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Re: algebraic numbers
*To*: mathgroup at smc.vnet.net
*Subject*: [mg106206] Re: algebraic numbers
*From*: Noqsi <jpd at noqsi.com>
*Date*: Tue, 5 Jan 2010 01:44:27 -0500 (EST)
*References*: <hhc7a1$2o2$1@smc.vnet.net> <200912300912.EAA17052@smc.vnet.net>
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
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