By all means take the concepts, but don't try to apply them like that. There are so many pitfalls that this doesn't help you identify, let alone avoid.
If it helps you grok stuff, great. I think I'm pretty clear on how to do a quadruple coronary bypass - I'll just figure out the details as I go along, shall I?
Yep, there is a fundamental tension between intuition and rigor. Unfortunately, I feel that rigor (and not even rigor -- rote details) are overemphasized, without giving any context, and more importantly without giving any inspiration to the student. It's like focusing on the spelling of a poem, not the content.
I do want to jump into the rigorous definitions in later posts (such as when limits exist, when they don't, why it's important) but I find intuition -> rigor is an easier path to navigate than rigor -> intuition [which often becomes rigor -> memorization]. If someone had just said "Hey, limits and infinitesimals help us solve the problem of 'How do we make an approximation which is still accurate enough?'" calculus would have clicked a lot faster for me :).
The problem is, as I'm sure you're aware, just how many traps and pitfalls there are for the insufficiently paranoid. Working "intuitively" has a tendency not to instill that paranoia.
And, as you say, working just on the rigor tends to result in rote work with no understanding.
I think only time and practice is the real solution. The article, though, is good for someone who already has the "rote" understanding, and needs to "grok" the whole thing.
Similar criticisms can be aimed at my stuff: http://news.ycombinator.com/item?id=940169
For the examples, my approach is definitely tailored for people who need to grok the subject after getting a rote understanding. It's not so much to become an expert but to realize what insights would help on the path to becoming an expert.
I figure there are thousands of other sites for the formal definition and practice problems :). I'll check out the article now, looks interesting.
In the one example given, sin(x)/x, the difficult bit is glossed over when it talks about sin(x) being more-or-less equal to x when x is "close to zero." The rigor there is missing entirely, because that's not the point.
but thought it would be too leading & tangential -- I wanted to show visually that most curves can in fact be modeled by a straight line over a small distance, without the sledgehammer of showing sine's expansion.
Similar arguments can show, for example, that acceleration in a circle is v^2/R. "Calculus" was, or course, done for decades, if not centuries, before "calculus proper" was (somewhat) formalised by Leibniz and Newton.
This whole discussion should probably be taken to email if it's to be taken further. The question is, how can we have this sort of discussion, and produce something worth publishing back onto HN?
Always happy to discuss on email (kalid.azad@gmail.com) -- I'm not sure if there's something to produce out of it necessarily, but I enjoy the discussion / investigation of personal philosophies & teaching approaches.
But if you supply rigor in an appropriate dose, it can really be eye-opening. For me, calculus was a nightmare because none of the terms used were well defined. I was just presented with all this highly developed theory and had to take much of it on faith. And I couldn't enjoy nor care about something I felt was just forced memorization of computational techniques for solving contrived problems. But if they'd done it rigorously, the classes would have begun with questions like "what is distance?", then answering that question would lead into "what is a space?", etc. This is rigor and I feel like exploring these types of questions is a lot more intuitive than forcing kids to memorize how to compute the volume of some cone.