One of the notational reforms we plan to introduce along our alternative technological math track (discrete math emphasis, though with full explication of N,Z,Q,R,C), is so-called Dot Notation.
Dot Notation is well nigh ubiquitous in industry and is used to represent Objects (say a house) with attributes (say windows) and methods (say door opening, shutting). Obviously state is involved (the current status of the house; occupied vs. unoccupied for example). And for those familiar with OO, you'll see this other advantage: a well-developed, consistent notation wherein it's already important to discuss abstractions versus special case instances of something.
In the early grades, Dot Notation will appear with reference to familiar objects of learning-to-read picture books. The dog Rover has a bark method. Most dogs do. The dog Fido is wagging its tail. Different special case dogs behave differently, yet have attributes and methods in common, as specified by the type or class Dog. By now we're moving into 6th grade and beyond. They've already been exposed to Sims (the genre). The House is behind us (an already-mastered concept). We're ready for Math Objects.
Math Objects begin with the lowly number types. We must emphasize that a lone number is meaningless without a game to play in, and it's the rules of the game which give numbers meaning [Wittengstein, IBSN 0-262-23080-2]. A number on a bus, associating the bus with a bus route, is involved in a different game than a number at the end of a needle on some scale or clock. The concepts of cardinality versus ordinality pertain [Gazale, ISBN: 069100515X].
The rational number type will be especially rich in clues and new concepts. Adding and multiplying them (with inverse operations) involves GCD and LCM, which links us to prime vs. composite, relatively prime, and totative. We're laying the groundwork for RSA by 12th grade (a crypto machine making use of number theoretic concepts).
By this time, students will be ready for something well beyond Dot Notation: an implementation language that uses it, seen from behind the scenes, as a programmer would see it. We're in a skeletal world by now, austerely lexical, but we assume the visual imagination is by this time well developed. A construction such as myhouse = House( someaddress ) now has years of memorable video clips to draw upon, other tactile experiences.
Any OO language that supports method overloading will do. Ruby would be a good choice for this kind of work. Build a rational number class (abstraction) such that individual special case fractions (1/3, 2/9) are able to combine using the familiar operators *, +, /, -. For example:
In studying the guts of the Rational Number blueprint (class template), students will learn Euclid's Algorithm, in preparation for later deployment of the extended and binary versions. Diophantine Equations and Continued Fractions become accessible, perhaps only as light subjects for the home theater (homework includes requiring video viewing, i.e. "for-credit movies" -- publishers are already scrambling).
Beyond the rationals we'll encounter the floating point (as distinct from the reals -- contrasting them will help reinforce concepts important to Algebra), and of course the complex (segment on Fractals). We'll be using extended precision rationals to converge on any real, by means of various algorithms. Partial sums and series are all part of this same segment (borrowed from pre-calc and now beefed up with programming -- much to the amazement of admiring parents).[NOTE]
The complex numbers on an Argand Diagram form a bridge to two key topics: vectors and trigonometry. Mathematical models needed spin (e.g. curl) and the complex numbers provided a great tool: e to a sequence of imaginary powers begets rotational motion over time. Historically speaking, the classic unit circle (a slice through a sphere) was now the site of a mini grand unification: the complex plane and the real plane of trigonometry had melded. And vectors as complex numbers could multiply. So how about in a 3-axis system (mutual orthogonals)? No problem: Quaternions. [Crowe, ISBN: 0-268-001189]
Again, per our earlier writings about decentralized curriculum planning, it sort of depends on the specific school how to mix it up once the basics have been covered. Given how we're using the Internet, versus mass-published text books, teachers have a lot of leeway to experiment, may have lots of chat windows going, lots of peer-to-peer. There's less sense of trying to get everyone "on the same page" at some micro level, given we're not control freaks.
So whereas one school might branch off into Maxwell's Equations as expressed using div, grad and curl, having developed a sense of spin and rotation through computerized play with vector cross products (the whole game engine and ray tracer business), another school might want to look at the pottery wheel and machinist's lathe, to get a better idea of groovy games and computations involving rotation. A third school might dive in to symmetry groups.
We're a lot higher in the tree here, and students are beginning to specialize, now that the have lots of video, video games, a computer language, and basic Math Objects in common.
The peer group I'm working with, a lot based in South Africa, is moving ahead with a Pythonic implementation of Math Objects (what I call pythonic mathematics). However, the same concepts of dot notation and class (abstraction) versus instance (concrete application or instantiation) work well with any SmallTalkish language. Our growing lesson plans database simply includes fields (blanks) for teachers to fill in, as to what computer language or other technology a student would need to complete the lesson.
In South Africa, not every TuxLab may have connect to the Internet straight off (wireless broadband is advancing, along with cell, but we're talking about a very large area -- Siberia is likewise big). A lot of the DVDs have to be circulated by land vehicle -- 'Warriors of the Net' for example (teaches basic TCP/IP, and which it's relevant to learn about even if there's an air gap betwixt your lab and the great outer world beyond the firewall).
Anyway, it all starts with Dot Notation. That's sort of a hallmark of whether you're on a Gnu Math track or not. If your teacher doesn't know what Dot Notation is, she or he is definitely not hip to GNU, nor probably EFF or OMSI either (Portland spin). You can't be considered techno-literate and not know this stuff. Our students consider that obvious (a given). And math minus techno-literacy just doesn't fly in the Silicon Forest. It's what we'd call a Bad Idea (a waste of valuable time).
So... if you're into supplying the market for techno-literate math stuff, check out our emergent Portland Tech District as an investment prospect. We're planning to get the jump on bigger cities like Detroit and New York, when it comes to gearing up for the next big thing in Education (e.g. "for credit movies"). We're maybe lagging behind LA though, in not having the sound studio infrastructure. There's no Universal Studios in our neck of the woods. Algebra City may need to pick up the slack on that one.
Sequences may be classified according to their convergent, divergent, periodic and aperiodic attributes (links to tiling and space-filling).
Early coding challenges will involve such sequences, such as polyhedral numbers [Conway and Guy, ISBN 0-387-97993-X] and the cuboctahedrals in particular [Fuller, ISBN 0-020-653204]