Expansion

Building from what students know to what we want them to learn

Here’s a question that is often hard for students.

Jenny earns $140 for 8 hours of work. She receives a 12% pay increase. What are Jenny’s hourly earnings, after the increase?

This is pulled from a set of sample standardized test questions for 7th graders. It’s the kind of question standardized tests love to ask: it combines multiple skills in a word problem. The individual skills aren’t particularly hard, but students typically struggle to put the pieces together. I also think this is a good question. It’s useful in the world, and reflects real skills we want students to have.

In the past, I might have made this question a lesson objective. We would practice multi-step problems with a unit rate and a percent increase. The lesson would be a bit of a slog. Some students would get it right away, some would struggle. And, inevitably, most would forget.

How can I teach students to solve problems like this one in a way that is more likely to stick? Better yet, how can I teach students to apply what they know in a variety of situations, without having to model every single possible variation?

The answer is three steps: preparation, expansion, and discrimination. Here goes.

Preparation

Before helping students with this specific skill, I need to break it into pieces and make sure students are comfortable with each piece. There are three major pieces here (each of which could be broken down further, but let’s keep it simple at three). First, students need to be able to find a unit rate. Second, students need to be able to find a percentage of a number. And third, students need to know how to find a percent increase. Each of those is a skill we practice in 7th grade. My first goal is to practice those skills individually — they’re part of my curriculum anyway, so no problem there. I don’t move on to the next two steps until students have decent fluency with each of those constituent skills. One important note about fluency is that it takes time. I can’t do a quick reteach of percentages at the start of the lesson and expect students to have the fluency they need for multi-step problems. That learning needs to happen with time to practice, consolidate, and build fluency before it’s needed for more challenging work.

Expansion

My next goal is to work gradually from what students know to the problem I want students to be able to solve. This is the expansion sequence: expanding what students know how to do, from the constituent skills in the preparation stage to the more challenging problems I want students to know how to solve. Here is a sample sequence of problems:

1. Jenny earns $40 for 2 hours of work. How much does Jenny make per hour?

2. Jenny earns $40 for 2 hours of work. What are Jenny’s hourly earnings?

3. Jenny earns $240 for 8 hours of work. What are Jenny’s hourly earnings?

4. Jenny earns $240 for 10 hours of work. What are Jenny’s hourly earnings?

5. What is 15% of 20?

6. What is 18% of 20?

7. Jenny earns $24 per hour. She receives a 15% pay increase. What are Jenny’s new hourly earnings?

8. Jenny earns $24 per hour. She receives a 12% pay increase. What are Jenny’s new hourly earnings?

9. Jenny earns $20 per hour. She receives a 12% pay increase. What are Jenny’s new hourly earnings, after the increase?

10. Jenny earns $160 for 8 hours of work. She receives a 12% pay increase. What are Jenny’s new hourly earnings, after the increase?

11. Jenny earns $140 for 8 hours of work. She receives a 12% pay increase. What are Jenny’s new hourly earnings?

The goal here is to remind students of what they know, and create really clear connections from one problem to the next. This approach takes a challenging problem, shows students that it actually isn’t that challenging if we break it down into pieces, and models how to do that.

When I do this well, students retain the learning much better than just teaching this as a standalone objective. The goal is to make really clear connections between what students know and what I want students to learn. When those connections are strong, students are much more likely to remember what they’ve learned and apply it in different situations in the future.

Preparation is key. The first few questions should feel familiar to students and build some confidence. If they don’t, we need to go back and do some more work. That’s the bedrock we’re building on. If I want students to remember how to solve a problem like this, the prior knowledge we’re building on needs to be secure.

Expansion sequences don’t always work. Sometimes it’s just too many steps for students to follow. Maybe we get to problem 7 and most students start to struggle. Great, that’s fine. Let’s stop there. Tomorrow, I’ll create a new sequence, starting a bit before where students got stuck, and we’ll see if we can get further.

For me, the biggest challenge in getting students to think about complex problems is that they take one look, don’t know how to solve it right away, and give up. The best antidote to this lack of perseverance is to build confidence with a bunch of questions students know how to solve. If students are getting discouraged working through the beginning of the expansion sequence rather than becoming more confident, I’m doing something wrong. Maybe we need more preparation. Maybe the expansion sequence needs to happen in smaller, more manageable steps.

Discrimination

Expansion sequences are great, but students also won’t have that type of scaffolding every time they’re asked a question like this. Expansion focuses on making connections between problems. Discrimination is where students practice telling the difference between problems. My goal in a discrimination sequence is to give students a few problems that look similar on the surface, but ask students to use different concepts. Here’s an example:

1. Jimmy earns $140 for 8 hours of work. He saves 12% of the money. How much does he save?

2. Jimmy earns $140 for 8 hours of work on Monday. On Tuesday he works for 12 hours at the same rate. How much does he make on Tuesday?

3. Jimmy earns $140 for 8 hours of work. He receives a 12% pay increase. What are Jimmy’s hourly earnings?

4. Jimmy earns $140 for 12 hours of work. He receives an 8% pay decrease. What are Jimmy’s hourly earnings?

Discrimination sequences are hard. Not all students will be successful every time. The goal is to spend our time thinking and talking about how to solve each problem, and focusing on what makes these similar problems different from one another.

If students are struggling with a discrimination sequence, I have a few possible solutions. One is to do more preparation. Maybe students need more confidence and fluency with the constituent skills. A second is to do more expansion. Maybe students need more clear connections from what they already know to these more challenging problems. And a third is more practice with discrimination. Maybe students need more practice telling the difference between different types of problems. It’s not always easy to tell which one students need, but at least I have three solid options to work with.

Expansion Everywhere

Expansion sequences are becoming one of my favorite tools. When I think about some of the tough questions, common on state standardized tests, that I want my students to be able to solve, an expansion sequence has become my go-to. But I also use them for pretty mundane, everyday skills. In everything I teach, I’m always looking for ways to start with something students confidently know, and build in small steps to something a little tougher. It’s how math works: we are always building on what students know to learn something new. This also gives me constant data about what students know, and what is too hard.

Like many teachers I’m under some pressure to improve standardized test scores. There’s a version of test prep that I think doesn’t work very well. The teacher takes a bunch of tough questions like the one at the top of this post. Then, they tell students they need to practice test questions, give students a bunch of those questions, and then go over the questions that seem hardest. In general, this results in a lot of students saying “idk” and feeling dumb. The goal of expansion is to practice hard questions in a way that shows students how those questions are connected to what they already know, and to give students more confidence approaching tough problems.

Expansion doesn’t always magically work. I wrote in my post last week about how I was having trouble helping my students extend their knowledge of one-step multiplication equations with whole numbers to fractions. We ended up going back and doing some more practice with fraction division, and then we’ll try again. And that’s one benefit of this system. If I have a tough skill students are struggling with or aren’t retaining, I have a few good tools to try: preparation, expansion, and discrimination. In this case they needed more preparation. In other cases it’s something else.

Repetition is an important part of an expansion sequence. Let’s say there’s a ladder of skills, ABCDE. It’s tempting to have students do ABCDE all in a row. That usually doesn’t work very well. It tries to take students too far too fast. Instead, I start with ABC. If that doesn’t go well, we’ll do ABC again, or maybe mix in some extra practice with A to build some confidence and a solid foundation. Then we’ll do BCD, or maybe ABCD to really build confidence. Then CDE, or BCDE, or ABCDE.

In expansion, each skill starts as a stretch, something tenuous that’s building on the limits of what students know. Then it becomes more natural and students gain confidence. Then it’s just another thing students are confident and fluent in, and we’re building for the next thing.