Problem solving is an important component of mathematics education, mainly because it provides an environment for students to reflect on their conceptions about the nature of mathematics, and develop a relational mathematical understanding (Lester, 1994). As students solve mathematics problems, they acquire ways of mathematical thinking, habits of persistence, and confidence in unfamiliar situations that serve them well outside the mathematics classroom (NCTM, 2000). Problem solving is given value in mathematics education as a skill to be taught, as a goal for mental development, and as a method for teaching (Brown, 2003). Given its importance, for the last three decades, there have been attempts in many countries around the world to make problem solving the focus of school mathematics rather than being an isolated part of mathematics curriculum. As the emphasis given to problem solving is widespread, it has become vitally important both for pre-service and in-service teachers to have an understanding about the place of problem solving in mathematics education. Teachers play a crucial role in changing the ways that teaching takes place (Handal & Herrington, 2003). Especially, their beliefs have a considerable effect on the nature of classroom practices (Wilkins & Brand, 2004; Ball, 1998), and have a close relation with their students’ beliefs, attitudes and performances (Grouws, 1996; Schoenfeld, 1992). The present study provides insight into how Turkish pre-service elementary mathematics teachers view problem solving in mathematics education. More specifically, the present study attempted to investigate what pre-service elementary mathematics teachers believe about (a) the importance of understanding why a solution to a mathematics problem works, (b) problems that cannot be solved by following a predetermined sequence of steps, (c) time consuming problems, (d) problems that have more than one way of solution, (e) the kind of mathematics instruction emphasized in the new Turkish elementary curriculum, and (f) technology usage while solving mathematics problems. In addition, the present study attempted to examine how pre-service elementary mathematics teachers assess a number of mathematics problems regarding their appropriateness in elementary mathematics education, and then to conclude whether their assessments about mathematics problems are in line with their beliefs about problem solving.

Sample of the present study consisted of 244 senior undergraduate students studying in Elementary Mathematics Teacher Education programs at 5 different universities in Turkey. All of the participants were chosen to be senior pre-service elementary mathematics teachers so that they have sufficient background in their subject area and pedagogy. Out of 244 pre-service teachers, 113 were males, and 131 were females. A belief survey was administered as the data collection instrument. This survey was prepared by the researchers mainly by making use of four previously developed instruments in the related field (Emenaker, 1996; Hart, 2002; Kloosterman and Stage, 1992; Zollman and Mason, 1992), with several modifications and additions in the light of literature review. The instrument consisted of two parts. In the first part, pre-service teachers were asked to evaluate five mathematics problems as either poor, average or strong according to their educational value and appropriateness in elementary mathematics education, and then to explain the reasons of these evaluations. In the second part, pre-service teachers were asked to indicate their agreements or disagreements to 39 questionnaire items, written on a five-point Likert scale. Data were collected in spring semester of 2005-2006 academic years

Data were analyzed using descriptive statistics. In general, the results of the questionnaire items revealed that pre-service teachers held positive beliefs about mathematical problem solving; that is, their beliefs were in line with the current reform movements in mathematics education. For instance, they gave importance to understanding why a solution to a mathematics problem works, and appreciated developing different ways of solutions to the same problem. However, it was also clear that they still held several traditional beliefs about problem solving. For example, most of the pre-service teachers tended to view problem solving as a reason for practicing computational skills, and supported following predetermined sequence of steps while solving problems. Similar to the findings of the questionnaire items, pre-service teachers’ judgements about mathematics problems were generally consistent with the theory and expectations cited in the literature. For example, they gave value to problems that are interesting in wording, offer a level of challenge, require mathematical thinking, and connect mathematics with real life and with other subject matters. However, the results also revealed that the pre-service teachers did not prefer problems that do not cover any objective in mathematics curriculum. Also, they did not like problems that have complicated numbers. Especially, they did not give value to problem that do not involve any numbers or require spending time. In short, the results of this study indicated that the pre-service teachers gave importance to solving problems in mathematics education. However, they considered mathematics teaching and learning as focused on applying the knowledge gained during the lessons, and mastering computational skills by solving mathematics problems. Moreover, they gave value to problems that require mathematical thinking and reasoning. However, at the same time they prefer problems that are directly related with mathematics curriculum, and do not require spending too much time. Although the current study was based on a sample in Turkish universities, the findings may have some degree of value for European level. Understanding pre-service teachers’ beliefs in different contexts and cultures could be a first step in improving practices of teacher education programs. In order to gradually challenge pre-service teachers’ negative beliefs about mathematical problem solving, adequate educational interventions should be planned and implemented in teacher education programs. Mathematics teacher educators need to examine their undergraduate courses both related to mathematics content and teaching, in order to offer courses more consistent with the research based understanding of the problem solving. In addition, instructors from different teacher education programs across Europe can professionally interact with each other on regular basis, and conduct seminars and workshops in order to share experiences and knowledge about teacher education. Engaging in these professional activities can be of great value both for teacher educators and pre-service teachers to challenge their beliefs about mathematics teaching, and become more aware of the current trends in mathematics education.