[Edu-sig] Concrete Mathematics

Kirby Urner pdx4d@teleport.com
Tue, 23 May 2000 09:39:01 -0700

>OK, sounds like "Concrete Mathematics" (CM) is my next 

Bought this book yesterday.[1]  Looks good -- like Tim 
says, not as "intense and telegraphic" as 'The Art of 
Computer Programming', but with Knuth an author, bringing 
the same expertise.  This is a text book for an actual 
course at Stanford.

My initial reaction on reading these books is pleasurable,
given how much it positively reinforces what I'd come 
up with in the 'Numeracy + Computer Literacy' series.[2]  

I'd sort of intuitively gravitated to Pascal's Triangle 
as a conceptual nexus, a "grand central station", and 
that's what these books do too, in the same context of 
talking about series.  Knuth's Volume 1 even shows the 
tetrahedral packing of spheres (exploded view), a 
geometric interpretation of one of the Pascal columns.

What's different about my Oregon Curriculum Network 
approach is that I develop this geometric connection 
more intensively, making use of Buckminster Fuller's 
concentric hierarchy as defined by 26 data points, 
plus a jitterbugging thereof -- which points I have 
the option to express in whole number coordinates, 
given the newfangled quadrays apparatus.[3][4]

In going with polyheda as paradigm objects (in the OOP 
sense), I'm bringing in Computer Game Programming 101
(e.g. rotation matrices, even quaternions -- the 
'Tomb Raider' engine is quaternion-based) which is 
more obviously relevant to a lot of kids -- as is 
spatial geometry in general (after all, we live in it).

Also, whereas CM is earmarked as grad schooler or upper
level, I'm pretty clear we have the option to alter the
mix in the lower grades (pre-college) as we follow the
CP4E thread where it most naturally leads (including
into the math classroom).  We already touch on a lot 
of these same topics at these earlier levels, but don't 
elaborate much (e.g. mention primes, but not Fermat's
"little theorem" or the link to RSA encryption) because 
everything is geared towards an intensive calculus 
experience, for which you need precalc.

Again, I have nothing against teaching calculus (was
a calculus teacher myself for two years), and CM is 
full of Riemann sums, right along with the SIGMAs.  And 
I didn't mean to say we should bleep over the chain rule 
or integration by parts, merely that it should be an 
option to NOT spend a _whole year_ doing calculus at 
the pre-college level AND to nevertheless be considered 
a top performing math student.  In other words, if 
you want to be a "star" in math, you might do something 
more along the lines of CM (and my essay, ahem), and 
less along the lines of today's conventional AP calc 

These kinds of "remixings" happen all the time.  Math
ed is like music -- you go through phases, things come 
into vogue.  I'm not saying we should get suckered by
the latest fads (a lot of what the traditionalists 
are fighting as "fuzzy math" seems pretty faddish to 
me), but on the other hand, we need to be aware that 
long term trends alter the landscape in math, as in 
every other discipline.  Change happens.

My view is that pre-college math is out of synch with
what kids could most use and benefit from.  Calculus
is over-stressed.  Fluency with calculus ideas should
be developed with an eye towards looking at other topics
(a proof of relevance), with more intensive drilling in 
this subject saved for those most likely to need it 

Too much other good stuff is falling by the wayside in 
the rush to cram calculus into the final year of pre-
college.  We need a more accelerated path (i.e. a 
"first pass" tour of duty) through that material that 
doesn't get so bogged down in the nitty gritty and 
leave too many students turned off math for life.

In making more room for other topics, we'll have the
opportunity to phase in computer languages, Python
included (it has a lot going for it, as do some of
the others).

Text books are too slow to match these needs.  My 
approach is going to be based in cyberspace as the
primary source medium, not text books.  Some of our
materials will be more like "finished goods", for 
teachers who want those (e.g. PDF handouts).  But I'm 
primarily interested in ideas, with teachers customizing 
content to fit their local needs and students.  A lot 
of this will be private sector and commercial, with 
curriculum supplies being sold via the same websites.  
Also working the DVD jukebox angle.

4D Solutions

PS: I see Knuth has his own ideas about calculus reform,
which I'm curious to read.  Just have to find a way to
get TeX files deciphered in Windows (or BeOS).

[1] Ronald L. Graham, Donald E. Knuth, Oren Patashnik, 
    Concrete Mathematics (2nd Edition), Addison-Wesley,
[2] http://www.inetarena.com/~pdx4d/ocn/cp4e.html#python
[3] http://www.inetarena.com/~pdx4d/ocn/oop7.html
[4] http://www.teleport.com/~pdx4d/quadrays.html