Thoughts on Changing Calculus
This post started as a comment on Nick Hamblet’s blog post about changing calculus. When I realized it was incomplete and almost 700 words long, I decided that I should probably put my thoughts up here instead and link to Nick’s post.
I have to agree with Robert Talbert’s comment that most of this dialog is taking place at liberal arts colleges. This is great for liberal arts colleges and their students. Unfortunately, the behemoths that educate the masses really need to get on board and realize that the way we’ve been doing it for generations is not necessarily the best way any more. I have seen some signs at Georgia Tech that it’s possible things may start to change at the top-tier research universities in the future. The Harvard physicist Eric Mazur (leading proponent of peer instruction and clickers in science courses) is a great example of someone who asked these same questions about his physics classes a while back (and did something about it!) and is also widely-respected as a scientist. We need more people like him in every discipline to create change at the Georgia Techs and UVAs of the world. Another problem with using liberal arts colleges as the proving ground for new instructional methods is that faculty at Big State U almost always say “That’s great in your class of 25. I have 250 and just can’t imagine it working in my class.” Well, the specific things being used might not work, but there are things that can be effective in that context, too.
What are my thoughts on what calculus should be and how we should accomplish this? I’ll admit their not fully-formed, but they’re getting further along as time passes. I’ll do my best at outlining them, but I should point out that I’m writing this at 34,000 feet on four hours of not-so-great sleep.
Foremost, every course needs designated learning outcomes. These should be broad and determined by consensus from within the department, the professional community, and the “consumer departments” that require our courses for their majors. For Calculus I or II, I don’t think “Be able to integrate a function using partial fractions.” is a good learning outcome. Unfortunately, this is the most common type of learning outcome written now to satisfy accreditation bodies. I’ve seen Georgia Tech’s ABET outcomes for the lower-division math courses, and they’re sadly almost all like this. Instead, learning outcomes should lay out how we expect our students to think differently when they leave the course. What are the concepts they should know about? For calculus, I’m much more concerned about students understanding the derivative as a rate of change and how that connects to the integral and area. Instead of caring if students can graph functions, we should care about if they understand what the derivative tells us about the graph of a function. I bet if you asked most mathematics professors how they want their students to think differently after their calculus course, you’d get similar answers. Unfortunately, too many people don’t understand that our teaching and learning activities, feedback and assessment mechanisms, and learning outcomes all need to be aligned. It does no good to want your students to be able to do these things and then just teach them a bunch of algebra tricks and test them on meaningless manipulation of symbols.
Learning outcomes at every level (from remedial courses to graduate courses) need to include communication (reading, writing, and oral) in the discipline. We cannot teach our students everything, so they will have to learn by reading in the future. This is easier if we help them when they’re getting started. If we genuinely care about concepts, they need to communicate their understanding in a meaningful way, often in the context of an application. We need to teach our students to critically evaluate their work to determine if it makes sense. How many times have we seen students compute a volume using calculus techniques and give a negative number as an answer without batting an eye? In evaluating my precalculus students’ work at the end of this semester, I realized that some of them still don’t have BS detectors when reviewing others’ (or their own) work. Students need to be able to take a result that they get (by hand or from a computer) and determine if it makes sense in the context of the problem. (In case of my class, when answering a question that asks for the total amount of interest paid over the life of a loan, a percentage is usually not the right thing to give.)
Once you’ve got your learning outcomes laid out, let people teach the course how they want to (even graduate students). Those of us who think that students should not be glorified calculators should be allowed to focus on concepts, writing, and communicating. The old guard who like meaningless manipulation of symbols can go ahead and do that. They’ll retire soon enough, so it’s just not worth our time to change them (although it may be worth the effort to keep them away from the froshlings). Common exams need to be eliminated, as I’ve only seen them leave everyone unhappy. (The only exception being when a course “czar” writes the exam. Then he/she is happy and everyone else is unhappy.) With learning outcomes and good assessment schemes, you just hold every instructor accountable for providing evidence that students met the learning goals. This may be through questions from the final exam, mastery exams, projects, papers, whatever the instructor believes documents it (and is generally considered valid assessment in the professional community).
A potential pitfall here is the issue of sequence courses. At a liberal arts college, this is not so much of a problem, I imagine. It’s likely that if you took Calc I from Prof. X, then Prof. X will also be teaching Calc II the next semester. At Big State U, this is less likely (and almost unheard of at science/engineering school like Georgia Tech). A student who comes out of a concept-driven calculus course will struggle immensely in the sequel if his/her instructor wants fast-paced symbol manipulation. I don’t think there’s a great solution to this problem. However, Duke has come close. They’ve instituted two calculus sequences. One is traditional (no computers, symbol manipulation, etc.) and taught primarily by senior, tenured faculty. The second is reform (lots of calculators, computers, labs, but concept-heavy) and taught primarily by non-tenure-track (but full-time and under long-ish term renewable contracts) and graduate students. They help the advisors from the various majors understand which sequence is better for which types of students. I think the idea is that as the old guard retires, the traditional course will go by the wayside. (I should note here that I’m not a big proponent of the reform movement. I think they’ve done a lot of things wrong politically and caused an artificial schism that makes young mathematicians feel like they need to choose a “side”.)
Yes, this is all pie in the sky at the moment. We’re probably decades away from this, and it may not work at all. However, I think we’re failing our students if we don’t try. This is also the heart of my career conundrum. I’ve been educated solely at medium-large public research universities and know them well. Apparently there is some belief that I can make a career of working at one (at least that’s what I’m inferring from the fellowship I just got). However, I question whether I can do the research I enjoy and need to do in order to get tenure at such an institution while also trying to create change in how we teach mathematics, even if it’s just in my own courses. Where does the time come from for the professional development required to keep current on these matters? Is there a reward other than a pat on the back (at a research university) for really doing something about undergraduate education? One of the reasons I feel like I should take that career path is to influence the future generations of mathematicians. No one in the Georgia Tech School of Mathematics is seriously evaluating how we teach introductory courses other than graduate students. We were the beneficiaries of a course from our Center for the Enhancement of Teaching and Learning that made us reevaluate our views. We’ve all changed how we teach, although the degree varies, because we’re fortunate to teach in a unit that gives us a lot of latitude. Many mathematics graduate students, however, seem to just be exposed to the status quo when it comes to teaching. Others learn about different approaches but are discouraged from pursuing them because of the time it takes the first time around.
Again, apologies for any incoherence. This is pushing 1500 words, so I should probably edit it. However, that’s no fun on an airplane, so I’ll leave it as is.
December 13th, 2009 at 1218
I don’t know about common exams being eliminated. We used them at Yale when I was a graduate student, and everyone was pretty happy. I enjoyed their system, actually.
First off, we had a bunch of small sections, rather than huge lectures (yes, if your school is dead-set on huge lectures this won’t work, but I don’t think huge lectures promote comprehension in the first place). A couple were taught by a tenured professor, more by postdocs, and the rest by graduate students (one section each). The professor set the tone and the pace, and we all worked to match him in content, though not in style. We wrote, administered, and graded exams as a group to further even the playing field. We found that if we didn’t have common exams within the small-section structure, students would raise all sorts of complaints when they got bad grades, blaming it on having one instructor over another.
The biggest benefit of having a bunch of closely-coordinated sections was in teacher training. My first semester teaching multivariable calculus, I had been in front of classes before as a T.A. (for linear algebra level courses) but I really didn’t have any hands-on experience with lesson planning or exam writing. I got to watch the professor, postdocs, and more experienced graduate students, and learn from them as I went. At first, I could just copy what I saw, but I could develop my own style as I felt more comfortable. When I moved to a different course (Calculus I), I could take some time to learn how we did things there as I got up to speed.
Contrast this with my first time teaching College Algebra as an assistant professor, where I was thrown the book a week before the class began, had to come up with syllabi and lesson plans from scratch, and all the other instructors guarded their own classes jealously from any outside observation or interference.
December 13th, 2009 at 1431
You raise some really interesting points here. I’m impressed with how well-informed and articulate you are about these matters, as a graduate student. That CETL course must have been a good one!
The point you made about common exams not working caught my attention. I’ve been a part of multi-section calculus courses where they’ve worked pretty well, although we took a different approach than the one you’ve outlined here. Instead of setting a few general course objectives and letting instructors teach the course largely as they wished, we set more detailed objectives and distributed suggested lesson plans to all the instructors. Instructors were welcome to modify or ignore these lesson plans, but since the common exams were aligned with these plans, at the very least they gave the instructors a clear sense of what was to be assessed so that they could align their lessons appropriately. Many opted to use the lesson plans as they were designed, which introduced these instructors (many with relatively little teaching experience) to methods of instruction designed to foster active learning.
You’ve identified the key problem with the “hands off” approach you outline–the sequential nature of calculus. If a student focuses on computations for Calc I but takes Calc II with an instructor focused on concepts and applications, the student is likely to have a rough time. The approach I’ve described largely avoided this given the shared understanding among instructors of learning objectives.
I think you’re trying to have it both ways: You want clear learning objectives for the calculus courses offered by a department, but you also want to give instructors freedom to teach calculus however they like. I don’t think you can have it both ways, and I come down on the side of clear learning objectives. And not only for individual calculus courses, but for the whole sequence. That, too, helps to avoid the problem with having different instructors for Calc I and Calc II.
Thanks for putting these ideas out there!
December 13th, 2009 at 2156
Thanks for the shout-out re: my comment over at Nick’s blog. I’ve been teaching calculus for 15 years at all kinds of institutions, and even at the ones where undergraduate education is done well (namely, Vanderbilt — a tip of the hat to Derek Bruff @1 there), getting calculus out of the mode of having symbolic manipulation at center stage is highly problematic. (Or at least it was that way when I was a grad student there from 1992-1997.)
Perhaps the x-factor in getting big universities to shift to the approach to calculus that you and Nick will ultimately come from technology. The arrival and slow but steady adoption of Wolfram|Alpha by students will, I think, eventually force universities of all sizes to come to grips with the axiom that calculus = symbol manipulation. A department simply cannot go on under this kind of assumption when there is free software out there, accessible to anybody with access to the internet, that will do nearly all the symbol manipulation you could want in calculus. Departments will have to either move symbol manipulation off center stage and change their ideas about calculus, or else start instituting draconian, locked-down assessments in their calculus courses with a view towards barring all such technologies from entering. The latter simply cannot be done indefinitely; the genie is out of the bottle.
I think it will also take accrediting bodies from client disciplines like ABET in engineering or from the medical sciences telling mathematics departments, as they seem to be doing quite vocally these days, that symbol manipulation is not what makes engineers, doctors, etc. effective, but rather problem solving, communication, critical thinking, and the kinds of things you’re mentioning. One of the big medical school associations recently released a report about the quantitative preparation of medical students and basically said calculus is good only insofar as students learn about dynamical systems; and that statistics was a lot more beneficial. One gets the sense that these outside professional bodies are getting tired of calculus courses training students to be really good at skills (such as arcane series tests or integration techniques) that in the end simply do not correlate with success or quality in those professions.
December 18th, 2009 at 1430
Thanks for chiming in. Sorry it’s taken me so long to get back with more comments, but I’m back in Fargo for the first time in several months, so I’ve had lots of people to see.
It seems that there’s a much bigger institutional factor to the common exams issue than I’ve experienced. Seems that Derek and John have been places with functional systems in this area. My most current perspective comes from an institution where we could never even consider them. In any given semester, we’re running 40 recitation sections of Calc I or Calc II. These sections (35 to 40 students each) are grouped together in lectures that combine anywhere from two to six recitation sections. The lectures are given by people ranging in experience from third year doctoral students to full professors with about seven or eight individuals having this role for a single course in a single term. If you had only one faculty member involved and the rest of the instructors were postdocs, instructors, or graduate students, I could see common exams being effective. However, GT prides itself on having faculty members from every rank take their turn in freshman courses. (This includes the folks with the biggest grants.) It’s hard to get most of these people to agree on much, especially what an exam should look like.
Derek makes a really good point about my post wanting things both ways. I agree it’s almost surely an unworkable system, and preference should be given wherever possible to clear learning objectives (specific, but not too specific I would say… they should be able to adapt to new technologies without needing any real revisions) that match up from one course in a sequence to another. I’d be interested to hear thoughts on what to do about such things at mega-research universities that can’t drop large lecture (we couldn’t afford it at GT). It seems that one option might be to stick to one tenure-track faculty member involved with each calculus course per term to oversee graduate students, instructors, and postdocs who teach the lectures.
I also have to say that I think Robert’s hitting the nail on the head regarding outside accrediting bodies being what will have to drive the change. I wish ABET would say something more meaningful about what students should learn in mathematics courses. I went looking when I took my CETL course last spring, and found only vague statements about mathematics in the ABET criteria. Maybe they’ve made statements that aren’t part of their criteria yet? If math departments don’t provide the education that constituent departments want, I think we’re going to lose a lot of our student base. Those departments will start teaching their own math classes to replace our classes.
February 4th, 2010 at 1830
“However, I question whether I can do the research I enjoy and need to do in order to get tenure at such an institution while also trying to create change in how we teach mathematics, even if it’s just in my own courses.”
It is a very good thing that you are asking this question now, before graduating. I not only asked that question, but I tried to move mountains by myself. I failed. Horribly.
You have a better pedigree than I, so you should end up landing a job in a better school. But in case you don’t, please don’t make the mistake of becoming what I’ve become.