Jay's definition of reflection as seeing alternatives for one's
ongoing actions or attitudes, and the various observations about
visual, kinesthetic, and other modalities besides the verbal with
which this can be done, are extremely useful insights into a tough
problem. That contemplation --which always is verbally based--
is used as a near-synonym for reflection in English, points to
the subtlety and difficulty of these concerns. As a mathematics
educator I find myself fighting against students' propensity to
engage in the manipulation of algebraic symbols based on subtle
spatial cues in the notational system by providing more verbal
access to the structure of equations and expressions. Perhaps
it is the subject matter that recommends this kind of verbalized
form of reflection to me. Perhaps, just my own biases.
David Kirshner
Louisiana State University