The philosophy of mathematics is the branch of philosophy whose task is to reflect on, and account for the nature of mathematics. This is a special case of the task of epistemology which is to account for human knowledge in general. The philosophy of mathematics addresses such questions as: What is the basis for mathematical knowledge? What is the nature of mathematical truth? What characterises the truths of mathematics? What is the justification for their assertion? Why are the truths of mathematics necessary truths? (Ernest, 1991)

The philosophy of mathematics is undoubtedly an important aspect of philosophy of mathematics education, especially in the way that the philosophy of mathematics impacts on mathematics education. Philosophy of mathematics education asks a very pertinent question by posing a trichotomy. Is the philosophical focus or dimension:

1. Philosophy applied to or of mathematics education?

2. Philosophy of mathematics applied to mathematics education or to education in general?

3. Philosophy of education applied to mathematics education? (Brown, 1995; Ernest, 2012)

Schwab's (1961) four 'common places of teaching' to the mathematics curriculum. These are the subject (mathematics), the learner of mathematics, the mathematics teacher, and the milieu of teaching, including the relationship of mathematics teaching and learning, and its aims, to society in general.

Ernest’s (2012) asks some questions about learning mathematics: What is the role of the learner? What powers of the learner are or could be developed by learning mathematics? How does the identity of the learner change and develop through learning mathematics? Does learning mathematics impact on the whole person for good or for ill? How is the future mathematician and the future citizen formed through learning mathematics? How important are affective dimensions including attitudes, beliefs and values in learning mathematics?

While teaching mathematics, infinity is one of the basic concepts for many mathematical concepts like numbers, which has been the center of attention for centuries (Kim, Sfard & Ferrini-Mundy, 2005). According to İşleyen (2013) center of mathematics and philosophy, concepts such as the concept of infinity, that occupy the human mind, are less common. In terms of students, the concept of infinity is a complex and abstract concept (Kolar & Cadez, 2012). Looking at the secondary school mathematics curriculum, there is no gain in the concept of infinity. Students use the concept of infinity more intuitively in these years (Özmantar, 2008). At the high school and university curriculum, infinity is located under the other subjects like limit, derivative, integral.

Why is it that, given the same content, preservice mathematics teachers learn so differently? While learning university mathematics at a university with preservice mathematics teachers, We realized that their thinks about the concept of infinitiy often varied. The main questions of this study are,

What do preservice mathematics teacher know about concept of infinity?

What are the preservice mathematics teachers’ philosophies of mathematics and mathematics education?


How are preservice mathematics teacher's philosophies of mathematics and mathematics education appear in learning practices? 

The aim of this study will be understanding how preservice mathematics teachers' philosophical perspectives are appear in their learning process. The study employs explanatory design that is one of the mixed method designs in which both quantitative and qualitative research techniques are used. While the quantitative stage of the research is designed in survey model, case study method is used in the qualitative stage. The sample of the quantitative stage consists of preservice mathematics teachers (n=45) who will take the calculus course in the faculty of education of a university in the spring semester of the 2018-2019 academic year. The 'Educational Beliefs Scale' developed by Yılmaz, Altınkurt and Çokluk (2011) scale will be used to determine preservice mathematic teachers’ educational philosophy.The sample of the qualitative stage consists of preservice teachers’ (n = 5) selected by the maximum variation sampling method among the same group. We will chose to do several cases to understand this relationship between philosophical perspectives and instruction. Following Merriam's (1988) suggestions for case study research, data were collected by a formal interview, and display collection.

Further analysis of data is ongoing. The results will have presented in conference.

Brown, S. I. (1995). Philosophy of mathematics education: Philosophy ofmathematics education newsletter,8. Ernest, P. (1991) The Philosophy of Mathematics Education, London: Falmer Press. Ernest, P. (2012). What is our first philosophy in mathematics education? For the Learning of Mathematics, 32(3), 8–14. İşleyen, T. (2013). Ortaöğretim öğrencilerinin sonsuzluk algıları. Kastamonu Eğitim Dergisi, 21(3), 1235-1252. Kim, D., J., Sfard, A., & Ferrini-Mundy, J. (2005). Students’ colloquial and mathematical discourses on ınfinity and limit. Paper presented at the Conference of the International Group fort he Psychology of Mathematics Education (29th, Melbourne, Australia, Jul 10- 15, 2005), 3, 201-208. Kolar, V. M., ve Cadez, T. H. (2012). Analysis of factors influencing the understanding of the concept of infinity. Educational Studies in Mathematics, 80(3), 389-412. Merriam, S. (1988). Case study research in education: A qualitative approach. San Francisco: Jossey-Bass Publishers. Özmantar, F. ( 2008). Sonsuzluk Kavramı: Tarihsel Gelişimi, Öğrenci Zorlukları ve Çözüm Önerileri. Eds. M.F.Özmantar, E. Bingölbali ve H.Akkoç. Matematiksel Kavram Yanılgıları ve Çözüm Önerileri (s.151-180). Pegem Akademi, Ankara. Schwab, J. J. (1961). Education and the structure of the disciplines. In I. Westbury & N. J. Wilkof (Eds.), Science, curriculum and liberal education: Selected essays of Joseph J. Schwab (pp. 229–272) Chicago: University of Chicago Press, 1978. Yılmaz, K., Altınkurt, Y. ve Çokluk, Ö. (2011). Eğitim inançları ölçeğinin geliştirilmesi: Geçerlik ve güvenirlik çalışması. Kuram ve Uygulamada Eğitim Bilimleri, 11(1), 335-350.