Let’s un/do1 with Husserl, Objectivism and Ethnomethodology! An invitation to Professor
Wolff-Michael Roth
Bal Chandra Luitel
Dear Professor Roth! It is my pleasure to offer some comments, although they may be very
insignificant, on your paper, Researching living/lived work of mathematics. Given the theme
of the conference, I have chosen to focus my writing on researching lived mathematical
experiences because your idea of ‘work’ appears to be somewhat limiting in relation to the
scope and the purpose of the DIME Research Group. In the process of writing this
commentary, I have chosen hermeneutical phenomenology, functions and forms of
language, views about the nature of mathematics and radical constructivism as my
theoretical referents. To me, hermeneutical phenomenology (Manen, 1990) offers the much
needed basis for a pedagogically oriented inquiry, taking various interpretive traditions at
the core of examining (i.e., generating, describing, interpreting and fostering) lived
mathematical experiences. Next, I am also using the notions that language cannot be
neutral; that it is constitutive of reality; and that it has significant roles in capturing and
analysing lived mathematical experiences in context (Hammersley, 1997). My next
referent, views about the nature of mathematics, enables me to foreground recent
developments in the field of the philosophy of mathematics in order to understand those
experiences (Ernest, 1994; Glas, 2006; Hersh, 2006; Luitel, 2009). You may agree with me
that (radical) constructivism as a post-epistemological philosophy, research paradigm and
learning theory can offer some empowering visions for capturing lived mathematical
experiences, especially elaborating research methodologies that are needed to capture,
legitimate and foster lived mathematical experiences (Glasersfeld & Ackermann, 2011).
Given these theoretical perspectives, I start this letter with key features of lived
mathematical experiences, thereby questioning dualisms that you promoted in the paper.
Toward the end of this letter, I would like to discuss possible methodological framework
for researching lived mathematical experiences.
Conceiving key features of lived mathematical experiences
Dear Professor Roth! Amongst plethora of meanings and definitions of lived experience
from within and without the traditions of phenomenology, I have come up with four key
definitional features – immediacy, intentionality, self-constitutive, hermeneutical -- that can
help us develop the conceptual and operational basis for researching lived mathematical
experiences. I have used the notion of immediacy so as to refer to the shortest possible
proximity of life-world and conscious experiences. Indeed, a more immediate engagement
(Tieszen, 2005) with mathematical phenomena (and objects) is the precondition for
generating lived mathematical experiences. The notion of intentionality refers to the
directedness of conscious experience towards things or objects in the world. However, the
Husserlian position of intentionality as directing our experience through the a priori
concepts, ideas, and thoughts has been questioned by other interpretive traditions and
non-Husserlian versions of phenomenology (Hopp, 2008). Perhaps, you may agree with
1
Throughout this paper, the symbolism of slash (‘/’) depicts dialectical relationships between sometime opposing
concepts and ideas.
me that another key feature of phenomenology is the primacy of self-constitutive
experience, although the notion of self is played out in a number of ways, ranging from self
as the expression authentic to self as a basis for unique subjectivity (Kupers, 2008). More
so, the hermeneutical feature is concerned with interpretive structures of experience in
which time, space and other meaning-producing mechanisms play significant roles in
understanding lived mathematical experiences (Dilthey & Jameson, 1972; Slattery, Krasny,
& O’Malley, 2007).
Can the Husserlian notion of what counts as mathematical experience be sufficient for us to
develop the frame for researching lived mathematical experiences for our times? Although
Husserl challenged psychologism and naturalism of his time, his articulation of the nature
of lived mathematical experiences is very close to Platonism (mathematics of eidos) and
Formalism (mathematics of materials) (Husserl, 1983). The idea of eidetic reduction as the
technology of identifying required (probably pre-existing) mental categories is less likely to
promote an emergent feature of lived mathematical experiences. Well, mathematics can be
conceived as a discipline constitutive of mental objects. Nevertheless, the notion of ‘mind’
cannot be taken for granted as an entity residing on the top of our body; rather we need a
more convincing and dynamic view of mind as an ‘organic’ network connected and
distributed throughout the body and beyond if we are to promote an ecological view of
mind, thus, of lived mathematical experiences. Here, the notion of ecology (see EsbjörnHargens, 2009) is to recognise complexities of and complementarities between different
layers and levels of lived experiences (for example: temporal, spatial, self-awareness, noself awareness) embedded in our streams of consciousness (Larkin, Eatough, & Osborn,
2011).
Questioning the dualism
Dear Professor Roth! As I go through your paper and phenomenological traditions available
to us, I have encountered of a number of contradictions and paradoxes in conceptualising
the nature and feature of lived mathematical experiences (see Dowling, 2007). These
contradictions are expressed, for example, via dualisms of subjectivity and objectivity,
beings and things, constructed and received, activity and passivity, and perception and
conception. In my view, if we are to capture authentic lived mathematical experiences, a
dialectical perspective is more appropriate than subscribing to a one-sided view of
anything including lived mathematical experiences (Wong, 2006). Perhaps, lived
mathematical experiences constitute both subjective and objective dimensions of
mathematics or, in some instances, these two categories may not represent some
dimensions of such experiences; there will not be categories such as objective and
subjective (Kauffman, 2007). You may argue here that the idea of objectivity and
subjectivity are helpful categories because they enable us to come up with the much
needed objective and pure mathematical essences. But, I am always suspicious of the
Platonist-Formalist grand design of projecting the essence of mathematics exclusively as a
body of pure (and objective) knowledge because what is pure comes from impure and vice
versa (Luitel & Taylor, 2008). Perhaps, you are aware of Hersh’s (1997) front-back analogy
on the nature of mathematical work in which the front analogy portrays the finished
mathematical product seemingly clean, objective and unproblematic, whereas the back
analogy hints us at the messiness of mathematics-in-making. What do you mean by lived
mathematical experiences — reified and a priori mathematical concepts or more emergent
and dynamic mathematical constructs?
Arriving at this stage, I argue that lived mathematical experiences are constitutive of
sometime opposing and paradoxical categories embedded in our streams of consciousness.
Following this we need to open to the views (a) that partly mathematical knowledge is
received or given and partly it is constructed; (b) that the given makes better sense through
the made and vice versa and, (c) that which is ‘mathematical’ arises from ambiguous and
complex relationships with which is ‘not mathematical’ (Glas, 2006; Kolmogorov, 2006;
Lerman, 1990; Rotman, 2006). Thus, the exclusive promotion of transcendental nature of
mind is likely to neglect the experiential dimensions of mathematics in the particular web
of time and space. More so, I am critical of the proposition of collapsing mathematical
objects (things) into the stream of consciousness (being) because such collapsing does not
account for complex relationships between un/known variables, such as context, person,
culture, language, so on and so forth. I guess this collapsing can often take us very close to
an absolutist view that mathematical objects have no existence in our everyday worlds,
rather they are part of the network of abstract eidos (Ferrarin, 2008). Consequently, I am
prepared to argue here that a transcendental phenomenological perspective alone may not
be sufficient for explaining the complexity enshrined in the nature of lived mathematical
experiences.
Un/clearing the myth of objectivity
Dear Professor Roth! You have mentioned in your paper that you are referring to an
objective construct like geometry whilst illustrating the nature of lived mathematical work
(sic). I am not quite sure what you mean by ‘objectivity’ in your deliberations. As you refer
to an example of eye movement as the sole basis for determining objectivity, does it not
give an impression that there is only one form of objective geometry that is Euclidean
geometry which can be seen by our two eyes? I beg to disagree on this because your
proposition of privileging our two eyes as a basis for generating lived experiences of
something like geometrical objects, such as triangle, line and quadrilateral is an utterly
futile suggestion. If this was the case, either Riemann and Lobachevski would have different
types of eyes or their non-Euclidean geometries would not have been conceived through
lived mathematical experiences.
Following Douglas (2004) and Rorty (Bagni, 2008) I can say that there are at least five
distinct meanings of objectivity (and objectivism), such as objectivity as consensus,
intelligibility, persuasiveness, reasonableness, fixity and unchanging universality. However,
many of us seem to have been guided by the conventional view of objectivity as fixity and
unchanging universality. Likewise, in the philosophical traditions of mathematics
objectivism is often used to deliver an unchanging, infallible, positivistic (cf. hermeneutical)
view of the nature of mathematics. Cannot an exclusive emphasis on the conventional view
of objectivity help us grasp multiple dimensions of lived mathematical experiences? How
can we bracket ourselves as phenomenological (sic) researchers if we continue to
un/wittingly promote a narrow view of geometry (and mathematics) as exclusively
objective discipline, where objectivity is meant to represent its conventional
interpretation? More so, if we aim to follow the conventional view of objectivity as the
quality standard of our research, we will be excluding the large sections of
phenomenological and other interpretive traditions because an exclusively conventional
objectivist view is not helpful for us to capture personalised, self-constitutive and
contextual nature of lived mathematical experiences (Kriegel, 2009). Of course, there are
several meanings of self; some regard it as local and personal, whilst others tend to regard
it as collective and global (Rao, 2002).
Arguing for an alternative-holistic epistemic space
I argue here that the methodology of capturing lived mathematical experiences needs to be
consistent with the historical evolution and contemporary applications of different
mathematical concepts developed from within the civilisations of the East, West, North and
South. Be it the development of vector algebra or rubber sheet geometry or quadratic
equations, the primacy of lived experiences of the inventors cannot (should not) be ruled
out. How can we ensure an emergence of newer mathematical constructs whilst
researching lived mathematical experiences? Is our research on lived mathematical
experiences to merely confirm pre-existing mathematical concepts, structures and objects?
Whilst raising these questions, I am hinting at the insufficiency of ethnomethodological
approach in this regard because it does not enable researchers to study the making of new
categories; instead it looks at how pre-existing categories come into play in different
contexts (Garfinkel, 1986). Perhaps, it is okay to use ethnomethodology to study lived
mathematical experiences from the vantage point of confirming what is already t/here.
Nevertheless, this method seems to be less useful for generating ‘ever new’ mathematical
concepts, ideas, objects and structures embedded in the lived reality of students,
mathematics teachers and mathematicians.
You may be thinking right now that I am merely criticising your ideas without offering any
alternatives. As I said in the opening paragraph of this letter, I use a number of
philosophical and epistemic traditions so as to construct a space possibly appropriate to
investigate the complex, ambiguous and multidimensional nature of lived mathematical
experiences. Owing to such a nature of lived mathematical experiences, I envisage the need
to develop a multi-paradigmatic design space via a number of research paradigms (Taylor,
Settelmaier, & Luitel, in press). Thus far, I have used a blend of interpretive, postmodern
and critical research paradigms so as to (a) describe lived mathematical experiences that
constitutes the self or selves of the research and research participants, (b) represent
several dimensions (mythic, rational, intelligible, ineffable) of lived mathematical
experiences via multiple genres and logics, (c) foster such lived mathematical experiences
to envision empowering pedagogies (Luitel et al., 2009; Luitel & Taylor, 2007). This does
not mean that there is no role of the paradigm of positivism, rather it will be less useful for
capturing lived experiences of anything including of mathematics.
Conceiving viable quality standards
Aligning with the notion of viability (Steffe, 2010), I would like to talk about possible
alternatives to conventional research standards of validity, reliability and objectivity. The
notion of viability is about conceiving and developing frameworks and criteria by which to
judge truth claims generated via different paradigmatic standpoints. Whereas the
conventional standards are aligned with a singular view of lived reality, the idea of viability,
thanks to Ernst von Glasersfeld, offers standards which enable us to work with multiple
epistemic (paradigmatic) dimensions of lived mathematical experiences. Whilst taking
different paradigmatic stances, we can think of multiple and complementary standards
with which to judge our research processes and products. For example, van Manen’s idea of
pedagogical thoughtfulness is useful to judge whether or not our accounts evoke readers
with their pedagogical sensibilities. Similarly the quality standard of illuminating is helpful
for us to judge whether the account of lived mathematical experiences is constitutive of
subtleties, vividness and complexities. Similarly, the quality standard of incisiveness is
about establishing the clarity of focus on the issue of study (Barone, 2006, 2007; Barone &
Eisner, 2006). In Van Manen’s term, a set of four key quality standards are useful for any
research on lived experiences, which are orientation (How clearly oriented is the study?),
richness (how experientially rich is the study?), strength (how strong is the presence of
self?), and depth (how deep (cf. sallow) is the engagement in the research issue?).
Describing alternative genres and thinking
Dear Professor Roth! You talked about the impossibility of accounting for lived
mathematical experiences in their state of purity. You argued that once we document our
experiences they become ideated; this is what Husserl said in relation to the transcendental
dimension of our experience (Gurwitsch, 2010). Nevertheless, I am interested in the
hyphenated relationship between the transcendental and local so as to capture a holistic
form and content of lived mathematical experiences. If we are to rely on a positivistic, nonporosic and reductionist language and hypothetico-deductive logic, it will be almost
difficult to represent such a hyphenated dimension of lived mathematical experiences. On
contrary, I have envisaged different forms of genres and logics which can be applied to
account for multiple dimensions of lived mathematical experiences. As per van Manen, “...
researching, thinking and writing are seen to be closely related and practically inseparable
pedagogical and research activities” (Van Manen, 1990, p.4, my emphasis). More so, during
the last three decades many social and educational researchers discussed the tension
between text and context under the auspices of triple crisis (Denzin & Lincoln, 2011),
thereby offering insights into and examples for newer forms of expressions and logics for
capturing lived mathematical experiences. Associated with these are recently developed
expressions (or genres) and thinking which can radically shift our conventional ways of
researching lived mathematical experiences to a more holistic space.
I would like to propose here that we can employ storied, narrative-reflective, poetic and
non-textual forms of expressions so as to capture several dimensions of human experiences
(Kearney, 1998). A storied genre can bring forth a unified stream of consciousness via the
temporal succession of events embedded in our life world. A story brings uniqueness and
commonality (see Davies, 2008) that make our lived mathematical experiences
illuminating, meaningful, incisive and contiguous. Perhaps, you are aware of the fact that
mathematics itself is a kind of story embedded in our personal-collective consciousness.
You may argue here that the Husserlian idea of objective science does not fit well with
portraying lived mathematical experiences via storied genres. For me, the Husserlian
description of pure consciousness is yet another form of story, perhaps a realist one
waiting to be re-storied in a more illuminating form.
Another way of expressing our lived mathematical experiences is narrative-reflective
genre. By playing out the fusion of context, time and people through personalised
narratives, we can engage as directly as possible with the stream of consciousness
(Cumming, 2007). Our narratives of lived mathematical experiences become incisive if we
are to bring reflective (and pre-reflective) qualities to our expressions. Nevertheless, the
categories of ‘reflective’ and ‘pre-reflective’ are not so distinct from each other; the
existence of pre-reflective mode of experiencing is inferred via the causal (co-)existence of
reflective mode of experiencing (Legrand, 2007). It can be through a blend of narrative and
reflective genres that lived mathematical experiences produce their contextual-logical and
pedagogical essences. However, these essences are not entirely permanent, rather they are
constitutive of both transitory and enduring.
Probably, you may agree with me that poetic genres are useful for capturing those lived
mathematical experiences which are difficult to capture via a normal academic language. In
the Eastern Wisdom Traditions the poetic language is considered to reach further than the
sunrays, thereby helping us to feel and see otherwise unfelt and unseen (Christie, 1979).
Don’t you agree with me that not all experiences are expressible to us and to others? How
can we communicate ineffable experiences with us and others? For me, an ineffable
dimension of our experiences can be expressed well via the poetic genre, the genre for (or
of) alchemist, the natural language of homo sapiens and the language for lived experiences
(Lamarque, 2009). I argue here that our lived experiences are largely poetic because our
stream of consciousness is constitutive of rhythm, meter and stanzas. I don’t mean to say
that the stream is free from anarchy and confusion; rather anarchy-order and confusionclarity are the key attributes of our consciousness and thus are of poetic nature.
Elsewhere (e.g., Taylor, et al., in press), we have mentioned that non-linguistic genres (such
as images, creative designs) can also be helpful for bringing forth our lived mathematical
experiences. The inscriptions of verbal expressions are often mediated via the conventions
of the normal academic texts. You may be well aware that the standard academic language
can be a constraint for experiencing and representing something (Cooper, 2010). And, this
does not mean that our lived experiences are and should be free from constraints, rather it
is better to develop mechanisms to unpack those constraints so that we can capture
subtleties and ambiguities associated with lived mathematical experiences. More so,
standard linguistic expressions alone may not be sufficiently illuminative for us to express
otherwise unexpressed dimensions of anything including lived mathematical experiences.
In this regard, images, imageries and visual metaphors can help us express incisively our
motives, deeply-rooted structures of our thinking and conditions in which we experience in
a particular way (Butler-Kisber, 2008).
Arriving at this stage, I would like to invite you to share my views about the logic (i.e.
thinking) of inquiry, which facilitates our research on lived mathematical experiences. In
my mind, the hypothetico-deductive logic of objective sciences (sic) is not sufficient for
capturing multiple dimensions of lived mathematical experiences. This does not mean that
it will be totally discarded, rather it will be employed as one out of many. Alternative logics,
such as metaphorical, dialectical, poetic and narrative are recently available thinking tools
at our disposal (Luitel, et al., 2009). More so, metaphorical thinking helps us go beyond
literalism, thereby offering possible ways to analyse lived mathematical experiences from a
number of possibilities. Metaphorical logic can enable us to engage in developing a creative
analysis of lived mathematical experiences, using elastic correspondence between
conflicting schemas, in order to capture the complexity enshrined in those experiences
(Lakoff & Johnson, 1980, 1999; Lakoff & Nunez, 2000). Dialectical thinking helps us to
understand the very nature of dynamism generated through sometimes opposing
categories embedded in our lived mathematical experiences (Basseches, 2005).
You may agree with me that poetic logic has been considered to be fundamental to human
species because of their ability to produce meanings. Drawing on the ideas of MerleauPonty and Vico, I argue that our meaning structures are the poetics of texts and contexts,
inscription and gestures, symbols and objects, and so forth (Collins, 2010; Marshall, 2011).
Perhaps, you agree with me that poetic logic helps us realise otherwise neglected (and
possibly unexplored) relationships between sometime opposing, contradictory,
paradoxical and mutually co-existing categories. Next, narrative logic promotes mythoscentric thinking, an approach to accounting for cultural-contextual nature of lived
mathematical experiences (Leonard & Willis, 2008). Similarly, narrative logic is helpful for
promoting post/reductionist thinking that transcends the hegemony of reductionism by
integrating place, people, action and time in generating lived mathematical experiences.
Importantly, narrative logic offers a much needed diachronic vision, a means for conceiving
lived mathematical experiences in terms of the chronological evolution of events (Luitel,
2007).
Concluding
Dear Professor Roth! The views expressed in this commentary come from what I believe I
have known thus far. I admit here that the dimension of my un-knowing is far bigger than
what I have claimed to have known. However, due to the very nature of the ego-prone
Samsara (i.e., the material world, according to Buddhist and Vedic philosophies (Panikkar,
1977)) we often try to work through our little projects, ideas and perspectives projecting
our differences and commonalities in relation to that of others. Our commonalities unite us
as humans and our differences demonstrate our unique ways of approaching to issues and
ideas we encounter. More often than not, we make better sense of differences on the basis
of commonalities and vice versa; commonalities and differences co-arise dependently just
as the concepts of day and night do! Given this inseparability of sometimes opposing
constructs, the moral of this commentary is to explore more holistic and organismic
approaches to research on lived mathematical experiences.
Researching lived mathematical experiences is no more a sacred business of
transcendental phenomenology and of functionalism’s second order derivation,
ethnomethodology. These traditions need to be valued as historical necessity rather than
‘all time true’ standpoints. As it goes well with post-epistemology, we need an eclectic
design space so as to capture most possible (if not all) dimensions of lived mathematical
experiences. Such an eclectic design is constructed via a meaningful mixing of research
paradigms regulated by a host of quality standards, thereby employing recently developed
genres and logics so as to capture possible dimensions of lived mathematical experiences.
Let’s join together in this process of conceiving and translating an eclectic epistemology
into the act of our research, shall we?
Best wishes from
Bal Chandra -;)
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