“I Understand” Talk in Script Writing: A Case from Euclid’s Elements

The pursuit of understanding in communication between interlocutors of different knowledge is one of the most distinct characteristics of teaching as a professional occupation. Consequently, mathematics teacher preparation programs should provide opportunities for the prospective teachers to put themselves in the shoes of more and less knowledgeable individuals who interact on mathematical matters. Our study is on one of such opportunities: a task of writing a script for a dialogue between fictional characters of historically different backgrounds, who discuss a particular proof from Euclid’s Elements, in the context of number theory. Twenty-three prospective secondary teachers responded to the task. We enquired what do the script writers identify as presumed points of difficulty in the given proof and what explanations are constructed by the participants to result in “I understand” claims. Three types of potential difficulty were attended to: number theory concepts and terms, claims within the proof, and the central idea of the proof. The points of difficulty were explained in various ways, such as producing procedural definitions and translating Euclidian terms into the modern terms. Implications for research on script writing and on the use of historical sources in mathematics education are drawn.