Knowing the context of application and using calculation skills are part of the mathematical problem-solving process (M(Montague et al., 1993). In the mathematical problem-solving process, individuals should understand and interpret the information in the problem. They should also make a choice about the operations they will perform and decide on the application (Özkubat & Özmen, 2020). Because the problem-solving process is described as a decision-making and implementation process, individuals engaged in this process must have a belief in mathematical problem solving. The concept of metacognition plays an important role in the development of belief in mathematical problem solving. Metacognition includes the skills required for an individual to be aware of the process of completing a task successfully, monitor themselves, and control their own performance (Flavell, 1979). Metacognitive awareness includes the individual's knowledge about a task and how, when, and where to perform the task (Schraw & Moshman, 1995). In other words, the knowledge about metacognitive skills is described as metacognitive awareness. Knowledge and skills involved in an individual's metacognitive awareness affect behaviors in the mathematical problem-solving process (Gelbal, 1991). Beliefs in problem solving are important because they influence behaviors. In the current study, which is based on this conceptual framework, mathematical problem-solving beliefs are explained with metacognitive awareness. The findings will show the relationship and effect between these concepts.

This study aims to investigate pre-service primary school teachers' metacognitive awareness and beliefs about mathematical problem solving by employing the correlational survey model, which is one of the quantitative research methods. The correlational survey is a research model that aims to reveal the existence and degree of covariance between two or more variables (Karasar, 2014). In this context, the study aims to investigate pre-service primary school teachers' metacognitive awareness and beliefs about mathematical problem solving according to different variables. The study also discusses the findings obtained in light of the relationships that emerge between metacognitive awareness and beliefs about mathematical problem solving.

Participants

The participants in the current research include 284 pre-service primary school teachers enrolled in the Department of Primary Education at an education faculty of a state university in the Aegean Region during the spring term of the 2016-2017 academic year. Table 1 presents the demographic features of the participants. As shown in Table 1, 193 (68%) of the 284 pre-service teachers are female, and 91 (32%) are male. Among the participants, 80 (28.2%) are first-year students, 70 (24.6%) are second-year students, 75 (26.4%) are third-year students, and 59 (20.8%) are fourth-year students. When the grade point averages of the pre-service teachers are examined, 25 (8.8%) have a grade point average of 2.00 or lower, 182 (64.1%) have a grade point average between 2.01 and 3.00, and 77 (27.1%) have a grade point average of 3.01 or higher.

In the analysis of the collected data, a statistical program was used. During the analysis phase, we used both descriptive statistics and inferential statistics. Descriptive statistics were used to reveal the participating students’ level of metacognitive awareness and beliefs about mathematical problem solving. Inferential statistics were used to examine the correlations between groups.

Before starting the analysis of the data in the study, the researchers conducted normality analyses to check whether the data were normally distributed. Figure 1 and Figure 2 show the Normal Q-Q Plot Graphs regarding the normality of the scores obtained from the Metacognitive Awareness Inventory and the Scale of Beliefs about Mathematical Problem Solving.

When Figure 1 is examined, the data are collected around the line standing at an angle of 45 degrees. This shows that the scores obtained from the Metacognitive Awareness Inventory follow a normal distribution.

When Figure 2 is examined, the data are collected around the line standing at an angle of 45 degrees. This result indicates that the scores obtained from the Scale of Beliefs about Mathematical Problem Solving show a normal distribution.

As a result of the analyses, the researchers decided that the data set would be suitable for parametric analysis. They used a t-test to determine whether the scores from the scales vary significantly depending on gender. The researchers interpreted the obtained results at the 0.05 significance level. They conducted Pearson Product-Moment Correlation analysis to determine whether there is a correlation between the participants’ metacognitive awareness and beliefs about mathematical problem solving, as well as to determine the direction of the correlation, if any. When the correlation coefficient is 0.30 or lower, it indicates a “low” correlation. When it is between 0.30 and 0.70, it indicates a “medium” correlation. When it is 0.70 or higher, it indicates a “high” correlation (Büyüköztürk, 2012). Finally, we used a simple linear regression analysis to determine the extent to which the pre-service teachers’ metacognitive awareness predicts their beliefs about mathematical problem solving.

The score obtained by dividing the inventory score by the total number of items indicates the respondent's metacognitive awareness level. A score lower than 2.5 means the metacognitive awareness is "low." A score higher than 2.5 means the metacognitive awareness is "high" (Akın et al. 2007). For the Scale of Beliefs about Mathematical Problem Solving, a score between 0 and 1.67 indicates a "low" general belief. A score between 1.68 and 3.36 indicates a "medium" general belief, and a score between 3.37 and 5.00 indicates a "high" general belief.

Research Question 1: What is the level of pre-service primary school teachers’ metacognitive awareness and beliefs about mathematical problem solving?

We calculated the means, standard deviations, minimum and maximum values, and levels for pre-service teachers’ metacognitive awareness and beliefs about mathematical problem solving. Table 2 and Table 3 present these results.

When Table 2, which shows the descriptive statistics regarding the metacognitive awareness of the 284 pre-service primary school teachers, is examined, the data show that the pre-service teachers’ mean metacognitive awareness score (X = 3.56) is high. The mean scores for the main dimensions of the inventory are also high, with X = 3.60 for knowledge of cognition and X = 3.54 for regulation of cognition. In addition, the mean scores that the pre-service primary school teachers obtained from the sub-dimensions are high as well.

WhenTable 2, which shows the descriptive statistics regarding the metacognitive awareness of the 284 pre-service primary school teachers, is examined, the data show that the pre-service teachers' mean metacognitive awareness score (X = 3.56) is high. The mean http://journals.ums.ac.id/index.php/jramathedu scores for the main dimensions of the inventory are also high, with X = 3.60 for knowledge of cognition and X = 3.54 for regulation of cognition. In addition, the mean scores that the pre-service primary school teachers obtained from the sub-dimensions are high as well. As a result of the current study, the pre-service primary school teachers' metacognitive awareness was found to be high. This result concurs with existing research on pre-service primary school teachers' metacognitive awareness (Baysal et al., 2013)(Erbaş et al., 2018);(Memnun & Akkaya, 2012)). (, 2020) also found that pre-service primary school teachers have metacognitive awareness above the average. Moreover, other studies that include preservice teachers from other branches, as well as pre-service primary school teachers, show that the metacognitive awareness of pre-service primary school teachers is higher than that of pre-service teachers from other branches (, 2020). Studies conducted on pre-service teachers, although not reporting specific results for pre-service primary school teachers, also found high metacognitive awareness (Baş et al., 2016). A high level of metacognitive awareness indicates that pre-service primary school teachers can control learning processes, and it can be described as a desirable situation. The reason why pre-service teachers' metacognitive awareness was found to be high in existing research may be that they discovered their own learning processes because the lessons they took during their education were learning-teachingfocused.

When Table 3 is examined, the data show that the pre-service teachers’ mean score (X = 3.22) on the Scale of Beliefs about Mathematical Problem Solving is medium. When the mean scores from the sub-dimensions of the scale are examined, the data show that the mean scores for mathematical ability (X = 3.76), understanding the problem (X = 3.47), and importance of mathematics (X = 3.60) are high. In addition, the mean scores for the sub-dimensions of place of mathematics (X = 2.24) and understanding the problem (X = 3.31) are medium.

In the current study, the pre-service primary school teachers' beliefs about mathematical problem solving were found to be at a medium level. Studies conducted on pre-service primary school teachers by (Deringöl, 2018) and (Toptaş & Gözel, 2017). (Hacıömeroğlu, 2011) also found that pre-service primary school teachers' beliefs about mathematical problem solving were at the "Agree" level for the sub-dimensions of mathematical ability, understanding the problem, and importance of mathematics. For the sub-dimension of place of mathematics, their beliefs were at the "Disagree" level, and for the sub-dimension of problem solving skill, their beliefs were at the "Undecided" level. In studies with pre-service math teachers, beliefs about mathematical problem solving were found to be positive and high (Gülten & Soytürk, 2012)(Kayan & Çakıroğlu, 2008)(Ünlü & Sarpkaya Aktaş, 2016)(Yavuz & Erbay, 2015). Pre-service primary school teachers generally graduate from high school departments where Turkish-centered courses are emphasized as much as math-centered courses. This situation can explain why their beliefs about mathematical problem solving are at a medium level. The medium level of belief in problem solving also indicates that pre-service primary school teachers are in a position that can be developed toward understanding and solving problems.

problem solving also indicates that pre-service primary school teachers are in a position that can be developed toward understanding and solving problems.

Research Question 2: Do pre-service primary school teachers’ metacognitive awareness and beliefs about mathematical problem solving vary significantly depending on gender?

The results of the t-test conducted to determine whether the pre-service primary school teachers’ metacognitive awareness varies significantly depending on grade level are given in Table 4.

Table 4 shows that pre-service teachers’ metacognitive awareness does not vary significantly depending on gender (t₍₂₈₂₎ = -.94; p>.05). However, the mean metacognitive awareness score for male pre-service teachers ( = 3.61) is higher than that for female pre-service teachers ( = 3.54). When examining the mean scores from the main dimensions of the scale, the mean scores for knowledge of cognition do not vary significantly by gender, but the score for male pre-service teachers (t₍₂₈₄₎ = -.61; p>.05) is higher than that for female pre-service teachers ( = 3.63). Similarly, the mean scores for the regulation of knowledge dimension do not vary significantly depending on gender (t₍₂₈₂₎ = -.84; p>.05). The mean score for male pre-service teachers ( = 3.58) is higher than that for female pre-service teachers ( = 3.52). When examining the scores from the sub-dimensions of knowledge of cognition, the mean scores for declarative knowledge (t₍₂₈₄₎ = -1.40; p>.05), procedural knowledge (t₍₂₈₂₎ = -.26; p>.05), and conditional knowledge (t₍₂₈₄₎ = -.07; p>.05) do not vary significantly by gender. For these three sub-dimensions, the mean scores for male pre-service teachers (_{declerativeknowledge} = 3.71; _{proceduralknowledge} = 3.54; _{conditionalknowledge} = 3.65) are higher than those for female pre-service teachers (_{declarativeknowledge} = 3.60; _{proceduralknowledge} = 3.51; _{conditionalknowledge} = 3.64). When examining the scores from the sub-dimensions of regulation of knowledge, the mean scores for planning (t₍₂₈₂₎ = -1.36; p>.05), monitoring (t₍₂₈₂₎ = -1.34; p>.05), evaluation (t₍₂₈₂₎ = -.24; p>.05), debugging (t₍₂₈₂₎ = -.38; p>.05), and information management (t₍₂₈₂₎ = -1.34; p>.05) do not vary significantly by gender. The mean scores for these four sub-dimensions for male pre-service teachers (_planning = 3.65; _monitoring = 3.56; _evaluation = 3.51; _{information management} = 3.65) are higher than those for female pre-service teachers (_planning = 3.54; _monitoring = 3.45; _evaluation = 3.49; _{information management} = 3.54). However, the mean score for the debugging sub-dimension is higher for female pre-service teachers ( = 3.58) than for male pre-service teachers ( = 3.55).

The current study found that pre-service primary school teachers' metacognitive awareness does not vary significantly by gender. Tunçer (2013) also concluded that preservice primary school teachers' metacognitive awareness does not vary significantly depending on gender. Similarly, (Dilci & Kaya, 2012) reported the same finding in their study on pre-service primary school teachers. Other studies also found that gender is not a factor affecting metacognitive awareness. These include research by (Kışkır, 2011) on preservice teachers, (Özsoy & Günindi, 2011) on pre-service primary school teachers, (Deniz et al., 2014) on pre-service math teachers, and Gül, Köse, and Yılmaz (2015) on pre-service biology teachers. However, some studies report significant gender-based differences in pre-service teachers' metacognitive awareness levels (Belet & Güven, 2011)(Baykara, 2011)(Kana, 2015)Öztürk & Serin, 2020). In light of these results, pre-service teachers' gender does not appear to affect the formation of metacognition or awareness of it. Metacognition can develop at the same level in all pre-service teachers, regardless of gender.

Table 5 presents the results of the t-test conducted to determine whether the pre-service primary school teachers’ beliefs about mathematical problem solving vary significantly depending on gender. We employed an independent samples t-test for this analysis. Examination of Table 5 shows that the pre-service teachers’ beliefs about mathematical problem solving vary significantly depending on gender (t₍₂₈₂₎ = -3.25; p < .05). The mean score of the male pre-service teachers ( = 3.35) is significantly higher than that of the female pre-service teachers ( = 3.17).

The study also investigated whether the mean scores from the sub-dimensions of the scale vary significantly by gender. The results show that the mean scores for the subdimensions of mathematical ability (t (282) = -1.65; p > .05), place of mathematics (t (282) = -1.05; p > .05), and understanding the problem (t (282) = -1.69; p > .05) do not vary significantly by gender. However, the mean scores for the sub-dimension of importance of mathematics vary significantly by gender (t (282) = -2.35; p < .05). The mean score for male pre-service teachers (X = 3.76) is significantly higher than that for female pre-service teachers (X = 3.52). The mean scores for the sub-dimension of problem-solving skill also vary significantly by gender (t (282) = -2.51; p < .05). The mean score for male pre-service teachers (X = 3.35) is significantly higher than that for female pre-service teachers ( X = 3.17).

In the study, the pre-service primary school teachers’ beliefs about mathematical problem solving varied significantly by gender, with male pre-service teachers scoring higher than female pre-service teachers. One possible reason for this difference is that male pre-service teachers may encounter more problem situations in society. (Soytürk, 2011).

also found that beliefs about mathematical problem solving varied significantly by gender among pre-service primary school teachers, but in favor of female pre-service teachers. Other similar studies on pre-service primary school teachers found that gender had no significant effect on beliefs about mathematical problem solving ((Toptaş & Gözel, 2017). Studies on pre-service math teachers show results about the effect of gender that are not similar to the result of the present study ((Baş et al., 2016); (Gülten & Soytürk, 2012); (Kayan & Çakıroğlu, 2008); (Sağlam & Dost, 2014); (Yavuz & Erbay, 2015)).

Research Question 3: Does pre-service primary school teachers’ metacognitive awareness predict their beliefs about mathematical problem solving?

Table 6presents the results of the Pearson Product-Moment Correlation analysis, which examines the relationship between pre-service primary school teachers’ metacognitive awareness and their beliefs about mathematical problem solving. Researchers conducted a simple linear correlation analysis to determine whether a correlation exists between these variables. The results show a positive, medium correlation between metacognitive awareness and beliefs about mathematical problem solving (r = .35; p < .01).

There was a positive, weak, and significant correlation between the sub-dimension of declarative knowledge and beliefs about mathematical problem solving (r = .28; p < .01). The sub-dimension of procedural knowledge showed a positive, medium, and significant correlation with beliefs about mathematical problem solving (r = .34; p < .01). The subdimension of conditional knowledge also had a positive, medium, and significant correlation with beliefs about mathematical problem solving (r = .30; p < .01). There was a positive, medium, and significant correlation between the sub-dimension of planning and beliefs about mathematical problem solving (r = .32; p < .01). The sub-dimension of monitoring showed a positive, medium, and significant correlation with beliefs about mathematical problem solving (r = .36; p < .01). The sub-dimension of evaluation had a positive, medium, and significant correlation with beliefs about mathematical problem solving (r = .34; p < .01). There was a positive, weak, and significant correlation between the sub-dimension of debugging and beliefs about mathematical problem solving (r = .24; p < .01). The sub-dimension of information management showed a positive, medium, and significant correlation with beliefs about mathematical problem solving (r = .33; p < .01).

The data show a positive, medium, and significant correlation between the sub-dimension of mathematical ability and metacognitive awareness (r = .30, p < .01). The data also show a negative, low, and significant correlation between the sub-dimension of place of mathematics and metacognitive awareness (r = -.20, p < .01). There is a positive, medium, and significant correlation between the sub-dimension of understanding the problem and metacognitive awareness (r = .42, p < .01), as well as between the sub-dimension of importance of mathematics and metacognitive awareness (r = .39, p < .01).

The data further show a positive, weak, and significant correlation between the sub-dimension of problem-solving skill and metacognitive awareness (r = .26, p < .01).

Table 7 presents the results of the regression analysis conducted to determine the extent to which pre-service teachers’ metacognitive awareness predicts their beliefs about mathematical problem solving.

Table 7 shows that the regression analysis results indicate a statistically significant relationship between the independent variable, metacognitive awareness, and the dependent variable, mathematical problem solving [R = .35; R² = .13; F_{(1. 282)} = 40.47. p < .05]. The metacognitive awareness independent variable explains 13% of the total variance in the variable of beliefs about mathematical problem solving. The F_(1.282) = 40.47 value shows that pre-service primary school teachers’ level of metacognitive awareness significantly predicts their beliefs about mathematical problem solving.

Researchers found a positive, medium, and significant correlation between preservice primary school teachers' metacognitive awareness and their beliefs about mathematical problem solving. Metacognitive awareness explains 13% of the total variance in beliefs about mathematical problem solving. This finding suggests that improving the metacognitive awareness of pre-service primary school teachers can contribute to the development of their beliefs about mathematical problem solving. Therefore, it is important to develop the metacognitive awareness of pre-service primary school teachers. (Yıldırım & Ersözlü, 2013) found that metacognitive awareness predicts university students' problem solving levels significantly and explains 45% of the variance in their problem-solving levels. (Baş et al., 2016) also found a significant and positive relationship between metacognitive awareness, attitudes, and beliefs towards problem solving. They found that beliefs and attitudes towards problem solving explain 8% of metacognitive awareness. Moreover, researchers should take into consideration the effect of metacognitive awareness in the formation of beliefs about mathematical problem solving.

This research aimed to reveal the metacognitive awareness and problem-solving beliefs of pre-service primary school teachers. The data show that pre-service primary school teachers have high metacognitive awareness and medium beliefs about mathematical problem solving. The data also show that metacognitive awareness does not http://journals.ums.ac.id/index.php/jramathedu vary significantly by gender. However, beliefs about mathematical problem solving vary significantly by gender, with male pre-service teachers showing higher beliefs. There is a positive, medium, and significant correlation between metacognitive awareness and beliefs about mathematical problem solving. The data show that metacognitive awareness significantly predicts beliefs about mathematical problem solving.

Future research can examine different study groups, such as primary school teachers and primary school students. In the context of primary school math classes, activities that support the development of metacognitive awareness can be conducted to increase primary school students' beliefs about mathematical problem solving. During undergraduate education, pre-service teachers can have opportunities to encounter activities that support the development of their metacognitive awareness. These experiences can positively affect their beliefs about mathematical problem solving.

We express our sincere gratitude to the pre-service primary school teachers studying in the Department of Primary Education in the education faculty who participated in this research. We partly presented this research as an oral presentation at USEAS 2017, International Limitless Education and Research Symposium, held in Alanya, Antalya, Turkey, on April 24-26, 2017.