*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 three:*

I believe strongly in how math should be taught, and even more strongly in how math should

*not*be taught. Nevertheless, when I am involved in teaching it as I believe it should be taught, I feel vaguely guilty, as if I am doing something against the rules and perhaps even wrong. That’s how groupthink works. It is an acculturation process.

I am reminded of a job I had as a nighttime janitor at the University of Michigan Medical School the summer between my sophomore and junior years. The janitors put up with the college kids who worked with them, but they also could give us a hard time. On my first day, the supervisor told me to get a broom from the store room. This was an initiation rite. No matter which broom I laid a hand on, someone piped up “That’s mine!” In fact all the brooms had been claimed except one which belonged to someone who was not there. That one was off limits as well, but the supervisor finally said with an air of reluctance, “Well you may as well use that one. He probably won’t be coming back.” And true enough, he never did and the broom was mine. Several weeks later, another “new guy” joined the ranks and he was told to find a broom. Though I had found this initiation procedure ridiculous, when the new guy put a hand on my broom, to my horror I heard my voice booming “THAT’S MINE!”

My algebra classes used a book published by Holt (referred to as Holt Algebra). The team of authors include a math professor (Dr. Edward Berger) and a math reformer (Steven Leinwand). The book predates the Common Core and is fairly traditional, as evidenced by something the math department chair had said during the teacher workday I talked about earlier. Sally, the person from the District office had been telling us about the Common Core approach to teaching math—more open ended problems, more discussion, more working in groups, more problems that have multiple right answers. The math department chair brought up the point that it's hard to do all this because the books they use just don't have those types of problems in them. “Most of the problems can only be solved one way,” he lamented.

Nevertheless, the book does cater to some of the current groupthink trends in math education. When teaching the first unit for the first year Algebra 1 course, I wanted to focus on how to express certain English statements in algebraic symbols; for example, “4 less than a certain number” can be written as x - 4. While Holt Algebra does do this, it tends to focus more on the other way around—taking an algebraic expression such as 4/x and translating it into English. While most algebra books do this (as did mine from 50 years ago), the good ones tend to focus more on going from English to algebra. Holt Algebra spends more time going from algebra to English. In addition, it asks students to find two ways of expressing it, thus satisfying the “more than one way to solve a problem” motif that supposedly builds “deep understanding”.

“How are we supposed to find two ways to say this? What does this mean?” a girl named Elisa in my 6th period class asked me. She had told me on the first day that she was bad in math and requested to sit in front so she could see better and not be distracted.

“How would you say 4/x in words?” I asked. No answer. “What are you doing with the 4? Multiplying by x? Dividing by x?”

“Oh, dividing,” she said. “OK, so ‘4 divided by x’?”

“Yes.”

“But what’s another way to say this?”

I told her to look at the page with the examples. In fact, for every student who asked, the dialogue was nearly the same, and I advised them to look on the page with the examples. (For those who are curious, the answer is “The quotient of 4 and x.”)

The problems in the book that asked for translation into algebraic expressions didn’t exactly start off with the easiest ones. In fact, Holt Algebra seemed to do this all the time—starting with complicated problems right away. One such problem that was stumping the students was: "William runs a mile in 12 minutes. Write an expression for the number of miles that William runs in m minutes."

The book offers the following hint: "How many groups of 12 are in m?” The hint seemed to confuse them more. It confused me. They needed a problem with a softer ramp-up, I decided. I told them to write an expression for the number of feet in m inches. While this also elicited blank stares, I then followed up by asking how many feet are in 24 inches. My student Elisa knew this at once. “How did you do it?”

“What do you mean?” she asked. This was going to be harder than I thought.

“I mean how did you solve it?” She thought a minute. “I just figured 12 times 2 is 24,” she said.

“Ah, so you divided 24 by 12,” I said. Asking students to explain something that seems obvious to them can present difficulties; I have no problem instructing them how to do that. (I realize however that her explanation in terms of multiplication could be viewed by some as superior to “I divided 24 by 12 because it shows her “deep understanding” of what division is.) I have not acculturated to that.

I then extended the questioning; try 36 inches, 48 inches. She repeated the pattern. What about if it’s m inches? Pause of uncertainty. “Uh, m/12?”

“Absolutely,” I said. For the next class, I devised my own worksheet with problems requiring translation from English to algebraic expressions. But despite my belief that I was doing the right thing, I still felt like I wasn’t quite legit. While I usually find it fairly easy to resist the acculturation process in math teaching it’s hard to escape it entirely. So I made sure they could also do the problems in the book that required the translation from algebra to English—in two different ways. I reasoned that such questions would be on the quiz and test that my teacher had devised. Then again, if anyone ever came in to my classroom to monitor me, they would see me teaching my students multiple ways of answering questions and think I was teaching them deeper understanding.

As it turned out, no one ever came in to observe me.

## 1 comment:

Understanding is a fuzzy idea, and there are different levels of understanding. Also, there is no simple path of understanding first and then skills; an idea which pervades a lot of modern math education pedagogy. Also, words can get in the way. A student may know how to do something but won't know how to put it into words - actually, to know what words might make a teacher happy. I see too many reform or discovery approaches that focus too much on words and not on whether the student can actually do the problems. Understanding is not tested by words, but by whether the student can do the problems.

There are also too many problem variations and words get in the way. How can you show understanding of manipulating fractions with just words? Worse, what happens when you get to more complicated rational expressions? Students have to practice, practice, practice. Understanding will come as skills develop. Some pedagogues see what they think is rote learning and decide to give up and rely on fuzzy words.

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