Interactions between defining, explaining and classifying: The case of increasing and decreasing sequences

This paper describes a study in which we investigated relationships between defining mathematical concepts|increasing and decreasing infinite sequences|explaining their meanings, and classifying consistently with formal definitions. We explored the effect of defining, explaining, or studying a definition on subsequent classification, and the effect of classifying on subsequent explaining and defining. We report that 1) student-generated definitions and explanations were highly variable in content and quality; 2) explicitly considering the meaning of the concept facilitated subsequent classification, and giving a personal definition or explanation had a greater effect than studying a given definition; 3) classifying before defining or explaining resulted in significantly poorer definitions and explanations. We discuss the implications of these results for the teaching of abstract pure mathematics, relating our discussion to existing work on the concept image/concept definition distinction and on working with examples.