1. Delineations of Culture in Mathematics Education Research Tony Brown

2. How do we conceptualise the boundaries and frontiers of our work in mathematics education research? How do we broach these boundaries and frontiers?

3. Premises of many research questions: How children could learn better How the teacher could assist them How the researcher can alert the teacher to possible strategies But such perspectives rest on certain assumptions: That the child is deficient in relation to a particular ideological perspective (e.g. raising standards, reform math, problem solving, performance on SATs, motivation, etc) That the researcher focuses primarily on the local classroom interactive level (rather than on socio-economic factors, policy instruments, structural adjustment, etc) That the teacher could understand the research provided and could/ would change their practice (rather than this being done by the school board, local authority, government, etc).

4. Generalisations are key to mathematical learning. Generalisations are culturally defined objects that cannot be seen as things in themselves. They are necessarily a function of surrounding discursive environment and are imbued with the power relations of that environment

5. In Radford, Bardini and Sadbena’s model the acquisition of generalization is depicted as a developmental process progressively encountering mathematical phenomena through a range of perspectives and filters over time. In this hermeneutic process successive frames are introduced to eventually encapsulate an emerging sense of generalization through identifying features common to each perspective.

6. Radford tracks generalisation of 23 cercles; 11 en haut, 12 en bas, 203 cercles; 101 en haut, 102 en bas -different stories/ encapsulations are possible that affect what we see

7. How might we conceptualise the teacher’s role? Gallagher identifies four versions of hermeneutics each identified with a particular teacher student relationship Conservative Moderate Critical Radical

8. Conservative Hermeneutics Reproductive -The teacher has the right idea -The pupil ’ s task is to understand it in those terms -The goal is clear for the teacher, but, generally speaking, not for the students (Radford et al)

9. Moderate Hermeneutics (e.g. Gadamer) tradition is not fixed, but rather it is transformed through an educative process. The discussion between teacher and student produces a new conception “ideas and objects are conceptual forms of historically, social, and culturally embodied reflective, mediated activity” (Radford et al)

10. Critical Hermeneutics (e.g. Habermas) Education is seen as a transformative process concerned with the emancipation from the ideological structures that govern our actions. Teacher -agent of oppressive/collective structure or emancipator Student -task to resist oppressive imposition

11. Symbolic/systemic violence that affect shape of maths: student performance read against pass/fail categorization; compulsory education fixes choices; differential access to different social groups; insecure teachers reducing the power of student mathematical engagement, perhaps through overly didactic approaches and the closing down of exploration; international curriculum criteria being applied in specific local contexts; the resistance of adolescents to adult guidance; the external imposition of perceptual schema (e.g. privileging teacher constructs of social objects; etc).

12. Radical hermeneutics All my books … are if you like, little tool boxes. If people want to open them, or to use this sentence or that idea as a screwdriver or spanner to short-circuit, discredit or smash systems of power, including eventually those from which my books have emerged … so much the better. (Foucault) “ a more powerful voice for the students in a situation where differing views and forces collide awaiting nonetheless new forms of divergence and resistance ” (Radford, Miranda and Guzman)

13. Foucault and Habermas each resort to some sort of idealistic aspiration to makes things better: Habermas - better life is achieved through more rationality and living according to some agreed moral code. Foucault - humans strive to transform themselves to attain prized states, e.g. happiness, purity, wisdom, perfection, immortality, ie a better life through personally resolving individual need with external demand with “ as little domination as possible ”. BROWN, T & ENGLAND, J. Revisiting emancipatory teacher research: A psychoanalytic perspective, British Journal of Sociology of Education, February 2004 25(1), 67-80.

14. Additional step - Althusser- Lacan Full picture unavailable in any quest for ideals Gap - divided self - between what we do and what we think we do Partly forced to use available discourses Partly not served by them But partly unable to see self outside those frames

15. Policy apparatus specific to mathematics: For schools; National Curriculum, National Numeracy Strategy, Standardised Attainment Tests, Standardised training programme for teachers administering the Strategy, Government Inspections; For training colleges; National Curriculum for Initial Training, Numeracy Skills Tests for teachers, Government Inspections for training colleges.

16. Teacher ’ s perspective I appeared to be uncomfortable with the ideology proposed by the government. I wanted to strive to be a better teacher. I felt I had it within my power to work out the best routes to take. Now I was to be told which route to take and how to travel it. How would I accommodate the mismatch between my perceived reality and this proposed ideology? Would it entail a change in my ideological stance? Or could I work with a mismatch? BROWN, T. Desire and drive in researcher subjectivity: The broken mirror of Lacan, Qualitative Inquiry, 2008, 14(3), 402-423.

17. Teacher ’ s perspective One of my roles was Evaluation Coordinator in which I was regarded as very efficient. I had computer files full of graphs recording and predicting results in tests. In interviews with the inspector responsible for target setting satisfaction with the analysis of results was always forthcoming. Official reports of visits always left me with a feeling of satisfaction. I felt that I was doing a good job. Indeed the document contained a number of health warnings, such as “ it is important to recognize that data analyses provide few, if any, answers ” and “ despite all the fuss about targets and the use of data, it is important to remember that children do not grow taller by being measured ”. My cynicism was probably softened by these remarks. I actually enjoy playing with numbers and data, so after voicing the negative aspects to the course leader I completed the necessary work satisfactorily. (i.e intellectual critique but practical compliance)

18. Teacher ’ s perspective The thing with government policies is really whether you agree with them or not and you think they ’ re beneficial or not, you ’ ve got to adapt and change to go with them, so it ’ s just a case of experimenting with them, trying them out and then adapting them to suit you, so... I mean, you ’ ve got to use them, so if you can adapt it to suit you then it ’ s going to be beneficial (final year trainee) BROWN, T. & MCNAMARA, O. New teacher identity and regulative government, The discursive formation of mathematics teacher education. New York, Springer 2005

19. (I)n different historical circumstances it would undoubtedly have subversive effects; today however, in the era of cynicism, ideology can afford the secret of its functioning … without in the least affecting its efficiency (Zizek).

20. How do we enable the future?

21. Delineations of culture Frameworks for subjectivity How do we construct culture? How do we identify with culture? How much can we live outside dominant ideologies? Can we understand ourselves as outside them? How do we develop maths for new ways of life?

22. How might we resist reproduction? How might we distribute responsibilities for cultural renewal? Can we enable a frame beyond teacher’s conceptions? Can we enable access to truth with less interference from cultural/ideological layers?

23. How might we understand: The notion of Truth or mathematical truth The growth of the mathematical universality The aggregation of mathematical knowledges We need to be aware of the culturality/ specificity of the perspectives we assume and of the values they assume.

24. Issues for maths ed research Autonomous professionals or tell teachers what to do? -problems with supply of teachers. Turbulence of governmental policies e.g. NNS massive start up costs but abandoned after 5 years How does the profession protect itself from being controlled like this? Can research help? How do we avoid setting up unfulfilable ideals? Split: Research ideals /attainable goals How much is education about cultural renewal?

25. BROWN, T. Lacan, subjectivity and the task of mathematics education research, Educational Studies in Mathematics, (2008), 68: 227-245. BROWN, T. Signifying “learner”, “teacher” and “mathematics”: a response to a special issue, Educational Studies in Mathematics, (2008) 69: 249-263. PRESMEG, N.& RADFORD, L. On semiotics and subjectivity: aresponse to Tony Brown’s “signifying “learner”, “teacher” and “mathematics”: a response to a special issue”, Educational Studies in Mathematics, (2008) 69: 265-276.

26. On going discussions (e.g. Radford): How do we define supply of teachers, teacher education content? E.g. how much do we suppose teachers can intellectualise educational processes/read research etc? How much are maths generalisations set on teacher’s vision? (Vygotskian welcome to old culture or trigger to new?) Gadamer or Foucault or Lacan, - historical continuity, renewal of growth parameters or split? Choice allows research multiple conceptions of improvement and a broader analytical armoury.