Wednesday, August 6, 2014

Conversations on the Rifle Range 5: Division by Zero, a Burning Question, and the Ocarina Player

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 five:

Discipline was getting to be a big problem, particularly for my sixth period class. So I confided in my across-the-hall neighbor Mrs. Rodriguez that I was having some problems with discipline. She had taught for many years and had a no-nonsense attitude. She advised me to threaten to hold the class in for a minute after the bell if they continued with whatever bad behavior was going on. “Tell them that if they don’t understand what you’re saying you can arrange to have the principal come in and explain your policy to them. That ought to do it,” she said.

I had occasion to try this in my fourth period class—first year Algebra 1. This class was generally well-behaved except for a few students, but certain students could be very rude. While writing something on the board, I heard someone pop their gum. There is a “no gum” policy in the school rules, and my teacher, in her classroom rules, has added to that by placing a ban on “gum popping” as well.

Without turning fully around I said “There is no gum popping in this class. In fact there is no gum in this class, so I advise you to either spit out the gum, or just stop the gum popping.” No one spit out their gum, so I went on with my lesson.

I heard the pop again. "OK," I said. "First of all there's to be no gum in class. You have elected not to spit it out. So if I hear you pop the gum again, the whole class will stay in one minute after the bell. And if you need the principal to come in and explain this rule I can arrange that. Is that sufficiently clear?"

A boy named Kenny raised his hand. He had a tendency to speak in a very precise fashion. “I have a question,” he said.


“How do you pronounce your name?"

Not quite what I was expecting, but I answered his question and, hearing no further questions or gum popping, decided to continue on with the lesson.

Kenny was a good student—one of the few students on the Junior Varsity football team who was doing well in the fourth period class. I gauged what I could or couldn’t teach in fourth period by what Kenny could understand. If he had difficulty, the rest of the class would, and I needed a different approach. Of my two first year Algebra 1 classes, the fourth period class was the only class where I could go into more detail and take mathematical “side trips.”. And if the fourth period class didn’t understand something, I knew I had to slow it down and teach it differently in sixth.

The day of the gum popping incident I showed that division by zero is impossible, among other topics related to multiplication and division. When I first learned about division by zero in my Algebra 1 class, fifty years ago, I found it remarkable—not because it is impossible, but because I wondered how I had gone so long without ever noticing this. My teacher, Mr. Dombey, then showed us that there was no number by which you could divide a non-zero number and get zero. He wrote 1/x on the board and asked us what the answer was if x was 1/2, then 1/4, then 1/10, then 1/100, and so on. Then he went the opposite direction; he made x larger and larger, showing the resulting quotient growing increasingly smaller—approaching zero. As x approaches zero, he explained, the quotient gets infinitely large. And as x gets infinitely large, the quotient approaches zero. The entire discussion took about ten minutes. There was no “Let’s explore and get into groups and written an essay about it.” And despite the lack of collaboration, group work and student-centered activities, the discussion had opened my eyes to aspects of math I hadn’t been aware of. One year later I decided I would major in math.

I used the same approach that Mr. Dombey had for my fourth period class. I showed why division by zero is impossible (something that critics of traditional math say is not done in traditional math) and then I thought I’d see how far I got with additional discussion. As Mr. Dombey had done, I asked the class what is 1 divided by 1/2, then 1/4, 1/10, 1/1000. Kenny and perhaps two other students supplied the answers which told me that the class’ overall facility with fractional division was weak. But for the most part, the class got the point. I knew I would not get that far with my sixth period class. I announced there would be a quiz next week. The class suddenly got nervous, especially the football players. They needed to maintain a C average in all classes or they could be suspended from the team, so the first quiz of the semester posed an immediate threat.

“Get started on your homework,” I said and started circulating around, answering questions. A boy named Matt who sat up front stared listlessly at his paper.

“Are you going to do your work?”

“Yeah, I guess,” he said. He was a nice boy but didn’t like to do his work in class. He told me on the first day that this was the second time he was taking the course, adding that he wasn’t very bright.

“Is that an ocarina?” I asked, pointing to a flute/whistle-like instrument on his desk.

“You know what an ocarina is?” he asked, his amazement no less than mine had been when I was learning about division by zero. “No one knows what an ocarina is!” he said.

“I’ve seen them before.”

“Can I play it for you?”

“I want you to do at least four problems in this set and show me your work and then I’ll decide,” I said.

A few minutes later he called me over and showed me that he had done some problems. I let him play. He played Greensleeves. I savored the moment, knowing that my sixth period class coming up later was not going to offer anything this nice.

It’s hard to know if anyone’s curiosity was stirred that day in that class by anything other than the question of how to pronounce my name. But then again, Mr. Dombey had no idea that his little discussion stirred so much curiosity on my part either.


Ze'ev Wurman said...


SteveH said...

I think my first instrument was the "sweet potato".

It appears that engagement in one area does not translate to another, but finishing your vegetables before dessert does. My son also found that his love of a topic affected few others, so teachers can't assume that the "active learning" and discovery they see by a few in a group is universal.

I liked starting homework in class because it forced (most) to get over the initial learning hurdle. If they waited until they got home, they would give up much more easily. Another thing I would do is to start the homework problem set on the board, have them do it - individually - and then lead them carefully over the initial hurdle. They liked the fact that I was helping them do part of their homework for them. I liked one textbook I used because the problems in the homework set would give references back to the page numbers where the problem type was explained. Unfortunately, modern educational pedagogy is stuck on the unreliable use of encouraging engagement and active learning and trusting the spiral in spite of clear feedback that it isn't working. That's why it took until fifth grade before my son's Everyday Math teacher could not ignore the amazing lack of mastery of basic math facts by bright kids.

S Goya said...

Encouraging engagement and active learning is not the problem. I would say that Mr. Dombey's students were engaged and actively learning as they watched both division patterns progress. The problem is "... trusting the spiral..." Another big problem is getting trapped in false dichotomies like active learning vs memorization of math facts as if they were mutually exclusive. By the way, we need to be clear what we mean by "math fact." For example, the idea of moving the decimal point when we multiply or divide decimals numbers is NOT a math fact. It is a math trick, that works because of the math fact behind the trick. Perhaps the biggest problem of all is that it is rare to meet elementary math teachers who know what they are doing.

SteveH said...

"Encouraging engagement and active learning is not the problem"

It is if you don't do anything else, and that is the meme of modern education. I wasn't complaining about Mr.Dombey.

"false dichotomies like active learning vs memorization of math facts as if they were mutually exclusive"

My son's Everyday Math schools loved to talk about balance, but it never happened, so it's not so much an issue of false dichotomies, but of believing what they say.

"the biggest problem of all is that it is rare to meet elementary math teachers who know what they are doing."

I'll agree with that. As soon as my son got to seventh grade (with subject certified teachers), the fuzziness started to disappear.

lgm said...

If you believe engagement is not an issue, you need to spend a day in a fully included classroom.