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