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Topic: algebraic numbers
Replies: 17   Last Post: Jan 8, 2010 4:16 AM

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DrMajorBob

Posts: 1,448
Registered: 11/3/08
Re: algebraic numbers
Posted: Jan 6, 2010 6:02 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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@noqsi.com> wrote:

> On Jan 4, 4:00 am, DrMajorBob <btre...@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...@noqsi.com> wrote:

>> > On Jan 2, 3:05 am, DrMajorBob <btre...@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...@noqsi.com> wrote:
>> >> > On Dec 31, 1:16 am, DrMajorBob <btre...@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...@yahoo.com

>>
>> --
>> DrMajor...@yahoo.com

>
>



--
DrMajorBob@yahoo.com




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