Absolutely. It is an art and a tool in not so different a way from how a language is an art and a tool.

And like learning a foreign language, learning Mathematics is not a straight path. We would love to build up a sense of mathematics from first principles in a perfect, coherent way, but that is no more realistic than learning French by studying the etymology of every word from the get-go. No, you learn French by listening to it, by speaking it, by making mistakes, without necessarily knowing how it evolved. Later, once you are more fluent in it, you begin to read more sophisticated literature, you begin to be interested in the development of the language, and then you say, "Ah... so that's how things come to be."

A student may fully appreciate "the transcendental nature of the trigonometric functions", but what good would that do if he cannot bother to memorize (yes, MEMORIZE) the double angle formulas. How would he understand later on a real life application of Calculus, where it is taken for granted that he is fluent in the language of trigonometry.

It's funny that Lockhart uses the practice of visual arts as a metaphor. Fact is, there is a lot of dry, uninteresting stuff that went into the training of an artist. The myth that Da Vinci started out painting eggs probably isn't too far from the truth. You think Picasso painted things in the style of Guernica when he first started? Doing the dry non-interesting stuff allowed Picasso to express his artistic vision with technical facility. So what if he had the "vision" of Guernica, if he can't even handle paint competently?

From my own experience as a pure mathematician, I can tell you that my own learning curve is far from linear. When one learns topology, one has to learn all the formal definitions of open sets, compactness, and so on. Of course, one tries to motivate these definitions with intuitive notions, but ultimately, a lot of my appreciation of "the language of topology" is obtained from seeing how it is applied. One can talk about donuts and coffee cups all they want at the beginning, but that doesn't even begin to capture the beauty of it (Try talking about cups and donuts in the context of p-adic topology on a p-adic field). It's a back and forth process. Most often the person coming up with the definitions isn't him/herself fully aware of the full implications. But that's the beauty of it.

Differential / integral / multivariable Calculus, ordinary differential equations.

No, and I don't know why someone would think that.

who do not allow calculators. Part of my rationale is that if I allow calculators, then those who have the fanciest equipment would have an unfair advantage over those who don't. And I hate to have students feel that they must buy expensive equipment in order to stay competitive in the class.

So, this WolframAlpha might actually be a good thing, for it could level the playing field (The majority of my students do have internet access). I am sure one could design math problems in a way that still tests a student's mathematical aptitude and knowledge, while taking into account the availability of WA.

Think about this the other way round: If WA doesn't exist, and some $1000 calculator can do what WA does, then the rich students who could afford to buy the calculator would have an unfair advantage over those who couldn't.

From TFA:

"Despite all their marketing suggestions to the contrary, no manufacturer has ever invented a shoe that is any help at all in injury prevention."

And here is the keyword: Injury, as opposed to: Pain. I have been injured multiple times running in running shoes, but I have only run barefoot (7 miles on wet sand) once. So no meaningful statistics on injury can be drawn from my personal experience. I can, however, say with absolute certainty that running with shoes helps prevent pain.

I do believe that with perfect form one can run barefoot with less chance of injury than with shoes. But:

1. Your average consumer does not have perfect form. It's hard to develop perfect form.

2. About building up those callouses? Your average consumer in a 1st world country does not have the will to build up the callouses. And no one should blame them, given that they wear shoes to work and to school all their lives.

The shoe manufacturers are merely giving their consumers what they want. The consumer wants a pair of shoes to prevent the pain, and he/she is all too willing to accept the cover story that those fancy shoes prevent injury. People are much more open to say they shun injury, than to say they are too pampered to endure pain.

Actually, I need to correct myself. If people in China jump so high that they escape Earth's gravitational pull, Earth might very well be knocked out of orbit, albeit by a very tiny amount.

You are confusing the issue with the myth that the Earth will be knocked (permanently) out of orbit if all people in China jump up together (I am Chinese, so I can say this ;) ).

Theoretically, a big *enough* explosion CAN knock Earth out of orbit. How do you think bullets are fired from a gun?

I concur. It really isn't that hard. I think the most intimidating part is all the preamble stuff like \documentclass \includepackage, etc, etc. So, just get someone else's latex file, and replace whatever's between \begin{document} and \end{document} with whatever you want.

As you use it often enough, eventually you would know what the things in the preamble are for, and you can streamline your latex file. From a practical point of view, you don't have to make a most streamlined latex document from day 1. Chances are your computer is powerful enough to render the any difference in compilation time insignificant.

I personally find writing equations and symbols in LyX highly inconvenient. Moving my hand back and forth between keyboard and mouse is annoying.

But then again I am just speaking for myself, who only writes documents on mathematics and not other subjects.

Your father should let you decide for yourself whether it's anti-feminist, and also let you decide whether the *mathematical* ideas contained within are in anyway affected by any anti-feminist view (or the lack thereof).

From your response I see that I was not being clear in my race car analogy. I do have a tendency to be lazy and try to use as few words as possible, and expect others to fill in the gaps.

I was referring to a situation where the criteria for world record is covering every inch of a fixed length race track in the least amount of time. A car could start at the starting point of the track and stop at the finishing point by traveling at a sub-world-record speed through a shortcut (over the grass or something), in a time less than everyone else who actually sticks to the track. I wouldn't say that driver has broken a land speed record.