Approaches to algebra

In the last two decades, the growing interest in algebra learning and teaching has instigated an international discussion on what we believe (school) algebra to be and what we believe it should be. Contemporary researchers have identified kernal characteristics of algebraic reasoning and algebraic language – such as generalizing, formalizing and symbolizing – which are related to different aspects of algebra (Kieran, 1989, 1990, 1992; Filloy & Rojano, 1989; Sfard, 1991, 1995; Sfard & Linchevski, 1994; Herscovics, 1989; Herscovics & Linchevski, 1994; Linchevski & Herscovics, 1996; Bednarz, Kieran & Lee, 1996; Kaput, 1998). A few months ago the Twelfth ICMI study ‘The Future of the Teaching and Learning of Algebra’ raised issues like ‘why algebra?’, ‘approaches to algebra’, ‘language aspects of algebra’, ‘early algebra education’, ‘technological environments’, and more. Meanwhile it has become clear that there is no agreement on what algebra is or what it should be; each classification has its strong and weak points. Therefore, instead of trying to establish what algebra is, one might consider algebra in terms of its roles in different areas of application instead.

Bednarz et al. (1996) distinguish four principal trends in current research and curriculum development of school algebra: generalizing, problem solving, modeling and functions. These different roles of algebra can be associated with the various ways in which the authors conceive algebra, and which characteristics of algebraic thinking they believe ought to be developed in order to find algebra meaningful. A fifth perspective presented by Bednarz et al. is the historical one, not as an alternative way to introduce algebra at school but as a valuable pedagogical tool for teachers and educational researchers.

The same researchers recognize that the classification is oversimplified and incomplete, and that various approaches have not yet been adequately researched: “The separation into four approaches to ‘beginning algebra’ is artificial; all four components are needed in any algebra program. (...) Some other possible approaches have probably been omitted.” (ibid., p. 325). Still, Bednarz et al. observe that their classification

has helped to structurize their discussion on essential issues of school algebra. roles of letters Some years earlier, Usiskin (1988) proposed a slightly different categorization of perceptions of algebra: as generalized arithmetic, as a study of procedures for solving problems, as a study of relationships among quantities (including modeling and functions) and as a study of structures. In each of these approaches to algebra Usiskin identifies different roles of the letter symbols: pattern generalizer, unknown, argument or parameter, or arbitrary object respectively. One might argue that this list is not complete; other meanings of the concept of variable that are mentioned regularly are those of placeholder (a symbol in an arithmetical open sentence such as 3 + • = 5), letter not evaluated (like and e) and label (letter to abbreviate an object, or a unit of measurement). A variable that varies (as argument or parameter) is considered to be of a higher level of formality than the variable as generalized number or unknown, which is again more formal than the placeholder; at the top end we find the arbitrary symbol. This subtle variation of meanings of letters has been identified .

A number of other characterizations of algebra can be found in the literature. For instance, the National Council of Teachers of Mathematics (1997) identifies four themes for school algebra: functions and relations, modeling, structure, and language and representation. Kaput (1998) has listed five forms of algebraic reasoning: generalizing and formalizing, algebra as syntactically-guided manipulation, algebra as the study of structures, algebra as the study of functions, relations and joint variation, and algebra as a modeling language.

In the present study we do not take an explicit position on what is the best classification of algebra. It is only relevant that we recognize which aspects of algebra are relevant for the proposed learning program. The algebraic activities that we have developed can be described as ‘advanced arithmetic’, with a large component of problem solving and studying relations . We have no clear preference for one classification or the other; it is only for the practical reason of having a framework that we have made a choice. 

If algebra is construed as a product of generalizing activities, its main purpose is to grasp generality, for instance by expressing the properties of numbers. The Oxford Dictionary exemplifies this perspective by defining algebra as the ‘study of the properties of numbers using general numbers’. Algebraic skills are directed at translating and generalizing given relationships among numbers. This approach to algebra stands a better chance if the learner’s intuitive base for the structure of algebra is already nourished in arithmetical activities. For example, Booth (1984) has suggested that if a student is to perceive an expression like a + b as an object in algebra, he or she must be able to view the sum 5 + 8 as an object in arithmetic, rather than as a procedure leading to the outcome 13.