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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 2, 2010 9:46 PM
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On Feb 2, 7:06 am, "T.H. Ray" <thray...@aol.com> wrote:
> 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
>


I was talking about the instantaneous weather report on the Weather
Channel or CNN: At this moment, the temperatures are:

New York -> 30 degrees, snow
San Francisco -> 55 degrees, rain
Los Angeles -> 69 degrees, sunny

>
> New York, 78 degrees, sunny
> San Francisco, 65 degrees, rain
> Los Angeles, 85 degrees, partly cloudy
>
> A function necessarily transforms...
>


What do you mean by "transforms"? Does the weather report transform
the set of cities into a set of temperatures? I would use the term
"assigns" instead of "transforms".

>
> ... one set of data to
> another.  The above is an example of a continuous
> function;
>


That would depend on your domain. If your domain is the Earth' surface
- then the current temperature is continuous. if your domain is the
set of 100 big cities - there can be no notion of continuity.

>
> a discontinuous function might be the kind
> found in non relativistic quantum mechanics, where
> time = 0 and change is binary.
>





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