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