As in all fields, mathematics has its own concepts (such as number, proportion, equation, etc.). In mathematics, definitions of these concepts are as important as these concepts. Because definitions help to form the concept appropriately, to distinguish it from other concepts and to express mathematical ideas (Çakıroğlu, 2013). The concept definition is a collection of words used to describe the concept, also the definitions are divided into two as informal and formal (Tall and Vinner, 1981). Informal definitions refer to expressions in which individuals explain their ideas and experiences about the concept. Formal definitions mean the definitions accepted by mathematicians and included in the books (Vinner, 1991). However, Tall and Vinner (1981) state that most of the definitions used by individuals are not formal definitions. In other words, individuals use the representations in their minds when defining the concept (Cornu, 1991; Vinner and Hershkowitz, 1983). However, some of the representations put forward by individuals in this defining process may not be compatible with the formal definition of the concept (Tall and Bakar, 1992).

When the studies conducted with teachers and prospective teachers in the context of the concept definition in the literature were examined, it was seen that researchers generally focus on limit (Domingos, 2009), derivative (Duran and Kaplan, 2016), functions (Vinner and Dreyfus, 1989), solid objects (Ubuz and Gökbulut, 2015) and quadrilaterals (Okazaki and Fujita, 2007) etc. In these studies, it was revealed that most of the teachers and prospective teachers could not define the mathematical concepts correctly. Unlike other studies, the concept definition the number pi was focused in this study.

When the literature was examined, it was seen that there were studies focusing on the number pi (Archer and Ng, 2016; Kurtuluş, 2015; Tavşan and Pusmaz, 2019). In his study, Kurtuluş (2015) examined the views of students and teachers about the competition which included questions about number pi. In this context, he determined that students learned what they did not know about number pi through the competition; also teachers put forward such as contributing to the learning because of the fun and it allows to consolidate the learned knowledge. In their studies, Archer and Ng (2016) focused on discovering what is number pi through mathematical modeling for 6th and 7th grade students. As a result of this study, they stated that students had the opportunity to see what kind of relationship there was between circumference and diameter of the circle and that acquired knowledge could be transferred to daily life. On the other hand, Tavşan and Pusmaz (2019) concluded that none of the expressions put forward by the 8th grade students in the context of definition clearly explain the number pi. However, it observed that some of the studies were mostly conducted in the context of irrational numbers, but in part of these studies number pi was mentioned. Ercire (2014); Haifa and Saikaly (2016); Sirotic and Zazkis (2007); Temel and Eroğlu (2013) and Zazkis and Sirotic (2010) determined that students handled number pi as 3; 3.14 or 22/7 . Adıgüzel (2013); Çevikbaş and Argün (2017) found that prospective teachers thought the value of pi as 22/7. Erdem and Man (2018) ; Güler (2017) concluded that mathematics teachers perceived number pi as 3.14 or 22/7. It was seen that in these studies mostly were focused on the numerical value of pi.

Shulman (1986) states that the mathematics educator aiming to teach a particular concept should know the definition of the relevant concept correctly to avoid misleading students. Because the teachers who know the definitions of mathematical concepts correctly and assimilate these definitions are confident in themselves and they can give satisfactory answers to the questions asked by the students (Cohen, McLaughlin and Talbert, 1993). Accordingly, it was thought that it is important to reveal the current situation of prospective teachers in the context of the definition of the number pi and to carry out studies aimed at completing their deficiencies, correcting them if they have misunderstandings, to help in their career. Taking into account all of these, the aim of this study was to examine the concept definitions made by prospective elementary school mathematics teachers in the context of number pi.

In this study, a case study method which is one of the qualitative research models was used. Participants of the research were selected by the easily accessible sampling method. The study group consisted of 103 prospective elementary school mathematics teachers studying at a state university in the Black Sea Region in the spring term of 2018-2019 academic year. 32 of the prospective teachers were in 1st grade, 30 of the prospective teachers were in 2nd grade, 26 of the prospective teachers were in 3rd grade and 15 of the prospective teachers were in 4th grade. The test consists of an open-ended question, such as ―Define what is the number pi.― was used as the data collection tool. After this process completed, one-to-one interviews were conducted with four prospective teachers who used different definitions in accordance with the principle of volunteering in order to examine the descriptions made by prospective teachers in more detail. The definitions made by the prospective teachers were categorized by thematic analysis approach. The stages suggested by Boyatzis (1998) were applied during the analysis of the data. Accordingly, firstly the code list was obtained with the help of the data and, if necessary, the codes obtained in previous studies were added to the list. In this context, it was seen that the code of “ratio of circumference to diameter in the circle” obtained in the study was the definition of number pi accepted by mathematicians (Çağlayan, Dağdelen and Korkmaz, 2018) and this code was considered as formal definition code. Then, the compatibility of the codes obtained with the data was reviewed and codes were arranged and categories were formed. Later, findings were presented through direct quotations obtained from the interview processes. In addition, the frequency and percentages of the prospective teachers' responses were calculated to determine the distribution of their definitions.

As a result of the study, it was seen that the majority of prospective teachers defined the number pi as “ratio of circumference to diameter in the circle”. However, it was determined that the distribution of this definition increased from 1st grade to 4th grade. In addition to these, it was found that some of the prospective teachers defined number pi as “ratio of circumference to radius”, “ratio of radius to circumference”, “an irrational number”, “mathematical constant‖. Also, it was determined that some of the prospective teachers defined the number pi by using numerical values as “the number that is 3.14”; “the number that approximately value is 3.14”; “the number that goes on as 3.14...”.

In this study, it was focused on the concept definitions made by prospective elementary school mathematics teachers in the context of the number pi. Since knowing the definition of a concept cannot reveal fully understanding of the concept (Vinner, 1991), different studies can be conducted about what else prospective teachers know about the number pi and how they relate this information.