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