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).