On Nov 25, 2013, at 6:07 PM, Gary Tupper <firstname.lastname@example.org> wrote:
> Actually, the gap would be smaller than 'tiny' since its width would be zero. We should remember that any physical representation of a mathematical object is a model. The objective of our model is essentially to impart information - the open circle on a curve is interpreted to mean that a single point is missing.
This might be your definition, but that would include algebraic expressions and d/dx and everything written. I prefer some granularity. By my definition, to be a physical model, or just model for short, it must appeal to our innate physical senses. If we put two segments next to each other and one is longer than the other, that is sensed, without any pre knowledge of anything needed.
Symbols and notation on the other hand are not innate at all. If I showed you a line with a hollow dot, and you had no previous instruction, you wouldn't figure out what that meant in a million years. Some people confuse "symbols that make sense" with innate sense. Hollow dots existed long before someone serendipitously came upon the idea to represent closed points with solid dots and open points with hollow dots. In effect, they invented a use of symbols that already existed, and that part makes sense. But until you are told what they are for, they make no sense at all.
> Take, for example, the graph of y=x^2 /(x-1) : > Usually drawn as a parabola with a hollow dot at (1,1).
I don't think that expression is a parabola, it is a hyperbola, but let's go with your visual, a parabola with a hole at (1,1). The position of the hole (its location) satisfies my definition of "model" because there are several (physical) properties that the student can feel innately. If (1,1) is a minimum then it will be felt as being at the bottom. But the hollow dot satisfies only my definition of symbol. Without being told what a hollow dot stands for, the student will never get it. The whole graph itself is a diagram and contains elements that model and elements (symbols) that communicate but only with pre agreed upon meaning. Similarly, drawing hashes across congruent segment in a geometric figure (diagram) are symbols. They communicate, but only after being told what they mean. That two sides "look" equal in length doesn't require any pre-instruction at all.
> BTW, although we are usually interested in the graph of the solution of y=x^2/(x-1) over a region of the plane, and therefore draw the hollow dot at (1,1), we should realize that the division by x-1 affects the whole plane - ie the whole line x=1 is 'removed'. Maybe the easiest way to visualize this is to ask about 'truth': > So, for y=x^2/(x-1), what about (2,4)? It satisfies the equation - ie 'true', What about (2,3)? - it generates 'false'. What about (1,1)? And what about (2,1)?
Now you are talking about mental visualization, not physical. Other than that, I agree that this is one of the mental exercises you should go through. It plays off the notion of sets and we know how well that helps sort this stuff out. But I would not call this or any single method the "easiest" way to visualize. Some methods are good and some very bad, and this may be one of the best, but no single method, model or analogy covers enough of the many facets of these constructions well enough alone.
The mapping is very easy to state formally, as is the case in most of mathematics. y = x^2 for all x except x = 1. But to comprehend and fathom its consequences takes time and thinking about it in different ways.