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Topic: Sixth grade math
Replies: 115   Last Post: Feb 15, 2010 5:36 AM

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ostap_bender_1900@hotmail.com

Posts: 681
Registered: 2/1/08
Re: Sixth grade math
Posted: Feb 1, 2010 7:56 AM
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On Feb 1, 3:58 am, "T.H. Ray" <thray...@aol.com> wrote:
> Ostap Bender wrote
>
>
>

> > On Jan 31, 11:34 am, Bill Dubuque
> > <w...@nestle.csail.mit.edu> wrote:

> > > "Ostap S. B. M. Bender Jr."
> > <ostap_bender_1...@hotmail.com> wrote:
>
> > > > On Jan 28, 11:05 am, Bill Dubuque
> > <w...@nestle.csail.mit.edu> wrote:
> > > >> "Ostap S. B. M. Bender Jr."
> > <ostap_bender_1...@hotmail.com> wrote:
> > > >>> On Jan 26, 4:48 pm, Bill Dubuque
> > <w...@nestle.csail.mit.edu> wrote:
> > > >>>> "porky_pig...@my-deja.com"
> > <porky_pig...@my-deja.com> wrote:
> > > >>>>> On Jan 25, 8:52 pm, eratosthenes
> > <rehamkcir...@gmail.com> wrote:
>
> > > >>>>>> I am currently tutoring at a community
> > college and one of my students
> > > >>>>>> is a sixth grade teacher going back to
> > school because she needs a
> > > >>>>>> calculus credit to complete her
> > certification or something, but that
> > > >>>>>> is beside the point.
>
> > > >>>>>> She asked me if I could give her a way to
> > explain what a mathematical
> > > >>>>>> function is to a sixth grader other than the
> > standard explanation
> > > >>>>>> about it being a machine that you put one
> > number into and receive
> > > >>>>>> another out of. She said that this does not
> > work.
>
> > > >>>>>> I tried explaining how I learned long ago:
> > As a map where the
> > > >>>>>> equation is the directions or something like
> > that. She was also
> > > >>>>>> dissatisfied with that.
>
> > > >>>> How does one make any sense of "map where
> > equation is the directions"?
>
> > > >>>>>> Any thoughts?
>
> > > >>>>> If f: R -> R, where R is a real line, then
> > thinking of f as a machine
> > > >>>>> that you put one number into and receive
> > another out is adequate.
>
> > > >>> Sort of like changing hands in draw poker? You
> > discard a card and
> > > >>> get a new one from the deck?
>
> > > >>> To me, if we are talking about  R -> R maps,
> > this "black box" model is
> > > >>> highly anti-intuitive, as it hides the
> > continuous nature of R. To me,
> > > >>> real functions are usually associated with
> > graphs (as in "plots").
>
> > > >> But we're talking about abritrary functions, not
> > just continuous ones.
>
> > > > Very few children in 6th grade are familiar with
> > many R -> R functions
> > > > that aren't continuous almost everywhere.
>
> > > But we're talking about arbitrary functions, not
> > only (continuous) real.
>
> > > >>>>>  In  fact, I can't think of anything better
> > than that. There may be some
> > > >>>>> simple formula associated with matching the
> > input with the output, or
> > > >>>>> not. I don't know if they also need to know
> > what is injection,
> > > >>>>> surjection and bijection, but that comes
> > after the basic definition.
>
> > > >>>> The problem with "thinking of f as a machine"
> > is that it is far too
> > > >>>> intensional to convey the modern set-theoretic
> > extensional concept
> > > >>>> of function, i.e. as a single-valued total
> > relation between sets.
>
> > > >>> You are right. No mathematician, other than a
> > logician or a set
> > > >>> theorist, would think of real functions as
> > black boxes.
>
> > > >> You're confused. The set-theoretical reduction
> > of the notion of function
> > > >> is as I said. Whether or not that counts as a
> > "black-box" definition
> > > >> I can't say since you haven't defined what you
> > mean by that vague term.
>
> > > > How does the statement "No mathematician, other
> > than a logician or a
> > > > set theorist, would think of real functions as
> > black boxes" contradict
> > > > your statement: "The set-theoretical reduction of
> > the notion of
> > > > function is as I said"? I didn't say anything
> > about set-theoretical
> > > > reduction, did I?
>
> > > I didn't say it did. "You're confused" refers to
> > you apparently
> > > believing that I agree with what you wrote after
> > "you are right".
>
> > > >>>> Indeed, said "definition" doesn't even specify
> > what it means for two
> > > >>>> functions to be equal, so it does not make it
> > clear that the concept
> > > >>>> of function is independent of any particular
> > representation (e.g.
> > > >>>> analytic, rule-based, computable, etc). The
> > concept of a relation
> > > >>>> and its associated properties of being total,
> > single-valued, etc
> > > >>>> are certainly elementary enough that they
> > could easily be taught
> > > >>>> at an early age, and motivated with many
> > concrete examples.
>
> > > >>> The only definition of a function f: A -> B
> > that I am aware of, is
> > > >>> that of a set of tuples (relations) in AxB,
> > where each element of A
>
> > > >> You mean "a relation" not "relations".
>
> > > > Yes.
>
> > > >>> occurs exactly once. Of course, this is not the
> > most intuitive
> > > >>> definition of the real functions either.
>
> > > >> Again, we're talking about general functions,
> > But, out of curiosity,
> > > >> what rigorous definition of a real function do
> > you think is more
> > > >> "intuitive than the standard set-theoretical
> > definition?
>
> > > > Please remind me what you call "the standard
> > set-theoretical
> > > > definition". I am not a set theorist.
>
> > > I said that above: "a single-valued total relation
> > between sets"
>
> > To be honest, at this point I have lost the reason
> > what we are arguing
> > about, given that I agree with you that the
> > definition of a function
> > is:

>
> > A binary relation f between two sets A and B is a
> > subset of A × B.
> > A function f: A -> B is a single-valued total binary
> > relation between
> > sets A and B.
> > That is, each element of A occurs exactly once in f.

>
> > My main point was that under this definition, a
> > function doesn't look/
> > feel to me like a "black box".

>
> "Black box" simply means that we only concerned with
> the input and output of a function, not the operations
> that produce the output.  The term can only be
> meaningful in the context of mechanical calculation,
> where we understand how the arithmetic works, and
> so can ignore the complicated maps produced by sub-
> operations, in favor of the two parameters that
> define the black box function.
>


Well, I guess that having received high school education in Russia, my
intuition works a little differently, that's all.



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