Encouraging versatile thinking in algebra using the computer

Abstract

In this article we formulate and analyse some of the obstacles to understanding the notion of a variable, and the use and meaning of algebraic notation, and report empirical evidence to support the hypothesis that an approach using the computer will be more successful in overcoming these obstacles. The computer approach is formulated within a wider framework of versatile thinking in which global, holistic processing complements local, sequential processing. This is done through a combination of programming in BASIC, physical activities which simulate computer storage and manipulation of variables, and specific software which evaluates expressions in standard mathematical notation. The software is designed to enable the user to explore examples and non-examples of a concept, in this case equivalent and nonequivalent expressions. We call such a piece of software a generic organizer because it offers examples and non-examples which may be seen not just in specific terms, but as typical, or generic, examples of the algebraic processes, assisting the pupil in the difficult task of abstracting the more general concept which they represent. Empirical evidence from several related studies shows that such an approach significantly improves the understanding of higher order concepts in algebra, and that any initial loss in manipulative facility through lack of practice is more than made up at a later stage.

What Is the Object of the Encapsulation of a Process?

“…(Gray & Tall, 1994) This notion of procept proved to be present throughout a large portion of mathematics. Tall & Thomas (1991) had already noted that, for many children, an expression such as 2 + 3x may be conceived as a process which cannot be carried out until the value of x is not known -a reinterpretation of the notion of "lack of closure" discussed by earlier authors. Gray and Tall (1994) also noted the peculiar case of the limit concept where the (potentially infinite) process of computing a limit may not have a finite algorithm at all.…”

“…Focusing on both operational processes and the properties of objects-either in turn or at the same time-gives a versatile approach (Tall & Thomas, 1991). This proves particular valuable when computer software is available to carry out the processes internally, allowing the individual to focus either on the study of the processes, which they may carry out, or program, for themselves, or on the concepts produced by the computer (Tall & Thomas, 1989).…”

“…This proves particular valuable when computer software is available to carry out the processes internally, allowing the individual to focus either on the study of the processes, which they may carry out, or program, for themselves, or on the concepts produced by the computer (Tall & Thomas, 1989). Versatile approaches have proved successful using visual properties, such as viewing the "local straightness" of graphs to complement the process of symbolic differentiation to give a derivative function (Tall, 1985), and also using process/concept, such as studying the evaluation of expressions (carried out by the student) separately from the properties of (equivalent) expressions, evaluated by the computer (Thomas, 1988).…”

Duality, Ambiguity, and Flexibility: A "Proceptual" View of Simple Arithmetic

“…However other research can also be re-interpreted in proceptual terms. We have evidence that the lack of formation of the procept for an algebraic expression causes difficulties for pupils who see the symbolism representing only a general procedure for computation: an expression such as 2+3x may be conceived as a process which cannot be carried out because the value of x is not known (Tall & Thomas, 1991). We have evidence that the conception of a trigonometric ratio only as a process of calculation (opposite over hypotenuse) and not a flexible procept causes difficulties in trigonometry (Blackett 1990, Blackett & Tall 1991.…”

Designing Educational Software: The Case of Grid Algebra

“…The influential Cockcroft report in the UK (DES 1982, p. 60) indicated that "algebra is a source of considerable confusion and negative attitudes among pupils" and as teachers we want to make sure that "algebra is not a meaningless game with 26 letters" (Freudenthal 1973, p.290, his emphasis). The difficulties include how the equals sign is viewed (Kieran 1981), the meaning given to letters (Küchemann 1981), the reading of formal mathematical notation (Kirshner 1989) and the need to treat an algebraic expression as an object as well as a process (Booth 1984;Tall and Thomas 1991).…”