revamping K-12 math curriculum

[Kirby Urner]

     Exploring Integrated Math Topics + a Programming Language 
          with an Eye towards Developing the Content of a 
           Standard Pre-College Mathematics Curriculum

                          by Kirby Urner
                          March 15, 2001


     [slightly revised from first draft on math-teach at
                    the Swarthmore Math Forum          ]


I. CURRICULUM AS CONNECTED NETWORK OF "FABRICS"

I've used this forum (math-teach) as a scratch pad for evolving 
some curriculum segments.  A curriculum segment is a patch of 
related topics and activities that might be revisited at 
different levels of difficulty and sophistication.  Another, 
perhaps better, word for a segment is a "fabric" or a "weave".  
There's an emphasis on achieving a kind of seamless blend of 
themes and patterns, which assists students wanting a sense of 
the big picture (overview is what seems to be lacking in a lot 
of current math teaching).[1]

One example of a fabric is the whole patch of topics surrounding 
sequences and series, figurate numbers, polyhedra and Pascal's 
Triangle.  Conway & Guy develop this weave in 'The Book of Numbers' 
(Springer-Verlag, 1996), whereas I'd independently hit on a similar 
approach in my 'Numeracy and Computer Literacy' series.  I add 
the Buckminster Fuller element, with mention of the jitterbug 
transformation, geodesic spheres, viral protein sheaths and 
buckyballs.[2]

Another example of a fabric blends topics from cryptology, 
probability and group theory.  Here we develop the idea of a 
simple 'clubhouse code' based on letter substitution, and use 
this as a segue to the group theory concept of permutations, 
and multiplication as the composition of permutations. This 
framework allows us to discuss the properties of a group, in 
contrast to those of semi-groups, rings and fields.  The next 
step is to use the new concepts to develop more sophisticated 
enciphering strategies, wherein the substitution dictionary 
continues to change throughout the encryption process.  This 
provides a useful segue to historical threads, such as the 
storylines involving German U-boats, the Enigma code, Alan 
Turing and Bletchy Park ala Neal Stephenson's bestseller 
'Cryptonomicon'.[3]

Both of these fabrics contain material we could be phasing in 
as early as first or second grade...

Fabric #1:  Number and Geometry

Sphere packing and polyhedra (built with tooth picks and little 
marshmellows, for example) are a popular topic with smaller 
kids (I know from experience -- and not just because of the 
marshmellows -- modeling clay works too).  Euler's Law (V+F=E+2) 
makes sense to slightly older kids, as does Descartes' Deficit 
(720 degrees).  The idea of triangular and square numbers, 
which may be modeled by packing spheres (as in 'Pool Hall Math' 
-- see math-teach archives), makes sense early too, with 'flat 
shaped numbers' becoming 'spatial' (my 'beyond flatland' theme) 
rather early in the game (ala 'The Book of Numbers'). Fuller's 
10 F^2 +2, for the number of spheres in a cuboctahedral shell,
identical to the number in an icosahedral shell, fits into 
this context.  That the number of spheres is the same for both
shapes is shown by the jitterbug transformation, which hyperlinks 
to phi (the cuboctahedron's square cross-sections transform 
into the icosahedron's golden rectangles) and provides a segue 
to virology, buckyball chemistry, crystalography and architecture.[4]

Fabric #2:  Permutations and Cryptology

The idea of a 'clubhouse code' starts to make sense as soon as 
children have some mastery of the alphabet, and an understanding 
of why you might want to keep a message secret and therefore 
indecipherable by anyone but the intended recipient.  Games of 
chance, ideas about probability, enter the picture here, 
explorable with polyhedral dice (not just hexahedra) -- note 
the hyperlink to Fabric #1 here.  Using permutations to discuss 
the composition of functions makes sense at the middle school 
level, where we also should be developing the idea of modular 
arithmetic e.g. 240 mod 13.

II.  BEYOND CALCULATORS:  PYTHON AS 'FIRST COMPUTER LANGUAGE'

Aside from 'beyond flatland', another major theme promulgated 
by my Oregon Curriculum Network (a kind of Oriental Rug Factory 
for curriculum fabrics -- math-related especially), is 'beyond 
calculators'. Since the 1980s, programming languages have 
matured considerably, and since the 1990s we've had the option 
of teaching Python as a first language.  The synergetic blend 
of procedural, functional, and object-oriented programming 
styles available through Python provides a good nucleus of 
concepts which will serve a student well when branching out to 
other languages later, whether in the direction of C/C++/Java, 
LISP/Scheme, or any of several other well-traveled pathways.

Having some knowledge of a generic programming language, 
developing familiarity with it by coding around interwoven math 
topics, is going to yield many benefits later on in life.  
Early exposure (but not too early) lowers the chances of 
developing debilitating phobias later on, and gives students a 
vehicle for turning math topics into opportunities for 
exploration and portfolio-building (programs may be saved, 
revisited and improved over time). Using programs written in 
earlier classes to tackle topics in later classes helps provide 
a sense of continuity and relatedness -- something students 
need, but don't always get, in the current hodge-podge.

In 'Numeracy + Computer Literacy', I use Fabric #1 to develop 
some graphical output capability, by synergizing Python with a 
ray tracer (I use POV-Ray and VRML plug-ins, but many other 
graphics applications would be suitable, plus there's the 
VPython option, permitting an even more interactive approach 
based on OpenGL).  This link to computer graphics is likewise a 
segue to what I'll call Fabric #3, the weave on concepts 
relating coordinate systems, vectors, and geometric transformations 
as implemented by matrices or quaternions or some other apparatus 
(e.g. Clifford Algebra).  

Fabric #3 is where we use object-oriented programming to explore 
the Gibbs-Heaviside vector operations, bringing in trigonometric 
functions, rotation matrices and so on -- all of which makes 
more sense and is a lot more fun when one is rewarded with 
colorful, shadowed, perspective renderings, such as POV-Ray 
(or other software) provides (such colorful renderings might 
be transferred to T-shirts for those so inclined).

III.  TRAVERSING THE NET: FROM CRYPTOGRAPHY TO CALCULUS

With Fabric #2, students will be able to encipher and decipher 
text files, using their newfound/maturing computer language 
skills.  Given the opportunity is to encipher random passages, 
we have more opportunities to jump outside the math domain and 
capture topical paragraphs from other points in the curriculum.  
For example, we might choose to encipher Lincoln's Gettysberg 
Address, or a speech by Martin Luther King.  If students are 
on a network, they can practice passing ciphertext and secret 
keys back and forth.

Some educators may object to this focus on cryptography at an 
early age, suggesting that students will misuse this 
knowledge.  However, my experience is that many in 8th grade 
and above develop this interest on their own, picking up memes 
from popular culture, and, since this material is "avoided" in 
school, they develop the impression that cryptography is 
something the "establishment" is against, i.e. this knowledge 
takes on a "subversive" spin.  This is unfortunate, and it 
would be better if the schools worked in a more empowering 
mode, to help students develop skills and knowledge which is 
part of the mainstream commercial environment and net-based 
ecosphere.  Cryptography is not inherently "subversive" -- 
unless we craft an environment which makes it seem that way 
(which, perhaps inadvertently, is where we are today, in 
many school districts).[5]

The group theory aspect of permutations (as applied to 
cryptography) segues nicely to other kinds of group, such as 
sets of postivie integers less than and relatively prime to
some modulus (this is why modulo arithmetic was important). 
We can talk about CAIN (closure, associative, inverse, 
neutral element) and Abel(ian) groups.  The notion of coprimes 
segues to Euler's totient function, and Euler's Theorem, of 
which Fermat's Little Theorem is a special case -- more 
topics typically covered in group theory with applications 
to cryptography, in that huge primes are hard to "crack" 
into factors (what makes RSA and some other public key 
systems effective).  

Thanks to Python, we have access to big numbers (long 
integers) and don't have to suffer the trade-offs that 
plagued earlier "math through programming" forays into 
this realm (i.e. we don't trade away access to big numbers 
just because we choose to use a generic programming language, 
and not a strictly math-focussed or number theoretic 
package).[6]

We can loop back to Fabric #1 and talk about symmetry groups 
and polyhedra, or the group properties of quaternions under 
multiplication.  All of these concepts will have been developed 
concretely, in a hands-on, interactive context, using a 
computer language (I prefer Python) by the end of high school. 
The result will be a more sophistacted and non-math-phobic 
mindset among students, who have a well-connected set of math-
related concepts and a feel for the bigger picture.

I haven't forgotten the calculus, and will simply point out 
that we have a good basis for exploring limits starting with 
Fabric #1, with phi and Fibonacci numbers in particular (with 
a link between phi and the fractal-like, recursive nature of 
phi-based five-fold symmetric geometries). Fabric #1 was a lot 
about Sigma and Pi (summations and products of a sequences 
respectively), and about differences between successive terms 
in a sequence, the discrete math analog of the differential.  

With Fabric #3 (graphics/vectors), we have the standard 
graphing calculator topics of (x,f(x)) and might explore the 
calculus of the catenary for example (suspension bridges and 
electrical wires have this shape), which will include mention 
of hyperbolic trig functions.  I can talk more about Fabric #4 
in another post.[7]

Kirby

[1] these 'fabric' terminology is borrowed from memetics as 
embodied in Fluidiom, which is also an elastic interval geometry
application (http://www.fluidiom.com/)

[2] http://www.inetarena.com/~pdx4d/ocn/numeracy0.html

[3] http://www.inetarena.com/~pdx4d/ocn/clubhouse.html
    http://mail.python.org/pipermail/edu-sig/2001-March/001077.html

[4] Jay Kappraff's, Connections, the geometric bridge between
art and science (McGraw-Hill, 1996) is also good on the 
root(2)-based vs. phi-based aesthetics.  For an animated GIF
of the jitterbug, see "Getting Inventive with Vectors"
http://www.inetarena.com/~pdx4d/ocn/numeracy1.html

[5] Teachers might want to use the relevance/importance of a
cryptographic "fabric" as an argument for bringing more powerful
computing platforms into math classrooms, vs. "making do" with
less suitable calculators for this purpose.  For more along
these lines, see my post to k12.math.teach:
http://www.mathforum.com/epigone/k12.ed.math/glaterdphum/

[6] here's a post to math-learn quoting an earlier curriculum 
writer who had to contend with this trade-off:
http://groups.yahoo.com/group/math-learn/message/217


[7] or see http://www.inetarena.com/~pdx4d/ocn/catenary.html