Dual nature of mathematical conceptions · Dual nature of mathematical conceptions by Anna Sfard...
Transcript of Dual nature of mathematical conceptions · Dual nature of mathematical conceptions by Anna Sfard...
Dual nature of mathematical conceptions
by Anna Sfard
Presented by Francesca Deserti
• How does mathematical abstraction differ from other kinds of abstraction
in its nature?in the way it develops?in its functions and applications?
• Language similarities• Unlike material objects: many representations
Structural & operational conception of a notion
• Structural:objectsstatic, instantaneous, integrative
• Operational:processes, algorithms, actionsdynamic, sequential, detailed
Structural vs procedural:a duality
• not exclusive• complementary
the ability of seeing a notion both as a process and as an object is
indispensable for a deep understanding
The process ofconcept formation
• Operational conceptions generally precede the structural
historical development (p.13)
individual learning?theoretical argumentpersonal experience
• Interiorizationcarry out through mental representations
• Condensationprocess as a whole
• Reificationnew objectinteriorization of higher-level concepts begins
The necessity of Structural Conceptions
• What is it that necessitates this dual outlook?
• Can anybody talk about computational processes performed on… nothing?
(cognitive schemata)“experiment”
• Operationally conceived information cannot be easily processed
• Meaningful learning• Retrieval processes : faster• Deeper confidence
Reification increasesproblem-solving
and learning abilities
• Skillfulness at performing algorithms in order to attain a good idea of the objects involved
• To gain full technical mastery, one must already have these objects, since without them the processes would be meaningless and thus difficult to perform and to remember
lower-level reification and higher-level interiorization are a prerequisite for each other