Many research studies showed that the mathematical understanding pre-service teachers bring from schooling and university mathematics courses was inadequate for teaching elementary school mathematics (Ball, 1990; Ma, 1999; Toluk Uçar, 2009).  To become a mathematics teacher preservice teachers need to develop profound subject matter knowledge, pedagogical content knowledge and knowledge of students’ cognition (Shulman, 1986; Ball, 1990; Carpenter, Fennema and Franke, 1997; Ma, 1999). Although the profound

subject matter knowledge does not assure an effective teaching and learning environment (Ozgun-Koca, and Sen, 2006), teachers’ understanding of concepts influences how they teach them (Dematte ve Furinghetti, 1999; McLeod, 1994; Thompson, 1984). In addition, as Ma (1999) pointed out, without a solid knowledge of what to represent, a teacher cannot produce a conceptually correct representation even if he/she has a rich knowledge of different teaching methodologies or student development. According to Ball (1990),  teachers’ knowledge of concepts and procedures should be correct; they should understand the underlying principles and meanings, and teachers must appreciate and understand the connections among mathematical ideas. Besides, teachers need to appreciate and understand mathematics as a coherent structurally organized discipline. Teaching mathematics in such a manner first of all requires an emphasis on the coherent picture of the numbers system including its hierarcical structure (Fischbein et al., 1995). In elementary school mathematics, understanding of irrational numbers is critical for the extension of the concept of number from the system of rational numbers to the system of real numbers (Sirotic and Zazkis, 2007). However, many studies well documented that both teachers and students have deficiencies in understanding rational and irrational numbers. According to Tirosh et al. (1998), many preservice teachers based their conceptions of numbers almost entirely on their experience with natural numbers. Fischbein, Jehiam, and Cohen (1995) reported that high school students and preservice teachers were not able to define correctly the concepts of rational, irrational, and real numbers. Many students could not even identify correctly various examples of numbers as being natural, integer, rational, irrational, or real. The present study focuses on the preservice teachers’ mathematics knowledge related to rational and irrational numbers. The purpose of this study was to investigate preservice mathematics teachers’ definitions of rational and irrational numbers, and to determine their success in determining the membership of a given number in diverse classes of numbers (natural, integer, rational, irrational, and real number).

Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. The Elementary School Journal, 90(4), 449–466. Carpenter, T. P., Fennema, E., and Franke, M. L. (1996). Cognitively Guided Instruction: A Knowledge Base for Reform in Mathematics Instruction. The Elementary School Journal, 97(1), 3–20. Dematte, A. & Furinghetti, F. (1999). An exploratory study on students’ beliefs about mathematics as a socio-cultural process. In G. Philippou (Ed.), Mavi-8 Proceedings: Research on Mathematics Fischbein, E., Jehiam, R., Cohen, D. (1995). The Concept of Irrational Numbers in High-School Students and Prospective Teachers. Educational Studies in Mathematics, 29, 1, 29-44. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Erlbaum. McLeod, D. (1992). Research on affect in mathematics education: a reconceptualization, in Grows, D. A. (Ed.), Handbook of Research on Mathematics Teaching and Learning , New York: Macmillan, 575-596. Ozgun-Koca, A. And Sen, A. I. (2006). The beliefs and perceptions of pre-service teachers enrolled in a subject-area dominant teacher education program about ‘‘Effective Education’’. Teaching and Teacher Education, 22(7), 946–960. Shulman, L. S. (1986). Those Who Understand: Knowledge Growth in Teaching. Educational Researcher, 15(2), 4-14. Sirotic, N. and Zazkis, R. (2007). Irrational numbers: the gap between formal and intuitive knowledge. Educational Studies in Mathematics, 65, 49–76 Thompson, A. G. (1984). The relationship of teachers’ conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15, 105-127. Tirosh, D., Fischbein, E., Graeber, A. And Wilson, J. (1998). Prospective elementary teachers’ conceptions of rational numbers’, Retrieved December 15th, 2012 from the World Wide Web: http://jwilson.coe.uga.edu/Texts.Folder/Tirosh/Pros.El.Tchrs.html. Toluk-Uçar, Z. (2009). Developing pre-service teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25(1), 166–175.