UC Berkeley

Mathematical definitions are central to learning and doing mathematics. Research has uncovered significant differences between how mathematicians and non-mathematicians construct, reason about, and refine mathematical definitions. Various strands of research provide insight into the development of definitional practices, yet an integrated approach to addressing foundational questions of development is lacking. Further, existing approaches often fail to account for the conventional nature of mathematical definitions and the social nature of defining practices. In the first paper of this dissertation, I present a sociocultural framework in which mathematical definitions are treated as cultural forms that emerge to serve specialized functions in collective practices. Building upon Saxe’s (2012) treatment of the development of mathematical ideas, I describe the development of definitional practices as constituted through processes of micro-, onto-, and socio-genesis of definitional form-function relations. In the second and third papers of this dissertation, I present two empirical studies that utilize the framework described in the first paper to examine similarities and differences in undergraduate mathematics and humanities students’ approaches to defining mathematical terms related to rational number. Study 1 examines participants’ conceptualizations of the function and forms of mathematical definitions. Study 2 examines participants’ process of defining, with a focus on the formulation and iterative refinement of definitions for three foundational mathematical ideas – fractions, multiplication, and division. Results provide evidence of two distinct definitional practices. Mathematics students tended to define for communication and proof, as they constructed and frequently revised definitions aimed at setting a standard for how a word form could be used across a disciplinary community. Humanities students tended to define for learners, constructing definitions aimed at advancing understanding and providing procedural instructions, often defining by example. Findings imply that while college mathematics majors are being enculturated into mathematical practices of defining for a disciplinary audience, work needs to be done to provide pathways for more students to engage in mathematically authentic practices of defining.