Re: Re: algebraic numbers

*To*: mathgroup at smc.vnet.net*Subject*: [mg106263] Re: [mg106206] Re: algebraic numbers*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Wed, 6 Jan 2010 06:02:31 -0500 (EST)*References*: <hhc7a1$2o2$1@smc.vnet.net> <200912300912.EAA17052@smc.vnet.net>*Reply-to*: drmajorbob at yahoo.com

I've always told people, "I test smarter than I really am," and now I see... I was right! But not because I worked hard or my parents got involved in my schoolwork, as the New Yorker article suggests. At least, I didn't think so, until I thought about it some more and came up with some factoids: a) My grandmother bought me comic books... and I READ them. b) I participated in summer reading programs at the local library (voluntarily). c) My mother coached me for spelling bees, twice. d) She took dictation for my history notebook one summer when I (voluntarily) went to summer school. e) Nobody told me math was hard, that I can remember. f) Comics led me to science fiction, which I read like a house on fire. So the article makes more sense than I originally thought. Highly recommended. Thanks for the link! Bobby On Tue, 05 Jan 2010 00:44:27 -0600, Noqsi <jpd at noqsi.com> wrote: > On Jan 4, 4:00 am, DrMajorBob <btre... at austin.rr.com> wrote: >> > The issue here is >> > whether the student has enough common culture with the test writer to >> > find the same answer. And that's *always* an issue. >> >> So those are cultural conformity questions?!? > > One might not need to conform, but one must at least understand the > culture. Mathematics is a human cultural artifact, and students are > going to need to understand some things about that artifact and its > expression to be successful in college. > > Specifically in this case series are often presented as specific terms > and ellipsis, judged to be easier to comprehend in some ways than a > formula, so the student should be able to comprehend that form. > > And this continues into professional life. Today I'm looking over the > specs of a megapixel image sensor. The drawings that document its > structure contain "..." in a number of places: it's not practical to > show every pixel! I can, of course, think of all kinds of perverse and > stupid ways to misunderstand what's omitted, but that wouldn't be > helpful in any way. > >> >> That's even worse than I thought! > > It's still worse. The intentions behind the widespread adoption of the > SAT didn't really address the need to establish that the student could > comprehend the academic cultural context: instead, they were > consciously bigoted. > > http://www.newyorker.com/archive/2001/12/17/011217crat_atlarge > >> >> Bobby >> >> >> >> On Sun, 03 Jan 2010 02:40:36 -0600, Noqsi <j... at noqsi.com> wrote: >> > On Jan 2, 3:05 am, DrMajorBob <btre... at austin.rr.com> wrote: >> >> When I clicked on the link below, the search field was already >> filled = > >> >> with >> >> the sequence >> >> >> target = {1, 2, 3, 6, 11, 23, 47, 106, 235}; >> >> >> Searching yielded "A000055 Number of trees with n unla= > beled >> >> nodes." >> >> >> I tried a few Mathematica functions on it: >> >> >> FindLinearRecurrence@target >> >> >> FindLinearRecurrence[{1, 2, 3, 6, 11, 23, 47, 106, 235}] >> >> >> (fail) >> >> >> FindSequenceFunction@target >> >> >> FindSequenceFunction[{1, 2, 3, 6, 11, 23, 47, 106, 235}] >> >> >> (fail) >> >> >> f[x_] = InterpolatingPolynomial[target, x] >> >> >> 1 + (1 + (1/ >> >> 3 + (-(1/ >> >> 12) + (7/ >> >> 120 + (-(1/ >> >> 60) + (1/144 - (41 (-8 + x))/20160= > ) (-7 + x)) (-6 + >> >> x)) (-5 + x)) (-4 + x)) (-3 + x) (-= > 2 + x)) (-1 + x) >> >> >> and now the next term: >> >> >> Array[f, 1 + Length@target] >> >> >> {1, 2, 3, 6, 11, 23, 47, 106, 235, 322} >> >> >> But, unsurprisingly, the next term in A000055 is 551, not 322. >> >> >> A000055 actually starts with another three 1s, but that doesn't >> change >> >> things much: >> >> >> target = {1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235}; >> >> >> FindLinearRecurrence@target >> >> >> FindLinearRecurrence[{1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235}] >> >> >> (fail) >> >> >> FindSequenceFunction@target >> >> >> FindSequenceFunction[{1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235}] >> >> >> (fail) >> >> >> f[x_] = InterpolatingPolynomial[target, x] >> >> >> 1 + (1/24 + (-(1/ >> >> 40) + (1/ >> >> 90 + (-(1/ >> >> 280) + (1/ >> >> 1008 + (-(43/ >> >> 181440) + (191/3628800 - (4= > 37 (-11 + x))/ >> >> 39916800) (-10 + x)) (-9 + = > x)) (-8 + x)) (-7 + >> >> x)) (-6 + x)) (-5 + x)) (-4 + x) (-3 + x) = > (-2 + x) (-1 + >> >> x) >> >> >> Array[f, 1 + Length@target] >> >> >> {1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, -502} >> >> >> So I ask you, from the data alone: what's the next term? >> >> > It's the sort of question where one might expect a specialist to >> > recognize a familiar sequence. It's all context. >> >> > Consider that in a narrow mathematical sense, spectroscopy is an >> > utterly ambiguous, "ill conditioned" problem. But show me a gigagauss >> > cyclotron spectrum, and I'll recognize it as such (see the >> > acknowledgment at the end of arxiv.org/pdf/astro-ph/0306189: the >> > authors were struggling to contrive an interpretation from atomic >> > physics before one of them showed the spectrum to me). But I expect >> > very few could do this, since few have the background. >> >> >> If one had the Encyclopedia of Integer Sequences handy, those SAT >> >> questions could be interesting. But they'd still be nonsense. >> >> > No they are not. Remember that the SAT isn't about the ability of a >> > student to function in some ideal abstract world of infinite >> > possibility. In the real world of academia, every single question they >> > will encounter will be ambiguous in some sense. The issue here is >> > whether the student has enough common culture with the test writer to >> > find the same answer. And that's *always* an issue. >> >> >> Bobby >> >> >> On Fri, 01 Jan 2010 04:32:58 -0600, Noqsi <j... at noqsi.com> wrote: >> >> > On Dec 31, 1:16 am, DrMajorBob <btre... at austin.rr.com> wrote: >> >> >> >> This is a little like those idiotic SAT and GRE questions that ask >> >> >> "What's >> >> >> the next number in the following series?"... where any number >> will = > >> >> do. >> >> >> Test writers don't seem to know there's an interpolating >> polynomial= > >> >> (for >> >> >> instance) to fit the given series with ANY next element. >> >> >> > Explanations in terms of epicycles may be mathematically adequate >> in= > a >> >> > narrow sense, but an explanation in terms of a single principle >> >> > applied repeatedly is to be preferred in science. The ability to >> >> > recognize such a principle is important. >> >> >> > And my mathematical logician son (who's looking over my shoulder) >> >> > directed me tohttp://www.research.att.com/~njas/sequences/for >> >> > research on this topic. When he encounters such a sequence in his >> >> > research, he finds that knowledge of a simple genesis for the >> sequen= > ce >> >> > can lead to further insight. >> >> >> -- >> >> DrMajor... at yahoo.com >> >> -- >> DrMajor... at yahoo.com > > -- DrMajorBob at yahoo.com