Introduction The evolution of the concept of function goes back 4000 years; 3700 of these consist of anticipations. The idea evolved for close to 300 years in intimate connection with problems in calculus and analysis. (A one-sentence definition of analysis as the study of properties of various classes of functions would not be far off the mark.) In fact, the concept of function is one of the distinguishing features of ‘modern’ as against ‘classical’ mathematics. W. L. Schaaf ([24], p. 500) goes a step further: The keynote of Western culture is the function concept, a notion not even remotely hinted at by any earlier culture. And the function concept is anything but an extension or elaboration of previous number concepts—it is rather a complete emancipation from such notions. The evolution of the function concept can be seen as a tug-of-war between two elements, twomental images: the geometric (expressed in the form of a curve) and the algebraic (expressed as a formula—first finite and later allowing infinitely many terms, the so-called analytic expression ). (See [7], p. 256.) Subsequently, a third element enters, namely, the logical definition of function as a correspondence (with a mental image of an input-output machine). In the wake of this development, the geometric conception of function is gradually abandoned. A new tug-of-war soon ensues (and is, in one form or another, still with us today) between this novel logical ( abstract, synthetic, postulational )conception of function and the old algebraic ( concrete, analytic, constructive ) conception.

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Evolution of the Function Concept: A Brief Survey.

This article describes a research project that examined the effects of the Computer-Intensive Algebra (CIA) and traditional algebra curricula on students' understanding of the function concept. The foundation for the research is a proposed conceptual framework that describes function knowledge in terms of component competencies. The results indicated that the CIA students achieved a better overall understanding of functions and were better at the components of modeling, interpreting, and translating. No significant differences were found for reifying, which emerged as the most difficult component in the proposed function model. Further, the CIA students showed significant improvements in their attitudes toward mathematics, were less anxious about mathematics, and rated their class as more interesting. A higher percentage of students successfully completed the CIA course. In response to calls for reform in the teaching and learning of mathematics (National Commission on Excellence in Education, 1983; National Council of Teachers of Mathematics [NCTM], 1980), an innovative curriculum, Computer-Intensive Algebra (CIA), was developed as an alternative approach to traditional algebra (TA). CIA is a function-oriented curriculum that is characterized by (a) a problemsolving approach based on the modeling of realistic situations, (b) an emphasis on conceptual knowledge, and (c) the extensive use of technology. Whereas previous research (Boers-van Oosterum, 1990; Matras, 1988) had shown the beneficial effects of CIA on students' problem-solving abilities and on their understanding of variables, the project described here was designed to examine the curriculum's effects on students' understanding of functions. A natural prerequisite to the investigation was a careful delineation of students' knowledge of the function concept. Furthermore, this definition of function knowledge had to be in the format of a working model that could be operationalized and assessed. These considerations led to the development of a conceptual model for describing an understanding of functions, a model that established the basic foundation for this research.

Computer-Intensive Algebra and Students' Conceptual Knowledge of Functions.

(1942). An Analysis of Techniques for Diagnostic Vocabulary Testing. The Journal of Educational Research: Vol. 36, No. 3, pp. 206-217.

An Analysis of Techniques for Diagnostic Vocabulary Testing

A Manual of Greek Mathematics. By Sir Thomas L. Heath. Pp. xvi, 552. 18s. 1963 - A History of Geometrical Methods. By J.L. Coolidge. Pp. xviii, 451. 18s. 1963. - The Mathematics of Great Amateurs. ByJ.L. Coolidge. Pp. viii, 211. 12s. 1963.

The present paper describes an attempt to alleviate some of the difficulties and misconceptions about the concept of function, by integrating a computer implemented learning environment with specifically designed activities in a curriculum covering functions This curriculum-the TRM curriculum integrates open-ended activities with a “conventional” approach to the concept of function, and “computer” sessions with normal classroom activities Typical problems and comments on student activities are given Comparative investigations led to a cognitive classification of functional reasoning Some practical issues arising from teaching about functions and pedagogical engineering are described at the end of the paper

William L. Schaaf

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