Mathematics Education and the Future: a Long Wave View of Change PETER GALBRAITH Mathematics education is a concern ofthe whole commun- ity in that the generation of wealth requires a workforce equipped with an appropriate level of scientific abilities On the other hand sub-standard levels of numeracy con- tribute to problems of unemployment and hence to a drain on welfare systems. Enquiiies such as [1,2] reflect the importance with which mathematics continues to be regarded at national levels. One problem with mathematics education is its lack of a macro theory. In consequence, while much research and development occurs this tends to be piecemeal and uncoor- dinated. As evidence of this, hom time to time state of the art documents appear [3,4,5,6] which attempt to summar- ize what has been achieved across a wide variety of contexts and to suggest key areas for renewed effort But the search for a set of principles to explain the past and to direct future efforts remains unfulfilled The future is usually viewed within some context of extrapolation from the past and present This paper is an attempt to contribute to the development of guiding principles for mathematics educa- tion by taking a frame of reference outside mathematics education itself It can be viewed as the development of a scenario that has its basis in the literature of the economic long wave [7,8,9,!0,11,12,13] This paper examines the implications of long wave theory for mathematics educa- tion. It addresses some of the tiaditional issues associated with educational research, interprets some historical events as outcomes rather than causes and points some directions towards which efforts should be directed In this sense, and from a provocative perspective it attempts to identifY prior- ities for future developments in mathematics education. l. Cyclic patterns and mathematics education The waxing and waning of distinct influences on mathe- matics education has been explicitly noted or implied by a variety of writers. Higginson [14] envisaged the four com- ponents of mathematics, psychology, socio-cultural fac- tors and philosophy as combined into a tetrahedral model to which each influence contributed a face The movement of the "centre of gravity" of the tetrahedron would reflect the changing "weight" exerted by the respective faces. He sees in both the United States and Western Europe the turn of the century (the Peny movement) and the Sputnik era (the new mathematics) as times of content domination, a view shared by Howson [6,15] Higginson sees the psycho- logical dimension of teaching since 1960 as having under- gone an almost complete cycle from didacticism to discovery and back again Influenced by changes in the social climate variations in philosophical rationale have been seen to vary from intrinsic reward to a largely utilitar- ian view. In the words of Howson [6] Educators in the first two decades of this century faced similar problems to those which we do nowa- days In their case a period of frenzied activity was followed by a period of retrenchment and consolidation Robitaille and Dirks [16] observed in summarizing eleven case studies that the type of social and political system operating in a given place at a given time influences "not only the educational system in general, but, in particular the mathematics cuniculum" Lucas, in Steen [I 7], under- lines this view when he writes, "the commonly held view that mathematical directions are somehow independent of the stresses acting on them is a naive one" Howson [15] further confirms this thinking by noting the contrast between the sixties when governments welcomed the pleas of educators for mathematics for all and the recent past which has seen disillusionment set in with talk of basics and minimum competencies Richardson [18] reviewed the his- tory of mathematics curricula in the U.S.A between I 893 and I 935 He identified the early years of the century as a time galvanised by issues of content which saw, for exam- ple, the birth of the College Entrance Examinations Board and the memorable address byE H Moore [1902] on the foundations of mathematics Economically the country was in the process of becoming predominantly industrial and the needs of such a society were clearly influential in shaping curriculum change. Moving forward in time to 1918, Kilpatrick's project method was identified as influenced by Dewey in its con- cern for personal engagement in the learning process and the primacy it gave to the interests of the individual, while the publication of the Cardinal Principles of Secondary Education [1918] further emphasized the "rise in concern for individuals and the fall of the discipline concept" This concern for the pupil found an echo in the sixties as, for example, Begle [3] noted: The preparation of special instruction passages, including teacher training, as well as curriculum units, for students doing poorly in mathematics received a great deal of support in the sixties This For tht Learning oj 8, 3, (November !988) H M Publishing A.ssociation Montreal Quebec Canada 27