Approaching Proof in a Community of Mathematical Practice
Kirsti Hemmi
PhD thesis, Stockholm University, Department of Mathematics, Sweden
Abstract
This thesis aims to describe how students encounter proof in a community of
mathematical practice at a mathematics department and how they are drawn to
share mathematicians' views and knowledge of proof. Considering the department
as a community of practice where the joint enterprise is learning mathematics
in a broad sense made it possible to perceive the newcomers as active participants
in the practice. The combination of a socio-cultural perspective, Lave and Wenger's
and Wenger's social practice theories and theories about proof offers a fresh
framework in understanding and describing the diversity of the culture involving
such a
complex notion as proof. Proof is examined from historical, philosophical and
didactical points of view and considered as reification and as an artefact from
a socio-cultural perspective. The metaphor of transparency of artefacts that
refers to the intricate dilemma about how much to focus on different aspects
of proof at a meta-level and how much to work with proof without a focus on
it, both from teacher and student perspectives, is a fundamental aspect in the
data analysis. The data consists of transcripts of interviews with mathematicians
and students as well as survey responses of university entrants, protocols of
observations of lectures, textbooks and other instructional material. Both qualitative
and quantitative methods were applied in the data analysis. A theoretical model
with three different teaching styles with respect to proof could be constructed
from the data. The study shows that the students related positively to proof
when they entered the practice. Though the mathematicians had not an explicit
intention of dealing so much with proof in the basic course, students felt that
they were confronted with proof from the very beginning of their studies. Proof
was there as a mysterious artefact and a lot of aspects of proof remained invisible
as experienced by students when they struggled to find out what proof is and
to understand its role and meaning in the practice. The students who proceeded
further experienced a mix of participation and non-participation regarding proof
depending on their capacity to follow lectures and on how much they invested
themselves in the negotiation of meaning of proof. The first oral examination
in proof seems to be significant in drawing students to the practice of proof.