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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Filed under Calculus, Critical thinking, Math, Problem Solving, Teaching

4 responses to “A problem with “problems”

  1. peter m. gruhn

    I wonder if this has something to do with students not liking word problems. By making a word exercise, it looks like the chapter ends in word problems but the students don’t have to think. Two ideas come to mind: maybe this is a clever transitional phase to trick the student into doing word problems (“Hey, the first three were easy. Let me try the fourth.”); maybe this is a clever way to write a textbook that is popular and helps students get better grades (“We used JLDobbs’s text and our test scores went up 15%!”)


  2. Yeeks. Lousy word exercise. “r” is the rate of increase, expressed in bees per week, right?

    Nothing on that graph is data, or even pretend data. We might have calculated the rate. But then we would have started with actual bee censuses (censi?), and only a complete idiot would integrate when the real number is available.

    I know it is boring to do tons of velocity vs time problems, but, you know, it gives us graphs that calc students can analyze, and , and they make sense.

    It’s also a strange curve. Do bees really reproduce like that? If the context is garbage, and all we want to do is assign exercises, then why not ditch the context?


    • Yeah, there are a lot more issues with this problem than just the one(s) I am bringing up here. How does one measure the rate of growth in a bee population, anyway? Some population rates seem feasible to measure — populations of cities, populations of bacteria, populations of people attending a basketball game… but bees?

  3. Bert Speelpenning

    Great ugly example!
    When you suggest that student A might think “Is it OK to just guesstimate the area by counting boxes” you would conclude that the student at least understands area. Student B who knows how to turn the crank on the midpoint rule may not understand anything about area at all. Neither may be clear how are relates to integration, and neither may be clear how any of this relates to bees. Of course, it is pretty clear that the problem designer doesn’t know or care how it relates to bees either.
    A test question like this is more like a vocabulary test. Student C would fail the test even if she knew the method but had forgotten its name.