Out in Left Field proudly presents the sixth in a series of letters by an aspiring math teacher formerly known as "John Dewey." All personal and place names have been changed to protect privacy.
Letter from Huck Finn: My Teacher Leaves, and I Become a Gunslinger
Things were looking pretty rosy by the third week in October. I was at the halfway point in my student teaching and had made it through two evaluations and was getting the hang of things. But then my trip down the river took an unexpected turn. I was getting the first period class through the morning routines on a Monday when Tina’s cell phone rang and she went outside to answer it. I could hear her voice through the door; it sounded like she was crying. She came in and told me she had to leave. “It’s an emergency. My father’s in the hospital. Can you handle the class?”
“I’ve got it,” I said.
“They’ll try to get you a sub. I’ve got to go.” She grabbed her purse and ran out the door. I found myself without my teacher, facing 30 or so students waiting for me to do something. I now had to explain zero and negative exponents and get my first period class to follow along with the explanation the text book had presented. This was clearly not going to be easy.
The way I learned was based on the rule for dividing powers: am/an = am-n. Suppose you have a3/a3 with a not equal to 0. Then it equals 1, because any number divided by itself is 1. But using the rule for dividing powers, it also equals a3-3 which is a0. So a0 = 1. Similarly, for negative exponents, something like a2/a3 is easily shown to be (a·a)/(a·a·a). By cancelling a’s, you’re left with 1/a, and by the rule for dividing powers, a2-3 is a-1 which equals 1/a. This all made perfect sense to me and I liked the idea that the application of one rule led to another. But the textbook I was using for teaching took a different approach. The authors seemed to think that deductive reasoning didn’t teach the right “habits of mind” given the prevailing belief that “math is about patterns”.
The explanation amounted to using a place value chart (thousands, hundreds, tens, ones, tenths, hundredths, etc). I first showed that the thousands column can be written as 10 x 10 x 10 or 103. “Moving to the right of the thousand’s place, what’s next?” I asked. A few students responded: “One hundred”. I wrote 10 x 10 = 102, and pointed out that this was 1,000 divided by 10. “And the number next to 100?” I asked. “Ten!” a few more said. “And look at our exponents. We had 103, and 102, what’s this next one going to be?” “Ten to the one” they said. I then asked what the number next to the ten’s place would be. “One,” they said. I pointed to the pattern of exponents, and said “Three, two, one…what’s the next exponent?” Someone said “Zero.” And there you have it: by the pattern, 100 is 1.
Continuing in that fashion, I showed how when we move further right we get into negative exponents: 10-1 is 1/10, 10-2 is 1/102 or 1/100 and so on. From this, students were to take it on faith that this pattern applied to powers other than ten, like 5-2 is 1/52. I was fairly certain the explanation wasn’t going to stick with them for more than two minutes—if that. So I had them write down the rules and relied on what can either be called an axiomatic definition or what the authors probably felt they were avoiding: rote.
My next challenge came when I saw a paper airplane fly across the aisle. Judging by where it landed, I had it narrowed down to one of two boys: Eliseo or Cesar. The class was silent, watching to see what I would do. I knew I had to do something. Up until now I had never made anyone “do a card”—a punishment in which a student had to copy what was written on a colored card kept in a folder at the side of the room: a treatise about the value of education. Increasing amounts of card punishments could get students a referral to the office, a parent-teacher conference, or even a suspension.
I’m not sure what prompted me to do this, but I walked down the aisle between Eliseo and Cesar in the manner of a gunslinger—slow, sure, and looking for danger. The closer I came to Cesar, the more nervous and fidgety he became. I made my choice. “Cesar, do a card,” I said. He fairly jumped out of his seat and took to the task; he even looked relieved. The class seemed impressed with my feat. They seemed much quieter after that.
The day moved on. By third period—my last one—I was still on my own. The class followed the exponent lesson, but Manuelo, one of the brightest students in that class, asked the question I had asked long ago when I first saw zero powers: “How can anything be raised to the zero power? And why would it equal one?”
The class looked at me like my first period class did when Cesar threw the paper airplane. “Well,” I said, “sometimes in math we just say something is a certain way because it fits the pattern.” “OK, but I still don’t get it,” Manuelo said. The axiomatic approach was not sitting well with him. Nevertheless, he worked with the concepts of zero and negative exponents with the rest of the class.
During the “prep” period (fourth period), the phone rang. It was Tina.
“How are you doing?” I asked.
“I’m OK. My father’s had a heart attack. It’s extremely serious.”
“I’m very sorry,” I said.
“So I may be out for at least another week. I talked to the principal; you’ll get a sub tomorrow. What did you do today?”
I told her about the exponent lesson. “Oh, we never do it the way the book does it. We teach them the rules for multiplying and dividing powers; you know--32/32 equals 1 and also 30?”
“I’m glad we’re having this little chat,” I said.
“Sorry,” she said. “There wasn’t time.”
“I know, I know. You don’t have to worry. I’m doing fine.”
“I have full faith in you,” she said. “I know you can do it.”
We both knew we were lying to each other, but sometimes that’s all you’ve got to get you through the day and night.