Concrete models and dynamic instruments as early technologytools in classrooms at the dawn of ICMI: from Felix Kleinto present applications in mathematics classroomsin different parts of the world

Abstract Most national curricula for both primary andsecondary grades encourage the active involvement oflearners through the manipulation of materials (eitherconcrete models or dynamic instruments). This trend isrooted in the emphasis given, at the dawn of ICMI, to whatmight be called an experimental approach: the linksbetween mathematics, natural sciences and technologywere in the foreground in the early documents of ICMI andalso in the papers of its first president, Felix Klein. However,the presence of this perspective in teaching practice isuneven. In this paper, we shall reconstruct first an outlineof what happened in three different parts of the world(Europe, USA and Japan) under the direct influence ofKlein. Then, we shall report classroom activities realized inthe same regions in three different research centres: theLaboratory of Mathematical Machines at the University ofModena and Reggio Emilia, Italy (http://www.mmlab.unimore.it); the pedagogical space of Kinematical Modelfor Design Digital Library at Cornell, USA (http://kmoddl.library.cornell.edu/); and the Centre for Research onInternational Cooperation in Educational Development atTsukuba University, Japan (http://math-info.criced.tsukuba.ac.jp/). They have maintained the reference to concretematerials (either models or instruments), with originalinterpretations that take advantage of the different culturalconditions. Although in all cases the reference to history isdeep and systematic, the synergy with mathematical modellingand with information and communication technologieshas been exploited, not to substitute but to complementthe advantages of the direct manipulations.