For 12 × 29, Jemea thought about twelve 30s and then subtracted 1 for each of 12 groups.

360 – 12 = 348.

For 36 × 17, Duane thought of 36 bowls, each holding 17 cotton balls. Ten bowls hold 170 cotton balls, and there are three groups of ten bowls—510 cotton balls. The other six bowls hold 60 + 42 cotton balls. All together, that makes 612.

To solve 36 × 17, Thomas rounded each factor up to the nearest 10; 40 × 20 = 800. Then he subtracted the 4 he added to 36 and the 3 he added to the 17, and got 793.

After students worked in small groups, explaining their strategies to each other, there was just a short time for discussion. Liz asked Thomas to present his strategy, and used it to present a challenge to the class. The clip begins as Thomas finishes explaining his strategy:

Liz reported that the class engaged in three days of powerful discussions about this challenge. It would have been a joy to witness those discussions, but they were not videotaped. Just the few minutes from the initial lesson, though, have given me lots to think about in at least three different areas.

**Teachers exploring mathematics content**

The challenge Liz gave to her students is a great one for adults, too! What happens when both factors of a multiplication expression are increased? We know we have to take into account groups of things, but how many groups of what size? Try it out with images of groups and by drawing an array. Create a story problem for 36 x 17. How does the story change when 36 becomes 40 and 17 becomes 20? What has to be subtracted to get back to the original problem? If you write out Thomas’s initial steps with arithmetic symbols and apply the distributive property multiple times, how do the symbols match the drawings and the story?

**Students making sense**

Although there is an error in Thomas’s strategy, there is also an aspect of his reasoning that makes sense. After all, when adding 36 + 17, you can round up to 40 + 20 to get 60, and then you subtract 4 and 3 to get the correct answer of 53. Thomas got me thinking about how much confusion students can encounter if they expect multiplication or subtraction to behave like addition. Many common computation errors arise when students mix up rules for the operations, and these problems carry over into their study of algebra, as well.

Notice that Jemea and Duane explained their strategies in terms of the meaning of multiplication. Jemea talked about groups and Duane described an image of bowls that each held the same number of cotton balls. Such images, when associated with the arithmetic symbols, can become tools for reasoning. They allow problem solvers to keep the distinct properties of each operation clear.

**Teachers establishing a classroom culture**

Liz’s classroom illustrates essential components of a culture in which students learn important mathematics. First of all, students engage in reasoning. Students are responsible for explaining how they approach and solve a problem, and classmates are responsible for following and/or challenging the explanation. Sometimes that means sharing not yet fully formed ideas or strategies everyone knows are incorrect. After all, a partially formed or incorrect idea can provide an opportunity for the class to build new ideas or figure out where a strategy went awry.

In classrooms like these, teachers’ attention focuses on student thinking. They listen to their students’ explanations and ideas in terms of the content they are teaching. Liz recognized how Thomas’s error could become a learning opportunity not only for him, but for the entire class. She leveraged his incorrect strategy to challenge all of her students and extend their understanding.

This lesson took place more than twenty years ago. Jemea, Duane, and Thomas have graduated from college and started their careers. But I still think about them as fifth graders, and continue to reflect on what their ideas and their teacher’s actions can show me about principles of learning and teaching.

The scenarios described in this post, and the video embedded in it, come from:

Schifter, Deborah; Bastable, Virginia & Russell, Susan Jo. Developing Mathematical Ideas: Building a System of Tens. Reston, VA: National Council of Teachers of Mathematics, 2016.

- “What Was Thomas Thinking?” - October 16, 2017