This year, I encountered the world’s worst calculus class, a mutant-frog specimen of undergraduate mathematics: UC Berkeley’s Math 16B. It’s an exercise in cynicism; a master-class in spite; a sordid and cautionary tale of everything that can go wrong in curriculum design.
16B is my blood-born nemesis. Neither can live while the other survives.
It comes as the second semester of a year-long sequence. Its predecessor is the respectable-enough 16A: a light, trig-free presentation of calculus. Accessible and coherent, it has a conceptual focus and lots of friendly graph-based questions. I enjoyed tutoring it.
But then comes 16B: a hasty meal assembled using whatever mismatched leftovers the fridge happened to have. The mathematical equivalent of Jello-olive tacos and bowls of mustard.
Simply consider 16B’s spaghetti-logic calendar of topics, presented here in precisely the order they’re taught. (No worries if you don’t know what these are—neither do the students completing the course, really.)
Like a blind driver on a six-lane highway, 16B veers madly from topic to topic, grabbing material from half a dozen different standard courses. Namely, it cannibalizes pieces of the following:
No discernible logic governs this sequence. No underlying principle unites the topics. 16B is merely Franken-calculus, stitched together from the corpses of actual classes.
As the semester wore on, the two students I tutored grew increasingly frustrated with the course’s haphazard path. “What are we even doing?” one asked me. “So… none of the chapters really relate to each other?” asked the other. They both wanted to know: “What’s the point of all this? How does it fit together?”
The exercises and homework questions rarely gave them trouble. Their biggest problem was the Kafkaesque purposelessness of the class itself.
A good course in any subject, from art history to chemistry, ought to feel gratifying and self-contained. It ought to build and develop, like a well-written novel or film. This goes double for calculus, whose beauty lies largely in its unity, its coherence, the harmony of its constituent parts.
16B possesses none of that. It just rambles on, like a shaggy-dog story.
This problem engenders another. In sprinting through so many disparate topics, the course can do justice to none of them. At every turn, it settles for computational shortcuts at the expense of conceptual depth.
Students learn the second-derivative test for maximizing functions of two variables, without even a whispered mention of the mystical formula’s origins.
When they encounter the normal distribution, it’s in a setting purged of calculus. Makes sense for a calculus class, right?
And don’t get me started on the class’s sham presentation of Taylor Series, so watered down that it’s practically homeopathic.
Though they fully understood the course’s futility, the students kept slogging. They had little choice. After all, 16B satisfies the university’s pre-med and pre-business requirements, despite offering absolutely nothing of value to any future doctor or entrepreneur. Accustomed to hoop-jumping and arbitrary tasks of academic strength, the students showed little outrage at being forced through this particular series of rat-mazes and hamster wheels. Hey, they’re Berkeley kids—school is what they do. Graduate first; ask questions later.
The best you can say of 16B is that its students have been “exposed to” something. That passive construction is the only appropriate phrase. “Exposed,” as if a man had thrown open his trench coat before their eyes. “Exposed,” like wheelbarrows left to rust in the elements. “Exposed,” that last excuse given by educators whose lessons leave no fingerprints, who justify wasted weeks with the remark, “Well, they may not seem to understand much of anything, but at least they’ve been exposed.”
You know the yearbook cliché: “Shoot for the moon; that way, even if you miss, you’ll land among the stars”? With 16B, Berkeley takes the opposite approach: “Shoot for the mud; that way, you’ll never miss.”
It all raises the obvious question: What the heck is the point of Math 16B?
As best I can tell, 16B allows someone, somewhere, to rattle off a long list of mathematical topics and say, “Isn’t impressive that all our pre-meds and pre-business students know this stuff?” It’s a class that serves the needs not of the students, nor even of the teachers (indeed, I pity the poor professors conscripted into teaching 16B), but of some distant curricular architect indifferent to them both.
I believe that good education comes from a compassionate expert crafting a course, like fine artisanal handiwork, to suit the needs of the students. In that sense, 16B is the unholy Manichean opposite of good education.
To their credit, UC Berkeley Math seems to be learning its lesson. One new course (Math 10, tailored to biology majors) is rightfully the pride of the department, a thoughtfully compiled toolkit of techniques and concepts. I only hope that someday soon they’ll revisit Math 16B and breathe real life into that rotting jack-o-lantern of a class.
The curious non-math student in me wonders: Are there schools that tailor aspects of their math curricula specifically to other disciplines (for instance, having set courses specifically for chemistry/business/bio/whatever students, in which they learn what they will need for those classes and little more)? I wonder if such courses would be valuable or limiting (for there is surely some benefit to learning MORE math than you need)…
I think many departments do offer courses like this: “service” courses for “client” departments (or programs) like engineering, pre-med, business, bio, and econ.
I agree there’s a benefit to learning more math than necessary, although I think that students generally appreciate a class well-tailored to their needs. Berkeley’s Math 10 seems like a good example of this sort of service class done well.
Nice. It reminds me of “Writing Across the Curriculum,” and how writing teachers and tutors might be called in to support students and to help faculty better integrate writing into their courses.
Having jumped into MA 50A as a freshman lo so many years ago, I remember helping many a dorm-mate with this class. I suspect what happened is that they had a whole bunch of ideas on what should go into the non-engineering Calculus class, made MA 16A make sense and stuck everything else into the second semester because if the students survive MA 16A, now we can give them the hard stuff.
It also looks like a course put together by people who have not yet admitted that Calculators can do more than add/subtract/multiply/ and divide. What they really need to do is hand the course over to the Applied Mathematicians.
Yeah, I think you’re right about 16B being the catch-all for stuff that didn’t fit in 16A. I’d speculate that it may also have been designed to meet the disjoint needs of two separate departments. (“Pre-med wants to cover these 7 topics? Pre-business wants to cover these 8 other topics? Great, we’ll throw ’em all together and call it a class.”)
Sounds like most elementary classrooms. “Put your books away. Okay, get out your English books now.” Context and relationships help form understandings. Keep working at it. Start small. If you approach instruction using concepts and connections in 16a, you’ll eventually get there in 16b.
I taught a 9th-grade classroom on that model once: Geometry/Earth Science/English back-to-back-to-back, same classroom, same teacher (me). Transitions were, as you say, a little tricky.
Sounds like the college version of the spiraling Everyday Math curriculum our elementary schools use.
Yeah, I hadn’t thought of that analogy but I think you’re right.
It’s funny, despite the bad reputation in most circles, my experiences at an Everyday Math school were quite good, although I think our teachers tended to supplement the curriculum with their own materials.
As a parent and math club coach, my main beefs with ED (at least the way it’s implemented in our district) is that it doesn’t have a lot of meat for the advanced kids, who are able to just race through the lessons in order to read, or draw, or do whatever else they’d rather be doing instead of lattice multiplication, while at the same time it moves too quickly for the kids who need more time to understand the concepts. So I’m seeing super smart middle school math club kids who rely on calculators to do fractions, because they never learned all the cool ways to manipulate them, and I hear about kids in the high school Algebra II class who never properly learned how to do long division.
In my calculus class, I got ODEs in one week immediately after Taylor series. The first class went something like this:
Ha, what a mess. As Steven Strogatz has pointed out, the interesting thing about ODEs is that most of them CAN’T be solved.
Sometimes a little compression is necessary, but it’s important always to be honest and what you’re compressing, and that you’re compressing it.
Well, yes, to a mathematician. But most people who take calculus classes are not mathematicians, and what they need is “tricks for solving ODEs they encounter” rather than “all about ODEs”.
It’s got to be the product of a committee.
I’d imagine. Apparently it’s been around for a while.
I taught 16B for a couple of summers in grad school, and with the right attitude, it’s got a lot going for it. I presented it as “a bunch of neat stuff you should hear about before quitting math classes,” and I think that my students enjoyed it (for the most part: I got great evaluations but one scathing RMP review) and that they left a little more comfortable with math than when they arrived. It’s not meant to be a conceptually deep class; it’s just a bunch of neat stuff crammed together so that they don’t get scared if they encounter any of it in the wild. They don’t leave understanding it deeply, but when the course is well taught, the students become familiar with the material, which is a bit further and more useful than “exposed.” Furthermore, the grab bag nature of the class means that difficulty in one area doesn’t necessarily translate into struggles in others, so it’s easier to keep the students from getting frustrated – so they’re less likely to leave what is likely their last math course feeling bad about math.
It’s not the course that I would design to follow 16A, but I can think of worse!
Thanks for the response, Meghan! That’s probably the best defense I’ve ever heard for the course. (I asked around among friends in the department and never got anything more supportive of the class than a shrug.)
I think your characterization of the class–a field guide of some various neat techniques you may encounter “in the wild”–is nicely honest and positive, and if you can sell that narrative to the students, I can see it helping them make sense of the jumps from topic to topic.
I wonder also if the accelerated 8-week summer schedule might, paradoxically, work in 16B’s favor. It would go from overstuffed to comically overstuffed, to the point where kids can’t help but see it as a survey course, a grab-bag toolkit, a sort of friendly math boot camp.
Anyway, I still think it’d be quite possible to design a course that has the benefits you describe (breadth, not-too-cumulative sequence, interesting ideas) without the incoherence and sloppiness of the 16B syllabus. But I appreciate your speaking up in its defense!
Sounds like a quick and dirty way to take more tuition in return for nothing
Well, Berkeley has a much more efficient way to accomplish that goal: Keep raising tuition! (To be fair, cuts in the state education budget have kind of forced their hand on that front.)
They can claim budget cuts all they want, but if you look closely you’ll see plenty of money. We pay between five hundred to one thousand dollars every term in purely extraneous, unnecessary fees here at U if Iowa.
I’ll leave off there because this is not the place for a discussion about university politics and the screwing of students. There’s enough to worry about on here with all this crazy math!
I liked this post though for what it said about the disconnected way math is taught in some schools. My elementary math education was a lot like this. They didn’t want you connect it all, just plug and chug into the formulas, pass, move on.
Yeah, 16B sounds like a disaster, but I think there might be ways to make it fun and interesting without having to wait for the university to revisit the curriculum.
For example, what if you came up with an interesting problem that required several topics to solve? If it could be some sort of real-world-ish problem, it would help ground the topic spasm a bit.
Obviously, you’re not going to find a single problem for all the topics, so maybe you could break the course into sections, and come up with a problem that would link two or three of the topics at a time. Basically, you’re grouping disparate topics into little narratives.
In some ways, 16B makes this approach easier. If your entire course were just trig (for example) the continuity is inherently there, but putting a narrative to a single topic is a lot harder. It’s that crazy zig-zagging that makes the potential for interesting mini-stories a little better.
Yeah, I’d be very interested in a more problem- or case-study-driven approach to 16B. So far as I know none of the professors has tried that – and it may be hard to find good problems that employ the techniques taught without exceeding the computational skills of the students – but it’s an interesting approach.
I’m late to the party, but that course sounds frustrating indeed. Coming from a liberal arts perspective, I prefer my students to develop at least some appreciation for mathematics as a field of study rather than a mere toolkit for other subjects, and so I don’t like to design courses based only on “what they need”. That being said, every course regardless of content should strive to have a narrative. Our service courses don’t compete with their counterparts in other disciplines when it comes to telling a story.
Yeah, I agree – it’s best for students to appreciate both math’s applied side (the toolkit) and its pure side (the intrinsic beauty/structure). Focusing on the toolkit aspect is sometimes necessary, but it does often leave those service courses a little narrative-less.