Trends Index

The failure, in the 1970s, of back-to-the-basics mastery approaches to generate quality mathematics teaching and learning environments stirred leading educators to try to do something which would improve the situation. What could be done to make school mathematics more relevant to everyday life, while at the same time providing a better preparation for those who wanted to proceed to further mathematical studies? In 1980, in the United States, the National Council of Teachers of Mathematics (NCTM) stated categorically, in An Agenda For Action: Recommendations for School Mathematics of the 1980s, that the major challenge for those concerned about the quality of school mathematics was to develop curricula and to provide professional development programs for teachers which would make problem solving the main focus of mathematics teaching and learning (NCTM, 1980). In the United Kingdom, the Cockcroft Report argued along similar lines (Cockcroft, 1982).

Such are the forces of colonialism that An Agenda For Action and the Cockcroft Report precipitated a major surge of interest in problem solving in school mathematics around the world. However, it quickly became apparent that it was difficult for mathematics curriculum developers, teachers, and researchers to define exactly what constituted a mathematics problem, or exactly how a problem-solving mathematics curriculum could be designed, implemented and evaluated. Numerous references were made to generic heuristics, as expounded by the mathematician George Polya (1973), but education researchers and teachers found it difficult to decide whether generalised problem solving skills could be effectively planned for and taught in mathematics classrooms. Furthermore, the richness of the problem-solving processes evident in children's thinking raised doubts on whether the standard inferential statistical, and developmental psychological research paradigms were of much use in research on problem solving.

The new emphasis on whether learners were capable of monitoring their own thinking about problem solving forced mathematics educators to ask questions about what was taking place in children's minds as they tried to solve problems. As language factors, the social and cultural milieu of learning, metacognition, and the use of imagery, for example, quickly came to be recognised as important components of any holistic account of problem solving, at the same time the old research paradigms were seen to be inadequate for carrying out the needed research.

The impetus for new approaches to mathematics education research generated by the problem-solving thrust was one among many forces which led to a redefinition of what constitutes acceptable mathematics education research. In fact, it could be argued that although in the early 1990s PME remains a vibrant organisation, the last decade has witnessed a major shift away from predominantly psychological, experimental cognitive research on the one hand, and clinical developmental research on the other, towards far more eclectic approaches to research, in which social, cultural and linguistic dimensions are assuming ever greater importance. We now summarise some of the new paradigms which are transforming the directions and methodologies of contemporary mathematics education research.

The Relativist Challenge and the Growth of Constructivist Ideas

The Relativist Challenge

The realisation, around 1980, that the back-to-the-basics, skill-oriented, hierarchical approaches to mathematics teaching and learning were producing a generation of school leavers with very mechanical, barren views of the nature of mathematics, caused leading mathematicians and mathematics educators to reflect on the images of mathematics they wanted mathematics programs to project.

As it happened, around the same time, questions about the nature of mathematical knowledge, which had always been important in philosophy, had taken centre stage in the world of mathematics. That this was indeed the case is supported by the following quotation from the Preface to Morris Kline's (1980) aptly titled book Mathematics: The Loss of Certainty:

Many mathematicians would perhaps prefer to limit the disclosure of the present status of mathematics to members of the family. To air these troubles in public may appear to be in bad taste, as bad as airing one's marital difficulties. But intellectually oriented people must be fully aware of the powers of the tools at their disposal. Recognition of the limitations, as well as the capabilities, of reason is far more beneficial than blind trust, which can lead to false ideologies and even to destruction.

Such uncertainties about the nature of their subject had not been evident among mathematicians at the beginning of the twentieth century, when the Platonic idea that mathematical knowledge, concepts and relationships existed "out-there," independent of human thought and culture, and of physical reality but were nevertheless reflected in the laws of nature and in the logico-mathematical structures of the human mind, had been propounded by intellectual giants such as David Hilbert and Bertrand Russell.

Hilbert, Russell and others attempted to lay the foundations of mathematics solidly as a formal system, where all truths could be proved and only truths proved (see Russell, 1974). However, this formalist philosophy of mathematics was shown to be unattainable by Kurt Gödel who, in 1934, proved that, even within such a basic structure as first order predicate calculus, together with axioms sufficient to model fully the natural numbers, there are statements which cannot be proved, even though they were constructed so that they are true (see Rucker (1982) for detailed comments on Gödel's celebrated theorem).

Gödel's revelation led to a deep questioning of the nature of mathematics: Gödel, himself a dedicated Platonist, was quoted as saying that either "mathematics is too big for the human mind, or the human mind is more than a machine." In a similar vein, Herrman Weyl is reputed to have said: "God exists because mathematics is undoubtedly consistent and the devil exists because we cannot prove the consistency" (quoted in Herlihy, 1986, p. 15). Other statements such as "mathematics is the only branch of theology that has a proof that it is a branch of theology" were made (quoted in Herlihy, 1986, p.15).

In 1930, shortly before Gödel announced his results, Hilbert, when giving a lecture on the nature of human reason, had electrified his audience by exclaiming "We must know! We must know!" Absolute knowledge was the goal of the Hilbertian formalist school. But soon after Gödel "dropped an atom bomb on mathematical foundations" (Nickel, 1985), Herrman Weyl was led to say that while "the question of the ultimate meaning of mathematics remains open, we do not know in what direction it will find its final solution, nor even whether a final objective answer can be expected at all" (quoted in Kline, 1979, p. 1207).

Wittgenstein, Popper, and Lakatos

Wittgenstein, a contemporary of Gödel, after commenting that "there is no religious denomination in which the metaphysical expression has been responsible for so much sin as it has in mathematics" (quoted in Shanker, 1987, p. vii), went on to say that he believed the twentieth century would witness a rejection of the logic of Leibniz, Gottlob Frege, and Bertrand Russell. For Wittgenstein, the Austrian/English philosopher, mathematics was not something entirely independent of reality, and no statement was true a priori. Western mathematics was merely a product of history, of social transmission processes at work, and had been more or less shaped by a "survival of the fittest" evolutionary process. Forms of mathematics that were accepted, developed, and utilised by a powerful group were passed on not only to children in the group but also to other groups. When a new kind of problem arose, an extension of existing mathematical knowledge was called for, in order that a solution might be obtained; this, in its turn, was socially transmitted, and came to be recognised as "mathematical truth" (Del Campo & Clements, 1990, p. 52). So, reducing Wittgenstein's thesis to its simplest form, although physical reality and our biological dispositions impose constraints on the conventions we develop and include within what we call mathematics, there is no mathematical reality that guarantees the results we get (Wittgenstein, 1956, p. 190).

Wittgenstein's prediction that, as the twentieth century progressed, mathematics would come to be seen as a relative rather than as an absolute form of knowledge proved to be right. Karl Popper's fallibilist philosophies, which became increasingly popular from the 1940s onwards, suggested that a scientific theory can never be proved, only refuted. These ideas were taken up in mathematics by Imré Lakatos (1976) who, in his book Proofs and Refutations, described a long history of disputes within mathematics about the properties of polyhedra, and argued that many mathematicians, in defending the view that mathematics is a form of absolute knowledge, have defined and redefined the term "polyhedron" to fit their goals. In this sense, as Wittgenstein noted, mathematics became a language game used to prop up the myth that in some genuine sense it is a priori, standing apart from other relative states of knowledge (Rizvi, 1988; Watson, 1989).

By 1980, then, it became commonplace for philosophers of science and education to view the practice of mathematics and mathematicians as not fundamentally different from human thought as embedded in other domains. Indeed, mathematics was no longer seen as involving the discovery of truths existing outside the realm of human activity, but rather as domain-specific, context-bound, and as procedurally rooted as any other form of knowledge. Such ideas were not common among mathematics teachers and educators, however, until they were taken up by so-called "radical constructivists."

And so it has come to be that in the 1990s, mathematics educators, and especially those identified with constructivist ideas, tend to label the Platonist notion that "mathematical objects somehow exist independently of human experience" as a common misconception. Mathematics educators are now calling for mathematics programs that present mathematics as a socially constructed body of knowledge (Lakoff, 1987, p. 354), which has accumulated over the years and "can be found in books, in journals, and in the exchanges in the many different communities of mathematicians" (Bergeron & Herscovics, 1990, p. 125). They are also using the philosophical framework provided by Wittgenstein to explain why children, left to themselves, cannot be reasonably expected to develop "mathematical" concepts that parallel those found in mathematics textbooks (Clements & Del Campo, 1990; Watson, 1989).

Psychology Culture

Trends Index