Mathematical Sense, Mathematical Sensibility: The Role of the Secondary Geometry Course in Teaching Students to be Like Mathematicians.

Abstract

How can the secondary Geometry course serve as an opportunity for students to learn to be like a mathematician — that is, to acquire a mathematical sensibility? In the first part of this dissertation, I investigate what might be meant by "mathematical sensibility". By analyzing narratives of mathematicians and their work, I identify a collection of dispositions that are characteristic of mathematical practice. These dispositions are organized as a set of dialectical pairs; collectively, they comprise a partial model of mathematicians' practical rationality. In the second part, I analyze data from a corpus of study group meetings among experienced Geometry teachers. In these meetings, teachers were confronted with representations of hypothetical practice — classroom scenarios, in the form of animated vignettes, in which teachers and students engage in mathematical discourse around the work of solving Geometry problems. The ensuing discussions among teachers touch on a wide range of topics, many of which correspond to the mathematical dispositions identified in the first part of the study. I analyze these records and show that, while teachers sometimes acknowledge the relevance of the mathematical dispositions, they generally do not hold themselves accountable for teaching those dispositions to their students; rather, they cite institutional factors (time, curricular mandates, etc.) as constraints on their ability to teach students a mathematical sensibility. In the third part of the dissertation, I turn to a collection of exam questions written by a Geometry teacher who made an avowed effort to teach students to "think like a mathematician". The questions are coded after the dispositions identified in the first part of the study, and it is shown that many (but not all) of the elements of the mathematical sensibility indeed played a prominent role in the work students were held accountable for. Moreover, a longitudinal analysis shows that the structure of assessment and the nature of the assessment items evolved over a three-year period. I use the notions of cohesion and adaptation to identify these changes, and to account for them as evidence of how practice adapts in response to feedback from the milieu of teaching.