CBMS Issues in Mathematics Education Volume 8, 2000 Teaching and Learning Calculus: What Can Be Learned from Education Research and Curricular Changes in France? 1 Michele Artigue ABSTRACT. Research on mathematics teaching and learning has been stimu- lated by the persistent difficulties that students have when they are introduced to the conceptual field of analysis and from general feelings of dissatisfaction with calculus courses. After sketching the main research results, we analyze the methods adopted throughout the 20 t h Century in preparing students for the baccalaureat in France2 in order to overcome these difficulties. We also con- sider what the historical evolution of the curriculum can teil us about current teaching. Introduction It is not easy for students to get into the conceptual field3 of elementary anal- ysis (i.e., calculus). Education research developed in this area over the past 15 years shows this very clearly (Artigue Ervynck, 1992 Farfän, 1993 Tall, 1991). Furthermore, this research has allowed us to better understand the nature of the dif- ficulties and the obstacles students encounter, as well as the reasons for the failure, as much of traditional teaching strategies, which reduce analysis to algorithms for algebraic calculations, as of the theoretical approaehes and formalisms developed· in the context of modern mathematical reform. Everywhere in the world new programs of study and new curricula are being set up in attempts to find ways to introduce this conceptual field that are, at the same time, rieh in meaning and accessible. Intuitive approaehes based on the use of Information technology, calculators, and Computers appear to be the most generally 1 Translation by Ed Dubinsky of Ensenanza y aprendizaje del anälisis elemental: «i,que se puede aprender de las investigaciones didäcticas y los cambios curriculares?, Revista Latinoamericana de Investigacion en Matemdtica Educativa, 1998, 1, pp. 40-55. 2 France has a national curriculum. In the Prench educational System, the secondary level begins with grade 6, when students enter what is called "college," and lasts for four years. At grade 10, students enter one of several types of lycees for three years at the end of which time they can take an exam called the baccalaureat. If they pass, they can go on to the university. We will use this terminology throughout the article. ï By "conceptual field," educational researchers mean "the central ideas of a content area, understood conceptually, along with connections among them, so that the area is understood as a coherent whole." ©2000 American Mathematical Society 1 http://dx.doi.org/10.1090/cbmath/008/01
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