Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)


Curriculum & Instruction

Committee Chair

Signe Kastberg

Committee Member 1

Rachael Kenney

Committee Member 2

Jill Newton

Committee Member 3

Jacqueline Sack


In this manuscript, I describe a study of pre-service secondary mathematics teachers’ (PTs) understanding of geometric reflection in terms of a motion and a mapping views. PTs often have a motion view of geometric reflection based on their understanding of reflection line, domain, and plane. A motion view is a preliminary perspective developed prior to the construction of a mapping view. PTs need a mapping view of geometric reflection, and to be conscious of sub-concepts of a mapping view involved reflection line, domain, and plane. However, there is no clear evidence documenting how a learner’s motion view evolves to produce a mapping view. A clinical interview methodology was used to describe how mental structures occur in the movement between PTs’ motion view and the mapping view. Also, factors critical to the transition from a motion view to a mapping view were explored. Four case studies were constructed from transcript audio records, videos, and written works. Ongoing and retrospective analyses using Dubinsky’s action, process, object and schema (APOS) framework were used to examine PTs’ mental structures. The results indicated that the motion view transforms into the mapping view through the development of mental structures associated with three important sub-concepts of geometric reflection. These three sub-concepts are reflection line, domain, and plane. The results further indicated that there are series of factors that impact the development from the motion view to the mapping view. These factors are perpendicularity and equidistance properties, the role of reflection line, type of figures (circle, semicircle, interior and exterior points of the figures), the operational definition of the plane, and relations between figure and plane.