The Ana
Sfard influential article published in Educational Studies in Mathematics in
1991 on the dual nature of mathematical conceptions is still important to read.
I recently came back to this paper while working on the conceptions of function
using of the modelling framework cK¢.
Indeed there is a difference in our approaches since her approach of Ana
Sfard, defining a conception as the mental counterpart of a concept, the latter
being the official form of a "mathematical idea". This meaning of
"concept" seems close to the usual meaning of the French word " savoir",
and far from the one adopted for cK¢ -- but this is another discussion. On the
other hand both approaches have in common the recognition of methodological
constraints: we have no choice in order to make sense of the formation of
abstract (mathematical) objects but to describe them in terms of such external
characteristics as student’s behaviours, attitudes and skills (Sfard 1991 p.19).
Anna Sfard
distinguishes two types of conception, operational and structural. The former
is characterized in terms of processes, algorithms and actions, while the
latter is "treating mathematical notions as if they referred to some
abstract object" (ibid. pp.3-4). The methodological constraints gives an
advantage to evidencing operational conceptions but make it delicate for
structural conceptions. Following Anna Sfard, a critical indicator of the
presence of a structural conception is the capacity to recognize an idea
"at a glance" and "to manipulate it as a whole, without going
into details" (ibid. p.4). This
emergence of a structural conception would be empirically reflected by the
"attempts at translating
operational intuition into structural definition" (ibid. p.15). Anna Sfard sees the most achieved state of
development of the conception of function in "the now widely accepted,
purely structural Bourbaki's definition. This simple description presented
function as a set of ordered pairs and made no reference whatsoever to any kind
of computational process." (ibid. p.15)
The initial
ambition of the founders of the Bourbaki group [1], was to write a treatise for
the teaching of calculus (incidentally claimed to be accessible to a not so
smart student obliged to work alone [2]). There is no question about the
structural character of the Bourbaki’s conception of function; however its characterization
by Anna Sfard (ibid. p.5 Fig.1) as "Set of pairs (Bourbaki 1934)" is
a bit short. Indeed, ordered pair should have been written here instead
of pair, but there is more to say. The definition of function appears in
the Set theory book (ST) where it emerges, so to say, from the definition of functional relation which is a
restriction of the definition of relation:
"Let R be a relation in C [equalitarian
theory]. The relation "(ⱻx)R and there exists at most one x such that R" is
denoted by "there exists exactly one x such that R". If this relation
is a theorem in C, R is said to be a functional relation in x in the theory C." (ST p.48)
Then
function is further defined as a set of ordered pairs under a specific
condition:
"A graph F is said
to be a functional graph if for each x there is at most one
object which corresponds to x under F (Chapter l, § 5, no. 3). A
correspondence f= (F, A, B) is said to be a function
if its graph F is a fonctional graph and if its source A is equal
to its domain pr1F. In other words, a correspondence f
= (F, A, B) is a function if for every x belonging
to the source A of f the relation (x, y)∈F is functional in y (Chapter l, § 5, no. 3); the
unique object which corresponds to x under f is called the value of f
at the element x of A, and is denoted by f(x) (or fx,
or F(x), or Fx)." (ST p.81)
By the way,
the contemporary teacher may interpret the graph as a curve, and the condition
as the perpendicular line criterion which is often associated to the
characterization of function in Anglo-Saxon curricula. In an informal way,
Bourbaki accepts here to aggregate functional
relation and functional graph in
one single concept: "Throughout this series we
shall often use the word "function" in place of "functional
graph"." (ST p.82).
Eventually,
Bourbaki comes back to all definitions in the “Summary of results” of the Set
theory book, with the idea of fixing terms which will be used in the remainder
of the series of the treatise. He adds the following caveat as a footnote: "The reader will not fail to observe
that the "naïve" point of view taken here is in direct opposition to
the "formalist" point of view taken in Chapters 1 to IV. Of course,
this contrast is deliberate, and corresponds to the different purposes of this
Summary and the rest of the volume." (ST p.347). The following
definition of function is proposed in this context:
"Let E and F be two sets,
which may or may not be distinct. A relation between a variable element x
of E and a variable element y of F is called a functional relation
in y if, for all x∈E, there exists a unique y∈F, which is in the given relation with x. We give the
name of function to the operation which in this way associates with
every element x∈E the element y∈F which is in the
given relation with x; y is said to be the value of the function at the
element x, and the function is said to be determined by the given
functional relation." (ST p.351)
This
definition bridges the naïve (in the Bourbaki sense) understanding of function
with its formal characterization. However, the word "variable", which
didn't appear before in the book, is here an adjective which meaning is fixed
in the first section of the Summary of results:
"variable element" means "arbitrary element" (ST p.347).
By denoting "the operation", the word "function" keeps some
contact with what Anna Sfard (1991 p.15) refers to as "its intuitive
origin". It is close to the prototypical example of operational conception
she gives in the [Fig.1] of her article, quoting Richard Skemp: "well
defined method of getting from one system to another" (or computational
process).
The Bourbaki
construction provides an example of an explicit link between the Anna Sfard
structural and operational conceptions of function. From a different
perspective, it illustrates well the claim that “the terms "operational"
and "structural" refer to inseparable, though dramatically different,
facets of the same thing." (Sfard 1991 p.9). In this quick record of the
Bourbaki enterprise to define "function", we see the coherent and
explicit integration of different connotation: functional relation,
functional graph, operation. In naïve words, they are facets of an object which are unified by
the formal construction. This notion of
object can be easily related to that of high-level interiorization proposed by
Anna Sfard.
