I’ve written a few times about how I use worked-out examples (and mistake analysis) in my teaching:

- Worked Examples and Loop-de-Loops
- Worked Examples and Geometry Proof
- Worked Examples and Focused Feedback
- Worked Examples and Feedback on a Whole Quiz

The resource I’ve returned to again and again is Algebra By Example, which is really well-done. (And get excited for Math by Example, their expansion pack for 4th and 5th Grade!)

I realize that there are two things that I haven’t yet shared: some research on how to design these things well, and my routine for using these materials in my teaching.

### The Research

Whenever you get a bunch of people talking about research in education, somebody eventually mentions how sad it is that research in education usually happens without the input of educators. Wouldn’t it be grand if researchers and educators were more collaborative?

Yes! This indeed would be grand, and it should happen more often. But sometimes it does happen, and those partnerships can produce really wonderful things. Algebra by Example is the result of one of these partnerships.

The project was led by Julie Booth of Temple University, and Booth has written a very nice piece in an NCTM journal* *about the design process her team uses and the research behind it:

McGinn, K. M., Lange, K. E., & Booth, J. L. (2015). A worked example for creating worked examples.

Mathematics Teaching in the Middle School,21(1), 26-33.

Booth’s piece is very nice. It cites a more research-y piece that more carefully trawls and summarizes the worked example literature, and that is also an interesting (though more research-y) read:

Atkinson, R.K., et al. (2000). Learning from examples: Instructional principles from the worked examples research.

Review of educational research, 70(2), 181-214.

Though it’s tough to summarize these pieces, here are a few important principles when designing example tasks:

- Try to put everything in one place…
- …but also try to visibly separate different steps.
- Switch up correct examples with incorrect work.
- Ask kids to explain what’s going on.
- Ask kids to try it on their own.

Try to put everything in one place. That’s a NO for splitting the problem and the solution with a not-so-useful line of text. Keep words at a minimum; ideally, you can see the whole example at once.

It’s useful (according to Booth’s research and also me) to switch up correct with incorrect examples.

Can you spot the error?

I like the way this incorrect work has some good thinking (I like the regrouping idea) and that it also clearly distinguishes each step.

From experience, I’ll say that people new to writing examples often try to do too much with the space they have. The goal really is to eliminate all but the most important thinking from the example, and it’s OK to use multiple examples to get at different aspects of the thinking. Overstuffing one of those little boxes can be a sign that it’s time to break this up into two smaller problems.

So much for the visual design of these materials, which is really a huge subject in of itself.

Some of the most interesting bits of those reviews are about self-explanation. Here’s a bit from the more research-y review:

Research on explanation effects suggests that self-explanations are an important learning activity during the study of worked examples. Unfortunately, the present research suggests that most learners self-explain in a passive or superficial way.

“Passive or superficial”: YES! All too often this is how students interact with an explanation (or with feedback).

So how do examples help at all, given the tendency of students to just passively say “OK fine yep I get it” when they see an example? I use a routine to really make example activities hum.

### The Routine

Whenever I use an example activity with students, here is my routine.

**Get ready**: I show the problem, covering up everything besides the problem itself. I usually use a document camera, so I mean I literally cover it up with my hand. Sometimes I ask students to begin thinking about how they’d approach the problem before I reveal the student work.**Read:**I ask everyone to read the student work to themselves. I ask students to offer a quiet thumb to let us know that they finished reading. I usually tell students to put a thumb up if they’ve read each line, even if they don’t understand everything fully yet.**Discuss:**I assign partners (really I’ve already done this) and ask students to discuss the example until each person understands every line of the student work. (I focus my class on spreading understanding, not on solving problems.)**Explain:**Every Algebra by Example problem has prompts for student explanation. When I make my own materials, I*always*include such prompts. It’s at this stage in the routine that I make sure everyone tries to answer the prompts. Then I usually call on a student to explain.**You Try:**This is where we help students try it on their own. There’s a whole art to picking good problems for students to try here — they should change the surface-details in some way, while keeping the underlying ideas constant. This is where we try to keep kids from making false generalizations, and it’s another level of protection against superficial understanding.

Using the Algebra by Example materials, the routine flows from left to right:

It’s not so easy to design these prompts, either! Here are some suggestions, from Booth’s piece:

If you’re in the mood for a bit of practice, you might think about how you’d fill in the rest of this example task:

Example analysis works especially well as a feedback routine. There was a problem that kids especially had trouble with? Write an example that focuses on how to improve, and then follow the routine with some time to revise and improve the quizzes (or a re-quiz, or etc.).

All of this — the examples, the prompts, etc. — is worth your time, I think. It’s a lot of fun for kids to puzzle out how someone else is thinking, and the format allows us to really focus on a whole idea. It’s a bit of a myth, I think, that these types of problems are boring.

It’s also a myth that this *kind* of math is boring. It’s not, because there’s something to be understood here. And fundamentally, understanding *is* interesting.