*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 four:*

When it comes to teaching math, I am a drill instructor. I say this without apology. I believe that practice is essential in mathematics; it results in automaticity which ultimately allows students to take on increasingly complex tasks. It is part and parcel to the much ballyhooed concept of “understanding”. Yes, I put this word in quotes.

I am also—again in terms of teaching math—the equivalent of a rifle range instructor. I’ve never served in the military but from what I’ve heard, in basic training the instructors on the rifle range are not the same drill instructors that wreak havoc on the recruits. According to a friend who served, on the rifle range the instructors were very patient, talked in quiet tones and gave the recruits advice and encouragement in learning how to shoot their rifle. It is in this spirit—one-on-one encouragement as opposed to the enmasse drilling techniques—that I use this term.

Out of necessity, I start out as drill instructor. Prior to the start of any class, when confronted with a sea of faces, people talking, not wanting to start class (particularly true of my 6th period class), my lesson plan disappears from my mind to be replaced with this solitary and recurring question: “What the hell am I going to do now?” I then revert to a technique I learned when I was student teaching. Clipboard in hand, I stand at the front of the room and shout: “In your seats and homework out, please!” This technique serves two purposes. It gives me the illusion that I am in control. Secondly, it tells the students what to do, and even if they don’t do it, they at least know what it is they should be doing.

My routine after that is something I’ve seen written up in articles that decry the traditional classroom (sometimes referred to as the “I do, we do, you do” technique) for its dull, staid, predictable and uninspiring drills. It is also something that I have heard people say will disappear once Common Core kicks in full tilt. Thus, I am a vanishing breed soon to be relegated to the world of buggy whips, slide rules, 8-track tape players, and well-written math textbooks. While I check in homework, they are to work on the warm up questions that I have on the screen. We go through the warm-ups, I put up the answers to the homework, answer any questions they may have on the homework and then embark on the lesson.

My rifle range instruction occurs mostly when I circulate around the room after I assign the problems they are to work on. The conversations can be personal as well as instructive, and it is a time for me to get to know them. My second period class is algebra, 1 second year. This means that the students have passed the first year of the 2-year sequence of algebra 1—most of them anyway—and are older and somewhat more mature and well behaved than the first year students.

In this class I had about six football players on the junior varsity team, almost all of them struggling with the material. I was working on solving systems of linear equations. The chapter started out by solving by graphing—always a frustrating topic because the main intent of the lesson, it seems, is to show how inaccurate and unreliable graphing is as a means of solving equations. I suppose it’s also to show that the solution of a system of equations exists at the point of intersection. And then there are the typical “real world” problems that describe two health club plans or two long distance plans, or any two plans that involve 1) a membership fee (which translates to the y-intercept) and 2) a monthly, weekly, or daily rate (which translates to the slope) and asks at what point (days, weeks, or months) will the plans cost the same. I have nothing against such problems but, like most problems in Holt Algebra, little is ever varied so that once you learn how to solve one such problem, you’ve solved them all. As it was, many of my students were struggling even with basic types of non-word problems.

During one of my rifle range tours, one girl in that class, worried about a quiz upcoming in the next week, asked “Will the quiz have any division on it?”

“You mean will you have to know how to divide? Yes, of course,” I said. She looked crestfallen. “I mean, it won’t have a

*lot*but can you divide things like 24 by 8 and things like that?”

“Oh yes,” she said, looking relieved. “It’s just the double and triple digit division I have a hard time with.” For some reason I didn’t find this very assuring.

My fourth and sixth period classes were also a mixture of abilities. Some students asked more questions than others. In sixth period it was Elisa, the girl who told me she had trouble with math. I found out she lived with her aunt, and had just moved from Colorado. She had had a tough time with math in Colorado and said she developed a stomach ulcer because of her last math class. The teacher wouldn’t answer any of her questions. Maria, a Mexican girl, was another who asked many questions and, like Elisa, said her previous teachers in math didn’t answer her questions. I don’t know if their teachers were of the philosophy that less teaching means more learning but both expressed gratitude to me for answering their questions.

Although I would explain the concept behind the problem, in most cases it always came down to telling students the procedures. Maria, for example, asked me how to solve -7 -3. “Maria, if you lost 7 dollars and then lost 3 more how much have you lost?”

“Ten dollars” she said, counting on her fingers.

“OK. So what you’re really doing is adding two negatives. It’s really (-7) + (-3).” I showed this on a number line.

“So if you have two negative numbers, you just add them and put a minus sign in front?” she asked.

I found myself thinking “What the hell do I do now?”

Knowing she was right, but also knowing and not overly concerned that she didn’t understand the “why” of the procedure, I answered her question, guiltily confident in my belief that procedural fluency leads to understanding. “Yes,” I told her. “That’s what you do.”

And that’s what she did.

## 5 comments:

“OK. So what you’re really doing is adding two negatives. It’s really (-7) + (-3).” I showed this on a number line.

“So if you have two negative numbers, you just add them and put a minus sign in front?” she asked.

Actually, applying the distributive property, she's right. (-7) + (-3) = (-1)(7) + (-1)(3) = (-1)(7+3) No worries!

I know she was right for the reason you pointed out. She probably did not understand why in the context of the number line explanation and certainly not in the context of the distributive propery. I don't care that she did not understand why, which is why I was "shakily, guilty" confident that procedural fluency leads to understanding.

Negative numbers are sneaky things that cause all sorts of problems. In the college algebra classes I used to teach, I made a point of showing how, like CT, a minus sign is really a factor of minus 1. I would have students draw circles around each term and each factor of expressions and equations and have them put the negative signs in as factors of the terms. This clears up many problems for them. If the term was a fraction or a rational expression, I would show them how that minus one factor could be put in the numerator or the denominator. Putting a stray minus sign in the middle of a fraction (at the dividing line) was very confusing for them.

I would show them that:

-3/4 = (-1)(3/4)

-3/4 = ((-1)/1)(3/4)

-3/4 = ((-1)(3))/4

-3/4 = (3)/((-1)(4))

How about

-3/4 = (3)/((-1)^-1(4))

Procedural fluency leads to understanding because fluency implies success and success in all problem variations implies understanding.

Algebra II is very hard for students because they have to expand their simple algebraic understandings to ones that can handle the manipulation of any complex situation, as with minus signs. When they try to apply their simple understandings of algebra and fail, many see it as a sign of rote learning and use that to justify all sorts of silly educational ideas.

I've told our kids that a lot of prealgebra and algebra are about learning how to keep track of your minus signs.

This is great. It explains why the series is called "Conversations on the Rifle Range." It shows how a really good lesson can be structured. And it portrays a successful teacher in action in a real world setting. Pleasure to read.

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