Leren vermenigvuldigen met meercijferige getallen

Abstract

The reason for this study lies in the fact that efforts to create a learning trajectory for the domain of multidigit multiplication based on the realistic approach of mathematics education and building on children’s own informal strategies with the aim of transforming these strategies into efficient notation-supported calculation strategies, have not been quite successful. That this is the case, can not only be concluded from an analysis of common learning trajectories in recent Dutch mathematics textbooks, but also from a number of personal experiences of the researcher with the development of one of these textbooks. Moreover recent research in the field of multidigit multiplication in The Netherlands, shows that the success rates being obtained by 11-12 year old Dutch students with solving problems in this field such as the one below, have dropped over the last ten years substantially from about .75 to .60. Research also shows that the number of students that use mental strategies without supporting those strategies with appropriate auxiliary notations, is increasing considerably. With these facts in mind a research was conducted that addresses the following questions: (1) How can a learning trajectory for multidigit multiplication be shaped that builds on students’ own, informal strategies and that helps them to develop theses strategies into efficient, notation-supported calculation strategies in the domain of stylized mental computation? (2) How do the solutions and the notations of the students develop under influence of such a learning trajectory? And to what extent does this learning trajectory provide also weaker students with sufficient support to develop the intended strategies? (3) What are the key elements of a local instruction theory that constitutes the rationale for such a learning trajectory? The research fits in with the tradition of developmental research that has been established in The Netherlands as a type of research in which, on the basis of an overarching domain specific theory (known as Realistic Mathematics Education, RME), an experimental learning trajectory is developed together with a corresponding local instruction theory that is being put to the test in a classroom teaching experiment. The whole process of designing, trying and revising is recorded and analyzed as carefully as possible in order to be able to develop an empirically grounded instruction theory. An important background of the research was constituted by the fact that in recent documents on curriculum development in The Netherlands, there is clear tendency to pay less attention to the traditional standard written algorithms in education, and to focus more on topics like number sense, mental calculation and measurement. In the light of this tendency it is important to dispose of a well outlined learning trajectory that is primarily aiming at stylized mental computation and that can offer also weaker students that do not arrive at the ultimate goals of most abbreviated procedures, sufficient opportunities to arrive at satisfactory achievements. The results of the research primarily consist of a description of the outline of a learning trajectory with a corresponding local instruction theory for multidigit multiplication. Within this theory special attention is paid to the level structure of the learning process. Next to this, a number of more general findings and recommendations is presented that refer to actual aspects of the RME-approach and the corresponding process of progressive mathematization.