Publication Date


Date of Final Oral Examination (Defense)


Type of Culminating Activity


Degree Title

Master of Arts in Education, Curriculum and Instruction


Curriculum, Instruction, and Foundational Studies

Major Advisor

Jonathan Brendefur, Ph.D.


Michele Carney, Ph.D.


Philip P. Kelly, Ph.D.


Students’ misunderstandings of the equal sign have been well documented in children as young as kindergarten. Misconceptions of the symbol (=) hinder students’ relational thinking and impede access to algebraic contexts. Symbolic equations (e.g., 4+2=__+3) have been widely used to test students’ understanding and communication of equivalence. The purpose of this study was to explore how first grade students communicate their understanding of equivalence when instruction involved using a bar model versus symbolic equations. It used a two-case study approach to compare an atypical instruction design (the bar model) to a traditional design (symbolic equations). Distinct patterns in communication and changes in students’ understanding were found in both cases. In addition, the bar model group used specific, concrete methods of communicating their thinking earlier and more frequently, which suggests using the bar model along with symbolic representations may be an effective instructional design.