### Introductory Remarks

I’m starting three sections of Mathematics 148: College Algebra on Monday; I intend to comment on the process in this blog to a much greater extent than I did in my Calc III blog last Spring.

So. First of all, the title: “really” Math 148-150 are two parts of a single course: Precalculus. There’s a sort of “grab-bag” structure more-or-less built in since there are typically several essentially unrelated topics needing further development (or remediation) before the target audience can be considered “ready for Calc I”.

But there are some “themes” and we might do well to consider ’em. First of all, then: Functionology. The central object of study in Precalculus is the function. In earlier courses (102-103-104 at my home campus; “Basic Algebra” hereafter), one makes considerable use of functions—in particular, one uses them to compute certain numbers—also certain graphical properties of functions are developed (the one thing that every Basic Algebra student always remembers is the “vertical line test”, for example). But in Precalc, we’ll look at functions in much greater depth.

The distinction I’m waving my hands at here can be made very vivid (for me) by invoking a notational issue. Suppose we decide to let $f$ denote “the squaring function”: $f(x) = x^2$. Notice that I said that $f$ is the function; I said it because I meant it. It’s common to say instead that $f(x)$ is the squaring function, but careful authors know that $f(x)$ is the name of a (variable) number. Just as $f(3)$ denotes 9 or $f(-2)$ means 4, $f(x)$ means $x^2$—and, again, this is a number, not a function. We can think of things this way: $f$ is the name for the entire graph, whereas $f(x)$ is the name of the y-co-ordinate of a (variable) point of the graph. In a Basic Algebra class, students believing that I’m making an awful fuss about what just must be a pretty trivial point might very well be justified; not so here. Definitions and notations matter and Precalculus is long past time to try to make that as clear as we know how.

Anyhow, the point is that we’ll consider functions as objects in their own right and develop an “algebra of functions” that generalizes the “algebra of numbers” (for example, just as the operations of addition and multiplication combine numbers to produce numbers, the composition operator combines functions to produce functions).

Meanwhile, our 148-150 class breaks down (very) roughly along the lines of Discrete (148) versus Continuous (150)—the Continental Divide of Higher Math. The focus is on algebraic properties of functions (polynomial functions in particular) in the first class and on their geometric properties in the second.

And so on. I’m not really making a point here and am inclined to delete the whole thing. Probably I won’t have even this much Big Picture stuff in Monday’s introductory lecture. Anyway, I really just wanted to serve notice that I’ll be using notations like ${\Bbb C}[x]$—as opposed to consistently writing out long strings of English like “the set of polynomials (in the single variable x) over the Complex Numbers”—and to offer some justification for this choice; also to put a good bashing on “synthetic division”. But my syllabi are printed out and there’s no compelling reason not to go home and enjoy the rest of the break starting now. I’ll get to these topics soon enough I suppose. These computers will be much busier next week (and throughout the quarter); sometime soon I’ll need a personal connection if I’m really going to keep up with “liveblogging” the quarter.

I’m unlikely to post again here before the first class; students finding their way here are more than welcome to comment. I hereby advise such students not to use their real names (and request that they e-mail me some notice of their in-class names; I promise not to match these to their online avatars/handles/noms-de-class in any public way [indviduals may of course choose to share their blog identities with their classmates]).

1. somebody else

“the central object of study in precalculus is the function”.

(i’d’ve made a note of this if somebody *else*’d said it
& so i’ll take the blame and make a note of it now.)

2. somebody else

Definitions and notations matter and Precalculus is long past time to try to make that as clear as we know how.

yep. i used to know what i was talking about alright.

PS… i forgot i was actually *called* “somebody else”.
i’m not there, i’m gone.

PPS sign this kid up.

3. somebody else

one more thing. (PPPS)

this turned out to be one of my
favorite classes of all time.
blogging rocks. or anyhow
used to.

• ## (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)