Points, Lines, and Their Representations.

Abstract

Students come to mathematics lessons with some exisiting concepts about the topics being presented. These concepts have been constructed by the students from information encountered in everyday situations, including lessons in school. Each student's set of concepts is an individual construction and is likely to be different from that constructed by any other student. The nature of these concepts significantly determines what is learnt and how it is learnt by the student. New information presented in school is likely to be interpreted differently by different students and so each student's concepts will be changed differently and perhaps unpredictably by the same instruction. Since existing knowledge is the student's own, often constructed over years of experience of mathematical idea, it will probably be resistant to marked change, unless new information is seen by the student as intelligible, plausible, and more fruitful in explaining the relationships being investigated than the student's existing set of concepts [Hewson, 1981]. Part of the student's exisitng knowledge will be constructed on the basis of the ways in which words with specialized mathematical meaning are used in everyday situations. For example, student will have encountered the word 4(line in common figures of speech such as deadline, a ticket line, neckline, and of sight. The ways in which line is used in common expressions like these differ subtly from the ways in which mathematicians use this word. The concepts constructed by students are therefore likely to differ from the concepts used by mathematicians.