I met with a student for extra help. Call her K.
K is in 9th grade this year, is new to the school, and has returned her last two math quizzes blank. To her credit, this is her third week in a row visiting me on a Friday afternoon for help. K maintains a healthy distance from her effort, using every cue she can muster to let me know she would rather be elsewhere. Nonetheless, she is here, along with a classmate of hers who is similarly struggling. I’m thinking of two episodes from our 40 minutes together.
First, we were working through this equation,
0 = 9x + 18
and we had established what to add and when, resolving the statement to
-18 = 9x
At this point K answered by saying “x is -2.” She explained that she recognized immediately that 9 times 2 is 18, and since -18 was our goal, the 2 also had to be negative.
When I introduced the way in which she had implicitly used division, she wondered why we would even talk about division here, as she read the whole thing as a multiplication problem. Of course, she is right. I explained the power of algebraic method to work in every situation, even when the numbers are beyond our immediate arithmetic access. I think I offered an equation like 3.181740 = 9.1308x, which you might run into on a lazy Sunday afternoon, as one does. She remained skeptical, which I’m starting to understand is a strength of hers.
Second, we were looking at a somewhat more complicated equation, which it turns is the same equation as the first one (I had my reasons):
4x + 10 = -5x – 8
This task would be filed under “solving a multistep equation” in a textbook or an online math tutorial. K and her classmate are both stumped about a next step here. The classmate proposes “combining like terms” and “moving” terms about, in such a way that the equality is not maintained. I try to explain that this is a priority, though I’m not sure they recognize the urgency behind that particular value.
K makes an attempt and suggests adding 10 to both sides. This would result in
4x + 20 = -5x +2
I respond with what I believe will show them the most efficient route to solving the equation, by which I mean generating a statement of “x = -2”. It’s boilerplate algebra teacher stuff: we want to “get all the x’s to one side and all the numbers to the other side” by “doing the opposite operation of what’s there” and finally “getting x alone.”
The algebra we’re doing here is review, so in class I am less interested in exploring the philosophical implications of the algebra. This is all the more true with students who are struggling to demonstrate their understanding on quizzes and show up at extra help. However, because I am the teacher that I am, and I am generally unsuccessful in having them just “do the math”. And as it turns out, this is the kind of thing kids often do like talking about, when they forget their dyspeptic attitude toward school math for a moment. For instance, both of my 9th grade math classes have already grappled with the ambiguity around, for instance, “-5x – 8” being the same as “-5x + -8” and the implications of that fact for algebraic manipulation, without much prompting from me. And as we’ll see here, extra help doesn’t turn off anyone’s meaning-making capacities.
K regards the problem after I’ve done my bit and says, “But isn’t adding 10 still correct?” Had I been sitting I would have fallen out of my chair, because of course she’s right. Again.
I suspect the word she was looking for was “true” rather than “correct” — it’s true to say that adding any value to both sides of the given equation will maintain the relationship described. The high school textbook name for this is the “additive property of equality”, which if you’ve ever tried to teach directly you know is a slog, because students are both like “what” and “duh” and “why even bother stating that” — all valid responses. But as truth has a tendency to do, it’ll come up in our work no matter what, sometimes to our chagrin.
That she wanted to describe it as “correct” is interesting, since I’m sure that not a few math teachers would respond by saying “no” — simply because adding 10 to both sides of the given equation will get you no closer to resolving the statement to its simplest expression. That my only purpose in working with algebra and equations is to find “the answer” rather than exploring the qualities of number itself is worth some self-reflection.
I try to capture much of this by saying, “Yes! That is still a true statement that maintains the relationship here. Good point. And also, our goal here is to find a value for x. Let’s talk about how to do that.” In my minds eye, I see her shrug wryly as if to say, “what if we just called it a day?”
Conversations with my math students, especially in small groups, often put me in mind of what I learned when I was training as a chaplain. There is this idea that as much as possible we don’t want to impose our own theological perspective on the patient, and instead let them tell their story themselves — a story that on one hand, they can’t help but communicate, but on the other, can be brought to conscious reflection if a proper space is opened for it. Much of the work of chaplaincy is waiting patiently for subtle cues that suggest that you have a green light to explore further, without steamrolling anyone with your agenda.
I was never much good at this. In the lingo, I had trouble stepping into my “pastoral authority”. But I can see that my own struggles with that patience and guidance live on in my work as a math teacher — where listening is so easily bypassed in favor of telling, and where students have stories to tell about how they understand so much more math than our narrow schoolish exercises could possibly bring to the surface.
In math teaching, as in chaplaincy, we have a responsibility to cultivate that meaning-making. For math teachers, perhaps more than chaplains, the struggle is even in recognizing that the meaning-making is happened. But the kids can’t not tell their stories.
