**Editor’s note:** This is the third piece in a series called “Out on Good Behavior: Teaching Math While Looking Over Your Shoulder” by Barry Garelick, a second-career math teacher in California. He has written articles on math education that have appeared in *The Atlantic*, *Education Next*, *Education News* and *AMS Notices*. He is also the author of three books on math education. Says Mr. Garelick: “At its completion, this series will be published in book form by John Catt Educational, Ltd.” Chapters 1 and 2 can be found here and here.

**3. Understanding, and Outliers in a
Sea of Outliers**

During my second week of school at St. Stevens the principal, Marianne, called me in to her office to tell me some good news. “I just want to let you know that we heard from Mary’s mother and that Mary said she is really happy in your class; she says that “Mr. Garelick really wants us to understand.”

I was glad to hear that Mary’s mother was pleased, and while I haven’t taught for very long, I knew enough not to believe I was any kind of miracle worker—particularly during the first few weeks at school when everything is new and has the cast of a halo over it. At my previous school during back-to-school night one year before, a parent of one of my students in my seventh grade class said something similar. “My son said that this is the first time in any math class that he actually understood the math.”

In both cases, it didn’t hurt that the word “understand” was used in conjunction with my teaching, although the word has different meaning for me than what others in education think it means. I want students to be able to do the math. That’s pretty much what students mean when they say they understand. It isn’t something I obsess over.

Mary was one of two girls in my eighth grade math class (Math 8), who had to come in twice a week for intervention help for half an hour before classes began. The other student was Valerie who had been classified as special needs since the lower grades. They were both very animated girls; Mary was very outgoing and friendly with me. Valerie was more guarded. In her world of Smart Phone, songs, reality TV, I felt she viewed me as frightfully out of touch with what was really important. Math was certainly not on her list.

My Math 8 class was similar in some ways to my last year’s seventh grade class. My prior school, like St. Stevens, had two seventh grade math classes—one accelerated, the other not. I taught the non-accelerated group who considered themselves “the dumb class”. Their doubts were compounded by their last year’s math teacher who was not popular with parents or students, and was finally let go by the school. My Math 8 class similarly knew they didn’t make the cut for the eighth grade algebra class (which I was teaching). Their previous math teacher was similarly unpopular—and also let go by the school.

In a school as small as St. Stevens, there weren’t enough students to form a remedial class by itself. As a result, in the midst of a class in which the students already doubted their abilities, Mary and Valerie felt they were outliers. I worked with them as best as I could. I called on them infrequently in the main class, and focused on them during my intervention time.

At first, I tried to get them up to speed with what the rest of the class was doing. During one of my sessions with them, I went over one-step equations. I asked them to solve the equation 6x = 12. I had reached the point where neither one was trying to subtract the 6 from 6x to isolate x. But while Marie understood that 6x meant 6 multiplied by x , Valerie could not see that; nor could she see that solving it meant undoing the multiplication by division.

“How do we solve it?” I asked.

“You put the 6 underneath both sides,” Valerie said.

That is:

Putting one number underneath another meant divide to Valerie which is as procedural-minded as you can get. If ever pressed to justify my acceptance of her level of understanding to well-meaning doing-math-is-not-knowing-math types I could say that her method at least incorporated the concept that a fraction means division. No one ever asked, but just to make sure, I said “And what do we call the operation when we put another number ‘underneath’ another?”

She thought a moment.

Mary whispered in Valerie’s ear: “Division”.

“Oh; it’s division,” Valerie said.

Over the next few weeks, I worked with the two girls privately while trying to keep them on track in the Math 8 class. I realized that their deficits were so significant that to hold them to the standards of Math 8 would result in failure. Katherine, the assistant principal agreed with me, and said to focus instead on filling in the gaps and to base their grade on their mastery of those.

I leveled with them one morning when they came in for their intervention.

“The Math 8 class must be extremely painful for you,” I said.

Valerie for the first time let down her guard. “I just don’t understand what’s going on.”

I found her statement true in a number of ways. One was just the fact that she admitted it. But more, it brought home the issue of understanding. Of course she didn’t understand—she barely had the procedural and factual tools that would allow her even the lowest level of understanding of what we were doing.

While there are those who would say “Of course they didn’t understand; traditional math has beat it out of them,” such thinking is so misguided that, in the words of someone whose name I can’t remember: “it isn’t even wrong”. In the case of Valerie and Mary, they needed more than the two half-hour interventions every week. They needed someone who specialized in working with what is known as dyscalculia. But qualifying as a special needs student doesn’t guarantee the student will get the kind of help to deal with a disability.

They had finished what I gave them to do early that day, and Mary being in a celebratory mood said the two were going to make a drawing for me. They giggled while they drew a rather strange looking bird laying eggs and eating blueberries among other odd things going on and presented it to me. I put it on the wall where it remained for the entire school year. Sometimes other students would ask what the drawing was, and they would explain it, excitedly. The excitement was partly due to them having drawn it, and partly due to my keeping it on the wall for all to see.

Betty Peters says

I enjoyed reading this–it reminds me of one or two math classes I substituted in years ago. I remember a 4th grade class in which I stopped to help a girl who was confused. When I finished explaining, she said, “Oh, I get it. No one has ever explained it that way before.” What wonderful words to hear! I walked on air the rest of the day. When I read your articles, I feel sorry that teachers are not being taught how to teach math properly anymore and even if they were, the curricula and assessments being used preclude their use of anything but the prescribed Progressive methods. This has been going on since at least 1989 in most schools.