Hermeneutics can be understood as the manifestation and restoration of meaning that an individual makes sense of in a personal way. Conversely, it can be understood as demystification, as a reduction of illusion. Are these two perspectives mutually exclusive? It seems that the first is indicative of an encoding process; something only exists if it is socially constructed, whereas the second is a decoding process, something already exists, through varying reasons (e.g., they may have been socially constructed) but the understanding is the unraveling of the layers. Discourse allows the learner to decode other (including expert) viewpoints that reveal their understanding (i.e., enculturation). It seems there is some common ground between these two perspectives. If we argue that everything is individually constructed, what is it that brought the individual to the point of readiness? It is the vast prelude of experiences and commonalities of understanding that have been previously negotiated; that is, the enculturation of the individual into those aspects that influenced the individual’s perceptions.
Meanwhile, Brown has sought to soften the individual/social divide with a phenomenological formulation that has an emphasis “on the individual’s experience of grappling with social notation within his or her physical or social situation” (Brown, 1996, p. 118). This is consistent with Vygotsky’s view of learning as the internalisation of social processes, but also sanctions Piaget’s emphasis on the individual and his notion of equilibration, if we consider the “grappling with social notation” as part of attending to an action, problem or interaction. Piaget uses assimilation and accommodation as vehicles for how existing schema or pre-conceptions evolve. A moderate hermeneutic perspective would see social processes and interpretation as inextricably immersed in those practices.
Hermeneutics, like education, is complex and one might argue that attempts to locate them both in specific philosophical positions only detracts from essential ambiguity. “In every case interpretation involves something that is less than absolute; it is always something imperfect and incomplete” (Gallagher, 1992, p. 348). He contends that interpretation is always a balance of constraint (with tradition) and transformation (of tradition). This balance is more the process of balancing rather than reaching a point of absolute balance. There is a play between familiar and unfamiliar horizons. The notion of the hermeneutic circle allows for Piaget’s ongoing process of multiple disequilibria and equilibration (Piaget, 1985) while remaining consistent with Vygotsky’s view of a long series of developmental events transforming interpersonal into intrapersonal processes, on the way to becoming individual knowing. There is a suggestion of an ongoing cyclical process oscillating between existing perspectives and new events.
Individual interpretation is also implicit in these perspectives; interpretation that is filtered by existing frames or discourses. As Brown so succinctly contends: “The social world is accommodated by focusing on the perspective the individual has of this and the possibilities open to them within the world they see” (Brown, 2001, p. 251). The place of dialogue has likewise been emphasised throughout this discussion, and the moderate hermeneutic perspective enables us to situate this within both theoretical positions. The role of phenomena, possibly evoking tension or perturbation, is also realised when we reconcile the approaches
through a moderate hermeneutic perspective, either directly as per Piaget, or through the stimulation of dialogue. The relevance of this, and the emphasis on environmental factors, likewise highlights the position of the teacher, and pedagogical media in the facilitation of learning.
Understanding emerges from both individual and collective interpretations of mathematical phenomena. It develops through social activity and dialogue, with all the historical, political, and cultural influences that implies. The mathematical activity is inseparable from the pedagogical device as it were, derived as it is from a particular understanding of social organisation, and hence the mathematical ideas developed will inevitably be a function of this device. Such pedagogical devices should be regarded as worthy objects of mathematical learning insofar as school mathematical learning is largely carried out in support of the student’s later engagement in mathematically-oriented social activity (Brown, 2001). Attending to these fundamentally different perspectives of the learning process through the lens of moderate hermeneutics allows some reconciliation of their basic tenets, while also enriching the moderate hermeneutic position as a way to enhancing understanding of the learning process. There is no absolute truth waiting to be discovered, but an evolution of socially and historically situated individual ways of knowing.
In concluding, the rationale that supports the data being viewed from a moderate hermeneutic perspective is briefly outlined. Firstly, hermeneutics is the theory of interpretation of meaning. In the educative sense, this is implicit to understanding. Interpretation, of text in the broad sense, of associated reflective dialogue and action, of any of the diverse range of communications and phenomena that permeate the rich milieu of the classroom, is how a shared understanding is manifest. This shared understanding may have conceptual, processing, emotional or physical elements. Understanding, and by inference learning, is central to what education is.
For those who embrace a socio-cultural viewpoint (e.g., Lerman, 2006), the dilemma of enculturation, as opposed to personal construction of understanding, is reconciled by the notion that personal understanding emerges in a social
context. A moderate hermeneutic perspective acknowledges the historically situated, socio-cultural discourses that the learner brings to the learning ‘situation’, while also accepting the political and institutional influences that pervade these discourses. It allows for a personal interpretation of a social or linguistic interaction, containing elements of socio-cultural learning theory while recognizing the personal individual assembling or structuring associated with this. Fundamental to this perspective is the notion of the hermeneutic circle, which describes the learning process, and sits particularly comfortably with the learning trajectories that evolve in the investigation of mathematics phenomena.
Complicit to this process is the notion that understanding is filtered through prevailing discourses, the pedagogical medium and active participation, including language. This has particular resonance with the purpose and questions of this thesis. The notion of a concept as an evolving process, and acknowledgement of the trust imbued in language to broker consensus, are also aspects that enhance the interpretation of the learning process that this research is situated within. The moderate hermeneutic perspective also gives recognition to the idea that socio- cultural influences are reproduced through the educative process in a transformative manner, rather than the educative process being used to fragment power and authority as per radical interpretive theories.
The nature of hermeneutics, and its broadening from the classical viewpoint of understanding derived from written language, to one cognisant of the notion of discourse (Gadamer, 1975), and its mediation with a phenomenological viewpoint (Ricoeur, 1981), was considered. A moderate hermeneutic perspective frames the interpretive approach taken in this study. The study contends that this approach also reconciles several key aspects of acquisitional and participatory theories, used by mathematics researchers and practitioners to examine how mathematical understanding evolves.
Philosophers as diverse as Foucault and Habermas discuss how power hierarchies, or tradition, might shape understanding through limiting the nature of the dialogue (Giddens, 1985; Philp, 1985). This is also consistent with the notion of understanding being situated within the social context that initiates the
learning process. The discourses that evolve in various pedagogical contexts are determined by language that is implicitly shaped by previous users and the community from which it derives. This dialogue is constrained by societal norms for the structure of language in that particular context (Brown, 1996). It seems a logical implication then, that varying the pedagogical lens will evoke different linguistic phenomena, and thus the negotiation of meaning will likewise vary. By examining the participants’ dialogue as they engaged in the tasks through the pedagogical medium of the spreadsheet; by observing their actions; and by analysing their reflections, it was intended that insights be gained into the ways investigating mathematical problems with a spreadsheet might influence their mathematical understanding.
This prefaces the next chapter in which the methodology and the approaches employed to obtain the data are described and examined.