Abstract
Secondary-tertiary transition issues are explored from the perspective of ways of doing mathematics that are constituted in the implicit aspects of teachers’ action. Theories of culture (Hall, 1959) and ethnomethodology (Garfinkel, 1967) provide us with a basis for describing and explicating the ways of doing mathematics specific to each teaching level, according to the “accounts” provided by the teachers involved in this research project. To borrow from Hall (1959), the “informal” mode of mathematical culture specific to each teaching level plays a key role in attempts to better grasp transition issues.
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Notes
Espace Mathématique Francophone (EMF) 2003, 2006, 2009, 2015; PME 2011; PME-NA 2012; CERME 2010
For example, if at the secondary level, letters represent essentially real numbers; at the tertiary level, particularly in the linear algebra course, letters represent various mathematical objects: real numbers, complex numbers, matrices, geometrical vectors, algebraic vectors, polynomial expressions, etc. This makes the formal mathematical language difficult to understand and use for many students (Corriveau & Tanguay, 2007)
Tertiary mathematics teaching has been the subject of several research studies (see the “How we Teach” project reported in Jaworski & Matthews, 2010). Some studies bearing more directly on transition issues have focused more specifically on the perspective of teachers regarding transition (Hong et al., 2009) or that of students regarding their teachers (De Guzman, Hodgson, Robert, & Villani, 1998), but they shed light partially on specific issues linked to transitional issues.
Students had to solve the following system of equations for all values of the constants a and b, with the help of the teacher (in linear algebra)
$$ \begin{array}{l}x+y+z=5\hfill \\ {}x\hbox{--} y+az=3\hfill \\ {}2x+y+z=b\hfill \end{array} $$Cégep is a French acronym for Collège d’enseignement général et professionnel (referred to in English as General and Vocational College). In Quebec, the “cégep level” lasts 2 years (grades 12 and 13, students of 17 to 19 years old) in the case of pre-university programs and 3 years in the case of technical/vocational programs. This level is, like the university level, part of the province’s system of higher education. “Cégep” institutions are independent of both secondary institutions (grades 7 to 11) and universities and lead to a degree specific to that level. Teachers receive formal training in a given discipline (e.g., masters in mathematics) and have access to research grants (in mathematics education), but they are not required to do research. So mathematics teachers at the Cégep are not doing mathematics for the purpose of scholarship and publication in mathematics.
Please note that while the general focus of our research concerns the secondary-tertiary transition, for the purposes of clarity, we shall describe those teachers and institutions participating in our study as “postsecondary”—in reference to the specific level of the Quebec higher education system concerned.
These ways of doing mathematics are situated in the context of teaching and therefore include thinking about and planning how to do mathematics with students and how to represent mathematical concept for the purpose of teaching.
Ethnomethodology refers to “ethno-methods” and “-logy”—i.e., the study of the methods used by a particular sociocultural group in its everyday activities for understanding and producing the social order in which its members live.
Ethnomethodology is rooted in this reflexive and interpretative capacity of each social actor, inseparable from action: « Le mode de connaissance pratique c’est cette faculté d’interprétation que tout individu, savant ou ordinaire, possède et met en œuvre dans la routine de ses activités pratiques quotidiennes (…) procédure régie par le sens commun, l’interprétation est posée comme indissociable de l’action et comme également partagée par l’ensemble des acteurs sociaux… » (Coulon 1993, p. 15)
“Territory” is an evocative metaphor serving to convey the idea of a “land” that people organize so as to be able to “live” in it (Raffestin, 1981). This space is continually undergoing organization.
The secondary teachers were Sam, Serge and Scott and the postsecondary (cégep) teachers were Colin, Colette and Corinne.
However, when the topic at hand is limit, f and g are instead the letters used to symbolize the associated functions.
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An erratum to this article is available at http://dx.doi.org/10.1007/s10649-017-9766-3.
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Corriveau, C., Bednarz, N. The secondary-tertiary transition viewed as a change in mathematical cultures: an exploration concerning symbolism and its use. Educ Stud Math 95, 1–19 (2017). https://doi.org/10.1007/s10649-016-9738-z
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DOI: https://doi.org/10.1007/s10649-016-9738-z