Relations between disciplinary practices and conceptual understanding in mathematics: Leveraging definitional practices to support students' fraction understanding

Amelia Farid

Student understanding of arithmetical operations with fractions is foundational to success in mathematics, yet the domain is notoriously hard-to-learn and hard to teach. At the same time, documents like the Common Core State Standards in Mathematics point to student engagement in disciplinary practices as important components of learning. I argue that students may benefit from engaging with disciplinary practices, like constructing, refining and using definitions of foundational mathematical ideas, toward conceptual understanding and vice versa. In my dissertation research, I use a design-based research approach to support students in a productive dialectic between their development of mathematical practices of defining and their development of conceptual understanding of fraction arithmetic. To understand this dialectic, I study and contrast 9th grade and undergraduate students' participation in an intervention in which they posit and refine of mathematical definitions related to fractions. Mixed methods analyses aim to illuminate the micro-processes by which conceptual understanding and disciplinary practices mutually support each other in student activity, as well as to uncover the effects of the intervention, both on students' conceptual understanding and definitional practice. Beyond contributing to productive instructional approaches for developing conceptual understanding of a critical and foundational mathematical domain, the research will help us better understand the coherence between mathematical content and practice goals ? goals that are emphasized but often treated as distinct in instructional standards and educational research.

About Amelia Farid

Amelia Farid is a doctoral candidate in the Graduate Group in Science and Mathematics Education (SESAME) at the University of California, Berkeley. Her research centers on reconciling dichotomies ? both theoretical and pedagogical ? between mathematics learning as engagement in disciplinary practices (such as proving, problem solving, and defining) and mathematics learning as developing conceptual understanding of rich content. Taking a design-based research approach, her dissertation uses both qualitative and quantitative methods to explore the ways in which mathematical practices of constructing, refining, and using definitions can be leveraged to develop content knowledge of fraction arithmetic, and vice versa. Her work is informed by her experience teaching mathematics at the secondary and tertiary levels, in both formal and informal settings in China, Colombia, and the United States. She has also contributed to the development of robust mathematics curricula aimed at the progress of rural communities in Latin America with the Foundation for the Application and Teaching of the Sciences (FUNDAEC). As part of her dissertation work, she has mentored a number of undergraduate researchers in educational research theory and methodology. Amelia holds a B.A. in mathematics from Columbia University and an M.A. in mathematics from the University of California, Berkeley.