The articulation principle for making long-term sense of mathematical expressions by how they are spoken and heard: Two case studies

The Articulation Principle explains how mathematical expressions can be given a clear and unambiguous meaning by the way in which they are spoken and heard. Leaving short pauses in speech, sub-expressions can be interpreted as operations to be carried out in time or as mental objects that can be manipulated at a more sophisticated level. This offers a fundamental foundation for the growth of meaningful mathematical thinking at all levels from young children to the wide array of adult mathematics. This paper sets the Articulation Principle in a wider long-term learning framework and provides empirical evidence for its use in meaningful interpretation of mathematical expressions. In this paper, we investigate how two teachers who have learned to operate routinely with expressions react when they are presented with expressions spoken and written in different ways. This reveals how the experience enables them to give meaning to mathematical expressions that previously had only been learned by rote. The Articulation Principle has the advantage that it can be introduced at any level to help teachers and learners make sense of symbolism in a manner appropriate to their experience and level of development. It has wide ranging implications in addressing the development of meaningful long-term curricula in the challenge of “math wars” between communities of practice with differing beliefs and needs in a complex society