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.

I’m curious where the kids were at coming in, how much they knew and then how they responded to this structure? Also what models did you use when you reached dividing by non unit fraction?

By the time they get to camp, every kid has learned how to divide by a non-unit fraction. My assumption is that pretty much every kid has learned the “keep-change-flip” trick or some equivalent, even if they are rusty on it. I have a lesson where I review the technique (frame it as “turning division into equal multiplication) and then play a game to practice using it. But I don’t teach it from scratch. I do spend time on common denominators with division — a few kids told me their teachers taught them this during the school year, which surprised me. I didn’t know it was at all standard practice.

The kids seemed happy with the structure, it felt seamless. One thing I know is that my class doesn’t create these BIG HUGE problems that are SO HARD and that the kids are SO PROUD of having solved. It’s a chiller vibe, and while that’s by design I sometimes feel a bit guilty about that.

I am surprised that you had multiple students familiar with common denominators! Maybe it’s sweeping NY.

I have used an interview question about common denominator division for middle school math teacher/specialist candidates in my district. Actually, we had to stop using it because it flopped. It was something like “A sixth grade student in your class looks frustrated. She tells you, “I can’t remember how to divide with fractions! Ugh! I remember that multiplication with fractions is easy: you just multiply across. Does that work with division, too? Divide across?”

I was looking for candidates to at least show curiosity. A successful candidate would say something like, “hmm! Let’s try a few examples and see.” Or even eliciting more of what they remember about multiplication and why it works, maybe draw a visual representation. Instead, every single candidate EVER said something like, “oh, that’s great that you remember how to multiply, but, no, you’re wrong about division. Watch me.”

And it made me sad, so we scrapped the interview question.

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First of all, I love teaching with puzzles. It’s an opportunity for students to make connections, and practice, and extend their thinking — sometimes all at once. (At least on a good day?)

I connected to this line:

“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.While it does not always work perfectly, there are often opportunities in puzzle work and the kind of novel practice that you are suggesting to be both more accessible and more challenging. Where I struggle here is making certain that I am circulating effectively through the room to continue to push student thinking for kids that are getting stuck, either unsure of how to start or that they are looping an inefficient idea. No matter how pleased I feel at the end of class, I know there are opportunities (and students!) I missed. I wrote about one instance here. Of course, because it’s a blog post with a purpose, I did not dwell on questions about the students who I didn’t get to talk to. What evidence do I have of their learning? (I mean, I have their worksheets, but what of that?)

“I’m aiming to approach familiar ideas in unfamiliar ways.”I think that's just good teaching. We want to help foster student sensemaking, and that means helping them make connections. One great way to do that? Approaching familiar ideas in unfamiliar ways. Besides, if everything is always in a familiar format, you still run risks. There's a risk that students won't generalize or make connections LATER. There's a risk that some students think it's too easy — whether we agree with their assessment or not — and that they do not engage enough with the material.

I am a person who likes to be in control. I also like to listen to students deeply and to help them grow. Sometimes these desires feel like they are in conflict, and it can be a helpful exercise for me to give up 100% of the control on what students are thinking about. It allows that divergent thinking student to come up with a cool connection I hadn't thought of. (It also means sometimes the divergent thinking student doesn't practice the thing I thought they would — a risk you already named — and that can be hard.)

This is all to say that you're going to take risks somewhere, and I think something that is interesting and increases engagement sounds great. You expressed some concern in a comment that your students may be missing out on the experience of the BIG HUGE problem with the BIG HUGE payoff. Maybe there's more opportunities to diverge from the structure for a few days when you're planning for a full year course — at least more opportunities than you have in a short summer course.

Oh, and I can't wait to check out all of the fraction materials! I love the mobile problems you gave here. I'm such a sucker for well aligned tasks and well designed curriculum. 🙂

“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. ”

One day, when I grow up, I want to be a master of this engineering.