Introduction   The following extracts from the curricula One Two and Three and Plus One form an example of teaching the long-term subject fractions according to principles developed in recent research in Mathematics education.   The units dealing with fractions in these series are based on three theoretic sources: Piaget taught that children at the age of concrete operations develop mathematical ideas as a result of reflection upon their own activity in the concrete world (Piaget, 1975, 1976).   Nesher's perspective of Learning Systems (Nesher, 1988, 1989) presents mathematics learning in elementary school as a series of Learning Units, each consisting of a well defined Knowledge Domain. Elementary school students are expected to construct the mathematical ideas of such a Knowledge Domain with the help of a specially designed concrete Exemplification Domain. The Exemplification Domain, in addition to being concrete, is expected to be as isomorphic as possible to the Knowledge Domain.   A Learning System according to Nesher Dubinsky's perspective of Action-Process-Object-Schema (APOS) is represented in these series through the main idea that the development of a new mathematical idea begins with an action (Dubinsky, 1991). In the case of elementary school students, combined with Piaget's theory described above, the action which is expected to give birth to a new mathematical idea should be concrete (Arnon, 1998).   Action - Process - Object At the Age of Concrete Operations

According to Nesher's perspective, story-problems (real-life situations) should be dealt with only after students have already developed (to some degree) the mathematical ideas and structures by means of which the situation can be described and solved.   Additional Real-World Situations in a Nesher Learning System We will show how the teaching of fractions is dealt with in our series according to the mentioned theoretical perspectives. Most of the ideas are taught within Nesher-like learning units, each consisting of a well defined Knowledge Domain and using a carefully designed Exemplification Domain. Almost every idea is presented as a concrete action. Activities are used to enable students to interiorize each action, develop it into a more formal (abstract) idea, and assimilate it into their previous body of knowledge. Only then are these ideas applied to real-life situations. The application is based upon the analogy between the occurrences in the situation and the actions that served as sources (definitions) for the mathematical concepts and structures. The Exemplification Domains used in these series were tried out with elementary school students, to make sure that most students do interiorize the concrete actions and eventually develop the appropriate abstract mathematical concepts. some of the operations and concepts concerning fractions are not taught as concrete actions. These include fraction reduction, fraction division, and the arithmetic operations on decimal fractions. The reasons for that deserve a separate discussion.   References   Arnon, I. (1998). In the mind's eye: how children develop mathematical concepts -extending Piaget's theory. Unpublished doctoral dissertation, School of Education, Haifa University.   Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95-123). Dordrecht/Boston/London: Clair Academic Publishers.   Nesher, P. (1988). Beyond constructivism (Learning mathematics at school). In A. Borbas (Ed.), Proceedings of the Twelfth Annual Conference of the International Group for the Psychology of Mathematics Education I, (pp. 54-74). Veszprem, Hungary.   Nesher, P. (1989). Microworlds in mathematical education: A pedagogical realism. In L.B. Resnick (Ed.), Knowing, learning & instruction (pp. 187-215). Hillsdale, N.J: Lawrence Erlbaum Associates.   Piaget, J. (1975). Piaget's theory (G. Cellerier & J. Langer, trans.). In P.B. Neubauer (Ed.), The process of child development (pp. 164-212). New York: Jason Aronson.   Piaget, J. (1976). The grasp of consciousness (S. Wedgwood, Trans.). Cambridge, Massachusettes: Harvard University Press. (Original work published 1974).