We are a commune of inquiring, skeptical, politically centrist, capitalist, anglophile, traditionalist New England Yankee humans, humanoids, and animals with many interests beyond and above politics. Each of us has had a high-school education (or GED), but all had ADD so didn't pay attention very well, especially the dogs. Each one of us does "try my best to be just like I am," and none of us enjoys working for others, including for Maggie, from whom we receive neither a nickel nor a dime. Freedom from nags, cranks, government, do-gooders, control-freaks and idiots is all that we ask for.
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Tuesday, September 11. 2012
The comments on that post show some eddication, too. Math is the best test of intellectual rigor because, up through Calc 3, it doesn't require talent. Just IQ (ie pattern-recognition, pleasure in the application of logic, and ability to handle abstraction), some humility in witholding one's own precious opinions and bullshitting skills learned in high school as in soft courses, and good study habits.
That's why tough colleges use Calc as "weed-out" courses for so many majors, and why so many grad school entry exams include Calc and advanced math (eg Med School, Engineering, Accounting, Computer, Biz School, Bio, the hard sciences obviously, etc). It's about capacity for mental discipline.
However, many Prep Schools and large High Schools in the US today offer Calc 2 and even Calc 3 so brainy kids don't have to waste college time on them and can jump right into real "Higher Ed."
The sad thing is that many bright kids' brains do not mature at the same pace (due to myelinization and other things), and sometimes can process things easily at 20 which they could not have done at 16.
As they say, eddication - like sex - is wasted on the young. I have friends who have taken up higher math, Chinese, and accounting in adulthood, just for the enjoyable challenge and for life-enrichment. Even writing books about obscure 16th Century Dutch artists. Reading fiction and watching TV and movies do not suffice for the active, adventurous mind. In fact, I have a 65 year-old (not retired) golfing buddy who is taking up Sanskrit. I admire people like that, and do not particularly admire people who do not have serious intellectual, artistic, or religious pursuits, regardless of their age.
I might like or love them, but don't admire. Precious grey matter should not be put out to pasture, because it is a gift. Very rare and fortunate are those who can combine vocation with avocation.
If your kid doesn't know Calc 1 or Stats 1 in High School, your government is ripping you off. That does not happen in India, China, or Singapore.
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I like the way you have described Math in your first paragraph: "Math is the best test of intellectual rigor because, up through Calc 3, it doesn't require talent. Just IQ (ie pattern-recognition, pleasure in the application of logic, and ability to handle abstraction), humility in the lack of the need to apply one's own opinions, and good study habits." That certainly describes me perfectly during my freshman year in college (which was also my last year of taking math in college). I made it through multi-variable calculus and earned an A in the course without much difficulty. But I decided to stop there because it was clear to me that I had no real comprehension what I was doing. I did all the problem sets regularly and went to tutorials when I had questions, but while I could recognize, understand and apply the relevant patterns and guidelines, I was baffled as to where they came from, and I certainly lacked the spatial imagination to really grasp the implications of derivatives and integrals in three dimensions.
Same issue with me. Lacking in the abstract, spatial conceptualizing even though the premises and calculations were entirely do-able.
There are many fields in which people can excel until they reach the point at which talent is a limiting factor.
x^0 = 1 is a rather sophisticated result. One first needs to define what it means, it isn't as simple as, say, x^2. Then there are several ways to go from there, one way is to extend exponents to positive rationals, show that raising to a power is continuous on that set, and extend to the closure in the reals. Another approach is to define x^a = exp(a*ln(x)), but then one needs to prove the properties of exp and deal with the fact that ln is a multivalued function and that a cut is needed in the complex plane to make it a true function, after all, even x^1/2 has two possible values ;)
Most of this is done in a good introductory analysis course, which should probably also cover the construction of the reals using Dedekind cuts, but I don't think you will see it in the usual undergraduate calculus sequence.
I majored in engineering and the first thing you had to do as an engineering student was pass the two semester freshman calculus course. To be an engineering major you need a 650 on your mathsat. Math majors need a 750.
"How all occasions do inform against me,
And spur my dull revenge! What is a man,
If his chief good and market of his time
Be but to sleep and feed? a beast, no more.
Sure, he that made us with such large discourse,
Looking before and after, gave us not
That capability and god-like reason
To fust in us unused."
The sad thing is that many bright kids' brains . . . can process things easily at 20 which they could not have done at 16.
Me. Sorta. I mean, I could get an A in High School Algebra, but I was serious number-phobe. Of course, Algebra had letters, which helped!! But I didn't really have any idea what I was doing, or what the math was really for.
So I avoided math all through college and then after college suddenly I found myself doing math in my head without concentrating hard and even spontaneously applying algebraic concepts to practical applications. Which very much freaked me out, when I realized what I was doing, because I was supposed to be bad at math. Suddenly I wasn't such a math dunce.
On Being Self-Educated
Edurbation: Spreading wide the pages of a lovely book. Pages the color of cream,
Pages soft as silk. Spreading them open wide,
and with eyes and fingers together, fondling its words and ideas.
Then bringing up the book to the nose and burying your face in it,
breathing deep that musky tang of printer’s ink and pulp.
Then laying it back in your lap, to caress, to sigh over.
Drinking in knowledge. Sweet joy.
For the sake of pure pleasure and never for duty.
Back in the day, I remember Calculus 1 being offered in high school. The only reason I know is because I took it as a 7th grader, by the time I was a Senior in high school, I was taking Differential Geometry at Salem State - it's my only talent. Well, other than inserting my foot into my mouth from time to time and playing really lousy blues on my quitars.
The problem, as I see it anyway, is that not everybody is suited to Calculus. One can certainly learn basic calculus, but it is not a discipline that one can readily adapt to everything. Normally, a sound and complete background ( Algebra, Geometry, Algebra 2, Geometry 2, Trigonometry, Pre Calculus) is more than adequate for most professions - including engineering.
However, your point about keeping mentally active is spot on. Get curious about something or try to learn something useful or new everyday. My latest endeavor is birds of Lake Murray and their various calls.
There are a lot of birds around here I'll tell you what.
I remember Alg II as having a lot of clever stuff, but I doubt I use it. I certainly don't remember any specifics.
Calc on the other hand is very, very handy. Understanding what the 1st and 2nd derivative mean, the Mean Value Theorem, and integrals as the "area under the curve" allows a rapid understanding of all sorts of data, and provides a bases for making sound decisions.
I played the quitar too. It's like playing the guitar, but harder. Which is why I quit.
Speaking of statistics: There is a little slight of hand that happens when the stats show America is falling behind and India, China, or Singapore is eating our lunch in math and science. We count all the kids, the good, the bad and the ugly. In China or India or almost any country that gets compared with us well over half the children are not counted and compared statistically. They magically fall off the chart. If you aren't smart or your family cannot help you get a good education you don't go to high school. For example in India with 1.2 billion people probably less then 10% have ever even seen calculus never mind taken it in high school. In China with 1.3 billion people the numbers are even worse. Most of their 18 year olds didn't graduate from high school, aren't counted in the statistics and will never in their lifetime even hear about calculus. In general in these broad comparisons we compare 100% of our students with the best 10% or 20% of the other country's students.
Quite true. Most conservatives and most liberals believe it can't possibly be so, but American schools teach more to more children. We do better than anyone in the world. at least, our students do better, so let's assume the schools have at least something to do with that.
We are multiracial and have many students whose first language is not English/immigrants whose first years of schooling were substandard. If you break the PISA scores down by group, America outscores the world: African-Americans outscore all Africans, Caribbeans, and African-descended in other countries; American hispanics outscore all Latin- American countries; Americans of European or Middle-eastern descent outscore all parent countries (Finland may be an exception); American Asian students outscore all Asian countries.
I started as a math major, but dropped it for similar reasons to those above. I found I liked numbers, but not much else. Had I believed there was little other way to become employed, I would have stayed with it, I suppose. I still play with numbers and statistics.