Saturday, June 27, 2009

About Algebra

[first published to Math Forum, typo fixed, links / pictures added]

There're a lot of comparisons one might do between any two Lower48 textbooks, I agree, although perhaps it'd make more sense to define what we mean by "algebra" and what learning it means.

Of course there's simplification of algebraic expressions and finding unknowns through the rules of equality, algorithms pretty well automated by now, even on some calculators, yet we do want to teach some of this stuff. But why? What's our context?

At the core of algebra is this notion of sets of things or objects "of the same type" i.e. all integers are connected in being members of the set Integer whereas all rational numbers, a superset of Integers, defined as p/q where p,q are Integers, comprise another type, a more inclusive type in that every integer p is likewise p/1 i.e. a member of Q.

So we want the notion of sets, some set notation probably, and a strong notion of types. The "type" discussion is necessary to define "closure" i.e. when you do an operation (unary or binary) on members of a set, do you get another member of the same set, or do you fall out of the set, perhaps to an object of another type? For this kind of thinking to make sense, you need that all important notion of type.

:: investigating types, note pi ::

We also have this notion of operation, which we combine with "function" i.e. 2 + 2 and add(2,2) or (+, 2, 2) are different ways notations express addition. You have these specialized operators (+, -, /, *) -- here already converted to their more standard computer-signified equivalents -- but under the hood you can simply think of "feeding the fish" i.e. a function is like swimming in a fish tank (namespace) and you feed it arguments (objects, sometimes other functions, as when feeding the "return a derivative" fish). If you feed it arguments of the wrong type, the fish (function) may "barf" (and we have hours of interactive slogging (hard fun!) to discover what that means in practice (might use more Java at this juncture as Python's duck typing means barfing at runtime if there's barfing at all and often there isn't, nor should there be)).
(scroll down for "guards at the gate" scenario, a traditional YouTube motif wherein kids themselves get to act, in the tradition of Monty Python skits i.e. it's not always about making cartoons, we also use live action to communicate math concepts, just like any teacher does).

With operations come identity operations, those which leave the arguments unchanged, or identity members of the set, such that "assert __mul__(a, 1) == a" generally evaluates to True in Python, unless you've done something perverse with your operation's definition. In a formal algebra, we expect group, ring and field properties to be present or not present i.e. we wish to think in these terms.

In building up a notion of "types", which so many computer languages are intrinsically strong in, we develop an "algebraic sense" among our students. Plus we connect to all the traditional topics in using trigonometric functions, whatever computer algebra systems (I favor writing a lot of low level object definitions, e.g. for rational numbers Q, for vectors V -- not taking too "black boxy" an approach, not when first learning the ropes). We cover N, Z, Q, R, C as consecutively concentric sets i.e. each is a superset of the one before. We also do a lot with finite groups, such as the totatives of a number (closed under multiplication modulo that number), a way of reinforcing prime vs. composite and setting the stage for RSA.

Of course students running through all of the above are going to out-perform most analog math track students per the criteria we care about in the Silicon Forest, e.g. familiarity with at least one computer language. Although we believe in consulting multiple textbooks (PDFs), the idea of buying truckloads of Lower48 poopka and wasting kids' time with that would be an anathema to our well-to-do, thinking parents. There'd be instant action and the school would go away, replaced with a charter with "the right stuff" (as we like to call it).

But I understand demographics differ around the country, from zip code to zip code, and some math tracks still insist on using calculators (har!). We tend to feel sorry for those poor slobs, know they won't have an engineering-related job in our region so easily, but there's always remedial college work. Nice if you can get it in high school though, on a competent digital math track (which does include some calculus, as our web sites make clear -- might even use some Mathematica at this point, depending on budget).

Also, your algebra needs to bridge the lexical with the graphical in some way, obviously through vectors and the polyhedra you might build with them, but the devil is in the details. My stickworks package, offered free to Portland schools, pretty much solves the problem of how to get colorful rotating objects on screen, a must in any math lab worth beans. What I don't do much about is the music or audio track components, needed for editing the final results (student work -- a lot of it headed for YouTube you'll see). Other teachers help me where I'm weaker. We collaborate, form voluntary associations with federal agencies e.g. VOA, and private companies e.g. 4D.

The school I teach in is called Saturday Academy although I'm off at the moment, busy lobbying, as I think we shouldn't have so many backward schools in Rose City especially, nor in the rest of the state. Seattle isn't really my purview, although I regard Silicon Forest to sometimes extend that far north (to Northgate, where our Math 'n Stuff sells Huntar CubeIT!, that thing we use to show MITEs, or minimum tetrahedra -- but that's going back to like 3rd grade so I'm getting off topic).

Stash of MITEs
We also band together as tutors over the summer, as many a "self scholar" is full of curiosity and wants to study interesting stuff, once the day care service is no longer available. That's where the coffee shops come in as well, as most of this is available through wifi (though it helps to have a guide).

I mostly tutor other adults, or run workshops for everyday math teachers ready to launch a digital math track through their school, an increasingly popular idea as teachers notice that once you throw away those calculators and start using Google Earth 'n stuff, the students perk up, say "why weren't we doing this earlier" (often there's no good excuse, as "spending too much money on dead tree textbooks" is more a confession than a reason for anything, an admission of corruption).

Here're some web pages from PPS/Winterhaven, a geek hogwarts I taught at. Compare this to the muggle educations you get in the rest of Lower48. If you wanna be a geek, maybe move to Portland, as that's not the training you'll get in other places probably. They're really slow out there, and not because less intelligent in any way, just docile, herd-like (the midwest is all about herding, whereas the east is all about thinking Europe is ahead in some way (snicker)).