By Kasi Allen, posted April 11, 2016 —

When creativity and risk taking become the norm in a math classroom, students show passion for their mathematical ideas because they have a new sense of ownership. Under the right circumstances, they might even be willing to disagree with a classmate for the sake of their own thinking.

A few years ago, when I was conducting a research project in a middle school math class, I noticed that the students sometimes argued in their small groups. When I asked the teacher about this, he smiled and told me that the ultimate sign of a successful lesson was “a good math fight.” At the time, I laughed nervously, but I now know that the teacher was on to something. I can recall only a few times in my own teaching when students have engaged in an authentic mathematical argument. More commonly, one student suggests an atypical strategy for solving a problem and passionately makes the case for its validity.

This happened the other day in an algebra class that I am teaching, designed to help elementary school teacher candidates develop a deeper understanding of key algebra concepts. My students had completed the problem shown below as a Problem of the Week. They had also read an article I had published in the November 2013 Mathematics Teacher entitled “Problems before Procedures: Systems of Equations”.

The problem was stated this way:

One student looked frustrated as she gave an explanation to her small group. I walked over to listen. I saw three equations written in her journal:

She looked up and said: “I am trying to use the idea that if everything in this equation is also in that equation, then I should be able to subtract this one from that one to make a new equation.” Another student chimed in: “I told her she needs to multiply the second equation by 5 first.” She interrupted: “I could, but I don’t think I have to. I really want to make this work, because I like the way that student in the article was thinking.” I asked a few probing questions: What was the goal here? What did the student in the article do with a similar situation? “Wait!” My student smiled. “I can just keep subtracting x + y = 200, can’t I?” And she began writing rapidly as her peers looked on:

“So, x = 120, and y = 80!” she nearly squealed. “And look, I subtracted the x + y = 200 equation five times!” She glanced over at her colleague: “That’s where the 5 comes from!”

Other groups were now listening, asking her to present the method to the class. She hopped right up, saying, “Thinking like a kid is so much more fun than thinking like an adult.” Not quite a math fight but a lesson I will never forget. I don’t think she will either!

KASI ALLEN, kasi@lclark.edu, has worked in mathematics education for nearly thirty years as a teacher, researcher, and scholar. For the last decade, she has served as a professor of mathematics education in Oregon, teaching math content and methods to preservice K–12 teachers. Kasi loves helping people of all ages experience the power of having their own mathematical ideas. She is a math activist who studies math trauma and promotes teaching mathematics for social justice.