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).
Cardinality versus Ordinality
Two different versus two the same
Of zip codes and phone numbers
Equality, greater than, less than
A Biotum Class (Python)
A Snake Class (Python)
Numbers and Bases
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
Python's dictionary structure
Lore: from secret key to public key crypto
Injection, Surjection, Bijection
What's a Relation?
On Growth and Form
Functions for Gnomon Growth
Lore and Proof: Gauss summing 1..100
Proof: Sum of consecutive squares
Lore: Fibonaccis and Phi
Generating Polyhedral Numbers
Generating Pascal's Triangle
Triangular and Tetrahedral Numbers
Prime and Composite Numbers
Euclid's Method for GCD
Primes versus Composites
Totatives and Totients
Functions and Relations (Part 2)
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
Group Properties (CAIN and Abelian)
Addition: What is it
Rings and Fields
Preview of Future Topics
artifacts and encoded geography
random number generators (Python)
Deck and Card classes (Python)
Sphere packing and the Octet Truss
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
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.
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
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).