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The Mountain View library has been pretty useful to me so far. During my last trip there, I found a copy of Martin Gardner's aha! insight, which is on the recommended list for tech interviews at techinterview.org. I checked it out, plus a book called The Advent of the Algorithm by David Berlinski. I have already read some parts of this book. Depending on how much I like the rest of it, I may buy it. (I've seen it on sale at BookBuyers.)

You may recall that I posted some time ago about a book my eleventh grade math teacher gave me that covered, among other things, the relative sizes of various infinite sets. As it turns out, I knew something about this long before then. I first learned about it in a book called Realm of Numbers by Isaac Asimov. The Mountain View library has a copy of the book in the children's section. Reading parts of it brought back a few fond memories. I think I read that book when I was in either the second or third grade, not too long after my family moved into our first house. I don't remember what I thought of infinite sets; it is unlikely I attributed any significance to them. I sort of vaguely recall wanting to be an astronaut sometime around then, as many kids did who were in grade school during the 1960s, what with the Apollo space program and all. (I later found out I couldn't be an astronaut because of my nearsightedness.)

So, I'm still somewhat puzzled why classes like 6.045 gave me so much trouble, seeing as I was familiar with the basis for the classes already. Maybe it was because I went so long without thinking about those subjects that I had to relearn them in the classes, whereas other students had been thinking about them all along. I was reading soc.college.admissions, and there is an article where the original poster (OP) suggests that students who are interested in math (or subjects that use math) take some time off between high school and college to study it independently, because there isn't enough time in a normal curriculum to cover everything in depth for someone who isn't already mathematically mature. In my case, I doubt that would have made much difference. No one in my family is into higher math, so I wouldn't have had anyone to guide my learning. I would've had to go to some kind of college, and since MIT accepted me, it made sense to go there.

The person following up to the OP made an analogy between students studying math and music. The suggestion was that a student who is (already a) virtuoso is more likely to get a lot out of college than someone who goes to college to learn how to be a virtuoso. That got me to thinking: how does admission to a place like MIT differ from admission to a music college of the same caliber? One major difference is that you have to audition to get into a top music college, and the pieces you perform are part of a standard repertoire that is universally recognized by music academics. AFAIK, you don't have to audition to get into MIT or anyplace like it. Lots of students have already taken a fair amount of college (or at least AP) level math, biology, chemistry, etc., but it isn't a requirement to do so. The admissions committee will accept students who show the potential do well at MIT, even though they may not have the same background as students who've already satisfied several MIT requirements before they go there. This is a very controversial subject, especially among students who don't understand why they didn't get into MIT even though they have done well in many AP classes, in addition to having SATs, ACTs, etc. that are at least on par with other admits. But how would this work in music schools? Do music schools ever admit students who show the potential to be virtuosos, even though they may not have completed the same amount of music training as other students who already are? What kind of experience is a student likely to have who does not have that level of music training? At MIT, if you haven't taken AP classes, you can still take classes that do not assume AP-level knowledge, but there is a much steeper learning curve for these students than those who already come in with AP-level knowledge. Having the first semester as pass/no record helps somewhat, and there are some students who are able to make up the difference. How possible is this at a music school?

Speaking of music, time to go work on my sonatina.


( 7 comments — Leave a comment )
Nov. 1st, 2004 01:25 pm (UTC)
I can only give you the answer from the singer's point of view, which is a bit different from other musicians': Singing, for instance, is a life-long thing, so one can enter college/music school behind other musicians, and pick up the things one didn't know, and by the time one's voice is mature, come out the other end with a balanced approach. Most singers should start singing "late."

Secondly, at least from my point of view, there is a whole sub-class of student who just "mature late," and by that, I don't mean vocally. I mean, in general. They get good grades in High School, but don't come into their best learning until later - and when they do, things just "click." I was one of these, and was also lucky to be a musician, luckier still to be a singer. Nothing came together for me until college. I think it was the way I was "drawn." I didn't get advanced mathematics until I was in college - just didn't understand them, and then, hey, presto! Something turned on, learning-wise, in my brain.

Finally, back to the music school analogy: there is no pass/fail, so there is an assumption of basic ability that needs to be honed. They just assume that not everyone is going to come in at a virtuosic level, but that everyone comes in with a specific set of gifts. That's why some people don't go to music school, have a career, and then come back for more training - because they have the playing down, and a basic musicological understanding; why others come to music school as an undergrad, and leave performing - because they came in with the chops, expanded them, and learned what they needed to know; and why others stay all the way through - they're running "behind," but can "stall."
Nov. 1st, 2004 10:54 pm (UTC)
Hmmm ... When I was a grad student at UCLA, I met someone who decided to switch majors to music. I don't think she had voice training prior to that, although she had some flute experience. She worked really hard to bring herself up to the level of other voice majors, and it seems to have paid off.

One thing I noticed among the UCLA voice majors was that virtually all of them had a private teacher. But I don't know of any MIT or UCLA CS students who had private CS teachers. I suppose one difference between music and CS education is that CS education isn't nearly as standardized. Also, people don't specialize the way they do in music unless they pursue PhDs (although that is changing somewhat).

Just curious ... what types of advanced math do/did you know? Are you interested in what I'm writing about (e.g. infinite sets)?

Nov. 2nd, 2004 12:56 pm (UTC)
Music, standardized? Gargh. The courses definitely are, but you need to look at those, with the exception of theory, as generalized courses - they're mostly historical, for instance. But training - training instruments, whether voice or an outside instrument, is definitely NOT standard. For instance, at the school I teach, there are 7 voice teachers. Only two of them teach the same (so they say) - and they do, because they're doing a "method." The other of us have very different approaches. I'm very physical in my teaching, leaning toward breath control as one of two main components to singing, leaving direct manipulation of the larynx alone, while the Estill folks are heavily into laryngeal manipulation, and hardly mention breathing.

You want to look at private teachers are an apprenticeship, not a course, though we do get course credit for it in today's settings. The reason for individual instruction is to bring different levels of different gifts out. Since most of what the aim of music is about is to bring people to a performance level, and since, even in a group, one performs alone, it makes sense to study privately - there are other reasons, too, particularly in voice - and, actually, especially in voice, these reasons are more compelling.

Re: math - well, not advanced to *you* - advanced to me, like basic algebra. You have to understand - my mathetics background is so horrid that when I did the SATs, I scored a perfect verbal (which I did again on my GREs) and my SAT scores were still so bad I didn't use them to get into college - I used the ACT, which was geared toward the verbal.

I'm not proud of that...
Nov. 2nd, 2004 09:41 pm (UTC)
I was thinking that music tends to be more standardized than CS from a standpoint of what you're supposed to study. There are certain pieces you need to know (such as the ones listed at the ABRSM site). While there might be debate on what the best method is for learning/performing a piece, there's a fair amount of agreement about what pieces constitute a certain level of accomplishment. Whereas in CS, there isn't much consensus about what needs to be learned or what's the best way to learn it.
Nov. 2nd, 2004 10:23 pm (UTC)
I'm beginning to see why some people don't think singers are musicians - because I've never seen a "core competency" list for voice...
Nov. 3rd, 2004 12:17 am (UTC)
Hmmm ... I checked the ABRSM singers exam list and it seems pretty comprehensive. What do you think? I worked on some of those with my last voice teacher -- we did Summertime and Vergin, tutto amor right before I moved away. Interestingly enough, there's a guy in my chorus who's taking a class that uses the book of Italian arias that includes the latter. He's been trying to get me to go, but I don't have time right now.

What criteria do you use for selecting pieces for your students to work on, especially those who want to study voice at a music college or conservatory?
Nov. 3rd, 2004 03:12 am (UTC)
There's definitely some material that is supposedly "the bomb" for beginning students - the Italian book you mention is one of those things - but in my humble (but well-educated ;-) opinion, I believe that it's more about covering eras and areas of vocal interest.

If, say, for instance, one is a soprano (like me) with a bigger voice (like me) one wants to be able to have on one's repertoire list the things that one will expect to use at the stage in one's career one is in - so, for me, there would be a list of Puccini arias/roles, a list of modern-opera repertoire, and a list of various art-songs, in various languages. It would be a sort of "pick 10 pieces from this category, and 5 from this, and..." thing. This is how it was for me - a list of things that fit my voice, but weren't universal. If I were musical theatre, I would have a list of musicals in my "fach" that would work, and so on.

The reason I don't think it works as well for voice, and find the list a little bogus, is that each voice is slightly different, and needs to be treated differently. When I pick music for my students, or help them pick music, I'm looking for things that a) they can achieve now in a satisfactory performance; b) things that they can grow into (these may be the same items, or not); and c) things that suit their current range - not a projection (until they're advanced/old enough to know what that possible projection might possibly be). I don't rely on the standard rep for everyone. In the case of one of the young ladies I saw today, for instance, we head toward more sophisticated musical choices, with less vocal weight. She's too young/untrained to do big vocal tour-de-forces (like "Summertime," for instance) but too musically mature (having studied piano for quite some time) to enjoy the early Italian repertoire. We're doing lighter Faure. It suits the criteria - she can achieve a good performance of it in the now, but it's a little challenging. If she continues to work on voice, she can use it in the future as well, and find different and wonderful things to do with it - and it suits her vocally and musically.

My repertoire list, and the lists of the people I consider to be most successful, isn't a formula - but it suits me well.

I find that the "formula lists" work well with other instruments - like "method books" for piano, etc. Voice people search for that method (like those at my institution who do Estill) but I feel that it's a bad choice for voice, because "one size" does NOT "fit all."

And there's so much regional variation, too, as to what a voice type is - some people can argue that I'm a lyric soprano, others will argue that I'm a lyric-spinto, and the Boston area will even come up with "mezzo-soprano" from time to time, based on the sound-preference and local comparisons. If that can happen to me (I'm pretty vocally stable and rather bland) it can happen to more interesting voices out there.

Whereas a piano is strictly about skill and artistic impression, not about color, timbre, etc. as well as skill and artistry.
( 7 comments — Leave a comment )

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