Back to my roots – middle school math

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…


Teaching first- and second-hand knowledge

I’m still ruminating on themes of “practice” and “mastery” I touched on in some previous posts. This blog is starting to serve as a “parking lot” for ideas that I hope to weave together into a more coherent form someday.

I’m not sure what exactly sparked off this latest thought, but I’ve been noticing the distinction between teaching first-hand and second-hand knowledge.  Teaching something we know first-hand is pretty obvious – I can teach a child how to tie a shoelace, or a colleague how to specify a statistical model. I don’t have to draw on external resources to provide the content, although teaching aids (pictures, text books, etc) can help embellish an explanation.

Secondary knowledge is something I don’t have direct experience with, and here it gets interesting.  Most of us adults (who aren’t professional historians) know something of the founding of the United States, the framing of the Constitution, the Revolutionary War, etc.  We can also tell these stories to our children, but how sure are we about the knowledge we’re imparting?  Telling any sort of cultural myth generally entails passing along a story or knowledge that one has not directly experienced; we serve as conduits for a communal story.

Okay, so far no problem – there are things we know directly (and teach/coach) and other things we pass along (such as historical narratives).  Now think of middle school science teachers.  Are they teaching primary or secondary knowledge? It’s an interesting question.  Many are probably generally well-versed in textbook knowledge (they may have even majored in a science), but to what extent have they internalized the knowledge as their own?  I could probably do a decent job of teaching a basic physics class, but I was never really a practicing physicist, and know little beyond the first year college course.  So what would I be doing, other than interpreting/explaining what was already in a text book? What would I add of my own?  And does it matter?

A while back a friend of mine asked me how to get rid of these whitish rings that had appeared on his dining table. I knew I’d read about those in a book on finishing, and sure enough I was able to look it up and respond with a remedy – try mild heat, and perhaps gentle abrasion with steel wool.  But I also warned him that I had never actually tried any of these remedies myself, so couldn’t vouch for them or for unforeseen consequences.  It was an unsatisfying experience.

In a similar vein, I’ve certainly read about wood movement (the tendency of wood to expand and contract along particular dimensions with varying humidity) and how to design for it, but I’ve never actually experienced, say, a panel blowing out of a frame or a drawer getting stuck in its casing.  On the other hand, I’ve both read and experienced what can happen if an off-cut catches the back part of a spinning table saw blade, or what a bowl feels like as it flies off of a lathe and into my face shield. I know that you shouldn’t brake around corners on a fast mountain bike descent, and I know why you shouldn’t do so (having done it and experienced the consequences).

So back to my question: when does it matter that a teacher possesses first-hand, experiential knowledge of a subject, versus largely second-hand, “received wisdom?”  We intuitively prefer the former, and I suspect there are varying degrees of the latter. That is, I was never a math major, but have a reasonable intuition about some aspects of middle- and high-school math.  I certainly know real-number algebra inside-out.  But – although I took a course in abstract algebra as an undergraduate – I know I don’t have the deeper connection to theories of algebras, how systems of domains and operators come together coherently. So I can help a high school freshman struggling with his or her algebra homework, but there are limits to what I can teach.

I’m starting to examine my own balance between first- and second-hand knowledge, both at work and in general life. It’s starting to feel like exploration – we go out and cover some territory, come to know it well, but hear from fellow travelers about what lies over the next hill.  Perhaps we even pass that folklore along to others, along with our own hard-won knowledge of familiar terrain. And we make judgments about risks – the consequences of mis-informing a fellow traveler about the safety of that frozen river, for example, could be catastrophic.

As I wrote, this will become a thread I’ll broaden and deepen; it weaves through a number of areas of my life/experience.