Tag Archives: calculus

Tales from Zombieland, Calculus Edition, Part 2

The comments on part I have been fascinating. I want to reiterate that my math zombie’s teacher is not encouraging this behavior; I have no idea if she lectures or teaches using a more “progressive” style, but she certainly doesn’t believe that “procedural fluency leads to conceptual understanding”. A commenter also argues that “We Are All Math Zombies”. No. “Zombie” doesn’t mean “ran into the math ability wall”, nor does it mean someone who struggles with a topic and decides to forge through an obstacle, putting a black box around the difficulty to be returned to later, with more experience. I refer readers to the Brett Gilland definition of “math zombies” who “who can reproduce all the steps of a problem while failing to evidence any understanding of why or how their procedures work”.

Back to it–we are now into the “rules” questions, 3 through 8. She did question 3 easily. Please remember that my knowledge of calculus is being pushed to the limit in this entire sequence. I found this nifty derivative calculator so non-calculus folks can see how much rote algebra my zombie was doing, mostly correctly, again with no understanding.

Problem up: question 4: g(x) = (x2 + 1)(x2 – 2x)

She began by just taking the derivative of both terms and multiplying them.

“Um, no.”

“You don’t just multiply them?”

“Didn’t you do a bunch of rules? Product, Power, Chain, Quotie….”

She looked vague, but I was pretty firm on this point. “Look, you have to stop being so helpless. This math hasn’t been imposed on you by some fascist regime. Turn back a page or two in the book again.”

And then, a page or two back, when she spotted the product rule, “Oh, yeah.”

And she instantly started into the procedure.

“Stop. STOP!!! What the heck are you doing?” She looked at me in confusion.

“You’ve done this before. You have no memory of doing this before. Now you’re all oh, yeah, mindlessly working a routine you didn’t even recognize 30 seconds ago. Your next two years are going to be a case of lather rinse and repeat if you don’t start forging some memories, some connections.”

“I’ll just forget it again.”

“Then stop making yourself crazy and go take actual pre-calc.”

“I don’t even think that exists in my school.”

“Then listen up. What you know how to do is find derivatives of individual terms added together. First step is to realize that multiplying, dividing, or exponentially changing functions is more complicated. So there are separate rules that build on the easier, basic task of finding derivatives of individual terms.”

I wish I could say I broke into her drive for “just do something”, but at least she slowed down a bit. “But I wrote it down.”

“You did that the first time. So let’s try something different. Repeat this. The Product Rule: multiply the derivative of the first term by the second. Add it to the derivative of the second term times the first.”

“Yeah, I wrote it down.”

“No, you wrote down an abstraction. Say it.”

“What, like in words?” I looked at her sternly.

“Okay, I take the derivative of the first term. Then I…multiply it…”

“Stop. You’re into memorization, so memorize. But words, not symbols. The Product Rule: multiply the derivative of the first term by the second term. Add it to the derivative of the second term times the first.”

She repeated it patiently; I made her do it two more times.

“Okay, now you can work the problem.”

(I have no evidence for the notion that auditory/oral repetition helps, but intuitively, it seemed to me that the many rules are easier to remember by focusing on what the actions are, rather than what they look like. I lunched a few days later with my friend the real mathematician and department head, who told me that he requires his students to write–yea, write, Barry and Katherine!–a description of the product, quotient, and chain rules in addition to the algorithms. “Whenever I had to recall them in college, I remembered them verbally first.”)

Did you know there were online derivative calculators? So for those who want some kind of idea what she did, I’ll link these in.

“I always wondered if you can just distribute the product and use the power rule,” I mused, scratching through the steps. “Looks like you can. (x2 + 1)(x2 – 2x) expands to x4-2x3+x2-2x which…has a derivative of 4x3-6x2+2x-2.”

“That’s what I got. But why would you multiply it out when you can use the Product Rule?”

“Oh, I dunno. Maybe some people forget the Product rule temporarily. But if they actually understood the math, they could just think hey, no problem. I’ll just expand the terms until I can look up the rule. Or until it occurs to me to look up the rule, since you were stuck on that step until I showed up.”

She allowed as that was true. “But you can’t do that with the quotient rule.”

“I’m not good enough at this to know for sure. But most of the time you’d have a remainder, which would be expressed as a quotient, so it’s kind of reiterative. Question 5 is a fraction that is, I think, always going to be less than 1, so I’ll take a crack at doing the division on question 6 while you work out the quotient rule on both problems.”

“But how can I find a derivative of a cube root?”

“Gosh, wouldn’t it be great if there were a way to express a root as an exponent?”

“Oh, that’s right.” And she set to work on some rather complicated algebra and then stopped. “How do you know that this will always be less than 1?”

“Well, look at it. I’m dividing the cube root of a number and dividing it by its square. So think about taking the cube root of, say, 8? which is 2. Then dividing it by 8 squared + 1, which is 65. Even if x is less than 1, I’m adding 1 to the square of the fraction, so that sum will always be greater than the cube root of a positive fraction less than 1. I think, anyway.” Her eyes had long since glazed over, but I confess–I graphed it just to brag.

cuberootdividedbysquare

“I finished question 5, but it doesn’t match the book.”

I looked. “No, you didn’t drop the power on the cube root. It’s going to be negative two-thirds, which will move it to the denominator.”

She redid the problem while I did long division on problem 6, getting -1 with a remainder of -2x+2. Since the derivative of the constant was zero, I then had to take the derivative of the remainder (divided by x2-1).

“It just occurred to me I could use the Chain Rule here, too. Huh. I wonder if that means all quotient derivatives could be worked with the chain rule.”

Our answers to number 6 matched up, and my student was mildly interested. “So I can find derivatives with more than one method?”

“As is usually the case with demon math. But file this away with ‘repeat the processes verbally’ as a means of survival strategy.”

She worked her way through the next group, enduring my comments patiently but with little interest. I kept plugging away, trying to get her to think about the math–not because I wanted her to share my values, but I thought the conversations might create some memory niches.

So when she worked the derivative for problem 10: “hey, that’s interesting. That graph will always be negative, which means the slope at any point on the original graph will be negative.”

“What? How can you tell?”

“No, you can figure this out. Look at it closer.”

“It’s negative 8 divided by…oh, I see. Squares are always positive. So it’s a negative divided by a positive.”

“So that means that no matter what point we put in…” I prompted.

“Wait. Every slope is negative? No matter what?”

“I wonder if it’s always true for reciprocal functions. Huh.”

“Is that a reciprocal or a hyperbola.”

“Huh. I….think… they’re the same thing? Or a reciprocal is a type of hyperbola? Not sure. Good question. A hyperbola is a conic, I know, and I’m more familiar with transformations than conics.” (Answer is yes, a reciprocal function is a rectangular hyperbola.)

Then, when we got to problems 11 and 12: “Look, you need to remember that a square root function will in all cases turn into some sort of reciprocal function. You keep on messing up the algebra and aren’t catching it because you aren’t thinking big picture.”

“I don’t see why it’s a negative exponent.”

“What do you always do with exponents in derivatives?”

“You subtract….oh! I’m always subtracting 1 from a fraction.”

“Bingo. And negative exponents are..”

“they’re reciprocals, you’re dividing. Okay.”

“But look at the bright side. You actually understood this question.”

“I do! You really have helped.” I beamed. And she was able to work problem 13, finding a derivative given a graph, without help when an hour earlier she couldn’t. Progress, at least in the short term.

Problem 14 was interesting. “Determine the points at which the graph of f(x) = 1/3x3 – x has a horizontal tangent line.”

“Should I use implicit differentiation?”

“What? No. Well. I don’t really grok implicit differentiation, but that’s not what this one is asking. What does a horizontal line have to do with slopes?”

“Horizontal lines have a slope of zero. So the rate of change is zero? It’s asking where the rate of change is zero? The derivative is….x2 – 1.”

“Which factors to (x-1)(x+1). Hmmm.”

“So it is implicit differentiation?”

“No. Look, I don’t know what implicit differentiation is specifically, but it always involves y. This is….I’m just confused, because the point at which this parabola has a slope of 0 is the vertex, which is x=0.”

“Yeah, the slope of the parabola isn’t what I’m looking for, right? That means the slope of the other graph is 0 and I should plug in 1 and -1.”

I looked at her, impressed. “My work here is done.”

“What, I’m wrong?” She quickly worked the problem. “It’s positive and negative 2/3. That’s what the book says, too.”

“You’re not wrong at all. I was the one who was confused and you spotted the problem. Very good!”

“But why couldn’t I have used implicit differentiation?”

“Look, you need to talk to your teacher about that because it’s at the edge of my knowledge. I know that working the math of implicit differentiation is easier than understanding it. But at 90,000 feet, what you need to remember is that you use implicit differentiation when you can’t isolate y, so your equation has two variables. Circles and ellipses, for example. Or some of those other weird circular graphs. Look at problems 16-19, for example. Anyway, the derivative on this one was simple. The crux of the question was the link between the zeros of the parabola and the rate of change on the original graph.”

And with that, our ninety minutes were up. I tried to talk the mom out of paying me, since I’d learned a lot and wasn’t an expert, but she insisted.

Some observations:

She was capable of some pretty brutal algebra without any real understanding of what she was doing, time and again. That’s the zombie part–that and the fact that she really didn’t much care about anything other than plowing through. She wasn’t ever really interested but hey, all this stuff the tutor was saying seemed to help, so play along.

I learned a great deal, in ways that will further inform my pre-calculus class curriculum. Can’t wait to try it out. I also wrote out a lot of equations and may have made typos, so bear with me. And yeah, that’s how I remember implicit differentiation–it’s the one with “y”. I get the basics–normally it’s just x changing, this is saying they both change with respect to each other, or something. Implicit differentiation is the point at which I start to realize that the algebra of the differentiation language (dy/dx) has its own logic and wow, a chasm of interesting things of which I know nothing about opens and threatens to swallow me up so I look away.

I’ve really increased my understanding in advanced (high school) math over the past few years, and going back into calculus armed with that additional knowledge has led me to think—really, for the first time—about the lunacy involved in high school calculus instruction. I am starting to understand how math professors could be dismayed at the total ignorance demonstrated by students who scored 5 on the BC Calc test.

Finally, consider that this student is taking pre-calculus. Her transcript reflects pre-calculus. Yet the content is clearly calculus. Meanwhile, I teach a lot of second year algebra with an analytic geometry spin in my pre-calc class. Most schools fall somewhere in between. This is why I laugh when people propose doing away with tests and using grades and transcripts. I still believe in good tests, despite my increased awareness of cheating and gaming.

This enormous range of difficulty and subject matter reflects the bind faced by high schools kneecapped by our education policy. We must offer all students “college level” material, and our graduation and class enrollments are scrutinized closely by the feds and civil rights attorneys ever in search of a class action suit. So we have to move kids along, since we can’t fail them and can’t offer them easier courses. So we have to try and teach good, solid math that isn’t too much of a lie. That’s what I do, anyway.

Maybe things will change with the new law. I’m not counting on it.


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.”

calcex1graph

“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.”

calcex1graph2

“But how does this work in problem 2? They don’t give me an equation but they want me to find a derivative.”

calcexfig2

“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.