In this spirit, the concept image is the total cognitive structure associated to the concept containing all the mental pictures and related properties and processes, including conceptions and structural elements. Specifically, the CID framework highlights the distinction between the mathematical concept as formally defined and the individual’s total cognitive representation for that concept. Research dealing with cognitive development is based essentially on the frames of Concept Image and Concept Definition (CID) of Vinner and Hershkowitz ( 1980) (see also Tall and Vinner 19), on the theory of Register Semiotic Representation (RSR) of Duval ( 1995), on the Action-Process-Object-Schema (APOS) theory of Dubinsky ( 1991) and on the Three Worlds of Mathematics (TWM) of Tall ( 2004). Much of this research deploys constructs from well-defined theoretical frameworks in the field of mathematics education others act from a more empirical point of view. This section aims to give a global vision of research on learning and teaching Calculus, we use the main issues investigated in recent research as a filter in order to structure it. Specifically, this overview of research includes a description of the main theoretical frameworks used in the field of Calculus education descriptions of punctual evolutions approached through the main trends in the field, with a particular attention to the concepts of limits, derivatives, and integrals a description of the state of Calculus instruction from both the European and American perspectives a brief summary of the research progress and some new issues initiated by this progress.Īs a complement to the main text, an extended bibliography with some of the most important references about this topic is included. The most important trend is related to Calculus design, which puts forward several considerations to build and to implement alternatives by taking into account the results of existing research in the whole field. The research in the field of Calculus education covers almost all of the general issues investigated in the area of mathematics education. These evolutions are approached with regard to the main trends in the field of mathematics education such as cognitive development or task design. This “ICME-13 Topical Survey” aims to give a view of some of the main evolutions of the research in the field of learning and teaching Calculus, with a particular focus on established research topics associated to limit, derivative and integral. In this spirit, the research highlights the complexity required to cope with two problematic situations: (1) the actual gap between students’ prior knowledge and the mathematical foundation of Calculus concepts (2) the ultimate reliance of Calculus on formal definitions and proofs. This formalism supports the mathematical existence of Calculus concepts the first step in the teaching of Calculus is to contextualize such existence. The research results, in spite of their variety, revealed a cornerstone issue that is strongly linked to the formalism of Calculus concepts and to the difficulties it generates in the learning and teaching process. We therefore focus on the main trends in the field in order to detect punctual evolutions that permit us to go beyond this survey and to put forward new research questions. The diversity of the research in the field of Calculus education makes it difficult to produce an exhaustive state-of-the-art summary.
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