> Q1: When we first talk in K-4 about numbers between > the whole numbers, shouldn't we be thinking of them > and presenting them as real numbers (not as > rationals)?
K-4 lacks overview, really OK to preview and do a heads up. So yeah, we'll be doing these concentric sets, somewhat recapitulating phylogeny (teachers' notes):
N < W < Z < Q < R < C
Those'd be naturals, wholes, integers, rationals, reals and complex.
Of course in gnu math we tend to stick with what we've got on tap as actually implemented types, e.g. instead of reals we have maybe context-precision Decimal and Float types, but OK to save that discussion for 7-10 maybe.
Also these are not the only types of course. We have integers modulo N ("modulo numbers" OK), vectors, will even count Polyhedra as "types", as well as "strings" i.e. the concept of "type", being well-defined in say the Python language, is going to take us a lot further a lot faster pre-college, because now everything is more concrete and yes, manipulable. It's a constructivist heaven, but that doesn't get you off the hook from actually learning your heritage -- in history class if they refuse you in math class (no Euclid's Method for GCD? -- ask your history teacher, insist).
> Q2: Might it not be a good idea to have kids in K-4 > do linear algebra with natural number coefficients > and capital letters standing for quantities other > than whole numbers? >
You're maybe not tracking the tilt towards Gattegno in the UK: use whole number lengths (color coded) and algebraic names (i.e. letters), and introduce the four operations as algebraic expressions tightly coupled to these Cuisanaire bricks. I know lots of teachers think those bricks are passe, but that's because they never got to see the Gattegno curriculum that went with 'em. Evidence suggests kids learn a way of reading this way, that helps with both linear algebra and coding algorithms down the road a ways.
> I believe that the answer to both questions is yes, > but I also believe that mine is a minority opinion. > And, although I have been thinking about these > e topics for twenty years, I don't claim anything > like proof. >
The goal is to encourage experimentation, not accept the tired saw that some inner circle "knows" i.e. the pedagogical questions are settled. I'm not sure if Haim really believes his extreme position or if he just likes to rattle cages with extreme positions, I expect the latter, as no one really thinks pedagogy is all settled, not with Internet, TV and computers still in their early chapters. And those aren't the only change vectors, either.
> I don't mind heated discussion and a few pointed > remarks, but I think it will help that we start any > discussion with everyone knowing I have enough > background that I don't need basic stuff explained to > me -- though of course I may have to be reminded of > things, and I do make mistakes. >
You may not be used to the level of diversity on this list. Although I've been a classroom math teacher, I'm also fairly experienced in publishing, so I follow different market indicators than some others here, who know more about early childhood education from the point of view of overcoming disabilities while taking advantage of whatever gifts and strengths (everyone is a mixed bag on that score, happy to admit many weaknesses, that's why I tend to work in teams).
> I wrote a dissertation in model theory in logic > (about 45 years ago), so I have background knowledge > of the semantics of languages like those used in K-12 > algebra and in the foundations of mathematics. About > 25 years ago, I was co-PI on an NSF grant in > mathematical linguistics, and I have co-authored > papers relating to the syntax of these languages with > a linguist who is now in the National Academy and > another who is a past president of the Linguistics > Society of America. I helped set up the first > undergraduate program I know of in cognitive science. > I then did a post doc in math education and got a > a secondary math credential (33 years after my PhD!). > I have taught highly gifted high school students and > d middle school kids who were two or more years > behind in math. > > Bill >
There's a rather strong contingent in Silicon Forest that's impatient with Paper Tiger Logic as we call it, i.e. if the adherents and practitioners haven't done the work to make it run on hardware, then maybe it's "2nd tier"? That's our bias. We think Russell-Whitehead laid some good groundwork, but since then we've had lambda calculus (Alonso Church et al), the OO revolution, and the APL family (which eat hyperdimensional arrays for breakfast). These Machine Logics (aka programming languages) deserve a *much* bigger footprint in K-12 is our reading. Teach SQL and "how things work" or don't even pretend you're an accredited high school, would be our leaning, not saying we actually have legislation up our sleeves, just a way of sharing our thinking via well understood marketing channels, i.e. we're skilled with media (witness OLPC.xo campaign, really quite high caliber).
> BTW, if anyone knows of discussion groups that > include psychologists interested in number concept > development, I hope you can let me know.
I'm sure you know to search on Piaget. Reams of stuff, probably 5 miles a minute of new verbiage, streaming onto the Internet from somewhere. Go crazy!