One of the advantages of my job is the incredible scheduling. We finished the fall semester the second week of December and my first class of the spring is next Tuesday! The downside is that this gives me way too much time to plot, scheme, doodle, dabble, think, rethink, and overthink. In the end, I usually settle on a plan that is far too ambitious, pedagogically impossible, philosophically suspect, and utterly indefensible. Thus, I bring you my plan for calculus this semester.

I read the wonderful article, *Putting Differentials Back Into Calculus, *which argues for using differentials in a way closer to their original creation than the way they are employed in modern textbooks. As a huge fan of Thompson’s *Calculus Made Easy*, this suggestion didn’t seem half bad. Considering that only a fifth of my students are math majors and that, for the rest, using calculus outside of their physics class is unlikely, why not make things as easy as possible. I’m going to push to teach this same group next fall in calculus II, so the only teacher I can hurt is myself, right? But then I started thinking: the reason the differential approach will work so well with these students is that they will always be using differentiable functions. What reason is there to mention limits and continuity? These are technical issues that won’t help them at all in understanding calculus or how to apply it in their field of inquiry.

Oh dear, so here I am with essentially two months of material (this includes learning to differentiate any elementary function and using this to solve the standard problems). What will I do for the last month and half? I quickly remembered to add Taylor series because I love teaching that in calculus I. Then I added in the obligatory introduction to antiderivatives and integration. I even sprinkled in some partial differentiation at the end so that I could show the students the totally-awesome-implicit-differentiation-trick that would save them five minutes on the final exam. Grr…still two weeks left. These are precisely the two weeks that I usually spend on limits in the beginning. Now I remember why I always do this. It perfectly fills in the semester calendar. And then it hits me.

Any subject can be made repulsive by presenting it bristling with difficulties.

—Silvanus P. Thompson

Limits sure confuse the heck out of students. Why in the world are we leading calculus off with limits, especially to non-math majors? For the purpose of rigor?

You don’t forbid the use of a watch to every person who does not know how to make one? You don’t object to the musician playing on a violin that he has not himself constructed. You don’t teach the rules of syntax to children until they have already become fluent in theuseof speech. It would be equally absurd to require rigid demonstrations to be expounded to beginners in the calculus.

—Silvanus P. Thompson

So I decide I just won’t do it. No limits for us. We will just do some extra exploratory work. There is a great article on math in medicine we could read together. Okay, time for sleep.

But sleep does not come.

Toss.

Turn.

Toss.

Turn.

All right. All right. I’m up.

Why can’t the limits just die? Why do I feel the compulsion to put them back in? They’re like the tell-tale heart beating under my floor. What will become of my math majors if they don’t see limits? No, I can’t do that to them. What a cruel joke to play: send them to real analysis without having used limits. Back in they go. Of course, those biology majors are going to be completely turned off and once you lose *them*, they’re gone for good. Argh, out they go. On the other hand, if I get audited by the department, they are sure going to ask questions. With my review coming up, I can’t afford that kind of chatter. Put them back in…

…but later. Huh? Put them back in, but later. Yes, of course. Later. The course description says I need to cover limits, but it doesn’t say when! What if we introduced everything in a reasonable way and, only after the students know what is going on and why any of this is important, then showed them those funky limit do-hickies? Hmm. Interesting. And that is my explanation for the following calendar and skills list:

I personally agree with the limit idea being introduced later. It actually worked that way in history anyway: Newton and Leibniz (I had to check on spelling for that one; I never remember it) figured out the intuition first, and THEN Weierstrass figured out the limit thing. If I recall, there was about 150-200 years difference between Leibniz’s formalization of calculus and the foundations of analysis.

When limits are introduced, I also think it needs to be done intuitively first (perhaps with a diagram or through an analogy or something like that.) Limits are not that hard to understand now that I think about it, but the abstraction is what is hard: tying deltas and epsilons to something concrete. That’s the challenge in teaching it.

Comment by Zach — January 12, 2011 @ 5:21 pm |

Strichartz has an analysis book where he replaces epsilons and deltas by 1/N’s (for a positive integer N). This took out some of the challenges of learning proper limit definitions for me.

Comment by Adam Glesser — January 13, 2011 @ 9:49 am |

And, as a thought, it might be better to bring up integration at the beginning too, since the concept of integration is a summation of small parts. Then, towards the end, the rules can be explained and used with greater ease (and perhaps even derived!). It allows for the big ideas (derivatives and integrals) to be presented first and then details to be covered throughout the course. I’m not sure, but I’ve always thought that might help newcomers.

Comment by Zach — January 12, 2011 @ 5:26 pm |

That is how Thompson starts things. I might go a little further than him, though. I showed my mother (math background = close to none) and even she understood that (I didn’t mention “” to her). My student should not have too much trouble understanding what means from that.

Comment by Adam Glesser — January 13, 2011 @ 9:52 am |

Leaving limits for later sounds like a great idea. I’m curious to see how it’ll turn out.

If you’re starting off discussing math in medicine, you could mention how someone recently got a published article by reinventing the trapezoid rule:

http://www.stat.columbia.edu/~cook/movabletype/archives/2010/12/reinventing_the.html

http://www.johndcook.com/blog/2010/12/03/you-can-be-a-hero-with-a-simple-idea/

and how calculus is really not so scary — it’s just a way of tying together commonsense solutions like this, and putting them in a good framework that helps you generalize them and thus solve harder problems.

Comment by Jerzy — January 13, 2011 @ 9:40 am |

Hi Jerzy,

I saw that article recently and only considered how ridiculous it was that it got published. It hadn’t occurred to me to use as a pedagogical tool. Thanks for pointing that out.

Comment by Adam Glesser — January 13, 2011 @ 9:57 am |

The same debate comes up in computer science all the time: whether to teach programming bottom-up (starting from machines and assembly language and gradually building up to more ideas that are more complex to implement) or top-down (starting from easy-to-use high-level languages like Python and gradually showing how the magic is made to work). Both approaches have proponents, with computer engineers and electrical engineers often favoring the bottom-up approach and AI researchers favoring the top-down approach.

The support often reveals something about what the proponents care about: how things work or what you can do with them. Both are interesting, worthy things to teach, but many students will only be interested in one of them.

Incidentally, limits are no longer essential for providing a rigorous basis for calculus. There is now a rigorous basis for the more intuitive approach using infinitesimals. Unfortunately, the rigorous approach is much more difficult to understand than limits, so you have the choice of easy-to-use, hard-to-prove infinitesimals or hard-to-use, ok-to-prove limits.

Comment by gasstationwithoutpumps — January 13, 2011 @ 9:42 am |

Great analogy. In some ways, this is a false dichotomy. Given the right classroom structure, one could introduce all of these concurrently. This makes sense because very often the rigor informs the practice and vice-versa. In fact, we offer an honors calculus course—in the form of an additional class meeting time—designed to provide the math majors additional time to spend on limits. I’m not teaching it this semester and so I’m trying to act independently of it.

And no I won’t be introducing superstructures to give students a rigorous foundation for infinitesimals. 🙂

Comment by Adam Glesser — January 13, 2011 @ 10:24 am |

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