ABSTRACT
Realistic can be misinterpreted easily. Obviously its first meaning signifies that the mathematics to be taught is linked up firmly with reality, or rather in reverse: reality serves both as a source of the envisaged mathematics and as a domain of application. This chapter contains the description of four building blocks for a course on fractions. All of them reflect this aspect of realistic albeit at different levels. Moreover fractions can evolve as a mathematical reality for the learners in this way. This means realistic also refers to the manner in which the learners realize (their) fractions in the teaching-learning process. For bridging the gap between concrete and abstract they need to develop tools such as visual models, schemas, and diagrams. These are the vehicles of thought for students that enable them to enter mathematics and to make progress within it. An extended description of a long-term, individual learning process illustrates this. It reflects the attempts to integrate the processes of teaching and learning fractions. This is what developmental research will result in: courses that deal with both teaching and learning.
Mathematicians from Klein to Freudenthal and psychologists like Piaget and Davydov have concerned themselves explicitly with the educational problem of learning about fractions. Many others have continued to address the challenge represented by fractions in mathematics education (Hilton, 1983; Usiskin, 1979). In the context of the current concerns regarding education, it is time to focus attention on those questions about fractions in mathematics education that are of primary importance in learning to think mathematically.