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

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Bill Dubuque

Posts: 1,739
Registered: 12/6/04
Re: Sixth grade math
Posted: Jan 28, 2010 2:05 PM
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"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
Read Sixth grade math
rehamkcirtap@gmail.com
1/25/10
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amzoti
1/25/10
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Andrew Usher
1/25/10
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1/25/10
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1/25/10
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rehamkcirtap@gmail.com
1/25/10
Read Re: Sixth grade math
rehamkcirtap@gmail.com
1/26/10
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1/26/10
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Porky Pig Jr
1/26/10
Read Re: Sixth grade math
Bill Dubuque
1/26/10
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Dave L. Renfro
1/26/10
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Robert Israel
1/26/10
Read Re: Sixth grade math
Porky Pig Jr
1/26/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
1/27/10
Read Re: Sixth grade math
Andrew Usher
1/27/10
Read Re: Sixth grade math
Andrew Usher
1/28/10
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1/27/10
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achille
1/27/10
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hdbanannah
1/27/10
Read Re: Sixth grade math
rehamkcirtap@gmail.com
1/28/10
Read Re: Sixth grade math
T.H. Ray
1/28/10
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umumenu@gmail.com
1/28/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
1/28/10
Read Re: Sixth grade math
Frederick Williams
1/28/10
Read Re: Sixth grade math
Bill Dubuque
1/31/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
1/31/10
Read Re: Sixth grade math
Bill Dubuque
2/1/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/1/10
Read Re: Sixth grade math
T.H. Ray
2/1/10
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
Read Re: Sixth grade math
T.H. Ray
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
T.H. Ray
2/6/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/6/10
Read Re: Sixth grade math
Frederick Williams
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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2/12/10
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2/13/10
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2/6/10
Read Re: Sixth grade math
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2/3/10
Read Re: Sixth grade math
Jack Markan
2/4/10
Read Re: Sixth grade math
T.H. Ray
2/4/10
Read Re: Sixth grade math
rehamkcirtap@gmail.com
2/5/10
Read Re: Sixth grade math
T.H. Ray
2/5/10
Read Re: Sixth grade math
Jack Markan
2/5/10
Read Re: Sixth grade math
T.H. Ray
2/5/10
Read Re: Sixth grade math
Jack Markan
2/5/10
Read Re: Sixth grade math
T.H. Ray
2/6/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/6/10
Read Re: Sixth grade math
T.H. Ray
2/6/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/7/10
Read Re: Sixth grade math
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2/8/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/8/10
Read Re: Sixth grade math
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2/10/10
Read Re: Sixth grade math
Michael Stemper
2/10/10
Read Re: Sixth grade math
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2/10/10
Read Re: Sixth grade math
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2/10/10
Read Re: Sixth grade math
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2/10/10
Read Re: Sixth grade math
T.H. Ray
2/11/10
Read Re: Sixth grade math
victor_meldrew_666@yahoo.co.uk
2/11/10
Read Re: Sixth grade math
T.H. Ray
2/8/10
Read Re: Sixth grade math
Marshall
2/8/10
Read Re: Sixth grade math
T.H. Ray
2/8/10
Read Re: Sixth grade math
Marshall
2/9/10
Read Re: Sixth grade math
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
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
2/10/10
Read Re: Sixth grade math
T.H. Ray
2/10/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/11/10
Read Re: Sixth grade math
T.H. Ray
2/11/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/12/10
Read Re: Sixth grade math
T.H. Ray
2/12/10
Read Re: Sixth grade math
Brian Q. Hutchings
2/12/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/12/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/13/10
Read Re: Sixth grade math
victor_meldrew_666@yahoo.co.uk
2/14/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/15/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/12/10
Read Re: Sixth grade math
Brian Q. Hutchings
2/9/10
Read Re: Sixth grade math
Brian Q. Hutchings
2/9/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/10/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/10/10
Read Re: Sixth grade math
Brian Q. Hutchings
2/10/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
2/10/10
Read Re: Sixth grade math
ostap_bender_1900@hotmail.com
1/28/10
Read Re: Sixth grade math
Frederick Williams
1/28/10
Read Re: Sixth grade math
Jack Markan
1/28/10
Read Re: Sixth grade math
Dave L. Renfro
1/28/10
Read Re: Sixth grade math
Porky Pig Jr
1/28/10
Read Re: Sixth grade math
Jack Markan
1/29/10
Read Re: Sixth grade math
Michael Stemper
1/29/10
Read Re: Sixth grade math
Jack Markan
1/28/10
Read Re: Sixth grade math
Jack Markan

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