FROM OSCILLATIONS TO NORMAL MODES: AN EDUCATIONAL PATH FOR THE UPPER SECONDARY SCHOOL

The present thesis work starts from the assumption that harmonic oscillations and normal modes are key physical concepts. They are fundamental in quantum physics, in electromagnetism (especially in treating coupled oscillating circuits and electromagnetic waves), in acoustics and in mechanical systems. The conceptual and practical importance of normal modes emerges also clearly from the fact that every small and sufficiently smooth oscillation of a complex system is given by a linear superposition of its normal modes. The notion of normal modes is thus a powerful conceptual organizer. Nevertheless, in teaching practice (at least in Italy), only short time is devoted to harmonic motion, rarely coupled oscillators are treated and, in secondary school text-books, normal modes are usually not even present. The purpose of this thesis work is to develop an effective path on scillations for the upper secondary school that leads to the normal modes of oscillations. To do this, an educational reconstruction of the concept of harmonic motion has been necessary as the harmonic motion is a fundamental prerequisite for the understanding of normal modes. The introduction of normal modes is, for upper secondary school students, complicated by the complexity of the mathematics involved. In our path we propose to overcome the mathematical difficulties through an experimental approach and the use of different tools such as video and picture analysis, also in slow motion, data logging and data analysis techniques and applet simulations, with the goal of being as simple as possible from the mathematical point of view but without losing the advantages that mathematics (even at simple level) can provide. In this perspective, a multiple representation approach has been used. The path on oscillations that we present here is the result of a Design Based Research on normal modes with Italian upper secondary school students. The complete path has been proposed to three classes of 11th grade students during curricular lessons. A version of the sequence has been proposed also to other three classes (one of grade 11th and two of grade 12th) during afternoon extra-curricular lessons, and a version with university-level formalism has also been proposed to a group of undergraduate students in mathematics during the third year course “Preparation of Didactical Experiments”. A reduced version of the path has also been proposed to a number of classes of 12th grade students within the one-shot lessons on oscillations (afternoon extra-curricular activities) in the framework of PLS (Piano Lauree Scientifiche) activities. The one-shot lessons have been attended, over time, by about six hundred students. The all path is based on a number of activities in which we start from a real experiment or a video or else an applet simulation to introduce and discuss a limited topic. The general purpose is to identify, among the oscillations, those that give rise to a peculiar kind of motion, the harmonic motion, and determine the conditions under which such motion can be obtained. A number of significant situations of harmonic and anharmonic motions are investigated and criteria to establishing the harmonicity/anharmonicity of the oscillation are discussed. An important tool for the analysis of the data is then introduced: the Fast Fourier Transform. The FFT is introduced as a tool and not discussed through mathematics. Then the concept on resonance is introduced in a phenomenological way through experiments and exploring related videos in the Internet videos database. The next step is the introduction of the coupling between two oscillators and the discovery of particular motion configurations: the Normal Modes of Oscillation. We then extend the experiments to three, four, five....many coupled oscillators until we arrive to the continuous case; first in one dimension with the string and then in two dimensions with the Cladni plates and we study the normal modes of such complex systems. The structure of the thesis is as follows: an introduction to the motivations, a description of the state of the art and the formulation of the research questions. Then a brief description of the methodological framework mainly based on the Design Based Research approach and the Model of the Educational Reconstruction. The reconstruction of the Harmonic motion at university level follows; the translation of such a reconstruction into upper secondary school level is developed in chapter five. In chapter four normal modes for a system of two, three, N coupled oscillators are treated with the proper formalism; also in this case the translation into upper secondary school level is developed in chapter five. Then the very core of the thesis follows , namely the developing of the path, as briefly described above. The next chapter reports a path developed with undergraduate students as an implementation of the study of oscillations. It is the study of the modes of oscillation in the interesting case of a parametric oscillator were there is a non-linear coupling between modes of oscillation. In the last section, the main results of the experimentation of the path with threes classes of 11th grade students are briefly presented. These results are based on questionnaires (pre-test and post-test), discussions and interviews.