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.

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Source for image is fractiontalks.com

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.

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This is a classroom poster with a version of my explanation. Below it is the next mental strategy I teach in the course. 

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.)

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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.

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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.

Two classroom things

#1

Ask anybody: I’m not some sort of god teacher in the classroom. But the other day a teacher complimented me on my classroom management, in particular my ability to hold attention without using my voice. And I was like, yeah! That’s something I do very much on purpose.

I forget if I came up with this on my own or (more likely) picked it up from something I read, but I try very very hard not to interrupt the flow of the class with a redirection to a kid. None of that “I’ll wait” or “settle down” or “I need all eyes up here.” Not if I can help it, at least.

The thought is that the content of the class shalt not be interrupted, and I undermine this premise when I use my voice to interrupt. Plus, it’s just confusing to kids who are following along to hear this other thing, and then to dive back in. PLUS, it directs everyone’s attention to some kid and their shenanigans. PLUS PLUS, it creates a little pocket of energy in the room for a few moments and that can get filled with nonsense or accidental attention-grabbing things.

So I use looks, pauses, taps on the shoulder, etc. I know (I think) a lot of other teachers do this stuff, but I do it too and maybe it’s worth sharing.

#2

Very frequently there are kids that aren’t feeling confident and you can sort of feel them withdraw from the classroom. Very frequently there is an implicit contract these kids have with their teachers: I will be silent and confused, but you will not mind because I will not cause you an ounce of trouble.

Forget that! Cause me trouble.

So there’s this thing that sometimes happens in my classrooms. It’s a bit awkward, but I think it works out OK, most of the time. Which is that sometimes I’ll see a kid who is settling into that withdrawal, and so I’ll cold call on them.

Putting a kid on the spot in front of the class — especially one who is having a rough time in class — is a tricky proposition, and my purpose is not to embarrass a kid for not engaging. It’s just that they’ve got this contract, and they’re trying to figure out if I’m on board or not. But I’m not — I’m going to invite you in to whatever our class is doing, and I’m going to keep on doing that.

“Sarah, do you agree that 40 divided by 4 is 10? Why does that make sense, do you think?”

I’d be curious to hear the recording of that conversation. In that moment I felt myself trying very hard to balance two contradictory stances: a jokey, non-threatening, supportive teacher who is very persistently insisting that you share your thinking.

I also felt myself very much caught up in another contradiction, which was not signaling to Sarah (or the rest of the class) that I was “dumbing things down” for her, while also trying to find a mathematically interesting way for her to engage. This was also a high-wire act, even if it only lasted a few seconds. In these moments one of my moves is to just produce more and more on my own, but to be insistent that the student take on why.

“So from skip-counting we know that ten 4s makes 40. Why would that help us solve 40 divided by 4?”

Sarah ended up saying something nice, which was that 4 x 10 made 40 and that this gave you the answer to the division problem. (Did I mention yet that this is in my 3rd Grade class? I should’ve.)

Having survived this ordeal, I saw her withdrawing again, not looking at the board, not hearing other people, the sorts of things that raise all of my alarms as a teacher.

So I did something unexpected, which is for the next problem I cold-called on Sarah again. And we did that same thing, the same conversation. DEFINITELY a gamble — this is a risky teaching move. Looking at her face, I started getting very nervous…I thought tears were a distinct possibility. But what could I do? I had to keep going. So I turned back to that last problem.

“You had said 4 x 10 makes 40 to help with the last problem. So how can we think about 48 divided by 4?” 

She made it through that conversation. Thank god for her friend, Kya, who gave her a big high-five after that conversation wrapped up. Sarah put on a big smile.

This was a big risk, but at least this time I feel that it paid off. Sarah is pretty good at skip-counting, and she was able to use that to solve some problems on her own without any prompting from me. I was able to give her a high-five also, to compliment her on thinking to do that all on her own.

And though certainly it could have gone bad, the whole point of this to me was that, no, I do not accept the terms of that contract.

What can “I’m not good at this” mean?

Here’s a quick note from the field.

Class today was largely about some fraction arithmetic, and one student was having trouble with it. In class, a couple times, this student said “I’m just not good at this” or “This doesn’t make sense” and “I’m bad at calculating stuff.”

But, all through class, the student stuck with it — I mean that they tried problems on their own and asked questions. And there was definite progress. The student was becoming able to handle this type of question on their own, and starting to make sense of things.

At the end of class I told this student that I thought they were getting the hang of this. I saw a sort of relief pass over this student. Then the student told me:

  • They had a hard time with this topic
  • But this student sort of blames previous teachers
  • This student recently signed up for extra math that is more like our class’
  • The math of our class is the “only math they’re good at”

So what did this student mean when they said “I’m not good at this” etc. in class? In retrospect, it was a statement of fear and anxiety precisely because it was the exception to the rule, at least the rule in our class. This student feels generally competent in our class, so much so that they’ve signed up for similar, optional math next year, and was feeling nervous that this good thing was in danger.

This doesn’t contradict the standard story that people tell about kids saying that they’re “bad at math.” (Especially because this student’s statements weren’t that they were bad at math, in general. It was more specifically about arithmetic and algebra.) But I do think that it illustrates one way that these statements can be an attempt to express something subtler than what they’re usually credited for.

Three things seem important to me about this case:

  1. The student clearly didn’t think that “being bad at arithmetic” was a fixed quality, as this student was simultaneously asking for help. This goes against the “fixed mindset” interpretation, I think.
  2. The student clearly doesn’t, in general, think that they’re bad at all math. These expressions of frustration can be about anxiety that a student’s good thing is under threat. So while it’s never a good thing to hear a kid say that they’re bad at something, it can be a local, specific issue rather than a global one about their status in math class.
  3. The student didn’t think that it was socially OK to be bad at arithmetic or bad at math. This wasn’t a student being proud of being bad at math, and it wasn’t a sign of anything but their own frustration.

I’d be curious to hear other people’s stories about students who make “I’m bad at…” statement in class, to learn more about the different contexts in which kids say this. I’m sure, at the end, we’ll find that they always come at moments of frustration. But I suspect that if a kid is saying that they’re bad at math in class, there’s more to that story, and maybe it’s a sign that (paradoxically) some fundamental things are working for that student in class.