Rethinking Cultural Research

1. Rethinking Cultural Research Kay Owens Charles Sturt University, Dubbo, Australia Glen Lean Ethnomathematics Centre, University of Goroka, PNG

2. Regeneratingthe forest

3. Younger researchers • Pockets of knowledge • Regenerated before they are lost • The mathematics within the cultural context • The researcher working with older generations

4. Content Analysis – Lean’s Counting System Analysis • The words and the connections between words e.g. 8=2x4, 9=2x4+1 or 8=2 before 10 • The analysis of the system e.g. 2, 5, 20 cycles; body-part tallies • Summaries of diverse data sets (quickly changing languages and student knowledge) • Similarities and differences between neighbouring or similar languages • Building on limited data sets • The mathematics within culture e.g. counting within exchange ceremonies • The intuitive knowledge e.g. counting beyond 6

5. Measurement • Mounds of sweet potato; • semi-perimeter lengths for sizes of gardens; • using lengths in building round or rectangular houses; • estimating sizes of saplings for bridges or houses; estimating amounts of palm or sapling for rope for a house

6. Length as a concept • In which activities is length used? • How systematically is it used? • What are the similarities and differences in units or ways of estimating length in different activities? • Do they use an instrument? A unit? A composite unit? • Do they use comparisons or ratios e.g. lengths of bamboo to water area of garden (Matang) • Bamboo bands

7. Content analysis A place to begin • attributes that are measured • activities, • instruments, • units, composite units, • degree of accuracy • processes • structures e.g. area arrays

8. Critical and Reflexive Research Cultural expectations of • the researched phenomena, • the researchers and • the research (Foucault, 1971)

9. Cultural expectations of measurement • Historical impact e.g. the tape measure on building • Status – importance of the fathoms of shell money, the number of mounds of equal size, the size of the house, the space and position of a house within a village or of sleeping space within a house • Embodiment – counting with the fingers or body parts, gestures • Visualisation skills developed by the physical and cultural environment (Bishop, 1988; Mitchelmore, 1984)

10. The participant supplying the importance of the researched phenomenon • How do you do that? (Fioriti & Gorgorió, 2001) • How do you know it is accurate enough? • How can the situated mathematics be used for the mental schema of the mathematics? • How important is the mathematics in power relationships and when? What else is important e.g. the distribution of the mounds of sweet potato

11. Gaining tacit knowledge • Tacit knowledge is hard to express in any language (Frade, 2005) • To what extent are the observations, the descriptions showing tacit knowledge – the importance of multiple data sources • What to pay attention to, what is significant, what are the power relationships? Members of the community need to express these.

12. Expectations of the researcher • Younger, educated member of the society • Cross-cultural researchers • Power of the knowledge owner • Relationship and description of the knowledge • Taken-as-shared knowledge among the experienced • Reciprocity for the research; a mine-field

13. Cultural understanding of research • Benefit of research – traditional researchers by oral records and sharing, valued • Benefit for education • Benefit for communities outside the immediate community • Limitations on opportunities to re-engage in traditional measurement activities • Earlier research with mixed results; respected or not respected; monetary privilege; clash of cultural values; • Change in attitudes to sharing traditional or village knowledge • Wary of who will benefit from research • Continuation of ownership, for money? • Importance of going back to the communities with the research knowledge

14. Cultural Approaches to Data • Obtaining data • Observation, survey, interview • Narrative research, story telling, centring in place-based research when writing. Writing as a research tool. • Interpreting data • Using different lenses to interpret data (Pamphilon, 1999); cultural and school based lenses • Fluidity and change impact on interpretation • Level of linguistics – morphemes, gestures, facial expressions, morpheme connections, metaphors • Tacit knowledge of experience • Implicit use in action, explicit use in explanation, and validation (Brousseau, 1984)

15. Preparing the researcher • Knowing when to ask and when not to ask within the community (Stake, 2005) • Asking within the community context, varying the way of asking and when to ask • Maintaining traditional knowledge in context • To be used not as an exhibit but compared with other systems • how to recognise and record tacit knowledge about concepts, procedures, attitudes and dispositions • care with filling the gaps

16. Plan • Start: Plan your entry into the community. • Hold preliminary discussions, observations. Record what you already know about measurement in the society. • Interview significant people in the community using the questionnaire as a basis. Record the words, metaphors and parts of objects that signify the object, gestures, looks, demonstrations and discussions between various members of the community

17. Reflect on this data in terms of your previously recorded data, your own understanding of school measurement mathematics. Where are there similarities or differences? • Share your understandings with your co-researcher. • Refine your findings and keep a record of your initial and later decisions.

18. Prepare questions for a second interview. Work out questions that will help you to clarify your preliminary analyses, check with the participants that you have a correct recording . Gain a sense of change in society over time. Time will act as a reflective device to establish validity and to recognise pressures on the measurement knowledge that has compromised it over time. • Do you know when not to ask questions? • Are you aware of the power and governance relationships present in the community during the research?

19. Deconstruct the situation by considering the power relationships between the participants and co-researchers. Record any issues that arise from these considerations. Consider the genealogy and governance of the mathematical activities as reported. Should you make a change to your summaries? • What metaphors are being used and what is different to school mathematics? • What areas do you consider need further investigation? • What was unexpected? • Have you used near and far lenses to look at the data? • Is the knowledge implicit and evident in action, made explicit and/or justified? • Centre yourself in the research situation and write about the place and your body in the place at the time. Select key words, connect them, draw them. What do you notice that is relevant about your research?

20. Have your summaries translated back into language and have them checked by the participants. • Consider your exit from the community. What will be future contact? • What benefits can you describe to the community? • Reflect on your findings with other researchers from neighbouring areas. • Consider any new findings that you have about mathematics.

21. Ethics • How to gain permission from the community, how to explain the value of the research, how the community decides on who should be interviewed (relationships are important here), how the community wants their permission recorded, how they want to be recognised for the information, where the gaining of permission may cause conflicts • Importance of data gathering techniques especially by recording • Importance of cross-checking data • Check on language translations • Cross-community and researcher checks; checks with school mathematics but • Room for the Unexpected and rich cultural diversity