>I couldn't disagree more. As an example, linear algebra is something >that has been bothering me for years, long after my college math >courses. One of its central topics is eigenvectors and eigenvalues. >Every textbook I can remember would demonstrate these with >pictures of a sheared rectangle (or a sheared box in 3d). That's fine, >but it isn't fun and its not motivating, so no surprise that the concept >never clicked.

>Then very recently, while browsing wikipedia I came across an >excellent demonstration of the concept: "eigenfaces" used in facial >recognition software. Its fun, useful, and it clicked.

I couldn't disagree more with your disagreement. Eigenvectors and eigenvalues are abstraction of things observed in many places and studied in one place so you can apply it to everywhere. Eigenface is simply an instance of linear space dimension reduction, whose implementation, by the way, is based on eigenvectors' connection to SVD and covariance matrix. Eigenface won't help with Taylor series definition of matrix exponential, nor solution of linear ODEs, nor Jordan canonical forms. Can anyone say those are not important topics of eigenvalues and eigenvectors?

If you truly want to understand something, there is no shortcut. You have to dig deep, look at and learn related topics, think hard about how and why scientists developed the subject this way. It takes time and concentration, a lot of them. Then you will gain something, and you need to keep at it to master it. No one said knowledge is easy, especially deep knowledge.

>You could draw a picture of a circle, emphasizing the line of the circle >as the boundary of the set. The invertible matrices are the interior of >the circle, and they complement the singular (non-invertible) matrices >which are represented by the boundary of the circle.

This demonstrates the pitfall of facile visualization, because the suggestion is wrong. There are dense sets with no boundary. The simplest example I can think of is rational numbers on the real line. It is dense on the real line, yet between every two rational numbers there is an irrational number, and between every two irrational numbers there is a rational number.

>But more importantly, the teacher should first explain why this might >be useful to know in the most concrete way possible, whether its >theory behind an applied technique in engineering or a lemma used >for an important abstract theorem (no handwaving - tell what's >important or significant about the abstract theorem).

This is easier said than done. At best, it is impractical; at worst, fantasy. Who is going to spend a week of lectures to explain one application in a possibly obscure engineering field. What about the fact no everyone is from an engineering department.

Sometimes theorems are useful for proving other theorems and then for proving other theorems. Gershgorin circle theorem is mainly useful for proving bounds about eigenvalues. That is it. I couldn't motivate more than that. It has application in numerical linear algebra, but it amounts to another proof and you need to go pretty deep in matrix analysis to appreciate it.

>Students (American or otherwise) who are willing to just put their >heads down and drill rote symbol manipulation are doing themselves a >disservice - it generally does not lead to much insight or >understanding of what they are doing (though it may in the >exceptional cases of very smart students), especially as the math gets >more advanced. Moreover, being adept at calculating is not useful in >the age of computers. You only need to do a calculation once (as an >algorithm in a computer program).

This I strenuously disagree. Math, like every human endeavor, requires practice and lots of it. If you cannot recall a pertinent theorem at will, then you will not be able to use it to prove it. You don't have to remember every theorem for all time, but when you need it you better. And practice does develop insights and understandings, which I can personally attest to. Advanced math especially requires a familiarity with basics, for no computer will prove for you a countra-positive.

I am opposed to rote learning. Who isn't? Specifically, I am opposed to Chinese teachers' mind-numbing deluge-of-exercises approach. All the proofs are nothing more than bags of tricks and they seem to take special delight to confuse students by not explaining things fully. I so detest that mindset. The U.S teachers are much better. There are good teachers and bad teachers, of course, and I suspect college professors can be a lot better if they actually put the necessary time in. American textbooks are leagues ahead. But one thing I have learned since is that in the end you have to remember the theorems and tricks because: THEY ARE MATH.

>More important is to gain insight and understanding into the nature >and limitations of the subject matter. Then the student is more likely >to recognize the cases where it can be applied after they're done with >the plug-and-chug problem sets at the end of the chapter (which >plague even advanced math textbooks in the form of boilerplate >theorem proofs). Of course, simply showing how to plug-and-chug is >easier for teachers so they praise obedient students who are easily >motivates and ask easy questions.

This I agree in general. It is after you gain insights and understanding can you innovate and advance. It is remarkable how many Ph.D.'s never master their fields. But it is hard and time consuming. My ideal of math textbook is Richard Courant's "Introduction to calculus and analysis." It is a perfect blend of mathematical rigor and insightful intuition. It is sad that I didn't have it as my introductory calculus textbook. It used to be the standard intro textbook in the West.

I suspect, and I could be wrong, the reason you blame teachers so much is that they didn't teach deep enough and you are smart enough to realize there is more. They had to be "easy" because it is already hard enough for some students. And this may be the ultimate problem with public education: it needs to teach the everybody but it can only do so by dumbing down the curriculum. That and it is expensive to teach but we want to do it on the cheap.