A couple weeks ago, I met with a charming math zombie who I coach for the SAT. “Could you help me study for a pre-calc test instead?”
I took one look and skidded to a stop.
“What the hell…heck. This is calculus.”
The mother sighed. “Yes, they cover calculus in pre-calculus so that everyone is ready for AP Calc next year.”
Huh. Remember that, folks, the next time you hear of a school with a 100% AP pass rate. They are teaching the kids some of the calculus the year before.
“OK, I can maybe help you with this but before we start: I don’t usually work in calculus. I’m pretty good conceptually, and my algebra is awesome, but at a certain point I’m going to have to send you back to the teacher.”
“That’s fine; I really need any help I can get.”
First up. “Use the limit process to find the derivative of f(x) = x2 – x + 4.”
“What on earth is the limit process?” I turn back in the book, leafing through the pages.
“I have no idea.”
“Well, you must have worked the problem before.”
“I don’t know how.”
“Maybe they mean the definition of a limit, the slope thingy.” I look at the next problem, which also focuses on slope, and decide that must be it.
“So you know the definition of a limit, right?”
“No, not really. I know the derivative of this is 2x-1.”
“Yes, but what is the derivative?”
“I don’t know. I don’t understand this at all.”
“Um, okay. The derivative of any function is another function, that returns the slope of the tangent line for any given point on the original function. The tangent line represents…um, .not just the average rate of change between two points, but the instantaneous rate of change at that point.” (I am not using math terms; whenever mathies get together and talk about the “intuitive” definition of a derivative I want to slap them. I checked a few places later, like this one, and I think I’m on solid ground.)
“Yeah, but why do we care about the rate of change?”
I should mention here that her teacher and I went to ed school together, and I’m certain she (the teacher) explained this multiple times from various perspectives.
“You say you know the derivative is 2x-1, yes?”
“Right. You’re saying that’s the slope of the line?”
“Almost. The derivative is the means of finding the slope of a tangent line to any point on the function, with various caveats I’m going to skip right now. Remember, most functions do not change at constant rates. You can find the average rate by finding the distance between any two points, and finetune that average by picking two points closer and closer together. The slope of the tangent line, which means the line is intersecting only at one point, is the….” I can see she doesn’t care, and her understanding is definitely ahead of where it was just five minutes earlier, so I stopped for the moment.
She sighed hopelessly. “Look, can’t I just find the derivative?”
I scrawled something like this:
“Oh, I remember that. Okay.” And she plugged it all in and calculated rapidly. “How come I have an h left over?”
I was a tad flummoxed, but then remember. “Oh, h approaches 0, so it’s basically negligible. I think that’s right, but check with your teacher. Now, what does this represent?”
“I have no idea.”
“Suppose I ask you to find the derivative when x=1, or at the point, um, (1,4).”
“I plug 1 in for x in 2x-1, which is 1. Then I write the equation y-4=1(x-1).”
“So graph that.”
“I don’t know how. It’s a line, right?” She thinks a bit, then converts the equation to slope intercept. “Okay, so it’s y=x+3.”
“Now, graph the parabola.”
“Um…” I sketched it for her, and marked (1,4). “Now sketch the line.”
“See how it just intersects at the point, perfectly tangent? That’s what a derivative does–it returns the slope of the line through that point that will intersect at just one point.”
“Yeah, I saw this before.”
“And it made quite an impression. Stop waving this off. You want to feel less hopeless about math? This is why you have no idea what’s going on. So gut it up and focus.” She nodded, somewhat chagrined.
“The slope of the line at that point indicates the slope of the original function at that point, which is the instantaneous rate of change. Remember: most functions don’t change at a constant rate. Finding the rate of change at a single point is an essential purpose of calculus. So pick another point and try it.”
“OK, I’ll try -1. What do I do first?”
“What do you need to know?”
She looked at the graph. “I need to know the slope of the line….which I get from plugging in -1 to the derivative 2x-1, which is….-3. And then I—”
“Stop for a minute. Say it. What did you just find out?”
“The derivative for x=-1 is -3, which means…the slope of the line where it meets the graph is -3?”
“Slope of the tangent line. And what does that represent?”
She frowned in concentration and looked at the sketch I’d drawn. “That’s the rate of change at that point. But where is that tangent line intersecting? Oh, I need the plug that in…” She did some work. “So the point is (-1,6), and the slope is -3, and that’s why I use point slope, because I have a point and a slope.”
“And remember, you don’t have to convert from point slope to slope intercept. I just do it because I find it easier to sketch roughly in y-intercept form.”
“But how does this work in problem 2? They don’t give me an equation but they want me to find a derivative.”
“You can find the equation from the graph.”
“Oh, that’s right. But I checked the answer on this, and it’s just -1, which makes no sense.”
“Sure it does. Graph the line y=-1.”
She thinks for a minute. “It’s just a horizontal line.”
“And the slope of a horizontal line is…”
Pause. “Zero. But does that mean the derivative is 0?”
“Which would mean what?”
“The rate of change is zero?”
“How much does a line’s slope change?”
“It doesn’t.” I wait. “You mean a line has a zero change in its rate of change?”
“There you go. And doesn’t that make sense?”
“So….because a line has a slope, which is the same between every point, its derivative is zero. So the derivative is….oh, that’s what you mean when you say other functions don’t change at a constant rate. OK. So lines are the only functions whose derivative is zero?”
“Um, yes, I think. But a derivative can return zero even if the function isn’t a line. ”
She sighed. “It’s much easier to just do the problem.”
I’m going to stop here, because I want to go through several of the conversations in detail so I’ll do a Part 2.
In my last post, I pointed out that Garelick and Beals and other traditionalists are, flatly, wrong in their assertions that procedural competence can’t advance well in front of conceptual understanding.
At the risk of stating the obvious, here is a nice, charming, perfectly “normal” calculus student who understands how to find a derivative, how to work the algebra to find a derivative, and yet has absolutely no idea or caring about what a derivative is—and complains in almost identical words to the middle school girl in G&B’s article. She just wants to “do the problem.”
Our entire math sequencing and timing policy is based on the belief that kids who can do the math understand the math. Yet increasingly, what I see in certain high-achieving populations is procedural fluency without any understanding.
In case anyone wonders, I’m not engaging in pointed hints about East Asians (I tend to come right out and say these things), although they are a big chunk of the zombie population. The other major zombie source I’ve noticed is upper income white girls. I have never met a white boy zombie, or a black or Hispanic zombie of any gender, although perhaps they are found in large numbers elsewhere. But the demographics of my experience leads me to wonder if culture and expectations play a big part in whether a student is willing to put the time and energy into faking it. Or maybe it’s easier for people with certain intellectual attributes (a really good memory, for example) to fake it.
Anyway, I’ll do a part 2, and not solely to reveal zombie thinking. I was planning on writing about this session before the G&B piece appeared. Not only did I enjoy the chance to work with calculus, but I also have really started to understand how unrealistic it is to teach calculus in high school. I’m moving towards the opinion that most kids in AP Calc don’t understand what the hell’s going on, thanks to the unrealistic but required pacing.
Oh and yes, I don’t know much calculus. Forgive me if my wording isn’t correct, and feel free to offer better in the comments.