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Tall D. Psychology of advanced mathematical thinking (Kluwer, 2002)(400dpi)(T)(C)(310s).djvu |
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of successful transfer of such meta-mathematical knowledge from one context to another;
no more, in any case, than of successful transfer of mathematical knowledge...
It is true that learning takes place, but the precise mechanism by which it
occurs is unknown...
This seemingly innocent
definition proved to conjure up all kinds of logistic and epistemological problems, which
incredibly, were often addressed explicitly in some school curricula...
All techniques, which form a major
component of school and university mathematics courses, can be discussed from a
functional approach...
Level 1 is concerned with a macro-organization of knowledge at a schema level;
for example understanding that hi the equation of the line L: y=mx+b, the m and
b represent the slope and the y-inlercept
Level 2 deals with compiled knowledge, macro-entities and entailments; for
example if m>0 the line rises, if \m\ is large, the line is steep, the point @,fc) is
the y-mtercept of the line, etcetera...
The lists of prerequisites which
must be mastered for even the simplest of the tasks get unwieldy; the topic gets over
atomized - and the whole seems to be much more than the sum of its parts...
One finally masters an activity so perfectly that the question of how and
why [students don't understand them] is not asked anymore, cannot be asked anymore and is not
even understood anymore as a meaningful and relevant question...
Today, this conception problem seems
to have been settled hi favor of the analytic description, at least mathematically speaking...
Ninety percent
of them were able to do this for the algebraic case with 55% being able to justify why their
procedure worked...
Those in favor of visualization considered that functions should be thought of graphically
whereverpossible and when applicable, operationson them should be thought of in this way
too...
But abstraction is what mathematics is all about, at least according to
Hilbert, the Bourbakists and most university professors...
Likewise,
visualizing functions in parametric form, also proves to be intolerably difficult, especially
when the representation moves from two dimensions to three...
Anxiety, the
nature of mathematics and the loss of meaning through the "intellectual" process of over
ambitious use of symbolism and abstraction have been discussed above...
Aline Robert A982a,b) has studied different models which students may hold of the
notion of the limit of a sequence...
Paulis apotentialist: To think of some whole, a set or a sequence, one has to run in thought through
every element of it...
In the initial stages of learning, we therefore see spontaneous conceptions arising which a
often in conflict with the formal definition...
) Is the limit attained ornot?
This is a debate which has lastedthroughout the history ofthe concept...
Although
Robinson thought that his neat logical solution would solve the three hundred year conflict,
this proved not to be so...
Although such meanings may be avoided in
written texts, they can passed on inadvertently from generation to generation as the teacher
tries to "simplify" the complexities to "hety" the students...
In particular, they were
made aware of the different meanings of the words which they were going to use...
It survived also in
applications, especially in physics, with a role approaching its original status under Leibniz
- as an infinitesimal increase - a useful tool for substitution in equations, but suspected of
being based on less rigorous practices...
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