Trends Index

INTERNATIONAL TRENDS IN MATHEMATICS EDUCATION RESEARCH

Nerida F. Ellerton and M. A. (Ken) Clements

After providing some historical perspectives on the rise and fall of the New Math(s), this paper summarises and critiques recent international trends in mathematics education research, paying particular attention to the implications of this research for mathematics teachers and curriculum developers. The worldwide movement away from the pervasive problem-solving and metacognitive psychological thrusts of the early 1980s, towards the more philosophical "constructivist" and sociological "situated cognition" thrusts of the 1990s, are described, as is the corresponding move from quantitative, experimental research towards qualitative, interpretive research.

The pertinence of radical constructivist and situated cognition research for mathematics education in Southeast Asian contexts is discussed. In particular, the paper focuses on recent attempts to develop mathematics teaching approaches and mathematics curricula which take greater account of prior knowledge and the personal worlds of individual learners. Other important issues, arising from recent research in mathematics education and relevant to mathematics educators in Southeast Asia, are also raised.

The Rise and Fall of the "New Math(s)"

The late 1950s heralded the "New Math(s)" era. Exactly where, when and why the New Math(s) movement began is open to debate: some say it began in the United States following the launching of Sputnik in 1957; others say it was in France as a result of the efforts of the Bourbaki Group; and others point to the efforts of Geoffrey Matthews and Bryan Thwaites in the United Kingdom. There is probably a basis for each of these claims, and certainly there were similarities in the New Math(s) movement as it expressed itself in various countries around the world in the 1960s and 1970s. Whatever the origins, there can be no doubt that a New Math(s) reform, characterised by the introduction of elementary and secondary mathematics curricula which were qualitatively different from previous curricula, gathered momentum in Western nations throughout the 1960s, and in many other countries during the 1970s. Furthermore, the same distinguishing features in the New Math(s) curricula introduced in Western nations in the 1960s could be recognised in the New Math(s) curricula introduced in other countries during the decade.

We shall argue that the New Math(s) curricula at the elementary school level tended to have different origins from the New Math(s) curricula at the secondary school level. At the elementary school level, the major influence came from developmental psychology, and educationists such as Jerome Bruner, Jean Piaget, Zoltan Dienes and Caleb Gattegno, but at the secondary level, the major influence came from professional mathematicians in tertiary mathematics departments.

Secondary School New Math(s) Curricula

Advocates of the "New Math(s)" usually believed that mathematics was a single entity which should not be presented to learners as compartmentalised areas of knowledge known as arithmetic, algebra, geometry, trigonometry, and calculus. In keeping with this notion of mathematics as an integrated whole, overriding coordinating themes and language forms were introduced; set language and set theory were regarded by many as almost synonymous with New Math(s) in some countries, and integrating concepts such as function and algebraic structure received considerable attention. At the secondary level, in particular, New Math(s) courses were characterised by a heavy use of symbolism, and an axiomatic approach to the teaching of number. Somewhat paradoxically, there was a swing away from Euclidean geometry in its purest form towards "transformation" geometry which, although still based on Euclid, seemed to require a different approach so far as teaching and learning was concerned (Clements, Grimison, & Ellerton, 1989; Moon, 1986; Pitman, 1989).

Secondary school New Math(s) programs were invariably introduced by centre-to-periphery curriculum development models (Ellerton, Clements, & Skehan, 1989; Popkewicz, 1988) by which new courses were prescribed by some central authority. Their implementation in schools was made compulsory, and textbooks guided much of the New Math(s) instruction and learning that took place in schools. The importance of the new symbolism and approaches was emphasised in externally set examinations. If professional development for practising teachers responsible for teaching the New Math(s) was provided then it tended to emphasise content - it was thought to be of paramount importance that teachers themselves knew their mathematics and mathematical symbols well, in order that these could be transmitted accurately to learners.

The decision to introduce New Math(s) programs into school systems was usually made by education ministry officials who had been influenced by mathematicians and politicians. The development and implementation of the programs was left largely in the hands of mathematicians and educators. Teachers in schools were rarely involved in the developmental phases, but once the decision had been made to introduce the courses, they were expected to work hard to understand not only the new mathematical topics (such as probability and statistics), but also the new teaching approaches (such as transformation geometry and the axiomatic treatment of number) and the new symbolism (such as set language and function notation).

In fact, the task of learning new content and trying to use new teaching methods simultaneously, usually without any reduction in normal teaching load, proved to be too much for most teachers of mathematics. Most of them continued to teach new content in old ways,with set symbols and function notation being used to create the false impression that something new was being done. However, a proportion of teachers responded positively to the New Math(s): for possibly the first time in their careers, they were asked to reflect seriously about issues associated with the teaching and learning of mathematics, and they took up the challenge. A new era in which mathematics education would be regarded as an important area of investigation in its own right had dawned, and this era gave birth to a new group of education researchers - those with a special interest in mathematics education.

Elementary School New Math(s) Curricula

The effects of the New Math(s) movement, which was inspired by a desire to link school and tertiary mathematics curricula, could be seen in mathematics curriculum statements of both elementary and secondary schools. Interestingly, at the elementary school level but not so much at the secondary school level, the New Math(s) movement was also strongly influenced by the theories of education psychologists. From the 1950s, elementary school mathematics curriculum developers took account of the ideas of educationists and psychologists such as Maria Montessori, George Cuisenaire, Caleb Gattegno, Zoltan Dienes, and Jean Piaget, and in the United States, the ideas of Jerome Bruner strongly influenced elementary school mathematics curricula.

The educators and psychologists mentioned in the previous paragraph emphasised that students should not be asked to study material unless they were developmentally "ready" to do so. Some, like Montessori, Gattegno, Cuisenaire and Dienes, went so far as to develop specially structured materials that, in their view, embodied important mathematical concepts and would assist learners to acquire these concepts. In fact, in some countries, elementary school mathematics curricula were developed which were almost entirely based a particular type of material - with Dienes blacks and Cuisenaire rods, in particular, being widely used. The common denominator of the theories espoused by all of these psychologists and educationists was that young children learn mathematics best through active physical and mental involvement.

Education Colonialism and the Decline of the New Maths

Critics of the New Math(s) movement often alleged that teachers concentrated more on teaching children the names of the various field laws for numbers (like, for example, the commutative law for addition, and the distributive law), than on fundamental mathematical knowledge, skills and principles. The spirit of the moment was captured in the popular song by Tom Lehrer which alleged that with the New Math(s), students did not know that 3 + 2 = 5 but did know that addition was commutative and that 3 + 2 was equal to 2 + 3. While such a caricature was unfair, it was nevertheless widely accepted. It was not surprising, therefore, that in the 1970s, the New Math(s) gave way to a "back-to-the-basics" curriculum thrust in school mathematics.

Although tertiary mathematicians had usually played a prominent role in the development of New Math(s) curricula for secondary schools, support for the new curricula was by no means universal among tertiary mathematicians. Indeed, early in the 1960s many leading mathematicians in the United States and in other countries made it clear that they did not support the New Math(s) reforms (Clements, 1989), at either the elementary or secondary level. Nonetheless, in the late 1960s and the 1970s, when many of the secondary school New Math(s) curriculum reforms which had been introduced in the United States were being discarded, similar New Math(s) courses were still being introduced in other countries around the world - often with the support of local mathematicians who believed that the mathematics taught in their schools needed to be brought into line with developments in the rest of the world.

It seems that such were the forces of colonialism that when curriculum innovations in "advanced" countries such as the United States and the United Kingdom were introduced, these innovations were mimicked in other countries after a 10 to 15 year lag period. In using the term "colonialism" we are not necessarily accusing the "advanced" countries of forcing their education ideas on others - rather, there was a tendency among the education leaders of many of the other countries to accept such innovations virtually without question. There were good reasons for this. Leading educationists from "advanced" countries often secured consultancies (through World Bank, UNESCO, etc), and they tended to recommend the curricula and approaches of their own countries to education authorities in the nations where they were serving as consultants; also, when leading educationists from "developing" countries undertook masters and doctoral programs of study abroad, they were likely, on their return home, to take steps to "modernise" curricula in their own countries.

Behaviourism

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