T.H. Ray
Posts:
1,107
Registered:
12/13/04
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Re: Sixth grade math
Posted:
Feb 1, 2010 6:58 AM
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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.
Tom
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