Algebra involves relationships between quantities, the use of symbols, modeling of phenomena, and the mathematical expression of change (Carraher et al., 2008). Therefore, ensuring that activities used in algebra instruction incorporate the essential components of algebra can contribute to students' algebraic thinking. The activities that teachers use in their lessons play a crucial role in the development of students' algebraic thinking skills. Algebraic thinking is essential for students to understand mathematical concepts and establish relationships between them (Kieran, 2022). Thus, teachers’ knowledge of mathematics instruction is significant in fostering students' algebraic thinking skills (Magiera et al., 2017). Hill et al. (2008) found that teachers’ pedagogical knowledge levels directly influence students' mathematical performance. Additionally, Hiebert and Grouws (2007) demonstrated that the teaching strategies used by mathematics teachers contribute to the development of students' problem-solving and conceptual learning skills. Therefore, pre-service teachers' knowledge of algebra instruction can directly impact their pedagogical approaches and their evaluations of students' mathematical thinking processes.

The use of activities in mathematics instruction makes the learning process more meaningful by allowing students to explore and deeply understand mathematical concepts (Griffin, 2009). The integration of activities into instructional processes not only enhances students’ mathematical thinking skills but also increases their active participation in learning (Kieran et al., 2015; Van Dooren et al., 2013). Moreover, the role of teachers’ pedagogical knowledge in selecting and evaluating activities has also been emphasized (Kilpatrick et al., 2001). Consequently, the way pre-service teachers evaluate activities is directly related to their level of pedagogical knowledge and instructional perspective. Research by Liljedahl et al. (2007), Güzel et al. (2021), and Van Dooren et al. (2013) indicates that limited studies have focused on the design and evaluation of activities in mathematics education. Liljedahl et al. (2007) emphasized that the evaluation process of activities requires a recursive and iterative approach based on reflective analysis and adaptation steps. Bozkurt et al. (2022), on the other hand, developed a more comprehensive model by evaluating activities not only through a process-oriented approach but also based on their components.The Mathematical Activity Design and Implementation model considers the mathematical potential of activities within the framework of depth, complexity, and mathematical focus components. Depth refers to the relationship between mathematical concepts and generalizations, complexity involves establishing connections from different perspectives, and mathematical focus represents the clarity of the mathematics embedded within the activity (Bozkurt et al., 2022).

Smith & Stein (1998) evaluated activities based on their mathematical cognitive demands. The cognitive demand levels of mathematical activities are a crucial factor in determining the types of thinking processes students engage in during learning. Activities with low cognitive demand are mostly limited to mechanical procedures, whereas those with high cognitive demand encourage students' conceptual thinking, problem-solving skills, and higher-order reasoning. Considering the cognitive demand levels of activities, how pre-service teachers evaluate these activities provides important insights into the teaching approaches they may adopt in future classroom practices. However, there is insufficient information on how pre-service teachers with different levels of algebra teaching knowledge assess algebraic activities with varying cognitive demand levels. This study aims to examine how pre-service teachers with different levels of algebra teaching knowledge evaluate algebra activities with different cognitive demand levels. Accordingly, this research seeks to answer the question: "How do pre-service teachers with different levels of algebra teaching knowledge differ in evaluating algebra activities with different cognitive demand levels based on their mathematical potential?"

This study aims to examine the evaluations of pre-service elementary mathematics teachers with different levels of algebraic knowledge for teaching on algebraic activities with different cognitive demand levels. To this end, the study is a multiple case study. The sample of the research consists of 60 pre-service teachers enrolled in the "Activity Development in Mathematics Teaching" course in the elementary mathematics teaching program of a state university in Turkey. Before the data collection tool was applied, the Algebraic Knowledge for Teaching test, adapted into Turkish by Bozkurt & Özmusul (2024), was administered to the participants. Based on the test results, the participants were categorized into good (29-25 points), moderate (24-20 points), and low (19-15 points) categories, with 20 participants in each category. The data collection tool in the study consists of 4 activities from the 7th-grade textbooks of the Ministry of National Education (MEB), which cover different cognitive demand levels. These 4 activities were selected according to the cognitive demand framework of Smith & Stein (1998). According to Smith & Stein (1998), the cognitive demand levels of the selected activities are classified as lower-level (memorization and procedures without connections) and higher-level (procedures with connections and doing mathematics). One activity was selected from each cognitive demand level. As a result, the data collection tool consists of 2 activities from lower cognitive demand levels and 2 activities from higher cognitive demand levels. This data collection tool was reviewed by two experts in the field of education. Based on the experts' feedback, necessary adjustments were made, and the data collection tool was finalized. In the study, the mathematical potentials of the activities were subjected to descriptive analysis based on the evaluation framework presented by Bozkurt et al. (2022) regarding activity design and implementations. In this context, the mathematical potentials of the activities were analysed through three components: depth, mathematical focus, and complexity. Each activity was scored on four different levels: very low (0 points), low (1 point), medium (2 points), and high (3 points) for each of these components. The reliability process of the data analysis was carried out independently by three researchers. Then, a consensus was reached on a few cases where differences were observed. This triangulation process ensures the validity of the qualitative study.

It has been observed that pre-service teachers' algebra knowledge for teaching influences their evaluation of the mathematical potential of algebraic activities. Pre-service teachers with low levels of algebra knowledge for teaching tended to assign similar scores to both lower- and higher-level algebraic activities in terms of mathematical potential. Additionally, it was found that those with low algebra knowledge for teaching provided similar justifications when scoring the depth and complexity components. However, while the depth component refers to the ability to organize/generalize information in an activity, the complexity component requires establishing relationships between different representations of concepts within the activity. Their confusion between these subcomponents can be associated with weak pedagogical content knowledge. Pre-service teachers with a medium level of algebra knowledge for teaching displayed variability in their reasoning when scoring the mathematical potential components of activities. However, they also exhibited errors in their evaluations. This suggests that, while they possess pedagogical content knowledge, they experience difficulties in evaluating activities accurately. In contrast, pre-service teachers with a high level of algebra knowledge for teaching provided accurate justifications when scoring the mathematical potential components of activities. In conclusion, it has been observed that pre-service teachers’ algebra knowledge for teaching plays a crucial role in their evaluation of different algebraic activities. Understanding pre-service teachers' pedagogical content knowledge in mathematics education more deeply will contribute to the improvement of teacher education programs. Furthermore, this study aims to determine the factors influencing pre-service teachers’ evaluation processes of activities with high cognitive demand.

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