“Can I teach integral calculus before differential calculus?”

In answer to the above question, Ted Alper writes:

Not only can you, but David Bressoud argues in his book, Calculus Reordered: A History of the Big Ideas, that you should teach integral calculus first, or — maybe a little more precisely — that the entrenched educational ordering of subjects in a first year calculus course — starting with limits, then derivatives, then integrals, then series — may have a certain logical appeal in organizing everything economically in a [somewhat] rigorous package, but lacks motivation and clarity for novices, and he suggests following a path closer to the historical order in which the subjects were studied, roughly (1) Accumulation (Integration) (2) Ratios of Change (Differentiation) (3) Sequences of Partial Sums (Series) (4) Algebra of Inequalities (Limits).

One quote from the preface to the book:

The progression we now use is appropriate for the student who wants to verify that calculus is logically sound. However, that describes very few students in first-year calculus. By emphasizing the historical progression of calculus, students have a context for understanding how these big ideas developed.

I took a look at the book. It’s cool, but it’s clearly written more for mathematicians who are curious about the history than for students who want to learn the ideas of calculus—fair enough, given the title, but still. For teaching, I think it makes sense to go through examples of accumulation, rates of change, etc., from simple physics and economics, rather than starting with tricky things like the area of a sphere. Although I can see that if you’re a mathematician you’d rather start with the sphere because the simple examples are too boring. It’s kinda like when I teach probability, I can’t bear to bring up the “Do any two people in the room have the same birthday?” problem because it’s too damn trivial; I prefer to give examples such as, “How do they estimate the proportion of identical and fraternal twins in the population?”, which is not mathematically deep but has statistical subtleties. This is always a peril of teachers, that we want to jump to the fun stuff.

For a quick read of the history of calculus, with examples from the ancient Greeks, I recommend the chapter from Aaron Strauss’s book from 1973. That whole book is fun.

But, yeah, to get back to the title of this post, it does seem to be just fine to teach integration first. The real point, I guess, is to teach the concepts and the methods and then get to the idea of limits at the end. That makes a lot of sense to me.