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Learning and Teaching Mathematics with Simulations
Plus 2000 Examples from Physics
The idea to write this digital book was born during discussions among a circle of friends 1 of the following questions:
Market for PhysicsWhy does the large majority of first year students throw away the very good subject related and professional career opportunities in these subjects?
Already in the final stage of the schools that lead to the Abbot in Germany, mathematics and physics are considered as hard subjects. The universities grudgingly accept, that the knowledge in mathematics of many school leavers is insufficient for taking up subject studies and has to be improved in bridging courses.
A shockingly large part of the students already fails in the first two semesters. For our society this will have serious consequences, since we urgently require a sufficient number of well qualified young professionals in the scientific and technical jobs to succeed the current generation of scientists and engineers for the future welfare of our society.
It is easy to understand, why the young generation rather chooses those soft subjects at university for which they feel better equipped and where they see better chances of success. That the ”material” concerns of finding a job later are not considered as crucial for choosing the subject can actually be considered as positive.
Why is it, that mathematics and physics are considered as so difficult? In fact these should benefit from not being route learning subjects: if one has understood a specific physics or mathematics problem within its context, one can forget the small details, since they can be reconstructed from the larger context.
It is obvious that in our schools one often does not manage at all to reach this state of understanding and insight into the mathematical structure and laws of nature; then the instruction cannot provide the wonderful experience to have understood something. Thus, physics can indeed become a route learning subject full of incomprehensible and disconnected formulas and tedious calculations and mathematics an art of calculation that is build on memorization, which increases in complexity from the times tables up to integration, while the fundamental ideas and deeper connections never become clear to the learner.
How the PISA-study showed in 2003, this dilemma has developed in the last decades to such an extent, that the level at German schools in mathematics and physics has declined from an earlier assumed top position to a now “ measured” weak mediocre level. Teachers
What is the reason for this problem? We think one of the important reasons can be found in the subject-specific education of the future teachers at the universities! Studies to become a teacher have been considered as an stripped down appendix of the education of scientists and were treated as such. The teachers at the schools determine in their respective subjects the quality of the education and the interest of the next generation! Their very important role in society as multiplicators was neglected in relation to the education of the future researchers representing the subject. The resulting lack of recognition for students preparing to become teachers certainly contributed to not having enough young teachers to fill all open positions in physics and mathematics.
Two developments in the immediate past worsened this development and made it clear, that a turn by 180 degrees was necessary:
The attempt to serve “old wine in new skins “ , i.e. to cram the time-tested degree programs and the content, that had grown due to the scientific progress into the shorter bachelor degree, lead to partial chaos and a general unhappiness with the study conditions.
In 2005 Siegfried Grossman came in discussions with the author to the conclusion, that it is a fundamental mistake to mix the subject specific education of teachers with that of researchers. He demanded specially developed Sui Generis curricula for the studies preparing for the teaching professions, that are directed at the future teaching job and that take into account the actually available time in the studies for teachers, which in Germany is tightened by future teachers having to study two different subjects.
Understanding the subject and the connections should be paramount, not detailed knowledge and specialized skills. In 2006 the DPG produced a careful analysis and documentation and thus made the sui generis studies
DPG-Memo a general demand of the colleagues organized in the DPG.
In order to realize this vision it would be counterproductive to base our actions for school and scholars on past conditions or wishful thinking. We should accept today’s conditions but also the technical possibilities in a positive spirit. The gymnasiums ( German schools leading to the “Abitur”) are not any more institutions for the elite, but will in future lead half of all children to the “Abitur”. Our children and grandchildren grow up in world with many stimulations and diversions, but have media skills that neither their parents or grand parents had, for example the knowledge and playful dexterity with the media and devices of information and technology. This digital book is the attempt to put the above mentioned studies on a foundation that makes use of these skills and dexterity. In this book an important subset of the mathematical foundations is embedded in a systematically evolving text and presented with the help of numerical simulations and visualized in many ways. The PC takes care of the often tedious calculations. Thus the user can concentrate on understanding the subject matter, the context and the algorithms used.
Since all the simulations are interactive and can in many cases also be used for scenarios that are totally different from the given ones, the students are thus given a quasi “experimental” access to mathematics. We make use of the fact, that a visual impression is more intensive and permanent than a heard or read one, and that experience based on ones own action results in deeper understanding than the mere reception of someone else’s knowledge.
In addition playfulness is given free range to visually experience and grasp the intellectual stimulation and aesthetic beauty of mathematical structures. The books provides colleagues in physics and mathematics with a thesaurus of simulations for the development of their own curricula.In addition to textbooks this thesaurus gives Physics students the possibility of deepened understanding of fundamental mathematical notions and physical phenomena’s. Future teachers experience the potential of modern media for the realization of interactive lessons in mathematics already during their schooling. Interested students can attempt a playful introduction to a higher level of mathematics; they will probably have less trouble with the techniques used than some older people. For the simulations the package Easy Java Simulation (EJS) is used, that provides a simple fast-tracked introduction to the development of simulations in Java. The files produced with EJS are very transparent, can be easily changed and reused as building blocks for ones own developments. The author considers EJS as prime candidate to become the standard program for didactically oriented simulations.
The authors of EJS, Francesco Esquembre and Wolfgang Christian, allowed me to supplement the text that is dedicated to the introduction to parts of mathematics with a systematic appendix with more than 500 simulations from physics, g for which I owe them many thanks. Francesco Esquember has also assisted me personally in many ways with creation of the mathematical simulations. I also thank Eugene Butikov for the possibility, to include his wonderful cosmological simulations. I want to give many thanks to Siegfried Gro�mann for the dedication and care that he applied to critically reading the text and the simulations and for the many valuable hints, that contributed to the final version. Ernst Dreisigacker, the general manager of WEH-foundation, has supported me with the careful correction of details and lively discussions.
In the last three years I had many deep discussions with Werner Martienssen about a book of two volumes with a similar goal, i.e. to assist reforming and improving the physics education of future teachers, which is supposed to be published as soon as possible. The first volume of this book will represent the fundamental connected structure of physics, while a second volume will present the state of research in individual fields of interest, written by a prominent representative of the respective field. The idea to write this digital introduction to mathematics came up during these discussions.
I want to thank my wife Doris for the loving understanding, with which she tolerated my absentmindedness during the time, when this work was written. I promise improvement!
15.12.2009 Dieter R��