A problem with “problems”

I have a bone to pick with problems like the following, which is taken from a major university-level calculus textbook. Read it, and see if you can figure out what I mean.

This is located in the latter one-fourth of a review set for the chapter on integration. Its position in the set suggests it is less routine, less rote than one of the early problems. But what’s wrong with this problem is that it’s not a problem at all. It’s an exercise. The difference between the two is enormous. To risk oversimplifying, in an exercise, the person doing the exercise knows exactly what to do at the very beginning to obtain the information being requested. In a problem, the person doesn’t. What makes an exercise an exercise is its familiarity and congruity with prior exercises. What makes a problem a problem is the lack of these things.

The above is not a problem, it is an exercise. Use the Midpoint Rule with six subintervals from 0 to 24. That’s the only part of the statement that you even have to read! The rest of it has absolutely nothing with bees, the rate of their population growth, or the net amount of population growth. A student might be turning this in to an instructor who takes off points for incorrect or missing units, and then you have to think about bees and time. Otherwise, this exercise is pure pseudocontext.

Worst of all, this exercise might correctly assess students’ abilities to execute a numerical integration algorithm, but it doesn’t come close to measuring whether a student understands what an integral is in the first place and why we are even bringing them up. Even if the student realizes an integral should be used, there’s no discussion of how to choose which method and which parameters within the method, or why. Instead, the exercise flatly tells students not only to use an integral, but what method to use and even how many subdivisions. A student can get a 100% correct answer and have no earthly idea what integration has to do with the question.

A simple fix to the problem statement will change this into a problem. Keep the graph the same and change the text to:

The graph below shows the rate at which a population of honeybees was growing, in bees per week. By about how many bees did the population grow after 24 weeks?

This still may not be a full-blown problem yet — and it’s still pretty pseudocontextual, and the student can guess there should be an integral happening because it’s in the review section for the chapter on integration —  but at least now we have to think a lot harder about what to do, and the questions we have to answer are better. How do I get a total change when I’m given a rate? Why can’t I just find the height of the graph at 24? And once we realize that we have to use an integral — and being able to make that realization is one of the main learning objectives of this chapter, or at least it should be — there are more questions. Can I do this with an antiderivative? Can I use geometry in some way? Should I use the Midpoint Rule or some other method? Can I get by with, say, six rectangles? or four? or even two? Why not use 24, or 2400? Is it OK just the guesstimate the area by counting boxes?

I think we who teach calculus and those who write calculus books must do a better job of giving problems to students and not just increasingly complicated exercises. It’s very easy to do so; we just have to give less information and fewer artificial cues to students, and force students to think hard and critically about their tools and how to select the right combination of tools for the job. No doubt, this makes grading harder, but students aren’t going to learn calculus in any real or lasting sense if they don’t grapple with these kinds of problems.

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Calculus reform’s next wave

There’s a discussion going on right now in the Project NExT email list about calculus textbooks, the merits/demerits of the Stewart Calculus textbook, and where — if anywhere — the “next wave” of calculus reform is going to come from. I wrote the following post to the group, and I thought it would serve double-duty fairly well as a blog post. So… here it is:

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I’d like to add my $0.02 worth to this discussion just because (1) I’m a longtime Stewart Calculus user, having used the first edition (!) when I was an undergrad and having taught out of it for my entire career, and (2) I’m also a fairly consistent critic of Stewart’s calculus and of textbooks in general.

I try to see textbooks from the viewpoints of my students. From that vantage point, I unfortunately find very little to say in favor of Stewart’s franchise of  books, including the current edition, all of the previous five editions, the CCC version (which is almost exactly the same as the non-reform version of the book but with less clarity in its language), or the “Essential” calculus edition. Stewart has a relentlessly formalistic approach to calculus that, while admirable in its rigor, renders it all but impenetrable to students who are not used to such an approach, which is certainly nearly every student I teach and I would imagine a large portion of the entire population of beginning calculus students.

If you don’t believe me, go check out his introductory section on the definite integral (Section 5.2 in the sixth edition). Stewart hopelessly confuses the essentially very simple idea of the definite integral by hitting students with an avalanche of sigma-notation right out of the gate. Or, try the section on exponential functions (1.5), in which Stewart for some reason feels like it’s necessary to explain how it is we can define an exponential function at rational and irrational inputs. This is all well and good, but does the rank-and-file beginning calculus student need to know this stuff, right now?

As a result, I find myself having to tell students NOT to read certain portions of the book, and then remixing and rewriting large parts of the rest of it. But that leads to the ONE thing I can say in the positive sense about Stewart, which I can’t say about many “reform” books: Stewart is what you make it. The book does not force me to teach in a certain way, and if I want to totally ignore certain parts of it and write my own stuff, then this generally doesn’t cause problems down the road. For example, at my college we don’t cover trigonometry in the first semester. In most other books we’ve examined, trig and calculus are inextricable, and so the books are unusable for us. With Stewart, though, given a judicious choice of exercises to omit, you can actually pull off a no-trig Calculus I course with very little extra work on the prof’s part.

I can also say, regarding Stewart CCC, that the ancillary materials are excellent. The big binder of group exercises that comes with the instructor edition is much better than the book itself.

I don’t think that I have yet seen a calculus book that is really fundamentally different from the entire corpus of calculus textbooks, with the possible exception of Hughes & Hallett. They all cover the same topics in the same order, more or less, and in the same ways. If you’re looking for the next wave of calculus reform, therefore, you’ll have to find it outside the confines of a textbook, or at least the textbooks that are currently on the market. Textbooks almost by definition are antithetical to reform. Perhaps real reform will come with the rejection of textbooks as authoritative oracles on the subject in the first place. That could mean designing courses with no centralized information source, or using “inverted classroom” models utilizing online resources like the videos at Khan Academy (http://www.khanacademy.org) or iTunesU, or some combination of these.

Actually, more likely the next wave of reform will be in the form of reconsidering the place of calculus altogether, as the CUPM project did several years ago. Is it perhaps time to think about replacing calculus with a linear combination (pardon the pun) of statistics, discrete math, and linear algebra as the freshman introduction to college mathematics, or at least letting students choose between calculus and this stat/discrete/linear track? Is calculus really the best possible course for freshmen to take? I think that’s a discussion worth having, or reopening.

Enjoy,
Robert Talbert
Franklin College
Peach Dot 1997-1998

Piecewise-linear calculus, part 3: Integration

This is probably the last of three articles on how piecewise-linear functions could be used as a helpful on-ramp to the big ideas in calculus. In the first article, we saw how it’s possible to develop some of the main conceptual ideas of the derivative, without much of the technical notation or jargon, by using piecewise-linear functions. In the second article, we saw how to use the piecewise-linear approach to develop an alternative limit-based definition of the derivative of a function at a point. To wrap things up, in this article I’ll discuss how this same sort of approach can help in students’ first contact with integration, again by way of a hypothetical classroom exercise.

When we took this approach with derivatives, we used the travels of three college students from their dorm rooms to the cafeteria. Each student had a different graph showing his position as a (piecewise-linear) function of time. From these we could get instantaneous velocities. Now let’s consider the reverse situation. A fourth student, Dominic, is traveling from his dorm room across campus, and we have this graph that shows his velocity (in meters per second) as a function of time (in seconds):

Question: How far did Dominic travel in the two-minute span shown here? This is easy, of course, and students get this right away: He traveled at 1.5 meters per second for 120 seconds, so that’s 120 x 1.5 = 180 meters. Distance equals rate times time.

Well, it turns out Dominic has a roommate, named Eric. Eric is leaving his dorm room for a walk too, and his velocity graph looks like this:

Same question: How far did Eric travel in two minutes? There’s a small amount of thinking to be done this time, but it’s still easy: He went 0.5 meters per second for 60 seconds, which is 30 meters; and then 1.5 m/s for 60 more seconds, which is 90 meters. Grand total: 120 meters.

A simple but very important question can be posed here: How come we couldn’t just use distance = rate x time to calculate Eric’s distance travelled? The answer is simply that Eric was not going the same velocity all the time. He had a “piecewise-constant” velocity, so we can use d = rt on either of the two time blocks we want to calculate distance; but we can’t use it globally because his speed changes. In other words: A nonconstant speed requires a kind of “local” d = rt calculation but we cannot use d = rt globally because the r isn’t a single number all the way through.

Now consider Frank, who is following both Dominic and Eric around but whose velocity graph is:

I’ve added the dashed vertical lines just to show where the graph breaks. How far did Frank go in two minutes? Still easy, but this time more work: Total distance = (0.5)(30) + (1.0)(30) + (1.5)(30) + (1.0)(30) = 120 meters.Related question: What does this calculation compute in terms of Frank’s velocity graph? With the dashed lines added in, students pretty quickly see that the sum they did is just an area sum, which we are using because we are doing four local d = rt calculations.

At this point students can stop and think about a few things they are learning:

• Calculating the distance traveled by a moving object cannot be done by calculating d = rt if the velocity changes.
• Instead, we have to “localize” the d = rt calculation by breaking up the time interval into chunks on which the r is constant. Do this on each chunk and then add up the resulting distances to get the total distance.
• This “chunk-wise” calculation is really just finding the areas of a bunch of rectangles.
• “Chunk-Wise” would be a very good name for a rock band. But we digress.
• This is really exactly the opposite sort of thing we did for derivatives. With derivatives, we were given a position function that was piecewise-“straight” and found velocity. Here we are given velocity graphs that are piecwwise-“straight” (actually constant) and finding positions (actually displacements).

Now comes the twist in the problem. We realized, when studying derivatives, that human beings cannot change velocity in an instant. So in the case of Eric above, he cannot possibly go from 0.5 meters per second to 1.5 meters per second without some kind of acceleration in between. His velocity graph is more likely to look like this:

Question: How far did Eric travel now?

Just like when the twist in the problem came for derivatives, I like just to throw this question out there to students and see what they come up with. Most will get the distance travelled on the 0-30 second and 90-120 second interval correct because those are the places where d = rt is in effect. But the 30-90 second interval in the middle doesn’t have constant velocity, so we can’t do that here. I find students do one of three things:

1. Transfer the idea that distance traveled = area under the velocity graph, then use geometry to calculate the area from t = 30 to t = 90.
2. Split the middle interval up into subintervals (usually two of them) and do some kind of rectangle approximation.
3. Average the heights of the endpoints of the middle line segment — that would be a height of 1 m/s — and do a d = rt calculation based on that average.

Each of these three approaches contains a lot of right ideas. The first and third will give them the exact results, and the second one might if they pick the approximations wisely. But any way they go at it, they acquire the right ideas: (1) Distance travelled = area under the velocity graph, and (2) when the velocity graph is not constant, we either approximate or use geometry to find the distance. Note also that if they get this far, they can do any displacement problem like it as long as the graph is piecewise-linear, because they have geometry on their side. For fun, throw in a graph where one of the pieces is below the t-axis and see what they do with it. It goes back to the idea from derivatives that the sign of velocity indicates direction — an idea they will carry with them if their intuition is sufficiently built up at first.

From here it’s an easy jump to start students thinking about non-piecewise linear velocity graphs. Give them one, and ask them to find the distance traveled. The natural thing to do based on their previous work is to try and approximate with piecewise-linear or piecewise-constant graphs. The latter approach is what we call a Riemann sum, and it’s very intuitive to students that more piecewise-constant “chunks” gives better results.

Some ways I think this approach is an improvement on the way calculus textbooks usually do integration:

• The usual approach starts students off with “the area problem” — find the area under the graph of a function, above the x-axis, and between x = a and x = b. There is no real reason given to the students to care about this problem, and the all-important connection between areas and displacement is relegated to the tail end of the section. Instead, here we are developing the notion of area as a necessary tool for calculating distances traveled by objects whose velocity isn’t constant.
• Because the usual approach buries the connection between areas and displacement, by implication it also buries the connection between derivatives and antiderivatives. By contrast, here we are making the connection between velocity and position via areas the focal point of the problem. There will be no surprises once we get to the Fundamental Theorem of Calculus.
• The usual approach presents Riemann sums as the solution to the area by fiat. It’s just “the way we do it”. Here, we build the idea of Riemann sums as a refinement of an intuitive idea, namely that of breaking up the non-constant parts of the velocity graph into constant chunks. Riemann sums are something that the students would have come up with themselves if they’d just been given the chance and the motivation to do so.

As always, I’m interested in your thoughts and criticisms of these three posts. Leave those in the comments.

Four things I used to think about calculus, and what I’ve replaced them with

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I’ve been teaching calculus since 1993, when I first stepped into a Calculus for Engineers classroom at Vanderbilt as a second-year graduate student. It hardly seems possible that this was 16 years ago. I can’t say whether calculus itself has changed that much in that span of time, but it’s definitely the case that my own understanding of how calculus is used by professionals in the real world has developed, from having absolutely no idea how it’s used to learning from contacts and former students doing quantitative work in business amd government; and  as a result, the way I conceive of teaching calculus, and the ways I implement my conceptions, have changed.

When I was first teaching calculus, at a rate of roughly three sections a year as a graduate student and then 3-4 sections a year as a newbie professor:

• I thought that competency in calculus consisted in the ability to think through difficult mechanical calculations. For example, calculating using multiplication by the conjugate was an essential component of learning limits.
• There were certain kinds of problems which I felt were inseparable from a proper understanding of calculus itself: related rates, trigonometric integrals, and a few others.
• I thought nothing of calculus that didn’t involve algebra. I’m not saying I held a low opinion of numerical or graphical calculus problems or concepts; I’m saying I didn’t even have them on my radar screen. I spent no time on them, because I didn’t know they were there.
• Mechanical mastery was the main, and in some cases the sole, criterion for student learning.

Since then, I’ve replaced those criteria/priorities with these:

• I care a lot less about mechanical fluency in algebra and trig, and I care a lot more about whether a student can read a problem for comprehension and then get an optimal solution for it in a reasonable amount of time and using a reasonable method.
• I don’t think twice about jettisoning any of the following topics from a calculus course if they impede the students’ attainment of the previous bullet point: epsilon-delta proofs of limits*, algebraic limits that involve sophisticated algebra tricks that students saw five times three years ago, formal definitions of continuity, related rates problems, calculation of integrals using limits of Riemann sums, and so on. I always want to include these, and I do it if I can afford to do so from the standpoint of managing class time and maximizing student learning. But if they get in the way, out they go.
• I care very much about whether students can do calculus on functions of all shapes and sizes — not only formulas but also tables of data and graphs — and whether students can convert one kind of function to the other, and whether students can judge the relative pros and cons of doing calculus on one kind of function versus another. The vast majority of functions real people encounter are not formulas — they are mostly evenly split between tables and graphs — and it makes no sense to spend 90% of our time in calculus working with formulas if they are so rarely the only option.
• I don’t get bent out of shape if a student struggles with u-substitution and the like; but it drives me up the wall if a student gets the units of a derivative wrong, or doesn’t grasp that a derivative is a rate of change, or doesn’t realize that the primary purpose of calculus is to quantify what we mean by “rate of change”. I guess that means my priorities for student learning are much more about the big picture and the main ideas than they are the minute, party-trick algebra/trig calculations.

Perhaps the story would have been different if I’d remained tasked with teaching calculus to an all-engineer audience. But here, my classes are usually 50% business majors, about 25% biology or chemistry majors, and 15% undecided with only a fraction of the remaining 10% being declared majors in mathematics (which includes students in our 3:2 engineering program). But that’s the story as it is, and I’m sticking to it.

* Technically I never have to omit these, because we don’t do them in our intro Calculus class here.

Deconstructing dx

Asking the following question may make me less of a mathematician in some people’s eyes, and I’m fine with that, but: How do you explain the meaning of the differential dx inside an integral? And more importantly, how do you treat the dx in an integral so that, when you get to u-substitutions, all the substituting with du and dx and so on means more than just a mindless crunching of symbols? 

Here’s how Stewart’s Calculus does it: 

• In the section introducing the definite integral and its notation, it says: “The symbol dx has no official meaning by itself; is all one symbol.” (What kind of statement is that? If dx has “no official meaning”, then why is it there at all?) 
• In the section on Indefinite Integrals and the Net Change Theorem, there is a note — almost an afterthought — on units at the very end, where there is an implied connection between in the Riemann sum and dt in the integral, in the context of determining the units of an integral. But no explicit connection, such as “dx is the limit of as n increases without bound” or something like that. 
• Then we get to the section on u-substitution, which opens with considering the calculation of (labelled as (1) in the book). We get this, er, explanation: 

Suppose that we let u be the quantity under the root sign in (1),  . Then the differential of u is du = 2x dx. Notice that if the dx in the notation for an integral were to be interpreted as a differential, then the differential 2x dx would occur in  (1), and, so, formally, without justifying our calculation, we could write …

So, according to Stewart, dx has “no official meaning”. But if we were to interpret dx as a differential — he makes it sound like we have a choice! — then using purely formal calculations which we will not stoop to justify, we could write the du in terms of dx. That is, integrals contain these meaningless symbols which, although they have no meaning, we must give them some meaning — and in one particular way — or else we can’t solve the integral using these purely formal and highly subjunctive symbolic manipulations that end up getting the right answer. 

Er, right. 

To be fair, my usual way of handling things isn’t much better. I start by reminding students of the Leibniz notation for differentiation, i.e. the derivative of y with respect to x is dy/dx. Then I say that, although that notation is not really a fraction, it comes from a fraction — and that much is true, since dy/dx is the limit of as the interval length goes to 0 — and so we can treat it like a fraction in the sense that, say, if then and so, “multiplying by dx”, we get . But that’s not much less hand-wavy than Stewart. 

Can somebody offer up an explanation of the manipulation of dx that makes sense to a freshman, works, and has the added benefit of actually being true?