From 4/21/98
This is a first draft…I suspect more will be forthcoming from me tomorrow….
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The study of chaos arose as scientists attempted to better understand how non-linear systems function. Linear systems had long been understood, that is the mathematics and physics of linear systems were well within the grasp of those who studied linear systems. Linear systems are orderly and predictable. For example, in the linear system/variable of predicting the mass of certain objects, the researcher only needs to know that mass of the object under investigation to predict the mass of other combinations of that object. Specifically, if a researcher weighed a brick, the weight of that brick could accurately be used to predict the weight of various combinations of that brick (2 of the same thing, 3 of the same thing, etc.). If this example was describing a non-linear system/variable, then knowing the mass/weight of the one brick would not necessarily help a researcher to predict the mass/weight of other combinations of that object.
In non-linear systems, researchers can make certain predictions as long as they know the pattern of the variable, as opposed to the quantity of the variable, within the non-linear system. The measurement of sound is a good example of this. If a sound is emitted and it is measured with a sound level meter to be 30dB SPL (sound pressure level,) a researcher knows that sound is non-linear (in the decibel scale) and simply adding two similar sounds together would not yield a sound of 60dB SPL. In fact, the resultant sound would be only 36dB SPL. Researchers know that because when they studied sound, they discovered a pattern. The pattern is this: when a sound is “doubled” (as in the case of playing two sounds of 30dB SPL at the same time) the resultant sound is an increase of +6 over the SPL of the two sounds. Along those lines, if the two sounds are played that yield a sound of 36dB SPL and then they are doubled (another sound of 36dB SPL is added) the resultant sound would be measured at 42dB SPL. It is in this sense that non-linear systems/variables differ from linear variables. Scientists found the pattern by experimentation and by knowing that pattern they could accurately predict future occurrences of the phenomena.
While scientists were immensely successful in discovering the patterns to a number of non-linear systems/variables (as in the case of sound previously mentioned), one system/variable continued to elude their understanding. This system was that of turbulence and the pattern of regularity remained invisible to them for a number of decades. This is where I believe we are in the point of our discussion with regard to learning patterns and coming to know. There is a point in the process, and while I believe we have focused our discussion on the issue of MOO’s/technology I would suspect that what we discover will be applicable to learning in general, where I don’t understand or recognize the pattern any more. This is the point we are trying to label. At the beginning of the learning process, learning IS linear. The learner moves in a rather straightforward path toward some goal and along the way combines and incorporates a variety of data sets to move the learner forward in the journey. It is during this time that learners acquire a number of facts, figures, and pieces of data. Ultimately, for those who persist, the process changes at some point and learners achieve new levels of knowing without accumulating new facts, but by seeing old facts in new ways.
This is the point – the point where a learner achieves a new level of knowing. My readings on the science community’s attempts to uncover the mystery of turbulence, a particularly elusive non-linear system, lead me to believe that what the scientists were seeking and have discovered about turbulence is similar to what I’m seeking and to what I sense.
Mitch Feigenbaum, in his studies of the turbulence problem, made the most significant advances in his attempt to explain the pattern of turbulence. This occurred in the 1970’s although previous work of other scientists contributed greatly to his ultimate success. He developed a “theory of universality.” This theory describes the pattern that Feigenbaum discovered in his work that was totally unexpected and, once known, explained why scientists had previously had such difficulty understanding what happens when a system changes from stable (linear) to turbulent (non-linear). He discovered that of least importance was the specific variables under manipulation and measurement. What was important was the number that resulted when he compared the changes in numbers for each experimental case. This number remained constant without regard for the variable he was examining. Specifically this meant that if he measured changes in velocity, pressure, or mass, a single number emerged that reflected the changes for each of the variables he measured. The number explained the changes that occurred across ALL variables, and that was the pattern that had remained obscured for so many years. He, in later years, explained, ” In the end, to understand you have to change gears. You have to reassemble how you conceive of the important things that are going on.“