
Re: Sixth grade math
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...@mydeja.com" <porky_pig...@mydeja.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 antiintuitive, 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 settheoretic extensional concept >> of function, i.e. as a singlevalued 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 settheoretical reduction of the notion of function is as I said. Whether or not that counts as a "blackbox" 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, rulebased, computable, etc). The concept of a relation >> and its associated properties of being total, singlevalued, 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 settheoretical 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

