University of Exeter, United Kingdom
Why teach mathematics? What are the purposes, goals, justifications and reasons for teaching mathematics? How can current mathematical teaching plans and practices be justified? What might be the rationale for reformed, future or possible approaches for mathematics teaching? What should be the reason for teaching mathematics, if it is to be taught at all? These questions begin to indicate the scope of what Niss (1996) has termed the justification problem for mathematics teaching.
Before discussing the aims of teaching mathematics there are three theses that I wish to assert as having an important bearing on this discussion. These concern, first of all, the lack of uniqueness and multiplicity of school mathematics; second, the current overestimation of the utility of academic mathematics; third, the socially and societally embedded nature of the aims of teaching and learning of mathematics. Acknowledging these claims means that the discursive space to be occupied differs from that in many traditional discussions of the aims of mathematics education.
First of all I want to argue that school mathematics is neither uniquely defined nor value-free and culture-free. School mathematics is not the same as academic or research mathematics, but a recontextualised selection from the parent discipline, which itself is a multiplicity (Davis and Hersh 1980). Some of the content of school mathematics has no place in the discipline proper but is drawn from the history and popular practices of mathematics, such as the study of percentages (Ernest 1986). Which parts are selected and what values and purposes underpin that selection and the way it is structured must materially determine the nature of school mathematics. Further changes are brought about by choices about how school mathematics should be sequenced, taught and assessed. Thus the nature of school mathematics is to a greater or lesser extent open, and consequently the justification problem must accommodate this diversity. So the justification problem should address not only the rationale for the teaching and learning of mathematics, but also for the selection of what mathematics should be taught and how, as these questions are inseparable from the problem.
Second, I wish to argue that the utility of academic and school mathematics in the modern world is greatly overestimated, and the utilitarian argument provides a poor justification for the universal teaching of the subject throughout the years of compulsory schooling. Thus although it is widely assumed that academic mathematics drives the social applications of mathematics in such areas as education, government, commerce and industry, this is an inversion of history. Five thousand years ago in ancient Mesopotamia it was the rulers need for scribes to tax and regulate commerce that led to the setting up of scribal schools in which mathematical methods and problems were systematised. This led to the founding of the academic discipline of mathematics.
the creation of mathematics in Sumer was specifically a product of that school institution which was able to create knowledge, to create the tools whereby to formulate and transmit knowledge, and to systematize knowledge. (Høyrup 1987: 45)
Since this origin, pure mathematics has emerged and has sometimes been internally driven, either within this tradition (such as scribal problem posing and solving in Mesopotamia and Ancient Egypt) or outside of it (such as the Ancient Greeks separation of pure geometry explored by philosophers from practical logistic). Nevertheless, practical mathematics has maintained a continuous and a vitally important life outside of the academy, in the worlds of government, administration and commerce. Even today the highly mathematical studies of accountancy, actuarial studies, management science and information technology applications are mostly undertaken within professional or commercial institutions outside of the academy and with little immediate input from academic mathematics.
However the received view is that academic mathematics drives its more commercial, practical or popular applications. This ignores the fact that a two way formative dialectical relationship exists between mathematics as practised within and without the academy. For example, overweight and underweight bales of goods are understood to have given rise to the plus and minus signs in medieval Italy. However it was the acceptance of negative roots to equations in renaissance Italy that finally forced the recognition of the negative integers as numbers.
The mathematization of modern society and modern life has been growing exponentially, so that by now virtually the whole range of human activities and institutions are conceptualised and regulated numerically, including sport, popular media, health, education, government, politics, business, commercial production, and science. Many aspects of modern society are regulated by deeply embedded complex numerical and algebraic systems, such as supermarket checkout tills with automated bill production, stock control; tax systems; welfare benefit systems; industrial, agricultural and educational subsidy systems; voting systems; stock market systems. These automated systems carry out complex tasks of information capture, policy implementation and resource allocation. Niss (1983) named this the formatting power of mathematics and Skovsmose (1994) terms the systems involved, which are embedded in social practices, the realised abstractions. The point is that complex mathematics is used to regulate many aspects of our lives; our finances, banking and bank accounts, with very little human scrutiny and intervention, once the systems are in place.
Furthermore, individuals conceptualisations of their lives and the world about them is through a highly quantified framework. The requirement for efficient workers and employees to regulate material production profitably necessitated the structuring and control of space and time (Taylor 1911) and for workers self-identities to be constructed and constituted through this structured space-time-economics frame (Foucault 1976). We understand our lives through the conceptual meshes of the clock, calendar, working timetables, travel planning and timetables, finances and currencies, insurance, pensions, tax, measurements of weight, length, area and volume, graphical and geometric representations, etc. This positions individuals as regulated subjects and workers in an information controlling society/state, as consumers in post-modern consumerist society, and as beings in a quantified universe.
In the era of late- or post-modernism a new mathematics-related ontology or root metaphor (Pepper 1948), has become dominant in the perceptions of the public and powerful in society. In particular, my claim is that the accountants balance-sheet and the world of finance has come to be seen as representing the ultimate reality. Although elements of such a social critique are well anticipated in Critical Theory (e.g., Marcuse 1964, Young 1979), this perspective has not so often been turned around and used to critique mathematics itself.
My claim is that the overt role of academic mathematics that which we recognise as mathematics per se in this state of affairs is overplayed. It is management science, information technology applications, accountancy, actuarial studies and economics that are the source for and inform this massive mathematization on the social scale.
This has important consequences for the justification problem, for it means that although there is undoubtedly an information revolution taking place, increased mathematical knowledge is not needed by most of the population to cope with their new roles as regulated subjects, workers and consumers. More mathematics skills beyond the basic are not needed among the general populace in industrialised societies to cope with these changes, if to cope means, as here, to serve rather than to critically master, which is discussed below. Thus national success in international studies of mathematical achievement is not the creator of economic success, unless having compliant subjects and consumers is what is needed. There is of course a need for a small elite who control the information systems and mechanisms, and a group of specialist technicians to service or programme them. These need to be present in all industrialised societies. But this group represent a tiny minority within society and their very special needs should not determine the goals of mathematics education for all. In addition, if this analysis is correct, it is not academic mathematics which is so very useful and needed for the information revolution. It is instead a collection of technical mathematised subjects and practices which are largely institutionalised and taught or acquired in practice outside of the academy.
In summary, my claim is that higher mathematical knowledge and competence, i.e., beyond the level of numeracy achieved at primary or elementary school, is not needed by the majority of the populace to ensure the economic success of modern industrialised society. Although other justifications for school mathematics can be given, and indeed will be given below, the traditional utilitarian argument is no longer valid. Most of the public do not need advanced mathematical understanding for economic reasons, and the minority who do apply mathematics acquire much of their useful knowledge in institutions outside of academia or schooling. This has been termed the relevance paradox, because of the simultaneous objective relevance and subjective irrelevance of mathematics in society (Niss 1994: 371). Society is ever increasingly mathematised, but this operates at a level invisible to most of its members.
Thirdly I want to claim that the aims of mathematics teaching cannot be meaningfully considered in isolation from their social context. Aims are expressions of intent, and intentions belong to groups or individuals. Educational aims are thus the expression of the values, interests, and even the ideologies of certain individuals or groups. Furthermore the interests and ideologies of some such groups are in conflict. Elsewhere, building on Raymond Williams (1961) seminal analysis, I distinguish five interest groups in the history of educational and social thought in Britain and show that each has distinct aims for mathematics education and different views of the nature of mathematics (Ernest 1991). These groups and their aims are summarised in Table I.
Table I: Five interest groups and their aims for mathematics teaching
1. Industrial Trainers
Radical 'New Right' conservative politicians and petty bourgeois
Acquiring basic mathematical skills and numeracy and social training in obedience (authoritarian, basic skills centred)
2. Technological Pragmatists
meritocratic industry-centred industrialists, managers, etc., New Labour
Learning basic skills and learning to solve practical problems with mathematics and information technology (industry and work centred)
3. Old Humanist Mathematicians
conservative mathematicians preserving rigour of proof and purity of mathematics
Understanding and capability in advanced mathematics, with some appreciation of mathematics (pure mathematics centred)
4. Progressive Educators
Professionals, liberal educators, welfare state supporters
Gaining confidence, creativity and self expression through maths (child-centred progressivist)
5. Public Educators
Democratic socialists and radical reformers concerned with social justice and inequality
Empowerment of learners as critical and mathematically literate citizens in society (empowerment and social justice concerns)
These different social groups were engaged in a contest over the National Curriculum in mathematics, since the late 1980s (Brown 1996). In brief, the first three more reactionary groups managed to win a place for their aims in the curriculum. The fourth group (progressive educators) reconciled themselves with the inclusion of a personal knowledge-application dimension, namely the processes of Using and Applying mathematics, constituting one of the National Curriculum attainment targets. However instead of representing progressive self-realisation aims through mathematics this component embodies utilitarian aims: the practical skills of being able to apply mathematics to solve work-related problems with mathematics. Despite this concession over the nature of the process element included in the curriculum, the scope of the element has been reduced over successive revisions and is currently being totally eliminated.
The Public Educators aim, concerning the development of critical citizenship and empowerment for social change and equality through mathematics, has played no part in the National Curriculum (and is absent from most other curriculum developments too). Thus although progressives see mathematics within the context of the individuals experience, the notion that the individual is socially located in an unjust world in which citizens must play an active role in critiquing and righting wrongs plays no part.
The outcome of the historical contests and processes is that the National Curriculum may be said to serve three main purposes. First of all, much of the National Curriculum in mathematics is devoted to communicating numeracy and basic mathematical skills and knowledge across the range of mathematical topics comprising number, algebra, shape and space (geometry and measures), and handling data (incorporating information technology mathematics, probability and statistics).
Second, for advanced or high attaining students the understanding and use of these areas of mathematics at higher levels is included as a goal. Thus there is an initiation into a set of academic symbolic practices of mathematics for the few (e.g. General Certificate of Education advanced level studies for 16-18 year olds).
Third, there is (or rather was, as it is soon to be radically reduced) a practical, process strand running through the National Curriculum mathematics which is intended to develop the utilitarian skills of using and applying mathematics to real world problems.
Each of these three outcomes is to a greater or lesser extent utilitarian, because they develop general or specialist mathematics skills and capabilities, which are either decontextualised equipping the learner with useful tools or which are applied to practical problems. The slant of this outcome comes as a surprise to no-one, because the whole thrust of the National Curriculum is recognised to be directed towards scientific and technological competence and capability. New Labour education policy has maintained this thrust.
In technology education, curriculum theorists distinguish between developing technological capability, on the one hand, and appreciation or awareness, on the other (Jeffery 1988). In brief, technology capability consists of the knowledge and skills that are involved in planning and making artefacts and systems. Technology appreciation and awareness comprises the higher-level skills, knowledge and judgement necessary to evaluate the significance, import and value of technological artefacts and systems within their social, scientific, technological, environmental, economic and moral contexts.
An analogous distinction can be applied to mathematics which suggests the following question. Is school mathematics all about capability, i.e. 'doing', or could there be an appreciation element that was overlooked in the National Curriculum? There is a well-known view that mathematics is not a spectator sport, that is, it is about solving problems, performing algorithms and procedures, computing solutions, and so on. Except in the popular domain, or in the fields of social science or humanities which comment on mathematics as opposed to doing mathematics, nobody reads mathematics books, they work through them. Furthermore the language of both school and research mathematics are full of imperatives, ordering the reader to do something, rather than follow a narrative (Rotman 1993, Ernest 1998). Thus the capability dimension of mathematics, and of school mathematics in particular, is dominant and perhaps universal.
Of course if mathematics is to be given a major role in the curriculum, as it almost invariably is, some large capability element is necessary, for unquestionably knowledge of mathematics as a language and an instrument does require being able work and apply it. Furthermore, a minimal mathematical capability is essential, a sine qua non, for the development of mathematical appreciation. But is capability enough on its own? Has any published curriculum addressed anything other, such as appreciation? Would the development of mathematical appreciation be a worthwhile and justifiable goal for school mathematics? If so, what is mathematical appreciation and how could appreciation be addressed?
The first issue that needs to be addressed is what the appreciation of mathematics means. In my view, a provisional analysis of what the appreciation of mathematics understood broadly, might mean, involves the following elements of awareness:
1. Having a qualitative understanding some of the big ideas of mathematics such as infinity, symmetry, structure, recursion, proof, chaos, randomness, etc.;
2. Being able to understand the main branches and concepts of mathematics and having a sense of their interconnections, interdependencies, and the overall unity of mathematics;
3. Understanding that there are multiple views of the nature of mathematics and that there is controversy over its philosophical foundations;
4. Being aware of how and the extent to which mathematical thinking permeates everyday and shopfloor life and current affairs, even if it is not called mathematics;
5. Critically understanding the uses of mathematics in society: to identify, interpret, evaluate and critique the mathematics embedded in social and political systems and claims, from advertisements to government and interest-group pronouncements;
6. Being aware of the historical development of mathematics, the social contexts of the origins of mathematical concepts, symbolism, theories and problems;
7. Having a sense of mathematics as a central element of culture, art and life, present and past, which permeates and underpins science, technology and all aspects of human culture.
In short, the appreciation of mathematics involves understanding and having an awareness of its nature and value, as well as understanding and being able to critique its social uses. The breadth of knowledge and understanding involved is potentially immense, but many learners leave school without ever having been exposed to, or thought about, several of these seven areas of appreciation.
My purpose in contrasting capability and appreciation in mathematics is to draw attention to the neglect of the latter, both in theory and practice. To be a mathematically-literate citizen, able to critique the social uses of mathematics, which is the aim of the public educator position summarised above, would go part way towards realising mathematical appreciation, if it were implemented. However there would still be a further element lacking, even if this were to be achieved. This is the development of an appreciation of mathematics as an element of culture, and of the inner culture and nature of mathematics itself. Despite the love for mathematics felt by most mathematics teachers, educators and mathematicians, the fostering of mathematical appreciation, in this sense, as an aim of mathematical teaching, is not promoted. It might therefore be said that mathematics professionals both undervalue their subject and underestimate the ability of their students to appreciate it.
To summarise, four main aims for school mathematics have been discussed above.
1. To reproduce mathematical skill and knowledge based capability
The typical traditional reproductive mathematics curriculum has focused exclusively on this first aim, comprising a narrow reading of mathematical capability. At the highest level, not always realised, the learner learns to answer questions posed by the teacher or text. As is argued elsewhere (Ernest 1991) this serves not only to reproduce mathematical knowledge and skills in the learner, but to reproduce the social order and social injustice as well.
2. To develop creative capabilities in mathematics
The progressive mathematics teaching movement has added a second aim, to allow the learner to be creative and express herself in mathematics, via problem solving, investigational work, using a variety of representations, and so on. This allows the learner to pose mathematical questions, puzzles and problems, as well as to solve them. This notion adds the idea of creative personal development and the skills of mathematical questioning as a goal of schooling, but remains trapped in an individualistic ideology that fails to acknowledge the social and societal contexts of schooling, and thus tacitly endorses the social status quo.
3. To develop empowering mathematical capabilities and a critical appreciation of the social applications and uses of mathematics
Critical mathematics education adds in a third aim, the empowerment of the learner through the development of critical mathematical literacy capabilities and the critical appreciation of the mathematics embedded in social and political contexts. Thus the empowered learner will not only be able to pose and solve mathematical questions, but also be able to address important questions relating to the broad range of social uses (and abuses) of mathematics. This is a radical perspective and set of aims concerned with both the political and social empowerment of the learner and with the promotion of social justice, and which is realised in mainstream school education almost nowhere. However, the focus in the appreciation element developed in this perspective is on the external social contexts of mathematics. Admittedly these may include the history of mathematics and its past and present cultural contexts, but these do not represent any full treatment of mathematical appreciation.
4. To develop an inner appreciation of mathematics: its big ideas and nature
This fourth aim adds in further dimension of mathematical appreciation, namely the inner appreciation of mathematics, including the big ideas and nature of mathematics. The appreciation of mathematics as making a unique contribution to human culture with special concepts and a powerful aesthetic of its own, is an aim for school mathematics often neglected by mathematicians and users of mathematics alike. It is common for persons like these to emphasise capability at the expense of appreciation, and external applications at the expense of its inner nature and values. One mistake that may be made in this connection is the assumption that an inner appreciation of mathematics cannot be developed without capability. Thus, according to this assumption, the student cannot appreciate infinity, proof, catastrophe theory and chaos, for example, unless they have developed capability in these high level mathematical topics, which is out of the question at school. The fourth aim questions this assumption and suggests that an inner appreciation of mathematics is not only possible but desirable to some degree for all students at school.
The justification problem in mathematics education is problematic, partly because any so-called solution can only be a partial set of arguments concerning the role of mathematics teaching and learning for a certain clientele (the learners), in certain countries, during a certain time-frame, satisfying the supporters of one or more viewpoints. Thus part of the problem is its shifting and relative nature. Another part of the problem is that mathematics is simultaneously undervalued and overvalued in modern western society. It is overvalued because first of all, its perceived utility is misunderstood to mean that all persons need maximal knowledge and skills in mathematics to function economically. However the mathematics underpinning the functioning of modern society is largely embedded and invisible. Secondly, mathematical attainment is mistakenly identified with intelligence and mental power and used to grade and select persons for various forms of work, including professional occupations as well as in terms of suitability for higher education. Because of this role, mathematics serves as a critical filter and has been implicated in denying equal opportunities to many (Sells 1973).
Mathematics also is undervalued because most justifications in support of its continued central role in education are based on extrinsic arguments framed in terms of utility and instrumentality. As an intrinsically valuable area of human culture mathematics it is rich in intellectually challenging and exciting concepts including infinity, chaos, chance, etc. It is an imaginary realm and domain of knowledge with its own aesthetics and beauty. Mathematics also has a central part to play in philosophy, art, science, technology, information technology and the social sciences. Appreciation of this is surely part of every learners entitlement, while they are studying mathematics.
The mention of student entitlements raises an as yet unaddressed question. Should mathematics be taught throughout the years of compulsory schooling and should the same curriculum be followed by all? Requiring learners to study mathematics throughout the years from age 5 to 16 years is less easy to justify if mathematics is not as useful as is often assumed. Furthermore, if it is an unhappy learning experience for almost half of the population as research suggests, should not learners themselves be given some say in the matter, perhaps after have acquired basic mathematical competency? Should not the changing personal preferences, career interests and vocational development plans that emerge in students during adolescence be accommodated, by a differentiated mathematics curriculum or by allowing students to opt out altogether? If education is to contribute to the development of autonomous and mature citizens, able to fully participate in modern society, then it should allow elements of choice and self-determination. However, in the space available here I can only raise these crucial issues, rather than treating them thoroughly.
Finally, let me add a remark on the gap between the domain of discourse on aims and the practical domain in which the impact of educational practices is experienced. However noble, high-flown, or otherwise intentioned the aims of mathematics teaching may be, they need to be evaluated in the light of their impact on individuals and society. Any consideration of the mathematics curriculum requires that three levels must be considered (Robitaille and Garden 1989). These are the levels of first the intentional or planned curriculum, second that of the implemented or enacted curriculum, and third that of the learned curriculum including learner outcomes and gains (including affective responses). The extent to which goals of mathematics education are implemented and realised in classroom practice is a major determinant of the nature of the mathematics teaching in classrooms. Teaching is an intentional activity and ideally there should be a strong relation between the expressed aims and the realised practices of mathematics education. Where this link fails to obtain there is an area of disequilibrium and inconsistency which creates stresses for teachers and students. Of course this can be reactionary in a site where traditional conceptions and practices subvert well justified curriculum plans. However it can also be a site of resistance where aims locally deemed unworthy or unpopular are subverted. This, once again, raises the issue of which groups values and views are dominant in determining the aims of teaching mathematics, and who gains and who loses.
Brown, M. (1996) The context of the research - the evolution of the National Curriculum for mathematics. In D. C. Johnson and A. Millett (Eds.) Implementing the Mathematics National Curriculum: Policy, Politics and Practice. London: Paul Chapman Publishing Ltd., pp.1 - 28.
Davis, P. J. and Hersh, R. (1980) The Mathematical Experience, Boston: Birkhauser.
Ernest, P. (1986) Social and Political Values, Mathematics Teaching, No. 116, 16-18.
Ernest, P. (1991) The Philosophy of Mathematics Education. London: The Falmer Press.
Ernest, P. (1998) Social Constructivism as a Philosophy of Mathematics. Albany: New York: SUNY Press.
Foucault, M. (1976) Discipline and Punish, Harmondsworth, Penguin
Høyrup, J. (1987) Influences of Institutionalized Mathematics Teaching on the Development and Organisation of Mathematical Thought in the Pre-Modern Period. Bielefeld. In Fauvel, J. and Gray, J., Eds, The History of Mathematics: A Reader. London: Macmillan, 1987, 43-45.
Jeffery, J. (1988) Technology Across the Curriculum: A Discussion Paper. Unpublished paper, Exeter: University of Exeter School of Education.
Marcuse, H. (1964) One Dimensional Man. London: Routledge and Kegan Paul.
Niss, M. (1983) Mathematics Education for the Automatical Society. In Schaper, R. Ed. (1983) Hochschuldidaktik der Mathematik (Proceedings of a conference held at Kassel 4-6 October 1983). Alsbach-Bergstrasse, Germany: Leuchtturm-Verlag, 43-61.
Niss, M. (1994) Mathematics in Society. In Biehler, R., Scholz, R. W., Straesser, R., Winkelmann, B. Eds. (1994) The Didactics of Mathematics as a Scientific Discipline. Dordrecht: Kluwer, 367-378.
Niss, M. (1996) Goals of Mathematics Teaching. In Bishop, A. J. Ed. The International Handbook of Mathematics Education. Dordrecht: Kluwer Academic, Volume 1, 11-47.
Pepper, S. C. (1948) World Hypotheses: A Study in Evidence. Berkeley, California: University of California Press.
Robitaille, D. F. and Garden, R. A. Eds (1989) The IEA Study of Mathematics II: Contexts and Outcomes of School Mathematics, Oxford: Pergamon.
Rotman, B. (1993) Ad Infinitum The Ghost in Turing's Machine: Taking God Out of Mathematics and Putting the Body Back in. Stanford California: Stanford University Press.
Sells, L. (1973) High school mathematics as the critical filter in the job market. Proceedings of the Conference on Minority Graduate Education, Berkeley: University of California, 37-49.
Skovsmose, O. (1994) Towards a Philosophy of Critical Mathematics Education. Dordrecht: Kluwer.
Taylor, F. W. (1911) The Principles of Scientific Management, London and New York.
Williams, R. (1961) The Long Revolution. London: Penguin Books.
Young, R. M. (1979) Why are figures so significant? The role and the critique of quantification. In Irvine, J.; Miles, I.; Evans, J. Eds. (1979) Demystifying Social Statistics. London: Pluto Press, 63-74.
© P Ernest 2000
This chapter is published in Why Learn Maths?, edited by John White and Steve Bramall, London: London University Institute of Education, 2000.