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) = (x ^{2} + 1)(x^{2} – 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. (x^{2} + 1)(x^{2} – 2x) expands to x^{4}-2x^{3}+x^{2}-2x which…has a derivative of 4x^{3}-6x^{2}+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.

“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 x^{2}-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}/_{3}x^{3} – 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….x^{2} – 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.

December 7th, 2015 at 4:06 am

Feh, the Quotient Rule. You have to know the chain rule. You have to practice with the chain rule. It’s complicated and unintuitive and important (assuming, you know, that you’re taking or doing calculus).

You have to know the product rule too. It’s not so tricky.

The Quotient Rule has no purpose except satisfying zombies. You never need it, for any reason, because d/dx of f(x)/g(x) is the same thing as d/dx of f(x)*[g(x)^{-1}], and the chain rule will give you the derivative of g(x)^{-1}. I passed calculus, got a 5 on the AP calc BC, and majored in math all without ever learning the Quotient Rule.

December 9th, 2015 at 12:31 am

The Quotient Rule has no purpose except satisfying zombies. You never need it, for any reason,I used to think that, but you are wrong.

You don’t strictly need quotient rule for finding the derivative. But then most questions in context only use the derivative, they don’t just differentiate and stop.

Trying to find points of inflection using only product rule for a function with a division is hopelessly confusing. You get a long expression which you have differentiate again and set to zero and then solve. If you quotient rule you get to ignore the bottom line when you set the second differential to zero, thereby simplifying it greatly. I found this out the hard way, by trying to not use quotient rule.

There are some actual functions that are much easier to differentiate by quotient rule as well, because otherwise you have to set up complicated chains in the differentiation e.g. f(x) = 3x / ln (3x).

As a rule teachers don’t often teach things for generations like zombies. (My one exception is algebraic long division, which is not required in this era of calculators.)

December 7th, 2015 at 5:00 am

“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.”

-I (White male) have a very strange anecdote of non-correlation of mathematics test scores. I scored very well on the math ACT (32). I did only 30 minutes of light studying for the ACT, and only for the math portion. Yet, I did very poor in Pre-Calc (the vast majority of the grade was tests), getting a C 1st semester and a D second semester. The White dude who was at the top of the class (A+) got an ACT math score of, IIRC, about 27, and claimed to have done quite poorly on the math SAT. He also didn’t do many (any?) APs. I took 8 AP tests; got an easy 5 in all of them. In Calc AB, I got a 5, as did 80+% of the class (here, too, most of the grade was tests). Yet, in terms of Calc AB grades, I was squarely at the bottom of the class (D first semester, E second). Thankfully, I thus never had to take Calculus in University while getting my Bachelor’s.

What is your diagnosis of my problems and those of the student at the top of the class in my Precalc?

December 7th, 2015 at 5:05 am

I’ve seen it before. I remember working with an engineer who used calculus in his job daily, had taken well beyond calculus in college, and yet couldn’t get over a 550 on the GRE to get into grad school. Meanwhile, even before I really understood math, I was able to fake my way to a 640 on the GRE. I did poorly in math in high school, too–worse than you, even though I got through AP Calc.

I don’t know what that’s about. I haven’t seen it recently, but that’s because math expectations are so different these days.

December 7th, 2015 at 5:49 pm

Speaking as someone who was in that situation–an engineer who used calculus quite a lot and yet barely got high enough on the GRE–I should point out that the GRE covers a lot more than just straight calculus.

To be honest, I was best prepared to take the GRE at the end of my sophomore year of college. The engineering GRE covers general engineering knowledge, and the end of my sophomore year was when I’d just had the last of my general-engineering-knowledge courses. After that it was two years of field specialization and my general knowledge fell away quite a bit (electrical engineering nearly did me in–thank god most of it turned out to be similar to the control-theory stuff I’d just done my final in!)

December 7th, 2015 at 6:10 pm

Oh, I think we were just talking the regular GRE, not the subject one. That’s brutal. I was pleased a few years ago to score a 540 on a practice test, just because I had so little advanced math.

December 10th, 2015 at 3:09 am

The SAT/GRE math don’t really measure math but puzzle-solving logic.

December 7th, 2015 at 5:35 am

[…] Education Realist on Math Zombies (read the comments; he and I are not so different!). […]

December 7th, 2015 at 7:46 am

[…] Source: Education Realist […]

December 7th, 2015 at 11:21 am

You wrote that problem 14 involves 1/3 x^3+x, but I believe that should be 1/3 x^3-x. You should correct that: readers could get confused.

December 7th, 2015 at 1:59 pm

Yep. Fixed.

December 7th, 2015 at 2:49 pm

What struck me most about this entry (as you allude* to when trying to beg off from taking money) is how much the back and forth was like a student study group.

“What does a horizontal line have to do with slopes?”

This sort of statement clearly makes me thing of the implications product of Gillford’s Structure of Intellect model. (Incidentally, I share this product formation as a bias, which is why I think I like your blog so much.) Other than the tendency/desire to create Zombies of the procedure-in-place-of-understanding route (e.g. assuming evaluative understanding will come from memory operations) I have long wondered whether these differences in textbook/instructor/student preferred product generation is to blame for some of the failure to learn/failure to teach. Though I have not taught, I know from experience that it is difficult to translate an understanding known in one product to the others (especially a polar-opposite product).

December 8th, 2015 at 2:56 am

I am struggling to understand why you take this as a lesson in why it is crazy to expect/allow students to learn AP Calculus (at least AB) in HS. You give a great example of exactly how much even a reluctant student can learn and begin to grok in a relatively short amount of time. The lesson that I take away from it is that too many students are trained to be good hoop jumpers by constant rewarding of hoop jumping over time.

Take away the rewards for hoop jumping and add in rewards for thought and you end up with students who actually begin to think a bit (after they and their parents scream bloody murder). As you point out, one good way to accomplish that is to work hard on writing tests that reward thought instead of hoop jumping, though this is very hard and the grading becomes much more intensive.

Iterate this process over a MS and HS career and you will start to produce students that think through things, many of whom- but not all- will begin to see value in this thinking thing. And AP Calculus isn’t really an impediment to that, in my experience. Their test comes a hell of a lot closer to testing understanding than any other standardized test I have ever seen. I am sure you can brute force it, but it is a hell of a lot easier to pass if you work on understanding instead.

BTW, writing a post outlining the meaning of math zombie soon, since so many seem to misunderstand it. We will see if that helps.

December 8th, 2015 at 3:42 am

I wouldn’t go so far as to say all AP Calc instruction is futile. I would say we’re digging too deep into the population.

“You give a great example of exactly how much even a reluctant student can learn and begin to grok in a relatively short amount of time.”

I tried to make it clear that, despite my best efforts, she was just focusing on getting the answers. Her teacher had already taught her once, focusing on concepts. I don’t have much faith she’ll know much more after that.

This isn’t a story of a successful teaching experience. Just a story of what it looks like to teach a kid who isn’t learning because she has no interest in doing so.

Agree with third paragraph.

I thought most people did understand it, then I read the two on my thread. Then Greg Ashman who seems to think it doesn’t exist.

December 8th, 2015 at 4:01 am

I tend to take the IB/AP approach to who should take the courses: anyone who wants to try, but you don’t water down the material and you don’t sacrifice quality instruction. Now, if you think people are pushing too many kids INTO the classes and THROUGH the classes in order to boost status, well people always try to cheat to attain results they haven’t earned (teachers and students and parents and administrators and… everyone). I am not sure there is really a way to stop that process, especially among those with significant capital (of all varieties).

And while I am sure the process was frustrating, not least because of student disinterest with actually thinking, I cant help but see that some growth has taken place. And I take you at your word that your friend is pushing this stuff in class as well, but also take you at your word that their exams probably don’t push the issue enough. Also, iteration and long term system effect have a huge impact that is hard to overturn very quickly.

December 8th, 2015 at 5:35 pm

I’m with Brett!

The fact that a zombie can so quickly start thinking like a non-zombie (even if it’s just for an hour) suggests that “zombie mode” is not primarily about IQ. It’s about sitting in a classroom of 25 versus having an hour of one-on-one time with a very good tutor like you. It’s about motivation, incentive, and what vision of mathematics they’re operating under. On a basic level, it’s about the structure of our educational institutions. This seems to be your conclusion, too: kids act like zombies not because they’re incapable of thought, but because we’ve built a system that rewards such behavior.

So what does “digging too deep into the population” have to do with it?

December 8th, 2015 at 11:59 pm

Well, thanks for the kind words about my tutoring–and I absolutely agree that zombie mode is nothing to do with IQ. In fact, I’d argue you have to have a pretty solid IQ to be a reasonable zombie.

“This seems to be your conclusion, too: kids act like zombies not because they’re incapable of thought, but because we’ve built a system that rewards such behavior.”

Yes.

“So what does “digging too deep into the population” have to do with it?”

Because we said, some 20 years ago, “Everyone has to go to calculus”, we’ve taken the agency out of it. So kids are pushing themselves way beyond their interest and, at a fairly early age, checking out of anything short of zombie thinking. Worse, if you try to convince them to leave zombie mode they want to know if it will hurt their grade.

I’d rather see us dramatically increase the difficulty and depth of English and history and ratchet back on moving forward into math quite so quickly. Leave calculus for the kids who are genuinely interested, with no harm coming to those who just get to precalc in high school.

December 9th, 2015 at 3:28 am

“Because we said, some 20 years ago, “Everyone has to go to calculus”, we’ve taken the agency out of it. So kids are pushing themselves way beyond their interest and, at a fairly early age, checking out of anything short of zombie thinking. Worse, if you try to convince them to leave zombie mode they want to know if it will hurt their grade.

…

with no harm coming to those who just get to precalc in high school.”

This seems to be the education equivalent of ‘1st World Problems’. I don’t know who the ‘we’ us here, but I have taught at 3 schools in 3 very different states (and talked with an awful lot of teachers along the way), and schools where more than 20% attempt calculus are, in my experience, very rare. I don’t doubt that they exist. I just haven’t seen this be near the problem you have found. At most schools I have experienced, the struggle is to get enough kids interested in pushing themselves to make a class.

As to whether zombie mode will hurt their grade, I think you addressed this earlier. With good questioning, you can actually cause zombie mode (and an unwillingness to switch out and think) to hurt the grade instead. Students who have been successful zombies in the past (and their parents) will scream bloody murder (at first). But change the game and you will change behavior, in my experience.

December 9th, 2015 at 4:50 am

“At most schools I have experienced, the struggle is to get enough kids interested in pushing themselves to make a class.”

Except they won’t push themselves, because they refuse to be zombies. It’s definitely not a first world problem. I don’t live in the first world.

December 9th, 2015 at 5:17 am

“Except they won’t push themselves, because they refuse to be zombies.” Well this is delightfully circular. I guess if you just have a bedrock assumption that the only way to be successful in calculus is to be a zombie, there is no shaking that. But it does seem a bit of begging the question to reject the possibility of students who would be able to succeed in calculus without being zombies and then attribute all disinterest in the class to kids rejecting this standard of success.

And if you ‘don’t live in the first world’ then I have no idea how you are running into schools where anything above 20% of the student body (and that is me being conservatively high in my estimate) are even attempting calculus in HS. If you aren’t encountering those rates of participation, then I am not sure what this even means:

“Because we said, some 20 years ago, “Everyone has to go to calculus”, we’ve taken the agency out of it. ”

Do you just believe that teachers/schools are somehow declaring the >80% of students who don’t do calculus failures? Again, I ask where this is happening, because in every school I have taught in, the majority of even our college bound students don’t take calculus and don’t seem to feel particularly bad about that fact.

December 9th, 2015 at 6:50 am

” I guess if you just have a bedrock assumption that the only way to be successful in calculus is to be a zombie, there is no shaking that. ”

I think you misunderstand, and definitely overstate my hostility to Calculus. I would hate to think most HS calculus students are zombies. However, I definitely know schools that are 80% Asian, and most of the kids are taking honors pre-calc as sophomores, and an alarming chunk of those students would qualify as zombies. To wit: they don’t know the basis of the quadratic formula, they don’t really understand what a derivative is, when asked the difference between a horizontal and vertical asymptote, they say “one goes up and down, the other goes left and right”.

In other schools, such as the one I teach at, relatively few get to AP Calc (although we have a non-AP Calc class). However, our school is under tremendous pressure, as are most schools, to get more kids to Calculus. I know the math teachers who teach the Calculus classes, I know what the kids tell me, and they are very much lecture and assign a problem set. I’m not saying that’s a terrible thing, but I am reasonably sure that a good number of them are zombies.

The number of 5s on the AP Calc tests is in the 45-50% range, which suggests it’s a fairly easy test to figure out and “game”. Recently, the AP Bio test was reworked after the College Board determined that too many kids were getting 5s. I wonder if the Calc test will be the same?

So imagine, for the moment, that the College Board reworks its test and it becomes much harder to get a 5. If you had to guess, would you think that the test became more or less amenable to zombie math?

“Do you just believe that teachers/schools are somehow declaring the >80% of students who don’t do calculus failures? ”

No, I know–not believe–that schools are evaluated based on how many of their kids are college ready at *worst*, and are taking calculus at best. Haven’t you been following the reform agenda? Haven’t you seem the outrage at Common Core because it supposedly doesn’t get kids ready for calculus?

I don’t think we’re communicating on the same wavelength, and it may not be possible to do so, but for now rest assured that I’m not convinced that high school should ban calculus.

December 9th, 2015 at 10:16 am

Ah, that makes sense.

I’m curious – do you think the push to calculus has been primarily top-down (“you kids need calculus”) or bottom-up (“I want to compete at good colleges so I’ll take calculus”)?

December 9th, 2015 at 2:27 pm

Both. At the top level, kids need to compete, so they take calculus. At the lower level, schools are pressured to put kids in AP classes because their principals are given bad evaluations for “achievement gaps” in advanced course population. (My last two schools, the principals have been dinged on exactly this point.)

AP Calc has always been more immune to this than other AP courses, particularly at schools with a wide range of abilities. You can’t just put kids in AP Calc, they have to have succeeded in the courses before that point. In contrast, you can just stick a kid into APUSH.

However, schools are also held responsible for how many kids have “college ready” transcripts, which in many states means algebra 2. So a lot of seniors who have very low skills but have managed to pass algebra 1, geometry, and usually discovery geometry are put into algebra 2. This dramatically increases the range of abilities in algebra 2.

All of these things place pressure on math teachers, who are more likely to find zombies unobjectionable in the first place.

December 9th, 2015 at 3:27 pm

These last two comments have helped me understand what you are saying quite a bit. I think YOU actually tend to overstate your hostility to calculus, but I now have a much clearer idea of why. The situation you describe definitely resonates and can be maddening. Thanks for continuing the conversation here. I think I am going to try and expand and respond at more length in a future post.

December 10th, 2015 at 11:20 pm

“The number of 5s on the AP Calc tests is in the 45-50% range, which suggests it’s a fairly easy test to figure out and “game”.”

In addition, the required score for a 5 is ~60% correct, so students can answer most of the easy questions correctly and miss most of the thinking questions

December 11th, 2015 at 12:06 am

Yeah, that’s what I’ve been told by others as well. I have no direct knowledge.

June 2nd, 2016 at 1:45 am

We might be digging too deep into the population. I wonder how some of these kids would do on a test from the early 80s. No graphing calculator crap.

December 8th, 2015 at 7:48 pm

“I tried to make it clear that, despite my best efforts, she was just focusing on getting the answers. Her teacher had already taught her once, focusing on concepts.”

Maybe I’m misinterpreting you, but you seem critical of this student for going through procedural motions and not making a more enthusiastic effort to understand calculus.

But…she’s enrolled in PRE-calculus, and the school doesn’t seem to offer an actual pre-calc class. It’s likely that this student and many/most of her fellow students lack the background necessary to understand calculus. It doesn’t matter if her teacher explained the concepts if she isn’t academically ready to understand them. Given the situation, her lack of enthusiasm is hardly surprising.

Here’s another way to look at this. A skating school has an “intermediate” class. The beginning class gets as far as skating backwards awkwardly in a circle; intermediate is about turning and “edges.” The school is worried, though, because lots of kids do poorly on the AP skating test. So it decides to teach advanced skills in intermediate to “reinforce” learning. “Advanced” means go fast backwards, turn, jump, do a half-turn in the air, and land backwards on one foot on the outside edge of your skate.

How enthusiastic will the kids be about this if class involves confusion and falling a lot? Will it help if the teacher gives them clear explanations about landing on their toe picks and dropping down to their outside edge? Or will they just stay confused and end up with poor form and bruises?

In terms of sports, this approach seems nuts. Yet it seems to be the norm in education — pre-algebra is algebra and geometry, pre-calc is calc. THE SCHOOLS DO THIS. Not the students. Not to mention that all the adults are telling kids that they HAVE to take these courses, lest they end up as total loser failures.

Yes, testing mania is out of hand, some kids are lazy, and too many are told they must to college. But these facts are irrelevant to expecting pre-calc students to understand material that relies on stuff no one has taught them.

December 8th, 2015 at 9:33 pm

Just to clarify, what are the elements of Calculus that you think students with adequate Algebra 2 mastery aren’t prepared for in the example above. I assume that limits were addressed earlier in this class (as they would be in a Calc class), so ask they be granted as a covered concept. What else are you uncomfortable with here?

December 15th, 2015 at 4:01 am

I’m not convinced that current algebra 2 courses cover material in sufficient depth to prepare students for calculus.

As just one example, I compared trigonometry coverage in 2 common Algebra 2 books: Prentice Hall Algebra 2 (first author Bellman) and Holt Algebra 2 (Larson) with one older precalc book (Brown).

Larson has 2 chapters of trig (13 sections); Bellman has 15 sections. Brown has 5 chapters of trig with 24 sections. Brown has lots of proofs as exercises. The other two have very few. Brown has a couple sections on polar coordinates; Bellman and Larson have nothing. Bellman and Larson present ideas in an odd order, like sandwiching complex numbers between two sections on quadratic equations and completing the square. Huh? Larson is an outline book: it’s light explanations and heavy on examples. All or most Larson books seem to be organized that way.

A lot of stuff in modern textbooks is out of order and presented to be memorized and regurgitated. It’s no wonder that students end up not understanding basic principles when they just aren’t taught them.

December 9th, 2015 at 12:40 am

I’m not convinced you have a “Maths Zombie” at all. Although without seeing the kid in action it is hard to tell, it has the classic signs of someone who just isn’t very good at Maths, so compensates for that by getting extremely rule based. Trying to teach understanding of abstract concepts to that person is likely to go awry. She isn’t interested in it because she can’t actually hold the concepts in her head for any length of time. It’s not willfulness. (Note I regard the line “anyone can do maths” as contradicted by the evidence of every classroom I have ever been in, current wishful thinking about how we are all equal notwithstanding.)

I am hopeless at music, literally tone deaf.When I learned an instrument it was entirely rule based. My teacher would say things like “can’t you hear that you are in the wrong key?” and I would answer “No”. I literally could not hear. I have a choice of being a “music zombie” or not playing an instrument at all. I was not being willful, despite what my parents thought. I just don’t have the mental equipment required. It was really very unhelpful when people suggested that “everyone can do a bit of music”.

December 9th, 2015 at 4:53 am

“it is hard to tell, it has the classic signs of someone who just isn’t very good at Maths, so compensates for that by getting extremely rule based.”

That’s kind of the definition of a math zombie, isn’t it?

I agree she’s not very good at math, but she usually gets As and Bs.

December 11th, 2015 at 2:36 am

Well, sort of the definition. I would only call someone a “zombie” if they become purely rule based even when capable of understanding. Such students are rare — I’ve only ever seen a couple.

My point is more that there sometimes isn’t a cure for “zombies” where is it is compensation for lack of native ability. You can tutor them, they seem to get it, then it just goes away, since they can’t hold multiple abstract concepts at a time.

The solution is that they shouldn’t take Calculus, and should choose a career that favours hard work over native Algebra ability — it’s not like there aren’t plenty of those, many of them quite good careers.

We’ve gone so far out pushing kids into STEM, and especially girls, that we are starting to get ones who believe the hype. They try to take STEM classes even when their abilities suggest other paths.

This isn’t a problem caused by Maths teachers, and it won’t be solved by us either.

December 11th, 2015 at 5:56 pm

Perhaps we are, in the words of my daughter, being too binary here–as if the world can be divided into two homogeneous groups: one of which is calculus zombies and one isn’t.

Ed seems to have a pragmatic definition: calculus zombies are people who get good grades in calculus, who can work the problems after enough practice, but don’t really understand what is happening when they do it or why it works.

Some commenters say this isn’t a useful definition, that reasons matter: some zombies may not be bright enough to understand, some may not care to understand, some may just be trying to fit “build a resume for college” into the limited waking hours of a life. This, of course, matters if the question is, “Can they be saved from zombiedom?”

Maybe it’s partly a matter of age. Just about everyone who will be able to “get” calculus at age 20 will not be able to get it at age 5. Maybe the same thing is true about ages 17 or 18. Which matters if we’re debating calculus in high school.

And, of course, zombiedom is rarely complete. Someone who understands limits may turn into a zombie when integration is covered later in the year. People who don’t understand derivatives as slopes of a curve may understand the slope of a straight line.

As an anti-Platonist, my feeling is always, “You can use whatever definition you want as long as you’re real clear at the beginning what you mean by the term.”

Obligatory GIF:

December 11th, 2015 at 6:18 pm

I’m thinking more of an overall approach. Zombies, in my experience, are always zombies. That is, someone who says “Look, I’m just faking it. I am following the rules but I’m not sure how it works. I’d like to.” is very different from someone who has been frantically following algorithm rules and forgetting everything not immediately relevant to a test for 6 years.

December 15th, 2015 at 12:05 am

I don’t see how anyone can “get” math without real-world application. You can use math jargon all day, but until you present students with a real-world, material problem that can only be solved with some given mathematical concept, then of course it will remain “zombie” knowledge for the vast majority of students. I could have multiplied matrices for you in high school, but I didn’t really understand what the hell matrices were all about until I found myself needing to use them in grad school to solve language processing problems.

January 2nd, 2016 at 7:31 am

[…] and The Test that made them go Hmmmm is an accurate representation of my classroom discussions. The zombie sessions with my private student capture my strength as an explainer. I also contined to build my […]

January 16th, 2016 at 6:53 am

[…] as a cheap way to win arguments. It was a term invented to describe the experience endured by teachers who stop and give a moment’s pause to sussing out which students are forming […]

June 2nd, 2016 at 1:05 am

You show a gap in not realizing that precalculus usually has SOME CALCULUS. Seriously, do a survey. This seems strange to you, but it is normal. And has been for a loooong time. And is a good idea. What else would they do with a year between alg2/trig and BC?

I think you can drill kids on the understanding too (Socractic questions, tricks, quizzes). That’s great. Just don’t assume they can’t do ANY procedures until learning some fundamentals perfect first. The mind doesn’t work like that. It actually puts a big unmotivated cognitive hurdle if you teach like that. Give them a basis intro conception that covers fundamentals (but don’t kill it for 3 weeks like some of your stuff please) and then move into procedures and just check and emphasize fundamentals as things go on.

June 2nd, 2016 at 2:11 am

“What else would they do with a year between alg2/trig and BC?”

Lots of stuff that they used to do until the 80s or so: Logic, sequences and series, lots of vectors including a vector treatment of analytic geometry, and proofs. Many proofs.

Precalculus today is more of a rushed mishmash review of algebra 2 (plus limits and basic derivatives), or a straight-on calculus course that gets repeated next year in calculus (as this blog pointed out).

The textbooks are designed around content on a high stakes test, not around challenging the kids to actually solve difficult problems. The university STEM profs complain about students not being able to work outside a script, and the focus on test content over problem solving is one reason why.

August 24th, 2017 at 1:02 pm

I went to a good school in the 80s. We had a course called “functions” first semester that hit inequalities, limits and basic derivatives, and then second semester course that was “analytic geometry” which contained the classic stuff: conics, translation/rotation of coordinates, as well as polar coordinates and graphing rational functions (that was cool).

If anything it is kind of a strange year. Alg 2/trig makes a lot of sense. Calc made a lot of sense. Even analytic geometry made sense. But functions seemed a bit like we didn’t need a full semester for it and definitely not if you cut out the baby calculus.

August 24th, 2017 at 1:08 pm

By the way, the Schaum’s Outline from 1958 for pre-calculus includes a basic calculus starter. So it is not that strange. If you are spending a whole year in between alg2/trig and calc, it makes sense.

In my experience, there was no review of alg2 or trig, by the way in my 80s HS in precalc. We were expected to know that from year before. Functions and analyt were not easy and had more of an adult expectation (nominally senior classes).

The tricky thing is not having a year of pre-calc and going directly to calc. I think this is possible as I was itching to do more calc when we did the baby stuff. But then again, I think it would be a train wreck for most students and would overload the calc course. Even though I thought all the theory fluff about functions was kind of a waste, it was probably good to have it solid. I don’t think I ever used that stuff about inequalities later though. That was just out of nowhere and went nowhere in future.