**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* – 2*x*
= –*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.

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Dylan Boone- 11/13/2017 2:47:00 PMI love this. Yes you can often times only have one correct anwer in mathematics but the methods in which to get those answers are all over the place. The standard algorithm for basic operations is hardly ever how any student actually thinks about the operation when they are preforming the operation mentally. Often times we break it down into more friendly numbers like your students explained. Relational thinking is huge and shows a deeper understanding of the operation than just preforming the standard algorithms.

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