The Role and Function of a Hierarchical Classification of Quadrilaterals.
Michael D. de Villiers · for the learning of mathematics · 1994
Introduction A well-known and useful distinction between different types of in mathematics is that of distinguishing between instrumental, relational, and logical [e.g. Skemp, 1976; 1977, Byers & Herscovics, 1977]. Where instrumental understanding (the author actually prefers the term proficiency) refers to the ability of an individual to correctly and efficiently manipulate mathematical content (e.g. by using algorithms, rules and definitions), relational and logical respectively refer to the conceptual relationships between content and the underlying logic upon which these relationships are based. A serious deficiency in this model is that no provision is really made for functional understanding, in other words, the role, function, or value of specific mathematical content or of a particular process [compare Human, 1989]. Extensive experience with children in interview and classroom contexts seems to indicate that many of their problems with mathematical content and processes often do not lie so much with poor instrumental proficiency nor inadequate relational or logical as in a poor of the usefulness or function thereof. It should be noted that this functionality is not confined to applications to the real world outside of mathematics but includes the relative values or functions of content and processes within mathematics. To a very large extent, it seems that the absence, presence, or level of an individual's functional determines that individual's motivation to study and learn mathematics. Without functional understanding, mathematics simply degenerates into a useless, meaningless and arbitrary subject, demotivating the learner from attempting to learn and explore it. The adequate development of functional is therefore an important criterion for evaluating any teaching approach. In this article different types of classification will be distinguished, as well as a theoretical analysis done of the role and function of hierarchical classification in mathematics. Lastly, some brief comments regarding the teaching of a hierarchical classification of quadrilaterals will be made.