Once upon a time, I was a high school math teacher. Twice upon a time, actually. Both times I flunked out. Well, not really flunked out – voluntarily withdrew from the profession. The first time I was 23 and teaching in a private school that turned out to be a bit of a cult (no exaggeration). Six years later I was in a public school with a “mentor” teacher who would leave the class with me when he needed to meet with his real estate clients (guess what his moonlighting job was). Sure, I could blame my early exits on the particulars of the situations, but the second go-around also taught me that I really didn’t want to work with adolescents so intensely day after day. I thought I did, but when push came to shove I preferred working with adults, which I’ve done ever since.
Now I live with two 7th graders. It’s homework time. The distributive property. Simplifying algebraic expressions. And everything I’ve been studying about middle-school mathematics teaching and learning is coming to the surface. There’s the basic “how do you teach algebra” question, but I had that pretty well down from the get-go. Then there’s the attitude question: “why do we have to learn this stuff?” Tonight’s sticking point – developing the metal habit of careful accounting for terms and signs. It’s not that the procedure for distributing terms is difficult, but it entails really understanding each component of the process, why a particular transformation is used, and careful book keeping. That’s tonight’s struggle.
An example: simplify 6b – 2(2b – 7) = 21. The tricky part here is keeping track of the minus signs. Actually, working through this has exposed some confusion in the student between unary minus (“eyebrow level minus”, as my old teacher called it) and binary minus (“belly-button minus”). So, what gets distributed here? There are a couple of ways to approach it. One is to treat the term 2(2b-7) as the basic unit to be unpacked, and you get:
6b – [2(2b) – 2(7)] = 21
Then you’ve got to deal with that minus sign before the expression in brackets – essentially distributing a (-1) again.
Or, you can try to distribute (-2) across the expression, and end up with
6b -2(2b) -2(-7) = 21
My student was not tracking the minus signs accurately, in part due to some odd (but not entirely incorrect) use of parentheses that obscured what was happening to the signs of each term.
Here’s where we hit the wall – I had started with a simpler version of this where all signs were +. Then I moved to just having a minus sign inside the parenthesis – no problem. Then I switched to a minus sign after the 6b but a + sign within the parentheses – trouble! That’s where the diagnostic flag went up. How to either distribute that pesky minus sign, or block out the whole expression in parentheses. Both were not making sense to the student. Blocking out the whole expression in parentheses (my first expansion example above) was a non-starter. But distributing -2 as a factor ran into trouble because “that’s not a negative 2, that’s a minus!” (in the way only a 7th grade girl can whine). In the end she worked enough examples that she’s learned to be aware of the -a(b-c) forms, and this expands to -ab + ac, but the learning is not robust. She hasn’t yet seen, for example, -a(-b-c) expressions, and I’m sure she’ll struggle as soon as she sees one.
Attitude. As soon as I tried to work an example I got the “just tell me if it’s right!” impatience. She’s not really trying to “get” it, but “get through” it. And – tonight at least – there isn’t a lot I’m going to do to change that attitude. That’s a longer term… I was going to use the word “battle,” but that frames the situation as one of domination. How to get her to take an interest in how algebra works? That’s the question. If she had some degree of curiosity tonight, I could steer her energy in a productive direction. But what I’m feeling is a sense of drudgery, that the homework is something to be gotten through and then moved past.
(Her sister, on the other hand, has an entirely different set of issues: she grasps the concepts quickly, is actually curious about how things work, aces tests, but is “rebelling” by not actually handing in her work and lying about what she actually has to do in a given night. That’s a later story…)
So what to do? This situation, right here, repeated day after day as a high school teacher, is where I hit a wall. If a student isn’t really latching onto the problem, how do I inspire? I had a high school senior once say to me “Mr. G., I know you’re trying hard, but really, I’m gonna take this course again in community college next year anyway, so I just don’t care.” Anybody who has ever found a passion – or even a modest interest – in life knows that feeling of “latching on.” One long-term question I’ve been really curious about is how to “transfer” that attitude of curiosity from situations where it occurs naturally to those where it might take a little work.
Then again, why *should* a student be deeply curious about the ins and outs of algebra? I was, but I wouldn’t claim that everyone *should* be. I was never that curious about British literature as a student, and still am not. But if I were in school right now with a general ed. requirement that included British literature, I would try to understand what the instructor saw in the subject matter. In fact, that’s one of the joys of getting to know somebody new – understanding what they’re interested in even if it’s not my own personal interest. But as a 7th grader, I was either interested in something or I wasn’t… but I have to wonder how my interests may have been shaped by talented teachers. I had a 6th grade social studies teacher who made world history fascinating for a year, and that was the only time I enjoyed a social studies class through 12 years of schooling.
I actually just asked the 7th grader about this – whether she was “interested” in understanding. She said she was actually interested when she started the homework, “but then it got hard.” She agrees that she’s not always interested in math, “not like that boy in class who gets excited whenever the teacher is about to do something new.” It makes me wonder, how much interest is enough?
To be continued…