Paper presented at the conference of the International Commission for the Study and Improvement of Mathematics Education (CIEAEM51), University College, Chichester, UK, July, 1999.
---------------------------------------------------------------------------------------------------------------------------------------------------------Values and mathematics education
Philip C Clarkson, Australian Catholic University
Alan Bishop, Monash University
Values in mathematics education are the deep affective qualities which education fosters through the school subject of mathematics. There is generally acknowledged to be a close relationship between values and attitudes, reaching back to Rokeach (1973), with values occupying a more central and deeply held position than attitudes, which are often considered to be reflected in our patterns of response to particular situations (Seligman et al, 1996). Previous research by Buxton (1981) and Fasheh (1982), indicate this in relation to mathematics teaching practices, while Martin (1997) shows how values can enter into the mathematical modelling process.
The recent developments in culture and mathematics, such as Bishop’s (1988) research on enculturation, Harris’s (1991) research with Aboriginal students, and Powell and Frankenstein’s (1997) overview on ethnomathematics and the politics of mathematical knowledge, have brought the issue of values into greater focus, raising awareness of non-Western mathematical ideas together with non-Western beliefs and values. It seems it is only since the realisation that there exist mathematical ideas other than those in the canonical mathematics curriculum of the West (Howson and Wilson, 1986) that there has been any concern about values teaching in mathematics. Until twenty years ago mathematics was considered a value-free and culture-free subject. That is no longer the case.
However, the realisation that mathematics teaching is as value-laden a school subject as any other has not meant that there are any clear ideas about how such values are taught. In fact we can state categorically that there is no empirical research to date on values teaching in mathematics. McLeod (1992) in one of the most comprehensive reviews of the affective research literature failed to find any research focussed on values. The tone of his discussion however makes it clear that ideas about both beliefs and attitudes towards mathematics do relate to the deeply held values of both teachers and students.
Wilson’s (1986) chapter is one of the rare writings about values in mathematics teaching and the book in which it appears contains many useful points. Science educators have been almost as remiss as mathematics educators in their failure to address values, but Poole’s (1995) book has made a huge contribution to knowledge. In our own work the enculturation book already noted (Bishop, 1988) has a chapter on the values underlying Western mathematics, and in Bishop (1991) values in the mathematics teaching process are analysed and discussed. Clarkson (1991) discussed issues of values embedded in different cultures and their implications for mathematics learning. There is a need to begin the difficult process of gaining some empirical basis for claims about values teaching in mathematics classrooms and about how to improve teachers’ knowledge in this area.
The preliminary question however is what values are mathematics teachers teaching? Our concern with this question is that although values teaching and learning go on inevitably in all mathematics classrooms, most of it appears from our preliminary studies to be done implicitly. Therefore there is only a limited understanding of what values are being taught, and of how much mathematics teachers are aware of what values they are encouraging (Abreu, Bishop and Pompeu, 1997). We believe that teachers’ knowledge needs to move from the implicit to the explicit.
We suggest three aims for future investigations that may move this discussion forward.
Aim 1: To investigate and document mathematics teachers’ understanding of their own intended and implemented values.
At present there is little knowledge about how aware teachers are of their own value positions, about how these affect their teaching, and about how their teaching thereby develops certain values in their students. Initial teacher education and in-service professional development need this kind of research basis in order to help change the situation.
We believe that the key to making development of values teaching possible is to investigate teachers’ understanding of their own values. Teachers’ intentions, and their actual teaching behaviours need to be studied in parallel. Values teaching happens both implicitly and explicitly and there is not necessarily a one-to-one correspondence between what is intended and what occurs. There is good evidence that teachers appear not to have consistent sets of attitudes to mathematics teaching events (Sosniak et al, 1991). Also, there is reason to believe that educational values are closely related to cultural norms.
Aim 2. To investigate the extent to which mathematics teachers can gain control over their own values teaching.
Research and development projects suffer from the same lack of knowledge as do teacher education institutions. There are important ideas which have been developed in the last few years which could have widespread benefits for mathematics learners around the world. In the areas of technology (see Noss and Hoyles, 1996), ethnomathematics (see Gerdes, 1995) and critical mathematics education (see Skovsmose, 1994) the role of mathematics teachers in teaching values is being critically examined.
But what is of particular interest about these developments, and their proponents, is that they imply a strong concern to change the values which they assume are currently being taught. This ignores the fact that not only do we not know what values are currently being taught but also that we have little idea of how potentially controllable such values teaching is by teachers. Therefore a question arises as to whether teachers have, or can gain, control over their values teaching. This would then theoretically enable them to teach other values besides those which they currently teach.
Aim 3. To increase the possibilities for more effective mathematics teaching through values education of teachers, and of teachers in training.
At present mathematics teaching focuses on the coverage of specific content and on the learning of particular skills and knowledge. Values are rarely considered, and indeed if one were to casually ask teachers about the values they are teaching in mathematics lessons they might well answer that they do not believe that they are teaching values. However it is our contention that improving and making values teaching more explicit in mathematics classrooms will make mathematics learning more effective.
Initial analyses reveal that there are three kinds of values which teachers intend to teach: the general educational, the mathematical, and the specifically mathematics educational. For example, when a teacher admonishes a child for cheating in an examination, the values of ‘honesty’ and ‘good behaviour’ derive from the general educational and socialising demands of society. Then when a teacher proposes and discusses a task such as the following: "Describe and compare three different proofs of the Pythagorean theorem" the mathematical values of ‘rationalism’ and ‘openness’ are being conveyed (see Bishop, 1988). However there are other values being transmitted which are specifically associated with the norms of the institutions within which mathematics education is formally conducted. For example, the values implied by the following instructions from the teacher: "Make sure you show all your working in your answers", "Don’t just rely on your calculator when doing calculations, try estimating, and then checking your answers", are about ‘examination-wiseness’ and ‘efficient mathematical behaviour’.
If these different values are considered important in good practice teaching then improving teachers’ knowledge of their values teaching will improve their mathematics teaching. Of particular interest are those values whose sources are either the culture of the mathematics itself, or the norms of the institutions of mathematics education. This leaves the general educational values, whose source is society. These may best be viewed from a cross cultural, international perspective.
Current written educational policies focus on improving achievement outcomes of students. Embedded in such statements is the intent of encouraging desirable values. For example, in The national statement on mathematics for Australian schools (Australian Education Council, 1991, pp.215-221), values such as ‘clarity’ are prominent in the non content oriented sections, but very few if any references to such values are made in the content sections, which are the sections that teachers tend to regard as the most important guides to what should be taught. This pattern of relegating statements on values to the periphery is repeated in most curriculum documents. In Victoria, the values such as ‘flexibility’ and ‘rationalism’, implied by highlighting the need for students to test different mathematical procedures, are simply not emphasized in those sections of the document which teachers turn to first when creating their own teaching situations (Board of Studies, 1995). The same pattern is also evident in documents from overseas. In the guide on assessment produced for teachers of mathematics in the USA, although ‘openness’ and ‘consistency’ make it to the glossary, very few other values are highlighted in these five pages (National Council of Teachers of Mathematics, 1995).
Clarke (1988, p.4) noted that alongside the assessment of various mathematical skills and content, there was also a need to assess "persistence, systematic working, efficient and effective organization, accuracy, conjecturing, ... creativity". Sadly in the rest of this valuable monograph, there is in comparison little emphasis on these values. We would contend that Clarke and others have got it right in that these qualities are absolutely fundamental to students’ achievement in mathematics. However we strongly posit that until they are given as much emphasis in the documents as the content and skills of mathematics, then not much will change. We further suggest that such a change is long overdue, but will not happen until our collective knowledge - that of policy makers, curriculum developers, systemic beureacrats, classroom teachers, and academics - about values teaching in mathematics improves markedly.
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