and overview.- Notes Introduction.- 1 Description of the fraction problem.- 1.1 Introduction and overview.- 1.2 Pointing out the problem.- 1.3 Description of the problem.- 1.3.1 Historical perspective.- 1.3.2 Outlining the problem.- 1.4 Overview and outlook.- Notes chapter 1.- 2 Currents-standpoint-proposal.- 2.1 Introduction and overview.- 2.2 Currents.- 2.2.1 The structuralistic approach.- 2.2.2 The mechanistic approach.- 2.2.3 Pragmatic structuralism.- 2.2.4 Intermezzo.- 2.2.5 The realistic approach.- 2.2.6 The empiristic approach.- 2.3 Standpoint.- 2.4 Proposal.- 2.5 Review and outlook.- Notes chapter 2.- 3 The context of the research.- 3.1 Introduction and overview.- 3.2 Starting from the construction principle.- 3.3 Development research.- 3.4 Design and method.- 3.5 The research group.- 3.5.1 Introduction.- 3.5.2 A closer description of the research group.- 3.6 The researcher's position.- 3.6.1 Roles.- 3.6.2 Between participation and distance.- 3.7 Conclusion.- Notes chapter 3.- 4 The course in theory and practice.- 4.1 Introduction and overview.- 4.2 Global outline of the fraction course.- 4.2.1 Foreword.- 4.2.2 Producing fractions.- 4.3 Producing fractions: the teaching experiment.- 4.3.1 Foreword.- 4.3.2' starting Fractions', or furnishing the concrete level.- 4.3.3 'Continuing fractions', or en route to higher levels. Seating arrangements and dividing in other than distribution situations.- 4.3.4 'Repeating fractions', or reviewing the rules.- 4.4 In view of mathematization.- 4.4.1 Foreword.- 4.4.2 The role of contexts.- 4.4.3 Manifold use of symbols, diagrams and models.- 4.4.4 Construction and production, or student contributions to education.- 4.4.5 Interaction.- 4.4.6 Intertwinement of the learning lines.- 4.5 Review and outlook.- Notes chapter 4.- 5 Individual portraits of the fraction researchers.- 5.1 Introduction and overview.- 5.2 Indicators for the (improvement in) quality of the individual learning processes.- 5.3 The student portraits.- 5.4 Review and outlook.- Notes chapter 5.- 6 Internal evaluation of the learning process.- 6.1 Introduction and overview.- 6.2 Yielding to and building up resistance to IN-distractors.- 6.3 Progression in schematizating through abbreviations.- 6.4 Models and diagrams.- 6.4.1 Flexible use of models and application of schemes and diagrams.- 6.4.2 Situation models.- 6.5 Acquiring representations fitting formally posed problems.- 6.6 Constructions and productions on a symbolic level.- 6.7 The framework of the level-theory and the course construction.- 6.8 Review and outlook.- Notes chapter 6.- 7 External or summative evaluation.- 7.1 Introduction and overview.- 7.2 Weighing the groups.- 7.3 Comparing and characterizing the programs.- 7.3.1 Introduction.- 7.3.2 NCR and NZR.- 7.3.3 Programs containing realistic features.- 7.4 The tests.- 7.5 Results.- 7.5.1 Introduction.- 7.5.2 The final test, a qualitative analysis.- 7.6 Summary and conclusions.- 7.6.1 Introduction.- 7.6.2 Summary of the results of the experimental group.- 7.6.3 Conclusions.- 7.7 Realistic and mechanistic: a difference in effects?.- 7.8 A look ahead.- Notes chapter 7.- 8 Theoretical review.- 8.1 Introduction and overview.- 8.2 Teaching and learning fractions.- 8.2.1 Introduction.- 8.2.2 Amply furnishing the first level.- 8.2.3 On to higher levels.- 8.2.4 General characteristics of vertical mathematization.- 8.3 Conclusion.- Notes chapter 8.- Epilogue: Sketch of a new course -fractions, an integrated perspective-.