"...free boys shall be taught calculation, a "purely childish" art, by pleasant sports, with apples, garlands etc. It makes men "more useful to themselves and wide-awake."
Having read that quote by Plato, I have to agree with Lou that algorithmic arithmetic was still in its infancy, with Plato's Academy offering a relatively superficial treatment of what we might call "business math" (accounting, bookkeeping, sums).
To this day, there's a sense among mathematicians (many of them) that "arithmetic" as we call it is a vocational skill that, if not orthogonal to mathematics, is certainly not its essential core.
Keith Devlin, whom we've mention a lot in this forum, has his book on the number sense in animals, their ability to dead reckon, omni-triangulate and so on. Whether we insist such skills must involve "counting" or not is a side issue when the definition of "numeracy" is suitably broad as to include birds flocking from point A to point B with pinpoint precision.
We don't accredit bird brains with any arithmetic ability, but again, in Lou's namespace, as in Devlin's, the arithmetical is far from synonymous with the mathematical.
Thinking of Sequence and Success in approaching cyber-world, that virtual space in which the imagination is more shared than it used to be, in daydreaming.
There's a way to start with
(a) Von Neumann architecture and discuss the CPU versus RAM versus ROM versus... the internal anatomy of a stored instruction set computer.
There's another way to start with
(b) the relationships between hosts, leaving the hosts themselves as "black boxes" for now and studying their network for telecommunications. This is where TCP/IP comes in, and kid-friendly movies like 'Warriors of the Net' (not war-like at all).
The student beginning in (a) will eventually expand her or his knowledge base to encompass (b) and vice versa. So do they both end up in the "same place"?
I would say "no" (that's my bias) because I think sequence matters and if you learn TCP / IP and networking and start thinking in terms of the whole globe, data centers, rack space, "Server Sky" (Keith Lofstrom) and so on, that you'll have a global vision on which to build. Other infrastructure weaves in. You're thinking more geographically, about the whole earth.
I'd say the younger you become fluent with a "whole Earth" point of view (happy with Google Earth and like tools, thinking in terms of a biosphere), the better suited you'll be for some kinds of later work.
(b) --> (a) and (a) --> (b) are not identical sequences. They're complementary but not the same.
Likewise, there are a great many sequences into math and STEM more generally.
Some (e.g. mine) start with Polyhedrons much earlier, others barely / never get to them. Some touch on computer languages or use them, pre end of high school. Others never do. These differences matter.
So do I think the job at hand is to sort through quasi-infinite possible sequences to figure out which ones would best lead to the best outcomes?
That's not my favorite focus. I'm more interested in what sequences *are* being used, which seem effective, which new ones are being tried, and what future sequences might be devised.
Consider the analogy with "religion": I'm not interested in what the "best religion" might be, I'd rather just have a clear picture of the many alternatives, thereby absorbing design patterns for starting some new ones (or rehabilitating old ones -- "fixer upper" religions (maybe Mithric?)).
Many people in the education field are working backwards from the tests they want to see passed. I'm kind of that way too, but my tests are so different.
"Are your young people developing smartphone apps to help deal with the problem of food waste in metro areas?" No? Well you might be failing, as a city or metro area, in my grade book then. You're being "left behind".
Or maybe the phone app is not necessary. We don't use one in my zip code, yet are addressing this issue.
Who I consider "faculty" in my school has a lot to do with what kind of track record I see. I realize there are many ethnicities out there, new ones forming all the time (old ones dissipating).
I do not attach meaning to the idea of "mathematics divorced of all ethnicity". I look for leanings towards architecture, urban planning. A love of maps, reading them, is important.
Are you directionally challenged? You'll need to work on that if wanting to rise through the ranks in my school system. Helps to have outdoor skills too, like those birds and bees in Devlin's book. If all you know how to do is sit on a chair in your cubicle and noodle numbers, then I don't think of you a "well educated" enough to be a good fit, probably.
If these sound more like STEM teachers than math teachers, that's intentional. Slogans like "math is an outdoor sport" are somewhat unique to my academy (not the same as Plato's).
Those people who collect personal data in large amounts and upload it into the cloud are behaving more like STEM teachers than the ones who think psychic and physical coordination are miles apart.
The Greeks did not (separate these forms of coordination) as much, so in that way I think I inherit from their Metaphysics / Olympics.
On Fri, Nov 2, 2012 at 1:47 PM, Robert Hansen <email@example.com> wrote: > > On Nov 2, 2012, at 3:19 PM, kirby urner <firstname.lastname@example.org> wrote: > > The > primacy of arithmetic is simply an artifact of a curriculum that > denies entry to those who haven't acquired proficiency at arithmetic. > > > This statement by Lou needs to be backed up with examples of arithmetic-less > children yet with mathematical development. The only way I can see doing > this is to not teach children to count, but I suspect that he is talking > about calculators. The problem with the belief that calculators hold some > special magic is that they have been with us for 30 years and there are > probably 2 or 3 in every house in the country. We probably have 5, not > counting computers, and my son's favorite possession is his calculator > watch. Ok, not his favorite possession, but its high on the list. In light > of those numbers, if there was any magic to calculators, we wouldn't be here > speculating about it. Wondering if calculators cure math woes is like > wondering if water cures cancer. It obviously doesn't. Everyone drinks > water. We still have cancer. > > Bob Hansen