There's something that I think most people are missing, although many of you will already know this.
People are saying that you need to develop the intuition, to develop the visualization skills, to develop the sense of what's happening rather than simply memorizing the formulas.
But to me, the visualization is not the point. To me, the sense of what's happening based on the visualization is not the point. To me, the point is the richness of understanding, the combination of many ways of thinking.
This doesn't come without effort.
The lunk-to item seems to suggest that by having the picture in mind one can avoid all the tedium of remembering the epsilon-delta limit arguments and can avoid the definition of lim_{e->0}(f(x+e)-f(x))/e and so on, but that's not true. The point is that the formula is tied up with the image, not that one subsumes the other.
Allegedly Euclid said King Ptolemy (in response to a request for an easier way of learning mathematics) that "there is no Royal Road to geometry".[1] Likewise there is no "Royal Road" to a mastery of calculus. Or indeed, to a mastery of any subject. That which can be mastered with little effort has long been surpassed, and work is required to gain the depth and breadth required to make these things easy.
But we do these things "not because they are easy, but because they are hard."[2] They are of value, and developing the mastery is satisfying in its own right, but also makes you a rare commodity.
You can certainly avoid all the tedium of the epsilon-delta arguments while mastering calculus. In fact there were over a hundred years between the invention of calculus and the formalization of the epsilon-delta definition! Calculus was not invented in the same way it is taught, and the early masters of calculus were much more intuitive than rigorous.
For references, calculus was first published in 1684:
You could perhaps try to learn calculus the way it was taught three hundred years ago, but I'm pretty sure it would be more difficult, not easier, than the standard presentation. In the early years calculus was considered black magic only the smartest few could master, probably not unlike the way, say, String Theory is seen today.
The most charitable interpretation of the lunk-to article is that it is asking you to develop your own deep understanding of the concept, from which your own visual picture will emerge.
In this retelling, the point is not the visual metaphor as an end product, it's the work needed to reduce the issue to its basic elements and their relationships.
People are saying that you need to develop the intuition, to develop the visualization skills, to develop the sense of what's happening rather than simply memorizing the formulas.
But to me, the visualization is not the point. To me, the sense of what's happening based on the visualization is not the point. To me, the point is the richness of understanding, the combination of many ways of thinking.
This doesn't come without effort.
The lunk-to item seems to suggest that by having the picture in mind one can avoid all the tedium of remembering the epsilon-delta limit arguments and can avoid the definition of lim_{e->0}(f(x+e)-f(x))/e and so on, but that's not true. The point is that the formula is tied up with the image, not that one subsumes the other.
Allegedly Euclid said King Ptolemy (in response to a request for an easier way of learning mathematics) that "there is no Royal Road to geometry".[1] Likewise there is no "Royal Road" to a mastery of calculus. Or indeed, to a mastery of any subject. That which can be mastered with little effort has long been surpassed, and work is required to gain the depth and breadth required to make these things easy.
But we do these things "not because they are easy, but because they are hard."[2] They are of value, and developing the mastery is satisfying in its own right, but also makes you a rare commodity.
[1] http://en.wikipedia.org/wiki/Royal_Road#Cultural_references_...
[2] http://er.jsc.nasa.gov/seh/ricetalk.htm