Advance Organizers in Learning Abstract Mathematics

The advent of modern mathematics, with its emphasis on the more abstract aspects of the subject, has forced the mathematics teacher to re-evaluate his methods of presentation. In teaching abstract mathematical material, one important variable is how the new material is introduced. Frequently, in the mathematics textbook or lesson, abstract material is introduced with a few paragraphs of history about the subject or its inventor, presumably to increase motivation. A well known algebra textbook, for example, introduces a section on group theory by relating the fact that Evariste Galois, an early mathematician instrumental in the invention of group theory, was killed in a duel in 1832.1 Another way of introducing such material is with concrete examples, models (e.g., Scandura, 1966c) or embodiments (e.g., Dienes, 1964) of the abstraction to be learned. The use of such illustrative materials is commonplace in mathematics instruction, but at the college level its introduction typically follows, rather than precedes, more formal presentation in terms of definitions and theorems.