A main concern of geometry education around the world is the continued unsatisfactory level of geometric thinking among teachers. For example, Gutierrez and Jaime (1999) found that many preservice teachers had the same poor concept images of the concept of altitude of a triangle as primary or secondary students. De Villiers (2010) states that very little progress in the quality of geometry instruction is likely to be achieved, unless teachers’ geometric thinking is improved. Teachers’ thinking and knowledge related to the mathematical definitions is another area of concern among the mathematics educators. Definitions play an important role in teaching and learning geometry. Mathematical definitions should distinguish a concept with certainty and be minimal. In other words, definitions may contain only necessary and sufficient conditions required to identify an example of the concept. Other critical attributes may be derived from the definition. Linchevsky, Vinner & Karsenty (1992) have reported that many preservice teachers do not even understand that definitions in geometry have to be economical and that they are arbitrary. Furthermore, Aytekin and Toluk Ucar (2011) found that most inservice teachers are not able to generate correct and economical definitions for quadrilaterals.

Van Hiele theorized that students’ geometrical thinking progresses through a hierarchy of five levels. According to the van Hiele theory, formal definitions are meaningful at Level 3, since students begin noticing the interrelationships between the attributes at this level. De Villier (1998) suggests three kinds of meaningful definitions for each of the first three Van Hiele levels. Visual definitions are characteristics of the first level since at this level, students use visual reasoning. At the second level, students begin to notice that different shapes have different attributes but relationships between the attributes are not understood and the resulting definitions are uneconomical. At the third level, correct, economical definitions are used because relationships between attributes are perceived by the students. Students' definitions at Level 1 and 2 would tend to be partitional,  whereas definitions at Level 3 are hierarchical. Tsamir, Tirosh and Everson (2008) point out that “one of our major aims, as educators, is to bring our students to use only critical attributes as the deciding factor in identifying examples and forming geometrical concepts.” (p.83). Therefore, definitions of mathematical concepts and the process of defining are fundamental parts of the subject matter knowledge of mathematics teachers, because teachers’ knowledge of mathematical definitions influences their curricular and pedagogical decisions (Zazkis and Leikin, 2008). Definitions generated by inservice and preservice teachers may serve as a tool for examining their understanding and geometrical thinking of the specific concepts involved. The purpose of this study is to investigate inservice and preservice elementary teachers’ understanding of square, rectangle, trapezoid, and parallelogram as reflected by the definitions they generate.

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