I was a physics and astrophysics undergraduate major in college. I love math and physics, couldn't get enough of it. On one exam, as a senior, we were allowed to bring in a 3x5 inch notecard with laws of physics on it. We didn't need the entire card. Physics is almost entirely approaches and derivations, generally using math, from very few basic laws.
After conceptually isolating and extracting a subset of the world as a model, the math involves labeling the parts of the model, symbolically manipulating the model into a new state, and then reasoning from the new state back to the real world to see if what you just derived actually works in practice. The math, to me, makes much more sense in a context where it is connected to reality at each end than where it is simply floating in space as something to learn on its own. Also, its quite relevant to realize that the process of model building and dissecting a model out of context is a crucial step to this actually being useful in the real world, can easily be done wrong, and is a part where system dynamics experience could be helpful.
Regardless, the math involved is primarily an activity of manipulating strings of symbols. This is not required by mathematics, but is typically what is used.
And therein lies the problem. If our practices had grown up after the development of image processing hardware, it's possible we would go about this a different way, and that battle rages in artificial intelligence today. Here's why: images are robust against noise, and symbol strings are not.
I can take a picture of George Washington on a $1 bill and randomly change half the pixels to black or white, and a human can still see George there. Images are, for the most part, quite forgiving of point-wise errors. They are also quite dense in information, or can be. ("One picture is worth 1000 words", etc.).
Symbol-string manipulation on the other hand is exquisitely sensitive to point-noise. If I randomly change one symbol in the middle of a derivation to something else, the entire derivation becomes instantly worthless. It is not even "mostly right" -- it is simply wrong.
So, in order to use symbol-string manipulation successfully, a huge amount of self-discipline, structure, and orderly behavior is required. This is not negotiable.
For whatever reasons and whatever you think of it, students in the USA today are not, in general, capable of such self-discipline. The culture as a whole not only does not support it, but is actively hostile to such behavior, unlike in Shanghai, to pick an example from the news. A long discussion of psychosocial determinants of "self-discipline" is out of place here, but I'll simply assert that much current thinking puts the determinants "outside the box" and outside the "student" and says that, in the absence of a strong cultural support system, an "individual" is simply not capable of intense self-control.
The point is, however, that there is no way that students, lacking self-control, can ever possibly utilize mathematical techniques of symbol string processing and get correct results, or get any kind of encouraging feedback. It won't work. It can't work. It can't be made to work. "Close" doesn't cut it.
In short, due to the nature of symbol-processing, "self-discipline" is a necessary pre-requisite for math, which is a pre-requisite for physics and other hard sciences. The discussion about HOW to teach math is therefore effectively irrelevant, to all except a few lucky teachers in classes with students who are, in fact, psychosocially and culturally prepared for the subject.
I pursue this line of thought somewhat further here:
http://newbricks.blogspot.com/
In short, I conclude there that nationally we are not likely to solve the self-discipline problem in the next generation, at the rate we are going, and if we wish to "catch up" with the Chinese we need to either figure out a new mathematics using image processing or some other technique that is, in fact, robust under noise, or we need to start working as small clusters or teams of individuals structured so that the team is noise-resistant, despite the error-proclivity of each member. Again that is a total shift in the focus of education, to see the intended "learner" is a team, not an individual.
Of course, teams have far more places to "put" learning than do individuals. The Institute of Medicine, interestingly enough, recently concluded that changes in health care practices should be focused on changing the behavior of "Microsystems" -- small teams of people who work as one to accomplish tasks -- not on trying to change individuals, based on increasing evidence that changing individuals doesn't really work. In safety engineering, we find results showing that, say, 74% of commercial aircraft "accidents" occur on the very first day that that TEAM of individuals attempted to work as a TEAM, despite their individual professional expertise. (Source:Bryan Sexton, Johns Hopkins). The real social leap would be letting a team, with a corporate identity, take over the slot called a "job" which we currently allocate only to "individuals". This may be facilitated by having the team teleworking, so that additional desks are not required. Massive social change is required for this to be implemented, although less, I think, than that required to restore a culture that supports discipline to two generations that reject it out of hand as a life-style to be admired.
Unless this problem, separable from "physics" and "math" is addressed, it seems purely academic to argue about how to teach "math" in the abstract. Students will continue to blame physics or math for being "hard" subjects conceptually, when in point of fact they are failing at a different point in the system. Physics, per se, is actually quite easy, if one is already fluent in math. And math, I'm asserting, is conceptually not that difficult either, in context, but is in practice essentially impossible due to lack of the required self-discipline.
Wade SchuetteAnn Arbor, MI
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