WORKING GROUP 3 Building structures in mathematical knowledge

Abstract: Mathematical objects are multiple connected one to each other, but also with non-mathematical objects and thus build up a network with manifold linkages, characterizing the structure of mathematics. During teaching and learning processes some parts of this network are carried over into pupils’ minds, changing their structure. This paper presents part of a study carried out in Germany, investigating the transformation of a network to the topic “sets of two equations of straight lines” during teaching and learning processes. Especially there are compared the network, that teachers stated to have taught in middle grade classes, and the networks, their students really learned. Overmore, there are provided findings on students’ abilities in using their network knowledge for problem solving. Keywords: mathematical network, network knowledge, connections, structure, network categories Introduction In the preface of the NCTM Yearbook 1995 it is pointed out that One of the four cornerstones of the NCTM Curriculum and Evaluation Standards for School Mathematics asserts that connecting mathematics to other mathematics, to other subjects of the curriculum, and to the everyday world is an important goal of school mathematics. Among recent reports calling for reform in mathematics education, there is widespread consensus that mathematics … must be presented as a connected discipline rather than a set of discrete topics … (House, NCTM Yearbook 1995 – Preface.) The notion that mathematical ideas are connected should, according to the NCTM Principles and Standards for School Mathematics 2000 (p. 64), permeate the school mathematics experience at all levels. These demands are not new, but they are expressed to an increased extend in the last few years. Especially in Germany, the call for a reinforced representation of mathe-matics as a network of interconnected concepts and procedures becomes louder, not at least because of the results of the TIMS-Study (Baumert & Lehmann, 1997; Beaton et. al., 1996; Neubrand et. al., 1998) that reveal a great failure in students’ problem solving abilities according to a lack of flexibility in thinking in mathematical networks. This failure was once again confirmed by the PISA–Study, where interconnections and common ideas were central elements (OECD, 1999, p. 48).