Why do we overcomplicate calculus like this?

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In the Stewart calculus text, which we use here, the first chapter is essentially a precalculus review. The second chapter opens up with a treatment of tangent lines and velocities, with the idea of secant line slopes converging to tangent line slopes and average velocities converging to instantaneous velocities taking center stage.

Calculating average velocity is just a matter of identifying two time values and two position values and then performing two subtractions and a division. It is not complicated. Doing this several times for shorter and shorter time periods is also not complicated, and then using the results to guess the instantaneous velocity is a little complicated but not that bad once you understand the (essentially qualitative, not quantitative) idea behind shrinking the length of the interval to get an instantaneous value out of a sequence of averages.

So I nearly hit the roof when a student came in this morning needing help understanding the Student Solutions Manual for the Stewart text on a problem where you had to find the average velocity of a moving object from 2 seconds to 2.5 seconds. A formula for position is given, y = s(t). The simple way to do this — the way that works, does not dumb the process down, and yet makes it understandable to the broadest possible audience and therefore sets  up general understanding of the more complicated idea of derivative calculations later — is to calculate s(2.5), calculate s(2), and then calculate \frac{s(2.5)-s(2)}{2.5 - 2}. Fifth-graders do this.

Instead, the Student Solution Manual does it like this:

  • Let h represent some positive number.
  • Calculate and fully simply the expression \frac{s(2+h)-s(2)}{h}.
  • Plug in h = 0.5.

This is crazy, absurd, and downright dangerous. It’s as if Stewart, and the person who wrote the manual, really believe that calculus is made up of algebra, and students who are in calculus are uniformly comfortable and skilled with algebra to the point that their way is just as transparent and simple as calculating distance divided by time — as if the algebraic work that ensues when you perform step (2) above were as natural as the concept of velocity itself and students spoke algebra like a first or second language.

Yes, the book’s approach works — and it closely mirrors what’s going to happen later when we want to get an exact value of the instantaneous velocity by letting h \rightarrow 0. But that’s not what students are doing right now. What students are doing is trying to understand the concept of average velocity. It’s not complicated. The complications should come, if at all, on the back end of the subject — where we are trying to make the concept of instantaneous velocity precise through limit calculations — but not on the front end when students are just trying to figure out what’s going on.

In the middle of typing this post out, another student came in, equally confused about the exact same problem. I told him to close his solutions manual. I asked him: What’s the definition of average velocity? He thought about it, and then gave me the right definition. “OK, then,” I said, “How would you get the average velocity from t=2 to t=2.5 here?” And he gave me an exactly right description of the process. The relief on his face was palpable. He understood this concept but the student solutions manual made it appear that he didn’t! How bad is it when you need a manual for the student manual?

Calculus is a really simple subject when you get to its core. I wish the book treated it that way.

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Filed under Calculus, Education, Math, Teaching, Textbooks

13 responses to “Why do we overcomplicate calculus like this?

  1. Susan

    I agree! Why let the algebra get in the way?

  2. arithmetic

    Could you simply choose a different textbook for next term? Maybe a different author had produced a solution manual with fewer design errors. Another alternative is to create/produce some of your own solution keys (but you probably do enough of this for the quizes and examinations for your courses).

    You might also want to ask various students to present how each managed their solution process that the exercise which you described – they could present what they did IN CLASS, and then you could comment on their steps and processes.

    • We’re looking at different books, but in reality it’s not so simple just to switch. The way we do our course makes it hard to just pick any off-the-rack book and use it — for example, we do no trigonometry in the first semester of calculus. Also, switching textbooks would level a pretty significant cost on our students due to not being able to find the old book used.

      And I’d love to do the student presentations you mentioned, but really there is just no time in which to do it. I’ll show you the schedule if you like.

  3. jedward706

    My suggestion:
    If the “problem” is primarily the “solutions” manual, then eliminate it from the textbook package and focus efforts on improving/developing peer recitation groups — you can always post answers to even questions (those not in the back of the text) if students need the confidence boost.

    For those problems, which you expect are difficult enough to warrant some problem solving “assist” — have a sharp student or TA post hints.

    One big problem with solutions manuals is that students assume the solution is THE way to solve the problem — this may or may not be the case — and tends to squelch sharp students from creative and “class comment worthy” approaches.

    • The thing is, the actual book does this problem the same way. Use a ton of algebra and then plug in h = 0.5. I think the manual was just following the book, but oddly it doesn’t *explain* the book but merely repeats it.

      • jedward706

        Ah, well —
        I suppose when you fret over enough of these issues with a particular text, you’ll find the motivation to write your own text — or — develop lecture notes sufficiently and find online resources to make the text unnecessary 🙂

  4. I think it’s great to use position as a function of time – but I recall hating finding average velocities. There was always something that was clearly not very useful about the answer…

    Different from what you are seeing and saying, but you reminded me.


  5. I’m just curious: In the problem you mentioned, is the student expected to find the average velocity over several different intervals, or only on the interval [2, 2.5]? If s/he were asked to find multiple velocities, I could understand why you would want to find the complex algebraic expression. If s/he only needs to find one average velocity, I agree that the book is just being ridiculous.

    This does, however, highlight what I tell my students whenever we start calculus: the hardest thing about calculus is usually the algebra.

    • They had to repeat this for several time intervals of decreasing length. So I can see the utility of an average velocity formula. But still I think it’s more instructive and confidence-building to do several straight calculations by hand without algebra first, and then fold it in so there is some real motivation to make the investment (namely, the algebra automates the calculations and therefore makes life easier) — rather then just pop the algebra on them at the outset as though the reasons for doing so were obvious to all.

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  8. amca01

    A very good point indeed – calculus is indeed made much harder than it needs to be, and elephantine books like Stewart (and all the others) don’t help. What the world needs is a sequence of short, straight-forward calculus texts, which start from what the students know (invariably not very much) and build up carefully from there. I use no books – can’t stand any of them – but my own notes instead.

  9. Mr.Robert, Rohedi believes after reading the above explanation your college students as soon as understand to distinguish between average velocity and instantaneously velocity. My question, how to prove that dS/dt=0 is correct for S=constant. Thanks.