Date: Apr 18, 1995 11:24 PM
Subject: regarding infinity

I have been followiing with interest the thread about showing fifth graders
all about infinity. (Let me set my bias right out front: I teach high school
and really am not terribly aware of what 5th graders can and cannot do
or should or should not be expected to assimilate). Anyway, I think we need to
be a tad careful about "showing them different sizes of infinity by considering
number lines." There ARE INDEED different "sizes" of inifinity, but to offer
the set of evens versus the naturals versus all integers really does them
a disservice, since all these particular sets are the SAME size of infinity.
(Even though at first blush it would seem "logical" -- dangerous word, that __
that there are "clearly "lots more" integers than even positive numbers.
ANd even moving to the set of all "fractions" (i.e. rationals) won't help since
that again is a set of the "same " size infinity as the previously mentioned.
It is not until you move to the set of REALS (say on the inetrval from 0 to 1
or any segment you might desire) that you have to move "up" to a truly
"larger" infinity....and I think at this point you have left 99.9999% of all
fifth graders (and probably many high school students) in your wake.

What you might do is to see if they can understand that there are the "same number of ppoints" on a segment of length 1 as on a segment of length 3...and not
deal directly with what that "number" is. You can do this by drawing the short segment
somehwat above the long segment and then drawing two lines which connect their
ends (and which will then meet in a point P somewhere above the two segments.
Now ask them if a particular point on the short one has a "mate" on the long -- yep, just run a line from P through your point until it meets the long and
that is the unique mate. How about the other way? No problem, do the same thing.
So every point on one segment has exactly one mate on the other segment and vice
versa....i.e., they can be matched one for one and so the two segments have the
same number of points. This is devilishly hard to convey via text without drawing
but I hope you get he idea.
Anyway, more power to you and when the fifth graders start clamoring to know
all about aleph-null and the number of the continuum, let me know..... :-)