Tales from Zombieland, Calculus Edition, Part I

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?”

She brought out her book, a hefty volume, and turned to chapter 4, page 320

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.

Understanding Math, and the Zombie Problem

I have been mulling this piece on the evils of explanations for a while. There’s many ways to approach this issue, and I highly recommend the extended discussion at Dan Meyer’s blog, as it captures experience-based teachers (mostly reform biased) with the traditionalists, who are primarily not teachers.

What struck me suddenly, as I was engaged in commenting, was the Atlantic’s clever juxtaposition.

All the buzz, all the sturm und drang about Common Core and overprocessed math has involved elementary school. The cute show your thinking pictures are from 8 year olds and first graders. Louis CK breaks our hearts with his third grader’s pain. The image in the Atlantic article has cute little pudgy second grade arms—with just the suggestion of race, maybe black, maybe Hispanic, probably male—writing a whole paragraph on math. The evocative image evokes protective feelings, outrage over the iniquities of modern math instruction, as a probably male student desperately struggles to obey meaningless demands from a probably female teacher who probably doesn’t understand math beyond an elementary level anyway. Hence another underprivileged child’s potential crushed, early and permanently, by the white matriarchal power structure unwilling to acknowledge its limitations.

And who could disagree? Arithmetic has, as John Derbyshire notes, “the peculiar characteristic that it easy to state problems in it that are ferociously difficult to solve.” Why force children to explain place value or the division algorithm? Let them get fluency first. Garelick and Beals (henceforth referred to as G&B) cite various studies finding that elementary school students gain competence by focusing on procedure first, conceptual understanding at some later point.

There’s just one problem. While the Atlantic’s framing targets elementary school, and the essay’s evidence base is entirely from elementary school, G&B’s focus is on middle school.

Percentages. Proportions. Historically, the bane of middle school math. Exhibit C on high school math teachers list of “things our students should know but don’t” (after negatives and fractions), and an oft-tested topic, both conceptually and procedurally, in college placement.

G&B make no bones about their focus. They aren’t the ones who chose the image. They start off with a middle school example, and speak of middle school students who “just want to do the math”.

But again, there’s that authoritatively cited research (linked in blue here):

Again, all cites to research on elementary school math. The researched students are at most fifth graders; the topics never move above arithmetic facts. G&B even make it clear that the claim of “procedure without understanding is rare” is limited to elementary school math, and in the comments, Garelick discusses the limitations of a child’s brain, acknowledging that explanations become more important in adolescence—aka, middle school, algebra, and beyond.

G&B aren’t arguing for 8 year olds to multiply integers in happy, ignorant fluency, but for 14 year olds to calculate percentages and simply “show their work”. And in the event, which they deem unlikely, that students are just going through the motions, that’s okay because “doing a procedure devoid of any understanding of what is being done is actually hard to accomplish with elementary math.” Oh. Wait.

Once you get past the Atlantic bait and switch and discuss the issue at the appropriate age level, everything about the article seems odd.

First, Beals and Garelick would–or should, at least–be delighted with math instruction in 8th grade and beyond. Reform math doesn’t get very far in high school. Not only do most high school teachers reject reform math, most research shows that the bulk of advanced math teachers have proven impervious to all efforts to move beyond “lecture and assign a problem set”. Most math teachers at the high school level accept a worked problem as evidence of understanding, even when it’s not. I’m not as familiar with middle school algebra and geometry teachers, but since NCLB required middle school teachers to be subject-certified, it’s more likely they profile like high school teachers.

G&B don’t even begin to make the case that “explaining math” dominates at the middle school level. They gave an anecdote suggesting that 10% of the week’s math instruction was spent on 2-3 problems, “explaining thinking”.

This is the basis for an interesting discussion. Is it worth spending 10% of the time that would, presumably, otherwise be spent on procedural fluency on making kids jump through hoops to add meaningless detail to correctly worked problems? And then some people would say well, hang on, how about meaningful detail? Or how about other methods of assessing for understanding? For example, how about asking students why they can’t just increase $160 by 20% to get the original coat price? And if 10% is too much time, how about 5%? How about just a few test questions?

But G&B present the case as utterly beyond question, because research and besides, Aspergers. And you know, ELL. We shouldn’t make sure they understand what’s going on, provided they they know the procedures! Isn’t that enough?

Except, as noted, the research they use is for younger kids. None of their research supports their assertion that procedural fluency leads to conceptual understanding for algebra and beyond. We don’t really know.

However, to the extent we do know, most of the research available in algebra suggests exactly the opposite–that students benefit from “sense-making”, conceptual approaches (which is not the same as discovery) as opposed to entirely procedural based instruction. But researching algebra instruction is far more difficult than evaluating the pedagogy of arithmetic operations—and forget about any research done beyond the algebra level. So G&B didn’t provide adequate basis for making their claims about the relative value of procedural vs conceptual fluency, and it’s doubtful the basis exists.

I’ll get to the rest in a minute, but let’s take a pause there. Imagine how different the article would be if G&B had acknowledged that, while elementary school research supports fact fluency over sense-making (and fact fluency seems to be helpful in advanced math), the research and practice at algebra and beyond is less well established. What if they’d argued for their preferences, as opposed to research-based practices, and made an effort to build a case for procedural fluency over comprehension in advanced math? It would have led to a much richer conversation, with everyone acknowledging the strengths and weaknesses of different strategies and choices.

Someday, I’d like to see that conversation take place. Not with G&B, though, since I’m not even sure they understand the big hole in their case. They aren’t experienced enough.

Then there’s the zombie quote, where Garelick and Beals most tellingly display their inexperience:

Yes, Virginia, there are “math zombies”.

In high school, math zombies are very common, particularly in schools with a diverse range of students and thus abilities. Experienced teachers commenting at Dan Meyer’s blog or the Atlantic article all confirm their existence. This piece is long enough without going into anecdotal proof of zombies. One can infer zombie existence by the ever-growing complaints of college math professors about students with strong math transcripts but limited math knowledge.

I’ve seen zombies in tutoring through calculus, in my own teaching through pre-calc. In lower level classes, I’ve stopped some zombies dead in their tracks, often devastating them and angering their parents. The zombies, obviously, are the younger students in my classes, since I don’t teach honors courses. Most of the zombies in my school don’t go through my courses.

Whether math zombies are a problem rather depends on one’s point of view.

There are many math teachers who agree with G&B, who rip through the material, explaining it both procedurally and conceptually but focus on procedural competence. They assign difficult math problems in class with lots of homework. Their tests are difficult but predictable. They value students who wrote the didactic contract with Dolores Umbridge’s nasty pen, etching it into their skin. They diligently memorize the cues and procedures, and obediently regurgitate the procedures, aping understanding without having a clue. There is no dawning moment of conceptual understanding. The students don’t care in the slightest. They are there for the A and, to varying degrees, play Clever Hans for math teachers interested only in correctly worked procedures and right answers. Left as an open issue is the degree to which zombies are also cheating (and if they cheat are they zombies? is also a question left for another day). For now, assume I’m referring to kids who simply go through the motions, stuffing procedures into episodic memory with nothing making it to semantic, all to be forgotten as soon as the test is over.

Math zombies enable our absurd national math expectations. Twenty or thirty years ago, top tier kids had less incentive to fake it through advanced math. But as AP Calculus or die drove our national policy (thanks, Jay Mathews!) and students were driven to start advanced math earlier each year, zombies were rewarded for rather frightening behavior.

G&B and those who operate from the presumption that math can easily be mastered by memorizing procedures, who believe that teachers who slow down or limit coverage are enablers, don’t see math zombies as a problem. They’re the solution. You can see this in G&B’s devotion and constant appeal to the test scores of China, Singapore, and Korea, the ur-Zombies and still the sublime practitioners of the art, if it is to be called that.

For those of us who disagree, zombies create two related problems. First, their behavior encourages math teachers and policy makers to raise expectations, increase covered material, accelerate instruction pace. They allow schools to pretend that half their students or more are capable of advanced, college level math in high school while simultaneously getting As in many other difficult topics. They lead to BC Calculus pass rates of 50% or more (because yes, the AP Calc tests reward zombie math). Arguably, they have created a distortion in our sense of what “college math” should be, by pretending that “college math” is easily doable by most high school students willing to put in some time.

But the related problem is even more of an issue, because the more math teachers and policies reward zombies, the more smart, intellectually curious non-zombies bow out of the game, decide they’ll go to a state school or community college. Which means zombie kids just aren’t numbered among the “smart” kids, they become the smart kids. They define what smart kids “are capable of”, because no one comes along later to measure what they’ve…well, not forgotten, but never really learned to start with. So people think it really is possible to take 10-12 AP courses and understand the material (as opposed to get a 5 on the AP), and that defines what they expect from all top rank students. Meanwhile, those kids–and I know many–are neither intellectually curious nor even “intelligent” as we’d define it.

The Garelick/Beals piece is just a symptom of this mindset, not a cause. They don’t even know enough to realize that most high school math is taught just the way they like it. They’d understand this better if they were teachers, but neither of them has spent any significant time in the classroom, despite their bio claims. Both have significant academic knowledge in related areas–Garelick in elementary math pedagogy, which he studied as a hobby, Beals as a language expert for Asperger’s—which someone at the Atlantic confused with relevant experience.

Such is the nature of discourse in education policy that some people will think I’m rebutting G&B. No. I don’t even disagree with them on everything. The push for elementary school explanation is misguided and wasteful. Many math teachers reward words, not valid explanations; that’s why I use multiple answer math tests to assess conceptual knowledge. I also would love–yea, love–to see my kids willing to work to acquire greater procedural fluency.

But G&B go far beyond their actual expertise and ultimately, their piece is just a sad reminder of how easy it is to be treated as an “expert” by major publications simply by having the right contacts and backers. Nice work if you can get it.

And the “zombie” allusion, further developed by Brett Gilland, is a keeper.