Last year I went to a PD at Math for America that was about approaching calculus from a geometric point of view. The presenter mentioned during it that, historically, the idea of the integral was developed first, followed by the derivative, and then the limit. Yet in many calculus courses, they are taught in the exact reverse order. I decided that, should I teach calc again in the fall, I’d do integration first.
Well, school is rapidly approaching, and so I’ve been thinking about it again. I did so searching and found this intense forum discussion (oh, old Internet), which pointed me in the direction of the Apostol’s Calculus 1 textbook, which starts off with integrals. The post also had a bunch of arguments about why I shouldn’t do it. One of the notable arguments was that in order to fully teach integration (including u-substitution and integration by parts), you need differentiation. But I actually view that as a benefit, not a downside, because it forces a more spiraled approach. I can start with integrals, then go to differentiation, and then tie them together.
In general, I feel like area is a much more approachable subject than slope. My years of teaching Algebra I to 9th graders certainly seems to support that claim. But I also think it’s easier to understand the linearity of integration than the linearity of slope. “If you add together two functions, the area under the new function is the sum of the areas under the old functions” seems much more evidently true than “If you add together two functions, the slope of the tangent line for each point of the new function is equal to the sum of the slopes of the tangent lines at the same points on the old functions.”
Of course, Jonathan has already worked to restructure his calculus course, and I plan on taking a number of cues from his more spiraled sequence – but still with integrals first.
Here’s what I’m thinking:
Q1 (Intro to Integrals) – (Sam’s Abstract Functions, Area Under Stepwise Functions/Definite Integrals, Properties of Integrals, Riemann Sums, Area Under a Curve, Power Rule for Integrals, Trig Integrals, some applications)
Q2 (Intro to Derivatives) – (Average vs Instantaneous RoC, Tangent/Secant Lines, Power Rule, Trig Derivatives, some applications)
Q3 (Fundamental Theorem) – FTC, Chain Rule/u-substitution, Product Rule/Quotient Rule/Integration by Parts, Curve Sketching/Shape of a Graph)
Q4 (More Applications) – Related Rates, Optimization, Volume, etc
How does that sound?
James Cleveland-Tran
The Roots of the Equation 
Comments on: "Integration First" (2)
I teach at a community college, and am required to use the department’s choice of textbook (Anton), so I have not thought moving integration first would make sense for my courses. (And, maybe I’m being stodgy, but I like having it at the end.)
But I do think limits come too early to be sensible, and have written a lot on my blog about rearranging the topics. I finally put together a calculus page for my blog after I presented the the Joint Meetings in January. Here: http://mathmamawrites.blogspot.com/p/calculus.html
I will look forward to hearing how your rearrangement goes. If you need materials, both Boelkins and Hoffman have good ones.
[…] up and get started on some chores – laundry and dishes. While the laundry goes I work on the blog post I wrote about teaching Integration first. Thinking through the post helped me solidify how I wanted to start the year in Calculus, so that […]