What to teach first: Riemann sums or anti-derivative? In some text books, I see that they teach Riemann sums first. In other texts, I see they teach anti-derivatives first. 
Is there any pedagogical preference? It seems to me that we should teach Riemann sums first to understand the point of integrals. 
Thoughts? 
 A: The historical order is the pedagogical order.


*

*Areas and any other relevant ideas from geometry, treated informally

*d(Area under graph of $f$) = $f(x)$ by the visual geometric argument.

*Antiderivatives are therefore useful.  Polynomials etc.

*Riemann sum as discrete approximation, as formalization of "area" concept, as motivation for "dx", and as method to compute limits of some finite sums by taking integrals.
A: The anti-derivative us the "opposite procedure" than that of the derivative, therefor it makes sense to do it after the derivative. The integral however is the area under a curve for the given interval and Riemann Sums explain why it is (amazingly) the difference between the values of the anti-derivative at the two endpoints of the interval. 
So before you can understand Integrals, you have to understand the anti-derivative, then Riemann sums as the approximate area under the curve. 
So it makes sense for me to do 


*

*Derivatives

*Anti-derivatives

*Riemann Sums

*Integrals


Well at least that's how I finally made sense of it when I was tutoring Calculus.
A: To take a contrarian point of view, there's merit in starting with antiderivates too. It all depends on which application you have in mind. For example if it's for physics the interrest of the area under the graph is often a distraction because what's actually the point is the solution of an ODE.
So if you put ODE's into the picture as something useful to learn you may want to introduce the problem of solving ODE's and see how antiderivates are part of the solution.
I noted that some complained that something didn't make any sense, but that may also be due to the road-map hasn't been conveyed yet. I think one problem may be that the student doesn't see the larger picture while doing Calculus, perhaps not even after taking the course. I think one should convey the roadmap before starting. If that's being done it's probably easier to alter the order without it loosing sense.
