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

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T.H. Ray

Posts: 1,107
Registered: 12/13/04
Re: Sixth grade math
Posted: Feb 3, 2010 6:48 AM
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Ostap Bender wrote

> 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
>

> >
The set of data is not a function, is it? The set is
generated by the broadcast network's function of
soliciting the data by all the sub-operations reporting
information to the point of your receiving it.

> > 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".
>

> >
No. The weather report is the result of the function
that transforms one set of data into another. It is
certainly a transformation, not an assignment of values,
because it is not arbitrary. There do exist arbitrary
assigments of values that are functions; e.g., ordering
some group of people by height. Even in such cases,
however, transformation applies--the order function
transforms a random distribution of values into an
ordered sequence.

> > ... 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.
>

> >
The domain for climate is, of course, the earth's
surface and atmospheres--what else would it be? The set
of cities is most certainly embedded in that surface
and their climate (by the fixed point theorem)is a
function of changes in that surface and atmosphere.
Brouwer, who gave us the fixed point theorem, also gave
us the deeper result: all real functions are continuous.

Tom

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

>
>



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1/25/10
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1/28/10
Read Re: Sixth grade math
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1/28/10
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1/28/10
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1/31/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
1/31/10
Read Re: Sixth grade math
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2/1/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/1/10
Read Re: Sixth grade math
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2/1/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/1/10
Read Re: Sixth grade math
T.H. Ray
2/2/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/2/10
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2/2/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/3/10
Read Re: Sixth grade math
T.H. Ray
2/4/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/5/10
Read Re: Sixth grade math
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2/6/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/6/10
Read Re: Sixth grade math
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2/6/10
Read Intuitionistic ignorance
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2/9/10
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ostap_bender_1900@hotmail.com
2/10/10
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2/11/10
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2/4/10
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2/6/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
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2/6/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
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2/8/10
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ostap_bender_1900@hotmail.com
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2/10/10
Read Re: Sixth grade math
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2/10/10
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2/10/10
Read Re: Sixth grade math
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2/10/10
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2/10/10
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Read Re: Sixth grade math
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T.H. Ray
2/8/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/9/10
Read Re: Sixth grade math
T.H. Ray
2/9/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
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Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/9/10
Read Re: Sixth grade math
T.H. Ray
2/9/10
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ostap_bender_1900@hotmail.com
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T.H. Ray
2/10/10
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1/28/10
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