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OOOP!! the earlier draft got out befor it was honed. I'd prefer using this one. SORRY! Above doorstops etc: a sky-pie AMPS
Posted:
Sep 7, 2010 1:24 AM
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Building on Alain's guidance, below:
"To create an organic whole it takes more than ..."[connecting topics]
Interesting perception. Indeed, the notion of an "organic whole" suggests a
program-design principle which long has been ignored in the American
core-curriculum, but which might well be imposed on any "ideal" program for
personal mathematical fitness (using a yet to be designed idealized Adult
Mathematical Preparation System).
As we consider how students' "living" personal mathematical theories evolve
within the
minds of various individual learners, might it be best to guide those
theories to evolve as progressively growing organic
wholes? If so, might it be safest to presume that, when a member first
enrolls into a fitness program, his or her mathematical theories consist
only
of fragmented elements or components -- and that any AMPS might do best to
lead the members to progressively integrate/assimilate those fragments into
evolving, cohesive (organic?) wholes? Would that mean that any IDEAL
fitness program should enable its members to reconstruct their mathematical
knowledge, from "scratch" (I.e. from some minimal presumed base)?
" .... cutting up of a subject matter into topics ... does kill the subject
..."
And that observation also suggests that constructive dialog toward designing
a sky-pie ideal had best entail ad hoc agreements on what its key terms are
to mean in that narrow context of instructional design. Common language
cannot suffice.
Example: the tactic of analyzing a body of mathematical knowledge into
"topics" might be
construed as meaning cutting the subject up into essentially non-overlapping
components
... perhaps a bit like the geographic separations of this nation's states.
Such has long been the American school-mathematics tradition of separating
arithmetic from algebra from geometry, etc. ... and of trying to separate
the (algebra 1) "topic" of linear equations from the "topic" of linear
functions. It has been argued that that kind of separation into "topics" is
not in the best interests of the learners. Indeed, the current push for
"connections" seems to be at least partly in response to the maladies of so
partitioning the curriculum into disjoint "topics". So, avoiding such
partitioning
into "topics" might be an appropriate guiding principle for the design
effort.
On the other hand, it is expedient (if not essential) for the program-design
effort to invoke a taxonomy that empowers its advisors to dialog about
various sectors of the subject matter ... e.g. the structure of scaled
rulers/tapes versus the structure of the Arabic numerals vocabulary for
lines of whole numbers. Every mathematical theory consists of concepts, and
theorems about those concepts --- the two kinds of elements of that
theory --- and a logic which connects those elements into a rationally
cohesive whole. The lattice-like structure of the theory generates an
associated topology of cognitive "neighborhoods" for each element or cluster
of elements --- the same neighborhoods that provide the "cognitive maps"
used by many researchers and by a few curriculum developers.
The program-development advisors must be able to
communicate about distinct neighborhoods within mathematical theories ...
and about the differences that distinguishes some from others. If calling
such differences, "topics" would misconstrue the nature of the design
effort, or distort the resulting fitness program, we had best find an
alternative term. But let's not be rejecting anything for the wrong reasons.
" ... goes directly against the trend in the way mathematicians try to
understanding mathematics, e. g. category theory."
Because it strongly pertains to how the learner personally abstracts
mathematical concepts and theorems, the word, "understanding", is another
crucial keyword --- whose commonplace meanings are too ambiguous for
purposes of instructional design.
Category theory is an extreme example of trying to "understand",
abstractly --to OVER-STAND, by achieving mathematical umbrellas. At the
other extreme, UNDER-STANDING entails knowledge of things from which the
mathematics is abstracted ... perhaps even of some real-world experiences
that can be mathematically interpreted. Every human searches both for such
a CONCRETE UNDERSTANDING of things, and an ABSTRACT UNDERSTANDING of
(perhaps even the same) things. But for purposes of instructional design,
the
psychomathematical difference is crucial.
To view that distinction through a context familiar to all
mathematics teachers (because it normally is ignored by American authors,
at the expense of their students):
The student who studies the functions, 2x+3 and -3x+7 and a few others soon
uses those functions as personal GENERATORS of the cognitive ABSTRACT whose
FORMULATION normally is the mx+b formula. Although abstracted from only
those few generators. that abstraction then REFLECTS its generators, by
using those
as EXAMPLES, to identify still other examples of the same abstract.
Contrary to many American textbooks, nothing can truly be an "example" of an
abstract that has not yet been generated. Several contributors to this
list's dialogs have
expressed that it often works best to abstract from real-world experiences,
to
their mathematical interpretations. The premature "colored box definition"
of a linear function as being one that "can be expressed as mx+b" ...
commonly laid onto
students who have not already constructed that abstract and reflected from
it ...
precludes the mx+b formula from being abstract within the students own
theory.
The device of subsequently providing "examples" might later enable the
already-confounded
student to belatedly abstract whatever was needed before the formulation.
But that humanly unnatural kind of learning totally misses the
"living stuff" of mathematically abstracting from empirical or conceptual
generators.
Once the learner uses concrete examples to achieve the mx+b abstraction
(even if
only belatedly doing the abstracting) that abstract (hopefully) is
GENERALIZED by extending its SCOPE to comprise additional examples of that
abstract ... in essence, by expanding the domains for the (m & b
coefficient) parameters or the domains of the functions . [Oddly, students
normally are expected to do so, without
being educated for doing so.]
The resulting, expanded class of examples that are
presently accommodated by the learner is his/her present CONCRETE
UNDER-STANDING of that abstract and of that formula --- often not as broad
and strong as teachers might casually presume. Without achieving such a
concrete understanding,
the mx+b formula is not at all abstract -- within the learner. It is merely
a character-string (which is how students commonly perceive it) ... which
they later are told to use as a template.
The difference between using concrete examples first as generators, later as
reflections ... versus using concrete examples only belatedly
as post-formulation applications ... can be the difference between
mathematics being rational and commonsensible to students, or being the
dictatorial regulation of data-processing procedures.
As umbrellas go, the learner's abstract understanding (or over-standing) of
the
mx+b formula might be pursued in many directions. Illustrative is that the
(Maclaurin) b+mx formula suggests that the function begins a (bib) and
progresses forward & backward at the rate of m-per-pos1 and neg-m-per-neg1.
Since each such b+mx formula is a concrete example of the (Taylor) k+m(x-h)
formulas ... for slope-m lines on (h,k) points ... that k+m(x-h) family of
formulas
provides one abstract umbrella from the mx+b formulas. [The American
fetish of equations has caused Taylor's point-slope formulas to be largely
ignored by algebra books.] Obviously the passage from the mx+b formula to
the k+m(x-h) umbrella is a very different kind of "understanding" than is
the passage from mx+b to 0.72x+(3/8).
"It seems ... necessary to have textbooks and ancillaries ... is
necessary ... that the textbooks respect the living stuff that mathematics
is."
Printed matter must have been one of the first media for enabling a
"distance learning" technology. But if each "textbook" consists of hundreds
of pages bound
together as a single document, their use has very severe limitations ...
including their possible causing of educational calamities.
To distort Lincoln: Not all textbooks are good for all students; not even
some are good for all students, some are good for no students; and only some
are good for some students. But apart from considerations of educational
quality is the problem of discerning just what media are optimal for use in
what situations. In some situations, textbooks might be necessary --- in
other situations, textbooks might not work, at all. From an
operations-research perspective, the "advisory team" for a sky-pie fitness
program must open-mindedly consider all possible combinations of all
learning media. But I, for one, could easily believe that the best results
will come from using one or more good mathematics books to unify the design
of whatever gallery of other media might be more directly used by the
learners.
So Alain's comments continue to help carve out the emerging visions of an
AMPS-based, mathematical fitness, community-education program. There still
is room for many volunteers to serve as advisers for that project.
Gratefully,
Clyde
--------------------------------------------------
From: "Alain Schremmer" <schremmer.alain@gmail.com>
Sent: Sunday, September 05, 2010 9:00 PM
To: "mathedcc list" <mathedcc@mathforum.org>
Subject: Re: Another Precalculus Doorstop, Another Migrane
>
> On Sep 4, 2010, at 3:09 AM, Clyde Greeno @.MALEI wrote:
>
> (1) To create an organic whole it takes more than
>
>> connect later topics back to earlier ones
>
> If one sets:
>
> Topic C = Connection between Topic A and Topic B
>
> one is teaching topics.
>
> But, while the cutting up of a subject matter into topics has a never
> waning appeal to "educators", it does kill the subject and, in fact, goes
> directly against the trend in the way mathematicians try to understanding
> mathematics, e. g. category theory.
>
> (2) In order
>
>> to create a working demonstration of a mathematics program
>
> it seems to me that it is necessary to have textbooks and ancilllaries.
> And the only a priori specification that is necessary is that the
> textbooks respect the living stuff that mathematics is.
>
> Regards
> --schremmer
>
>
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