Abstraction is inherent to mathematics. It is a must for mathematics teachers to know and understand what this process is and what its products are. Knowledge of it can enrich our reflection of our own practice as well us guide us and make us conscious of the type of learning activities we provide our students.
All the definitions below give emphasis about abstraction as a process. Note also that the direction of the abstraction is always from a set of contexts to an abstract concept. Abstraction is related to generalization which I discussed in another post.
- the omission of qualities from concrete experience – Aristotle
- the process of separating a quality common to a number of objects/situations from other qualities – Davidov (1972/1990, p. 13)
- the act of detaching certain features from an object – Sierpinska (1991, p 61)
- Abstracting is an activity by which we become aware of similarities … among our experiences. An abstraction is some kind of lasting change, the result of abstracting, which enables us to recognize new expereinces as having the similarities of an already formed class. … To distinguish between abstracting as an activity and abstraction as its end-product, we shall … call the latter a concept. – Skemp
Empirical vs reflective abstraction (Piaget et al)
- Empirical abstraction is based on superficial similarities and is the type of abstraction involved in everyday concept formation.
- Reflective abstraction is, according to Piaget, based on reflection one one’s actions. For example when one object and two objects are put together you always get three objects. This leads to recognition of invariance (later expressed as 1+2=3). These objects of invariance become concepts (the numbers 1, 2, and 3) and the invariant action becomes a relation between these concepts (addition). In reflective abstraction, concepts and relations are abstracted together.
Abstract-apart vs abstract-general (Mitchelmore et al)
- abstract-apart: concepts formed that exist apart from any contexts from which they might have been abstracted
- abstract-general: concepts that have been abstracted through the recognition of similarities between contexts. These concepts derive their general meaning from the set of contexts from which it has been abstracted
Stages of abstraction
- a cycle of interiorization-condensation-reification – by Sfard 1991
- generalization-synthesis-abstraction cycle – Dreyfus (1991)
Reference: The Role of Abstraction and Generalization in the Development of Mathematical Knowledge by Michael Mitchelmore – paper presented during 2nd EARCOME.
You may want to read my post about assessing understanding of function in equation form for an example of abstracting based on Sfard’s interiorization-condensation-reification cycle.