Communication and Construction of Meaning

Abstract

Construction of mathematical meaning often results from processes of communication. In this introduction, we identify and discuss conditions and aspects of communication which may influence mathematical meaning. For example, meanings constructed between mathematicians are rather different from those of students on account of their different goals and concerns. Mathematicians are concerned with solving new problems while students are concerned with learning. Similarly, authors of textbooks are concerned with consistency of presentation and uniformity of approach while students look for convincing arguments, neat explanations and strategies which are economic in getting their task completed. The nature of a community is not however the only determinant of mathematical meaning, which is a function of many factors emanating from diverse sources. The following vignette of a learning situation serves to illustrate this point. In the framework of a master class, in front of a public of mathematics educators, two 16-year-old students were asked to solve a geometrical task using dynamic geometry software. The situation was set up on a computer and can be described as follows: ABC is a rectangular triangle with the right angle at A, P is any point on BC and D and E are the orthogonal projections of P onto AB and AC respectively. The students explored the diagram by dragging the triangle and noticed how the relationships remained unchanged-that is that A was always a right angle, and PD and PE were always perpendicular to AB and AC. They were then asked to focus their attention on P and on DE. They were shown how to measure the length of DE and to display this value on the computer screen. Finally, they were asked to move P along BC and notice that the length of DE changed. A question was then posed to them: What must be the position of P on BC in order for the length of DE to be minimal? The students dragged P up and down BC watching how DE varied. They found a zone, a small segment of BC, minimising DE but were not able to characterise the location of P geometrically.