Please read the article with a critical eye, some of it is complete non-sense, for example:

CALCULUS: This course will explore the mathematics of motion, and the best ways to bury it
under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be
mentioned.

"Mathematics of motion", which makes it sound so simple, has in fact perplexed philosophers and mathematicians for centuries and continues to perplex a great many people even today, consider for example the Zeno paradox:

http://en.wikipedia.org/wiki/Zeno%27s_paradoxes

The ideas of Newton and Leibniz were hardly simple, they had some valid intuitions and managed to do formal manipulations that led to correct results, but in their day it was impossible to at all logically understand why what they are doing works, and not for some god knows how complicated things, but even for most elementary ones. You don't even have to go back to writings of Newton or Leibniz, just have a look at a 19th century textbook of calculus to see how noticeably strange and illogical the exposition of the subject was even then, with "infinitely small quantities" and an air of mysticism about it:

http://archive.org/stream/elementsofdiffer00woolrich#page/n1...

This kind of approach simply doesn't make sense, even though it happens to apparently produce correct results sometimes. Now, "function-based approach" is a weird phrase, but I guess he means the common modern exposition of elementary calculus using limits. This however wasn't developed in response to "various analytic crises". The only explanation of this statement I see is that he knows history of mathematics poorly and confuses the latter developments by Lebesgue, Jordan etc. that led to what we now call real analysis (inspired by considerations of nowhere continuous functions, continuous but nowhere differentiable functions etc.) with the earlier and more general lack of any decent understanding of how calculus works at all that was solved by Cauchy, Weierstrass and others. It is their introduction of what the author considers "unnecessary formalism" that made us finally really understand "mathematics of motion" and satisfactorily resolved things like the before-mentioned Zeno's paradox.

If it is only motivated appropriately, the concept of a limit is actually very interesting and powerful. There is a ladder of granularity with which you can treat computational problems, with the most elementary approach being always trying to get the exact answer. However, the class of problems that can be solved this way is very narrow. You can jump over this severe restriction by getting a bound, with inequalities for example, or you could try to get an equality in the limit (when n approaches y, the sought thing x approaches w*z). Unfortunately in school people almost exclusively learn to look for the exact answer, while in mathematics proper and in real world it is much more common to look for approximations and limiting behaviour. Furthermore, since the limit concept so powerfully extends the range of problems for which we are able to state anything interesting, there are lots of mathematical disciplines that rely on it to a great extent, for example probability theory (laws of large numbers, central limit theorem, ...). You won't understand almost any higher mathematics without learning limits first!

One can get an excellent and well motivated introduction to reasonably rigorous calculus using limits in Courant's "What is mathematics?" in less than a 100 pages, up to the point of understanding basic differentiation and integration, the exponential function, power series etc. The problem is not the formalism, but the teachers who can't motivate the material well enough both mathematically and physically and students who are not always mature enough to put in the amount of work necessary to understand calculus, which for most of them will be by far the most difficult thing they ever attempted to learn.