A Combination of Historical Physics Documents and Other Teaching Tools for the Instruction of Prospective Teachers in Chaos and Complexity

Abstract

This work presents and analyses a didactic methodology for teaching aspects of Chaos and Complex Systems in Physics to prospective Science educators. The methodology includes historical Physics texts and experimental instrumentation, as well as computer models and simulations. The objectives are mainly to help undergraduate teachers realise the way that Physics evolves through changes and standoffs, and the way in which scientists work, which is much related to teaching the Nature of Science (NoS). At the same time, through this teaching methodology, there is an attempt to instruct undergraduate students in basic elements of Chaos Theory and Complexity Theory, by avoiding heavy mathematical formalism incompatible with their age and their learning level and ability. This teaching sequence is intended to be applied in a pilot study involving undergraduate students of the Department of Primary Education, University of Athens, so as to generate initial qualitative and quantitative results.

20.6 Appendix

The exact excerpts from Poincaré and Lorentz, used in the teaching sequence.

1.1 20.6.1 Extract from Poincaré

…it seems that chance alone will decide. If the cone were perfectly symmetrical, if its axis were perfectly vertical, if it were subject to no other force but gravity, it would not fall at all. But the slightest defect of symmetry will make it lean slightly to one side or other, and as soon as it leans, be it ever so little, it will fall altogether to that side. Even if the symmetry is perfect, a very slight trepidation, or a breath of air, may make it incline a few seconds of arc, and that will be enough to determine its fall and even the direction of its fall, which will be that of the original inclination.

A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that that effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But, even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.” (Poincaré 1914, pp. 67–68)

1.2 20.6.2 Extract from Lorentz

“In Fig. 43 [note: corresponds to Fig. 20.2 of the chapter], we see a copy of fifteen months of the somewhat faded original output, divided for display purposes into three five-month segments.

The chosen variable is an approximate measure of the latitude of the strongest westerly winds; a high value indicates a low latitude. There is a succession of “episodes,” in each of which the value rises abruptly, remains rather high for a month or so, and then drops equally abruptly, but the episodes are not identical and are not even equal in length, and the behavior is patently non-periodic. At one point I decided to repeat some of the computations in order to examine what was happening in greater detail. I stopped the computer, typed in a line of numbers that it had printed out a while earlier, and set it running again. I went down the hall for a cup of coffee and returned after about an hour, during which time the computer had simulated about two months of weather. The numbers being printed were nothing like the old ones. I immediately suspected a weak vacuum tube or some other computer trouble, which was not uncommon, but before calling for service I decided to see just where the mistake had occurred, knowing that this could speed up the servicing process. Instead of a sudden break, I found that the new values at first repeated the old ones, but soon afterward differed by one and then several units in the last decimal place, and then began to differ in the next to the last place and then in the place before that. In fact, the differences more or less steadily doubled in size every four days or so, until all resemblance with the original output disappeared somewhere in the second month. This was enough to tell me what had happened: the numbers that I had typed in were not the exact original numbers, but were the rounded-off values that had appeared in the original printout. The initial round-off errors were the culprits; they were steadily amplifying until they dominated the solution. In today’s terminology, there was chaos. It soon struck me that, if the real atmosphere behaved like the simple model, long-range forecasting would be impossible. The temperatures, winds, and other quantities that enter our estimate of today’s weather are certainly not measured accurately to three decimal places, and, even if they could be, the interpolations between observing sites would not have similar accuracy. I became rather excited, and lost little time in spreading the word to some of my colleagues. In due time, I convinced myself that the amplification of small differences was the cause of the lack of periodicity. Later, when I presented my results at the Tokyo meeting, I added a brief description of the unexpected response of the equations to the round-off errors.” (Lorenz 2005b, pp. 134–135).