Barry Garelick, who wrote various letters published here under the name Huck Finn, is at work writing what will become "Conversations on the Rifle Range". This will be a documentation of his experiences teaching math as a long-term substitute. OILF proudly presents episode number 23:
It took me about three weeks to learn all the names of my students. Identifiable patterns of behavior took me a little longer. For example, Cindy, who is in one of the two algebra classes, tended to stop me in the midst of explaining a new procedure and say: "Wait, wait, I'm confused, I don't understand."
Over time it got so I could anticipate when this would occur. When showing the factoring of 9x2 –16, for example, I paused after writing down (3x + 4). As expected, I heard: “Wait. I’m so confused. Where did the 3x + 4 come from?” I knew it was Cindy.
“I just don’t understand why it works that way,” she said. I had started the lesson by having students multiply (x-y)(x+y) and other similar problems showing how the middle value drops out. It is not unusual for students to have difficulty extending the pattern of the x2– y2 form to one like 9x2–16. I explained how it worked. Students got it but Cindy persisted. Once she understood something, she got it, but until she did it was painful—particularly when she would get frozen and could not move on until she understood, which was the case here. Students who manage to get it groan when this happens. Someone told Cindy “Because it works out that way; just follow the rule and figure it out later.”
While that exchange tends to bolster my views on how procedures lead to understanding, I can also easily imagine how easy it would be for those in the “students-must-understand-or-they-will-die” camp to make an out-of-context smear campaign against traditional modes of teaching. They would film Cindy saying "Wait, wait, I don't understand!" Freeze frame of Cindy and cue announcer voice-over: "Rote learning is no way to learn algebra. Paid for by Friends of the Common Core.”
Cindy eventually got it as she did most things; she aced all her tests and often came to my classroom before first period to get help. And all-in-all, the algebra classes did fine on the factoring unit, and on both the quiz and the test, the class average was 90. They seemed happy with themselves and with me, and we moved on to quadratic equations. There I was able to have them use what they learned in factoring to solve quadratic equations, all the while continuing with worksheets that I made, drawing from older algebra textbooks. We went through how to “complete the square” and then used that technique to solve quadratic equations that could not be factored.
The day came for presenting the quadratic formula, which, in case you have forgotten, is
x=(-b±√(b2-4ac))/2a. It was fifty years ago when I saw Mr. Dombey present it in my algebra 1 class. I watched dumbstruck as I realized that the solution of a generic equation with tools we had been using yielded a much more powerful tool. For people who think that math education should be about “patterns” rather than “meaningless algebraic symbols that bore students”, I point to the derivation of the quadratic formula as an example of taking pattern-finding ability to the next level. Some problems can be solved from a few small examples, but solving every quadratic by completing the square is too time-consuming. That's where the magic of formalism comes into play. The intuition lets go and math does the work of creating a formula by solving ax2 + bx + c = 0 via completing the square. In my opinion it is a large part of what mathematics is about. I wanted to give at least a few students that same epiphany.
My classes were patient with the explanation and with the predictable interjections from Cindy of "I don't get it" and "I'm so confused". Were I to do it over again, I would start with showing the formula and how it’s used to solve any quadratic equation. But I prefaced my presentation by saying that those who could present the derivation on the next quiz would get 10 percentage points added to their score. This tended to focus concentration.
My presentation went fairly well, though I knew that it was the sort of thing that only a few truly followed, and others would put in the effort afterward, if only to learn it well enough to get the extra credit. Others wouldn’t bother. This is my version of differentiated instruction.
Pamela (one of the students who I suspected complained about me to the counselor) tried to negotiate for more. “Can’t you give the extra credit on the chapter test? You’ll be giving a quiz next and it won’t count as much.”
I gave what I felt was a measured response.
“Well, compared to the ultimatum I gave my daughter regarding the quadratic formula when she was taking algebra, I think this is all quite fair,” I said.
“What was that?” Pamela asked.
I described how in order to entice my daughter into learning the derivation, I told her that once she starts dating, her future dates would have to show me they can derive the quadratic formula. “Now this gives you a choice. You can either date a boy who knows how to derive it, or if he doesn't, you can learn how to derive it so you can show him how, and then he can demonstrate it to my satisfaction when he picks you up."
The classroom became strangely silent and Pamela looked at me in disbelief. “You actually told her that?” she asked.
“What did she say?”
“She made a fist, held it in front of my face and said ‘I will hurt you!’ ” The class was generally appreciative of this and someone said “Good for her!”
Lonnie, a bright boy, asked me how old my daughter is. I saw in his notebook, he had every detail of the derivation copied down. “Too old for you, Lonnie,” I told him. “Go for the extra credit.”
Some students asked what if they simply memorize the derivation? I suppose I could have told them they had to supply reasons for each step, but I decided not to. Anyone willing to put in the time to reproduce the derivation was going to pick up something. Even things one learns by rote represent the substrate, the raw material, of understanding. Not the popular view, I realize. In my case, Mr. Dombey didn’t require us to derive it. But his presentation of it fascinated me enough that I tried to reproduce the derivation on my own. It also played into my decision to major in math.