It isn't that I disagree with all of these neat discussions, but these discussions are meaningless without proper development.
What use is it to ask "What instruments and controls might we count as continuous versus discrete?" of students that haven't even fully developed the concept yet? You seem to think that students have some innate understanding of continuity, intermediate value theorem, discrete and analog, and have been just itching to discuss it with someone. These are great discussions for students that have already tackled these concepts, but not for students that haven't.
What you are doing here is simply thinking up good examples of continuity, but they are only good if you understand continuity. They are confusing to someone that doesn't. The kids would not know what you are talking about. Not without development.
Can you tell me how you picture the students' minds unravelling these examples into a generic theory of continuity?
On Nov 14, 2012, at 1:08 PM, kirby urner <email@example.com> wrote:
> On Wed, Nov 14, 2012 at 12:25 AM, Robert Hansen <firstname.lastname@example.org> wrote: > > << snip >> > >> Common Sense: Draw a curve without lifting the pencil. >> >> Formal Reasoning: What does that actually mean and why is it significant? >> >> Bob Hansen > > When I was in high school, they had this symbol, a little open circle > (not colored in) that you could put in a graphed (x,y) line to show a > discontinuity. "No defined value of y for this value of x" is what it > meant. So you could draw like f(x) = x * x (parabola) and then "shoot > it full of holes" by undefining the function for x = 0.71, 1.95 and -3 > -- if you felt like it. > > What instruments and controls might we count as "continuous" versus > "discrete"? There's a correspondence with our concepts of "analog" > versus "digital". Digital instruments are quantized. Most dashboard > instruments are "analog" if they have needles. Gas tank, speedometer > -- we usually think of these as "continuous" i.e. you can apply the > intermediate value theorem and say: if I went from 40 mph to 60 mph > then I must have passed through every possible speed in between, at > least briefly. > > In the case of what's in your gas tank, we could talk about Avogadro's > Number and how many moles of a polycarbon (refined crude oil) you are > sending through your cylinders. In metabolizing or combusting > molecules, you are engaged in a discrete activity and your gas gauge > could be said to be measuring a discrete number, just like grains of > sand come in discrete numbers. > > Computers are said to be discrete devices, but then so are films and > videos discrete. Around 24 frames / sec is sufficient to create a > sensation of continuity i.e. no gaps in the action. Frame rates of 30 > and above are common. The color value of each pixel is controlled by > some 32 on-off toggles, or 255 * 255 * 255, where each integer 0 <= x > <= 255 is represented by 8-bits. Red, green blue (RGB). > > Playing with RGB to control pixel colors is something to do early and > often in math class, if you have any access to recent technology. > > Pixels are another good example of discrete entities given the > appearance of continuity or "analog smoothness". This will inspire > some students to ask whether perceived reality might be considered > "discrete" at some level as well, and in terms of rods and cones, > neurons firing (or not) the answer is yes: there's usually a way to > take any analog phenomenon and model / represent it in digital terms. > > Kirby