Darmok at Tanagra Cunningham and Kehle at Bloomington Gauss With Chalk in Hand
This essay is the first of three short reflexive papers intended to identify the issues and implications that result from viewing mathematics education through a semiotic lens. By mathematics education I mean to include consideration of mathematics itself as a discipline of on-going human activity, the teaching and learning of mathematics, and any research that contributes to our understanding of these preceding enterprises. More specifically my current interests are in disentangling the confusion among the mathematics education community regarding the epistemological foundations of mathematics, the meaning and usefulness of constructivism as a theory of learning, and how these two issues are related to the learning and teaching of formal mathematical proof. Because I have found interdisciplinary approaches to the study of most anything both more fruitful and more enjoyable, I will employ such strategies in these papers. As a result, it may not always be clear that mathematics education is my main concern--please rest assured that it is and that if I gain insight of value in that domain I will do my best to render to Caesar what is his.
When Captain Picard and the Enterprise meet the Tamarians they encounter a communication problem that is eventually revealed by Data and Troi to be due to the Tamarians' "unusual", or as a less diplomatic Federation member might say "impaired", ability to use abstraction. Furthermore, as Raphael Carter points out on his WWW site, Data skates on even thinner ice when he concludes that a Tamarian's ego structure doesn't allow for what we think of as self identity. As a result the Tamarians communicate by citing highly specific incidents from their past that serve as metaphors in present situations. No wonder that the crew of the Enterprise has difficulty interpreting a language that consists solely of phrases such as: "Darmok at Tanagra", "The river Temarc, in winter!", "Shaka, when the walls fell.", "Kailash! When it rises." and "Kiazi's children, their faces wet." Picard, independent of Data and Troi, reaches an even more useful understanding of the language through a period of forced co-habitation on an alien planet with the captain of the Tamarian ship.
My first reaction to this episode was one of fascination with such a creative exploration of metaphor use in language. Even though I read, liked, and learned from Lakoff and Johnson's analysis of the role of metaphors in our lives, it was "Cunningham on the e-waves" who framed the important question: "HOW DIFFERENT IS THAT FROM WHAT WE (IN THE FEDERATION) DO?" In light of this challenge, I now have two specific issues to explore.
My initial reaction relates to my on-going quest to understand what it is we do when we say we are doing mathematics. Varela, Thompson, & Rosch (1992) and Kieren et al. (1995) seem to offer support, though my understanding of their work is very...