*Barry Garelick, who wrote various letters under the name Huck Finn and which were published here 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 11:*

Mrs. Halloran, the teacher for whom I was subbing, was known for her strictness. On the day I met with her, she had me observe some of her classes and she introduced me. She told the students “I expect you all to behave well with Mr. Garelick. He and I will be in contact with each other and if there is trouble with any of you, I will hear about it.” This was met with a reverential silence.

On my first day, I took advantage of the students’ association of me with their former very strict teacher. I started each of my classes on that day with a general introduction and my rules. “My name is Mr. Garelick,” I said. “Or you can call me Mr. G. I answer to both. Here are my rules; there aren’t too many and they’re fairly simple. Ask permission to leave your seat; ask permission to throw something away in the wastebasket. Do NOT try to throw it in basketball style. Walk it over and drop it in. Do not throw things in class. If someone asks you for a pencil, I don’t want to see it thrown across the room. Ask for permission to leave your seat and walk the pencil over to the person. As far as behavior goes, if you are disruptive, I will give you one warning to stop the behavior. The second time it happens you will get a referral. That’s it.”

Nice and simple. I only had one student ask a question: a boy named Jacob in my 4th period pre-algebra class. He was from Chile and from what I could see, he was either going to be a mathematician or a lawyer. His question: “You say there will be two warnings before we get a referral. Is that just during one day, or is it all year?” I told him it was just for the one day.

Over the course of the semester, my rules would slowly disintegrate, though some days were better than others. These were pretty good kids and they exhibited the normal range of misbehaviors one would expect at a middle school. And compared to the high school where I had subbed, this was like paradise.

The 8th graders in my Algebra 1 classes also listened politely. These were bright students and tended to be rather noisy when presented with new knowledge: many questions, and people shouting out answers. I had to add to my rules: “Do not blurt out answers to questions; raise your hands!”

We were now starting the chapter on systems of linear equations in two variables. They used the same book as the one the high school used: Holt Algebra. Not the worst book around but definitely not my favorite. The chapter on systems of linear equations was known for being rough on students; probably because, like most of the chapters, it throws in the kitchen sink of topics so that by the time the unit test rolls around, students are overwhelmed with what they have to know.

I prefer starting simultaneous equations with the elimination method because it becomes obvious what is going on. That is, in a system of equations like x + y = 7 and x – y = 5 it is easier to explain how you can eliminate one variable to then have one equation in one unknown. Holt starts out with the substitution method, however, and leads with something like y = x + 1 and y = 2x -5 and other similarly easy problems to start students off. Had I planned better, I would have used a different equation because a very bright student, Luanne, remembered from a previous course she had taken that you could set the two equations equal to each other since they both equal “y”.

That is perfectly true and so we come up to the eternal question that plagues algebra students: is it substitution or transitivity (i.e., if a=b and b=c then a =c). It became obvious to me that, aside from Luanne and some others, most saw the problem as neither substitution nor transitivity but as something you do when the equations are in that form. When I tried showing them that x + 1 = y = 2x + 5, thus appealing to the transitive approach and telling them to eliminate the “middle man y” this way, they would nod and someone next to them would whisper “Just set the two equations equal to each other.” Based on the whispering, most seemed to see it as a purely procedural approach.

This next problem will set them straight, I thought: 4x + y = 21, and y = x + 1. But Luanne raised her hand and said “Just solve for ‘y’ in the first equation and then set the equations equal to each other.” The room again buzzed with what they were now taking as the way to solve these problems. They were holding on to their procedure like a visitor to a new city uses a few main streets to get around before they discover alternate routes. I tried to decide whether I was seeing that 1) sometimes “understanding” just has to wait until they get to know their way around town a bit better, or 2) maybe some of them did have an understanding of transitivity without knowing that they did. Or 3) maybe my philosophy of “procedural fluency leads to understanding” was crashing upon the rocks.

I decided it was a combination of the first two.

“I just want to bring something to your attention,” I said. “When I solve problems, I go for the easiest way possible. You can solve these problems in the way that’s easiest for you. But what may seem easy now may not work as well as these problems get harder.”

The room quieted down. I gave them another problem: x - 6 = y and 2x + 3y = 27. I showed them that to solve the second equation for either x or y you would end up with fractions and it gets messy. Much easier to substitute x – 6 for y in the second equation. The second equation becomes 2x + 3(x-6) = 27.

Eventually, students began to see how substitution works, as evidenced by questions like “Do I plug it in like this?” and “Did I distribute this right after I plugged it in?” And with Common Core and its emphasis on “understanding” becoming the testing ground for whether teachers were with the program or not, I now had a story to tell: how I turned procedure into understanding. Or something along those lines.

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