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Re: Sixth grade math
Posted:
Jan 31, 2010 2:34 PM
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"Ostap S. B. M. Bender Jr." <ostap_bender_1900@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"
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