MATHEMATICAL EXPLANATIONS: IN-ACTION AND
AS RE-PRESENTATION
L. Gordon Calvert
University of Alberta
The purpose of this ongoing study is to explore the nature of
mathematical explanations and make comparisons between explanations that arise
in-action and those provided as re-presentations or summaries of formative
efforts either to the teacher or to the whole class. Some of the contrasts that arose during the study were in
relation to how explanations were posed or offered to others; the purpose or
need the explanation appeared to fulfill; the criteria used to accept or reject
an explanation; how that acceptance or rejection was signaled; and what was
hidden, lost or ignored as explanations were re-presented.
In
brief, explanations expressed in-action were offered to both oneself and to
others in the group in an effort to broaden understanding in and for that
moment, often with the assumption that they could return to the ideas later if necessary. The explanations were hesitant, incomplete
and viewed as plausible, viable and consistent with previous experiences with
that task and with previous tasks. The
incompleteness allowed other participants to add on and revise in the course of
interaction. Accepted explanations were
significant points in the path of activity as they allowed participants to move
on. The explanations were then
incorporated into subsequent actions and explanations. In contrast, explanations offered as
re-presentations were often posed to outsiders as complete arguments in an
effort to convince others of the correctness of the explanation. As such, they were generally not used to
initiate further investigation but as an endpoint to activity. Acceptance was signaled when no errors,
inconsistencies or disagreements were stated with the explanation presented.
An awareness of how explanations are altered in
content and intention has implications for understanding the generative
features of mathematical explanations in the relation to human perceptions of
the nature of mathematics.