A Small Shift with a Big Impact

How one teaching move can transform student understanding

First off, I want to say welcome to a handful of new subscribers! I seem to be getting subscribers from a comment I made on a post totally unrelated to education, so apologies if you didn’t realize what you were signing up for. I’m a teacher coach and consultant sharing ruminations on math education. I hope you’ll stay, enjoy what you read, and share it. And if you were hoping for something political, well, my political thoughts here are implicit rather than explicit. Then again, math is pretty political these days (but isn’t everything?).

While the math wars rage on, one thing I think (I hope!) we all agree on is that we want students to leave the classroom with more understanding than they had when they came in. Whether the new understanding is a particular skill or a deep mathematical connection, we want them to confidently – and correctly – approach similar problems in the future.

One small shift in instruction can make this happen, and I witnessed it last week.

The shift involves starting a discussion with a mistake rather than with the correct answer. We’re going to zoom in on the consolidation, the debrief, the “landing” that happens after students have been working on their own or in small groups. Many teachers – maybe this is our instinct as teachers – highlight the correct answer, asking a select few to explain how they got that answer. Maybe we highlight a common mistake that students made. We think that showing students the problem worked correctly will help them understand where they went wrong. Students nod as if they understand, but they make the same mistake the next time. Then we wring our hands and declare, “I don’t get it – We went over those mistakes. They should know this by now!”

But reviewing correct answers doesn’t produce understanding. Compliant students will nod along and respond affirmatively when you ask if they get it, but that doesn’t mean they do. Starting from a mistake, rather than the correct answer, has a much greater impact than revealing the answer.

I know there is a lot of debate out there over Building Thinking Classrooms, but, whatever your opinion on Liljedahl’s work, stay with me for a minute. This is a gem buried in chapter 10 of his first book, and I would argue it’s one of the most critical practices.

Liljedahl calls starting with the most common mistake “consolidating from the bottom” (as opposed to “consolidating to the top” by revealing the correct answer). This is a reworking of Smith and Stein’s idea of “sequencing” in a discussion, detailed way back in 2011 in 5 Practices for Orchestrating Productive Mathematics Discussions. Both books argue for the importance of building understanding from where students are. When we wrap up an activity by simply revealing the correct answer, we lose all of the kids who haven’t gotten there yet.

Think of understanding as a ladder. Misconceptions make students get stuck on a certain rung. If we want everyone to get to the top, we need to help them get to that next rung. By simply showing them the answer, we’ve placed them on the top rung, bypassing all the intermediate rungs. They’re wobbly up there; they probably feel like they’re going to fall. And we’re surprised when they don’t make it to the top on the next problem.

(To be clear, this isn’t an argument for explicit instruction over guided discovery, which is where we often see the ladder metaphor. The ladder is not meant to represent skills that build from year to year. Rather, it’s each person’s particular understanding within one specific topic, and they can climb that ladder in a number of different ways.)

Let’s look at how this can play out in a classroom. Last week, I was working with a high-school teacher who has adopted some, but not all, practices from BTC. That day, her Algebra 1 students were working on simplifying polynomial expressions. The teacher knew this wasn’t the most exciting lesson ever, but it was a necessary one, so she had her students stand and work in groups at whiteboards around the room. Every single student was engaged, discussing the exercises with their groups, taking turns doing the writing. As expected, some groups got to the correct answers, while others made calculation errors, almost always involving a negative sign.

She decided to focus on the first problem she had given them, which was something like this:

(2x³ - 3x² + 7x - 5) - (4x³ + 12x² - 8x + 17)

The teacher gathered the class around one of the boards that showed the problem worked correctly. She asked a few questions:.

“Let’s look at what this group did. What happened to that negative sign that was in front of the parentheses?”

“Should that be negative 8x or positive 8x? Why?”

“How does that change our final answer?”

When students had answered the questions satisfactorily, she tried to drive home the point where the other groups had erred:

“Do you see how this negative sign is like a negative 1 that gets distributed? So what was negative 8x becomes minus negative 8x, which is the same as plus 8x. Do we all get that? A lot of you made that mistake in your work, so make sure you check your negative signs carefully next time.”

Students listened and nodded, making a show of understanding. Then the bell rang.

I wasn’t convinced those who had made the error understood. In fact, I was pretty sure they hadn’t, and they would make the same mistake on the next problem.

The next period, several groups made the same mistake. Knowing what the teacher was probably going to do, I sidled over to her and whispered, “Let’s gather around one of the boards that shows the mistake and ask students if they can identify it.”

This was an easy instructional shift, and one of the most powerful ones the teacher could have made. When the students gathered around a board that showed the common error, she asked, “Where do you think this group made their mistake? Everybody take a minute to look at their work and compare it to yours. Where did they go wrong?”

The students stood quietly, looking at their boards and back at the one in front of them. After about half a minute, a boy standing next to me shot his hand in the air: “Oooh ooh, I know! I see it!” He was in the group whose work we were analyzing, and he was one of the first to spot their mistake. But the teacher waited. “I want to see more hands. If you haven’t found the mistake yet, keep looking.”

Another minute passed and several more hands went up. The teacher called on a student from another group – not the boy next to me who was literally jumping with eagerness to share his thoughts – who explained that they hadn’t distributed the negative. Other students chimed in, explaining what the next step should have been and how the group could have gotten the correct answer.

When the bell rang and students left the room, I felt fairly confident that nearly all of them understood the mistake and would solve a similar problem correctly. But I knew there was an even better way to have this discussion – a way that positions students as the experts in the room.

For the third Algebra 1 period, again seeing the same mistakes, we gathered students near two boards, one that had the correct work and one that didn’t. “Hmm, I see two different answers,” the teacher said. “How is that possible? Is one of these correct and one of them incorrect?”

A pause. Then several students tried to answer at once. “Ooh, this one’s correct, and I see where the other group made a mistake!” One student near the back snuck away to fix the same error on his group’s board across the room. It was a small class, and the teacher let the conversation flow organically. They pointed out where the group made a mistake and how to correct it. One girl asked a couple clarifying questions and her peers answered.

I felt fairly confident they all understood the mistake, but to extend their thinking, I asked, “What do you think would happen if the problem had looked like this instead?” I wrote a similar problem but switched a negative to a positive. The students discussed, asked clarifying questions, and came to a consensus on what the final answer would be.

This time I was confident every single one of them understood and would be careful not to make that mistake again. While they learned, they also got the chance to negotiate a peer discussion, to support each other, to ask any questions they had about the procedure. They normalized mistakes and proved they could answer each other’s questions.

I’m pretty sure we can all agree these are experiences we want students to have.