Re: Programming languages for the very young

From: Samuel Walters (swalters_usenet_at_yahoo.com)
Date: 01/20/04 

Date: Tue, 20 Jan 2004 09:34:20 GMT

| Darius said |
(On deriving the vertex formula using derivatives.
> Any form will do.
Good point.
It makes me wonder if teaching has stunted some of my "real" mathematical
skills. I tend to think first about how I would teach a concept to a
college freshman.

> One of the benefits, in my opinion, of the differentiation approach
> points out how I prefer(red) to handle math. The differentiation
> approach isn't particular to quadratics nor does it use techniques
> (somewhat) specific to quadratics, not even knowledge of the quadratic
> equation is necessary. I like(d) to have the more general ideas or the
> more basic building blocks to derive others, and I like(d) to know where
> things come from. I never memorized completing the square because I knew
> what the final form would look like so I just set that equal to what I
> was given and applied general algebraic manipulation. Getting better at
> a generally applicable technique, algebra or differentiation, made me
> better at special applications as well.

I, like you, find it much easier after knowing a "why" to the "what."
 
> In a nutshell, I try/tried to internalize (or sometimes memorize) the
> knowledge that gives/gave me the most bang for the buck and don't/didn't
> actively try with things that could be subsumed by other things.

It's true that the more powerful and deeper methods of mathematics should
subsume clever tricks. The vertex formula is a clever trick taught at a
very specific time for a few reasons. One is that it allows students in
college algebra to start tackling optimization problems. I think this is
important. At about that time they're whining about how math doesn't
apply to real life, and optimization problems demonstrate an important
spirit in mathematics. Another is that the vertex formula is a lead in
towards more rigorous proof-based methods. For many, the derivation is
the first time they've seen a real one. The third, and weakest, is that
it rounds out the endeavor into parabolas. It could be argued that this
should be delayed until pre-calculus. (I'm not arguing it! I just said it
could be argued!)

The other similar concept introduced in college algebra is the imaginary
number. This time, it's the first truly abstract mathematical entity
they've encountered. Despite the name, they are effectively unimaginable.
 Sure, we cope with them by inventing "the complex plane," but that only
works if you accept that imaginary numbers are a necessary blemish on
mathematics. They, too, round out the foray into parabolas. The problem
is that they present no utility to students at that level. It's touched
on very briefly, learned by process and rote, and then dropped. I
think that students would benefit if complex numbers were delayed until
they get a deeper discussion of polynomial roots in pre-calculus. At
least there, it connects to some contextual hooks.

Preemptively answering a question I hear asked often at work: I think
logarithms should stay right where they are. College algebra, with it's
focus on algebraic methods and the relationship between functions and
their inverses. The lead into logs is as smooth as it can be made, and it
forms an unbroken progression of ideas if you first introduce exponential
functions.

Sam Walters. 

-- 
Never forget the halloween documents.
http://www.opensource.org/halloween/
""" Where will Microsoft try to drag you today?
    Do you really want to go there?"""

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