
Re: Treatment of Negative Numbers
Posted:
Dec 4, 2012 6:12 PM



On Tue, Dec 4, 2012 at 2:01 PM, Clyde Greeno @ MALEI <greeno@malei.org>wrote:
> ** > Stay tuned! > > You quoted my whole thing, and Robert's, adding one line!
On some listservs, the moderators would never accept that (Wittgenstein list comes to mind, where S. Wilson runs a pretty tight ship).
I'm miffed in part because the archived web version has some typos fixed, whereas my emailed version, which you quoted, has some "my bads".
Anyway, to continue this thread:
In my STEM curriculum, we're interested in notions of convergence and divergence, oscillation, chaotic sequences.
These may be introduced through numeric series, including those that eat their previous terms. The OnLine Encyclopedia of Integer Sequences is of great interest in this respected.
Numeric Sequences (rule driven): Convergent Divergent Oscillatory / Periodic Chaotic / Aperiodic
Periodic / Aperiodic also relates to spacefilling and tiling.
Under the heading of Sequences we also have a miniunification of rulebased functional expressions e.g. algebra / prealgebra, with geometry. How? Through figurate and polyhedral numbers.
Figurate numbers are what they sound like: numbers easily represented as polygons such as growing squares, triangles, hexagons etc. We can prove such things as the sum of consecutive triangular numbers is always a square number.
Polyhedral numbers are what they sound like as well: growing tetrahedra, cubes, cuboctahedra, octahedra.
A great source: The Book of Numbers by Conway and Guy, and also Gnomon by Midhat Gazale.
Once you have a growing cuboctahedron (1, 12, 42, 92....) you're looking at something fairly omnisymmetrical, almost planetlike i.e. the polyhedrons are a bridge from Geometry to Geography. Start spinning a cuboctahedron around its axes: those through face centers, triangle centers, mid edges, vertices. You get 25 axes of spin. With the icosahedron, you get 31.
These grids may be superimposed to create networks with repeating triangular areas some call LCD triangles (120 lowest common denominator triangles tile a sphere). We're talking about subdividing a sphere now. Ed Popko's new book on Divided Spheres is a primer for higher level students (I'm listed in the bibliography).
http://www.routledge.com/books/details/9781466504295/
One outlet for this material is in San Antonio, Texas where Dan Suttin does a good job with younger students, helping them get the basics about Platonic and Archimedian families, space frames, spacefilling. I'm not sure if he teaches about duals (e.g. tetrahedron is its own dual), probably so.
http://youtu.be/ULJbXfLFr28 (Dan sharing with younger people about polyhedrons  refreshing change from the usual fare).
What I like about this "miniunification" is it bridges left brain lexical algebraic computer programming type thinking with right brain graphical visual type thinking, but without dwelling on any specific coordinate system at first. It's something we can start spiraling into quite early.
It's when we have a Planet Earth in view (ala Google Earth) that we can start talking about "number lines" (e.g. equator) and rulers and grids, for surfaces, for volumes. We don't need to stress "infinite" too much and we're not scared off by lines that curve, even corkscrew.
Helices are our friends (they're a VPython primitive after all). Helices are the ultimate chiral phenomenon and as we have seen, this curriculum is very invested in talking about left and right, mirror images and so on  much more so than New Math ever was.
Looking down on the north pole, we can see the Earth spinning. Clockwise or counterclockwise? Many adults have no idea as this was never a focus in school. She rolls towards the east. That's counterclockwise from this angle. Looking at the south pole, from out further, that's clockwise. You might need a globe to see this. That's geometry, not just geography.
So you can see how STEM is going: much better integration of solar system geography (astronomy) with geometry and geometry with numeric sequences. Numeric sequences become a first opportunity for learning computer languages in an algebraic / lexical context.
Once the polyhedrons are well developed, we have a much stronger basis for talking about symmetry and symmetry groups (based on rotation). We also get into topology with V + F = E + 2 (provable), Descartes' Deficit (720 degrees) and ratios relating to omnitriangulation.
My 'A Bhutanese Mathematics Curriculum' summed up some of this stuff. I wrote it while in Thimphu in the 1980s.
http://www.flickr.com/photos/kirbyurner/3859886616/
Kirby

