The mathematical definition of a quadrilateral is an expression that involves necessary and sufficient conditions for that quadrilateral. The concepts of mathematics often have several equivalent definitions. Different definitions can be used in different times. The choice of a definition useful for a particular task is decided by the context.

According to the van Hiele model of geometric development, learners move through five hierarchical levels of understanding named as: visualization, analysis, informal deduction, formal deduction, and rigor (van Hiele, 1986). As this model proposed, at level 1, learners perceive a figure as a whole shape and do not perceive its parts. They can identify names, compare and operate on geometric figures according to their appearance. At (the) 2^nd level, learners operate on certain objects, namely classes of figures and discover properties for these classes, but they cannot logically order the properties. At level 3, properties are logically ordered; one property precedes or follows from another property. But at this level, the meaning of deduction, that is, the role of axioms, definitions, theorems, and their converses are not understood. At level 4, learners prove theorems deductively and at level 5 different axiomatic systems can be understood by the learners. De Villers (1998) explained the levels operationally particularly for the concepts of the definition. As he explained, at first level visual definitions can be generated, for example a rectangle is a quadrilateral that looks like this (showing or drawing one) or describes it in terms of visualcharacteristics (such as like two long and two short sides). At 2nd level, learners can produce uneconomical definitions; for example they give many properties which are more than necessary to define the concept. For example for a learner at this level, a definition of a rectangle can be as a quadrilateral with opposite sides parallel and equal, four right angles, equal diagonals, half-turn-symmetry, two axes of symmetry through opposite sides, two long and two short sides, etc. When the learners reached the level 3, they can generate economical and correct definitions, as a rectangle is a quadrilateral with an axis of symmetry through each pair of opposite sides.

This study attempts to answer of the question: “how well 8^th grade students define a rectangle?” Definitions produced by 8^th graders may serve as a guide for studying their understanding of the meaning of definition, equivalent definitions, deduction and class inclusion between rectangle and parallelogram. Besides when the definitions they generated are examined, we can understand their van Hiele geometric thinking levels. The focus was the concept of rectangle, since it is one of the most familiar quadrilaterals in the school mathematics curriculum. The students faced with this concept since their preschool days.

Ahuja, O. P. (1996). An Investigation in the Geometric Understanding among Elementary Preservice Teachers Paper presented at the ERA-AARE Conference in Singapore, 29 November, de Villiers, M. (1998) To teach definitions in geometry or teach to define? Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education, 2, 248−255. van Hiele, P. M. (1986) Structure and Insight: a theory of mathematics education (Orlando: Academic Press).