By Adam Sarli, posted September 25, 2017 —

When I started teaching math, I definitely held this belief. If a student correctly solved for x, I would smile and let him or her know that the answer was correct. I would continue circulating around the room, letting students know who was correct while working with those who were incorrect. I judged a lesson to be successful after seeing how many students got the right answer. 

Over time, I realized how little my students seemed to “get it.” They stumbled whenever a problem’s context shifted, and they struggled to justify their answers. Many ended up forgetting the process they had used to find the correct answer in the first place. Framing my teaching around answers was not helping my students learn mathematics. 

I don’t want my students simply getting the right answers; I want them getting the underlying concepts and connections inherent in the mathematics. And the best way to encourage this process may be by focusing on the methods that students create to arrive at their answers. 

The authors of Making Sense: Teaching and Learning Mathematics with Understanding argue that math is indeed a subject with a right answer, but that students can intuitively create their own strategies for solving problems. Hiebert and colleagues (1997) urge teachers to frame their teaching around these student-created strategies, suggesting that “Engaging in open, honest, public discussions of methods is the best way to gain deeper understandings of the subject” (Hiebert et al. 1997, p. 39). 

Let's look at a unit on integers and explore how focusing on student solution methods can lead to a deeper understanding of mathematics. I have a few overarching goals for students in my seventh-grade integer unit. I want students to explore—

• the big ideas inherent in positive and negative space; and

• the idea that numbers can be decomposed strategically.

Take a simple integer problem like 4 – 8. To get both my students and myself caring about more than the answer, I decided to simply write the answer on the board. “The answer is –4. Does anyone want to share their strategy for finding it?” See Destiny’s method: https://flic.kr/p/XSYyDs

Destiny's method was the most common. Students thought of subtraction as “moving down,” and the 8 as meaning “8 times.” A pair-share question of “How did Destiny know what numbers would look like past zero?” got students talking about the big ideas in negative space.

I was determined to push for multiple strategies, so I asked, “Does anyone have a different strategy in which they didn't have to make 8 jumps?” See Malik’s and Rhian’s work: https://flic.kr/p/XSYyoN

Malik’s method made the act of decomposing numbers transparent for students like Destiny who were initially uncomfortable with the process. Such questions as “Why did Malik go down by 2 four times?” and “Why did he cross off 2s?” helped students get to the heart of this decomposition idea. 

Sharing Rhian’s method after Malik’s work was strategic. “Why did it take Malik 4 jumps, but Rhian only 2?” Student pair-shares, again, explored ideas of decomposition. Students were informally discussing how 4 – (4 + 4) = 4 – (2 + 2 + 2 + 2). They also called Rhian’s strategy “Going to a Friendly Number,” since going to 0 made the problem easier. 

At this point, Hymadou had a theory that he shared with the class: https://flic.kr/p/XSYyn5 

Hymadou was convinced that his strategy would work for any integer, so the class began challenging his theory by trying out problems like 5 – 10, 12 – 24, and 100 – 200. Many concluded that Hymadou was correct, but they struggled to put into words exactly why. They were informally exploring the conjecture that x – 2x = –x. 

In A Guide for Teachers Grades 6–10, Mark Driscoll calls Hymadou’s theory “extending,” since he “followed the lines of further inquiry suggested by a particular mathematical result” (Driscoll 1999, p. 97). Hymadou asked his own question and took the mathematics to an algebraic level I hadn't even anticipated!

Notice just how much mathematics there was to explore once multiple strategies and a few guiding questions were presented—all from the simple problem 4 – 8! This problem may have been buried with 50 others if I held the belief that the goal in math is simply to get the correct answer. 

Shifting the focus in math class away from answers and toward methods has huge implications for student learning. It prompts teachers to plan lessons around deep mathematical ideas and to ask questions that get students’ reasoning in focus. It encourages students to develop or try new strategies. It can even get students asking their own questions and justifying conjectures that hit at the heart of mathematics. To really help our students “get it,” I’d choose methods over answers any day. 

References

Driscoll, Mark. 1999. Fostering Algebraic Thinking: A Guide for Teachers Grades 6–10. Portsmouth, NH: Heinemann.

Hiebert, James, Thomas P. Carpenter, Elizabeth Fennema, Karen C. Fuson, Diana Wearne, Hanlie Murray, Alwyn Olivier, and Piet Human. 1997. Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, NH: Heinemann.

Adam Sarli is a middle school math teacher and math coach at MS331: The Bronx School of Young Leaders in the Bronx, New York. He has been teaching middle school mathematics for over ten years and is a Math for America Master Teacher. He is passionate about student-to-student discourse, student-created strategies, and problem-based math instruction. He tweets at @adam_sarli. His math music videos can be found on YouTube at youtube.com/user/Remikarli.