1

Research with multiple epistemic metaphors: Searching for wisdom in science and mathematics education

research

Bal Chandra LuitelBal Chandra LuitelKathmandu University, Nepal

Peter Charles TaylorPeter Charles Taylor Curtin University of Technology, Australia

Presented at the Annual Conference of Australasian Science Education Research Association (11th - 14th July 2007): Fremantle, Western Australia

2

3

This presentation includes

• My ongoing doctoral research • Illustrate use of Illustrate use of alternative logics alternative logics

– Metaphor– Dialectical logic – Poetic logic – Narrative logic– Vision logic– Non/dual logic

• Conclusion

4

My doctoral research Title: Culture, Worldview and Transformative Philosophy of Mathematics

Teacher Education in Nepal: A Cultural-Philosophical Inquiry

Inquiry Agendas

• In what ways are the Western Mathematical Worldview and Nepali Worldview similar and different in terms of their epistemologies and ontologies?

• In what ways can wisdom traditions of the East (e.g. Hinduism and Buddhism) contribute to the development of an alternative philosophy of mathematics teacher education in Nepal?

• How can mathematical knowledge for teacher education in Nepal be made holistic, ecologically balanced and discursive?

• What can a transformative philosophy of mathematics teacher education be for Nepal?

5

My doctoral research

Research Paradigms: Post-modern and Integral

Methodology: Arts-based auto/ethnography

Owing to the nature of my inquiry, I Owing to the nature of my inquiry, I cannot rely on only propositional, cannot rely on only propositional, deductive and rational-analytical deductive and rational-analytical logics of positivism. Therefore, I logics of positivism. Therefore, I need to cultivate need to cultivate alternative logicsalternative logics which help me to unpack various which help me to unpack various facets of the holistic (socio-cultural facets of the holistic (socio-cultural and spiritual) nature of mathematics and spiritual) nature of mathematics education, thereby helping me to be education, thereby helping me to be a a wisewise teacher educator. teacher educator.

6

MetaphorMetaphor Poetic logicPoetic logic

DialecticsDialectics

Non/dual logicNon/dual logicVision logic Vision logic

Narrative logicNarrative logic

Alternative logicsAlternative logics

7

MetaphorMetaphor

• According to Lakoff and Johnson (1980, p.5):According to Lakoff and Johnson (1980, p.5):

metaphor is understanding and experiencing one kind of metaphor is understanding and experiencing one kind of thing in terms of anotherthing in terms of another

• ExamplesExamples: : Life as journey, learning as constructing, Life as journey, learning as constructing, conversing as languagingconversing as languaging

• BenefitBenefit: A tool for exploring multiple meanings and : A tool for exploring multiple meanings and perspectivesperspectives

8

Metaphors• Epistemic metaphors help me to embrace multiple ways of Epistemic metaphors help me to embrace multiple ways of

knowing/inquiry: knowing/inquiry: knowing as storying, inquiry as currere, knowing as storying, inquiry as currere, knowing as reconceptualising self, knowing as generating knowing as reconceptualising self, knowing as generating wisdom, inquiry as talking to heart, knowing as creating, wisdom, inquiry as talking to heart, knowing as creating, knowing as being knowing as being (Miller, Karsten, Denton, Orr, & Kates, (Miller, Karsten, Denton, Orr, & Kates, 2005)2005)..

• I have used metaphor as a tool for exploring different I have used metaphor as a tool for exploring different images of mathematics that I have experienced as a images of mathematics that I have experienced as a student, teacher, teacher educator and researcher (Luitel & student, teacher, teacher educator and researcher (Luitel & Taylor, in prep a). Taylor, in prep a).

9

ExampleExample

““After leaving my brief career as a tutor in a teacher After leaving my brief career as a tutor in a teacher education college, I joined the University of Himalaya as education college, I joined the University of Himalaya as a mathematics teacher trainer. While working with a mathematics teacher trainer. While working with teachers of semi-rural schools, I continued to develop teachers of semi-rural schools, I continued to develop many (helpful) images of many (helpful) images of mathematics as storytellingmathematics as storytelling, , mathematics as cultural enactmentmathematics as cultural enactment and and mathematics as mathematics as languaginglanguaging.”.”

(Luitel & Taylor, in prep a)(Luitel & Taylor, in prep a)

10

Dialectical logic • Triad/process: Thesis, Antithesis and Synthesis Triad/process: Thesis, Antithesis and Synthesis • Opposites (A and ~A) co-exist within the same Opposites (A and ~A) co-exist within the same

phenomenon phenomenon • Conflict and contradiction is normal Conflict and contradiction is normal • Change is an internal process  based on internal Change is an internal process  based on internal

contradictions /frictioncontradictions /friction• Change is the unity of opposites   Change is the unity of opposites   • Dialectical thinking is about looking for dynamic Dialectical thinking is about looking for dynamic

alternatives and widening one’s horizonalternatives and widening one’s horizon• Promotes the notion of interdependence of opposites Promotes the notion of interdependence of opposites

(Wong, 2006)(Wong, 2006)

11

Example

““We wish to make clear that our intention is We wish to make clear that our intention is notnot to reject to reject conventional images of mathematics. Rather we subscribe conventional images of mathematics. Rather we subscribe to the integral perspective of to the integral perspective of de/contextualisationde/contextualisation which which represents a dialectical relationship between alternative represents a dialectical relationship between alternative and conventional images of mathematics. At times it might and conventional images of mathematics. At times it might be ok to teach ‘as though’ (i.e., metaphorically) be ok to teach ‘as though’ (i.e., metaphorically) mathematics is universalist in nature by, for example, mathematics is universalist in nature by, for example, focussing exclusively on manipulation of abstract symbolic focussing exclusively on manipulation of abstract symbolic expressions, whereas at other times and in other contexts expressions, whereas at other times and in other contexts it might be better for an alternative pedagogical focus.”it might be better for an alternative pedagogical focus.”

(Luitel & Taylor, 2007, p.13)(Luitel & Taylor, 2007, p.13)

12

Poetic logic • helps express layered meanings

जहाँ�� पु�ग्दैनन रवी� त्यहाँ�� पु�ग्छन कवी�जहाँ�� पु�ग्दैनन रवी� त्यहाँ�� पु�ग्छन कवी� !!

““a poet is able to reach the unreachable!”a poet is able to reach the unreachable!” • has the power to de-familiarize one’s experiences• helps cultivate aesthetic aspects of inner self(s) and

exterior realities • can help magnify my understanding of issues under

study (Cahnmann, 2003; Faulkner, 2007; Sri Aurobindo, 1972)

13

Example

He Never Quoted His FatherHe Never Quoted His Father

He produced a lecture. HeHe produced a lecture. Hetold us mathematics is difficult. Hetold us mathematics is difficult. He

positioned himself up there. Hepositioned himself up there. Helooked at us down here. Helooked at us down here. He

symbolised us as subjects. Hesymbolised us as subjects. Hequoted Western mathematicians. Hequoted Western mathematicians. He

never quoted his parents. Henever quoted his parents. He… …

14

Narrative logic Narrative logic promotes:

• knowing as storying -- knowing as storying -- narrative truthnarrative truth• diachronic (emergent) representational style diachronic (emergent) representational style • fragmented knowledgefragmented knowledge• performance of possibilitiesperformance of possibilities • personal voice and sustainability (reflectivity and personal voice and sustainability (reflectivity and

reflexivity)reflexivity)• crystallization – ‘moving from plane geometry to light crystallization – ‘moving from plane geometry to light

theory’– in my textual creation theory’– in my textual creation

(Ellis & Bochner, 2000; Richardson & St. Pierre, (Ellis & Bochner, 2000; Richardson & St. Pierre, 2005)2005)

15

Example I

“I crossed many minor and major borders while conducting this research. In the beginning, it was a shift from knowing as probing to knowing as storying and reflecting. Even after subscribing to such an epistemological standpoint, I tried initially to use a traditional epistemic structure for my research. However, as I moved towards the process of writing the research proposal and preliminary chapters, I realised that the traditional five-chapter structure does not help promote the notion of research as an emergent and evolving enterprise.”

(Luitel, 2003, p.117)

16

Example II

“Working with senior professors, who seemed to regard designing a mathematics teacher education program as though a mixmaking of ‘pure mathematics’ courses and education courses, put me in a dilemma because of not being able to fully translate my vision of culturally contextualised mathematics education. I could see that the déjà vu of mathematics-is-a-foreign-subject was occurring all over again as my colleagues put renewed emphasis on the same heartless and soulless mathematics that I aimed to refurbish. What does it mean to transform mathematics education in Nepal? Does it mean to promote unquestioningly the image of mathematics as a foreign subject? Does this mean to neglect the diverse cultural-rural realities of Nepal by its mathematics education programs?”

(Luitel & Taylor, in prep a)

17

Vision logic Vision logic Vision logic enables me to:

– develop a perspectival view of the field of mathematics education

– embrace and apply a futuristic approach to my textual creation – cultivate an holistic view of (mathematics) education – look for possibilities– cultivate critical, reflexive, reflective and imaginative thinking– operate through the principles of freedom, creativity and

uniqueness.

(Aurobindo, 1998; Chaudhuri, 1972; Wilber, 1996)

18

Example

“Primarily, we envision that teachers working within the context of culturally contextualised mathematics education endeavour to generate meanings of alternative natures of mathematics. Subscribing to the metaphor of teacher as awakened facilitator, they will recognise students’ cultural and individual differences, promote inclusive participation of students, create a caring and collaborative learning environment, thereby promoting meaningful mathematical acculturation by which students often cross the two-way borders of local and formal mathematics. Engaged in exploring connections between formal and informal mathematics, we would see Nepali students a) co-generating mathematics from their cultural contexts; b) linking their cultural experiences with formal mathematics; c) developing local classifications of mathematical ideas, based on their uses in local cultural contexts; and d) solving real world problems by using different forms of mathematics.”

(Luitel & Taylor, in prep b)

19

Logic of Non/dualismThe logic of non/dualism helps me to:

– explore an inclusive view of my identity – critique and transcend unhelpful dichotomies promoted by the

modernist and dualist worldview– envision an integral (and de/contextualized) view of mathematics

education through multiple and often contradictory natures of mathematics.

Nagarjuna: Nagarjuna: Without relation to Without relation to goodgood there is no there is no badbad, in dependence , in dependence on which we form the idea of on which we form the idea of goodgood. Therefore . Therefore goodgood is is unintelligible. There is no unintelligible. There is no goodgood unrelated to unrelated to bad bad; yet we form our ; yet we form our idea on idea on badbad in dependence on it. There is therefore no in dependence on it. There is therefore no badbad..

(Cited in Loy, 1997) (Cited in Loy, 1997)

20

By the help of ‘Self’ I explain ‘Other’

Fusing Self and OtherI generate a vision of Self-Other.

Some people say: Self-Other is not ‘Self’

Because it has Other within it.Self-Other is not ‘Other’

Because it has Self within it.

But I say:Self-Other is also Self

Because it is inclusive of Other Self-Other is also Other

Because it is inclusive of Self.

I start with two: Self and Other.They become three: Self, Self-Other and Other.Indeed, they are many as facets of the Being

which I aim to attain!Does this mean that I will be a wise teacher educator?

conclusion… just for now

21

List of references Cahnmann, M. (2003). The craft, practice, and possibility of poetry in educational research. Educational Researcher,

32(3), 29-36.Ellis, C., & Bochner, A. (2000). Autoethnography, personal narrative, reflexivity: Researcher as subject. In N. Denzin

& Y. Lincoln (Eds.), The handbook of qualitative research (2nd ed., pp. 733-768). Thousand Oaks, CA: Sage.Faulkner, S. L. (2007). Concern with craft: Using Ars Poetica as criteria for reading research poetry. Qualitative

Inquiry, 13(2), 218-234.Loy, D. (1997). Nonduality : A study in comparative philosophy. Atlantic Highlands, N.J.: Humanities Press.Luitel, B. C. (2003). Narrative explorations of Nepali mathematics curriculum landscapes: An epic journey

Unpublished Master's Project, Curtin University of Technology, Perth: Downloadable from http://pctaylor.com Luitel, B. C., & Taylor, P. (2007/in press). The shanai, the pseudosphere and other imaginings: Envisioning culturally

contextualised mathematics education Cultural Studies of Science Education 2(2).Luitel, B. C., & Taylor, P. (in preparation (a)). Blending inside-out and outside-in: An holistic approach to sustainable

mathematics education Luitel, B. C., & Taylor, P. (in preparation (b)). Defrosting the ideology of pure mathematics: Social justice and

contextualisation imperatives in mathematics education. Philosophy of Mathematics Education.Miller, J. P., Karsten, S., Denton, D., Orr, D., & Kates, I. C. (Eds.). (2005). Holistic learning and spirituality in

education : breaking new ground. Albany, NY: State University of New York Press.Richardson, L., & St Pierre, E. (2005). Writing: a method of inquiry. In N. Denzin & Y. Lincoln (Eds.), The Sage

handbook of qualitative research (3rd ed., pp. 959-578). Thousand Oaks: Sage.Sri Aurobindo. (1972). Collected poems. Twin Lakes, WI: Lotus Press. Available Online

http://www.poetseers.org/the_poetseers/sri_aurobindo/sritwo/ascent.Sri Aurobindo. (1998). Supramental manifestation and other writings (2nd ed.). Twin Lakes, WI: Lotus Press Wilber, K. (1996). A brief history of everything (1st ed.). Boston: Shambhala.Wong, W.-c. (2006). Understanding dialectical thinking from a cultural-historical perspective. Philosophical

Psychology, 19(2), 239-260.