A quantization exercise that I have always found stimulating is the
determination of the properties of "objects" in closed boxes, based on
indirect measurements of the mass of the boxes. I used machine nuts in
The idea was to start with _no_ assumptions about what was in the boxes,
but to begin massing them. With enough measurements a pattern emerges,
and a histogram might be suggested as a way of visualizing the data. Even
that requires consideration of bin width, and a thoughtful assessment of
what precision is really necessary for the experiment (i.e., students
always think that greater precision in the measuring instruments implies
better results - this just "hides" some of the pattern in this analog).
Anyway, I usually spike the samples so that a single object never seems to
exist, that there may be an odd gap in the clusters, and that there seems
to be a decreasing observation frequency with mass (according to a
specific law that I determine beforehand).
The point of this is to thoroughly explore student reasoning behind their
explanations that, for example, the objects are likely to be the same
because of the clusterings of observations and equal differences between
the clusterings (except for the missing ones!), their justification of and
determination of the mass of an object (do _not_ give them the box mass),
and their guesses as to what distribution may be behind the observation
frequency. Interestingly, to avoid going insane with effects that can
make the results less clear I had to make the "unit" mass = 3 machine
nuts, and this flies in the face of the simplest explanation ("unit" mass
= 1 object, Occam's razor)) but should arise in the discussion as a
possibility quite naturally.
It's fun, lively, and gets students thinking quite hard about
observations, quantization, models and explanations based on models.