## Classes, Laws, and Associations: Not JAWOPPA Anymore!

Posted by vlorbik on November 12, 2009

Let “Algebra 101” denote a generic

“remedial” algebra course for college

students.

For historical reasons, one typically finds

“introductory” material in Algebra 101

on Number Systems

(the Natural Numbers, the Integers,

and so on) and on such algebraic

“laws” as “The Associative Law

of Multiplication”… which is of course

the statement that for any

$x$, $y$, and $z$ (in any of our systems),

$x(yz) = (xy)z$.

For practical purposes however

this material is actually much

more {\it advanced} than most of the

rest of the course. However

{\it logically} prior these ideas

may be to {\it formal} discussions

about the material,

we will be duty-bound to count

ourselves lucky if we can see

some precise {\it calculations}

from our students, and insisting

on correct use of vocabulary

courts disaster (since, for example,

many instructors are themselves

incapable of work at this level).

So we throw the facts out there in a big pile

knowing it’ll baffle ’em; it’s easy enough

to be a whole lot easier to understand

than the textbook and indeed several

students in any decent-size class

will have prepared minds (this

amounts to a certain “cultural

literacy” consisting mainly of

a “college level” of ordinary

words-on-paper literacy) and

get quite a bit out of this work

in a pretty short time…

so it’s a long way from a total loss

classwide. But for many,

maybe most, it’s just blah blah blah.

And this is even more of a shame than usual.

I’ve taken the time… stolen it even…

from a few of these Algebra 101 classes

to work carefully with {\it one little bit}

of this “high-level” stuff: associativity.

Enough to know that one can easily

bring {\it the whole class} (or the subset

that actually shows up and works;

never forget these are college students)

to work out some pretty convincing

exercises.

{\bf 1.} Prove that $((ab)(cd))e = (a(bc))(de)$ showing

each application of the associative law as a separate “step”.

I’m going to start yelling now in the certainty that

I won’t be heard. This looks trivial and is;

that’s why it’s important. If it {\it isn’t} trivial,

you don’t have any idea what

associativity even means

in any way that matters

outside classrooms. It takes

about a week to get a class to do this

though and there’s just no time.

I’m claiming here that there’s no point

in continuing to present lists of algebraic laws

at this level if we actually seek

student understanding. I now go on

to claim that this is perfectly well-known.

These are weeder courses and

this is a way to make flowering young minds

into the weeds they’ve unwittingly

signed up to be.

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