Wednesday, May 7, 2008

Two approaches to learning about proofs: traditional vs. reform algebra

1. Doing them:

Use an indirect proof to prove that the conclusion is true.

If p is an integer and p2 is divisible by 2, then p is divisible by 2. (Hint: An odd number can be written as 2n + 1, where n is an integer.  An even number can be written as 2n.)

If a < b, then a + c < b + c.

If ac > bc and c > 0, then a > b.

(From McDougal & Littell's Algebra I, p. 703; a traditional text).

2. Writing about them; doing easy, concrete ones; and conducting interdisciplinary research:

Name and describe three forms of proof.

Prove that if 7x - 3 = 25, then x = 4.  Write a two-column proof.

Research  Edgar Allan Poe may be best known for his horror stories, including "The Pit and the Pendulum," and for his poetry, including "The Raven." He is also highly regarded as a lierary theorist and as a short story writer.
Edgar Allan Poe set out guidelines for the use of ratiocination in what was to become the modern detective story. Find out what he meant by ratiocination and how he used it in mystery stories such as "The Murders in the Rue Morgue."

(From McDougel and Littell's Integrated Math, p. 407; a reform text).

4 comments:

Anonymous said...

i'm not at all sure what
you're getting at in this post.
but it looks like as good a place
as any to make a few general remarks.

thanks for stepping up in defense
of the groups described in your sidebar.
i'll happily identify as a
math buff, overwhelmed by multi-media;
of *course* i prefer reading to partying;
that i shy away from large groups
and don't work well in groups
could be considered comical understatements.

and there's damn sure a lot of bias
*against* these attitudes ...
in the culture at large and in
mathematics education in particular
(yes, even against "math buffs" --
the party line is "remove math
from math classes; not everyone
is going to be a professional
mathematician, you know!" --
lefty is aware of this of course;
this bit's just to mention
that i'm aware of having seemed
to contradict myself).

moreover, your experience (lefty)
with autism in your own family
and professional experience in
working with autistics are not
to be dismissed; you're bringing
a lot of much-needed perspective
to the discussion in my opinion.

and you've even made it clear
that you're using "left brain"
metaphorically ... for which,
much thanks.

so just please *be careful*.
recent years have seen a huge growth
in bullshit-about-brains
and it seems to me that you're
in real danger of giving the enemies
of reason a pretty plausible excuse
for dismissing your arguments.

here's a post on the skolnick effect
from about a year ago; i've only recently
become aware that this phenomenon
has a name (and was sure relieved
so to discover).

meanwhile, keep it coming!
you're doing good work here ...

yrs in the struggle. vlorbik

lefty said...

vlorbik,

Thanks so much for your thoughts--esp. as a fellow "left-brainer"! And for your caveat about the use of this term....

Hmm... I've been hoping that the disclaimer on the side bar will prevent my detractors from using my metaphorical appropriation of brain terminology against me. But perhaps I'm being overly optimistic? Any suggestions on how better to cover my a**? Perhaps repeated links to the sidebar every time I use the term?

In general, it has often struck me that, no matter how careful you are, those who oppose you--particularly if they have no substantive argument--will find a way to quote you out of context, and I know I'm very vulnerable to this.

Thanks again for your words of support.

lefty said...

Thanks also, vlorbik, for the very interesting link to the Skolnik effect. I wasn't familiar with the term either, but it names an important phenomenon.

Now I'm wondering whether there's a term for the legitimacy conferred by mathematical terminology to works by literary & social theorists--e.g., that critiqued by "Fashionable Nonsense".

Or for the appropriation, by same, of linguistic terminology--e.g., "deep structure"?

Anonymous said...

isabel GPD posted
a few months ago about
"mathification" (citing _language_log_,
strangely enough [as i did upthread
on a different subject]).

we've recently seen "mathefication"
(spelled differently) at KTM.

neither of 'em is quite what
you're describing, though ...
maybe the field's wide open
("the OILF phenomenon").