Let’s Integrate Calculus

Calculus is a full-brained subject being taught by left-brained instructors to right-brained students. I see many posts lamenting the lack of student understanding resulting from the standard way that a calculus course is taught.

For example, Tevian Dray suggests:

“Skip the fine print. Emphasize examples…Emphasize the need to be fluent with multiple representations…[emphasize] geometric reasoning as the key to conceptual understanding.”

Another recent post by Katie Courage summarizes a study featuring the lack of understanding of calculus I (not a lack of performance in Calculus I!) as a major reason cited by women for dropping out of STEM majors.

So, in order to facilitate an engagement of our full-brains, I would like to encourage the use of imagination and do a better job promoting conceptual understanding.

My main effort in this direction resulted in the (free and open-access Journal of Humanistic Mathematics) paper entitled, “A Truly Beautiful Theorem: Demonstrating the Magnificence of the Fundamental Theorem of Calculus” co-authored with Darlene Olsen, who is a Homer L. Dodge award winner for University Teaching Excellence at Norwich University. Her efforts, and excellence as an expositor ensured the writing of that paper is of high quality!

Before reading it, you may ask yourself (or your students) why the derivative of the area of a circle should be the circumference of the circle, or if you really could convincingly explain the formula for the volume of a cone, or if you’re happy with how students absorb related rates. Do students think of the chain rule as merely a method for calculating derivatives, or can they use it backwards? Do they really get the First Part of the Fundamental Theorem of Calculus? Can you think of a useful classroom example that can be used in the first week of the course, which can illustrate all of the essential features of the course? Ok don’t wait—please read our paper now!

One thing we challenge is the following dogma:

“Limits must be taught before derivatives. Derivatives must be taught before integrals. Riemann sums must be taught before definite integrals.”

For this dogma is simply not true. It’s not true historically and it is very strange indeed that our method of teaching calculus is so far removed from the human thinking that led to its discovery.

In fact I see no advantage in hiding derivatives from students while teaching limits. I see plenty of advantages to revisiting the Fundamental Theorem of Calculus many times in different contexts throughout an introductory course. Our paper suggests very minor changes to an introductory course—so they would be very easy to implement for most experienced teachers. We believe such changes would substantially alter the perception that a student would have of calculus.

A few carefully chosen examples may help to connect limits, derivatives and integrals, leading to a more enriched view of the subject.

So instead of introducing the main topics strictly sequentially, please try—at least in a few more places—to introduce them simultaneously. In other words, let’s integrate calculus! Please also provide me with feedback—I’d love to know how it works out.

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4 Responses to Let’s Integrate Calculus

  1. ben says:

    I wouldn’t say its particularly strange or dogma. Our current conceptual model dates from Cauchy’s seminal works that unified and clarified the previous work by Newton and Leibnitz.

    Post-Cauchy its hard to conceive of Calculus without limits. That said, if being a little less linear is useful for developing understanding then that’s great.

    • daniel-mcquillan says:

      Great comment Ben. Love the historical perspective and value added. I agree that “Dogma” seems a bit harsh at first, and I meant no disrespect to Cauchy. In my defense, I use the word only to describe the standard order in which the subjects are taught; unlike other subjects, I think calculus is almost universally taught in the same way. More importantly, I am not suggesting any major changes to the way a calculus course is taught. I think my post was clear about that. I think we are actually in agreement… Earlier today, Viktor Blasjo (who I did not know before today) had a related post that you may also find interesting: http://intellectualmathematics.com/blog/a-criterion-for-deciding-if-something-is-worth-teaching/

  2. Jeremy Holden says:

    I completely agree.

    Limits are a technical device to avoid crisis and maintain consistency, and as such have limited usefulness for raw beginning students of calculus. The notation can also be quite intimidating. Taking derivatives, on the other hand, is simple, easy, builds confidence, and cements all the analogies like slope/position/rate-of-change/area in the mind.

    Deriving and building up modern methods from first principles is not only unnecessary but may even be considered monstrous.

    To make an analogy, teaching limits before derivatives is like refusing to show a child the real number line until they prove the existence of the real number line via Dedekind cuts. Absurd.

    The point of these technical devices like limits, sigma-algebras, etc, is to keep things working the way that the right-side of our brains think they should work. The left-brained counter-examples and crises that caused these devices to be invented requires experience and motivation to understand.

    The rules behind derivatives can be shown, and then concrete examples where the rules DON’T WORK can be introduced, so students can see where and why things break down, and exactly why these complicated and subtle notions like limits are needed. Ideally, these counter-examples would be put in the context of concrete, real world examples and analogies.

    They build intuition first, then they see the crises, then they tackle the technical devices LAST.

    As another example, I actually am of the opinion that neighborhoods are a far more intuitive and easier to understand concept than epsilon-delta arguments, and that they could actually be introduced in their place, but without all the formalism. Even the language of it is far clearer.

    The calculus curriculum is riddled with problems.

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