Here’s a problem I have with the way most calculus textbooks are written, and therefore by default the way most calculus courses end up being taught. Tell me if I am crazy or missing something.

We teach calculus from a depth-first viewpoint. That means that whenever we encounter a concept, we go as deeply as possible in that concept before moving on to the next one. There are some subjects where this makes sense, but in calculus we have a small number of main ideas that are made out of several concepts, and if we stop to attain maximal depth on every single thing, there’s a good chance that we never arrive at the main idea with any degree of understanding.

The big ideas of calculus — the rate of change (derivative) and accumulated change (integral) — are actually really simple if you consider them simply for what they are and what they were invented to do. Derivatives, for instance: You have a function, and it is changing in all kinds of ill-behaved ways. The object is to find out exactly how quickly it is changing at a given point. We quantify that rate of change by sticking a tangent line on the graph of the function at that point and measuring its slope. Really, that’s it. Slopes of lines. The rest are technical details on how to calculate this slope with some degree of accuracy, and those details range from graphical estimation to interpolation tricks to algebraic techniques.

But in Stewart’s Calculus book, the coin of the realm of calculus texts, here’s what students have to study before the derivative is defined: an entire chapter of precalculus review (a mind-numbing section 1.1 on functions and notation, mathematical models, families of functions, exponential functions, inverse functions and logarithms), then a chapter on limits in which students have to master finding limits from graphs, calculating limits using the Limit Laws, the epsilon-delta definition of a limit (mostly untaught these days), continuity, and limits at infinity.

*Then* there’s a section on “Tangents, Velocities, and Other Rates of Change” followed by two sections on the Derivative.*

This approach plays directly in to the greatest weakness of the average calculus student, which is algebra/precalculus content mastery and the ability to master technical details of calculations and theory. How likely is it, for the student who struggles to read mathematics or use algebra correctly, that this student will be in any shape to learn what a derivative is, and what one is for, by the time they get there?

You want students to master those technical calculations and theory, of course. But you also want those to be mastered in context, not just as mathematical tricks to be learned as parlor games. The few students who survive the onslaught of detail mastery and are still psychologically around to learn what a derivative is, often find it extremely hard to know what f'(3) = 2 actually means. All they know is that you bring the power down and subtract one, and maybe the Product Rule.

I’d prefer some kind of approach to calculus that is not depth-first but more like breadth-first, where students get a good grounding in the overall ideas of calculus and do some basic work before mining into the really deep details. Not all students really need those deep details, after all.

* OK, there is a section (2.1) where the ideas of tangent lines and velocities are briefly introduced. And then summarily ignored until the end of that chapter. The students typically ignore that material right along with the book.

I didn’t use Stewart for single-variable calculus, but I did use it for multivariable and liked it a lot. But my single-variable calc textbook (Demana-Waits-Kennedy-Finney) used a similar arrangement of material. My teacher simply skipped the precalc stuff and then briefly introduced both the concept of the derivative and of the integral in the first week of class. I suppose the idea behind teaching limits before derivatives is to allow the derivative to be introduced and defined as a limit, but I think it was helpful to have a roadmap of the course from the very start.

(We did have to learn the epsilon-delta definition.)

From the perspective of people who understand calculus, your descriptions make sense. The problem is that to go beyond the verbal, high-level, somewhat fuzzy descriptions in English, we have to dive in to some actual figuring. The level of technical skill that students have directly determines what kind of problems they will be able to solve. And it is through solving problems that a true understanding develops.

I hear what you’re saying: If we give students some context and an overview, perhaps their motivation and the confidence will improve, and that will give us better results. But that would entail making a couple of passes over the same theoretical material, at different depths of mathematical skill. So if a student’s understanding of, say, square roots or logarithms or fractional exponents is weak, then at some point that’s going to catch up with you. That might actually be the best approach!

To teach that way requires a significant refactoring of the calculus material based upon required prior math knowledge. I suppose that’s the point of your post. (I’m a bit slow, but I’m with you now.) It’s much easier to say to students, “Here’s a bunch of prerequisites you need,” and then proceed directly through the content. But just because it’s easier doesn’t mean it’s good teaching. It might be nice to divide up the algebra and trigonometry review and sprinkle it throughout the course. The only problem I can foresee is that by doing so, you limit the scope of possible exercises at the start of the course significantly, and I worry about students solving by rote. But hey, it’s certainly worth a shot! Let us know when your book comes out. š

@Ben Chun:Re: the last half of your last paragraph, I think that you might have to do different kinds of problems — e.g. problems where the derivative gets calculated by physically drawing the tangent line and calculate the slope using rise/run rather than some kind of Quotient/Product/Chain rule business — but I don’t see that so much as a limitation as it is just dealing with a particular kind of problem at one point in the course and another kind of problem at another point in the course.

There is a precedent for this approach in the various “Applied Calculus” courses one finds in some schools. I’ve taught a class like that before, and there was very little algebra in it, but lots of calculus, done mostly at the level of visual and numerical computation. And I think it was a really good course for the clientele it served (business and life science majors, most of whom didn’t have the greatest algebra backgrounds and forget about trig).

Which of the Stewart books are you using? (there are several)

We use Concepts and Contexts and I have been really pleased with our experience. I taught out of the Hallet-Hughes book at a previous school, so I did have lots of experience teaching from a “reform” text before taking a step back towards a more traditional/reform approach. In the Concepts and Contexts text there is an emphasis on understanding the concept and the detailed work always comes later. It does not go into epsilon-delta proofs. It sounds like you might be using the Stewart 6 Calc Early Trancendentals? If so, try the CC book – it might be just what you’re looking for.

@Maria: Stewart Calculus is starting to become like Microsoft Vista these days, isn’t it? Stewart Calculus Home Edition, Stewart Calculus Ultimate Edition…

Seriously, though, we are using the foreshortened version of the basic Early Transcendentals book. (It’s just chapters 1-6.) We farm out the epsilon-delta proofs as part of a special one-hour activity course for people going on to calculus II, so the business, etc. majors never see it.

I used C:C&C when it first came out back in 1997 in my very first teaching gig. If you ask me, there’s hardly any difference between C:C&C and Early Transcendentals. In fact large swaths of the exercise sets are identical in both books.

I wrote a post about the various Stewart incarnations last year.

I teach calculus out of Stewart, and for Calculus I, I don’t spend more than one and a half days on Chapter 1 (usually for the last 20 minutes of the second day of class, I give them a quiz over pre-calculus stuff). Other than a *very* brief overview of some of the concepts, I tell them that I *will* assume that they have a solid grasp of the material in Chapter 1, and that if they don’t, then it’s not disastrous, but they have a bit more catching up to do (followed by a reminder that they’re welcome to drop by during office hours, etc). The only exception to this are the sections on inverse functions, exponentials, and logarithms. I take about a day and a half to cover those just before covering the derviatives of exponentials and logs, later on in the course.

I was taught from a book by a guy named Thomas. I think they stopped making it since there were no pictures. /sarc.

There have to be books out there that do what you want. My step-father was trying to get me to review forever some tapes that he thought really did a nice job without getting bogged down in manipulation (I’ll call and ask the name, if I remember). And bigger schools offer several flavors of the course, including one for humanities folk.

My shelf is littered with books, but they reflect different eras, and all more or less follow the same outline. In fact, a “light” text from 45 years ago has much much more algebra and precalc up front, even as it advertises itself as the easy version… I also bumped into Marsden… one day I will open it.

My son’s high school teaches AP Calculus together with AP Physics. I think that’s brilliant — you can have your theory together with your application. When I was in high school, our physics did not rely on or reference Calculus. I took Physics the year before I took Calculus (which was a common pattern for the accelerated math kids, believe it or not) and so never really noticed the overlap. Only later when someone pointed out: look, the acceleration formula (which was ugly to just memorize with no context) is just the derivative of the velocity formula did I “get it”. I did get the big picture that the derivative was the rate of change, but I didn’t get what it might be used FOR. Even as I went on in a Math degree, Calculus was just symbol manipulation, for the most part. I’m hoping that when my son gets to AP Calc/Physics, the combination is helpful in letting the kids get the big picture much more than we ever did.

mm,

I had them at the same time, but separately. In physics we would have rough and ready explanations of why stuff worked, and then boom, let’s grab these formulae or principles and apply them. The calculus course was a college course masquerading as AB. The combination, the way the theory and application complemented each other, was wonderful.

Jonathan

Robert,

We’re teaching our “average” high school juniors to find the slope of a function at a point. To them it’s just another thing they’re doing with functions. They seem to have a pretty good understanding of slope as a rate of change. It’s kinda cool.

Have you looked at the Hughes-Hallett (“reformed”) texts? They de-emphasize the algebraic manipulations (somewhat) and do suggest links with other topics before the other topics are covered in depth. Also, lots of problems without a given formula (or even a formula to “find” as in traditional calculus story problems)– i.e. just a graph or a table of data or a descriptive story given. Many teachers don’t like the book but I thought it was very good for anyone except, say, college students in physical sciences, math, and engineering who need a lot of drill and coverage of all the symbolic methods.

@Ned Rosen: I’ve use the Hughes-Hallett multivariable calculus book for Calculus III before and I liked it a lot, and I’ve mined the exercise sets from the single-variable book for problems before. I can’t say that I’ve scrutinized the single-variable text apart from that.

@Jackie: Exactly what I had in mind for the first pass through calculus.

Regarding the Calculus and Physics Simultaneously comments, it’s a neat idea, but it wouldn’t work for us since we have no physics major and only a fledgling engineering program — most of our people are business, finance, etc. types.

I cannot believe you people can say this with a straight face — that calculus is taught depth first. Not even Harvard super-honors calculus for math majors is taught depth-first. The whole point of college freshman calculus is the diametric opposite and the antithesis of depth first. If you taught calculus depth first you would at a minimum teach something like the construction of the real numbers from the rational numbers. (In fact, you would probably have to teach the construction of the numbers in the first place since you wouldn’t be able to assume they get it from some place else.) You would then proceed on to a chapter on point-set topology, discussing metric spaces in some detail. Then, you wouldn’t skip straight on off to differentiation, but rather cover infinite sequences first and expand that treatment into infinite series. Then you would do limits and continuity of functions, and only then would you finally get to differentiation. You probably wouldn’t get to differentiation in the first semester. At that rate, with freshman, you probably wouldn’t get there in the first year. You may not get to integration until the third year. That’s a depth first approach to calculus.

And, I’m not talking about any additional level of generality in the material covered, here, either. You are literally covering the very same material that is covered in a normal calculus class plus only the material required for depth. Calculus is and has always been about trying to short cut all that material that it takes to really deeply understand the subject so that we can get some kind of “conceptual” understanding of instantaneous rates of change and so on — mostly so students can rush off and do a bunch of science with it (like physics, especially for engineering majors).

@Adrian: I believe the point is that we are sacrificing conceptual understanding of calculus for depth of mastery in technical sub-topics whose connection to calculus as a whole is not apparent to most calculus students. Of course, nobody goes COMPLETELY depth-first, not even in an honors calculus class.

Whether this sacrifice is the right one to make in intro calculus is what this discussion is all about.

This isn’t just a case of not going *completely* depth first. This is a case of not even remotely approaching the actual content of the subject matter. You’re saying that epsilon-delta proofs are the “technical details”. They ARE the concepts. I totally agree that most college freshman aren’t equipped for the Full Monty, but let’s not act like we aren’t completely bastardizing the material to teach it to them anyway.

What you are advocating is to approach the material even more heuristically than it already is. You are acting like the heuristics are “the concepts” as if they are the real reasons why these theorems are true or what this subject is “really” all about, anyway, or something like that. At best, this is a “harmlessly” fallacious way to teach the material. And, we are happy teaching it this way so long as students don’t go on to make too many mistakes in practice. I’m not disputing that this is “the way things are” to some extent, but let’s not act like what we are doing is anything different than what it actually is.

The whole “concept” behind calculus IS the limit. The fact that we don’t teach that well is precisely us trying to just “get through it” so that we can cover more topics that have imminent bearing on business, science and engineering. That is a breadth first approach not a depth first approach. And, as is no doubt evident form my comments, I certainly don’t agree with it — we should either give them their degrees without asking them to do it or we should insist that they be able to do it correctly. We should not try to pretend they are doing something they aren’t.

@Adrian: Nowhere am I advocating completely throwing out the subtopics of calculus which, like epsilon-delta limit arguments, are in the upper echelons of technicality. I am just saying that it’s silly to require students to master these before they ever see what a derivative is. Those epsilon-delta proofs exist, from the standpoint of calculus at least, for the sole purpose of making the idea of the derivative precise. How can we expect students to care about epsilon-delta proofs if they have no idea what they are for? How can we expect students to be motivated enough to stick with a difficult concept if they don’t see the big picture?

You can call that harmlessly fallacious if you want, but I’m telling you that my students, at least, need to see where they are going before they will ever commit to going there.

@Adrian: While I can understand your point of view, I have to take issue with your claim that epsilon-deltas ARE the concepts. This is precisely historically backwards. There were 200 years between Newton’s and Leibniz’s development of calculus and the development of the concept of a limit by Bolzano, Cauchy, and Weierstrass in the 1800s. Many people were able to develop important ideas using calculus without limits, let alone epsilons. While I agree that it was important to put calculus on a firm theoretical basis, there’s no need to know all of those details in order to use the results (although, I agree, it sometimes helps).

No — the epsilons and deltas are the concepts. People have all sorts of intuition about euclidean geometry, too. And, people speculated all sorts of things long before any of it was rigorously developed. That doesn’t mean that their fuzzy, poorly thought out heuristics are “the concepts”. They are just imperfect attempts at the precise and fully explicated concepts of the rigorous development. The reason Weierstrauss’ epsilons and deltas even do their job at all is because they completely capture what everyone had been (poorly) talking about up until that point.

For instance, your child might be thinking of some animal. It’s large, gray, has floppy ears, and a prehensile trunk. Those aren’t “the concept” — the animal she is thinking about. She’s thinking about an elephant — that’s “the concept”. What she is doing is just giving characteristics of the thing that she is thinking of without knowing exactly what it is. The “concept” isn’t just her poorly understood representation of it. It is that which the representation implicitly refers to — what she is trying to get at.

There are basically two kinds of ways the epsilon-delta derivative can end up not being the concept. One way is for it to not really capture what people were talking about. Perhaps people were originally conceiving of a function being able to have multiple limits or something like that. Then, the epsilon-delta version is not going to lead to that and it uses some other notion of the limit, instead. Had something like that happened, then we probably wouldn’t be talking about it right now. But, it can happen — it could be a more productive concept than what people were originally talking about. At any rate, that didn’t really happen like that and, in any case, we want to teach these more productive concepts, anyway.

A more meaningful way something can end up not being the concept is with something like the Radon-Nikodym derivative. or the measure theoretic version of calculus, in general. Those aren’t “the concepts”, either, because of their level of generality. They are a generalization of the concept. They are, in fact, better, more general tools for doing all the same things you would use regular classical real analysis for. But, if we were teaching that sort of thing, then I would see an argument that does go like “Hey, we need to just teach the concept so they know what they are doing first.”

But, what we are really talking about, here, is just imperfectly teaching the concepts, not teaching the concepts well at the expense of some power in our methods.

@Adrian: You seem to be ignoring the fact that calculus was used quite productively for a couple hundred years before anyway had thought of epsilons and deltas. Perhaps you’re just using the word “concept” differently from me, but to me that makes it clear that calculus doesn’t absolutely require the concept of “limit” in all its technical glory to productively use the concepts of “derivative” or “integral”. It would be next to impossible to be a professional mathematician without a decent familiarity with deltas, but the vast majority of people in calculus classes aren’t intending to be professional mathematicians. I’ve known many a successful scientist who uses calculus regularly but doesn’t care one whit about epsilons, and there’s nothing wrong with that.

That you can use some vague, poorly thought out ideas to come up with the right answer 80% of the time doesn’t even conflict with the fact that the concept isn’t your vague, poorly thought out idea, but the clear, precise one you are trying to get at. Coming up with some sort of weak rationalization for something that looks like it might be true does not constitute knowing what you are talking about. Without rigorous calculus, people were just wrong a lot of the time — too much of the time — and that’s why they had to fix it. No, it’s not some technicality. Yes, limits are the foundation of calculus. And, yes, to actually know what you are talking about, you have to do the epsilons and deltas.

Without limits, it’s not calculus, and without rigor, it’s nothing more than sophistry. What do you think sophistry is? Where someone makes some outlandish assertion and then proceeds to try to defend it? That’s not sophistry — that’s absurdity. Sophistry is where someone makes an altogether plausible assertion and tries to defend it with a persuasive, seemingly quite reasonable argument that is, nevertheless, fallacious. If you don’t use epsilons and deltas (i.e. do it heuristically with less rigor), then that is precisely what you are doing — giving persuasive, seemingly quite reasonable arguments that are, nevertheless, fallacious. And, the reason sophistry has such a bad reputation is because it is fake knowledge that, though not always immediately, does ultimately lead to actual false beliefs. No matter how much you think it “works”, it doesn’t.

@Adrian: Rigor makes for good mathematics, but it can also lead to absolutely terrible pedagogy. If you are trying to say that epsilon-delta proofs are in some sense what calculus is really all about, then you may be right from a mathematician’s point of view — or specifically an analyst’s point of view — but if you tried to teach a freshman one-size-fits-all calculus course centered around this idea, your course will simply crash and burn, no matter how pure or canonical your rigor may be. You may maintain the mathematical purity of the subject, but at the expense of most of the students who are trying to learn it. Pedagogically this is a deal that I, for one, cannot accept.

It is like saying that the main concept of algebra — high school algebra — is the list of axioms for fields. It is absolutely true that the axioms for fields are a rigorous, foolproof basis for what we do in algebra. Without a firm theoretical foundation, we can only pretend that algebra as we know it really works. Without a firm foundation for field theory, people were wrong a lot of the time about algebra. And to really “know what you are talking about” when it comes to algebra, you need to know what works for fields (and rings). But: Do you really want to teach 8th grade algebra that way?

Or it’s like saying that I should teach my 2-year old daughter about verb conjugations and active/passive voice before I teach her how to say her full name or ask for a cup of milk. Without a proper understanding of syntax and grammar, we cannot say that we truly understand language. But I am not trying to teach her to “understand language”; at least not yet. I’m just trying to get her to *talk*.

The question, at least the one I am posing, isn’t whether or not to drop the rigor in calculus. Rather, I question the pedagogical inclinations of textbooks to direct students to develop content mastery in subtopics of subtopics of subtopics while failing to develop any sort of thematic idea of what the subject is used for in the first place. Get the students in the game, show them what use calculus has, and then you will have some basis for introducing rigor. And then you should certainly do so. But all in due time!

The Thomas book is Thomas and Finney (http://www.amazon.com/Thomas-Calculus-Alternate-9th-George/dp/0321193636/ref=sr_1_12?ie=UTF8&s=books&qid=1202375826&sr=8-12) which many many years ago was a reasonable text for HS calc. It probably still is.

As far as students getting concepts like the derivative, I think it matters how you explain it. To say it is just the slope may mislead students who are familiar with equations like y=mx +b and understand the slope to be a single value, m. I think you need to make your students see that the slope is not a value, but a function. In this case one with a very small range. Explaining exp(x) and it’s derivative in words, can help illustrate this.

It’s not clear if you are telling your students that the functions they are taking derivatives of are “changing in all kinds of ill-behaved ways,” but you should make clear that both the equations and the derivatives are not ill-behaved. For the domain on which you are differentiating, you can alway draw a line between two points on the curve and you know that somewhere along that curve, the slope is identical to the slope of the line you drew. That’s pretty well behaved.

Make your students who are weak in algebra do algebra drills. Add it to every homework assignment until it is second nature. If they enter your class not knowing algebra, they may leave not knowing calculus well, but at least they will know algebra.

“Make your [calculus] students who are weak in algebra do algebra drills.”

This is college that we’re talking about, not high school. Students who are weak in algebra and have somehow made it to calculus need to relearn (or learn) algebra on their own time.

Right on, but it’s not going to change. The dumbing down of America is the goal of the powers that be. They want a task oriented workforce not a forward thinking society.

Great post, I agree with you 100%. I’ve noticed there are two kinds of teachers / text-book writers in this world.

The first kind has a passion for teaching. In presenting information either orally or written, their first and most important goal is that this information is understood by the student.

The second kind, usually found more in, say, college professors than in high-school teachers, presents their information to showcase their knowledge and to showcase their elegant mathematical rigor and succinctness.

I’m someone who has a passion for math. I’ve long since been done with school and I read the old textbooks for fun in the evening before I go to bed. Well…the good text books. Some of these math texts are just impossible to comprehend for those of us who aren’t seasoned and conditioned to reading mathematical rigor. This is all fine and good, but understand that the STUDENT is not the proper audience here, so don’t try or presume to be teaching us with this crap. Unfortunately, this is like 95% of math texts.