Toward theory advancement in mathematical cognition and teacher cognition

This thesis is concerned with mathematical cognition and teacher cognition, two of the subfields within mathematics education research. Within each there is a broad range of diverse theories that cultivate varied understandings of complex phenomena in mathematical thinking, learning, and teaching. However, the abundance and diversity of theories can polarize perspectives and foster the development of narrow and restricting theoretical accounts. This thesis uses existing theoretical tensions to stimulate the development of more powerful theoretical accounts by coordinating theoretical perspectives in mathematical cognition and teacher cognition. The thesis consists of three articles, which aim to blend opposing theoretical perspectives to reveal complementarity in the field of mathematical knowing and learning, challenge assumptions to reveal restrictions in the field of teacher knowledge, and portray some complex phenomena that cannot be accounted for using intuitive models of teacher noticing. These articles link apparently disparate approaches, revealing the complexity of the phenomena under consideration and the limitations of existing theoretical accounts for them. The first article blends theoretical perspectives from two local theories of mathematical cognition (abstraction-from-actions and abstraction-from-objects) to present a bi-directional, dynamic, non-linear view of mathematical concept formation. The second article examines teacher cognition, discussing existing conceptualizations of mathematics teacher knowledge, revealing their limitations, and offering alternative views that direct attention to underexplored issues. The third article examines teacher cognition from the perspective of the construct of teacher noticing, drawing on insights from cognitive science and the applied science of human factors to develop a model of teacher noticing which challenges intuitive assumptions and views individual and environment as interdependent and inseparable. It is hoped that these contributions add value to the field by advancing knowledge, providing links between previous conceptualizations, and offering fresh insights and theoretical views.