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numbers and succession: metamathematical reflections, historical and educational activities on a song Leopardi (4)

4. Looking to the historic teaching
explicit attention the operational concept, then, is the basis of some important attempts
formalization of arithmetic. Such an attitude is
constant in the history of mathematics, as stated by F. Arzarello, L. Bazzini and
G. Chiappini:
"The development of the concept of numbers you can see how the conduct of
a chain of transitions from the operational and structural conceptions. On the other hand
, even before the process of generating new numbers were regarded as objects
, mathematicians used them and combines them into operations (Arzarello,
Bazzini & Chiappini, 1994, p. 9).

Concludiamo osservando che l'annotazione storica secondo la quale molto
spesso l'aspetto operativo precede quello strutturale, assume una netta rilevanza
in numerose questioni di didattica della matematica (8).
A. Sfard, in una nota ricerca (1991), dopo avere sottolineato la sostanziale
astrazione che caratterizza la matematica (9), sottolinea la possibilità di
concepire (e di presentare) parallelamente i contenuti matematici in termini
strutturali (interpretandoli, dunque, come "oggetti") ed in termini operativi
(interpretation, therefore, as "processes")
"Being able to see a mathematical entity as an object
means being able to refer to it as a real thing, a static ... and manipulate
as a whole ... Interpret the notion as a process means considering
as potential rather than actual entities, which comes to light in the face of a sequence of
actions. So while the structural conception is static ... itananea
and overall the operation is dynamic, sequential and detailed
"(Sfard, 1991: 4).
The Sfard also extends this distinction to coding (and the author seems
here again ideally records Leopardi mentioned above):
'verbal encodings can not be captured' at a glance 'and must
be processed sequentially, so they seem more suited to present
calculation procedures. Thus, the internal representation is not iconic
relevant to operational thinking "(Sfard, 1991, p. 7, re:
Hadamard, 1949, p. 77).
Without claim to exhaust a subject very deep and sensitive, even
from epistemological point of view, we can therefore conclude that the introduction of operational
many fundamental concepts of mathematics (and,
these, the elements of arithmetic ) is a particularly important
and debated in a teaching environment.