Here's something I originally posted to another list, but one that

is white-space insensitive. Hardly a venue for sharing anything

Pythonic. Anyway, it's relevant here too.

This is more of a sketch than something I've poured over for

hours, just trying to give the gist of a possible Discrete Math

course, one among hundreds or thousands or... not trying

to write "the national standard" (blech).

Kirby

====

DIGITAL MATH:

Cardinality versus Ordinality

Naming things

Two different versus two the same

Of zip codes and phone numbers

Sorting things

Equality, greater than, less than

Hello World!

A Biotum Class (Python)

A Snake Class (Python)

Numbers and Bases

Positional Notation

Lore: Algorithms liberate Europe (Liber Abacci)

Lore: from ASCII to Unicode

Decimal versus Hexadecimal

Volume Bases: Tetrahedron versus Cube

Functions and Relations (Part 1)

Lore: the rise and fall of New Math

Mappings

Python's dictionary structure

Caesar Codes

Lore: from secret key to public key crypto

Inverse Functions

Injection, Surjection, Bijection

What's a Relation?

On Growth and Form

Functions for Gnomon Growth

Triangular Numbers

Lore and Proof: Gauss summing 1..100

Square Numbers

Proof: Sum of consecutive squares

Tetrahedral Numbers

Cubic Numbers

Lore: Fibonaccis and Phi

Generating Fibonaccis

Generating Polyhedral Numbers

Generating Pascal's Triangle

Triangular and Tetrahedral Numbers

Prime and Composite Numbers

Euclid's Method for GCD

Primes versus Composites

Strangers

Totatives and Totients

Functions and Relations (Part 2)

Permutations

Polyhedral Rotations (dice in Casino Math)

Composition of Functions (a kind of multiplying)

Abstract Algebra I

Multiplication: What is it

Python and "Modulo Numbers"

Vegetable Group Soup

Cayley Tables

Group Properties (CAIN and Abelian)

Addition: What is it

Rings and Fields

Preview of Future Topics

Supermarket Math

SQL

web frameworks

Neolithic Math

artifacts and encoded geography

constellations

Casino Math

random number generators (Python)

Deck and Card classes (Python)

Martian Math

Sphere packing and the Octet Truss

Tetrahedral Accounting

Pentagon Math

Lore: geodesic spheres and domes, radomes

Phi in Fuller's concentric hierarchy

Notes for Teachers:

Cardinality versus Ordinality --

Before we order or sort, we need to recognize which things or

objects are of which type. This course uses a type based

mathematical logic known as the Python computer language, so

awareness of types will be front and center from the get go.

Exercises will include querying objects as to their types.

Z-axis (depth dimension): if you've going through this in

a spiral with plans to go deeper each time, then at some

point your students may want to define their own classes

and implement meanings for __lt__ __gt__ __eq__, Python's

"ribs" (special names) for < , > and == respectively. However,

this course outline does not make too many assumptions about

which turn of the spiral one is in. Students will vary, as

will teachers.

Numbers and Bases --

This should feel like fairly easy review. I recommend playing

Tom Lehrer's 'New Math' from 'That Was the Year that Was' and

making sure students get that it's mathematically correct. This

is looking ahead to later lore, where we talk about the rise and

fall of New Math.

About Lore: this curriculum is premised on the notion that

storytelling is integral to passing on a culture, and that too

much time on a technical axis, to the exclusion of narrative

context, is either counter-productive or is an intentionally

applied filter aimed at testing student tolerance for

"in the dark" learning.

Functions and Relations (Part 1) --

A lot of this is standard Algebra 1. New Math helped writers

formalize their notion of function as distinct from a relation,

using set theoretic constructs. This may not be the right place

for a Python dictionary on a first pass. I was getting into

Caesar Codes again recently, relating them to permutations and

polyhedral rotations, and am freshly persuaded this is one of the

better routes to elementary group theory, just ahead.

http://mail.python.org/pipermail/edu-sig/2010-March/009867.htmlhttp://mail.python.org/pipermail/edu-sig/2010-March/009870.htmlOn Growth and Form --

Here's a way to connect the graphical and lexical without getting

into XYZ coordinates or vectors right off. Gnomon studies and

sphere packing keep the number sequences connected to the

visualizations. Influences and valuable resources here would

include 'The Book of Numbers' by Conway and Guy, 'Gnomon' by

Midhat Gazale, and certain passages from 'Synergetics' by R.B. Fuller,

with bolstering writing from H.S.M. Coxeter.

Lore: getting a sense of Coxeter's outrage on finding nature's

geometry had been patented by Fuller prefigures a generic distrust

of the private ownership paradigm when applied to common natural

heritage. In some classes, this might lead to a discussion of

the free and open source software movement. I didn't include that

in this particular segment though, as we're running low on time.

Look in Pentagon Math under Future Topics.

Abstract Algebra I --

Because Python makes our exercises much more concrete and hands

on, especially with scaffolding (pre-written / canned libraries or

modules), it's new feasible to get more abstract. 'Concrete

Mathematics' is an influence ('con' from continuous, 'crete' from

discrete). With this kind of groundwork in place, it becomes

easier to review topics such as dividing fractions, as we may now

talk about division as "syntactic sugar" for "multiplying by the

multiplicative inverse". Reviewing such basic concepts as the four

operations with Q (Rational Numbers) would not be out of place in

this segment.

Preview of Future Topics --

If this course or talk was used as a teaser or sampler, then here

would be another chance to look ahead.

I don't have anything on Pentagon Math at the Wikieducator site

(Heuristics for Teachers). That's because you can easily fold it

in with Neolithic and/or Martian Math. I split it out here because

I wanted to dig into some of the lore in the Siobhan Roberts bio

of Donald Coxeter.

Remember: we think lore is very important, as are exercises and

time alone with the Python interpreter (not all programming is

pair programming in these initial stages, nor even later on --

depends on the project).

For further reading:

http://www.4dsolutions.net/presentations/p4t_notes.pdfKirby

_______________________________________________

Edu-sig mailing list

[hidden email]
http://mail.python.org/mailman/listinfo/edu-sig