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Interacting With Implicit Knowing in the Mathematics Classroom Open Access


Other title
mathematics education
analogical reasoning
Type of item
Degree grantor
University of Alberta
Author or creator
Metz, Martina L.
Supervisor and department
Simmt, Elaine (Secondary Education)
Examining committee member and department
Glanfield, Florence (Secondary Education)
Martin, Lyndon (York University)
Eppert, Claudia (Secondary Education)
McGarvey, Lynn (Elementary Education)
Department of Secondary Education
Mathematics Education
Date accepted
Graduation date
Doctor of Philosophy
Degree level
This study explores Grade Seven students’ experiences of doubt and certainty in mathematics. During nine months of (bi-monthly) sessions, students responded to several mathematical prompts; their interactions with each other and with the teacher- researcher were video-taped, transcribed, and coded for learners’ evolving perceptions of what was (a) sufficient to define certainty (including what was experienced as intuitive or counter-intuitive and ways such certainty was interrupted), (b) relevant to the tasks (including understandings that initially dwelled on the periphery of awareness), and (c) mathematically connected. The study is conceptualized within an enactivist view of cognition that emphasizes autonomous, co-emergent, and embodied knowing (Thompson, 2007; Varela, Thompson, & Rosch, 1991), and classes were designed with these principles in mind. It became clear that doubt and certainty emerge from a broader, holistic understanding that is largely beneath ordinary awareness and is deeply implicated in what we experience as “repeatable context” (Bateson (1964/1972). An important aspect of the study was to bring more of this understanding to awareness. In doing so, Varela’s (Varela & Scharmer, 2000) notion of researcher as empathic coach and Gendlin’s notions of “felt sense” (1962, 1978) and “implicit intricacy” (1991; 2009a) assumed importance. By attending to the holistic sense that points to implicit understanding, it was possible to broaden the scope of what was deemed relevant in selected contexts. It was found that previously subconscious understandings nonetheless influenced learning. Once named (even broadly), implicit understanding co- evolved with language in developing mathematical understanding. By attending to external indicators of felt meaning, learners interacted with each others’ implicit understanding, thereby bringing it closer to consciousness and into conversation. Prematurely insisting on clarity and logic precluded awareness of the implicit.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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