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• Inquiry based mathematics education and the development of learning trajectories

  63-89

  Michèle Artigue

  Marianna Bosch

  Michiel Doorman

  Péter Juhász

  Ladislav Kvasz

  Katja Maass

  Views:

  876

  This article is based on the panel on inquiry based mathematics education and the development of learning trajectories held at the VARGA 100 Conference. After an introduction presenting the theme and organization of the panel, this article focuses on the diversity of conceptualizations of inquiry based education existing today in mathematics education and their influence on the vision and development of learning trajectories. More precisely, it considers the conceptualizations respectively associated with Realistic Mathematics Education, Genetic Constructivism, Tamás Varga’s educational approach and the Anthropological Theory of the Didactic, presented by the panellists, and also shows the efforts undertaken in European projects to reach consensusal visions.

  Subject Classification: 97C30Q, 97D10, 97D20, 97D30, 97D40, 97D50

• Teaching centroids in theory and in practice

  67-88

  Szilárd András

  Csaba Tamási

  Views:

  40

  The main aim of this paper is to present an inquiry-based professional development activity about the teaching of centroids and to highlight some common misconceptions related to centroids. The second aim is to emphasize a major hindering factor in planning inquiry based teaching/learning activities connected with abstract mathematical notions. Our basic problem was to determine the centroid of simple systems such as: systems of collinear points, arbitrary system of points, polygons, polygonal shapes. The only inconvenience was that we needed practical activities where students could validate their findings and calculations with simple tools. At this point we faced the following situation: we have an abstract definition for the centroid of a finite system of points, while in practice we don't even have such systems. The same is valid for geometric objects like triangles, polygons. In practice we have triangular objects, polygonal shapes (domains) and not triangles, polygons. Thus in practice for validating the centroid of a system formed by 4,5,... points we also need the centroid of a polygonal shape, formed by an infinite number of points. We could use, of course, basic definitions, but our intention was to organize inquiry based learning activities, where students can understand fundamental concepts and properties before defining them.

• Some Pythagorean type equations concerning arithmetic functions

  157-179

  Ildikó Kézér

  Views:

  61

  We investigate some equations involving the number of divisors d(n); the sum of divisors σ(n); Euler's totient function ϕ(n); the number of distinct prime factors ω(n); and the number of all prime factors (counted with multiplicity) Ω(n). The first part deals with equation f(xy) + f(xz) = f(yz). In the second part, as an analogy to x² + y² = z², we study equation f(x²) + f(y²) = f(z²) and its generalization to higher degrees and more terms. We use just elementary methods and basic facts about the above functions and indicate why and how to discuss this topic in group study sessions or special maths classes of secondary schools in the framework of inquiry based learning.

  Subject Classification: 97F60, 11A25