1.1. MM in Mathematics Education

While models are defined as a system incorporating rules, relationships, operations, and other components to explain and construct a complicated and difficult-to-understand system (^{Lesh & Doerr, 2003}), modeling refers to the process followed to create a model (^{Sriraman, 2005}). MM is the process of using math to construct a model that produces answers to individuals’ important real-life questions and then of testing and implementing this same model (see: ^{Niss, Blum, & Galbraith, 2007}). MM is one of the fundamental components of mathematics education in the world-wide accepted Program for International Student Assessment [PISA] 2012 Assessment and Analytical Framework. As such, the fundamental objectives and goals that the current mathematics education paradigm seeks to impart to students need to be evaluated from a MM perspective.

In mathematics education, four separate teaching objectives are considered important for MM: (i) behavioral objectives, (ii) process objectives, (iii) affective objectives, and (iv) cognitive objectives (^{Lesh & Doerr, 2003}). Behavioral objectives include knowledge, input-output, and provisory courses of action. For example, skills included in educational programs expected to be acquired by students are considered behavioral goals. From a MM perspective, however, since knowledge is likened to a living organism instead of to a machine, acquiring behavioral objectives is not a primary objective in and of itself. Process objectives focus on problem-solving strategies used in lessons, with these strategies being taught independent of content. In MM, however, problem-solving strategies reflect how students interpret situations requiring solutions. That said, affective objectives represent students’ beliefs about their lessons as well as their attitude, motivation, emotions, and values (^{Lesh, Zawojewski, & Carmona, 2003}). Finally, cognitive objectives include models and conceptual systems used to explain, construct, and test mathematically complicated systems. From a MM standpoint, emphasizing other goals without including cognitive goals renders mathematics education into nothing more than rote memorizing rules that are then used to find the correct answer without actually making any calculations. Just as these four teaching objectives must be a component of students’ experiences and mathematical interpretations, models must also be understood as approaches developed to aid students in making these interpretations (see: ^{Lesh & Doerr, 2003}).

On the other hand, the effective teaching of MM in a classroom setting is generally performed through the execution of relevant activities. Just as modeling activities aid students in using important mathematical concepts, they also offer teachers the opportunity to discover what their students think about mathematics and to shed light on their mathematical development (^{English, 2003}; ^{Mousolides, 2009}). Accordingly, it is essential that MM activities gain a foothold in educational environments.

1.2. Mathematical Modeling Activities

Mathematical modeling is an activity in which real-life problems are solved using mathematical tools. (^{Vos, Hernandez-Martinez, & Frejd, 2020}). However, not every activity in a real-life context can be a mathematical modeling activity. In the modeling activity, not all the data necessary to solve the problem are given in the problem. It is essential for the student to make some assumptions in order to solve the problem as in real life (^{Borromeo, 2018}). In these activities, calculation is not at the forefront, and all necessary thinking and planning skills are at the forefront before the calculations begin. In addition, Modeling activities not only help students use important mathematical ideas, but also help teachers uncover their students’ thoughts and provide an opportunity to develop an understanding of their mathematical development. (^{English, 2003}; ^{Mousolides, 2009}). The theoretical and philosophical structure underlying this thought is based on American pragmatists like Dewey, who defended obtaining information by transforming concrete experiences into abstract concepts through observation, thought, and new inferences, as well as on modern philosophers like Piaget and Vygotsky (see ^{Lesh & Doerr, 2003}; ^{Lesh & Sriraman, 2005a}, ^2005b). This is because this philosophical approach argues that the forms of thinking that are required always occur by using and integrating much more than a single discipline, single course book, or single subject area.

In the last 10-15 years, many empirical studies have been conducted to investigate the teaching and learning processes of mathematical modeling. Much of this work has focused on teaching students at the K-12 level. The most well-known of these is the work of ^{Lesh and Doerr (2003)} (^{Borromeo, 2018}). These studies argue that mathematical modeling education should begin at a very early age for students to understand how mathematics is needed in their real lives. Below, mathematical modeling applications are given from the perspective of students from the eyes of ^{Lesh and Doerr (2003)}.

1.3. Mathematical modeling for students

Modeling activities require students to develop mathematical models that allow them the opportunity to work in groups to make sense of their mathematical experiences and to focus on their structural characteristics, such as relationships between models, operations, and components (^{Chamberlin, 2004}). Considering this importance, modeling activities can be executed in a planned manner using basic mathematical concepts previously taught; yet, this requires students to first develop a mathematical model for themselves (^{Lesh, Carmona, & Post, 2002}). ^{Lesh and Doerr (2003)} express the principles that need to be taken into consideration while conducting modeling activities as follows: (i) The concepts and mathematical notions aimed to be taught need to be discussed with students beforehand, (ii) If necessary, a warm-up activity or activities seeking to increase the real-world application and meaningfulness of the problem’s context for students should be completed, and (iii) Activities pertaining to the usability of models developed by students should be done following the main activity’s completion (^{Lesh, Hoover, Hole, Kelly & Post, 2000}). Modeling activities following these principles are conducted with students in groups of three to five individuals. During group work, each student first offers his or her own interpretation of the problem, and these interpretations are then discussed in groups. During discussions, students are given ample opportunity to improve their communication skills. During these group discussions, students attempt to compromise on the most appropriate model. What is sought is a mathematical model able to be used to solve a problem identified in a given scenario or by an individual in need of an answer. So that their peers may benefit from the most suitable model possible, students must clearly describe their thought processes and explain their solutions (^{Chamberlin & Moon, 2005}). This way, group work is more than simply a social activity for students; it develops into a cognitive environment (^{Burkhardt, 2006}).

The stages that students go through during modeling activities are summarized by ^{Yoon (2006)} in Table I.

Generally speaking, MM activities’ in-class execution and student-teacher roles greatly resemble the environments recommended in the constructivist approach (^{Burkhardt, 2006}). The modeling process is realized in student-centered environments not only in which students are given the opportunity to learn from their mistakes in a constructive manner and are guided to consider different ideas and methods to solve problems but also in which teachers manage the entire process, including assessments, in an appropriate manner (^{Kunter & Voss, 2013}). It is also known that these learning environments improve students’ modeling skills (^{Blum & Borromeo, 2009}).

1.4. Mathematical modeling in terms of teachers and teacher candidates

There are few studies in the field of mathematical modeling for teachers and prospective teachers (^{Borromeo, 2018}; ^{Yang, Schwarz & Leung, 2021}). These studies report that teachers misunderstand the nature of modeling before receiving a training in mathematical modeling, and they also have difficulties in modeling processes (^{Manouchehri, 2017}; ^{Paolucci & Wessels, 2017}; ^{Yang, Schwarz & Leung, 2021}). ^{Blum & Borromeo (2009)} reported that teachers are an indispensable element for mathematical modelling. Accordingly, teachers should provide ample opportunities for students to connect, encourage students’ independence, and adopt an effective and student-centered classroom management. ^{Niss, Blum & Galbraith (2007)} report that the content of education designed for teachers should include modeling experiences that match what they are expected to teach. However, there are also studies suggesting an educational design that combines theory and practice. Accordingly, first the participants are given a theoretical training and they are provided to experience the current modeling activities. Then the participants design their own activities and apply it to the target audience. Finally, all applications are evaluated. (^{Borromeo, 2013}; ^{Borromeo, 2018}; ^{Borromeo & Blum, 2013}; ^{Lesh & Doerr, 2003}).

1.5. MM in the Turkish Mathematical Curriculum

Together with the introduction of a new mathematics education program, a constructivist approach in which student-centered activities where students produce solutions to real-life problems was adopted in 2005 in Turkey, which served to revamp the entire education paradigm and its practices (^{Ersoy, 2006}). Over the subsequent years, mathematics curricula were readjusted to reflect this change. In the Middle School Curriculum published in 2013 (^{Ministry of National Education in Turkey [MONE], 2013a}), for example, MM skills were included among the basic skills expected to be acquired by students. During the same period, an elective class called Mathematics Applications incorporating MM activities was added to the middle school curriculum (^{MONE, 2013b}). In the more recently updated 2018 Mathematics Curriculum, acquiring mathematical literacy skills, of which MM holds a central position, were included among the basic objectives of mathematics education (^{MONE, 2018}). Generally speaking, a strong emphasis on mathematics’ being an integral part of real life and a worthwhile endeavor pervades throughout the newly restructured mathematics curricula (^{Şen, 2017}).

2. Research Objective and Importance

Studies have concluded that MM activities are not performed at the desired level in classroom settings (^{Blum & Borromeo, 2009}; ^{Burkhardt, 2006}). Although performing MM activities is included as a basic objective of mathematics curricula, one of the most important reasons for their exclusion from being used in classroom settings is that it is difficult for both teachers and students to perform related activities. (^{Blum & Borromeo, 2009}; ^{Frejd & Ärlebäck 2011}). In any case, every step of the above-mentioned modeling process constitutes a an aptly-named cognitive hurdle that must be overcome by students (^{Galbraith & Stillman, 2006}). Upon comparison, the percentages corresponding to the number of correct answers to questions on the international PISA requiring MM skills were found to be lower than those of other questions on the same test (^{Turner, Dossey, Blum, & Niss, 2013}). Additionally, teachers also need to have acquired a variety of skills, like being able to properly guide students in completing complex modeling processes, have mathematical and non-mathematical knowledge, and be able to conduct activities that facilitate the acquisition of new ideas (^{Blum, 2015}).

Accordingly, teachers and students must know just what is required of them play an important role in activities so that activities may be effectively realized and gain real-life applications. In addition to this, it is important that preservice teachers (PTs) have worked with modeling activities and have gained valuable experience using them so that they may understand the difficulties involved in completing them and how they should direct their future students in their interactions with unfamiliar processes that incorporate open-ended questions and for which they must take into consideration numerous variables to solve. As such, the present study has examined three aspects of activities actively used in classroom settings: (i) the educator conducting the activity, (ii) the classroom teacher’s behaviors, and (iii) students’ behaviors. The outcomes of modeling activities offer important insight in helping educators identify potential problems while attempting to integrate them into teaching environments. In classes in which such activities are performed, determining the trends, behaviors, and opinions of students, teachers, and those PTs who have received MM training, including training about the approaches used to perform such activities will aid in making in-class adaptations of modeling activities more effective. It is therefore postulated that knowing how teach and integrate mathematical teaching and, even more importantly, MM activities into teaching environments will have a positive impact on teachers’ future practice.

The current study, therefore, seeks to identify primary school mathematics PTs’ opinions about performing MM activities. To this end, answers to the following question were solicited.

What are PTs’ views on performing MM activities?

3.1. Research Design

This study follows a case study methodology. Being qualitative in nature, case studies seek to gain further data on and a better understanding of a specific individual, group, event, or organization (^{Patton, 2014}). Accordingly, the case investigated in the current study is PTs’ opinions about creating modeling activities.

3.2. Participants

Participants consisted of 18 middle school mathematics PTs in their fourth year of study in the Faculty of Education of a state university located in Turkey’s Marmara Region. Criterion sampling, a type of purposeful sampling, was used to select participants. As such, all volunteer PTs in their fourth year of undergraduate study enrolled in the MM elective class participated in the study. Moreover, all participants had previously completed several content education and research methods classes like Analysis, Special Education Methods, and Scientific Research Methods.

3.3. Research Process

This study was conducted as part of the participants’ undergraduate MM class. PTs were first provided theoretical information on models, the definition of modeling and MM, areas of use, basic components, different MM perspectives, activities and the modeling process, and in-class applications of MM. During this process, PTs also completed their own literature review in groups in which they investigated the solutions of different modeling activities. PTs were additionally asked to use EDMODO, an internet-based classroom management program designed for educational use, to report their opinions about the subjects and activities they had covered during lessons to the researchers at the conclusion of every lesson. Accordingly, PTs wrote what they thought about the activities they had completed at the end of each class in the form of a diary, which they then shared with the researchers. These diaries were read by the researchers every week, and feedback was provided to the diaries’ owners if necessary. This way, participants were able to obtain sufficient theoretical knowledge about MM as well as ample experience solving related activities.

Traveling to the schools where they performed their practicum together with their fellow group mates, PTs video recorded the modeling activity that they had performed in class, which was shared with the other students in their modeling class. These activities consisted of activities thought to build relationships with middle school students’ real lives. PTs realized these activities in groups of four to five individuals. Additionally, the middle school students’ own mathematics teachers were included in three of the four activities conducted. Video recordings of the activities performed over a four-week period by four separate groups were watched in PTs’ MM class, which served to convey their in-class experiences with their classmates. Every week, PTs offered their opinions and assessments of the videos in question.

Before performing any activity in a classroom setting, PTs chose several activities from a larger pool of activities. Therefore, different factors (e.g., how appropriate activities were for students’ levels, activities’ ability to attract students’ interest, scheduling) related to the activities performed in PTs’ MM class were taken into consideration beginning from the very start of the semester. Subsequently, documents importing on the significance of making an appropriate lesson plan and on the necessity of devising an effective division of labor during preplanning were submitted to the researcher during the MM class. PTs’ preplanning documents were evaluated based on several criteria, including (i) how many individuals were to be included in groups based on class size, (ii) what concepts were mentioned during warm-up activities prior to moving onto the activity’s problem, (iii) how much time was to be given to solve problems, and (iv) the order that groups were to present their solutions. PTs were also provided feedback by the researcher. In order to prepare the participants for the activity, a number of adjustments were made, like reducing the number of in-class groups, providing participants with sufficient preliminary information about the activities to be performed, and increasing the amount of time they had to solve problems. Furthermore, PTs were also asked to include detailed explanations about the warm-up phase of the lesson, about how they had moved on from the activity’s introduction to the actual problem itself, about their experiences with other groups and offering guidance, and about how they were planning to share various in-class responsibilities, like obtaining a video camera for the class. This was done to prevent any potential chaos from occurring during class.

PTs showed the video recordings of the activities they had performed with their peers in their MM class. The students were asked to watch the videos and listen to their fellow classmates describe their experiences while conducting these activities. After sharing videos and experiences, students participated in class-wide discussions in which the activities performed, the PTs conducting them, the students involved, the classroom environment, and the classroom teacher’s attitude were all addressed from various angles. PTs listened to each other as they shared their experiences and even offered each other feedback based on the MM training they had received. Immediately following this, the researchers then used a semi-structured observation form to record what each PT thought about the video they had watched one day earlier, which was then uploaded to EDMODO. This was repeated for four weeks. Figure 1 gives a general overview of the research process.

Figure 1 General overview of the steps followed during research 

3.4. Data Collection

The following means were used to collect data for the current study: (i) lesson and execution plans for the activities PTs were to conduct, (ii) semi-structured observation forms, and (iii) notes taken by the researcher. Table II summarizes the activities that participants chose and implemented.

The activities in Table II are briefly given below.

Figure 2 Big Head Activity adapted from ^{Herget, Jahnke & Kroll (2001)}  

Figure 3 Largest Shoe Size in the World Activity (^{Herget, 2000}) 

Figure 4 Horse Race Activity (Swan, Turner, Yoon & Muller, 2007) 

Figure 5 Bales of Hay Activity (^{Blum & Leiß, 2007}) 

Generally, pre-service teachers stated that they paid attention to the fact that the activities were easy and interesting because the students were not accustomed to such activities. They thought that it would be fun for students to compare their own body measurements in activities related to ratio-proportion. In the activity related to probability, they reported that the students had difficulties in probability, so they chose an activity that explained the importance of knowing probability.

PTs were given semi-structured observation forms after watching videos. Some of the questions included in these forms are:

Additionally, looked at research took her own notes while PTs’ videos were being screened and while they shared their experiences with their peers. Figure 6 offers an example of such notes.

Figure 6 Example of notes taken by the researcher 

3.5. Data Analysis

The data generated from this study were subject to content analysis. After watching the video recordings of activities, all the observation notes made by PTs were collected and transferred to a qualitative data analysis program. Then, all statements were coded and appropriate categories were made. These categories were themselves grouped under two headings, namely Activity-Student-Teacher and Preservice Teachers. Table III depicts a short section of the data analyzed. The themes addressed by PTs were divided into various categories and subcategories. Specifically, the subcategories included under the category Classroom Environment were: (i) active participation, (ii) communication-collaboration, and (iii) modeling process whereas the subcategories included under the category Difficulties Experienced were: (i) understanding-modeling, (ii) group work, and (iii) expression-assessment.

3.6. Trustworthiness of Research

One means to increase the trustworthiness of qualitative studies is the triangulation technique (^{Denzin, 1978}). Accordingly, this study made use of PTs’ lesson and execution plans, semi-structured observation forms, and field notes to collect data. To further increase the data’s trustworthiness, peer assessments were employed. According to ^{Guba and Lincoln (1994)}, peer assessment constitutes an external control mechanism for trustworthiness. That being the case, the themes and categories created by the researchers were sent to a researcher with a PhD in mathematics education and whose comments were used to make amendments to six subcategories. For example, while the category Modeling Process had been considered a theme related to activities, it was decided that it should be considered a student-related theme because it reflected the classroom environment and students’ behaviors. Furthermore, we used participant confirmation to determine whether studies’ findings accurately reflected their own opinions (^{Merriam, 2013}). For this, the researchers conducted data analyses and, after the study had been completed, were reported and all data collected. These reports were then sent to participants, who were asked to indicate whether the analyses accurately reflected their own opinions and the research process.

4.1. Opinions regarding Activities

Figure 7 summarizes participants’ opinions on the activities conducted in class.

Figure 7 Participants’ opinions regarding activities 

Figure 7 illustrates how participants’ opinions about activities were grouped into four separate categories. This chart indicates that PTs considered activities not only to increase various thinking skills of students and to be interesting, and could be used in teaching the concepts at hand.

Regarding Thinking Skills, PTs discussed how they thought the activities had influenced students’ analytical, creative, and mathematical thinking skills, their sense of numbers, and their ability to observe relationships with real-life events and to consider issues from different perspectives. ‘I think it helps to understand the subject better and use it in daily life’ (Rukiye- pre-service teacher- for big head activity). In general, since all the comments made by PTs indicated that the activities attracted students’ attention, it may be concluded that activities in the form of games or that are related to real life attract students’ attention.

‘The activity was interesting. Most of the groups were willing to figure it out. It felt different because they had not done such activities before and they did not do group work. Several groups discussed among themselves and found good results and expressed themselves well on the board. They discovered estimation skills and how to use ratio in such a problem’ (Didem-pre-service teacher-for Largest Shoe Size in the World activity’

With regard to Concept Teaching, PTs discussed which activities could be used to teach which concepts. Appropriate answers were given to the designated target objectives in the first three activities; however, although the target objective in the final activity dealt with the Pythagorean relation, no PT mentioned it.

‘Even though the students understood the concept of number sense, the main purpose was not to give them the concept. Our friends should have given this concept as additional information. They put too much emphasis on it, and from this point of view, the students perceived this concept of number sense as if it was a subject covered in a mathematics lesson. While the concept to be given in this activity is the Pythagorean relation, no student in the class has even used this concept’ (Nurseda- pre-service teacher-for horce race activity)

In addition to stating that activities could be used to teach the concepts in question, PTs also articulated that activities actually had a greater impact on the other categories composing this theme than they did on concept teaching.

‘We saw that the students tried to reach a solution by estimating the length of the shoe given in the picture, the shoulder width of the man holding the shoe, the glasses, the waist width and proportioning it with the shoe. Thanks to this activity, students put forward their thoughts by making assumptions like the previous activity, contributed to their discussion skills by discussing their thoughts in the group, had the opportunity to demonstrate their group work skills, and used their previous mathematical knowledge. Here they gained a lot of skills rather than an activity on ‘ratio-proportion. This part was more important’ (Seda- pre-service teacher- for Largest Shoe Size in the World activity’).

4.2. Opinions regarding Preservice Teachers

Figure 8 depicts what participants thought about their peers after having watched video recordings of them conducting activities and listening to them share their experiences.

Figure 8 PTs Opinions regarding their Peers’ Execution of Activities 

Figure 8 indicates that PTs’ opinions regarding their peers were divided into three separate categories. Accordingly, PTs’ opinions regarding their peers’ behaviors during activities considered to be improper were entered in the subcategory Mistakes during Implementation. Those opinions on the behaviors PTs’ deemed to be constructive were entered in the subcategory Implementer’s Positive Aspects. Finally, statements made regarding the most difficult situations encountered during activities’ implementation were entered in the subcategory Difficulties Experienced.

Accordingly, the mistakes that PTs made during activities’ execution were further grouped into three separate subcategories, namely (i) meaningfulness, (ii) eliminating diversity, and (iii) misdirection. With regard to Meaningfulness, PTs stated that additional preliminary information could have been given or more concrete materials could have been used to make the activity more meaningful for students and that it was necessary to spend more time on the preplanning phase instead of moving on to the actual problem, as that would help students better understand problems. With regard to Eliminating Diversity, statements existed indicating that student groups attempting to use different means to address the problem were interfered with and that groups were persuaded to solve the problem the same way that PTs had. For example, an examination of the third activity (i.e., Horse Race) reveals that one group of students thought it was necessary to use dice in order to solve the problem in the activity and, accordingly, started to make a die out of paper. Upon seeing this, the PTs orchestrating the activity advised students that dice were a waste of time and made several other recommendations instead. During the fourth activity (i.e., Bales of Hay), students modeled the positions of the bales of hay without actually looking at them and made a link between their conclusion and objects that existed in their own environments (Figure 3). After all of the groups had finished their presentations, one student mentioned that it was necessary to consider the positions of the bales and that it was therefore necessary to make another, entirely different model. However, the PTs conducting the activity did not ask the students to find another solution after this student had realized what needed to be done and instead ended the lesson. Consequently, the PTs watching the video recording of this activity criticized its implementers, stating that it is absolutely necessary for different ideas like the one in question to be given a place in classes.

With regard to Misdirection, after PTs had watched video recordings of the fourth activity (i.e., bales of hay), they articulated that some of their fellow PTs had misguided students and that they had not achieved their stated goal. At the beginning of the activity’s execution, PTs’ made comments on the sense of numbers during the warm-up activity that caused students to address the problem using various forms of estimation instead of with mathematical methods. Students were unable to construct an accurate model of the bales of hay by placing them on top of each other and attempted instead to solve the problem building off of relationships with real-life objects and people’s heights. Figure 6 shows the solutions reached by all the groups in the class.

Figure 9 illustrates that students attempted to associate the total height of the bales of hay with people and objects found in their own environments.

Figure 9 Example of Activity 4 

In addition to offering constructive criticism to their peers, PTs also stated that some of the activities were effectively executed, and such opinions were included in the category Implementer’s Positive Aspects. Specifically, PTs successfully described all that had happened in the classroom, collaborated appropriately with their fellow PTs, and were able to manage the class effectively. These opinions may be considered evidence that PTs came prepared for their lesson, were enthusiastic, and successfully completed the activities.

The category Difficulties Experienced mostly includes statements pertaining to PTs’ experiences while conducting activities. Several PTs stated that they had insufficient time to complete some of the activities when responding to their peers’ criticism toward their implementation of activities.

‘ [..]We were very happy that a student finally noticed this. But we couldn’t say yes to the 30 people who enthusiastically held the event there. That would be to discourage them. That’s why we said your friends made the right point. But we said that our request from them is an estimated value, not a definite result. I do not think that we discouraged the student who realized this situation, and extinguished him. We congratulated and thanked him for noticing this. Our friends said that we should direct students to the right solution. Yes, then we could have achieved a tremendous result. But we didn’t have much time. We could only do so much in a limited time’ (Feyza-pre-service teacher - for bales of hay activity).

In fact, several articulated that the fact students were unfamiliar with this type of activities was an obstacle to their smooth progression and that the limited amount of time caused them stress. With regard to Classroom Management, statements indicated that students’ inability to fully comprehend the activity and other factors like classroom teachers’ attitudes caused PTs severe difficulties in managing the class. That said, PTs stated that they would have faced even more difficulties had they been required to work on their own instead of working in groups of 4-5. Finally, the classroom teachers working with PTs were more likely to exhibit behaviors that hindered the activity’s objectives from being accomplished than otherwise.

4.3. Opinions regarding Students

PTs’ opinions regarding the middle school students with whom activities were conducted were divided into two separate categories, as illustrated in Figure 10.

Figure 10 Participants’ opinions regarding students 

Figure 10 reveals that participants’ opinions regarding students are divided into two separate categories, namely Classroom Environment and Difficulties Encountered. Under the former category are the subcategories effective participation, communication-collaboration, and modeling process whereas under the latter category are the subcategories understanding-modeling, group work, and expression-assessment.

PTs’ statements indicating that students had approached the activity with a positive mindset and had thought over and discussed the problem for a significant amount of time in order to find an appropriate solution were included in the subcategory effective participation. Included in the subcategory communication-collaboration are statements by PTs indicating that students had discussed their solutions in groups, had communicated with others, and in short, had shared their ideas with their peers. In the subcategory modeling process are statements made by PTs indicating that students used math first to construct and then to solve the problem after understanding what was being asked of them. During all of this, however, PTs determined certain areas in which students struggled. One such area was understanding-modeling, in which students had difficulties both in understanding the activity and in constructing relationshipships between real life and mathematics. In an effort to remedy this, PTs resorted to numerous strategies to guide students. Moreover, since students were unfamiliar with group work etiquette, PTs articulated that the students struggled to work together in groups in an effective manner. PTs’ statements describing such difficulties were included in the category group work. Finaly, although several students reached a specific conclusion, they were unable to express themselves in a convincing manner. PTs’ statements indicating that students had difficulties deciding as to whether their conclusions were accurate or not were included in the category expression-assessment.

4.4. Opinions regarding Teachers

PTs’ opinions regarding the mathematics teachers partaking in the activities with them were included in the theme Mathematics Teacher. The three categories included in this theme are illustrated below in Figure 11.

Figure 11 Preservice Teachers’ Opinions regarding the Mathematics Teachers Participating in Modeling Activities 

Figure 11 depicts the subcategories into which PTs’ opinions regarding classroom teachers were divided, namely: misguiding-casual, authoritarian, and helpful. These categories included opinions of different teachers. For example, PTs’ comments on the teacher participating in the first activity (i.e., Big Head) were included under the subcategory misguiding-casual. PTs stated that they had difficulty maintaining control over the class as a result of this particular teacher’s overly casual attitude. The same teacher was also stated to have given hints to students without allowing them the opportunity to think about the problem for themselves. ‘The teacher wanted to help the students. But he misdirected. There was a more casual classroom because of the classroom teacher. They saw our friends who went to the practice not as teachers, but as older sisters/brothers’ (Sude-preservice teacher-for big head activity).

Under the subcategory authoritarian-traditional were comments made about the second activity (i.e., Shoe Number). The teacher who participated in this activity exhibited a very authoritarian attitude throughout the entire activity, which negatively affected the entire course of the activity ‘The groups were supposed to discuss together, and it was normal for noise to come out during this time. However, the teacher’s warning at every sound prevented the students from discussing and arguing’ (Nurseda-preservice teacher).

The third activity (i.e., Horse Race) included PTs’ statements about a teacher who would support them whenever they asked. This teacher would exchange ideas with the PTs about forming groups to ensure that they were not homogeneous and about which students should present their group’s findings to the class. Finally, since no mathematics teacher participated in the fourth activity (i.e., Bales of Hay), no category was created for it.

‘The teacher behaved exactly as he should have. He helped our friends with classroom silence and control. In addition, the students did not direct them while doing the activity. It is clear that he is much better than the teacher standing at the head of our friends last week’ (Şevval-preservice teacher).