate to each other) is huge and hugely important.

Would understanding the conventions of the equals sign and equations make a difference in the absence of experiences that help kids gain relational understanding? Do some kids start making connections on their own when they learn ways of writing equations? Does relational understanding instruction simply fail because kids don’t understand what the equations their teachers are using mean?

I don’t know.

## Dear Aunt Sally: The exit tickets are all over the place!

Dear Aunt Sally,

My students had a hard time with my lesson on solving equations. How do I know? I gave a short “exit ticket” and the results were…mixed. There were three questions:

• 3x + 5x = 56
• 3x + 7 – 5x = 45
• 6x – 5 + 11x + 17 = 63.

The main issue is combining like terms. (I’ve included some pictures of student work for you, Aunt Sally.)

That said, some students in each class definitely did understand the material.

One more issue. In most of my classes, students didn’t struggle with the first problem. But in each of my classes, some students did, and in one of my classes very few students got that first one correct either.

What do I do to help the students who didn’t understand the material, and what do I do with the kids who did?

Ryan in Florida

***

Dear Ryan,

You have described one of the perpetual struggles of teaching. How do I balance the needs of the many versus those of the few? I suppose back in Neanderthal times the fellow charged with teaching youth to spear a mammoth came home to the cave feeling similarly. “Some of those kids get it,” he’d say. “But what about the one who used the flat end of the spear? He’s going to starve to death. He would really benefit from getting back to the basics. Maybe I’ll split the group in half?”

Thank goodness we aren’t Neanderthals. We are modern teachers! This means we have access to the wonders of modern technology. I of course mean the blackboard and copy machine.

My favorite way to follow-up on these sorts of quizzes is with examples, so I put together three such activities for you:

And then some mixed-up practice, to take another step in the right direction.

A good example activity, in my experience, can seem so simple so as to hide its design. And in fact the actual design of the student work was mostly straightforward. The choices to make are mostly ones I made long ago — to prefer simplicity, to subtly use arrows and lines, and to make the work as much like a student’s as possible while making the work as clear as possible to read.

(The mistake, in particular, closely follows the design of an Algebra by Example mistake.)

Most of the work here is in narrowing in on a specific type of problem that is worth including in the examples. (The simplicity of the format works well with the more complex task of narrowing in.)

Here is a rule I tell myself while looking at a student’s mistake: Every mistake points to a family of situations very similar to the mistake that the student doesn’t yet know how to handle. Find that family, and teach it!

In this case I don’t assume the mistakes on the second problem (3x + 7 – 5x = 45) point not to issues “combining like terms” in general. I assume instead that the mistake points to issues where the like terms are separated visually in the equation and where subtraction is involved. Those things are distinctive and make the equation more difficult to solve correctly — that is the family that I focus in on for the example.

How the examples are used is your decision, Ryan, but here is how I would do it. Begin class by showing just the example, and ask students to silently study it. (They can let you know when they’ve read it all with their thumbs.) When finished, you can ask them to answer the explanation questions on their own or with a partner. If you want to interject with an explanation, by all means, but often students are ready to dive straight into the practice problems.

There are three examples/non-examples I’ve made. Use as many as your class would benefit from.

Then, there are four practice problems. Those are important because they are mixed practice. If we see each of these examples as pointing to a micro-skill, a small family of problems within the broad category of “two-step equations,” then these problems are interleaved in the practice set. That can be useful! Your students will have to think back and remember what they did with the examples.

Teachers of equation-solving know that there are always more mistakes that need to be addressed. The difference between good and bad teaching of this topic, as I see it, is whether the teacher can get specific and point to precisely what is hard for their students. Subtracting, different visual formats, handling a variable with coefficient 1 (x as 1x), and so many other little things — there will always be more of these. Some teachers just repeat and repeat and repeat without getting any more specific. Instruction should get more specific as practice gets more general.

So, keep at it! Pretty soon your young charges will be scattering the landscape with Woolly Mammoth carcasses, so to speak.

-Aunt Sally

## Quick direct instruction and interesting practice at math camp

Today is my first day of summer break, sort of. I spent the last six weeks working for a wonderful math camp where the teaching is so much fun.

My class is the closest that students come to school content during camp. It’s a fractions class that students get placed into based on an assessment. If the camp thinks that students could use more time working on fractions — that’s who I teach.

That said, the course content is tricky because I don’t want to simply repeat what they’ve seen in school. That would be boring for a lot of kids, and I’m aiming to approach familiar ideas in unfamiliar ways. I’m trying to work on skills, but from interesting perspectives.

Here’s a one-two-three sequence from my fractions course that I think worked particularly well.

First, I ask students to think about visuals. This was a focus of the previous lesson, but I want to make sure every students has it at the front of their minds.

I’m trying to give everyone a chance to figure out what fraction a piece is by multiplying. (“There are four pieces, this is divided into fourths, that would make sixteen in total.”)

I’m teaching this both because it’s a useful bit of visual fluency, but also because I want to use this as part of my direct instruction.

Next is the direct instruction. I’m trying to teach students a mental shortcut: if you’re dividing a fraction by an integer (e.g. 1/2 divided by 10 is 1/20) you can multiply the denominator by the integer because that’s simply making the pieces 10 times smaller. I use visuals to explain this.

I immediately give students a few chances to try out this new technique on some mental math problems. (Below is my little cheat sheet — this is what I ask students, but I don’t give them this paper.)

That’s the basics. But how are we going to practice it further? And how are we going to keep it interesting, and make sure students start using this technique in other contexts?

I then move to the third activity in this lesson, some mobile problems (designed by me on the EDC site). They’ve been carefully designed to give us a chance to use that mental shortcut we’ve just studied.

A lot of the lessons in my fractions course seem to follow something like this pattern: reminder, quick explanation, interesting practice.

What exactly is it that worked about this? I think this pattern of quick direct instruction followed by interesting practice is a useful one. Of course not every topic is amenable to quick direct instruction (some skills need to be taught in larger chunks) but some are. And after some quick “are we on the same page” questions, it was nice to follow it up with interesting practice. And what made it interesting? I think that it looked different than the direct instruction, but there was still the chance to use it frequently.

This is a way of engineering challenging classroom experiences around stuff that you want to just explicitly teach. I think a lot of people think of these things as incompatible, but they clearly aren’t. At the same time, for a lot of my groups during the year I am trying to make things more accessible — I’m not trying to make it more challenging.

Or maybe I should be? Maybe this pattern of instruction would work just fine in my school-year work. One issue during the year is that I’m much more cautious about whether the practice is actually going to help with the skill. There is a risk to practicing in a different context than instruction. It’s always possible that kids won’t make the connections, that it will be either too hard or students won’t actually practice the thing you thought they would.

So, I’m not sure whether this is something I’ve learned about teaching camp or teaching school or teaching math. Time will tell, I guess.

## Encore!

Another school year: done.

I love the dismantling that happens after the kids leave. For months this place has been made just so, and it takes just moments for the entire accumulation process to be set in reverse. Posters are first in the line of fire, but sooner or later everything is headed either to a drawer or the trash. The recycling bins were stuffed with homework this afternoon.

I’ve never been involved with anything theatrical (ok fine once) but I imagine it’s similar to what it feels like to strike the set the day after everything closes. Everything of significance has got to go. It’s all trash now, but just a few hours before the whole thing was whatever it is you call trash’s opposite. This stuff was indisposable by mutual assent. Now: nah.

And what that made me think of was the artifice of this whole enterprise. School is such a weird performance of the strangest kind of pretend — they call me ‘mister’! People sometimes point to the artificiality of schools as a critique: this isn’t what learning looks like in its natural state. And of course that’s exactly right, it’s not. It’s all fake. And that guy up on the stage — did you know he’s not really a wizard?

Teaching is weird, it’s fake. Teaching is not medicine. Medicine is someone is sick and you can help them, the most natural thing in the world. Driving a cab is not weird. People want to get to x, they pay you to take them there. Natural. Getting paid to cook people food is a reasonable transaction, it is not a weird. Lawyering, on the other hand: super weird. And as long as we’re on the topic: basketball player, musician, researcher, middle manager, writer, actor, teacher, weirdos all.

So it’s summer and the show is over and all but, come autumn, let’s do it again! I’ve got a school I love working at and I’m ready to play pretend for another year. Year nine is over.