Date: Jan 15, 2003 2:47 AM
Author: Kirby Urner
Subject: Another Alien Curriculum
While the Math Warriors duke it out on both sides of the reformer
vs. traditionalist debate, my role seems to be to lob my posts
from somewhere beyond the orbit of Mars (Osher is even further
out though, from at least beyond Pluto) -- I'm the guy who steps
off the UFO and doesn't really understand about the local food
fights.
So here's another take on a curriculum that might someday have
data based research to back it up, but right now is more just
based on some 44 years of real world personal experience. Sorry
I don't bring more to the table, but ya gotta start somewhere.
Here are my basics (as in "back to basics"):
Time
Space
Objects
Events
So with Time: we already teach how to read a clock. Have
you ever seen a 24 hour *dial face* clock? They exist.
Useful. But more important, adding hours as in "Thursday
10 PM + 3 hours = ?" (Answer Friday 1 AM) is good practice
and reminds us that we need to learn addition modulo other
than 10 (divide the clock into 360 later, when we get to
trig). Calendars. What was that Y2K thing all about
(sidebar)?
Space: includes our planet, with the latitude/longitude thing,
where adding hours also comes in as well, because distance is
of temporal significance (time zones, velocity, accelaration).
If it's 4 PM PDT in LA, what time is it in Tokyo right now?
Oops, that was time again. So how far is Tokyo from LA? In
any case, Planet Earth is a sphere (sort of), so if the diameter
is 8000 miles, what's the approximate circumference (yes, you
may use pi).
Objects: populate time/space (the planet is an object). We
think of real objects, yes, but also of more abstract entities
with attributes and invokable functions or operations. Objects
have "control panels" -- instrumentation for exposing their
potentials to the world (cockpit of an airplane). We'll be
talking about objects in such general terms. Because numeracy
is a subset of literacy, is a kind of language. Yes, you
*should* be able to write about math.
Events: stuff happens. Events occur along timelines (back
to time) and at places in space (how do we describe the when
and where of an event?: coordinate systems, grids, maps,
addressing schemes, time codes... objects may know how to report
their own positions). Events also occur between objects --
communications/messages get passed from A to B (encryption).
Or objects might collide (an event!). Particle physics (links
to graph theory -- Feynman diagrams).
What does any of this have to do with "real math"? Well, you
can map a lot of the current content, following some simple
guidelines. We divide up space and time. Volume, fractions,
area. Looking at the planet as a spherical topology sets the
stage for polyhedra in general -- a kind of graph theory approach,
with polys as wireframes (Euler's Law for Polyhedra, Descartes'
Deficit). We start in space and work down to a plane (Euclid),
because space is more experiential than flatland. Figurate
numbers (includes polyhedral numbers ala Coxeter, Conway and
Guy). Rule based sequences: a great way to start programming
(from Fibonaccis to Fractals).
The objects contain functionality, meaning they accept inputs
and return outputs. f(x) accepts x from the domain of f, and
presumably returns range value y. We need to know something
about the domain in terms of types. Because algorithms are not
just about numbers, and even if about numbers, might accept
lists or arrays (too few examples of this in traditional math).
x might be a character string (yes, math is about symbols
other than numbers -- we knew that). An object that draws
graphs might be able to take f (the function) as an argument,
along with a domain set -- passing functions as arguments to
other functions, gotta do it (the derivative function operates
on functions, after all).
Events happen at a certain rate. We have intervals, frequency.
Graphing time against another axis. Slope, rate of change, velocity.
Events contain/involve energy. Links to physics (momentum for a
distance, in a time = mvd/t = mvv = E = hf). Frames of action
(action per frame, as in a film). Faster film = more power
(energy/time).
Communications -- a type of event, or maybe they're codified as
objects (distinction between event and object not set in stone).
Permutations of 1s and 0s map to symbols -- maybe just 128 symbols
(ASCII), maybe 256 (extended ASCII), or maybe unicode. Casting
between types. Translation. Lookup tables. Data dictionaries.
Hash codes. Venn Diagrams. SQL.
Sure, there's lots of computer science going on here (OOP ideas
permeate), but not to the exclusion of physics (events), geography
(planet as object), history (timelines) or literature (symbolic
communications, records, string processing, databases and library
science). I'm a bit short on grocery store arithmetic. I never
liked retail-based story problems. Just my bias -- others can
add that in (a transaction is an event, and the global balance
sheet includes steady energy income from the sun).
We'll do graphing, talk about number types in relation to the
number sets Z, Q, R and C. There's implicit type/object unification
going on, i.e. a complex number object has certain properties, as
does a vector object, matrix object, derivative object or
polynomial object. They have different functionality. The
same operator (symbol) might mean something different depending
on whether x*y refers to two integers, matrices, complex numbers
or polynomials (many other types possible! -- lets not forget
boolean objects).
We'll do some abstract algebra stuff (group, ring, field) to point
out commonalities, and to talk about the timeline of number systems
themselves (Z before Q before R before C). Greek math sort of
stopped between Q and R (the irrational numbers quandry), and then
playing with polynomials eventually forced us into C.
An operator is sort of like a verb. Verbs are action words. Actions
denote events. Objects, involved in events, denote nouns. The
noun/verb distinction exists in math. f(x) -- f is a verb, x is
a noun (or might be another verb). We might even modify how a
verb acts on nouns (adverbs?). Shall we foray into J at this point?
I would. But then I'd have already started using Python in like
7th or 8th grade. Of course we have computer languages in this
curriculum -- it'd be unthinkable to do it without them (we might
not use calculators though).
Time for another outline:
Time
clocks
timelines
time codes
adding into the future, subtracting into the past
time scales (geological vs. human vs. pico)
rates of change (ratios)
Space
Coordinate systems (including latitude/longitude, degrees)
Girding/griding the planet (GIS/GPS)
Mapping/addressing schemes
Length and number (ratios)
Area and volume
Objects
Shapes
spatial networks (polyhedra)
plane figures (triangles especially)
Objects organized in sets
types of number
operations with number types
permutations and combinations (DNA)
Objects organized in hierarchies
subtypes (rationals a subtype of real a subtype of complex)
geometric hierarchies
biological taxonomies
APIs: exposing objects to users
Fractions, Decimals, Percents (rational number type)
Polynomials (solving, graphing)
Vectors
adding and subtracting
scalar multiplication
dot and cross products
Matrices (translation, rotation, scaling)
Events
Communications
Symbol systems and codes
Energy
kinetic energy (units, dimensions)
heat energy and temperature (conversion constants)
Frequency (links to time)
color / optics (the spectrum, visible and not)
Rates of Change
velocity vectors, acceleration, slope, gradient
derivative
anti-derivative (integral)
Event-triggered objects
mouse clicks and key presses
Probability and Permuations
This could be a roadmap to a multi-year course (in broad outline),
a single course, or an outline of a single PowerPoint presentation
(some overview/review for those already familiar with these topics).
Kirby