Several previous comments blamed how they had been taught, but I've found (I'm currently teaching a statistics course) that most students don't want the material to be made understandable. They want recipes to memorize.I've found that, too. But I've also found it to be true with humanities courses. I'm a college advisor, and I actually get a lot more students complaining about how humanities professors demand that they think than about how math and science professors do. I think that a lot of students come out of high school with an expectation that they'll be given answers to memorize, and they have trouble across the board in college when there's an expectation that they'll understand concepts and apply them to unfamiliar questions or problems.
Mathematics is the art of explanation.posted by Zephyrial at 10:43 AM on October 17, 2009 [27 favorites]
The saddest part of all this "math reform" are the attempts to "make math interesting" and "relevant to kids’ lives." You don’t need to make math interesting— it’s already more interesting than we can handle! And the glory of it is its complete irrelevance to our lives. That’s why it’s so fun!
Mathematics is viewed by the culture as some sort of tool for science and technology. Everyone knows that poetry and music are for pure enjoyment and for uplifting and ennobling the human spirit (hence their virtual elimination from the public school curriculum) but no, math is important.
At no time are students let in on the secret that mathematics, like any literature, is created by human beings for their own amusement; that works of mathematics are subject to critical appraisal; that one can have and develop mathematical taste. A piece of mathematics is like a poem, and we can ask if it satisfies our aesthetic criteria: Is this argument sound? Does it make sense? Is it simple and elegant? Does it get me closer to the heart of the matter?
Mathematics is the music of reason.
Math (and the basic hard sciences in general) is not something that can be debated, not something that has "two sides" that need explaining, not something that you can make up a bunch of woo-woo about to handwave away when you're not willing to think with precision.I can't imagine equating "coming to a predetermined correct answer" with "thinking with precision." It's true that some people can't deal with ambiguity and can't think creatively, and those people tend to like lower-level math and science, because there are right answers. At the top levels, math and science are all about ambiguity and creative thinking, and there aren't any clear answers, but most of us aren't capable of doing math or science at that level. I don't think you're doing math and science any favors, however, when you depict it as a refuge for people who can't deal with the complexity and open-endedness of human existence.
From all I hear, they tend to rote, which is not well regarded, and yet, said Asians are a cliche of math/science superiority, possibly deserved for all I know.I don't remember the details, but I read somewhere that it's a big myth that Asian countries teach math by rote and that places like Singapore focus much more on math concepts and much less on rote memorization than the U.S. does.
The semester I took Intro to Proofs, I and several others in my class became involved with research while we were still in the course. If that's not a low entry barrier, I don't know what is.The problem with this is that I decided that I didn't like math somewhere around the eighth grade. If students don't find out what's appealing about a subject until college, you're going to lose a lot of people.
49
x 15
------
245
49
------
735
To see if this is right, you can replace each of the 9s in the factors with 0s, then add them together:
49 => 49 => 4 + 0 = 4 15 => 1 + 5 = 6Then, multiply those numbers together:
4 x 6 = 24Then, add those numbers together (replacing 9s with 0s as before) :
4 x 6 = 24And add the numerals of those results together:
2 + 4 = 6Now, do the same (convert 9 to 0, add numerals together until it comes down to a single digit) for the result:
735 => 7 + 3 + 5 = 15 => 1 + 5 => 6So, the numbers match! You know that your original multiplication was probably correct. I think you can get false positives this way (e.g. if you came up with 6000, this method of verification wouldn't say you're wrong), but if you came up with something like 635 by not carrying a 1, it would tell you you did it wrong.
If you actually understand why that is, it's not really that interesting. Any time you raise e to an imaginary number, you end up involving trigonometric functions. here's an explanation
e^(pi*sqrt(-1)) = -1
If that doesn't make you feel something, either you don't understand the notation or you're dead inside.
Serre's goal in this section is to give a complete classification of the quadratic forms over the rationals. As preliminaries to reaching this goal, he introduces the reader to quadratic reciprocity, p-adic fields and the Hilbert Symbol. After these three, he spends the next chapter detailing the properties of quadratic forms over Q and Q_p (the p-adic field). The reason to work over Q_p is the Hasse-Minkowski Theorem (which says that if you have a quadratic form, it has solutions in Q if and only if it has solutions in Q_p). Using Hensels Lemma, checking for solutions in Q_p is (almost) as easy as checking for solutions in Z/pZ. After doing that, he spends yet another chapter talking about the quadratic forms over the integers. (Note: the classification goal is already achieved in previous chapter).posted by delmoi at 11:11 AM on October 29, 2009 [1 favorite]
posted by Namlit at 9:39 AM on October 17, 2009 [37 favorites]