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Topic: Easier and quicker methods for algebra - rote tricks? (was Re:
Why?)

Replies: 3   Last Post: Oct 28, 2012 1:02 PM

 Messages: [ Previous | Next ]
 kirby urner Posts: 3,690 Registered: 11/29/05
Re: Easier and quicker methods for algebra - rote tricks? (was Re: Why?)
Posted: Oct 27, 2012 10:40 PM
 att1.html (7.3 K)

On Thu, Oct 25, 2012 at 10:44 AM, Paul A. Tanner III <upprho@gmail.com>wrote:

>
> Case in point: One of my favorite evaluations is to ask a person to solve
> (you know what I mean - isolate the variable) for h in
>
> (ab)/(cd) = (ef)/(gh)
>
> and
>
> (ab)/(cd) = (ef)/(g[h-i])
>
> and get to the point of writing the final expression as quickly as
> possible.
>

What happens when a person does not evaluate well on a favorite evaluation
I wonder.

Have they lost all chance of being a favorite person? Hansen speaks of a
"club" (the inner circle of those who "got it").

We'd need to clear up possible ambiguities in your notation of course, as
some might see g[h-i] as subscripting g, whereas you're simply nesting two
kinds of bracket (not every notation wastes two kinds of grouping symbol,
each with a left and right that should stay balanced).

>
> h = (cdef)/(abg) + i.
>
> A good manipulative to show this at least for the multiplication/division
> context and practice visually solving for any randomly chosen of the 8
> variables - especially with a group of people - is to use those children's
> blocks with letters of the alphabet written on them and a device that has
> the four positions of upper left, lower left, upper right, and lower right
> to put the blocks on.
>
>

I'm lost. You have letters a - i and you're looking at blocks with six
sides each with colored letters on them... and we're supposed to do what
again?

> So again, I ask, in a different way, "Why should it be that those who need
> more help and especially the most help and who could and even would (since
> "would" has already been demonstrated to hold for so many in my experience)
> benefit from obtaining certain information be disallowed from obtaining
> that information?"
>
>

Where I'd like to go with this cohort is on a foray into notations more
generally, with some attention to prefix notations, and their isomorphism
with infix in some namespaces.

To be more concrete about it, we could boot Haskell, not because we're
planning to do college level functional programming, but because here's a
simple calculator-like device with an REPL (feedback loop) that will allow
prefix as well as infix notation around functions or operators like add.

3 + 2
(+) 3 2

could all mean the same think in that notation.

I like having them limbered up and alert to the fact that there's no fixed
notation, no "once and for all" symbolic problem solving language. There
are many. And they're partially overlapping. The whole is more than the
sum of the parts.

In the old days, you needed a big budget and computer center to offer
students any time at all using an REPL. Nowadays, you have so many free
options, with more in the pipeline. The economics have changed a lot.

I'm not saying your average 5th grade math teacher is in any position to
boot up and screen the Haskell interactive prompt and let students take
turns at the wheel. Above average math teachers are in such positions
though.

I'm not talking about weeks and weeks studying Haskell or Scheme or other
such language. Not in 5th grade. There'd be choices like that coming up
though. Students want to do websites, maybe do some things with maps.
Gotta learn what an API is, then find how your language talks to hosts.

That's what alpha-numeracy looks like today. In addition to Algebra and
Geometry, or in tandem with same. Geometry includes playing with POV-Ray,
VRML, maybe vZome (virtual Zome tool). That's what many of the adult STEM
teachers I know do in their spare time. OpenGL. VPython.

I've you've got the budget for it, Mathematica is another excellent
playground in which to experiment with the different ways of notating
functions and operators.

This may be exactly the kind of curriculum the Education Mafia is thinking
to implement at the reverse-engineered IBM clone school in New York that
Haim was ranting about (I don't think he wanted money committed to a school
for IBM clones).

The idea of a "polytechnical school" would like like a school for
"polymaths". They'd use tools like Sage. They'd study Euclid's Method for
the GCD. IBM would snap them up.

To anticipate an answer that would try to justify disallowing them such
> information: I utterly reject the idea that there is such a thing as
> harmful knowledge. Only ignorance can be harmful. Knowledge is power, and
> ignorance is weakness. (I'm of course excluding knowledge that can cause
> one emotional harm - and this well-known saying is not necessarily meant to
> cover knowledge that can cause one emotional harm.)
>

There's what both the GSTers and the Economists call "opportunity cost"
though.

The more time you spend getting ready to climb Calculus Mountain, so you
can impress a certain crowd, the more you may lag in your ability to
impress a different crowd that hopes you'll sling code and join them on
Github.

We don't sufficiently explain to students the unconscious choice that
they're making (or is being made for them), in agreeing to Obey (to do what
they're told).

I tell them to look for warning signs. No mention of TCP/IP? No talk of
electronic record keeping? No "how things work" with respect to
infrastructure, municipal water, sewer, electric?

Well then, you've got some legitimate reasons to be suspicious.

If you learn nothing of local history, how your neighborhood got to be the
way it is (timelines), then yes, you have every reason to question the
relevance of what the imperial masters have imposed upon you.

Sounds like you're getting the short end of the stick, whatever the deal is.

(This is the same message I share with overseas schools i.e. cultural
imperialism, which teaches you to ignore your own time and place as
relatively unimportant, is rampant, a legacy set of reflexes
(uncoordinated), and needs to be overcome if we're to have less than
awkward responses to our various challenging situations. Don't let them
*not* teach you about the ecosystem you live in -- it's not their place to
waste your time like that, you have rights as a human being to avoid
imprisonment by default i.e. unless you've been proved in violation of the
rules in some way that merits such treatment (we expect jury trials by
peers then, not simply "guilt by association" or "because he's a minor" or
because "you failed to pass the evaluation")).

Kirby

Date Subject Author
10/25/12 Paul A. Tanner III
10/27/12 Paul A. Tanner III
10/27/12 kirby urner
10/28/12 Paul A. Tanner III