Date of Award


Degree Name

Doctor of Philosophy



First Advisor

Dr. Jon D. Davis

Second Advisor

Dr. Steven W. Ziebarth

Third Advisor

Dr. Rose Mary Zbiek

Fourth Advisor

Dr. Christine A. Browning


Representational fluency (RF) includes an ability to interpret, create, move within and among, and connect tool-based representations of mathematical objects. Taken as an indicator of conceptual understanding, there is a need to better support school algebra students’ RF in learning environments that utilize both computer algebra systems (CAS) and paper-and-pencil. The purpose of this research was to: (a) characterize change in ninth-grade algebra students’ RF in solving problems involving linear equations, and (b) determine conditions of a CAS and paper-and-pencil learning environment in which those students changed their RF.

Change in RF was measured by comparing results from initial to final semi-structured task-based interviews using a specifically designed framework based on the SOLO taxonomy. Following a design research approach, an instructional theory was used as a lens and object of analysis to determine conditions of the learning environment that supported RF. This theory was posited prior to the study, tested during a five-week collaborative teaching experiment in which a ninth-grade algebra teacher taught all lessons, and revised during ongoing and retrospective analyses.

Each of three student’s performance on linear equation solving tasks posed in the symbolic representation type was initially characterized at prestructural, unistructural, and multistructural levels of RF. Two of the three students demonstrated relational levels of RF in the final characterization based on similar tasks. This change in RF is attributed to a specifically designed instructional intervention based on an instructional theory that includes: (a) an activity structure for representation-specific tasks and techniques, (b) a learning progression that emphasizes a multirepresentational approach to equivalence of expressions and solving linear equations, and (c) classroom expectations. A revised activity sequence that incorporates the Cartesian Connection earlier in the progression is proposed.

Results suggest that improving one’s RF may be connected to affect and disposition toward mathematics. Tasks and classroom discourse that were specifically designed to focus on reconciling differences between representations seemed particularly powerful. The use of a task-technique-theory framework might support research and practice efforts aimed at instructional design for learning environments that utilize a combined use of tools for doing and learning mathematics.

Access Setting

Dissertation-Open Access