Your, "I am not certain what you mean that it is not a matter of opinion" No issue: If "proportion reasoning" is a special case of "function reasoning", it follows that "function reasoning" is more useful than "proportion reasoning"... all opinions notwithstanding.
"... not teach functional reasoning..." Not at all what I meant. My own recommendation: teach the broader kind ... to impart the big picture ASAP ... to the extent that it is pertinent and can be fully common-sensible ... with emphasis on whatever special kinds warrant attention ... linearity, powers, exponentials, waves, etc. But too much detail, too early, forces haste, superficiality, alienation and withdrawal ... which is the prevalent mode.
From: Ed Laughbaum Sent: Monday, April 09, 2012 2:59 PM Cc: email@example.com Subject: Re: MLCS Webinar on April 24: Registration Open
Sorry I write in a manner that is confusing. I find writing a listserv post not demanding of too much time, and therefore I may not think through what I have said or what I want to say. It can then seem like "stream of consciousness" writing.
Ignoring your first three paragraphs, I would focus on my intention that (linear) proportional reasoning is a special case of (linear) functional reasoning. I am not certain what you mean that it is not a matter of opinion. Unless it is your way of saying that "why would you teach (linear) proportional reasoning when it is just a special case of linear functional reasoning, and thus one would teach linear functional reasoning?" Can we/should we stop there and not teach functional reasoning (with the implications that 2-year, 4-year, and 5-year college graduates be able to reason and process with a wide variety of real-number function types)?
Ed ====================================== On 4/9/2012 2:26 AM, Clyde Greeno wrote: ???!!!###
I can't make much sense of " ... focusing on proportional reasoning ... or whether we should be focusing on function...." Are we saying we better be focusing on functions *instead of* on proportionality? Or are we saying that we better expand our focus so that (often myopic) "proportional reasoning" should be attended as a special aspect of "function reasoning".
Mathematically, I take "proportion" to mean the set of all real-scalar multiples of any tuple of quantities or of numbers ... each of those tuples being a "ratio" within that proportion. In the case of tuples being ordered pairs, proportions are simply the mx functions ...[often called the y=kx "direct variation" functions, with (slope) k being its "constant of proportionality".]
In n-dimensional space, each proportion identifies with (n-1) functions. For example (from HS geometry), the "Pythagorean" proportion m*(3,4,5) [=(m3,m4,m5)] defines the first-place-controller function, x->y, where y= ((4/3)x, (5/3)x)) ... having (x,y) function-points (x, ((4/3)x, (5/3)x)). Likewise, m*(3,4,5) has a 2nd-place-conroller function and a 3rd place controller function
Since it appears that "proportional reasoning" is just a special case of "function reasoning", it is logically certain that "functional reasoning" is at least as useful as "proportional reasoning." That much is not at all a matter of opinion.
So, I find the dialog to be quite confusing.
From: wmackey Sent: Sunday, April 08, 2012 8:37 PM To: Guy Brandenburg Cc: Ed Laughbaum ; Clyde Greeno ; Clyde Greeno ; <firstname.lastname@example.org> Subject: Re: MLCS Webinar on April 24: Registration Open
I agree that functions are much more useful in everyday life.
Quoting Guy Brandenburg <email@example.com>:
-- Edward Laughbaum www.math.osu.edu/~laughbaum.6/ The Ohio State University 231 West 18th Avenue Columbus, OH 43210