Key Issues and Trends in Research on Mathematical Education
Professor Mogens Niss, Roskilde University, Denmark
Manuscript of plenary lecture delivered at ICME-9, Tokyo/Makuhari 2000
To appear in the proceedings of this congress
I. Introduction
Mathematics education as a research field is no longer in what Geoffrey Howson in the proceedings of ICME-2 (Howson, 1973) called 'the formative years', i.e. a 'child' or a 'teenager', but a 'young adult' whose development, character and achievements can now be considered and discussed. However, even though our field has reached a first stage of maturity, it is definitely not marked by coherence and unity, let alone uniformity, but by considerable complexity and diversification in perspectives and paradigms. Against this background it is not a trivial matter for a single individual to set out to identify and discuss key issues and trends in research on mathematical education. In fact, the more I indulged in preparing this talk, the more I have been forced to realise how impossible the task is. First of all, any attempt to deal with the issues and trends, in an exhaustive way, rather than with some issues and trends, is deemed to failure. Therefore, focal points have to be determined, choices have to be made, constraints have to be faced, all of which will, inevitably, reflect the limitations, tastes and biases of the person who is silly enough to embark on such a daunting endeavour. In other words, the deliberations I am going to submit to the reader's consideration will be of a fairly personal nature. However, it is not my primary intention to act as a critic, rather as a cartographer or an anthropologist. I shall, however, at the end of this paper, express some personal views of the future development in of our field.
A final remark before take off. In line with other researchers (e.g. Cronbach & Suppes, 1969, and Kilpatrick, 1992, and others), I adopt, here, a fairly inclusive and pragmatic definition of the term ‘research’ by taking it to mean disciplined enquiry, i.e. questions are asked and answers are sought by means of some methodology, the specific nature if which is not predetermined. Below, different types of disciplined enquiry concerning mathematical education are identified.
II. Focal questions and approaches
In attempting to identify, describe and discuss issues and trends in research on mathematical education, a number of different foci appear to be relevant to the analysis. The following focal questions, among others, lend themselves to such an analysis.
* What are the prevalent general issues and the specific questions that are posed and studied in research?
* What are the objects and the phenomena which are typically subjected to investigation?
* What are the predominant research methods adopted by researchers to deal with the questions they pose? What can be said about to the foundations, ranges, strengths, and weaknesses of these methods?
* What kinds of results are obtained in research, and what sorts of scope do they typically have?
* What are the current and emerging problems and challenges regarding research on mathematical education that we have to face?
All of these focal questions can be considered from a static, a kinematic or a dynamic perspective. A static perspective attempts to look at the state of affairs at a given point in time in order to produce a snapshot of this state. In contrast, a kinematic perspective tries to provide a description of how the state of affairs has changed over a certain period of time, whereas a dynamic perspective endeavours to uncover the forces and mechanisms that are responsible for the changes detected.
Furthermore, the issues can be viewed from either a descriptive perspective ("what is (the case?)") or a normative perspective ("what ought to be (the case)?") (Niss, 1999). It is worth emphasising that both perspectives can be pursued in an analytic manner.
The approach I have adopted to deal with aspects of the first four foci combines a dynamic and a descriptive perspective, whereas the last focus automatically implies some form of a normative perspective. The analysis will be based on sample observations obtained from probing into research journals, ICME proceedings and other research publications from the last third of the 20th century and, of course, on my general knowledge and perceptions of our field.
III. General issues and research questions, objects and phenoma of study
An overarching characteristic of the development of mathematical education as a research domain is the gradual widening of its field of vision to encompass more and more educational levels. In the beginning, the objects and phenomena of study primarily referred to school mathematics (primary school throughout the 20th century, secondary school since the 60’s) but soon aspects of post-secondary education were addressed as well. Teacher education and pre-school education gained momentum in the 70's and 80's, as did tertiary education in general (in the 80's and 90's). Today, undergraduate education for mathematics majors is a growing item on the research agendas all over the world. To my knowledge higher graduate education has not yet been included but that might well happen in a not too distant future. Non-tertiary mathematical education for adults returning to education, e.g. early school leavers, has been an expanding area of interest during the 90's. This widening of the scope of research to encompass virtually all kinds of mathematical education is, above all, a reflection of the fact that mathematical education at any level is being provided to a steadily increasing group of recipients. 'Mathematics for all' could be a slogan for this development.
When it comes to the objects and phenomena studied in research, mathematics educators in the 1960's and 70's were predominantly preoccupied with the mathematical curriculum and with ways of teaching it (for a similar observation, see Kilpatrick, 1992, p. 22). This was in continuation of the prevalent preoccupation in the first part of the century. Traditionally, the underlying, often implicit, issues were to identify, structure, sequence and organise 'the right content' and to devise and implement 'the right mode of presentation': What topics should be included, on what conceptual basis, and in what order? What should the teacher do, and when? What tasks and activities for students should be orchestrated? What properties should textbooks have? And what is the role of technical aids for teaching? Questions such as these, though seldom formulately explicitly in the publications, can often without too much difficulty be detected as the ones which actually drove the deliberations. Until the mid-70’s, in many publications the term ‘curriculum’ was given a fairly narrow meaning, namely that of ‘syllabus’, in which only the issue of mathematical content is on the agenda. During the 70’s, ‘curriculum’ gradually gained a much wider meaning, encompassing also aims, teaching approaches and assessment modes.
One manifestation of the preoccupation with curriculum issues in general is the series New trends in mathematics teaching that was prepared under the auspices of ICMI and published by UNESCO. These publications attempted to identify and describe curriculum trends throughout the world. For instance, Vol. III, which appeared in 1973 (UNESCO, 1973, edited by Howard Fehr and Maurice Glaymann), looked at the situation in primary school mathematics as a whole, whereas, for post-primary education, the focus was on mathematical topics, such as algebra, geometry, probability and statistics, analysis, logic, and applications of mathematics,. Besides, there were chapters on ‘trends in methods and media’, on ‘evaluation’, and on ‘research’. Also Vol. IV (UNESCO, 1979, edited by Bent Christiansen and Hans-Georg Steiner) spent the first five chapters on depicting curriculum trends at all educational levels. In addition to international trends, national or local trends were described in hosts of papers outlining the situation in a given country or in particular schools, most certainly meant to simply serve as a source of factual information and inspiration for people in other places. Papers on national and international trends are particularly plentiful in times of worldwide reform movements such as the emergence and dissemination of the ‘New (or Modern) Mathematics’. Thus the first volume (UNESCO, 1980) in UNESCO’s series Studies in mathematics education (established by Edward Jacobsen as the editor-in-chief, while Robert Morris edited the individual volumes) was devoted to national presentations of the situation in Hungary, Indonesia, Japan, The Philippines, The Soviet Union, and Tanzania, and more were published in the second volume (UNESCO, 1981) on the Arab States and the USA. It should be stressed that although the interest in curriculum issues and trends was particularly manifest in the 60’s and 70’s, this interest has never really vanished. This can be seen in the ICMEs, this one included, which have always had well-attended sessions dealing with curriculum issues at various levels, in response to the continued widespread interest amongst participants in learning about new developments and exchanging experiences and views.
Every so often mathematics educators have cultivated a deep interest in the ‘sub-curriculum’ of a specific mathematical topic. This can be seen, for instance, in the papers from the ‘Conference on the Teaching of Geometry’, Carbondale, Ill, USA , March 1970, published in Educational Studies in Mathematics in Volumes 3 and 4 (1971). Papers include ‘n-Gons’ (Bachmann, 1971), ‘Geometrical Activities for Upper Elementary School’ (Engel, 1971), ‘The Teaching of Geometry – Present Problems and Future Aims’ (Fletcher, 1971), ‘Topology in the High School’ (Hilton, 1971), ‘ Geometric Algebra for the High School Programme’ (Levi, 1971), ‘The Geometry Relevant to Modern Education’ (Menger, 1971), ‘The Introduction of Metric by the Use of Conics’ (Pickert, 1971), ‘The Position of Geometry in Mathematical Education’ (Revuz, 1971), ‘The Geometry and Algebra of Reflections (and 2x2 Matrices)’ (Room, 1971), ‘A Foundation of Euclidean Geometry by Means of Congruence Mappings’ (Steiner, 1971), ‘Learning and Teaching Axiomatic Geometry’ (Stone, 1971), ‘A Logical Approach to the Teaching of Geometry at the Secondary Level’ (Villa, 1971). It appears already from the titles of these papers that discussions were centrered around the role and position of geometry in the curriculum, the choice of basic approach and of which content to present to students, and the way to structure it once it has been chosen. Quite a few of the papers can actually be seen as (potential) textbook sections under the motto ‘look how this can be done!’. It is characteristic of all the papers mentioned that the focus is on mathematical subject matter and that the deliberations, if any, offered with regard to teaching are predominantly of a theoretical nature, only empirical to the extent that they draw on authors’ general observations and experiences with students and teaching . Besides, some of the approaches proposed had been used (successfully, we are told) by the author in his own teaching. So, the predominant objects of study in these papers are of a mathematical nature and are situated in the realm of (potential) teaching. A few papers from the conference took slightly different perspectives, however. Robert Davis in ‘The Problem of Relating Mathematics to the Possibilities and Needs of Schools and Children’ (Davis, 1971), insisted on the importance of real schools and real children, in addition to mathematics (geometry), and discussed issues of teaching and cognition while illustrating his points by interview episodes with young children. And Hans Freudenthal, in ‘Geometry Between the Devil and the Deep See’ (Freudenthal, 1971), offered an essay on the nature of geometry and on how mathematics is (not) learned.
Interests in fundamental particular topics gain momentum and fade away, gain momentum again, and so forth in waves. Geometry is a topic that enters the agenda every now and then. Thus a recent ICMI Study, ‘Perspectives on the Teaching of Geometry for the 21st Century’, was published in 1998 (Mammana & Villani, 1998). That study, however, was certainly not confined to considering curriculum issues only.
As soon as discussions on matters pertaining to curricula become non-technical and non-rhetorical they will inevitably involve issues of the goals and aims that should guide the selection of subject matter and the organisation of teaching, and the more specific objectives to be pursued. So, in the 70’s and early 80’s goals, aims and objectives became objects of debate and investigation: What should they be? And why? And what are their interrelations? Clearly, these questions touch upon the ultimate ends and the very purposes, roles and functions of mathematical education for different groups of recipients and hence implicate value-laden perceptions of society and culture. It follows that research in this category tends to include strong normative components. In New trends...III (UNESCO, 1973), for instance, goals were briefly touched upon from a combination of descriptive and normative perspectives, in terms of objectives to be pursued in the teaching of various topics, e.g. geometry, probability and statistics, logic etc. The common pattern is that these topics should be studied for three reasons: Their significance to the application of mathematics in other subjects and fields of practice, their role in relation to the world at large, and their position and function in a unified mathematics curriculum. Overarching goals were discussed in the chapters on ‘applications in mathematics’ and, in greater depth, in the chapter on ‘trends in methods and media used in teaching mathematics’. The emphasis given to identifying and discussing goals of mathematical education was increased during the 70’s. Thus a chapter on ‘overall goals and objectives’ (D’Ambrosio, 1979) was included in New trends IV... (UNESCO, 1979), in addition to being paid attention to in the other chapters of that publication, and the second volume (1981) in the UNESCO’s Studies in Mathematics Education series was mainly devoted to considering goals of mathematical education from a variety of perspectives. A few years ago, I attempted to provide a conceptual and historical analysis of aspects of goals of mathematics teaching (Niss, 1996).
One thing is to write papers for journals and books and to hold scholarly meetings to consider what the goals, content and organisation of mathematics curricula are, and perhaps reach agreement on what they ought to be. It is quite another thing to implement such ideas in the reality of the classroom. This often requires change that is not so easy to bring about because of obstacles present in various places in the educational system, e.g. with curriculum authorities, with teachers, or with society’s stakeholders in education like, for instance, employers and parents. This fact gave rise to an interest, since the mid 70’s, in mathematics curriculum development, as such, which in turn implied consideration of a spectrum of societal and institutional matters. A major chapter in New Trends...IV (Howson, 1979), ‘A critical analysis of curriculum development in mathematical education’ was devoted to examining these questions in quite a detail, by considering the meaning of curriculum development (including the forces that can initiate such development), identifying agencies, strategies, and difficulties for curriculum development, and by examing the role of the teacher (and of in-service education) and the transfer of materials and know-how.
The agencies and agents of curriculum development are different in different countries at different times. In some countries teachers are protagonists in the design and development of the mathematics curriculum, sometimes even as what Clarke et al. (1996) call ‘the curriculum maker’. In any country, however, the teacher plays a key role in the acceptance and implementation of the curriculum, whether or not that role is of an official and formal nature. If we remember that many mathematics educators are involved in pre- or in-service teacher education, it is only natural that issues related to the education and profession of mathematics teachers have been included in research agendas since the 70's.
It goes without saying that the luggage teachers are (to be) equipped with in order to exert their profession has always been a major issue in teacher training. In the past this issue was seen as boiling down to the issue of the pre-service - and in times of reform also in-service - curricula. In that way the discussion was parallel to the one concerning students’ curricula. What knowledge should the teacher of mathematics have, and how should it be organised and sequenced? How to strike a balance between pre-service and in-service training? (see, e.g. Scandura, 1971) During the 70’s it became clear, however, that matters other than the mathematics the teacher knows and the methods courses (s)he has taken are no less important to his or her functioning as a teacher. Not only was it no longer possible to pack the teacher’s mathematical suitcase once and for all, whence it was necessary to prepare the teacher for acquiring new knowledge, the role of the teacher in the classroom changed dramatically from one of transmitting a well-defined syllabus in well-established ways, to one requiring the teacher to orchestrate a variety of activities for the class as a whole and for students individually or in small groups, so as to allow for independent student work to explore situations, make discoveries, deal with problem solving and much more. The more multi-faceted the teacher’s job becomes, the greater is the impact of that job on the mathematical education of the students, in particular when the teacher is required to not just deliver a message but also to look into and after the learning of his/her students. These matters were studied already by the two groups which at ICME-2 (1972, see Howson, 1973) considered ‘the initial training of primary school teachers’ and of ‘secondary school teachers’, respectively.
Piling up demands on the teacher is not much of a research task in itself, but systematic reflection on the nature and conditions of the profession of mathematics teacher is. It seeems that Ed Begle’s paper ‘Teacher Knowledge and Student Achievements in Algebra’ (Begle, 1972) inaugurated the study of such issues. At ICME-3, the section on the theme of teacher education agreed on recommending to the Executive Committee of ICMI that ‘the education and professional life of mathematics teachers’ should be accorded a permanent place in all future ICMEs. A paper of exactly this title, written by Michael Otte and collaborators, in New trends ...IV (Otte, 1979) was one of the first to offer a framework for the conceptualisation and study of this theme. Since then this theme has indeed been on the agenda of all ICMEs – which was already markedly visible at ICME-4 – and in many other places. Two volumes in Studies in Mathematics Education were devoted to ‘the mathematical education of primary school teachers’ (Vol. 3) (UNESCO, 1984) and ‘the education of secondary school teachers of mathematics’ (Vol. 4) (UNESCO, 1985).