A good mathematical competence can be reached through “the sense of awareness and control over what to do and how to do it” (Lucangeli & Cornoldi, 1997, p.123). Therefore, metacognition supports the learners of mathematics to be flexible and strategic to complete complex tasks/problems. It allows learners to acquire new skills and to use existent skills effectively while solving challenging problems/tasks (Desoete, Roeyers, & Buyyse, 2001).

In order to create a mathematical learning environment where learners can benefit from metacognitive activities, the realistic problem-solving situations should be provided (Mayer, 1998). The problems posed should allow students to be metacognitively active from the beginnings to the final steps of problem-solving process (Desoete et.al. 2001). In addition, poor and good problem solvers can be recognized through their use of their metacognition; especially its components (Lucangeli & Cornoldi, 1997).  Problem-solvers need to manage their metacognitive skills and knowledge that result in knowing what, how and when to use various skills in problem solving process (Mayer, 1998). Therefore, the learners need to be metacognitive skillful that provides the repertoire of skills in management of problem-solving process (Veenman et.al., 2004). Moreover, metacognitive knowledge also contributes to problem-solving process through providing knowledge about how to use different strategies(Schraw,1998).

Given the importance on the role of metacognition in problem-solving, developing student metacognition is an important role for teachers. Teachers might provide the classroom activities that journaling, process reflection and self-reflections for problem-solving metacognitively (Darling-Hammond, Austin, Cheung, & Martin, 2008). Furthermore, modeling thinking strategies and scaffolding are provided as teachers’ impetus to active students’ metacognition. However, before providing certain sources to develop student metacognition, the teachers are expected to be a good problem solver who used their metacognition effectively (Vander Walt & Meree, 2007).  Therefore, teacher education should be arranged around teachers’ use of their metacognition during problem solving. Pre-service teachers should be given appropriate tasks/problems so that they can use their metacognition effectively during problem solving process (Demircioğlu, Argün & Bulut, 2010). Research showed that more challenging tasks (e.g. tasks requiring intelligent guess, drawing etc.) result in high frequency of metacognitive behaviors of pre-service teachers. In addition, the researchers explored the types of metacognitive behaviors to regulate problem solving process effectively (e.g. Demircioğlu, Argün, & Bulut, 2010; Yimer & Ellerton, 2010).

In the light of the literature, the aim of the study is to explore metacognitive activities of pre-service mathematics teachers (PMTs) employed during problem solving through Problem Solving Model (Yimer & Ellerton, 2010). There are certain studies conducted based on Problem-Solving Model (e.g. Truelove,2013; Yimer & Ellerton, 2010). This study is an extend study; it has been extended on prior studies in a way that in this study PMTs reflected on their solution process right after they solved the problems while watching themselves.  In addition, the study is delimited by PMTs the aim is to explore PMTs metacognitive activities during problem-solving before they start teaching and right after they completed their education.

The study is significant because to develop students’ metacognition teachers need to be metacognitively active problem solvers. Exploring PMTs’ metacognitive behaviors during problem solving process informs both in-service and pre-service teacher educators on where to start developing teacher metacognition, and how to contribute to teacher metacognition before entering to the field. 

The research question of the study is “What are the metacognitive activities employed by PMTs during mathematical problem-solving process?”

This study is a qualitative research adopting multi-cases study approach. The cases are three PMTs as Gökay, Kadir and Simay (pseudo names). I used purposeful sampling that is appropriate for qualitative research since the experience of the participants in the area is important (Creswell,2009). I chose the three participants because they all have similar educational backgrounds(e.g.taking same mathematics related courses) and they had quite experience in solving problems for the purposes of teaching and learning. I collected data during solving a two-parts algebra task provided by Truelove(2013) through interviews. In the algebra task, there was a three sequential pattern given. PMTs were asked to find the pattern through explaining, drawing or stating algebraically and to find the number of tiles in 20nd figure. In the second part, they were expected to find the figure that includes at least 10000 tiles. The reason why I chose the task from Truelove(2013) for the data collection purposes is that the pattern task can be solved through multiple ways. Audio-recorded interviews took 20-25 minutes. I conducted a think-aloud interview and a follow-up interview. First of all, I asked them to reflect on their solutions; tell me what they were doing and why. Then I started the interview before watching their video-recordings with “What did you think when you saw the problem?”, and I finished the interview through asking “Would you like to revise any of it?” and “Do you want to add anything?”. For data analysis, I used pre-assigned coding scheme to code the transcribed interviews. The problem-solving model prepared by Yimer and Ellerton (2010) was used as coding scheme. In Yimer and Ellerton (2010)’s model, there are phases and each phase is consisted of metacognitive behaviors/activities. The phases are engagement, transformation-formulation, implementation, evaluation and internalization respectively. The phases are coded as themes in this study. In addition, in each phase there are various metacognitive behaviors/activities (e.g. reflecting on the problem, formulating a plan, assessing the plan, assessing for reasonableness of results). These metacognitive behaviors/skills are coded as categories. First of all, I analyzed how PMTs followed the routes from engagement, transformation/formulation, implementation, evaluation and internalization; whether they followed suggested route by the Problem Solving Model. Then I analyzed what kinds of metacognitive behaviors/activities are used by PMTs while following the route and how the problem solving process is related to the use of metacognitive behaviors/activities.

The preliminary findings showed that each case used different ways to solve the algebra task. Kaan used two formulas which referred the same pattern; he chose one formula to complete the task. Simay found the formal formula and used it for two parts. Gökay found the formal formula in the second part of the task, but he did not use or define a formula for the first part. Kadir and Simay used a variety of metacognitive behaviors. They started with engagement phase, followed by transformation, implementation, and evaluation. They used certain phases again and followed the path provided by Yimer and Ellerton (2010). For example, Simay firstly engaged in the problem, formulated a plan and then implemented the plan. When she realized something is wrong, she assessed the plan for consistencies and possible errors, after evaluation, she went back implementation phase, and after implementing she evaluated her results with respect to consistencies and appropriateness of the results. For Gökay, although the frequency of metacognitive behaviors is high, he had a limited variety of metacognitive behaviors in his repertoire to complete the task. He did not monitor and evaluate most of his actions. He did the calculations through intellectual guessing. Gökay did not could not complete the task. All in all, the results showed that while solving the problem using a variety of metacognitive behaviors helps PMTs to complete the task. Although one could not reach the correct solution, certain metacognitive behaviors explored during problem-solving. However, the effective uses of metacognitive behaviors and the phases of problem solving can lead learners to correct solution (Yimer & Ellerton, 2010).

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