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One of the most important goals of mathematics education is to improve students’ problem solving skills, which can only be realized by teachers who are well-trained in this field. In this context, the purpose of the studies on the subject is to investigate the models used by elementary school teachers to solve verbal problems and their opinions in this process. A multiple-case study was conducted for this study which employs the Problem Information Scale comprised of eight open-ended questions. The study sample consists of a total of 100 elementary school teachers. Six of them were selected for individual interviews on the basis of theoretical sampling. The data obtained revealed that the elementary school teachers experienced problems with students aged from 7 to 11 in the use of models to solve verbal problems and they make mistakes by assigning values and replacing variables such as x, y, a with shapes such as !, ∆ in the problem equation, instead of using models to solve problems. Introdunction and Theoretical Background Improving students’ problem solving skills is one of the most important aim of mathematics education. While solving problems students not only use their mathematical knowledge they already gained but also improve their knowledge and understanding leading them to a better mathematical insight (Çamlı and Binbaşı, 2009; Okur, Tatar and İşleyen, 2006; Williams, 2003; Olkun and Toluk, 2002; Schoenfeld, 2002; Taplin and Chan, 2001; National Council of Teachers of Mathematics, 2000; Jonassen, 2000; Roth, and McGinn, 1997; Reusser and Stebler, 1997). Therefore, problem, the structure of problem solving, and improving success in problem solving have been investigated by many educators (Kılıç and Samancı, 2005). Recent curriculum reforms in some Australian states (e.g. Department of Education Tasmanian, 2002) have highlighted the importance of thinking, communicating and instilling deep understanding in our students and have seen a re-emphasis on mathematical problem solving as an important mechanism for enhancing these skills. Problem solving is, of course, not a new idea in mathematics education. Over half of a century ago, the importance of problem solving was recognized (Brownell, 1942, cited in Suydam, 1980) and its importance was emphasized strongly throughout the 1980’s (Suydam, 1980). Polya (1957) and others (e.g. Branca, 1980; Schoenfeld, 2002) maintain that problem solving is the goal of mathematics learning while the NCTM(2000) go further saying that problem solving “is not only a goal of learning mathematics but also a major means of doing so” (Beswick and Muir, 2004). As problem solving has increasingly become more important in Australian Journal of Teacher Education Vol 35, 4, July 2010 26 mathematics education, the importance of examining problem solving processes and teachers’ approaches toward these processes regard has also increased. The attempts at reforming curricula and mathematics education have frequently underlined the need to integrate problem solving into all grades and every mathematical subject. Therefore, teachers’ perceptions of problem solving processes and their beliefs about problem solving have become a significant research subject (Kayan and Çakıroğlu, 2008). Problem solving should not be simplified by reducing it to simply answering a mathematical question. Problem solving is a way of thinking; it requires reconsidering what is learned and using it in all mathematical activities (Barb and Quinn, 1997). It is asserted that the solution of a problem depends not only on the calculation skill, but also on specific types of knowledge (domain-specific knowledge). The studies in the literature define the types of knowledge as linguistic/factual knowledge, schematic knowledge, algorithmic knowledge, and strategic knowledge and underlines that individuals should possess these types of knowledge to solve a problem (Karataş and Güven, 2003). As only calculation skill, i.e. operational knowledge is highlighted in the process of problem solving; students are observed to fail in the problems that require comment and analysis. It is observed that students do not usually experience much difficulty with standard problems which can be easily solved using basic operations without the need for any strategic formulation, but they have difficulties with problems that cannot be solved quickly using mathematical operations and require mathematical models and interpretation (Soylu, 2007a; Soylu, 2007b). To ensure meaningful learning in mathematics, while teaching the lesson, we need to attach importance to the use of problems that could help students understand the concepts concerning the subject, see the operations taking place between these concepts, and establish connections between concepts and operations (Soylu and Soylu, 2006). As a matter of fact, for Cognitive theorists, comprehension and understanding occupy a significant place in problem solving (Slavin, 2008). When the presence of such difficulties is obvious in problem solving, studies have revealed that by clearly identifying the process of problem solving and applying suitable teaching techniques, children can acquire problem solving strategies (Koray and Azar, 2008; Özkök, 2005; Yazgan and Bintaş, 2005). Mathematics is a lesson involving abstract concepts and the relations between these concepts. Abstract concepts are usually difficult to be acquired by students in the concrete operational stage. However, this difficulty could be overcome by concretizing the abstract concepts of mathematics during instruction (Baykul, 1999). Petit and Zowojewski (1997) point out in their study that a teacher should employ the appropriate method and techniques to facilitate problem solving. A child’s cognitive development should be taken into account when determining these techniques. Primary school teachers should perform problem solving activities by taking into account their students’ cognitive development. They should use concrete activities and models more at primary school level since they teach students who are in the concrete operational stage (7-11 years) (Albayrak, 2000). A child cannot be directly introduced to a mathematical concept. Instead, s/he should be presented with the concept by way of using mathematical models. Thus, the child can construct this mathematical concept in his/her mind by performing some operations on these models. The model for a mathematical concept could be a picture, a drawing, a symbol or a concrete instrument that contains the relationship that this concept conveys. Teacher’ should duty to concretize abstract concepts by way of using various models. A multiple number of different models should be employed when concretizing abstract concepts. The highest level of conceptual understanding will be achieved when students learn a concept using multiple models. If we let students experience the same concept using different ways and under different circumstances, but within a similar structure, they will discover that the Australian Journal of Teacher Education Vol 35, 4, July 2010 27 concept does not depend on a single physical model and thus isolate the common characteristics of these experiences (Olkun and Toluk, 2004). However, many teachers complain that they cannot solve problems without using equations since they become used to solving problems using equations during their high school and university years. It is not sufficient for a teacher to solve a given problem using equations. He or she should also be able to explain the solution of the problem to his or her students by drawing shapes or schemas without using equations. It is not suitable for the cognitive development of elementary-level students to solve a problem using equations by replacing the unknown value with a variable such as x, y, a or b. Sometimes, teachers might reflect their own problem solving techniques to their students. Thus, teachers should get accustomed to problem solving methods that are suitable for the cognitive development of their students. It might at times seem impossible to a teacher to solve a basic operations problem without equations (without using algebra); however, it is a fact that all the basic operations problems at elementary school level can be solved by drawing suitable shapes or schemas (Tatar, Okur and İşleyen, 2005). In accordance with these studies, this study aims to identify the models and the approaches used by elementary school teachers while solving verbal problems. The motive behind conducting the present study has been to determine the positive and negative approaches displayed by elementary school teachers while solving verbal problems and whereby to inform them about these approaches. By informing teachers about it, it is aimed that they will replace their current negative approaches displayed when solving verbal problems with positive approaches. Thus, teachers’ use of such positive approaches when solving verbal problems will positively contribute to students’ abilities to solve verbal problems. Under this theoretical framework, this study attempts to answer the following research questions: • What are the approaches and models used by elementary school teachers to solve verbal mathematical problems? • Are the approaches and models used by elementary school teachers to solve verbal mathematical problems appropriate for students to achieve competency in this field that problem solving skills?

The Models Used by Elementary School Teachers to Solve Verbal Problems