Sep 22, 1986 - The overall model is related to children's conceptual ..... This list provides an indication of the extent to which transformations pervade .... his focus is the range of methods used by experts, including calculating prodigies,.
Br. J . educ. Psychol., 57,343-370, 1987
A MODEL OF THE COGNITIVE MEANING OF MATHEMATICAL EXPRESSIONS BY PAUL ERNEST (School of Education, University of Exeter) SUMMARY. The central focus of the paper is the nature of the mental representations of the meaning of the linguistic expressions of mathematics. An information-processing model for the construction of mental representations is presented. The model provides syntactical tree structures as meaning representations. It is argued that the major function of written mathematical language in school mathematics is in the initial presentations of pupil tasks. An information-processing model for the performance of routine mathematical tasks is proposed. The central feature of the model is the major role that transformations of mental representations play. The overall model is related to children’s conceptual development, and a series of stages in the acquisition of mathematical language is proposed. Finally, the model is shown t o be consistent with a range of current concepts, theories and observational data.
INTRODUCTION feature of mathematics is its characteristic formal symbolism. Children devote a great deal of their time in school to mastering the symbolism of mathematics. In understanding written mathematical expressions evidently children must form mental representations of these expressions, or at least of their meanings. However, this account raises a number of questions. What are the cognitive processes by which the meaning of a linguistic expression of mathematics is apprehended? What types of processes and representations are involved and what are their characteristics? Can the process by means of which the meanings of mathematical expressions are represented be modelled? And, finally, how are such meanings used by learners? In this paper partial answers to these questions are proposed in the form of a tentative model of meaning for compound mat hematical expressions. The model focuses on the mental representation of the syntactical structure of compound mathematical expressions. It leaves aside the issue of the meanings of individual denotative symbols which can include concepts, images, etc. The model assumes the syntax of formal mathematics, which is implicit in expressions such as 2 + 3, 7 x 1 1 = 77 and 7x + 2 = 16. The structure of languages suitable for expressing mathematics has been extensively researched, and there is a consensus that the simplest appropriate languages are the first-order predicate languages. These are treated in standard mathematical logic texts such as Church (1956), Lightstone (1964), Mendelson (1964), Schoenfield (1967), Smullyan (1968) and Bell and Machover (1977). A brief formulation of a first order predicate language L is as follows. The basic symbols of L consist of sets of: constants (e.g. one, 2, %, 0.7, e) individual variables (e.g. x, Y, 4 n-place functions (e.g. +, -, x ) n-place predicates (e.g. =,
