> > Kirby's generous note said a lot, inclluding: > > "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." > > I love this stuff. The first course I was allowed to > teach on my own, in 1966, was the foundations of the > number systems. I was told to use Stoll's quite good > book, but I went off in my own way. > > I started by working backwards from C > R > Q >Z > N, > where, for me, N included zero, so was what math ed > folk call W. I then started from N and went up, > first to the infinite cardinals and then looked at > them as special infinite ordinals. If I were to > teach such a course now, I would include quaternions, > infinitessimals, and non-standard real numbers.
I prefer chrono order and telling history as we go i.e. what pushed the envelope at each step in the game to enlarge the scope. Why weren't rationals sufficient? What motivated "imaginary" i.e. "complex" and so on. This helps keep Polynomial objects in play, even as we give more limelight to Polyhedra than before.
I'm fine with including other objects, have Quaternions going at Oregon Curriculum Network, no problemo, also some 4-tuple objects called Quadrays or Chakovians, like XYZ but no linear independence or negative numbers, lots of precedents in the literature ("simplicial coordinates").
Yes, very esoteric but students sometimes enjoy getting off the beaten path. grunch.net/synergetics/quadrays.html
We market through coffee shops, knowing most schools have no patience for private industry and its mathematics (way better than "schoolish math", much closer to real engineering, less mired in stultifying arcana). "Self school and come work for us" is our subversive message, but then we also run schools, have a footprint in the public system (as business associates, sponsors, para- teachers of various kinds).
> > For kids K-2, I'd do N and W "counting numbers", then > introduce real numbers. I'll say much more about > this on the measuring number thread.
Again, stories matter, so telling how Liber Abacci helped open Europe's eyes to the Abacus Way, a real leap from the imperial domination of idiocratic Roman Numbers, made the Renaissance possible. There's a temptation to prohibit math teachers from doing this segment, as they tend to botch everything to do with math. But that's to get tangle up in semantics. Of course math teachers teach math, by definition right?
I think the better way of saying it is we Amerish (a-MER-ish) speakers have no intention of waiting for some Math Czar in DC (a fantasy) to give us a green light.
We're already off and running, many laps into it, with students getting it about the DOM (Document Object Model), the importance of XML. Don't call it "numeracy" if it has "too many letters in it" (like algebra?) but do call it basic literacy (or how about "alphanumeracy"?), "what any kid should know" (if you want to play in places like CubeSpace -- generic office cubes I blog about (i.e. more websites, some more on Gattegno, other constructivists, Karplus especially (physics, more successful at reform, better managed in many ways))).
> > Kirby also says: "You're maybe not tracking the tilt > towards Gattegno in the UK..." > > I'd like to. If you have a few good websites, I'd > greatly appreciate getting them. I don't have easy > access to a good library. I'll more about what I am > learning about Gattegno's ideas in the early linear > algebra thread. The attachment to my first note > there will also mention some use of computers.
Yeah, definitely the Web is where it's at. Start with tizard.stanford.edu, a cosponsor of Python for Teachers at Pycon this year.