Abstraction in mathematics: Conflict, resolution and application

Everyday usage of the term "abstract" has been shown to lead to a conflict in which abstract mathematics is seen to be both easier and more difficult than concrete mathematics. A literature review undertaken to identify the source of this conflict has revealed that the term "abstraction" may be used to denote either a process or a product. Two meanings of "abstract" are also identified. The first meaning, called abstract- apart, refers to ideas which are removed from reality; the second meaning, called abstract- general, refers to ideas which are general to a wide variety of contexts. It is argued in this paper that, whereas mathematics is abstract- general, mathematics teaching often leads to abstract- apart ideas. The initial conflict has been resolved by noting that abstract-apart ideas are adequate when a mathematical problem can be solved within a single level of abstraction; such problems are relatively easy. On the other hand, abstract-general ideas are essential for the successful solution of problems which require links between levels of abstraction; these problems are relatively difficult. The concepts of abstract-general and abstract-apart have then been applied to re-interpret two research studies (on letters in algebra and rates of change). It is suggested that greater interest in abstraction as a process would be beneficial to both the theory and practice of mathematics education.