The study attempts to outline what are some of the major obstacles to learning the language of algebra. It is in two Parts. Part 1 is theoretical and is the outcome of a belief that the wrong question might often have been asked about the learning of mathematics, i.e. perhaps the important question to ask is not 'How do children learn mathematics'?, but 'why do pupils fail to learn mathematics'?. Chapters 1 and 2 suggest the reason for this might be due to the fact that mathematics is a system of language systems. As such it demands of the learner a sequence of conceptual adaptations to new meanings for concepts as new languages are introduced. With particular reference to the algebraic language itself it is suggested that a pupil might be conditioned early in life to think of a letter in arithmetic as an ordered entity with an unique numerical determination, and thus might 'carry over' this understanding into algebra itself. To comprehend the algebraic language, however, the pupil will need to develop an understanding that the letter is a numeral in its own right - i.e. it is used to convey the symbolic number concept. Part 2 is empirical. Tasks were specially devised to show that pupils would demonstrate two distinct, logically consistent, usages of a letter, the first matching that of the Mediaeval mathematician, and the second that of the contemporary mathematician. 144 pupils of ages 11-18 years from two grammar schools were interviewed in a structured situation, using the tasks as investigatory material. Responses to each task were arranged into hierarchies of algebraic sophistication, and these were used: (a) to study the development of the symbolic number concept, and (b) to generate three broad levels of algebraic activity. The results suggest that the symbolic number concept (i) is available to a small minority of pupils of ages 12-18, and (ii) might be associated with dynamic imagery.
|Date of Award