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The Characterization Problem
Published in Stephen Hester, David Francis, Eric Livingston, Ethnographies of Reason, 2016
Stephen Hester, David Francis, Eric Livingston
What makes a mathematical proof a naturally accountable proof of ordinary mathematical practice? Said differently and formulated in terms of the social circumstances where provers engage in proving for and among other provers, how do theorem provers recognize each other as theorem provers? What is it about their practices that identifies them, to other provers, as mathematicians and that identifies what they’re doing as proving-theorems-really? This, at least initially, in its most general formulation, is the characterization problem for mathematics. We’re looking for what proving is as a social activity; we want to discover what the social is in and as the work of proving mathematical theorems.
A combined application of APOS and OSA to explore undergraduate students’ understanding of polar coordinates
Published in International Journal of Mathematical Education in Science and Technology, 2020
Vahid Borji, Hedyeh Erfani, Vicenç Font
The OSA theory describes the mathematical activity from a personal and institutional point of view. In OSA, it is important to consider the objects involved in such activity. In this theory, the mathematical activity can be modelled in terms of the configuration of primary objects and processes that appear during the practice. A mathematical practice is considered in this theory as a sequence of actions, controlled by institutionally accepted rules, oriented towards an objective which is usually solving a problem. Languages, problems, concepts/definitions, procedures, propositions and arguments are considered as part of objects, the six mathematical primary objects. Connected together they form configurations of primary objects. Language refers to terms, expressions, notations, and graphs in their various registers such as written, oral, gestural, etc. Problems include extra-mathematical applications, tasks, exercises, examples, etc. Definitions/Concepts introduced by means of explanations or descriptions, explicit or otherwise such as straight line, point, number, function, derivative, etc. Propositions are statements about concepts. Procedures are considered as operations, algorithms, and techniques of calculation. Arguments can be considered as statements used to explain or validate the procedures and propositions, whether deductive or of another kind. In the following a priori analysis of the mathematical activities needed to solve the tasks in terms of practices and objects performed, based on OSA theory, is made. A priori analysis of the mathematical activities needed to solve the tasks in terms of practices and objects performed, based on OSA theory, is presented below.