# Rationella tal som tal : Algebraiska symboler och generella modeller som medierande redskap

Sammanfattning: In this study the teaching of mathematics has been developed in relation to rational numbers and towards a learning activity. At the same time topic-specific mediated tools have been studied. The iterative model for learning study has been used as research approach.The purpose of the study was to explore what in an algebraic learning activity enables knowledge of rational numbers to develop. The specific questions answered by the study are how an algebraic learning activity can be formed in an otherwise arithmetic teaching tradition, what knowledge is mediated in relation to different mediated tools and what in these tools that enable this knowledge.The result of the study shows how an algebraic learning activity can be developed to support the students to understand rational numbers even in an arithmetic teaching tradition. The important details that developed the algebraic learning activity were to identify the problem to create learning tasks and the opportunity for the students to reflect that are characteristic of a learning activity. The result also shows that the mediating tools, the algebraic symbols and the general model for fractional numbers, have had significant importance for the students' possibilities to explore rational numbers. The conditions for the algebraic symbols seem to be the possibilities for these symbols to include clues to the meaning of the symbol and that the same symbol can be used in relation to several of other mediated tools. The conditions in the general model consisted of that the integer numbers and the rational numbers in the model could be distinguished and that the students could reflect on the meaning of the different parts. The general model consists of the algebraic symbols, developed in the learning activity. The algebraic symbols make the structure of the numbers visible and the general model mediates the structure of additive and multiplicative conditions that are contained in a rational number.The result of the study contributes in part to the field of mathematics education research by examining Elkonin's and Davydov's Mathematical Curriculum in a western teaching practice and in part to a development of the model of Learning study as a didactical research approach by using an activity-theoretical perspective on design and analysis.

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