For anybody outside UTK who may still be reading here: M300 is `Introduction to Abstract Mathematics', essentially a language course, teaching sets, relations, induction, formal proof-writing and axioms for real numbers. --- We have local course notes, and the following are only footnotes to a tiny portion of the material. If you find any of this useful in any course and have feedback from students on it, I appreciate to know.
This one is on motivation: Why do they force you into a language course on
pre-Cal' stuff after having completed calculus already? (I'm not sure how
convincing this was, maybe I'll do better advertising next time :-)
Next time, I plan to try the following, and if anybody likes it, I give the
reference (complete material not here for copyright reasons).
The students have a lot of
trouble reasoning (putting if-then to gether and contrapositives).
To address this issue, nice material is available by Rev Charles Lutwidge
Dodgson (aka Lewis Carroll): see eg. `A selection from symbolic logic' in
Penguin's Complete Lewis Carroll. I quote Carroll's first example (slightly
adapted for our less formal purposes) for illustration:
(1) Babies are illogical
(2) Nobody is despised who can manage a crocodile
(3) Illogical persons are despised
Find the / a conclusion (using all info)!
In our course notes, we have a few warning examples that you can't
conclude from the validity of a statemant for a whole lot of instances
to its validity for all instances. Here is another example which
isn't in the course notes. Its advantage is that (provided I mention the
combinatorical meaning of the binomial coeffs later), I can then return
to the example and resolve the mystery.
We discuss the axiom of induction, but honestly, if I didn't know more than
what the audience is expected to know, I'd wonder why it is an axiom and
not a theorem. A few comments to help with this potential question.
Here is another nice example for induction. I advertise it, because it deals with non-mathematical objects (i.e., independent of mastering a formal symbol like the summation symbol, or technical algebraic skills) and displays the recursive nature of an algorithm parallel to the formal induction proof, thus (hopefully) illustrating the reasoning behind the induction. If you want to use this example, you have to work it out for yourself, for the time being, because it's tedious to typeset, and so typesetting it isn't my priority: Any 2n by 2n chessboard with one (arbitrary) square punched out can be tiled by tiles consisting of three squares arranged in an L-shape. (This was a problem in a German math competition for high school students a couple of years ago)