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More content change proposals
Posted:
May 13, 2005 12:41 PM
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So we've been discussing some variations on the traditional curriculum around factor trees, primes and composites:
http://www.mathforum.com/kb/thread.jspa?threadID=1149429&tstart=0
Recap: do more with Euclid's Algorithm, use sets as a segue to other data structures, apply these concepts in a computer language (I've circled Python as a strong candidate, but we could go with Scheme or some other -- I don't see a need to nail that down; just "some computer language" at this point (time to end the hegemony of the TIs)). Also, stay with the prime vs. composite topic long enough to get to coprimes and Euler's phi function (totient and totatives). You'll be laying a foundation for appreciating RSA before the end of high school.
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Now I want to talk about geometry. I think we should be doing more with linear algebra in K-12, but in conjunction with computational geometry.
There's currently too much emphasis on reducing matrices to solve simultaneous equations, but the story problems don't have much interesting content.
Computer graphics, on the other hand, is a topic a lot of kids have an interest in, even from an artistic point of view. So let's do vectors to define polyhedra (say) and then apply 3x3 rotation matrices to make them go around various axes.
Now that we've got access to a computer language (yay) it's not so tedious to apply the same 30 degree rotation to say 12 vertices of the icosahedron.
And how shall we introduce polyhedra? Typically, they're stuffed at the end of the text in a 10th grade geometry course, and maybe are never reached (stuff at the end rarely is). Once you get there, you get a few lame volume formulas, maybe some mention of conic sections.
And yet, the world we live in is not Flatland. Abbott's idea that we can imagine what it'd be like to live in 2D is bogus; we can't. Whereas I have nothing against studying Euclid, I think spending an entire year on a plane sucks the life out of this subject. A lot of the same theorems have application in space. We should get to the planar stuff *through* spatial geometry, and not vice versa. 2nd and 3rd graders should know their Platonic Five, learn them simultaneously with the concepts of equilateral, obtuse, acute, isosceles. Learn about polygons, sure, but then immediately show them as "faces" of polyhedra.
And with the polyhedra, there's a much richer assemblage we should be sharing, which involves relating them to each other, not treating them as standalones. Two tetrahedra inscribe inside a cube, as face diagonals, creating what Kepler called the Stella Octangula. The cube's dual is the octahedron, and the combination of both (edges meeting at 90 degree angles) is the rhombic dodecahedron (Kepler again), a space-filler. We need to *show* this stuff, and on future returns through this network, start computing angles, both on the surface, and subtended central angles. From packed rhombic dodecahedra, we get to lattices (and Bell's kites).
Vectors provide one way to connect polyhedra to algebra (I'm lumping XYZ coordinate geometry in with vectors -- same thing), but there's another, simpler way. Sequences and figurate numbers, polyhedral numbers. This is the approach taken by Conway and Guy in 'The Book of Numbers.' And so many entries in Sloane's Encyclopedia of Integer Sequences (on line) have a simple geometric meaning and interpretation. Another book in this tradition is 'Gnomon' by Midhat Gazale (also the author of 'Number'). These are sources to draw upon as we move to revamp the curriculum, to save it from becoming even *more* stale and stultifying. Plus I'm injecting much-needed content from Buckminster Fuller. The DVD clips will be great, and the IMAX movies even greater (if they get made that is).
Kirby
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