Editor’s note: This is the tenth chapter 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.” The previous chapters can be found here:
Ch 10: Not Making Sense, and a Conversation I Never Had
“Math doesn’t make sense.” This was the chief complaint that Lucy evinced when seeking help with algebra. She was a bright girl in my eighth grade algebra class at St. Stevens.
Lucy’s statement will no doubt serve as evidence for those who view me as an unbending traditionalist hell-bent on teaching procedures at the expense of “understanding”. While I do provide the underlying concepts to procedures, there are students, like Lucy for whom math had always come easy and the connection between procedure and concept was obvious. With algebra the level of abstraction ramps up and things were no longer as obvious. Lucy thought that if math didn’t come easy then either something was wrong with her, or math made no sense.
The range of abilities in the St. Stevens algebra class was much wider than my previous classes, and likely more typical of most schools. There were about five students at St. Stevens who were at the very top of the class. At the lower end there were about four or five. Lucy was starting to fall into that lower group. She made a good effort in my algebra class in the beginning but increasingly got caught in waves of confusion starting with multiplying and dividing powers.
She had begun to make a good comeback with factoring of trinomials such as x²+5x+6 into two binomials: (x+2)(x+3). She even volunteered to do a problem at the board. But the next day we had more complex trinomials like 6x²-5x-6. Students were having a hard time with these and Lucy was back to sitting with arms folded, answering questions I asked of her with a shrug and a response of “I don’t know” laden with teenaged insouciance.
I had taught this particular type of trinomial by using a trial and error method in which you try various factors like 2x and 3x, and 6x and x to get it to work. (If you’re curious, the factorization of 6x²-5x-6 is (2x-3)(3x+2).
There is another method, sometimes called the “diamond method”, which involves some steps that I won’t go into here, but results in the trinomial being expressed as 6x²-9x+4x-6. This can then be expressed as 3x(2x-3) + 2(2x-3). Since (2x-3) is a common factor, this now can be expressed as (2x-3)(3x+2). I’ve tried to teach this method in the past with mixed success; many find it difficult. Given the problems I was having with Lucy and others, I decided to stick with the trial and error method.
I allotted time in every class for students to start on their homework to allow me to offer help and guidance. She accepted my help grudgingly. After working through a problem I asked “Does it make sense now?”
She gave her usual response. “Sort of.” I took this as “no”.
Katherine would sometimes use that period to catch up on paperwork, and in so doing would observe what was going on in class. She never offered any criticism or comments on anything that happened unless I asked. (And when you get down to it, that’s how I like to be mentored.) When I saw Katherine later that day I told her “I’m at my wits end with Lucy.”
“I know,” she said. “One look at her body language tells you she’s given up.”
“I’ve tried everything,” I said. “I’ve communicated with her mother, let her know she can get help, but she doesn’t even try. I feel like saying ‘I’m bending over backwards for you; the least you can do is show some respect and make an effort.’”
“You should tell her that,” she said. “Just talk with her and tell her what you told me, and what your expectations are. She’ll be real honest with you, but you need to reach an understanding.”
I lost sleep that night, rehearsing how that conversation would go. I decided I would pull her aside when the rest of the class was doing their warm-up questions, have the talk. But when I arrived in the classroom, I was greeted by a very cheerful Lucy who offered to help me pass out the day’s warm-up questions to the class. She then excitedly told me “I found a way to do the factoring.”
She showed me. It was the diamond method I had decided not to teach because I thought it would be too confusing for her.
“Where did you learn this?” I asked.
“I looked it up on the internet. It’s really easy.”
“Fantastic,” I said. “Do you want to show the class how it’s done?”
She didn’t want to, so I demonstrated the method. There were the sounds of people getting it as I put some problems on the board for them to work. I left the class elated that Lucy had taken the initiative and was getting it.
I ran into Katherine after class was over and excitedly told her about Lucy’s miraculous turn-around. As it turned out, after Katherine had talked with me the previous day, she decided to talk with Lucy at the end of the day.
“That explains her change in attitude,” I said.
“I should have told you,” she said. “I’m sorry. But she was in the classroom getting something so I just talked to her.”
“What did you tell her?”
“I told her that her body language is telling us she’s given up.”
“I said ‘Mr. Garelick thinks you don’t like him.’”
I wished she hadn’t said that. “What did she say to that?” I asked.
“She said ‘Oh no, that’s not true.’ She felt bad.”
That evening, my wife, who was brought up Catholic, said this was part of Catholic guilt. I have chosen to remain agnostic on such matters.
In the end, the top students were able to work the diamond method, while the other students relied on the trial and error method. Lucy would forget the procedure she had found on the internet and even simple trinomials would elude her despite the fact that factoring trinomials doesn’t go away in subsequent lessons.
There is an advantage to continued practice should anyone have their doubts. It leads to proficiency and eventually can connect with the understanding and “sense” that Lucy felt was lacking.
She would continue to be a challenge. And I would learn to take my victories if and when they occurred.