1. Educating teachers about values in mathematics education: Alan J. Bishop Faculty of Education Monash University Melbourne Australia <alan.bishop@education.monash.edu.au>

2. Values and mathematics education Mathematics educators are increasingly being challenged about the goals to which mathematics education should aim: technology, societal demands, scientific development, economic growth…. To a large degree these goals are all about the values we should be inculcating in our students. Values are the deep affective qualities which education aims to foster through the teaching of mathematics. Values are the deep affective qualities which education aims to foster through the teaching of mathematics.

3. Values and mathematics education Values are not the same as beliefs, although the two constructs are related, and there is much research on beliefs in mathematics education but little on values. One way to relate them is to see values as ‘beliefs in action’, that is, one may hold several beliefs, but when one is faced with choices it is one’s values which determine which choice one accepts. Beliefs are the support or justification for one’s choices.

4. Values and mathematics education There is a widespread misunderstanding that mathematics is a value-free subject. Whereas it is relatively easy and common in the teaching of humanities, arts subjects and perhaps also the sciences to discuss the development of values, this is not the case at present in mathematics teaching.

5. Values and mathematics education Whenever teaching takes place, choices are made, which are based on, and therefore reveal, certain values. From a research perspective, there is only a limited understanding at present of what values are being transmitted, and of how effectively they are being transmitted. Perhaps this is because most values appear to be taught and learnt implicitly rather than explicitly in mathematics classrooms.

6. Values and mathematics education Therefore there are new research questions which need to be asked such as: What values are teachers of mathematics teaching? To what extent can mathematics teachers gain control over their own values teaching? What values are students learning from their teachers?

7. Values and mathematics education What values are implicitly and explicitly being transmitted or ‘shaped’ through curricula and textbooks? Is it possible to develop more effective mathematics teaching through the values education of teachers, and of student teachers? To what extent can teachers be helped to teach other values than those they currently teach?

8. Values in mathematics education are of three basic kinds Mathematical values: values which have developed as the subject has developed within the particular culture. General educational values: values associated with the norms of the particular society, and of the particular educational institution. Mathematics educational values: values embedded in the curriculum, textbooks, classroom practices, etc. as a result of the other sets of values.

9. White’s (1959) background research My research approach to these issues has initially been to focus on mathematical values, and I have found White’s (1959) three component analysis and terminology very helpful: Ideological values: ‘rationalism’ and ‘empiricism’ Sentimental values: ‘control’ and ‘progress’ Sociological values: ‘openness’ and ‘mystery’.

10. Valuing Rationalism means emphasising argument, reasoning, logical analysis, and explanations. It concerns theory, and hypothetical and abstract situations, and promotes universalist thinking.

11. Valuing Objectism means emphasising objectifying, concretising, and applying ideas in mathematics. emphasising objectifying, concretising, and applying ideas in mathematics. It favours analogical thinking, symbolising, and the presentation and use of data. It also promotes materialism and determinism.

12. Valuing Control means emphasising the power of mathematical knowledge through mastery of rules, facts, procedures and established criteria. It also promotes security in knowledge, and the ability to predict.

13. Valuing Progress means emphasising the ways that mathematical ideas grow and develop, through alternative theories, development of new methods and the questioning of existing ideas. emphasising the ways that mathematical ideas grow and develop, through alternative theories, development of new methods and the questioning of existing ideas. It also promotes the values of individual liberty and creativity.

14. Valuing Openness means emphasising the democratisation of knowledge, through demonstrations, proofs and individual explanations. emphasising the democratisation of knowledge, through demonstrations, proofs and individual explanations. Verification of hypotheses, clear articulation and critical thinking are also significant, as are a willingness to consider alternatives and the transparency of procedures and assumptions.

15. Valuing Mystery means emphasising the wonder, fascination, and mystique of mathematical ideas. It promotes thinking about the origins and nature of knowledge and of the creative process, as well as the abstractness and dehumanised nature of scientific and mathematical knowledge.

16. Findings from the VAMP research : www.education.monash.edu.au/research/ groups/smte/index.html www.education.monash.edu.au/research/ The teachers found it difficult to discuss values in relation to mathematics. The teachers had many goals in planning for mathematics lessons. The teachers hold their own values about mathematics and about mathematics education.

17. Findings from the VAMP research The teachers chose to make explicit certain mathematics or mathematics education values or they ‘showed’ them implicitly. It was easier for teachers to think about and recognise the values they were teaching, than to implement new values.

18. Teachers’ values and practices in mathematics and science This study involved both primary and secondary teachers who each taught mathematics and science. 13 primary and 17 secondary teachers volunteered to answer our questionnaires. The questionnaires involved ranking the six values, and also rating the teaching time spent on the values

19. Primary teachers’ values and practices in mathematics and science Concerning Ideology, they prefer Empiricism over Rationalism for both science and mathematics. Their reported practices also show this. For the Sentimental (attitudinal) dimension, Control is much less favoured than Progress for both subjects, but the practices are reversed.

20. Primary teachers’ values and practices in mathematics and science Another main difference is in the Sociological dimension where Openness and Mystery reverse their positions with the two subjects, the first being more favoured than the second in mathematics and the reverse in science. However this difference does not translate to the practices, with the science practices being ranked much more like the mathematics practices.

21. Secondary teachers’ values and practices in mathematics and science Concerning the Ideological dimension, they favour Rationalism for mathematics and Empiricism for science, disagreeing with their primary colleagues. However for the Sentimental dimension, the secondary teachers largely agree with their primary colleagues

22. Secondary teachers’ values and practices in mathematics and science For the Sociological dimension, they again agree with their primary colleagues favouring Openness for mathematics compared with Mystery, and reversing these for science. Indeed Mystery for science is ranked 2 and 4 by the secondary teachers and ranked 2 and 3 by the primary teachers, showing how significant they consider that aspect to be.

23. Some conclusions and implications. Rationalism, Empiricism and Control are strongly favoured in practice, but the other three values figure more prominently in the teachers’ preferences. However, we must remember that the data are from questionnaires and consist of teachers’ reported views of their preferences and their practices. We do not know the extent to which their rankings of these practice statements reflect their actual practices. But the data for science at the secondary level, where teachers emphasises other values than mathematics, indicates the usefulness of comparing subjects and their values emphases.

24. Implications Finally one can see that, if the data reported here are valid, the differences show that teachers’ values in the classroom are shaped to some extent by the values embedded in each subject, as perceived by them. This implies that changing teachers’ perceptions and understandings of the subject being taught may well change the values they can emphasise in class. Further if teachers wish to emphasise values other than those they currently emphasise, it is possible to learn strategies from their teaching of other subjects.