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Replies: 115   Last Post: Feb 15, 2010 5:36 AM

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 Bill Dubuque Posts: 1,739 Registered: 12/6/04
Posted: Jan 28, 2010 2:05 PM

"Ostap S. B. M. Bender Jr." <ostap_bender_1900@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.

>>> 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.

>> 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".

> 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?

> However, most children learn about functions before they understand
> the concept of a relation.

But there's no innate reason why that need be so, since both relations
and functions are quite elementary concepts with many concrete examples.

--Bill Dubuque

Date Subject Author
1/25/10 rehamkcirtap@gmail.com
1/25/10 amzoti
1/25/10 Andrew Usher
1/25/10 amzoti
1/25/10 Robert H. Lewis
1/25/10 rehamkcirtap@gmail.com
1/25/10 rehamkcirtap@gmail.com
1/26/10 Virgil
1/26/10 Porky Pig Jr
1/26/10 Bill Dubuque
1/26/10 Dave L. Renfro
1/26/10 Robert Israel
1/26/10 Porky Pig Jr
1/26/10 ostap_bender_1900@hotmail.com
1/27/10 Andrew Usher
1/27/10 Andrew Usher
1/28/10 Frederick Williams
1/27/10 achille
1/27/10 hdbanannah
1/27/10 rehamkcirtap@gmail.com
1/28/10 T.H. Ray
1/28/10 ostap_bender_1900@hotmail.com
1/28/10 Frederick Williams
1/28/10 Bill Dubuque
1/31/10 ostap_bender_1900@hotmail.com
1/31/10 Bill Dubuque
2/1/10 ostap_bender_1900@hotmail.com
2/1/10 T.H. Ray
2/1/10 ostap_bender_1900@hotmail.com
2/1/10 T.H. Ray
2/2/10 ostap_bender_1900@hotmail.com
2/2/10 T.H. Ray
2/2/10 ostap_bender_1900@hotmail.com
2/3/10 T.H. Ray
2/4/10 ostap_bender_1900@hotmail.com
2/5/10 T.H. Ray
2/6/10 ostap_bender_1900@hotmail.com
2/6/10 Frederick Williams
2/6/10 victor_meldrew_666@yahoo.co.uk
2/9/10 ostap_bender_1900@hotmail.com
2/10/10 victor_meldrew_666@yahoo.co.uk
2/10/10 ostap_bender_1900@hotmail.com
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2/12/10 ostap_bender_1900@hotmail.com
2/12/10 victor_meldrew_666@yahoo.co.uk
2/12/10 ostap_bender_1900@hotmail.com
2/12/10 Frederick Williams
2/12/10 victor_meldrew_666@yahoo.co.uk
2/12/10 ostap_bender_1900@hotmail.com
2/12/10 ostap_bender_1900@hotmail.com
2/13/10 victor_meldrew_666@yahoo.co.uk
2/13/10 Marshall
2/14/10 ostap_bender_1900@hotmail.com
2/12/10 ostap_bender_1900@hotmail.com
2/13/10 ostap_bender_1900@hotmail.com
2/13/10 Tim Golden http://bandtech.com
2/13/10 victor_meldrew_666@yahoo.co.uk
2/6/10 Marshall
2/3/10 Jack Markan
2/4/10 T.H. Ray
2/4/10 rehamkcirtap@gmail.com
2/5/10 T.H. Ray
2/5/10 Jack Markan
2/5/10 T.H. Ray
2/5/10 Jack Markan
2/5/10 T.H. Ray
2/6/10 ostap_bender_1900@hotmail.com
2/6/10 T.H. Ray
2/6/10 ostap_bender_1900@hotmail.com
2/7/10 T.H. Ray
2/8/10 ostap_bender_1900@hotmail.com
2/8/10 T.H. Ray
2/10/10 Michael Stemper
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2/11/10 victor_meldrew_666@yahoo.co.uk
2/11/10 T.H. Ray
2/8/10 Marshall
2/8/10 T.H. Ray
2/8/10 Marshall
2/9/10 T.H. Ray
2/8/10 ostap_bender_1900@hotmail.com
2/9/10 T.H. Ray
2/9/10 ostap_bender_1900@hotmail.com
2/8/10 ostap_bender_1900@hotmail.com
2/9/10 T.H. Ray
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2/10/10 T.H. Ray
2/10/10 ostap_bender_1900@hotmail.com
2/11/10 T.H. Ray
2/11/10 ostap_bender_1900@hotmail.com
2/12/10 T.H. Ray
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2/13/10 victor_meldrew_666@yahoo.co.uk
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2/12/10 Brian Q. Hutchings
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2/10/10 ostap_bender_1900@hotmail.com
2/10/10 ostap_bender_1900@hotmail.com
1/28/10 Frederick Williams
1/28/10 Jack Markan
1/28/10 Dave L. Renfro
1/28/10 Porky Pig Jr
1/28/10 Jack Markan
1/29/10 Michael Stemper
1/29/10 Jack Markan
1/28/10 Jack Markan