The purpose of this study is to see how preservice teachers understand mathematical definitions within a geometry context. Yet, within the collegiate mathematics coursework, many preservice teachers do have struggles with some of the basic geometric concepts. Consequently, this study specifically looks at the geometric definitions of quadrilaterals and how preservice teachers use those definitions to form a holistic understanding of quadrilaterals.

Using the Action-Process-Object-Schema (APOS) theory as the theoretical framework, the study proposes a preliminary genetic decomposition for the concept of the hierarchical properties of special quadrilaterals. Data is analyzed from interviews and class documents of twenty-six preservice teachers as to whether they used the constructions from the preliminary genetic decomposition or other constructions not considered.

Due to the importance of mathematical definitions in preservice teachers’ background preparation for future field work, this study proposes the following questions:

1) What are preservice teachers’ understandings of geometric definitions?

I. What are preservice teacher’s personal definitions for special quadrilaterals?

II. How do preservice teachers apply the distinction between necessary and sufficient conditions for a mathematical definition?

2) How does the understanding of geometric definitions contribute to preservice teachers’ understanding of special quadrilaterals?

I. Are preservice teachers able to perceive and use the hierarchical nature of special quadrilaterals?

II. Are preservice teachers able to discern equivalent definitions for special quadrilaterals?

Based on the results of the data analysis, the genetic decomposition is revised. Finally, the study concludes with pedagogical recommendations for teaching the concept of special quadrilaterals and suggestions for further research on this topic.