# Worked Examples and Loop-De-Loops

This is a Loop-de-Loop:

At camp, I taught a session on Loop-de-Loops, a mathematical object from Anna Weltman’s book. I had about 30 people in the room (15 counselors, 15 kids) looking for patterns and trying to figure out a bit about what Loop-de-Loops are all about.

I introduced Loop-de-Loops using Chris Lusto’s fantastic site, asking everyone what they noticed and were wondering after I hit “Show Me.” (Go try it; it’s a blast.)

This was perfect for us, as I wanted the focus of the session to be about asking questions. (Lord knows that the kids spend enough time solving problems in this camp. Asking questions is at least half of a full mathematical picture.) Lusto’s site makes it easy to quickly explore many different lists of numbers, generate theories and see if those theories hold up.

In fact, it’s too easy to generate these Loops with the software. For the sake of understanding, it’s always good to construct stuff by hand a couple times. Often we notice different things when we construct objects by hand than we do on the computer.

My first time running this session, though, I realized something: it’s hard to make a Loop-de-Loop!

Students and counselors struggled to draw these things — though, when they saw these objects on the computer, they were extremely confident that they understood how to make them. Surprise!

Today, I ran the session for another group of kids and counselors. This time, I came prepared. I wrote a worked example activity that aimed to help everyone better learn how to make these cool Loopy things:

I quickly made a handout right before class today. (Hence the marker.) I knew that I needed to include the three things I always include on one of these things:

1. The example, clearly distinguishing between the “task” and the “student work,” and trying to make sure to only include marks that contribute to understanding. (See those little arrow heads? I noticed that people had a hard time keeping track of directions while drawing these.) I tried to remove any distracting text or clutter — reading about the split attention effect helped me learn how to do this.
2. Prompts for noticing the most important stuff. Research suggests that students often don’t explain things to themselves, or do so superficially. (And experience totally confirms this.) Prompts, along with a clear call for students to spend a minute responding to the prompts, helps a lot with this I find.
3. A chance to try it out on your own, with the model nearby to help, if you get stuck.

I first read about this structure while reading about Cognitive Load Theory, but things didn’t click for me until I also saw the Algebra by Example project. Two other pieces have helped me better understand this bit of my teaching:

Here’s how I do these things. First, I put the “student work” up on the board and ask everybody to silently study it. (People need time to think before talking!) Then, I ask everyone to check in with a neighbor and to take turns making sure each can explain what’s happening. (This is usually where there are “ah!”s and “oh wait!”s.) Then, we talk about the prompts. After that, I tell everyone to try it on their own.

This part of the session went so much better today than my first pass. Everyone was still challenged by drawing these, and there were still a lot of mistakes. But the difference between these two sessions was precisely the difference between productive and unproductive struggle. Instead of flailing around when they got lost in the construction, everyone had something to go back to. Ah, OK, so this is how it’s done.

Part of my job is also to help the counselors support the students in their math work. A lot of the counselors tell me they struggle with knowing about how much to give away to a student when they are stuck. And while I totally know what they mean, I always tell them that their job isn’t to give away stuff or to avoid giving stuff away. Their job is to get the ball rolling for the student, get thinking happening, as quickly as possible — and then to step back and let that thinking happen.

Today I needed to get the ball rolling. It was a session about posing problems, and I started with asking everyone to notice and ask questions. But an important part of getting the ball rolling was a worked example.