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.
> > >>> 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?
> > >> 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? >
Please remind me what you call "the standard set-theoretical definition". I am not a set theorist.
> > > 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. >
I suppose so. But that would depend on what you consider to be the most intuitive sequence of presenting math to children, especially in the view of how they are being taught math in school. It could also depend on the individuality of the child. The concept of axioms and theorems is also elementary, but many schools teach them quite late.