T.H. Ray
Posts:
1,107
Registered:
12/13/04
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Re: Sixth grade math
Posted:
Feb 2, 2010 10:06 AM
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Ostap Bender wrote
> On Feb 1, 7:05 am, "T.H. Ray" <thray...@aol.com>
> wrote:
> > Ostap Bender wrote
> >
> >
> >
> > > 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.
> >
> > And probably better. :-)
> >
>
> I don't know about that, but to me, a function is an
> assignment of
> properties to the members of your set, like the
> weather report:
>
> New York -> 30 degrees, snow
> San Francisco -> 55 degrees, rain
> Los Angeles -> 69 degrees, sunny
>
> etc.
>
>
>
Ah, now I see the difficulty. No--the set of cities
with corresponding temperature and weather conditions
are not defined by a function until you define a
relation among them. A simple relation to define the
function is the season of the year; If these initial
conditions exist in winter, then in summer this set
of numbers will change at some particular time to
something like
New York, 78 degrees, sunny
San Francisco, 65 degrees, rain
Los Angeles, 85 degrees, partly cloudy
A function necessarily transforms one set of data to
another. The above is an example of a continuous
function; a discontinuous function might be the kind
found in non relativistic quantum mechanics, where
time = 0 and change is binary.
Tom
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