Thursday, September 8, 2011

Unknown Quantity

What is Algebra, exactly--especially this advanced variety we're currently studying?  Let's answer the first part of that question first, because it is, I think, the more interesting one.  Is it solving for x?  Graphing? Functions?  All of the above?  When I asked for a list of words/phrases/concepts that you talked about in your last Algebra class, you rather quickly--like, in under two minutes--came up with an impressively large list.  So thanks for that.

But what I really want to jump into your head when you think about Algebra, the word that should immediately lodge itself in your cerebral cortex, is structure.  Algebra defines the rules of the game for working with mathematical objects--in our case, (mostly) the real numbers.

For instance, you’ve been working with expressions like 3 + 4 basically since you started counting.  Then, many years later, Algebra started asking questions like, What happens when you add x + 4 instead?  What is the fundamental nature of adding 4 to something?  What does that relationship look/feel/taste like? You’ve done a lot of that (remember generating sequences of numbers based on recursive rules just like that one?).  In fact, you got way broader and more general than that.  What happens when you consider relationships like x + b instead of just x + 4?  How about relationships of the form mx + b?  Now you’re talking about all the relationships that we call linear, and those things have taken up a sizeable chunk of your algebraic brain over the past few years.  So really, when you think about it, looking at the graph of a line captures a whole lot of important information about addition, about the structure of addition with real numbers.  Such is the power of Algebra.

It’s tempting to think so, but Algebra isn’t the only important way to give structure to a set like the real numbers (when you move on in mathematics, you’ll learn about some other important ones like groups and topologies).  In fact, it’s not even the only way to put an algebraic structure on a set.  What we’ve been calling capital-A Algebra is really just one of a bunch of algebras, most of which are very mysterious and exotic indeed.  We’ll encounter a few of them, though, later in the year when we start getting a handle on complex numbers and matrices; they come with algebras--and structures--of their very own!

As for the second part...

You’ve been wending your way through this subject for at least a year now—probably way longer than that, actually—and suddenly you find yourself in a class called Advanced Algebra thinking, “Okay, so what makes this particular variety of Algebra so especially advanced?”  At least that’s what I’m thinking, as a person who has to spend a lot of his personal time forming professional-sounding thoughts about your current and future education.

Besides scaring underclassmen into quivering submission, the word advanced is fairly unimportant for our purposes, though it reflects an interesting change in the MN state math standards.  Approximately the first half of Algebra (up through all of the material about linear relationships) is now required for eighth grade. The class we used to call ‘Algebra,’ which has undergone something of an overhaul, now has to be called ‘Intermediate Algebra’ in order to distinguish it from the junior high stuff.  Finally, that leaves us with your (and my) current class, ‘Advanced Algebra,’ which really just means that it’s the third and final course in the overall Algebra sequence, and the second of the two we now offer here in the high school.  Last year we called it ‘Higher Algebra,’ and this year’s course is essentially identical.  New look, same great taste.  If it makes you feel less jittery, you can think of this class as Algebra III.

So welcome, again, to Algebra.