# 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.

#### 62 responses to “Tales from Zombieland, Calculus Edition, Part I”

• Polynices

I always thought calculus was useless without learning physics at the same time. I took both as a senior and it worked well — each made more sense because of the other.

• Ben Orlin

Your calculus is fine! The dy/dx expression you wrote represents the slope between a specific point with x-value “x,” and a generic “nearby” point with x-value “x+h.” We imagine the h shrinking down to zero, so that the two points are virtually on top of each other. This lets us calculate the instantaneous slope a particular point without confronting the weirdness of rise/run equaling 0/0.

As you say, this kind of thinking is typical of most calculus students. But to me, that doesn’t mean the concepts of calculus are inaccessible. As you saw, the right kind of questioning, instruction, and experience can help kids see the logic and meaning behind their blind procedures. Kids CAN understand. The problem isn’t with their cognitive abilities; it’s with the structure of educational institutions, which haven’t developed to support that kind of thinking.

For what it’s worth, I taught at a school that was 2/3 Hispanic and had tons of students with basically this approach coming into my Precalculus & Calculus classes. My first time teaching those courses, I did a pretty bad job helping them access the real concepts, but the second and third time went better, and I don’t think you’d have called them zombies by the end.

(Last thought: most kids have a zombie understanding of slope itself, too, and haven’t internalized it as a natural measure of the “steepness” of the line. When I meet a student who’s not “getting” the derivative, I usually turn towards strengthening those intuitions about slope.)

• educationrealist

But to me, that doesn’t mean the concepts of calculus are inaccessible.

I agree. But do you think the pace of instruction is reasonable–reasonable, that is, in trying to teach calculus over a year? It seems to me that kids simply can’t absorb that much material in such a short amount of time. In fact, it’s probably true that most college kids can’t absorb that much material in a short amount of time. That’s really my underlying issue, I think: we are dramatically increasing the ability range of kids who go onto advanced math, but not increasing the time needed to understand the material. I don’t think everyone can do advanced math, but I do think more can get at least a reasonable understanding if we slow it down so that procedural fluency along with conceptual understanding can be managed. Some will genuinely be able to get further in math, if I’m right, but even the rest will not feel as if they are just mindlessly going through material that has no point.

Because what is really the point if most kids are going to forget all the procedures quickly and never understood the concepts in the first place?

(and then, as I mentioned in the last post, many kids simply won’t go along with the joke, whether they are capable or not.)

Good to know that Hispanic zombies exist, too. (mild joke). I find that many kids who have been obediently plugging and playing really value getting more understanding, while a few are ferociously resistant.

Thanks for the explanation of h shrinking to zero. A lot of this is just hard for me to write out, since I so rarely teach it.

• Ben Orlin

Yeah, we’re largely on the same page. I think you’re right about the pacing, as it stands – too much crammed in. But we could streamline the curriculum; there’s no mandate that says “a year of calculus” has to look the way it currently does!

Personally, I’d cut lots of the semi-rigorous limits stuff (the Brits have already done so) along with some of the computational drudgery of integration (which symbolic computational software can handle for us now). Specifically I’d reduce or eliminate most of the following:

1. Ways a limit can fail to exist
2. Notions of “differentiable” vs. “nondifferentiable”
3. The mean value theorem
4. The intermediate value theorem
5. The squeeze theorem
6. Riemann sums
7. Hyperbolic trig functions
8. Differentiation and integration with inverse trig
9. Trig substitution for integration
10. Integration by partial fractions

Some of this stuff I love teaching, but I’d still favor cutting it in favor of a more manageable version of calculus.

Anyway, as I think you’ve said, the problem isn’t that kids decide to become zombies in 11th grade. It’s that they’ve been encouraged into zombie-dom since the beginning of their mathematical careers. They never understood slope, functions, or what it means to solve an equation. And learning how to think deeply about math while simultaneously learning calculus is simply too much!

As for Garelick’s argument, I think you’ve nailed it on the elementary vs. secondary thing. Asking kids to “explain” why their fraction algorithms work is questionably useful. But at the high school level, the idea that procedural fluency automatically endows conceptual understanding flies in the face of every experience I’ve ever had as a teacher.

• Brett Gilland

I would argue that my AP Calculus AB class is almost certainly the most reasonably paced course I get to teach in a year. 3 big concepts, all interrelated, along with various applications. That is much easier than what we ask our students to do in Algebra 2, for example.

AP Calc BC, on the other hand, is a ridiculous course. It has all the material of university Calc I and Calc II, but is actually paced quicker than the university course, owing to the test taking place in early may. There are more classroom hours devoted to the material, but less calendar time, which doesn’t give nearly enough time to process and struggle.

• educationrealist

I have a friend (the guy I mentioned above) who says the same thing about AB vs BC. But this girl is taking Pre-calc and learning calculus, so the pacing seems pretty ingrained. Maybe they are introducing it this year so it will be familiar next year.

Thanks for the mention of zombies. Very useful!

• Joseph Nebus

I know that I recognized right away what the ‘limit process’ was supposed to mean, although I’m not sure where I picked it up from. It’s a peculiar name to give the technique, almost like the namer just grabbed obvious keywords and tossed them together. On the other hand, I got what they referred to, too.

Figuring out when it’s safe to drop terms was the thing that most baffled me when I was learning calculus and limits. I’m not sure I ever got a clear explanation of when it was safe to let an h disappear.

• educationrealist

I asked a friend of mine who teaches BC calc and he said the same thing–didn’t recognize the name, but knew what it must be.

You’re not sure? Then there’s no hope!

• cthulhu

Telling someone, especially a high school student, to find the derivative by “the limit process” is pedagogical malpractice, unless the limit definition of the derivative – the equation shown in the post for dy/dx – is called out immediately previous as the definition of the “limit process”. And even then, there’s got to be a better way.

• nerdy.nel

This is a phenomenon that I observed back when professional tutoring was my main occupation (albeit at the college level, so not directly related to this). There were clear and distinct differences between those who sought understanding of a problem and those who merely wanted solutions to problems.

I bet you can guess which group of tutees showed the most long-term improvement.

I find it disconcerting that they’re sneaking actual calculus into pre-cal courses. Seems like “teaching to the (AP) test” on steroids to me – and I predict it’ll have the same result as the standardized testing rush that NCLB brought on – unmitigated failure.

• educationrealist

” Seems like “teaching to the (AP) test” on steroids to me – and I predict it’ll have the same result as the standardized testing rush that NCLB brought on – unmitigated failure.”

It’s definitely the first. But as for the second, over 50% of whites and Asians get a 5 on BC Calc (I suspect the pass rates for blacks and Hispanics is high, too, but I was only comparing whites and Asians in that post). The test isn’t easy, but it does lend itself to teaching to the test. A few years ago, the College Board redesigned the AP Bio test and dramatically reduced the scores, increasing conceptual knowledge and difficulty. In my opinion, the AP Calc tests need the same treatment.

• nerdy.nel

I don’t know if this issue was prevalent when I took the AP Cal BC exam in 2003 (an exam that I got a 4 on), but if it’s that easy to game, then the College Board should change it. Perhaps less multiple choice and more word problems?

• educationrealist

I am not knowledgeable enough to know what is best to do. I do know that, as mentioned, the College Board completely revamped the Bio test because too many 5s were handed out. I would think they have to wonder the same thing about the AP tests, which I also know college math professors aren’t terribly impressed by. But maybe the high pass rate works for them.

• surfer

For Bio, they changed the content (and deviated it from a traditional course) to be more conceptual and less descriptional. However, life sciences especially Bio ARE descriptional. A lot of basic content was cut out (e.g. Kreb’s cycle).

The scores were dropped by the curve and by tricky SAT style conceptual questions. Of course the content was moved away from a traditional college course, so they really are wandering off of the reservation. More and more AP is like advanced high school (of its own devising) vice college equivalent.

• Jesse Malkin

You are implying that Garelick and Beals and other traditionalists would defend the manner in which this girl was taught when clearly that is not the case. The girl has not mastered the material and is incapable of answering all but the simplest questions. I think you are attacking a straw man here.

• educationrealist

From G&B:

“Furthermore, math reformers often fail to understand that conceptual understanding works in tandem with procedural fluency. Doing a procedure devoid of any understanding of what is being done is actually hard to accomplish with elementary math because the very learning of procedures is, itself, informative of meaning, and the repetitious use of them conveys understanding to the user.”

Also, Barry Garelick repeatedly discussed his belief that procedural fluency leads to conceptual understanding in his book, or at least the excerpts:

1
I tried to decide whether I was seeing that 1) sometimes “understanding” just has to wait until they get to know their way around town a bit better, or 2) maybe some of them did have an understanding of transitivity without knowing that they did. Or 3) maybe my philosophy of “procedural fluency leads to understanding” was crashing upon the rocks.

2:
Students got it but Cindy persisted. Once she understood something, she got it, but until she did it was painful—particularly when she would get frozen and could not move on until she understood, which was the case here. Students who manage to get it groan when this happens. Someone told Cindy ‘Because it works out that way; just follow the rule and figure it out later.’

While that exchange tends to bolster my views on how procedures lead to understanding, I can also easily imagine how easy it would be for those in the “students-must-understand-or-they-will-die” camp to make an out-of-context smear campaign against traditional modes of teaching. They would film Cindy saying “Wait, wait, I don’t understand!” Freeze frame of Cindy and cue announcer voice-over: ‘Rote learning is no way to learn algebra. Paid for by Friends of the Common Core.’

3
“So if you have two negative numbers, you just add them and put a minus sign in front?” she asked.

I found myself thinking “What the hell do I do now?”

Knowing she was right, but also knowing and not overly concerned that she didn’t understand the “why” of the procedure, I answered her question, guiltily confident in my belief that procedural fluency leads to understanding. “Yes,” I told her. “That’s what you do.”

• Jesse Malkin

Thank you for your response.

Perhaps the favorite textbooks of “traditionalists” are the old Saxon texts. If you look at Chapter 19 of “Saxon Calculus with Trignometry and Analytic Geometry,” Second Edition, by John H. Saxon Jr. and Frank Y.H. Wang, you will see an explanation of the derivative that is similar to the one you offered to your tutee. It is not possible to master the material in this chapter without thoroughly understanding the concepts.

Garelick is willing to let a student get away with not understanding how a number line can be used to add two negative numbers, but I bet neither he nor any other traditionalist would let his Calculus students not understand something so basic as the definition of a derivative. This would be like trying to teach Geometry to kids who don’t know what an angle is. Nobody wants that!

So, yes, I still say you are attacking a straw man.

By the way, the first two examples you cited support Garelick’s argument, not yours. In both cases, the students eventually developed mastery by mechanically doing problems. There is no evidence of conceptual deficiencies, no evidence that these kids have been turned into math zombies, and certainly no evidence that these kids would have benefited from regurgitating Common Core-style “how I did this problem” essays.

• educationrealist

And one more thing: if you read my original zombie post, I made it clear that G&B are making assertions that hold for one age group that don’t hold for another, either from a research standpoint or from an experience standpoint. I’m not particularly advocating for one method of instruction. I’m just showing that a concept they completely reject–a zombie that can do math without understanding–is quite common, and assert that they reject the concept because they lack adequate experience.

• educationrealist

They’re the ones that asked the question. I provided the quote, then an answer.

Garelick is willing to let a student get away with not understanding how a number line can be used to add two negative numbers, but I bet neither he nor any other traditionalist would let his Calculus students not understand something so basic as the definition of a derivative.

Barry got limited experience teaching high school students. When faced with a high school student using a procedure that really didn’t show understanding, but got the right answer, he shrugged. You (and he) may assert that he won’t have a similar reaction if teaching calculus, but his stated principle is procedural fluency leads to understanding. And I’ve demonstrated that’s untrue.

In both cases, the students eventually developed mastery by mechanically doing problems

Wrong. In both cases, the students during that class period developed procedural fluency. If you think that’s mastery then you, too, haven’t taught for long enough. In fact, in other passages Barry was clearly shocked at how badly kids did on quizzes when they’d shown understanding earlier.

• educationrealist

Another comment to somethingyou (Jesse Malkin) wrote before (the comments processor isn’t really putting these in order):

You are implying that Garelick and Beals and other traditionalists would defend the manner in which this girl was taught

I mentioned in the piece that I know this girl’s pre-calc teacher. She’s excellent, and I’m absolutely sure that she’s explaining the material conceptually and thoroughly.

What she’s almost certainly not doing is testing in a way to kill zombies. And she’s probably not doing that because her mandate is to prepare kids for AP Calculus by teaching some of it in precalc. I don’t think AP calculus has any concern with conceptual understanding at all.

• DensityDuck

I’ll be honest, my understanding of derivatives is exactly as you describe here–“the instantaneous slope of a function, and here is the procedure you apply to a function to get the derivative”. All that pantsing about with limits and infinitesimals went completely over my head.

This may have contributed to my needing to take PDE’s twice in college, but once someone clearly stated “the derivative of e-to-the-X is just e-to-the-X”, I was set from then onwards.

• Jesse Malkin

I’ve been perusing your blog this morning — a lot of very interesting posts! — and came across this:

“So my epiphany was this: Working with computers had taught me how I learned. How I learned. Which was not like most people. When books don’t work as a learning tool, then I have to learn by a particular type of doing. Explanations won’t help. Learning in a vacuum won’t help. I need to learn by trial and error. And then, I learn like Wile E. Coyote traverses the desert; I just keep on going until something blows up in my face. Go this way? Boom! Okay, that way doesn’t work. File it away. Go that way? Two steps, yes, then BOOM! Okay, the two steps, file away, then don’t go that way because BOOM! how about this way? Tiptoe, tiptoe, try this, ha! It worked! Done. On to the next. Make sense of the chaos, bit by bit, understanding the rules by the reaction.

“When I’m learning something, I neither know nor care about why. Understanding will usually come.”

https://educationrealist.wordpress.com/2012/08/11/learning-math/

Sounds a lot like the approach advocated by Garelick!

• Roger Sweeny

It sounds different to me. Ed is searching, trying to find a way to get to the solution. Garelick advocates GIVING students a way.

• educationrealist

Yes, Roger is correct. But Jesse is correct as well, that I learned just by doing.

However, he missed something from that same link, immediately after that passage:

(Note: Ironically, as a math teacher, I am big on explaining why, but that’s because I’ve realized that most people aren’t like me.)

In other words, most kids do not eventually learn by doing. They either don’t bother doing without understanding, or they do endlessly without understanding–that is, they are zombies.

This is what I mean when I say Barry hasn’t taught long enough.

• Mr Small

There’s a series of videos on MIT OpenCourseware where Gilbert Strang elaborates on what he explains as “the opening, the introduction that tells what’s important here.” He feels a lot of students are plunged into the course without first seeing the main point of calculus. I found them really useful both for my understanding and now for answering the perennial “Why?” question.

http://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/index.htm

I don’t think the procedural focussed approach is particularly new, having ploughed into a Hades deep furrow 30 years ago when put on the further maths track. Dropping to the normal maths syllabus gave me much more time to understand what was going on, otherwise it was just jumping through hoops to pass an apparently endless series of exams.

• Roger Sweeny

The other major zombie source I’ve noticed is upper income white girls. I have never met a white boy zombie …

I have, but they are less common.

As far as I can tell, upper income (and upper middle class) white girls care a lot more about doing well in school than their male peers. But like most young people, they don’t have much intrinsic interest in the material, and so aren’t terribly concerned with deep understanding.

As an old person, I am constantly struck by how often smarter-than-average males are satisfied with mediocrity.

• educationrealist

Grade mediocrity in particular. I think it’s because grades are viewed as nothing more than compliance. And yes, exactly to the grade obsession for upper middle income (and middle class) white girls. Which is also, I think, why zombies are well-represented among East Asians.

Not as common among Indians–at least, not at the top end. Again, in my experience.

Lots of students would be perfectly happy to be zombies, but they’re not very good at it.

• Vijay

Untrue: everyone is a math zombie at some level. The level at which your inner math zombie is triggered depends on your IQ. The reason you have not seen the Indian math zombie is because you have not seen enough Indians or you have not taught higher math classes. Teach a second class in partial differential equations, and see your entire class turn into zombies.

• educationrealist

No, not everyone is a math zombie at some level. Zombie requires will to fake it. Many people simply won’t fake it. When I say I see fewer Indian zombies, I’m saying that Indian kids of my acquaintance are less interested in faking it to the extent necessary to zombify advanced math.

• liamtheboscure

Everyone is most certainly a math zombie at some level. Do you understand why 1+1=2? Or did you just memorize that fact at some point? Russell and Whitehead took 379 pages in their Principia before they could derive it:
https://en.wikipedia.org/wiki/Principia_Mathematica
There must be some level at which you black-box the things you don’t think it’s worth understanding or haven’t the time to fully grasp.
I would agree that you should be conscious of the level at which you are understanding things, but it’s hubris to ever say you understand it all the way down.

• educationrealist

First, like the previous commenter, you don’t understand the term zombie. If you’re going to make everyone a zombie, rather than just say that everyone at some point “hits a wall” in their understanding, then you don’t understand what a zombie is. I’m not going to let someone redefine the term just because they don’t know who George Romero is.

Second, in your specific example, I believe I understand why 1+1=2, in that if I move one unit to the right away from 1, I will end up at 2, which means that both sides of the expression end up at same point of the number line. Provided I understand that, I am not a zombie.

• DensityDuck

If you think “derivative means that thing where you knock all the powers-of-X down by one” is the same thing as “I guess I can’t explain why adding one to one results in two”, then you don’t understand what ED means by “math zombie” and shouldn’t be taking part in the conversation.

• cthulhu

I think that a big reason for apparent satisfaction with mediocre performance is that these late teenage males don’t really know what they want to do with their lives beyond making sure they still have time for skateboards and video games, and hopefully not having to move back in with their parents. Not being flippant or judgmental here; I’ve come to the realization that knowing what you really want out of your life in terms of day-to-day work is pretty rare in teens and twentysomethings, and that I did not suffer from this is one of the great blessings of my life.

That being said, it takes motivation to dig down deeply enough into calculus to truly understand what is going on. For me, I knew that I wanted to be an engineer and that my degree required a ton of math (nearly 30 credit hours by the time it was done, starting with Calc 1), plus I was taking Physics 1 (which fascinated me) at the same time, and it became clear that calculus was the language of physics. So I had to get the concepts of the math to understand the physics, and since the physics classes were the initial weed-out classes for engineering, they focused on setting up the problems and symbolic manipulation, not plug-and-chug. I don’t think that any zombies made it through that gauntlet, though many tried – the drop rate was about 40%.

Which leads me to my final point: high school is not the right place for calculus unless you’re extremely motivated. The basic concepts can be made fairly intuitive with real-world examples, but again, unless the motivation is there, it won’t matter. Hell, the epsilon-delta limit proofs are tough enough to understand for the motivated college student; how many high schoolers can really get there? The vast, vast majority of high school students would be better off with probability, statistics, and logic replacing precalculus and calculus in the curriculum.

Sorry for the rambling; the current drastic overemphasis of “high schoolers should learn calculus!!!” Is one of my pet peeves. Thirty years ago, it was uncommon for all but a very few students, and the university system didn’t expect it, and the world survived.

• Roger Sweeny

Lots of students would be perfectly happy to be zombies, but they’re not very good at it.

LOL

I’m a High School teacher and agree with your experience and how it seems to come from a certain demographic. I’ve shared my own synopsis of this article: http://mitadmissions.org/blogs/entry/picture-yourself-as-a-stereotypical-male with a few female students of mine as they prep for tests and have noticed short-lived changes in attitudes and outcomes. It’s definitely worth a read to have in your “helping zombies toolkit.” Of course, you and I both know a single article, conversation, or experience isn’t going to simply “unzombie” anyone. Our hope is that over time, experiences, like the one you shared, add up to some sort of change in mindset though. It certainly sounds like your interaction was at least a mini-catalyst in the right direction for this student! Keep fighting the good fight!

• Vijay

A common misconception is that math zombies can be taught to overcome perception issues. That is possible at AP Calc B level or may be all the way to differential equations. However, even a perceptive student would soon hit their internal zombie threshold at some higher level (say, integro-differential equations, method of variations or small perturbations). Each and every student has a zombie trigger level in calculus, beyond which no amount of tutoring can help you, and it is correlated to personal IQ. This note is not to discourage the teacher and the student, but to make one aware that math is a series of steeper mountains and the ability to attain peaks stops somewhere. The key is to realize and accept your internal zombie.

• Val

It’s hardly surprising that your student didn’t understand the concepts she was taught. Why should she? She was taking pre-calculus, not calculus. Students need a solid understanding of functions, trigonometry, etc. before they can make a realistic attempt at calculus. Worse, the table of contents for her book shows that there’s nearly no pre-calculus there: two chapters of bits and bobs of pre-calc and then…we go boldly to limits in chapter 3. This is ridiculous. The book doesn’t even cover trigonomentry, which is a sine qua non for calculus. IMO,this is the school’s fault, as it shouldn’t be putting a pre-calc sticker on a calculus class.

The thing is, it doesn’t matter if her teacher explained the idea of limits conceptually and thoroughly. If her students weren’t ready for the material, they probably only understood her in a vague way, making them unlikely to retain the information, and unable to apply it. The result, as you correctly pointed out, was that your student could apply the power rule to get 2x-1, but she had no idea what it meant. Why should she be expected to understand major concepts (and details) of a subject she’s not supposed to be taking in until next year?

The situation that gives us a phony precalc courses reminds me of the thinking that went into algebra-for-all-in-8th-grade: someone noticed that the kids who took algebra in 8th grade did better in college, and ascribed their success to WHEN they took algebra, rather than to other factors, such as high intelligence or being more advanced through hard work (or both). Now that algebra in middle school is falling out of favor (because, err, so many students failed), we’re pushing algebra to 9th grade, supposedly to teach the kids what they need to know to succeed in algebra. But doing so means that THEY CAN’T TAKE CALC IN HIGH SCHOOL! This situation is obviously a disaster (for reasons that aren’t clear to me, apart from the college admissions/AP-class arms-race perspective), and so someone decided to cram calculus into advanced math. The result is a replay of algebra for all in 8th grade: most of the students don’t understand the material and people wonder what went wrong.

Very few of the grownups seem to understand that the best way to prepare students for calculus is to create a solid foundation of advanced math topics. This is not the same as rushing them into calculus and then repeating the same material next year, when they still won’t understand it because…they still don’t understand functions and trigonometry (because they were busy learning calculus last year instead). Oh, this makes my brain hurt. Who makes these decisions?

That was long. Thanks for reading.

• educationrealist

Yes, I totally agree with you. Did you see my and ben’s conversation above? I really don’t know that it makes sense to teach Calculus in a year, and by pretending to, we are basically forcing the kids to fake it if they want to pass.

• Val

I agree that the pace of math courses is too fast — even for the students at the very top of ability. The primary problem is too much breadth, too little depth, and too much reliance on algorithms over understanding. It’s too much for most kids and the talented & hard-working ones who get the ideas are being cheated out of an opportunity to learn the material in a meaningful way (harder problems, proofs) in the rush to cover implicit differentiation and l’Hopital’s rule (because they’ll be on the AP test and proofs won’t).

Schools seem to try to compensate for the pacing problem by forcing students to self-teach two chapters of material during the summer. IMO, the vast majority of students won’t learn the material properly (especially because of the poor quality of the textbooks). So if their parents can’t help and they can’t afford a tutor, they’re going to be behind on day 1. Of course, the same problem applies in other subjects.

Again, I find the entire situation ridiculous. On the surface, it all seems to be driven by hysteria over high-stakes tests, but the problem goes a lot deeper: wretched textbooks, unrealistic expectations that are cruel to students, and a host of other problems. By “unrealistic expectations,” I’m referring not just to summer AP coursework, but the idea that everyone should be taking college-prep classes.

• Calvin Hobbes

Another great post.

It seems like we “teach” a lot of math to students who are incapable and/or not particularly interested in understanding it, and most of whom are almost certainly never going to need it later anyway.

Maybe one fundamental problem is that our education system is as much about credentialing and signaling as it is about learning. A college degree, for example, is a credential that signals that the holder was capable of jumping through a series of hoops, though not necessarily that the holder learned much of anything useful in the process. Often one of those hoops is passing calculus. For someone determined to get a degree that requires calculus, or determined to get into a college that requires AP Calc, the easiest (and maybe the only) way of jumping through that hoop may be the zombie way. And passing calculus the zombie way is not something anyone can do, so it does signal something.

It would be great if we could have a different credentialing and signaling process that is more informative and efficient. The one we have seems colossally stupid.

• Roger Sweeny

Your first paragraph applies to a lot more than math!

In high school, there is not enough time for students to actually learn all they are supposed to learn. But there is way more time than necessary for them to learn what they want to learn.

(Well, what they want to learn of what the school has to offer.)

• Jim

Very well and truly said!

• Retired84

What happens to these kids in college? Fake it thru? From experience I know they are often exposed and derided in the working world.

• Jim

To Val – L’Hopital’s Rule is an example of something which, while it has some value, is often emphasized in textbooks way out of proportion to its true value, simply because it is easy to test.

• Roger Sweeny

Hmmm. I wonder how many other things that can be said of.

• educationrealist

The bulk of high school, alas.

• Val

This is what a nation gets when it prioritizes industrial metrics like test scores over teaching people how to think. Our public schools have become factories — but the widgets that they produce are test scores, not graduates. The students and the teachers are just cogs in the great American school machine.

High-stakes testing mania isn’t the only problem in the American school system, but it’s a big one.

• Roger Sweeny

Val, I may be misreading you but it sounds like you think there was once a time when young Americans were taught to think in schools, forced to think in schools, and passed on to the next grade (and ultimately allowed to graduate) based on how well they thought–and not on how well they did on tests.

There is an obvious problem in my “sounds like” since teachers have always given tests and students have always been given credit for a class (and ultimately allowed to graduate) based on their test grades.

But perhaps you mean that old tests tested thinking while new tests test rote memorization. My experience as a teacher says that is not true. It is extraordinarily hard to develop tests that grade students on thinking (though I kept trying) and then it takes a long time to grade them. Fifteen minutes an exam for a hundred kids is 25 hours; that’s 3 1/2 days in a 9-5 office.

The difference is that there are now more tests which are developed by outsiders and graded by outsiders. They will inevitably miss some things that classroom teachers see. They would be pointless except for two offsetting strengths.

There is a tremendous conflict of interest when teachers grade their students. They both teach their students and then determine how good a job they have done teaching!

Also, teachers are generally nice people. They don’t want their students to fail. Nor do their department heads, or principals, or superintendents–who can tell them what bad teachers they are if they fail too many students. So lots of students are passed and graduated who do not have much in the way of thinking skills or basic academic skills, e.g. reading, writing, calculating.

Outsider high-stakes take away some of the ability to pass those people. Though all too often, when a lot of kids fail the high-stakes test, the cut score is lowered and next year’s test is made easier.

• Val

Adding something. I spent the last hour trying to understand why I could get two apparently very different antiderivatives from the same function. One was a single trig term and the other was a long messy string of different trig terms. I finally figured it out when I plugged numbers in and realized that the answers differed by 1/8. When I added 1/8 to the long messy antiderivative and plugged it into Wolfram Alpha, it equaled the expression with the single trig term. Funky.

This idea is implied in the basic idea , but the conceptual extension of the + C thing to what I wrote above is subtle, yet savage. This is another way that students can reproduce something basic that doesn’t stray too far from the script without really understanding what’s going on. The worst part is that they think they understand it.

I’m not saying that instructors should spoon-feed ideas like the one above to their students, but they (and the textbooks, especially) should be giving examples or assignment problems whose answers involve subtleties that should be discussed in class. I also realize that this sort of thing will only be accessible to top students, and that our everyone-can-go-to-college-and-be-an-engineer! mentality gets in the way of stretching our brightest minds. Which leads us back to focusing on the stuff that will be on the test and stripping courses down while increasing the homework load in the name of rigor.

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• Val

@Roger Sweeney, what I meant was that there was a time (before the mid-80s or so) when schools taught in a way that was more conducive to learning how to think. The system certainly wasn’t perfect then, but it was a lot better than it is now. There was less emphasis on multiple choice tests, for example, and more emphasis on learning grammar and other basics. The school days were also longer when I was a kid, and we had time for things that schools don’t have time for now.

Where I live, K-8 schools end at 2:30 pm except on Wednesdays, when school’s out a little after 1. The primary emphases are on language arts and math, with science and “just say no” as side courses, and classes in art, music, drama, etc. being rare or non-existent. The kids do their first practice sheets for May testing in August or September. IMO, the schools are teaching them how to take high-stakes tests. This is hardly surprising, given that the schools are judged on the results of high-stakes tests. But the thing is that if they just taught the basic skills and taught them well, and coupled this with appropriate pacing for quicker and slower learners, the test results would improve. But everyone is, I don’t know, too stressed out to think clearly.

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• surfer

1. Simple derivatives and antiderivatives have been part of pre-calc for eons.

a. My public school early 80s, had in between alg2/trig and calc a year of math: one semester of “functions” (mostly baby calc, limits, Archimedes method, an example epsilon-delta, and simple derivative/antiderivatives, plus max min. I think there was some stuff on domain, continuity, differentiability and inequalities…from a simple standpoint, conceptual…not real analysis). Then we got a semester of analytical geometry which LEVERAGED the baby calc (since analyt geometry is curve visualization, you need to know max/min points and asymptotes).

b. The 1958 Schaum’s Outline on first year college math contains a 30 page introduction to calculus. (along with “college algebra”, trig and analyt). https://www.amazon.com/Theory-problems-first-college-mathematics/dp/B0007DPVM2?ie=UTF8&*Version*=1&*entries*=0

2. Traditionalists (at least me, and one other I just read on the net today) don’t see procedural learning as GUARANTEEING conceptual understanding. They just see it as cognitively easier to acquire that deep understanding if you have already gotten the mechanical procedures down. Think about how easy your kid picked it up. Don’t you think her background in the procedures made it easier to pick up the fundamentals? Conversely think about having to learn calculus from scratch from a real analysis textbook!

3. Good work with the kid.

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