Elaboration Questions

Questions to get students thinking hard

There’s a principle of learning that I find important but also hard to articulate. It sounds something like, “we learn what we think about, so we should get students to think hard about important ideas.” Or, “learning is more durable when students think about the deep structure of a concept rather than its surface structure.” Or, “learning works best when students make connections between concepts and think about the meaning of what we want them to learn.”

One term cognitive scientists use to describe this type of thinking is “elaboration.”

The vast majority of teachers I’ve met understand this idea intuitively. We want students to think about big ideas. Walk into a random classroom and you might find teachers asking hard questions, prompting students to dig deeper.

A Non-Example

The hard part of elaboration questions isn’t the questions themselves. The hard part is getting every student thinking about the questions.

Here is a non-example.

One practice I’ve seen recommended in this general direction is for students to create their own problems. Teach a topic, then ask students to write their own problems. Maybe they trade with a partner, or we write them on the board and solve them together. The logic looks like this: generating problems involves deeper processing than solving problems, so it will cause students to think more deeply about what they are learning.

That’s probably true! Here’s the catch: when I’ve used this strategy I find that some students are able to generate their own problems, while others stare at me with blank expressions unsure of where to start.¹

There are two key qualities of a good elaboration question: first, it should require effortful thinking. Second, it should cause as many students as possible to do that thinking. Below are some examples of my favorite elaboration questions. These are my favorite because, in my experience, they are the best questions for getting reluctant students to engage with this type of thinking.

Elaboration Questions

Find the pattern

Give students a sequence of problems with some sort of repetition that reveals a pattern:

• Find 10% of 200.

• Find 25% of 200.

• Find 3% of 200.

• Find 4% of 200.

• Find 6% of 200.

• Find 60% of 200.

• Find 80% of 200.

Then ask: what patterns do you notice?

We can talk about the patterns, we can make predictions about new problems using those patterns, and we can extend the pattern in different directions — maybe finding percents of 300 or 400 to see what happens next. There’s a ton of math here. Students are practicing percents, while also creating an opportunity to think hard about why percents do what they do.

Expansion

Here is a sequence of problems that get gradually harder:²

• 5 pounds of cheese cost $50. How much does one pound cost?

• 2 pounds of cheese cost $8. How much does one pound cost?

• 3 pounds of cheese cost $11.40. How much does one pound cost?

• 3 pounds of cheese cost $12. How much does one pound cost?

• 3 pounds of cheese cost $6. How much does one pound cost?

• 2 pounds of cheese cost $6. How much does one pound cost?

• 1 pound of cheese costs $6. How much does one pound cost?

• 0.5 pounds of cheese costs $6. How much does one pound cost?

• 1/2 of a pound of cheese costs $6. How much does one pound cost?

• 1/3 of a pound of cheese costs $6. How much does one pound cost?

• 2/3 of a pound of cheese costs $6. How much does one pound cost?

The goal here isn’t that every student solves every problem. The goal is to increase the difficulty gradually, figure out where students get stuck, and use that information to help students learn something new. I often find that when I sequence questions in this way, students can solve much harder problems than if I just throw out the problem cold.

What If?

Students solve a problem, or we solve a problem together. We leave the solution visible, on paper or on the board.

Then I ask what if — here’s an example.

We solve the equation: 2x + 1 = 11.

What if the equation was -2x + 1 = 11?

The exact questions here will depend on what your students know and don’t know. This is a great way to get students thinking hard if they’re reasonably confident with the first equation, confident multiplying negatives, but don’t have much experience with negatives in equations.

Non-Examples

Students solve a problem, or we solve a problem together. Then I ask a question that is a deliberate contrast with the last problem, where students might overgeneralize a rule.

First question: distribute

2(3x + 10y - 5z)

Next question: distribute

2(3x + 10y) - 5z

A non-example helps to avoid the problem of going on autopilot, copying an example without considering the structure of that example. It often leads to great conversations!

Stepping Back

The last few examples of elaboration questions all have something in common. They are all designed to get students solving one problem or several problems accurately before doing some deeper thinking. That’s a great general strategy to get students thinking hard about elaboration questions: build confidence with a few things students know how to do, and then use that confidence as a springboard to tackle tougher questions. The details here will depend on your students. The initial questions need to be ones students can solve confidently, and the leap you’re asking students to make needs to be accessible but not too easy.

Ok, back to a few more examples.

Sentence Completion

Writing in math class can feel like a pain. I could say a lot more about this, but for this post I have one go-to teaching move to get students writing about math.³ Here’s what it looks like:

Complete the sentence:

To find the surface area of the prism, Lin found the areas of the three rectangles, added them, and multiplied by 2. She multiplied by 2 because…

This is one of those things students end up doing without really understanding. “Oh yea, I multiply by two for those problems…” Sentence completion is the best way I’ve found to get students thinking and writing about the why, without getting a bunch of blank stares and blank papers in response.

Numberless Word Problems

Word problems are a pain. This could also be a much longer post. But the short version: a good way to get students thinking hard about word problems is to take away the numbers. Without numbers, students can’t just grab numbers, smush them together with an operation, and move on. You can do this by covering up the numbers in a problem you already use, or writing your own. An example might look like this.

A runner jogs one lap around a circular track. How far does the runner jog?

Then, prompt students. What do you need to know to answer the question? What would you do with that number if I gave it to you? What if I gave you the radius instead?

Elaboration Questions

You can call these whatever you like. I’m partial to “elaboration questions” because it’s concise but you might like something different. You also might have lots of other strategies that work well! Let me know what works for you.

The key idea I want to emphasize: the goal of elaboration questions is not to get a handful of students thinking hard. The goal is to get every student, or as close as possible, thinking as hard as possible. Some of the examples I gave might seem simple. That’s often the case! We are teachers. We know a lot more than our students. It’s easy to overestimate what students know, ask a bunch of hard questions, and get a lot of blank stares in response. My priority when I ask elaboration questions is to get as many students thinking as I can. These strategies are the best tools I’ve found to do so. Even questions that seem simple on the surface can work well when we scaffold and sequence them in ways to get every student thinking.

1. I’m sure there are teachers out there who are successful building routines to get students creating their own questions. If it works for you, that’s great. I haven’t been able to make it work for more than a small fraction of my students, but don’t let me rain on your parade, do what works.

2. Worth clarifying for these problems that we are talking about a hypothetical world with a remarkable variety of cheese available for purchase by the pound, each priced differently.

3. I can’t resist the temptation of adding another non-example here. Since the Common Core math standards rolled out in the US, everyone loves to ask students to explain their reasoning. It’s explicit in the standards: “Construct viable arguments and critique the reasoning of others.” Most curricula interpret this in a really narrow way: tack “explain your reasoning” onto half the questions we ask students. Explanation becomes a tedious chore, an endless barrage of “I found my answer by multiplying” that does nothing to explain and prompts no substantive thinking. I find explanation works best when used sparingly, intentionally, and with scaffolds to make it accessible for students.