Optimally Challenging

@republicofmath tweeted a link to this article on Advanced Placement courses. This article nicely summarizes my feelings on AP calculus—very few students learn much calculus beyond the algorithms (I am not citing anything here because this is little more than my impression). Combining this with the push to make algebra as a required course for students as young as 7th grade, and we begin to see a pattern of the maxim “earlier is better.”

One particularly dangerous way this manifests itself is the in the corollary “harder is better.” In fact, I find myself doing that a bit this semester in my abstract algebra class—I find myself introducing graduate-level ideas to my students at times. This has mostly been fine, since my students have been really good. However, what is the cost of this? For every graduate-level idea I introduce, an undergraduate idea is left unsaid (roughly). But somehow I want to introduce these topics.

I do not think that I am alone in this habit of making courses more challenging than they need to be; in fact, I think that I far from the worst offender, since I am aware of this tendency and work hard to keep the material at the level of my student. I know of other professors who brag about making very difficult exams and homework assignments. I fall into this trap, too, when I am not consciously thinking about this issue. I cannot speculate about everyone else who prides themselves on being a “tough” teacher, but here are my best guesses as to why I am this way:

  1. I find the material more interesting. Since the course is for the students, this is not a terribly good reason. “Research” should be the outlet for the material I find most interesting.
  2. I find that my ego is stroked when I teach harder material. It is rarely a good idea to do things just for ego, though.
  3. I think that I am being a good teacher by challenging my students. However, this is not true. It is very easy to make a course that is too easy or too hard. Unfortunately, students do not learn much in these classes. It is relatively hard to create a course that is optimally challenging for the students, which is where they learn the most. Instead of aiming for “hard,” I should be aiming for “just hard enough.”
  4. I find that my colleagues respect me more. It seems like the fastest way for a professor to lose the respect of his/her peers is to gain the reputation of being an “easy teacher.” This is easily done, since many of my colleagues in this country think that anything short of “students killing themselves to make it through a math class” is too easy.

I do not want to make it sound like I have done a bad job this semester—I have actually been very pleased. I can merely point to occasions when I have introduced ideas that are too hard. If anything, I think that I might be developing the reputation as being “too easy” on my students (this will likely be the topic of my next post).


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2 Responses to “Optimally Challenging”

  1. Derek Bruff Says:

    I’m with you on the “just hard enough” approach. So is George Polya, for that matter. (His book, “How to Solve It,” has some great suggestions on working with students in one-on-one situations along these lines.)

    One problem is that “harder is better” often translates to “computationally harder is better” and not “conceptually harder is better.” I talked with a colleague recently who teaches in a math department that has common exams across the calculus sections. The course head essentially writes the common exams, and she loads them up with computationally difficult problems. The students need to know all the usual calculus procedures and several unusual ones, and they need to know them cold. Sure, those tests are hard, but they could also be hard if they focused less on procedures and more on concepts. My colleague finds it frustrating to teach in this settings, since he would prefer to focus more on concepts and a little less on procedures.

    Harvard physics professor Eric Mazur has reported widely that his students are quite capable of learning how to solve computational problems in physics without internalizing any of the associated concepts. Conceptual learning can be very difficult for students, but it tends to lead to deeper, more lasting learning than procedural learning, particularly in calculus.

    Meaningful conceptual learning is often “just hard enough” for students. There’s often no need to pile on the procedural learning to make things harder.

    • bretbenesh Says:

      Hi Derek,

      It is always a pleasure to hear from you, and I agree with you that sometimes teachers overuse “computationally hard” questions. I spoke with Paul Zorn about the calculus problems in his textbook, and he said something to the effect that he tries to think of “simple theoretical questions.” I liked this—think of questions that are really easy to answer if you understand, but cannot be answered without conceptual understanding.

      Here is another drawback of the “computationally hard” problems: they are the toughest problems to grade!

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