> Richard recently wrote:
>
> > [clip]
> > Look! The definition of Level 3 is not complicated; it's just that
> > it doesn't make sense to a Level-2'er. There's a mind mechanism
> > that has not yet been developed in the Level-2 mind. It has to do
> > with flexible modeling, position shifting, and consciousness of
> > purpose. The Level-2 mind thinks about this for about 3 seconds,
> > then goes into distinguish-and-discard mode, mapping it onto some
> > poor excuse for a Level-2 analogy and producing all the
> > contradictory "definitions" of Level 3 that have been "bandied
> > about on the list." Cut it out!
> > [clip]
>
> Could you please repost the 'real definition' of level 3? I have
> not been on the list terribly long, and have probably only seen
> the contradictory definitions. Anyway, I don't see why any old
> mind can't learn flexible modeling, position shifting, and
> consciousness of purpose, and consequently I don't see why we
> need a level 3 distinction meme.
>
>
> - JPSchneider
> - jschneid@hanoverdirect.com
JPSchneider, when I first saw this level-2/level-3 terminology: my initial
reactions were what yours appear to be now.
I don't see what you don't see above, either. I *do* observe it, in
every math class I teach recitation for.
If the apparent impossibility were not superficially real, the mathematics
exams at K-State would be *much* more difficult!
=====
Example:
One of the questions on a recent Calc I exam suffered a design error.
[Algebra is, of course, required to have a ghost of a chance.]
Instead of requiring knowledge of:
The "product rule" [whatever that is]
The rules for derivatives of "tangent" and "cotangent" [whatever
those are]
The "chain rule" [whatever that is], twice
It really required knowledge of:
Definition of "tangent" and "cotangent"
The constant rule: constant functions are always horizontal, so the
answer is 0.
Since only ~1% even *could have* exploited the design error, I conclude
that 'flexible modeling' has relatively low usage [although probably
higher than 1%!]. I have no reason to believe this semester's class is
unusual.
=====
In general, I have to work on unhamstringing the students I have to
teach. They usually are hamstrung by the "template" metaphor on the
first day of class:
"If I have not seen this exact problem type before, *PANIC*!"
This is classic Level-2 thinking. In the above example, the Calc-only
templates prevented using the much simpler trigonometric/Calc template,
which they had never seen and would have needed dynamic-rebuild to notice.
Certainly, for calculational details, this is a good metaphor. However,
at College Algebra and upwards, the emphasis starts shifting to include
how to *use* calculational details. Metaknowledge actually starts
becoming useful.
If the student fails to [for whatever reason] learn "flexible modeling,
position shifting, and consciousness of purpose" EARLY, they are doomed to
C or lower, no matter how much effort they put in. All I can do is show
them how it works; if they refuse it [the level-2 reject algorithm may go
off even though they are supposedly there to learn what I'm doing--the
dynamic rebuilding of solutions in real-time is at the border of
level-2/level-3, AND they may focus on the templates that I'm
accidentally creating], I really can't do much for them.
//////////////////////////////////////////////////////////////////////////
/ Towards the conversion of data into information....
/
/ Kenneth Boyd
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