Evaluation of the learning environment designed to develop student mathematics teachers’ mathematical modelling competencies

Evaluation of the learning environment designed to develop student mathematics teachers’... Abstract The purpose of this study was to examine the mathematical modelling competencies of student mathematics teachers within a learning environment that was based on a holistic approach to instruction. The participants were student mathematics teachers enrolled in undergraduate programmes at two different universities. In one of the university programmes, the participants took a mathematical modelling course that was taught according to a holistic approach, while the participants at the other university followed the standardized undergraduate programme that did not include a mathematical modelling course. The sub-competencies of mathematical modelling of the student mathematics teachers who participated in the mathematical modelling course were examined using an analytic rubric that was designed according to the holistic approach applied within the context of the study. In order to determine whether the observed changes were related to the experiences of mathematical modelling, these results were compared to the sub-competencies of mathematical modelling of the control group. The findings showed that experience within a learning environment based on mathematical modelling, as well as affective factors, supported the development of modelling competencies. On the other hand, it was observed that certain sub-competencies were difficult to develop; while many sub-competencies were enhanced by the modelling experiences, others were adversely affected. Furthermore, a certain group of sub-competencies was found not to be directly linked to modelling experiences. 1. Introduction Mathematical modelling plays a major role in solutions to daily life with mathematical view. In general, ‘mathematical modelling’ refers to a cyclic process of translating a real-life problem to a mathematical one with a view to developing mathematical solutions and implementing secondary processes to interpret these mathematical solutions in applied environments (Borromeo-Ferri, 2006; Lesh & Doerr, 2003). This process is crucial for solving real-life problems and requires individuals to understand the relationship between the mathematics learned at school and the real-life experiences. Mathematical modelling began to gain prominence in early 1980s, followed by the inclusion of mathematical modelling in school mathematics curriculum (Blomhøj & Kjeldsen, 2006) and emphasis on the issue among the skills to be developed within the framework of school mathematics teaching in various countries (National Council of Teachers of Mathematics [NCTM], 2000). There are also studies which emphasize the applicability of mathematical modelling from preschool years till the very last year of high school (Borromeo-Ferri, 2006) and the necessity to include mathematical modelling in school mathematics (NCTM, 2000). In Turkey, mathematics curriculum in elementary and secondary school levels also focuses on the raising of individual’s abilities of using mathematical concepts, developing models and associating such models with verbal and mathematical statements, as important objectives (Ministry of National Education [MNE], 2013). 1.1 Theoretical framework Examination of the studies concerning mathematical modelling reveals that the concept is interpreted in two distinct ways in educational programmes and applications. One of these interpretations considers mathematical modelling as a means to reveal and develop the relationships among certain mathematical contexts and to motivate students to work on real-life problems (Chinnappan, 2010). In the second interpretation, mathematical modelling is considered as a goal in and of itself, in terms of the achievement of the objectives of education rather than as a means to improve certain types of mathematical learning through educational processes. Both interpretations, however, consider mathematical modelling as a wider process covering certain elements such as formulation, mathematization, solution, interpretation and assessment (Stillman, 2012). In this context, as both perspectives emphasize the importance of the process of mathematical modelling, one should first try to define the sub-processes the students would be required to experience in the process of mathematical modelling, the competencies for the completion of such sub-processes and the sub-competencies they involve. In this study mathematical modelling competencies and sub-competencies approaches which are defined by Blum & Kaiser (1997, quoted in the study by Maaß, 2006) are adopted. 1.2 Studies on the development of mathematical modelling competencies Related literature indicates that it is possible to come up with rough categorizations of the studies with reference to certain characteristics, even though a widely accepted shared perspective regarding the development and evaluation of mathematical modelling competencies is lacking. A review of the studies considering mathematical modelling as an educational objective and as a learning environment to develop mathematical modelling competencies contains within itself two distinct approaches. These approaches are ‘micro-level approach’ and ‘holistic approach’ (Grünewald, 2012). Furthermore, there are also some studies which adopt a ‘mixed’ perspective, underlining the need to establish a balance of the micro-level and holistic approaches both (Blomhøj, 2007). The aim of learning environments which embrace a holistic approach is to ensure that the individuals go through the whole mathematical modelling process defined through a model-development activity. On the other hand, the learning environments based on the micro-level approach intend to enable individuals experience a specific sub-process of the mathematical modelling process. Therefore, in the holistic approach all competencies and sub-competencies expected through the mathematical modelling process are implemented through the activity (Maaß, 2006), while micro-level approach implements various sub-competencies within the framework of distinct activities (Crouch & Haines, 2004). It is possible to categorize the studies aiming to develop and evaluate mathematical modelling competencies with a perspective of mathematical modelling competency as the objective of mathematics education, with reference to their approaches to integration to the learning environment. The categorization is summarized in Table 1. Table 1 Learning environment approaches of mathematical modelling Approaches Learning environments approaches Examples of literature 1. Micro-level approach Sub-competency-oriented Crouch & Haines (2004) 2. Holistic approach Theoretical knowledge-based Kaiser (2007) Open Model Eliciting Activity [MEA]-oriented Focusing on the procedure to follow the steps of mathematical modelling Blum & Borromeo-Ferri (2009) 3. Mixed approach Contains both micro-level and holistic approaches Blomhøj (2007) (Balance of micro-level approach and holistic approach) Approaches Learning environments approaches Examples of literature 1. Micro-level approach Sub-competency-oriented Crouch & Haines (2004) 2. Holistic approach Theoretical knowledge-based Kaiser (2007) Open Model Eliciting Activity [MEA]-oriented Focusing on the procedure to follow the steps of mathematical modelling Blum & Borromeo-Ferri (2009) 3. Mixed approach Contains both micro-level and holistic approaches Blomhøj (2007) (Balance of micro-level approach and holistic approach) View Large Table 1 Learning environment approaches of mathematical modelling Approaches Learning environments approaches Examples of literature 1. Micro-level approach Sub-competency-oriented Crouch & Haines (2004) 2. Holistic approach Theoretical knowledge-based Kaiser (2007) Open Model Eliciting Activity [MEA]-oriented Focusing on the procedure to follow the steps of mathematical modelling Blum & Borromeo-Ferri (2009) 3. Mixed approach Contains both micro-level and holistic approaches Blomhøj (2007) (Balance of micro-level approach and holistic approach) Approaches Learning environments approaches Examples of literature 1. Micro-level approach Sub-competency-oriented Crouch & Haines (2004) 2. Holistic approach Theoretical knowledge-based Kaiser (2007) Open Model Eliciting Activity [MEA]-oriented Focusing on the procedure to follow the steps of mathematical modelling Blum & Borromeo-Ferri (2009) 3. Mixed approach Contains both micro-level and holistic approaches Blomhøj (2007) (Balance of micro-level approach and holistic approach) View Large As these studies focus on competencies, the sub-competencies leading to the failure in terms of a given competency may not always be clear. Taking into account the fact that the acquisition of certain competencies may be slower compared to others (Blum & Niss, 1991), the need to design learning environments to review individual sub-competencies leading to a failure in the development of a given competency deemed necessary for a successful mathematical modelling becomes evident. 1.3 Studies on the assessment of mathematical modelling competencies The consensus on the importance of mathematical modelling in education led to research studies aimed to assess mathematical modelling competencies. As a result of this trend, many studies recently provided means to assess mathematical modelling competencies (Borromeo-Ferri, 2006; Zöttl et al., 2011). The studies so far discuss mathematical modelling competence as a variable lending to assessment. Even a comprehensive agreement about the need of assessment in the literature, the actual means to the assessment of mathematical modelling competencies are still an issue of debate. As yet, the literature offers no categorization of approaches to assessment. However, a review of the studies on the assessment of mathematical modelling competencies clarifies that the assessment approaches can also be categorized with reference to ‘micro-level’ and ‘holistic’ approaches, in parallel to the categorization of learning environments. The approaches to the assessment of mathematical modelling competencies can be categorized as shown in Table 2. Table 2 Assessment approaches of mathematical modelling competencies Approaches Assessment approaches Examples of literature 1. Micro-level approach One or more sub-competence to deal separately Haines et al. (2001, quoted in Kaiser, 2007) 2. Holistic approach A. Competency evaluation a. Identification of existing competencies (identification of the steps implemented) Schwarz & Kaiser (2007) b. Identification of the actual level of implementation of individual competencies (steps) Borromeo-Ferri (2006) c. Establishment of the level of progress regarding the efforts so far (with reference to individual steps) Ludwig & Xu (2010) (Some studies have dealt with the process steps) d. Identification of individual sub-competencies implemented within the framework of competencies Tekin-Dede & Yilmaz (2013) B. Level determination Henning & Keune (2007) C. Multidimensional evaluation Niss & Jensen (2006, quoted in Jensen, 2007) 3. Integrated approach Balance of micro-level approach and holistic approach Zöttl et al. (2011) Approaches Assessment approaches Examples of literature 1. Micro-level approach One or more sub-competence to deal separately Haines et al. (2001, quoted in Kaiser, 2007) 2. Holistic approach A. Competency evaluation a. Identification of existing competencies (identification of the steps implemented) Schwarz & Kaiser (2007) b. Identification of the actual level of implementation of individual competencies (steps) Borromeo-Ferri (2006) c. Establishment of the level of progress regarding the efforts so far (with reference to individual steps) Ludwig & Xu (2010) (Some studies have dealt with the process steps) d. Identification of individual sub-competencies implemented within the framework of competencies Tekin-Dede & Yilmaz (2013) B. Level determination Henning & Keune (2007) C. Multidimensional evaluation Niss & Jensen (2006, quoted in Jensen, 2007) 3. Integrated approach Balance of micro-level approach and holistic approach Zöttl et al. (2011) View Large Table 2 Assessment approaches of mathematical modelling competencies Approaches Assessment approaches Examples of literature 1. Micro-level approach One or more sub-competence to deal separately Haines et al. (2001, quoted in Kaiser, 2007) 2. Holistic approach A. Competency evaluation a. Identification of existing competencies (identification of the steps implemented) Schwarz & Kaiser (2007) b. Identification of the actual level of implementation of individual competencies (steps) Borromeo-Ferri (2006) c. Establishment of the level of progress regarding the efforts so far (with reference to individual steps) Ludwig & Xu (2010) (Some studies have dealt with the process steps) d. Identification of individual sub-competencies implemented within the framework of competencies Tekin-Dede & Yilmaz (2013) B. Level determination Henning & Keune (2007) C. Multidimensional evaluation Niss & Jensen (2006, quoted in Jensen, 2007) 3. Integrated approach Balance of micro-level approach and holistic approach Zöttl et al. (2011) Approaches Assessment approaches Examples of literature 1. Micro-level approach One or more sub-competence to deal separately Haines et al. (2001, quoted in Kaiser, 2007) 2. Holistic approach A. Competency evaluation a. Identification of existing competencies (identification of the steps implemented) Schwarz & Kaiser (2007) b. Identification of the actual level of implementation of individual competencies (steps) Borromeo-Ferri (2006) c. Establishment of the level of progress regarding the efforts so far (with reference to individual steps) Ludwig & Xu (2010) (Some studies have dealt with the process steps) d. Identification of individual sub-competencies implemented within the framework of competencies Tekin-Dede & Yilmaz (2013) B. Level determination Henning & Keune (2007) C. Multidimensional evaluation Niss & Jensen (2006, quoted in Jensen, 2007) 3. Integrated approach Balance of micro-level approach and holistic approach Zöttl et al. (2011) View Large Against this background, the need to assess the students’ mathematical modelling competencies with special emphasis on sub-competencies is obvious. Such assessments would provide insights into specific sub-competencies causing a failure to achieve, at least on a sufficient level, individual competencies in learning environments designed to foster the development of mathematical modelling competencies. The awareness of specific sub-competencies blocking the development of mathematical modelling competencies, on the other hand, is crucial for it sheds light on the sub-competencies requiring micro-level attention in learning environments designed with respect to mathematical modelling competencies. Although there are studies (Galbraith & Stillman, 2006; Galbraith et al., 2006, 2007) that reveal the blockages—although not focusing in sub-competencies— that are experienced during the transition from one stage of the modelling cycle to the next passage, it is not known what kind of picture emerges when working to remove the blockages. Such an approach would provide insights to mathematics educators intending to teach mathematical modelling competencies to students, about which specific sub-competencies incline to resistance to development, and hence give them clues to revise the learning environment design. 1.4 Objective of the study The related literature highlights that students at various level of education have been suffering difficulties in the process of mathematical modelling and were failing to achieve expected level of mathematical modelling competencies (Bukova-Güzel, 2011; Maaß, 2006). However, the direction of the development of sub-competencies and the specific sub-competencies leading to a failure of the development of development-resistant competencies are still matters of debate. Against this background, the need to do away with the shortcomings identified in the literature and to design and assess learning environments to enable the development of mathematical modelling sub-competencies is obvious. From this aspect this research is a next step to deepening our understanding of the students’ development of mathematical modelling sub-competencies and on how to support and assess such development. On this background, the objective of this study is to establish how exactly the sub-competency-assisted learning environment process designed to develop mathematical modelling competencies through a holistic approach would contribute to the development of mathematical modelling competencies of student mathematics teachers. The review of mathematical modelling competencies in this study was based on the theoretical framework identified by Blum & Kaiser (1997, quoted in Maaß, 2006). Thus, the main research problem in this study can be stated as follows: How do mathematical modelling competencies of student mathematics teachers develop through the course providing mathematical modelling processes based on a holistic approach? The sub-problems discussed in this context are as follows: 1. How does the competency to understand the real problem and to develop a model based on the reality develop with student mathematics teachers? 2. How does the competency to create a mathematical model out of the model based on the reality develop with student mathematics teachers? 3. How does the mathematical model thus developed help improve student mathematics teachers’ competency to solve mathematical problems? 4. How does the student mathematics teachers’ competency to interpret mathematical results against real cases develop? 5. How does the student mathematics teachers’ competency to validate the solution develop? 2. Methodology Within the framework of the present study, a learning environment was designed by taking into account the shortcomings discussed in the literature and implemented as an undergraduate course in the Department of Mathematics Education in the Faculty of Education of Giresun University. In this context, it is possible to call the study an example of action research carried out as researcher and as teacher. Furthermore, it would not be far-fetched to call the study a qualitative one, given the qualitative approach to the processes experienced in the learning environment designed within the framework of the study. 2.1 Participants The study was carried out by two distinct groups of participants from two different universities, with reference to the research questions at hand. The first group consists of 40 sophomore-year student mathematics teachers who took the mathematical modelling course in the academic year 2013–2014 at the Department of Mathematics Education in Giresun University. The second group consists of three sophomore-year student mathematics teachers at the Department of Mathematics Education in the Education Faculty of Karadeniz Technical University. The second group did not have any experience of mathematical modelling and would be unable to contact the first group. These students comprise the second tier of the study group, providing a means of control. In the first group, 40 student mathematics teachers were asked to form groups of four to six as they wished. In response seven groups were formed. As the study was intended to be based on a detailed process analysis, two of such groups were selected on a voluntary basis. The initial participants of the study were nine student mathematics teachers who formed those groups. The ensuing sections of the study shall refer to these student mathematics teachers as K1, K2, K3,…, K9. No assessment to investigate existing levels of mathematical modelling competencies of the student mathematics teachers to take part in the learning environment was carried out in the beginning. Therefore, the prior mathematical modelling competency levels of the student mathematics teachers involved in the learning environment are not known. As a different MEA design was required for each student teacher who was not included in the learning environment, not to mention the need to carry out individual clinical interviews with each and every one of them, the participant count of the control group was limited to three. Throughout their secondary and higher education tenures, the participants had taken courses concerning mathematical concepts required to complete the MEAs discussed in this study. Not all participants of the study had formal experience regarding mathematical modelling. 2.2 Learning environment design Once the existing problem was identified within an action research framework, the relevant literature was reviewed to see potential solutions for the problem and a decision on how an environment reached the requirements. The literature survey revealed that experiences to help students go through all steps of the mathematical modelling process were required to help enhance their mathematical modelling competencies and sub-mathematical modelling competencies. On the bases of this background, the holistic approach was chosen to provide such an environment to support the development of mathematical modelling competencies and thus used as the basis of the learning environment design. The design was implemented in observation of the following principles: • The directives regarding sub-competencies should enable the development in individuals of the cognitive structure for the mathematical modelling process. • In the case of individuals who lack formal experience required for the development of models, the activities should be ranked from simple to complex and from structured to unstructured. • The activities should be designed to enable gradual transitions between the world of mathematics and the real world, after a start in the former. • Activities requiring different processes applied on different mathematical contents against different backgrounds should be employed. • A holistic structure should be provided through activities with associations in contextual and mathematical frameworks the students are experienced with. • With a view to increasing the level of interest and motivation on part of the students, materials about daily life such as videos, news, posters and so on should be used to enrich the learning environment with respect to the contexts of activity. • Opportunity to enable group and individual work inside and outside the classroom, presentation of such work and a discussion of modelling processes and models per se should be provided. The guidelines regarding the activities in the learning environment designed in homage to these principles were drawn up to offer students experiences regarding such sub-competencies and to regulate their studies. Examples of the guidelines intending to provide an experience concerning sub-competencies are provided in Table 3. Table 3 Examples of guidelines that are related to mathematical modelling competencies and sub-competencies A. Competencies to understand the real problem and to set up a model based on reality A1. Competency to make assumptions for the problem and simplify the situation What do you know? What do you want to learn? What could be the assumptions? A2. Competency to recognize quantities that influence the situation, to name them and to identify key variables Which variables are most important? Which variables you can ignore? A3. Competency to construct relations between the variables Can you create an initial model to elaborate later? A4. Competency to look at available information and to differentiate between relevant and irrelevant information — A. Competencies to understand the real problem and to set up a model based on reality A1. Competency to make assumptions for the problem and simplify the situation What do you know? What do you want to learn? What could be the assumptions? A2. Competency to recognize quantities that influence the situation, to name them and to identify key variables Which variables are most important? Which variables you can ignore? A3. Competency to construct relations between the variables Can you create an initial model to elaborate later? A4. Competency to look at available information and to differentiate between relevant and irrelevant information — Table 3 Examples of guidelines that are related to mathematical modelling competencies and sub-competencies A. Competencies to understand the real problem and to set up a model based on reality A1. Competency to make assumptions for the problem and simplify the situation What do you know? What do you want to learn? What could be the assumptions? A2. Competency to recognize quantities that influence the situation, to name them and to identify key variables Which variables are most important? Which variables you can ignore? A3. Competency to construct relations between the variables Can you create an initial model to elaborate later? A4. Competency to look at available information and to differentiate between relevant and irrelevant information — A. Competencies to understand the real problem and to set up a model based on reality A1. Competency to make assumptions for the problem and simplify the situation What do you know? What do you want to learn? What could be the assumptions? A2. Competency to recognize quantities that influence the situation, to name them and to identify key variables Which variables are most important? Which variables you can ignore? A3. Competency to construct relations between the variables Can you create an initial model to elaborate later? A4. Competency to look at available information and to differentiate between relevant and irrelevant information — However, no guidelines were developed with respect to certain sub-competencies. The lack on this front is perhaps associated with the perceived need to develop these sub-competencies with other guidelines and interactions through the process or the need to do away with instructions regarding such competencies. In the learning environment thus designed, the students worked on MEAs, as groups they themselves formed. In conclusion of the group work stage, each group presented the model it had developed, paving the way for discussion in the class. Furthermore, the students were given group- and individual-based project assignments. Such project assignments were again presented in the class, followed by discussions on the models developed. 2.3 Activities and design process In line with the objective of developing a ‘Learning Environment to Support Mathematical Modelling Sub-Competencies (LES-MMC)’, MEAs with embedded guidelines encouraging the implementation of sub-competencies expected to arise during the mathematical modelling were developed. In the development of MEAs, the principles of ‘model construction, reality, self-assessment, model documentation, effective prototype, model share-ability and reusability’ identified by Lesh et al. (2000) during model-eliciting activities as principles representing real-life problems were adopted. When designing the MEAs, special attention was paid to make them contain contexts student mathematics teachers are familiar but not extremely involved with. The objectives of MEAs are presented in Table 4. Table 4 Some MEAs implemented in the designed learning environment and objectives thereof Number* MEA Objective 1 Big Food (BF) The goal is to determine student teachers’ sub-competencies without any guidelines before any experience on mathematical modelling 5 Traffic Accident Statistics (TAS) A similar situation to the previous activity, however, ensuring that students decide which data and which model will be more meaningful. Different linear models may emerge. Now the goal is to reduce the guidelines and allow the use of the sub-competencies that were targeted in the previous lessons. 8 Individual Project Development (IP) Making student mathematics teachers realize real-life problems which can be solved through mathematical knowledge. Enabling solutions of the real-life problems they had identified, through the implementation of mathematical modelling sub-competencies. Number* MEA Objective 1 Big Food (BF) The goal is to determine student teachers’ sub-competencies without any guidelines before any experience on mathematical modelling 5 Traffic Accident Statistics (TAS) A similar situation to the previous activity, however, ensuring that students decide which data and which model will be more meaningful. Different linear models may emerge. Now the goal is to reduce the guidelines and allow the use of the sub-competencies that were targeted in the previous lessons. 8 Individual Project Development (IP) Making student mathematics teachers realize real-life problems which can be solved through mathematical knowledge. Enabling solutions of the real-life problems they had identified, through the implementation of mathematical modelling sub-competencies. *Numbers indicate the order of implementation. Table 4 Some MEAs implemented in the designed learning environment and objectives thereof Number* MEA Objective 1 Big Food (BF) The goal is to determine student teachers’ sub-competencies without any guidelines before any experience on mathematical modelling 5 Traffic Accident Statistics (TAS) A similar situation to the previous activity, however, ensuring that students decide which data and which model will be more meaningful. Different linear models may emerge. Now the goal is to reduce the guidelines and allow the use of the sub-competencies that were targeted in the previous lessons. 8 Individual Project Development (IP) Making student mathematics teachers realize real-life problems which can be solved through mathematical knowledge. Enabling solutions of the real-life problems they had identified, through the implementation of mathematical modelling sub-competencies. Number* MEA Objective 1 Big Food (BF) The goal is to determine student teachers’ sub-competencies without any guidelines before any experience on mathematical modelling 5 Traffic Accident Statistics (TAS) A similar situation to the previous activity, however, ensuring that students decide which data and which model will be more meaningful. Different linear models may emerge. Now the goal is to reduce the guidelines and allow the use of the sub-competencies that were targeted in the previous lessons. 8 Individual Project Development (IP) Making student mathematics teachers realize real-life problems which can be solved through mathematical knowledge. Enabling solutions of the real-life problems they had identified, through the implementation of mathematical modelling sub-competencies. *Numbers indicate the order of implementation. With a view to comply with the learning environment principles established through the literature survey, MEAs were ranked in the order of complexity, as well as from the guidelines concerning sub-competencies to guidelines concerning primary competencies, as well as activities without guidelines. In first MEA, there are many guidelines and it includes all mathematical modelling process. The other MEAs are sorted correctly from focusing some sub-competencies at the same time including all competencies to focusing all competencies, from use of known mathematical constructions to require the construction of new structures. In the Table 4 there are examples of objectives of some MEAs. Full information of all MEAs and their objectives can be found in the PhD thesis written by Aydin-Güç (2015). In order to prevent any influence student mathematics teachers who are not included in the designed learning environment may have on each other while working on MEAs, the decision to issue a different MEA to each one was taken and hence three distinct MEAs were designed. These MEAs are compliant with the criteria established through the research and the six principles representing real-life problems as defined by Lesh et al. (2000). In the first hour of the lesson, the student mathematics teachers were provided an introduction on the execution of the course and groups were formed. The group lessons carried out until individual works were as follows: groups worked on the related MEA, shared their works with all students, discussed insights and identified applicable and optimal mathematical models before starting the next MEA. During the last two weeks IP presentations and discussions were conducted. 2.4 Data gathering and analysis process Nine student mathematics teachers in the first group in the study have completed seven MEAs with their group. The process was concluded with the completion of an individual modelling cycle to develop a solution for real-life problems the student mathematics teachers chose individually. Both group work and the classroom debates were recorded on video. The competencies of the student mathematics teachers who were not included in the learning environment were discussed with reference to individual clinical interviews on the MEAs assigned. Each student teacher was interviewed for a total of three weeks, with one interview per week, discussing the solutions regarding the activities. Understanding the depth and scale of the students’ preparedness in terms of coping with the issues observed with each sub-competency is crucial in terms of the assessment of modelling competencies. Therefore, in this study, an analytical rubric to allow evaluation with reference to performance criteria was developed to ensure the assessment of mathematical modelling sub-competencies. The study opted to go with the monitoring of sub-competencies pertaining to competencies described in detail by Blum & Kaiser (1997) and quoted by Maaß (2006). Thereafter, performance indicators were defined. A pilot study with sophomore-year student mathematics teachers enrolled in Primary School Mathematics Teacher programmes comparable to the student mathematics teachers to take part in the actual implementation was carried out with a view to fine-tuning sub-competencies. The sub-competencies discussed in the analytical rubric are listed in Table 5. Table 5 Revised mathematical modelling competencies in consequence of pilot study Mathematical modelling competencies and sub-competencies A. Competencies to understand the real problem and to set up a model based on reality   A1. Competency to make assumptions for the problem   A2. Competency to simplify the situation   A3. Competency to recognize quantities that influence the situation, to name them   A4. Competency to identify key variables that influence the situation   A5. Competency to construct relations between the variables   A6. Competency to look for available/not available information to solve the problem   A7. Competency to differentiate the related/unrelated information to solve the problem B. Competencies to set up a mathematical model from real model   B1. Competency to mathematize relevant qualities/quantities   B2. Competency to mathematize the relations between the related qualities/quantities   B3. Competency to reduce the number of relevant quantities and simplify their quantities when necessary   B4. Competency to reduce the number of relations between related quantities and simplify the relations when necessary   B5. Competency to choose appropriate mathematical notations and represent situations graphics if necessary C. Competencies to solve mathematical questions within this mathematical model   C1. Competency to use heuristic strategies (division of the problem into part problems, establishing relations to similar or analogue problems, viewing the problem in a different form, varying the quantities or the available data, etc.)   C2. Competency to use mathematical knowledge to solve the problem D. Competencies to interpret mathematical results in a real situation   D1. Competency to interpret mathematical results in extra-mathematical contexts   D2. Competency to generalize solutions that were developed for a special situation E. Competencies to validate the solution   E1. Competency to reflect on found solutions   E2. Competency to critically check on found solutions   E3. Competency to review some parts of the model or again go through the modelling process if the solutions do not fit the situation   E4. Competency to reflect on other ways of solving the problem or if solutions can be developed differently Mathematical modelling competencies and sub-competencies A. Competencies to understand the real problem and to set up a model based on reality   A1. Competency to make assumptions for the problem   A2. Competency to simplify the situation   A3. Competency to recognize quantities that influence the situation, to name them   A4. Competency to identify key variables that influence the situation   A5. Competency to construct relations between the variables   A6. Competency to look for available/not available information to solve the problem   A7. Competency to differentiate the related/unrelated information to solve the problem B. Competencies to set up a mathematical model from real model   B1. Competency to mathematize relevant qualities/quantities   B2. Competency to mathematize the relations between the related qualities/quantities   B3. Competency to reduce the number of relevant quantities and simplify their quantities when necessary   B4. Competency to reduce the number of relations between related quantities and simplify the relations when necessary   B5. Competency to choose appropriate mathematical notations and represent situations graphics if necessary C. Competencies to solve mathematical questions within this mathematical model   C1. Competency to use heuristic strategies (division of the problem into part problems, establishing relations to similar or analogue problems, viewing the problem in a different form, varying the quantities or the available data, etc.)   C2. Competency to use mathematical knowledge to solve the problem D. Competencies to interpret mathematical results in a real situation   D1. Competency to interpret mathematical results in extra-mathematical contexts   D2. Competency to generalize solutions that were developed for a special situation E. Competencies to validate the solution   E1. Competency to reflect on found solutions   E2. Competency to critically check on found solutions   E3. Competency to review some parts of the model or again go through the modelling process if the solutions do not fit the situation   E4. Competency to reflect on other ways of solving the problem or if solutions can be developed differently Table 5 Revised mathematical modelling competencies in consequence of pilot study Mathematical modelling competencies and sub-competencies A. Competencies to understand the real problem and to set up a model based on reality   A1. Competency to make assumptions for the problem   A2. Competency to simplify the situation   A3. Competency to recognize quantities that influence the situation, to name them   A4. Competency to identify key variables that influence the situation   A5. Competency to construct relations between the variables   A6. Competency to look for available/not available information to solve the problem   A7. Competency to differentiate the related/unrelated information to solve the problem B. Competencies to set up a mathematical model from real model   B1. Competency to mathematize relevant qualities/quantities   B2. Competency to mathematize the relations between the related qualities/quantities   B3. Competency to reduce the number of relevant quantities and simplify their quantities when necessary   B4. Competency to reduce the number of relations between related quantities and simplify the relations when necessary   B5. Competency to choose appropriate mathematical notations and represent situations graphics if necessary C. Competencies to solve mathematical questions within this mathematical model   C1. Competency to use heuristic strategies (division of the problem into part problems, establishing relations to similar or analogue problems, viewing the problem in a different form, varying the quantities or the available data, etc.)   C2. Competency to use mathematical knowledge to solve the problem D. Competencies to interpret mathematical results in a real situation   D1. Competency to interpret mathematical results in extra-mathematical contexts   D2. Competency to generalize solutions that were developed for a special situation E. Competencies to validate the solution   E1. Competency to reflect on found solutions   E2. Competency to critically check on found solutions   E3. Competency to review some parts of the model or again go through the modelling process if the solutions do not fit the situation   E4. Competency to reflect on other ways of solving the problem or if solutions can be developed differently Mathematical modelling competencies and sub-competencies A. Competencies to understand the real problem and to set up a model based on reality   A1. Competency to make assumptions for the problem   A2. Competency to simplify the situation   A3. Competency to recognize quantities that influence the situation, to name them   A4. Competency to identify key variables that influence the situation   A5. Competency to construct relations between the variables   A6. Competency to look for available/not available information to solve the problem   A7. Competency to differentiate the related/unrelated information to solve the problem B. Competencies to set up a mathematical model from real model   B1. Competency to mathematize relevant qualities/quantities   B2. Competency to mathematize the relations between the related qualities/quantities   B3. Competency to reduce the number of relevant quantities and simplify their quantities when necessary   B4. Competency to reduce the number of relations between related quantities and simplify the relations when necessary   B5. Competency to choose appropriate mathematical notations and represent situations graphics if necessary C. Competencies to solve mathematical questions within this mathematical model   C1. Competency to use heuristic strategies (division of the problem into part problems, establishing relations to similar or analogue problems, viewing the problem in a different form, varying the quantities or the available data, etc.)   C2. Competency to use mathematical knowledge to solve the problem D. Competencies to interpret mathematical results in a real situation   D1. Competency to interpret mathematical results in extra-mathematical contexts   D2. Competency to generalize solutions that were developed for a special situation E. Competencies to validate the solution   E1. Competency to reflect on found solutions   E2. Competency to critically check on found solutions   E3. Competency to review some parts of the model or again go through the modelling process if the solutions do not fit the situation   E4. Competency to reflect on other ways of solving the problem or if solutions can be developed differently The performance indicators of the analytical rubric developed with reference to the sub-competencies listed in Table 5 are presented in Table 11, with reference to a specific MEA to serve as an example. When analyzing the data, it was seen that between each stage the student mathematics teachers move forth and back many times during the task solving. In this case, the sub-competencies of the student teachers were evaluated using analytical rubric. Although this assessment appears to be quantitative, it was originally made to see qualitative change. So the criteria were not taken as scoring; students’ full scores were not calculated. When a student teacher studies at this sub-competency again, the criteria of the study were redetermined. In a MEA, student teachers’ competencies had encoded considering their last study on a sub-competency. With a view to ensuring the reliability of data analysis, a part of the transcript concerning a group study was analyzed by two distinct researchers with experience on mathematical modelling competencies studies, using the analytical rubric developed previously. The consistency rate proposed by Miles & Huberman (1994) was found to be 0.89. Therefore, one can argue that the data analysis is reliable. Table 6 The studies on competency A and expected analyses Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) A. Competencies to understand the real problem and to set up a model based on reality A1 K8: [Reading the guidelines] Asks what are your assumptions? What are our assumptions? K6: What did we do? We assumed a consistent rate of fall each year: Other than that... What else did we do? K8: What can we assume? We assumed the gender to be the same. K6: That would not do. K8: OK whatever. Coming up with fairly robust assumptions to make the problem simpler; to present the case with justification. For example: Let’s assume that the deaths in traffic accidents occur only due to the physical impact. There may be other cases of death to take place after the accident, due to the fear factor or another condition triggered. A2 K4: Three thousand. 3835. For instance, assuming a given number of deaths in a specific number of accidents… K3: We could calculate the probability of increase in accident count. That would give us the number of dead. Something akin to an equation with two unknowns. K1: I believe it is not about the accident, but rather about the number of people involved. K3: Driver’s licence, etc. Do you suggest we take a look on these? K4: Then we will check the ones with driver’s licence. For, looking at the number of people alone does not help, as all are not drivers. In other words, if you say that this number of people died, the dead who died on their own may have died due to normal causes, rather than due to the accident. K3: But now there is the number of people who do not have a driver’s licence. And then there are who do not have a driver’s licence, but who still take the wheel. K1: But the number of dead here… K3: Then we should correlate it with the whole population. For, there are those who do not have a driver’s licence, but still cause an accident. Applying robust simplifications to be able to develop a model. For example: Deaths in traffic accidents can be associated with numerous variables. For example, there could be deaths caused by accidents involving drivers who do not have a licence. We need to know the exact numbers of such cases. This is an extreme case we cannot control for. A3 K3: I mean, the population is growing. Say, 3 out of every 5 person get a driver’s licence. The licences are obtained without thinking about the consequences. That’s why the accident rates are rising. Identifying, naming and providing justification for the quantities which affect the case. For example: the number of deaths in traffic accidents is affected by a number of variables including the measures taken, the state of the roads, the traffic volume, the requirements for a driver’s licence, traffic assistance and first-aid capabilities and the state of affairs regarding these variables through a number of years. A4 K4: I said, an increase was observed with respect to driver’s licence numbers, due to the increase in population. The increase in the number of driver’s licences, in turn, led to an increase in the overall number of accidents. However, given the fall in the number of deaths and the rise in the number of injured in the accidents with deaths, one can talk about an increase in the measures taken through the years. Researcher: What was the thing you were trying to estimate? [Group discussion is taking place.] K4: The number of dead. Researcher: How will you associate these variables in order to estimate the dead count? K4: With traffic accidents. Researcher: I mean, will you be estimating the change in dead count with reference to the number of traffic accidents? [Group discussion is taking place.] K4: Varies by the year, I mean, falls in parallel. Having a robust identification of the key variables and explaining them with proper basis. For example: We are asked to provide the dead count caused by traffic accidents in the next year. We should forecast the next year by going over the change occurring in dead count through years. A5 K7: The population had fallen. K8: The population is rising, so is driver’s licence count. K7: It is rising and falling; see.... K8: It fell just once. On other occasions it had been rising. K7: Driver’s licences are increasing. K8: Number of accidents is increasing. K7: It rose substantially. K9: The higher the number of people, the higher the accident count. K7: The number of deaths is falling. So perhaps the safety measures are increasing. K8: Yes. K7: The number of injuries grew. That means, the number of survivors is increasing. Having a robust identification of the relationships between the variables including the key variables and explaining them with proper basis. For example: If the requirements for getting a driver’s licence were made stricter and if the trainings are increased, the number of deaths caused by traffic accidents would fall. Therefore, in cases where the number of driver’s licences is increasing, the number of dead would fall even if the number of accidents rises. If the first-aid mechanisms were improved year over year, the number of deaths would again fall despite a possible increase in the number of accidents. A6 K7: The number of deaths is falling. So perhaps the safety measures are increasing. Focusing on fairly applicable and accessible information in order to solve the problem. For example: Embracing the idea that we cannot have the exact figure of those who took the wheel without a driver’s licence. Reaching across to the data for specified variables, if it had no such data been provided in this activity. A7 K8: 3835. For instance, assuming a given number of deaths in a specific number of accidents… K7: We could calculate the probability of increase in accident count. That would give us the number of dead. Something akin to an equation with two unknowns. K6: I believe it is not about the accident, but rather about the number of people involved. K8: Then we will check the ones with driver’s licence. For, looking at the number of people alone does not help, as all are not drivers. In other words, if you say that this number of people died, the dead who died due to natural causes, rather than due to the accident may be included. Having clear view of what is related and what is not for solving the problem. For example: The population of the country, the number of driver’s licences, total number of accidents, number of accidents with death or injury, the number of injured individuals are unrelated data associated with our assumptions. What is crucial for us is the number of dead varying through the years. Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) A. Competencies to understand the real problem and to set up a model based on reality A1 K8: [Reading the guidelines] Asks what are your assumptions? What are our assumptions? K6: What did we do? We assumed a consistent rate of fall each year: Other than that... What else did we do? K8: What can we assume? We assumed the gender to be the same. K6: That would not do. K8: OK whatever. Coming up with fairly robust assumptions to make the problem simpler; to present the case with justification. For example: Let’s assume that the deaths in traffic accidents occur only due to the physical impact. There may be other cases of death to take place after the accident, due to the fear factor or another condition triggered. A2 K4: Three thousand. 3835. For instance, assuming a given number of deaths in a specific number of accidents… K3: We could calculate the probability of increase in accident count. That would give us the number of dead. Something akin to an equation with two unknowns. K1: I believe it is not about the accident, but rather about the number of people involved. K3: Driver’s licence, etc. Do you suggest we take a look on these? K4: Then we will check the ones with driver’s licence. For, looking at the number of people alone does not help, as all are not drivers. In other words, if you say that this number of people died, the dead who died on their own may have died due to normal causes, rather than due to the accident. K3: But now there is the number of people who do not have a driver’s licence. And then there are who do not have a driver’s licence, but who still take the wheel. K1: But the number of dead here… K3: Then we should correlate it with the whole population. For, there are those who do not have a driver’s licence, but still cause an accident. Applying robust simplifications to be able to develop a model. For example: Deaths in traffic accidents can be associated with numerous variables. For example, there could be deaths caused by accidents involving drivers who do not have a licence. We need to know the exact numbers of such cases. This is an extreme case we cannot control for. A3 K3: I mean, the population is growing. Say, 3 out of every 5 person get a driver’s licence. The licences are obtained without thinking about the consequences. That’s why the accident rates are rising. Identifying, naming and providing justification for the quantities which affect the case. For example: the number of deaths in traffic accidents is affected by a number of variables including the measures taken, the state of the roads, the traffic volume, the requirements for a driver’s licence, traffic assistance and first-aid capabilities and the state of affairs regarding these variables through a number of years. A4 K4: I said, an increase was observed with respect to driver’s licence numbers, due to the increase in population. The increase in the number of driver’s licences, in turn, led to an increase in the overall number of accidents. However, given the fall in the number of deaths and the rise in the number of injured in the accidents with deaths, one can talk about an increase in the measures taken through the years. Researcher: What was the thing you were trying to estimate? [Group discussion is taking place.] K4: The number of dead. Researcher: How will you associate these variables in order to estimate the dead count? K4: With traffic accidents. Researcher: I mean, will you be estimating the change in dead count with reference to the number of traffic accidents? [Group discussion is taking place.] K4: Varies by the year, I mean, falls in parallel. Having a robust identification of the key variables and explaining them with proper basis. For example: We are asked to provide the dead count caused by traffic accidents in the next year. We should forecast the next year by going over the change occurring in dead count through years. A5 K7: The population had fallen. K8: The population is rising, so is driver’s licence count. K7: It is rising and falling; see.... K8: It fell just once. On other occasions it had been rising. K7: Driver’s licences are increasing. K8: Number of accidents is increasing. K7: It rose substantially. K9: The higher the number of people, the higher the accident count. K7: The number of deaths is falling. So perhaps the safety measures are increasing. K8: Yes. K7: The number of injuries grew. That means, the number of survivors is increasing. Having a robust identification of the relationships between the variables including the key variables and explaining them with proper basis. For example: If the requirements for getting a driver’s licence were made stricter and if the trainings are increased, the number of deaths caused by traffic accidents would fall. Therefore, in cases where the number of driver’s licences is increasing, the number of dead would fall even if the number of accidents rises. If the first-aid mechanisms were improved year over year, the number of deaths would again fall despite a possible increase in the number of accidents. A6 K7: The number of deaths is falling. So perhaps the safety measures are increasing. Focusing on fairly applicable and accessible information in order to solve the problem. For example: Embracing the idea that we cannot have the exact figure of those who took the wheel without a driver’s licence. Reaching across to the data for specified variables, if it had no such data been provided in this activity. A7 K8: 3835. For instance, assuming a given number of deaths in a specific number of accidents… K7: We could calculate the probability of increase in accident count. That would give us the number of dead. Something akin to an equation with two unknowns. K6: I believe it is not about the accident, but rather about the number of people involved. K8: Then we will check the ones with driver’s licence. For, looking at the number of people alone does not help, as all are not drivers. In other words, if you say that this number of people died, the dead who died due to natural causes, rather than due to the accident may be included. Having clear view of what is related and what is not for solving the problem. For example: The population of the country, the number of driver’s licences, total number of accidents, number of accidents with death or injury, the number of injured individuals are unrelated data associated with our assumptions. What is crucial for us is the number of dead varying through the years. Table 6 The studies on competency A and expected analyses Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) A. Competencies to understand the real problem and to set up a model based on reality A1 K8: [Reading the guidelines] Asks what are your assumptions? What are our assumptions? K6: What did we do? We assumed a consistent rate of fall each year: Other than that... What else did we do? K8: What can we assume? We assumed the gender to be the same. K6: That would not do. K8: OK whatever. Coming up with fairly robust assumptions to make the problem simpler; to present the case with justification. For example: Let’s assume that the deaths in traffic accidents occur only due to the physical impact. There may be other cases of death to take place after the accident, due to the fear factor or another condition triggered. A2 K4: Three thousand. 3835. For instance, assuming a given number of deaths in a specific number of accidents… K3: We could calculate the probability of increase in accident count. That would give us the number of dead. Something akin to an equation with two unknowns. K1: I believe it is not about the accident, but rather about the number of people involved. K3: Driver’s licence, etc. Do you suggest we take a look on these? K4: Then we will check the ones with driver’s licence. For, looking at the number of people alone does not help, as all are not drivers. In other words, if you say that this number of people died, the dead who died on their own may have died due to normal causes, rather than due to the accident. K3: But now there is the number of people who do not have a driver’s licence. And then there are who do not have a driver’s licence, but who still take the wheel. K1: But the number of dead here… K3: Then we should correlate it with the whole population. For, there are those who do not have a driver’s licence, but still cause an accident. Applying robust simplifications to be able to develop a model. For example: Deaths in traffic accidents can be associated with numerous variables. For example, there could be deaths caused by accidents involving drivers who do not have a licence. We need to know the exact numbers of such cases. This is an extreme case we cannot control for. A3 K3: I mean, the population is growing. Say, 3 out of every 5 person get a driver’s licence. The licences are obtained without thinking about the consequences. That’s why the accident rates are rising. Identifying, naming and providing justification for the quantities which affect the case. For example: the number of deaths in traffic accidents is affected by a number of variables including the measures taken, the state of the roads, the traffic volume, the requirements for a driver’s licence, traffic assistance and first-aid capabilities and the state of affairs regarding these variables through a number of years. A4 K4: I said, an increase was observed with respect to driver’s licence numbers, due to the increase in population. The increase in the number of driver’s licences, in turn, led to an increase in the overall number of accidents. However, given the fall in the number of deaths and the rise in the number of injured in the accidents with deaths, one can talk about an increase in the measures taken through the years. Researcher: What was the thing you were trying to estimate? [Group discussion is taking place.] K4: The number of dead. Researcher: How will you associate these variables in order to estimate the dead count? K4: With traffic accidents. Researcher: I mean, will you be estimating the change in dead count with reference to the number of traffic accidents? [Group discussion is taking place.] K4: Varies by the year, I mean, falls in parallel. Having a robust identification of the key variables and explaining them with proper basis. For example: We are asked to provide the dead count caused by traffic accidents in the next year. We should forecast the next year by going over the change occurring in dead count through years. A5 K7: The population had fallen. K8: The population is rising, so is driver’s licence count. K7: It is rising and falling; see.... K8: It fell just once. On other occasions it had been rising. K7: Driver’s licences are increasing. K8: Number of accidents is increasing. K7: It rose substantially. K9: The higher the number of people, the higher the accident count. K7: The number of deaths is falling. So perhaps the safety measures are increasing. K8: Yes. K7: The number of injuries grew. That means, the number of survivors is increasing. Having a robust identification of the relationships between the variables including the key variables and explaining them with proper basis. For example: If the requirements for getting a driver’s licence were made stricter and if the trainings are increased, the number of deaths caused by traffic accidents would fall. Therefore, in cases where the number of driver’s licences is increasing, the number of dead would fall even if the number of accidents rises. If the first-aid mechanisms were improved year over year, the number of deaths would again fall despite a possible increase in the number of accidents. A6 K7: The number of deaths is falling. So perhaps the safety measures are increasing. Focusing on fairly applicable and accessible information in order to solve the problem. For example: Embracing the idea that we cannot have the exact figure of those who took the wheel without a driver’s licence. Reaching across to the data for specified variables, if it had no such data been provided in this activity. A7 K8: 3835. For instance, assuming a given number of deaths in a specific number of accidents… K7: We could calculate the probability of increase in accident count. That would give us the number of dead. Something akin to an equation with two unknowns. K6: I believe it is not about the accident, but rather about the number of people involved. K8: Then we will check the ones with driver’s licence. For, looking at the number of people alone does not help, as all are not drivers. In other words, if you say that this number of people died, the dead who died due to natural causes, rather than due to the accident may be included. Having clear view of what is related and what is not for solving the problem. For example: The population of the country, the number of driver’s licences, total number of accidents, number of accidents with death or injury, the number of injured individuals are unrelated data associated with our assumptions. What is crucial for us is the number of dead varying through the years. Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) A. Competencies to understand the real problem and to set up a model based on reality A1 K8: [Reading the guidelines] Asks what are your assumptions? What are our assumptions? K6: What did we do? We assumed a consistent rate of fall each year: Other than that... What else did we do? K8: What can we assume? We assumed the gender to be the same. K6: That would not do. K8: OK whatever. Coming up with fairly robust assumptions to make the problem simpler; to present the case with justification. For example: Let’s assume that the deaths in traffic accidents occur only due to the physical impact. There may be other cases of death to take place after the accident, due to the fear factor or another condition triggered. A2 K4: Three thousand. 3835. For instance, assuming a given number of deaths in a specific number of accidents… K3: We could calculate the probability of increase in accident count. That would give us the number of dead. Something akin to an equation with two unknowns. K1: I believe it is not about the accident, but rather about the number of people involved. K3: Driver’s licence, etc. Do you suggest we take a look on these? K4: Then we will check the ones with driver’s licence. For, looking at the number of people alone does not help, as all are not drivers. In other words, if you say that this number of people died, the dead who died on their own may have died due to normal causes, rather than due to the accident. K3: But now there is the number of people who do not have a driver’s licence. And then there are who do not have a driver’s licence, but who still take the wheel. K1: But the number of dead here… K3: Then we should correlate it with the whole population. For, there are those who do not have a driver’s licence, but still cause an accident. Applying robust simplifications to be able to develop a model. For example: Deaths in traffic accidents can be associated with numerous variables. For example, there could be deaths caused by accidents involving drivers who do not have a licence. We need to know the exact numbers of such cases. This is an extreme case we cannot control for. A3 K3: I mean, the population is growing. Say, 3 out of every 5 person get a driver’s licence. The licences are obtained without thinking about the consequences. That’s why the accident rates are rising. Identifying, naming and providing justification for the quantities which affect the case. For example: the number of deaths in traffic accidents is affected by a number of variables including the measures taken, the state of the roads, the traffic volume, the requirements for a driver’s licence, traffic assistance and first-aid capabilities and the state of affairs regarding these variables through a number of years. A4 K4: I said, an increase was observed with respect to driver’s licence numbers, due to the increase in population. The increase in the number of driver’s licences, in turn, led to an increase in the overall number of accidents. However, given the fall in the number of deaths and the rise in the number of injured in the accidents with deaths, one can talk about an increase in the measures taken through the years. Researcher: What was the thing you were trying to estimate? [Group discussion is taking place.] K4: The number of dead. Researcher: How will you associate these variables in order to estimate the dead count? K4: With traffic accidents. Researcher: I mean, will you be estimating the change in dead count with reference to the number of traffic accidents? [Group discussion is taking place.] K4: Varies by the year, I mean, falls in parallel. Having a robust identification of the key variables and explaining them with proper basis. For example: We are asked to provide the dead count caused by traffic accidents in the next year. We should forecast the next year by going over the change occurring in dead count through years. A5 K7: The population had fallen. K8: The population is rising, so is driver’s licence count. K7: It is rising and falling; see.... K8: It fell just once. On other occasions it had been rising. K7: Driver’s licences are increasing. K8: Number of accidents is increasing. K7: It rose substantially. K9: The higher the number of people, the higher the accident count. K7: The number of deaths is falling. So perhaps the safety measures are increasing. K8: Yes. K7: The number of injuries grew. That means, the number of survivors is increasing. Having a robust identification of the relationships between the variables including the key variables and explaining them with proper basis. For example: If the requirements for getting a driver’s licence were made stricter and if the trainings are increased, the number of deaths caused by traffic accidents would fall. Therefore, in cases where the number of driver’s licences is increasing, the number of dead would fall even if the number of accidents rises. If the first-aid mechanisms were improved year over year, the number of deaths would again fall despite a possible increase in the number of accidents. A6 K7: The number of deaths is falling. So perhaps the safety measures are increasing. Focusing on fairly applicable and accessible information in order to solve the problem. For example: Embracing the idea that we cannot have the exact figure of those who took the wheel without a driver’s licence. Reaching across to the data for specified variables, if it had no such data been provided in this activity. A7 K8: 3835. For instance, assuming a given number of deaths in a specific number of accidents… K7: We could calculate the probability of increase in accident count. That would give us the number of dead. Something akin to an equation with two unknowns. K6: I believe it is not about the accident, but rather about the number of people involved. K8: Then we will check the ones with driver’s licence. For, looking at the number of people alone does not help, as all are not drivers. In other words, if you say that this number of people died, the dead who died due to natural causes, rather than due to the accident may be included. Having clear view of what is related and what is not for solving the problem. For example: The population of the country, the number of driver’s licences, total number of accidents, number of accidents with death or injury, the number of injured individuals are unrelated data associated with our assumptions. What is crucial for us is the number of dead varying through the years. Below, a model-eliciting activity is described, along with specific experiences concerning certain sub-processes throughout the modelling cycle during the execution of the activity, in the light of the group study and classroom debate. 2.5 An MEA example: TAS MEA’s characteristics: This MEA asks student teachers to determine whether the measures taken for traffic accidents are sufficient. This activity to be implemented through group work will allow estimations for future years, based on the data for past years. The data for past years, concerning the context, were available on Turkish Statistics Agency (TurkStat) website. Thereafter, the real-life problem to be dealt with within the framework of the activity was described alongside justification. In this activity, all data received from TurkStat were presented directly, without identification of key variables. The student mathematics teachers were asked to work with data for years 2006 on. The key variables in this activity were the year and the death count. A glance at the data for key variables revealed a linear direction. This data set is conducive to representation through a linear model. Student mathematics teachers’ experiences with the implementation of the TAS activity were analyzed through taking into account the rubric presented in the following tables. The studies on the competency A were analyzed through taking into account the rubric presented in Table 6. As illustrated in the Table 6, the work on A1 sub-competency reveals that K6 developed valid assumptions required to develop the model which could provide a solution for the problem, even if she did not provide justification, while K8 engaged in an unrelated assumption. K8 did not inquire K6’s critical view’s justification and duly abandoned her idea. That is why K6 is deemed to have sub-competency A1 at the level of criteria 1, while K8’s competency A1 is deemed to be at the level of criteria 0. Work on sub-competency A2, in turn, reveals that certain student mathematics teachers have reviewed the variables affecting the state of the problem on a wide perspective, while some others had a more realistic outlook, seeking to simplify the variables in turn. Yet, as the simplifications focused on incorrect key variables, student mathematics teachers K1, K3 and K4 were considered to meet only criteria 2 for these sub-competencies. A look at sub-competency A3 reveals that K3 voiced variables concerning population increase, the number of driver’s licences and the rate of accidents among all the variables to affect the picture. In this context, an increase in the number of deaths should have arisen. However, the measures taken and other variables such as first-aid measures contribute to a fall in the number of deaths. Against this background, K3 was observed to identify various variables but fell short of being able to interpret the state of affairs concerning those variables. Hence, K3 is assessed to meet criteria 2 for this sub-competency. With reference to sub-competency A4, K4 was observed to identify the linkages between various variables affecting the state of affairs but remained unable to take note of key variables. Hence, K4 is assessed to meet criteria 0 for sub-competency A4. The in-group debates executed with the counsel of the researcher paved the way for K4 to accurately identify the key variables. The discussions with the researcher and the group have led K4 to criteria 3 level for sub-competency A4. With reference to sub-competency A5, K7, K8 and K9 were observed to focus on the variables specified in the activity and try to establish the linkages between such variables. Yet, the relationships between other variables which would have an impact on the state of affairs but which were not covered in the activity, such as the measures taken, and the desired state of affairs had not been covered. Only K7 was observed to establish a link between the measures taken —the key variable— and other variables. Nonetheless, K7 established the said relationship not with the number of deaths with reference to the year —the key variable— but with the fall in the number of injuries. Taking all these into account, K8 and K9 exhibited the ability to establish connections between the variables, to the tune of criteria 1, while K7’s skills were assessed to meet criteria 2. In terms of the work on sub-competency A6, K7 was observed to focus on the causes of the fall in the number of deaths, without any attempt to check the availability of such data, instead focusing on the information provided. Hence, her competence level for A6 is assessed to meet criteria 0. A glance at the work on sub-competency A7 reveals that the student mathematics teachers did not associate the key variable they identified when working on the variables to affect the state of affairs, with the year variable entailed in the problem case, but instead tried their luck with other variables which are not related with the desired result. Therefore, these student mathematics teachers meet criteria 0 for sub-competency A7. The studies on the competency B were analyzed through taking into account the rubric presented in Table 7. Table 7 The studies on competency B and expected analyses Table 7 The studies on competency B and expected analyses The work on sub-competency B1 suggests that K5 is aware of the need for a mathematical expression of the qualities identified in accordance with the state of the problem, through an interpretation of data. However, a mathematical expression still eludes them. Therefore, K5 meets criteria 1 for sub-competency B1. The work on sub-competency B2 reveals that student mathematics teachers have expressed the equation to lead to the number of deaths with reference to the average rate of change. Hence, K6, who developed this idea, is assessed to meet criteria 3 for sub-competency B2. On sub-competency B3, K6 was observed to offer some means to simplify the quantities and to provide some remarks on the cases where the model would apply after simplifications. K7 too provides accurate remarks on why the approach could be the right one and confirms the case with real-life data. Therefore, the student mathematics teacher meets criteria 3 for sub-competency B3. With reference to sub-competency B4, the simplifications required to render the mathematical model usable can be observed to be effected. The simplifications render the model more usable, while remaining mathematically accurate. Therefore, student mathematics teacher K6 who effected such simplifications exhibits the competence to reduce the number of relationships between relevant quantities and to simplify the relationships, where necessary, to the tune of criteria 3. A glance at the work for sub-competency B5 reveals that student mathematics teachers had identified the relationships explaining the case and were able to come up with a mathematical model by choosing applicable mathematical expressions to match the mathematical operations applied for this purpose. Furthermore, it is evident that the student mathematics teachers fell short of using regression or least squares method which would provide a better representation of the data set, when developing the mathematical model to represent the data for the case. Therefore, K6, who chose the mathematical expressions used for the development of the model, exhibits the competence to choose applicable mathematical expressions and represent them graphically where necessary, to the tune of criteria 2. The studies on the competency C were analyzed through taking into account the rubric presented in Table 8. Table 8 The studies on competency C and expected analyses Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) C. Competencies to solve mathematical questions within this mathematical model C1 No work was carried out. Making robust use of the heuristic strategies to assist the solution of the problem; providing explanations with justification. For example: Developing solutions to the problem by taking part of the data into account; or using the model by establishing links with the previous activity which is similar in nature. C2 Making very good use of the mathematical information required to solve the problem. For example: Reaching to the required information in a mathematically correct form, by accurately noting the known data on the linear model developed for this purpose. Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) C. Competencies to solve mathematical questions within this mathematical model C1 No work was carried out. Making robust use of the heuristic strategies to assist the solution of the problem; providing explanations with justification. For example: Developing solutions to the problem by taking part of the data into account; or using the model by establishing links with the previous activity which is similar in nature. C2 Making very good use of the mathematical information required to solve the problem. For example: Reaching to the required information in a mathematically correct form, by accurately noting the known data on the linear model developed for this purpose. Table 8 The studies on competency C and expected analyses Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) C. Competencies to solve mathematical questions within this mathematical model C1 No work was carried out. Making robust use of the heuristic strategies to assist the solution of the problem; providing explanations with justification. For example: Developing solutions to the problem by taking part of the data into account; or using the model by establishing links with the previous activity which is similar in nature. C2 Making very good use of the mathematical information required to solve the problem. For example: Reaching to the required information in a mathematically correct form, by accurately noting the known data on the linear model developed for this purpose. Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) C. Competencies to solve mathematical questions within this mathematical model C1 No work was carried out. Making robust use of the heuristic strategies to assist the solution of the problem; providing explanations with justification. For example: Developing solutions to the problem by taking part of the data into account; or using the model by establishing links with the previous activity which is similar in nature. C2 Making very good use of the mathematical information required to solve the problem. For example: Reaching to the required information in a mathematically correct form, by accurately noting the known data on the linear model developed for this purpose. All student mathematics teachers were found to meet criteria 0 for sub-competency C1 regarding TAS activities. This is caused by the lack of any heuristic strategy, rather than by the use of inapplicable heuristic strategies. A glance at work on sub-competency C2 reveals that student mathematics teachers had made very effective use of their mathematical knowledge when solving the problem through the mathematical model they had developed and were able to reach to the result through the model. Therefore, K6 and K7 which carried out these procedures exhibit competence to use mathematical information to solve the problem, to the tune of criteria 3. The studies on the competency D were analyzed through taking into account the rubric presented in Table 9. Table 9 The studies on competency D and expected analyses Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) D. Competencies to interpret mathematical results in a real situation D1 K9 : But after some years, we will reach to a state... where deaths will be no more. Offering very robust interpretations of mathematical results in non-mathematical contexts and providing justifications. For example: If the result is negative, the case would refer to a negative dead count, which should not be interpreted to suggest unexpected births but rather as a reflection of our linear reduction perspective. D2 K8 : Asking is it generalizable? K6 : It can be generalized in cases of continuous fall. K7 : It can be generalized in cases where no huge increase occurs. K5 : It cannot be generalized in cases of excessive fall or increase. OK. Providing very effective generalizations of the solutions developed for special cases; providing justifications for doing so. For example: Realizing that the mathematical model thus developed can be generalized only in linear cases with a fall in the rate of change and developing a general mathematical model in this direction (for a linear change; estimated number of deaths = rate of change reflecting the data x time frame from the base year to the focus year + the number of deaths in the base year). Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) D. Competencies to interpret mathematical results in a real situation D1 K9 : But after some years, we will reach to a state... where deaths will be no more. Offering very robust interpretations of mathematical results in non-mathematical contexts and providing justifications. For example: If the result is negative, the case would refer to a negative dead count, which should not be interpreted to suggest unexpected births but rather as a reflection of our linear reduction perspective. D2 K8 : Asking is it generalizable? K6 : It can be generalized in cases of continuous fall. K7 : It can be generalized in cases where no huge increase occurs. K5 : It cannot be generalized in cases of excessive fall or increase. OK. Providing very effective generalizations of the solutions developed for special cases; providing justifications for doing so. For example: Realizing that the mathematical model thus developed can be generalized only in linear cases with a fall in the rate of change and developing a general mathematical model in this direction (for a linear change; estimated number of deaths = rate of change reflecting the data x time frame from the base year to the focus year + the number of deaths in the base year). Table 9 The studies on competency D and expected analyses Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) D. Competencies to interpret mathematical results in a real situation D1 K9 : But after some years, we will reach to a state... where deaths will be no more. Offering very robust interpretations of mathematical results in non-mathematical contexts and providing justifications. For example: If the result is negative, the case would refer to a negative dead count, which should not be interpreted to suggest unexpected births but rather as a reflection of our linear reduction perspective. D2 K8 : Asking is it generalizable? K6 : It can be generalized in cases of continuous fall. K7 : It can be generalized in cases where no huge increase occurs. K5 : It cannot be generalized in cases of excessive fall or increase. OK. Providing very effective generalizations of the solutions developed for special cases; providing justifications for doing so. For example: Realizing that the mathematical model thus developed can be generalized only in linear cases with a fall in the rate of change and developing a general mathematical model in this direction (for a linear change; estimated number of deaths = rate of change reflecting the data x time frame from the base year to the focus year + the number of deaths in the base year). Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) D. Competencies to interpret mathematical results in a real situation D1 K9 : But after some years, we will reach to a state... where deaths will be no more. Offering very robust interpretations of mathematical results in non-mathematical contexts and providing justifications. For example: If the result is negative, the case would refer to a negative dead count, which should not be interpreted to suggest unexpected births but rather as a reflection of our linear reduction perspective. D2 K8 : Asking is it generalizable? K6 : It can be generalized in cases of continuous fall. K7 : It can be generalized in cases where no huge increase occurs. K5 : It cannot be generalized in cases of excessive fall or increase. OK. Providing very effective generalizations of the solutions developed for special cases; providing justifications for doing so. For example: Realizing that the mathematical model thus developed can be generalized only in linear cases with a fall in the rate of change and developing a general mathematical model in this direction (for a linear change; estimated number of deaths = rate of change reflecting the data x time frame from the base year to the focus year + the number of deaths in the base year). A glance at work on sub-competency D1 reveals that K9 was able to interpret the results reached through the mathematical model and to reach to inferences regarding the actual context. However, no remarks were made on whether or not the lack of any deaths in traffic accidents was a realistic case against the actual state of affairs. That is why K9 is assessed to exhibit the ability to interpret mathematical results in non-mathematical contexts, to the tune of criteria 1. A glance at work on sub-competency D2 reveals that K6 and K7 provided remarks on the justifications on the cases where the mathematical method developed could be generalized. However, as various simplifications were applied to render the model practical, the mathematical model can be used to estimate the time frame 2000–2099. Furthermore, as no constant increase or fall took place in the actual state of affairs, in its current form, the model cannot be generalized. The student mathematics teachers are aware of the conditions required for the generalization of the model yet fall short of the generalization work by assigning variables to stand for actual data, so as to be able to compensate for such shortcomings. Taking all these into account, K6 and K7 exhibit the competence to generalize the solutions developed for a special case, to the tune of criteria 1. The studies on the competency E were analyzed like the other competencies. The sub-competency levels of each student teacher involved in group work were assessed separately. Individual assessments allowed the observation of the development of competencies on part of each student teacher through the process, as well as the combined interpretation of competencies to arise through the activities carried out as part of the group work, as well as through the individual activities. With a view to providing an example of how the analysis process took place, the rubric developed to assess the student mathematics teachers’ mathematical modelling sub-competencies and the examples of top performance indicators for the MEA at hand are presented in Table 11. The processes in MEAs were reviewed using the analytical rubric exemplified in Table 10, to identify the student mathematics teachers’ sub-competencies regarding each MEA. The number of student mathematics teachers with specific sub-competencies was established with respect to each MEA. As an example, the data set generated for competency B regarding activity BA is provided in Table 12. Table 10 Mathematical modelling competencies analysis rubric and examples of student work regarding TAS activity, at criteria 3 level No attempt or unacceptable performance (criteria 0) Insufficient performance (criteria 1) Imperfect yet acceptable performance (criteria 2) Perfect performance (criteria 3) B2 The relationships between the applicable quantity/quantities were not mathematized, or the mathematization implemented is not applicable or is irrelevant. Attempts were made to mathematize the relationships between the applicable quantity/quantities, but they are not sufficient for the development of a model. The relationships between the applicable quantity/quantities were mathematized; but some small insignificant errors exist or no justifications were provided. The relationships between the applicable quantity/quantities were mathematized quite well; justifications were provided to explain. For example: There’s a general fall. The data exhibit a linear fall. Stating the most applicable correct equation to represent the data such as [(y-y0)=m(x-x0)]. B3 The number of applicable quantities was not reduced even though such a reduction was in order; no simplification of quantities was applied, or the applied simplification is invalid. Some remarks were made about the simplification of applicable quantities were voiced; however the implementation of such remarks is not sufficient, or the simplifications are not suitable in terms of usability. Necessary simplifications were effected with respect to applicable quantities; however certain small and insignificant errors persist; or no investigation of the impact of simplifications on the model was applied. The simplifications concerning applicable quantities were carried out quite well; a review of their impact on the model was carried out, along with a discussion of justifications. For example: The average change per year is 159.6. Since we are talking about the number of individuals, rounding it up to 160 would have only a minimal impact on the error margins. For our estimations will certainly be in four digits. We can apply such a simplification. No attempt or unacceptable performance (criteria 0) Insufficient performance (criteria 1) Imperfect yet acceptable performance (criteria 2) Perfect performance (criteria 3) B2 The relationships between the applicable quantity/quantities were not mathematized, or the mathematization implemented is not applicable or is irrelevant. Attempts were made to mathematize the relationships between the applicable quantity/quantities, but they are not sufficient for the development of a model. The relationships between the applicable quantity/quantities were mathematized; but some small insignificant errors exist or no justifications were provided. The relationships between the applicable quantity/quantities were mathematized quite well; justifications were provided to explain. For example: There’s a general fall. The data exhibit a linear fall. Stating the most applicable correct equation to represent the data such as [(y-y0)=m(x-x0)]. B3 The number of applicable quantities was not reduced even though such a reduction was in order; no simplification of quantities was applied, or the applied simplification is invalid. Some remarks were made about the simplification of applicable quantities were voiced; however the implementation of such remarks is not sufficient, or the simplifications are not suitable in terms of usability. Necessary simplifications were effected with respect to applicable quantities; however certain small and insignificant errors persist; or no investigation of the impact of simplifications on the model was applied. The simplifications concerning applicable quantities were carried out quite well; a review of their impact on the model was carried out, along with a discussion of justifications. For example: The average change per year is 159.6. Since we are talking about the number of individuals, rounding it up to 160 would have only a minimal impact on the error margins. For our estimations will certainly be in four digits. We can apply such a simplification. Table 10 Mathematical modelling competencies analysis rubric and examples of student work regarding TAS activity, at criteria 3 level No attempt or unacceptable performance (criteria 0) Insufficient performance (criteria 1) Imperfect yet acceptable performance (criteria 2) Perfect performance (criteria 3) B2 The relationships between the applicable quantity/quantities were not mathematized, or the mathematization implemented is not applicable or is irrelevant. Attempts were made to mathematize the relationships between the applicable quantity/quantities, but they are not sufficient for the development of a model. The relationships between the applicable quantity/quantities were mathematized; but some small insignificant errors exist or no justifications were provided. The relationships between the applicable quantity/quantities were mathematized quite well; justifications were provided to explain. For example: There’s a general fall. The data exhibit a linear fall. Stating the most applicable correct equation to represent the data such as [(y-y0)=m(x-x0)]. B3 The number of applicable quantities was not reduced even though such a reduction was in order; no simplification of quantities was applied, or the applied simplification is invalid. Some remarks were made about the simplification of applicable quantities were voiced; however the implementation of such remarks is not sufficient, or the simplifications are not suitable in terms of usability. Necessary simplifications were effected with respect to applicable quantities; however certain small and insignificant errors persist; or no investigation of the impact of simplifications on the model was applied. The simplifications concerning applicable quantities were carried out quite well; a review of their impact on the model was carried out, along with a discussion of justifications. For example: The average change per year is 159.6. Since we are talking about the number of individuals, rounding it up to 160 would have only a minimal impact on the error margins. For our estimations will certainly be in four digits. We can apply such a simplification. No attempt or unacceptable performance (criteria 0) Insufficient performance (criteria 1) Imperfect yet acceptable performance (criteria 2) Perfect performance (criteria 3) B2 The relationships between the applicable quantity/quantities were not mathematized, or the mathematization implemented is not applicable or is irrelevant. Attempts were made to mathematize the relationships between the applicable quantity/quantities, but they are not sufficient for the development of a model. The relationships between the applicable quantity/quantities were mathematized; but some small insignificant errors exist or no justifications were provided. The relationships between the applicable quantity/quantities were mathematized quite well; justifications were provided to explain. For example: There’s a general fall. The data exhibit a linear fall. Stating the most applicable correct equation to represent the data such as [(y-y0)=m(x-x0)]. B3 The number of applicable quantities was not reduced even though such a reduction was in order; no simplification of quantities was applied, or the applied simplification is invalid. Some remarks were made about the simplification of applicable quantities were voiced; however the implementation of such remarks is not sufficient, or the simplifications are not suitable in terms of usability. Necessary simplifications were effected with respect to applicable quantities; however certain small and insignificant errors persist; or no investigation of the impact of simplifications on the model was applied. The simplifications concerning applicable quantities were carried out quite well; a review of their impact on the model was carried out, along with a discussion of justifications. For example: The average change per year is 159.6. Since we are talking about the number of individuals, rounding it up to 160 would have only a minimal impact on the error margins. For our estimations will certainly be in four digits. We can apply such a simplification. Table 11 The number of student mathematics teachers to meet the established competency levels, as observed through the evaluation of the work carried out in first activity (BF), using the analytical rubric Number of student teacher Sub-competencies Criteria 0 Criteria 1 Criteria 2 Criteria 3 Total B2 8 1 0 0 9 B3 9 0 0 0 9 Number of student teacher Sub-competencies Criteria 0 Criteria 1 Criteria 2 Criteria 3 Total B2 8 1 0 0 9 B3 9 0 0 0 9 Table 11 The number of student mathematics teachers to meet the established competency levels, as observed through the evaluation of the work carried out in first activity (BF), using the analytical rubric Number of student teacher Sub-competencies Criteria 0 Criteria 1 Criteria 2 Criteria 3 Total B2 8 1 0 0 9 B3 9 0 0 0 9 Number of student teacher Sub-competencies Criteria 0 Criteria 1 Criteria 2 Criteria 3 Total B2 8 1 0 0 9 B3 9 0 0 0 9 Once the data presented in Table 11 were established for each activity, tables pertaining to the criteria observed in activities for competencies were drawn up with a view to reviewing the development of each sub-competency within the framework of the process. For instance, one can review Table 13 indicating the distribution regarding sub-competency B3 to arise in all MEAs. Table 12 Distribution of B3 sub-competency levels regarding assumptions about the all MEAs Number of student teacher MEA Criteria 0 Criteria 1 Criteria 2 Criteria 3 Total 1 9 0 0 0 9 2 7 2 0 0 9 3 5 1 2 1 9 4 6 3 0 0 9 5 6 0 1 2 9 6 9 0 0 0 9 7 6 0 0 3 9 8 1 0 1 7 9 Number of student teacher MEA Criteria 0 Criteria 1 Criteria 2 Criteria 3 Total 1 9 0 0 0 9 2 7 2 0 0 9 3 5 1 2 1 9 4 6 3 0 0 9 5 6 0 1 2 9 6 9 0 0 0 9 7 6 0 0 3 9 8 1 0 1 7 9 Table 12 Distribution of B3 sub-competency levels regarding assumptions about the all MEAs Number of student teacher MEA Criteria 0 Criteria 1 Criteria 2 Criteria 3 Total 1 9 0 0 0 9 2 7 2 0 0 9 3 5 1 2 1 9 4 6 3 0 0 9 5 6 0 1 2 9 6 9 0 0 0 9 7 6 0 0 3 9 8 1 0 1 7 9 Number of student teacher MEA Criteria 0 Criteria 1 Criteria 2 Criteria 3 Total 1 9 0 0 0 9 2 7 2 0 0 9 3 5 1 2 1 9 4 6 3 0 0 9 5 6 0 1 2 9 6 9 0 0 0 9 7 6 0 0 3 9 8 1 0 1 7 9 Table 13 An example for the evaluation of development within the framework of the process Process Example for B2 sub-competency Development for participants (both group work and individual work) Sub-competencies for each MEA were evaluated on an individual basis, followed by the review of general change. Individual development (both group work and individual work) Sub-competencies for each MEA were evaluated on an individual basis, followed by the review of individual change. Without intervention The sub-competencies of student mathematics teachers who are not involved with learning environment. A general review of the work three student mathematics teachers who were not involved with the learning environment carried out with MEAs provided reveals that just one of the student mathematics teachers reached criteria 3 levels regarding sub-competency B3, while others remained at criteria 0. The student mathematics teacher who had reached criteria 3 levels is the same one who scored criteria 3 levels for sub-competencies B1 and B2. Student mathematics teachers with low levels of sub-competency for B1 and B2 failed to provide mathematical expression of the relationships between the variables, and hence did not get the chance to work on simplifications regarding quantities. This finding suggests that sub-competency B3 is affected by other sub-competencies. Process Example for B2 sub-competency Development for participants (both group work and individual work) Sub-competencies for each MEA were evaluated on an individual basis, followed by the review of general change. Individual development (both group work and individual work) Sub-competencies for each MEA were evaluated on an individual basis, followed by the review of individual change. Without intervention The sub-competencies of student mathematics teachers who are not involved with learning environment. A general review of the work three student mathematics teachers who were not involved with the learning environment carried out with MEAs provided reveals that just one of the student mathematics teachers reached criteria 3 levels regarding sub-competency B3, while others remained at criteria 0. The student mathematics teacher who had reached criteria 3 levels is the same one who scored criteria 3 levels for sub-competencies B1 and B2. Student mathematics teachers with low levels of sub-competency for B1 and B2 failed to provide mathematical expression of the relationships between the variables, and hence did not get the chance to work on simplifications regarding quantities. This finding suggests that sub-competency B3 is affected by other sub-competencies. Table 13 An example for the evaluation of development within the framework of the process Process Example for B2 sub-competency Development for participants (both group work and individual work) Sub-competencies for each MEA were evaluated on an individual basis, followed by the review of general change. Individual development (both group work and individual work) Sub-competencies for each MEA were evaluated on an individual basis, followed by the review of individual change. Without intervention The sub-competencies of student mathematics teachers who are not involved with learning environment. A general review of the work three student mathematics teachers who were not involved with the learning environment carried out with MEAs provided reveals that just one of the student mathematics teachers reached criteria 3 levels regarding sub-competency B3, while others remained at criteria 0. The student mathematics teacher who had reached criteria 3 levels is the same one who scored criteria 3 levels for sub-competencies B1 and B2. Student mathematics teachers with low levels of sub-competency for B1 and B2 failed to provide mathematical expression of the relationships between the variables, and hence did not get the chance to work on simplifications regarding quantities. This finding suggests that sub-competency B3 is affected by other sub-competencies. Process Example for B2 sub-competency Development for participants (both group work and individual work) Sub-competencies for each MEA were evaluated on an individual basis, followed by the review of general change. Individual development (both group work and individual work) Sub-competencies for each MEA were evaluated on an individual basis, followed by the review of individual change. Without intervention The sub-competencies of student mathematics teachers who are not involved with learning environment. A general review of the work three student mathematics teachers who were not involved with the learning environment carried out with MEAs provided reveals that just one of the student mathematics teachers reached criteria 3 levels regarding sub-competency B3, while others remained at criteria 0. The student mathematics teacher who had reached criteria 3 levels is the same one who scored criteria 3 levels for sub-competencies B1 and B2. Student mathematics teachers with low levels of sub-competency for B1 and B2 failed to provide mathematical expression of the relationships between the variables, and hence did not get the chance to work on simplifications regarding quantities. This finding suggests that sub-competency B3 is affected by other sub-competencies. Table 13 presents an example on the evaluation of the sub-competency development in line with these data. The sub-competence levels of the mathematics teachers who were involved and did not participate in the learning environment were compared. In this way the contribution to the development of the learning environment was perceived. A general review of the graph presenting the change student mathematics teachers, who were involved in the learning environment, went through in terms of their qualification levels for sub-competency B3 reveals that the student mathematics teachers did not get sub-competency B3 without receiving a training on mathematical modelling; that the level of qualification for this sub-competency developed, even if at a limited rate, compared to the models developed through the process; and that at the end of the process, the majority of the student mathematics teachers exhibited criteria 3 levels of competence in this context. When individual opportunities were extended (with IPs), the student mathematics teachers who did not engage in work on sub-competency B3 through the process were, strikingly, observed to exhibit the sub-competency B3 in their own projects. The sub-competency B3 scores in IPs were higher then group work scores. Because the student mathematics teachers needed to render their mathematical model practical so as to become able to generate solutions for their individual problems. They work together in the groups' mathematical models, so they do not need practical models. They overcome the non-practical mathematical models in the group works. (We can say that the learning environment is ‘Not Directly Influential (NDI)’ on B3 sub-competency and it is activated in individual work process.) A comparison of the student mathematics teachers’ competence levels for sub-competency B3 with the case of student mathematics teachers who were not involved with the learning environment leads to the conclusion that those who were involved with the learning environment were found to reach better levels at the end of the process. However, considering the view that the sub-competency arises with reference to the structure of the mathematical model thus developed and the existence of preceding sub-competencies, one can forcefully argue that sub-competency B3 is not directly associated with the mathematical modelling experience. (We can say that B3 sub-competency’s development is ‘Not Directly Dependent on Experience (NDDE)’ of mathematical modelling.) Only an example of analysis for B3 sub-competency is given in this section. Similar processes have been carried out for other sub-competencies. The development of sub-competencies of student mathematics teachers involved with the learning environment, to arise within the group, for each sub-competency, as well as their sub-competencies to arise in individual projects and the sub-competencies of the student mathematics teachers who were not involved with the learning environment were compared to establish the contribution of the learning environment to the development of sub-competencies. If a sub-competency has not developed to the expected criteria for more student mathematics teachers, we can say that the learning environment is ‘Influential (I)’ on this sub-competency. If a sub-competency is developed in both general and individual in high criteria, we can say that the learning environment is ‘Extremely Influential (EI)’ on this sub-competency and if student mathematics teachers who are not involved with learning environment have not this sub-competency, we can say that this sub-competency’s development is ‘Depends on Experience (DE)’ or if student mathematics teachers who are not involved with learning environment have this sub-competency even if low criteria, we can say that this sub-competency’s development is ‘Somewhat Depends on Experience (SDE)’. If many student mathematics teachers have a sub-competency in high criteria and this sub-competency appears in low criteria late in the process, we can say that the learning environment is ‘Negative Influence (NI)’ on this sub-competency and if student mathematics teachers who are not involved with learning environment have engaged in better studies regarding this sub-competency, we can say that this sub-competency’s regression is ‘Depends on Experience (DE)’ on mathematical modelling. 3. Results The objective of this study is to reveal how LES-MMC process contributes to the development of mathematical modelling sub-competencies. In line with this objective, the development of sub-competencies of participants involved with the LES-MMC process was analyzed for individual participants, as well as for the whole group. The findings were compared against the sub-competencies of the student mathematics teachers who were not involved with the learning environment, leading to the analysis of the impact of the designed environment on the development of mathematical modelling sub-competencies. Given the limitations of space, it is not possible to cover the sub-competency related findings for all participants as well as the student mathematics teachers who are not involved in the learning environment and is not within the realm of possibilities. Therefore, this section will compare the three data sets developed for each sub-competency to discuss the conclusion reached with respect to the development of sub-competencies in the light of the literature. Also only the interesting findings will be discussed in this section. The full PhD thesis written by Aydin-Güç (2015) can be reviewed for further results and discussion. The summaries regarding the findings concerning the competency to comprehend the real problem and to develop a model based on the reality (A) and the conclusions reached with respect to associated sub-competencies are presented in Table 14. Table 14 Development of sub-competencies associated with competency A, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature A1. Competency to make assumptions for the problem EI DE Bukova-Güzel (2011) A2. Competency to simplify the situation I SDE Blum & Borromeo-Ferri (2009); Şen-Zeytun (2013); A3. Competency to recognize quantities that influence the situation, to name them NI DE - A4. Competency to identify key variables that influence the situation I SDE - A5. Competency to construct relations between the variables I SDE - A6. Competency to look for available/not available information to solve the problem I SDE - A7. Competency to differentiate the related/unrelated information to solve the problem I SDE - Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature A1. Competency to make assumptions for the problem EI DE Bukova-Güzel (2011) A2. Competency to simplify the situation I SDE Blum & Borromeo-Ferri (2009); Şen-Zeytun (2013); A3. Competency to recognize quantities that influence the situation, to name them NI DE - A4. Competency to identify key variables that influence the situation I SDE - A5. Competency to construct relations between the variables I SDE - A6. Competency to look for available/not available information to solve the problem I SDE - A7. Competency to differentiate the related/unrelated information to solve the problem I SDE - Table 14 Development of sub-competencies associated with competency A, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature A1. Competency to make assumptions for the problem EI DE Bukova-Güzel (2011) A2. Competency to simplify the situation I SDE Blum & Borromeo-Ferri (2009); Şen-Zeytun (2013); A3. Competency to recognize quantities that influence the situation, to name them NI DE - A4. Competency to identify key variables that influence the situation I SDE - A5. Competency to construct relations between the variables I SDE - A6. Competency to look for available/not available information to solve the problem I SDE - A7. Competency to differentiate the related/unrelated information to solve the problem I SDE - Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature A1. Competency to make assumptions for the problem EI DE Bukova-Güzel (2011) A2. Competency to simplify the situation I SDE Blum & Borromeo-Ferri (2009); Şen-Zeytun (2013); A3. Competency to recognize quantities that influence the situation, to name them NI DE - A4. Competency to identify key variables that influence the situation I SDE - A5. Competency to construct relations between the variables I SDE - A6. Competency to look for available/not available information to solve the problem I SDE - A7. Competency to differentiate the related/unrelated information to solve the problem I SDE - It is clear that the designed learning environment is very influential in the development of A1 sub-competency. A comparison of A1 sub-competencies of student mathematics teachers who are involved with the learning environment against those of the student mathematics teachers not involved suggests that the development of this sub-competency depends on the learning environment. The literature is not bereft of other studies with similar results on the experience-based development of that competency (Bukova-Güzel, 2011). The development of the A2 sub-competency based on experience concerning mathematical modelling, albeit not at the desired level, suggests that the learning environment design influences the development of A2 sub-competency. Related literature suggests that the simplification competencies of the students leave some to be desired, given the fact that they engage only in crude simplifications, even though the studies do not investigate sub-competencies of mathematical modelling individually. Furthermore, these studies emphasize the fact that the competencies to make assumptions regarding the context are less than stellar (Blum & Borromeo-Ferri, 2009; Şen-Zeytun, 2013). The A4, A5, A6 and A7 sub-competencies of student mathematics teachers who were not involved with the learning environment, compared against the development registered by the student mathematics teachers who were involved, revealed that the latter group achieved higher competency levels in practices concerning such sub-competencies. This state of affairs suggests that the learning environment design influences the development of A4, A5, A6 and A7 sub-competencies and that such sub-competencies develop in parallel to mathematical modelling, albeit not at the desired level. Given the fact that the studies and the results of which are discussed in the literature do not discuss the sub-competencies separately, the findings concerning sub-competencies A3, A4, A5, A6 and A7 would certainly shed light on sub-competencies. The summaries regarding the findings concerning the competency to develop a mathematical model based on the real one (B) and the conclusions reached with respect to associated sub-competencies are presented in Table 15. Table 15 Development of sub-competencies associated with competency B, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature B1. Competency to mathematize relevant qualities/quantities I SDE - B2. Competency to mathematize the relations between the related qualities/quantities I SDE - B3. Competency to reduce the number of relevant quantities and simplify their quantities when necessary NDI- individual use NDDE - B4. Competency to reduce the number of relations between related quantities and simplify the relations when necessary NDI-individual use NDDE - B5. Competency to choose appropriate mathematical notations and represent situations graphics if necessary I SDE - Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature B1. Competency to mathematize relevant qualities/quantities I SDE - B2. Competency to mathematize the relations between the related qualities/quantities I SDE - B3. Competency to reduce the number of relevant quantities and simplify their quantities when necessary NDI- individual use NDDE - B4. Competency to reduce the number of relations between related quantities and simplify the relations when necessary NDI-individual use NDDE - B5. Competency to choose appropriate mathematical notations and represent situations graphics if necessary I SDE - Table 15 Development of sub-competencies associated with competency B, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature B1. Competency to mathematize relevant qualities/quantities I SDE - B2. Competency to mathematize the relations between the related qualities/quantities I SDE - B3. Competency to reduce the number of relevant quantities and simplify their quantities when necessary NDI- individual use NDDE - B4. Competency to reduce the number of relations between related quantities and simplify the relations when necessary NDI-individual use NDDE - B5. Competency to choose appropriate mathematical notations and represent situations graphics if necessary I SDE - Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature B1. Competency to mathematize relevant qualities/quantities I SDE - B2. Competency to mathematize the relations between the related qualities/quantities I SDE - B3. Competency to reduce the number of relevant quantities and simplify their quantities when necessary NDI- individual use NDDE - B4. Competency to reduce the number of relations between related quantities and simplify the relations when necessary NDI-individual use NDDE - B5. Competency to choose appropriate mathematical notations and represent situations graphics if necessary I SDE - Sub-competencies B1, B2 and B5 are found to have developed through the process. Yet another telling result is that upon the completion of the process the student mathematics teachers fall short of the maximum level of achievement concerning these sub-competencies. Furthermore, some student mathematics teachers who are not involved with the learning environment were found to engage in activities regarding such sub-competencies, even though at a limited scale. Still, a comparison of these sub-competencies in the case of student mathematics teachers involved with the learning environment against those who were not involved reveals that the learning environment had affected the development of these sub-competencies, and hence such sub-competencies had been somewhat dependent on the experience. Even though the studies on mathematical modelling had not yet discussed mathematical modelling competencies within the framework of individual sub-competencies, they often note the difficulties students experience during the transition from the real world to the mathematical one (Şen-Zeytun, 2013). In that transition, B1 and B2 sub-competencies are extremely important. In this context, one can possibly argue that this learning environment designed with a holistic approach focusing on sub-competencies is influential in the elimination of the difficulties faced during the transition from the real world to the mathematical one, as discussed in the literature. The B3 and B4 sub-competencies of student mathematics teachers who were not involved with the learning environment, compared against the development registered by the student mathematics teachers who were involved, revealed no significant differences in their sub-competency levels in this context. It is evident that the student mathematics teachers involved with the learning environment did not engage in activities concerning B3 and B4 sub-competencies with a view to making the model developed through group work more practical. However, the student mathematics teachers individually apply B3 and B4 sub-competencies in the model developed in line with the problem they had identified. Taking into account the importance of the practical applicability of the model, one can reach to the conclusion that the learning environments designed thus do not support B3 and B4 sub-competencies, which are crucial in terms of practical application. The findings of this study, on the other hand, reveal that experience is influential in the development of sub-competencies B1, B2 and B5, while sub-competencies B3 and B4 were not associated with experience. The summaries regarding the findings concerning the competency to solve mathematical problems through the mathematical model thus developed (C) and the conclusions reached with respect to associated sub-competencies are presented in Table 16. Table 16 Development of sub-competencies associated with competency C, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature C1. Competency to use heuristic strategies I SDE - C2. Competency to use mathematical knowledge to solve the problem I SDE - Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature C1. Competency to use heuristic strategies I SDE - C2. Competency to use mathematical knowledge to solve the problem I SDE - Table 16 Development of sub-competencies associated with competency C, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature C1. Competency to use heuristic strategies I SDE - C2. Competency to use mathematical knowledge to solve the problem I SDE - Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature C1. Competency to use heuristic strategies I SDE - C2. Competency to use mathematical knowledge to solve the problem I SDE - The C1 and C2 sub-competencies of student mathematics teachers who were not involved with the learning environment, compared against the development registered by the student mathematics teachers who were involved, revealed that the latter group achieved higher competency levels in practices concerning such sub-competencies. This leads to the argument that C1 and C2 sub-competencies have been developed based on experience concerning mathematical modelling, while the learning environment design influences the development of C1 and C2 sub-competencies albeit not at the desired level. The studies in the literature emphasize, in parallel to this finding, that mathematical knowledge influences mathematical modelling competencies, even if the sub-competencies are not discussed specifically (Eraslan, 2012; Grünewald, 2012; Maaß, 2006; Şen-Zeytun, 2013). The summaries regarding the findings concerning the competency to interpret mathematical results against real-life cases (D) and the conclusions reached with respect to associated sub-competencies are presented in Table 17. Table 17 Development of sub-competencies associated with competency D, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature D1. Competency to interpret mathematical results in extra-mathematical contexts I SDE Tekin-Dede & Yilmaz (2013); Blum (2011) D2. Competency to generalize solutions that were developed for a special situation I SDE - Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature D1. Competency to interpret mathematical results in extra-mathematical contexts I SDE Tekin-Dede & Yilmaz (2013); Blum (2011) D2. Competency to generalize solutions that were developed for a special situation I SDE - Table 17 Development of sub-competencies associated with competency D, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature D1. Competency to interpret mathematical results in extra-mathematical contexts I SDE Tekin-Dede & Yilmaz (2013); Blum (2011) D2. Competency to generalize solutions that were developed for a special situation I SDE - Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature D1. Competency to interpret mathematical results in extra-mathematical contexts I SDE Tekin-Dede & Yilmaz (2013); Blum (2011) D2. Competency to generalize solutions that were developed for a special situation I SDE - The D1 sub-competency of student mathematics teachers who were not involved with the learning environment, compared against the development registered by the student mathematics teachers who were involved, revealed that the latter group achieved higher competency levels in practices concerning that sub-competency. Therefore, the development of the D1 sub-competency based on experience concerning mathematical modelling, albeit not at the desired level, suggests that the learning environment design influences the development of D1 sub-competency. In parallel to this finding, there are also studies leading to the conclusion that the experience regarding mathematical modelling is influential in terms of understanding the real-life impact of the results (Blum, 2011). Yet, there are findings to the contrary as well. There are indeed studies which note the difficulties students may have in the interpretation of mathematical results against the real world, despite experience regarding mathematical modelling (Bukova-Güzel, 2011; Blum, 2011; Maaß, 2006; Şen-Zeytun, 2013). Zbiek & Conner (2006) regard low levels of interpretation competency as an act of subconscious and may be caused by lack of clear statements at all times. Tekin-Dede & Yilmaz (2013), on the other hand, argue that making student mathematics teacher’s work on MEAs by following mathematical modelling cycle through would lead to the validation competency. This study, however, reveals that experience gained in the order of hierarchy from structured to non-structured through guidelines taking sub-competencies concerning mathematical modelling process into account, without actually providing detailed theoretical insights into mathematical modelling, would contribute to the improvements in D1 sub-competency, even if not at the desired scale. Table 18 Development of sub-competencies associated with competency E, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature E1. Competency to reflect on found solutions I SDE - E2. Competency to critically check on found solutions I SDE Tekin-Dede & Yilmaz (2013) E3. Competency to review some parts of the model or again go through the modelling process if the solutions do not fit the situation NDI resistant to development NDDE - E4. Competency to reflect on other ways of solving the problem or if solutions can be developed differently NDI resistant to development NDDE Şen-Zeytun (2013) Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature E1. Competency to reflect on found solutions I SDE - E2. Competency to critically check on found solutions I SDE Tekin-Dede & Yilmaz (2013) E3. Competency to review some parts of the model or again go through the modelling process if the solutions do not fit the situation NDI resistant to development NDDE - E4. Competency to reflect on other ways of solving the problem or if solutions can be developed differently NDI resistant to development NDDE Şen-Zeytun (2013) Table 18 Development of sub-competencies associated with competency E, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature E1. Competency to reflect on found solutions I SDE - E2. Competency to critically check on found solutions I SDE Tekin-Dede & Yilmaz (2013) E3. Competency to review some parts of the model or again go through the modelling process if the solutions do not fit the situation NDI resistant to development NDDE - E4. Competency to reflect on other ways of solving the problem or if solutions can be developed differently NDI resistant to development NDDE Şen-Zeytun (2013) Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature E1. Competency to reflect on found solutions I SDE - E2. Competency to critically check on found solutions I SDE Tekin-Dede & Yilmaz (2013) E3. Competency to review some parts of the model or again go through the modelling process if the solutions do not fit the situation NDI resistant to development NDDE - E4. Competency to reflect on other ways of solving the problem or if solutions can be developed differently NDI resistant to development NDDE Şen-Zeytun (2013) The summaries regarding the findings concerning the competency to verify the solution (E) and the conclusions reached with respect to associated sub-competencies are presented in Table 18. A comparison of E1 sub-competencies of student mathematics teachers who are involved with the learning environment against those of the student mathematics teachers not involved suggests that the first group generally have more competencies in their endeavours in this front. This suggests that the learning environment thus designed is influential in the development of the E1 sub-competency. In the same vein, the development of the E2 sub-competency based on experience concerning mathematical modelling, albeit not at the desired level, suggests that the learning environment design influences the development of E2 sub-competency. Even though sub-competencies of the validate the solutioncompetency are not discussed separately, the literature is rich in studies suggesting that the students do not see a reason to verify their solutions confident in their accuracy and that they do not check for calculation errors and the accuracy of solutions regarding the state of the problem (e.g., Blum & Borromeo-Ferri, 2009; Maaß, 2006; Şen-Zeytun, 2013). Şen-Zeytun (2013) and Blum & Borromeo-Ferri (2009) note that this is caused by the students’ conviction that the validation of the solution is actually the job of the instructor. Even though emphasis on the review of the modelling process and the development of different solutions has not been lacking where the solutions are not consistent with the case in hand in the learning environment, the student mathematics teachers’ E3 and E4 sub-competencies were found to be extremely resistant to development. The comparison of the E3 and E4 sub-competencies of student mathematics teachers who were involved with the learning environment against those of the control group revealed that these sub-competencies were either non-existent or were weak and far in between in both groups. Therefore, one can argue that E3 and E4 sub-competencies are not directly associated with the experience for mathematical modelling and that the learning environment design does not contribute directly to the development of these sub-competencies. The literature, on the other hand, contains some studies suggesting that student mathematics teachers with mathematical modelling experience question the model and go through it once again where necessary (Tekin-Dede & Yilmaz, 2013). As the modelling cycle contains the step of ‘validating’ as well, going through the modelling cycle deemed a guideline may have led student mathematics teachers to carrying out studies concerning E3 sub-competency. Therefore, the E3 sub-competency applied by going through the guiding cycle does not necessitate the comparable application when working with free MEAs. Even though certain activities entail guidelines directing student mathematics teachers to envisaging different solutions, the student mathematics teachers chose to skip the guideline altogether, rather than responding to it. The student mathematics teachers consider a single route to solution sufficient and do not bother to seek different routes. This fact, on the other hand, indicates that the importance attached to mathematical modelling process influences the E4 competency. The literature is not devoid of studies which have reached to similar conclusions that different routes to solutions were not sought after when working on MEAs (Şen-Zeytun, 2013). 4. Conclusion and implications While some of the literature emphasizes supra-cognitive and sensory mathematical modelling competencies, we focused solely on cognitive mathematical modelling competencies as evidenced in the actions, speech and work of our participants. Our findings depend on comparing groups whose educational background was similar, rather than assessing competences prior to the LES-MMC. We have used qualitative methods that have enabled us to identify details and differences in performance between those who have and have not undergone the LES-MMC. The LES-MMC process appears to have influenced positively the development of sub-competency A1, that is the making of assumptions about the problem situation. By and large the student mathematics teachers had made progress with respect to most sub-competencies; they were more able to simplify the problem, identify variables and relate them, decide on appropriate information for the problem, mathematize quantities and relations between them, use appropriate representations, apply heuristics and mathematical knowledge and interpret and check findings. However, context appeared to sometimes lead to the identification of inappropriate variables that did not affect the situation mathematically, arising from mental imagery about similar situations, thus blocking the development of accurate representations. Furthermore, those students who were not involved with the learning environment were found to have done better regarding the associated sub-competency A3, identification of influential quantities, possibly because they had not been immersed in situations they could imagine from general knowledge. Sub-competencies associated with reduction and simplification of variables and relations, B3 and B4, and those associated with reviewing and developing the model, E3 and E4, did not develop in the study, possibly because the student mathematics teachers did not perceive a need to do this, since their aim was to produce a model, not to use it in reality. Our assessment of mathematical modelling through this holistic approach revealed that the development of the competencies is not linear. Therefore, a holistic approach is more advisable than written tests to assess such competencies. Assessment based on written documents alone would not suffice to show a true picture of mathematical modelling competency. It would be advisable for future studies to develop assessment approaches that make it easier to assess mathematical modelling competencies throughout the process. As the present study discussed sub-competencies on an individual basis, it has been possible to identify some resistant ones. It would be advisable for future studies to design learning environments at a similar micro level, providing more experience regarding those sub-competencies which had been influenced negatively in this study and those which had developed at a level less than expected or which appeared resistant to development. Acknowledgements This study was produced from PhD thesis titled ‘Examining mathematical modelling competencies of teacher candidates in learning environments designed to improve mathematical modelling competencies’. Funda Aydin-Güç is an Assistant Professor at the Mathematics and Science Education Department in Giresun University, Giresun, Turkey. She received a PhD in Mathematics Education from Institute of Education of Karadeniz Technical University in 2015. Her research interests are teaching and learning mathematics. Adnan Baki is a Professor at the Mathematics and Science Education Department in Karadeniz Technical University, Trabzon, Turkey. He was awarded the degree of MD in Mathematics Education in 1989 from the University of New Brunswick in Canada. He received a PhD from the Institute of Education of the University of London in 1994. He publishes papers on teaching and learning mathematics. References Aydin-Güç , F. ( 2015 ) Examining mathematical modelling competencies of teacher candidates in learning environments designed to improve mathematical modelling competencies . Ph.D. Thesis , Karadeniz Technical University , Trabzon . Blomhøj , M. ( 2007 ) Developing mathematical modelling competency through problem based project work - experiences from Roskilde University . Philosophy and Science Teaching Conference . [Avaible online at: http://www.ucalgary.ca/ihpst07/proceedings/ihpst07%20papers/125%20blomhoj.pdf], retrieved on October, 05, 2011, pp. 1–6. Blomhøj , M. & Kjeldsen , T. H. 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Evaluation of the learning environment designed to develop student mathematics teachers’ mathematical modelling competencies

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© The Author(s) 2018. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please email: journals.permissions@oup.com
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Abstract

Abstract The purpose of this study was to examine the mathematical modelling competencies of student mathematics teachers within a learning environment that was based on a holistic approach to instruction. The participants were student mathematics teachers enrolled in undergraduate programmes at two different universities. In one of the university programmes, the participants took a mathematical modelling course that was taught according to a holistic approach, while the participants at the other university followed the standardized undergraduate programme that did not include a mathematical modelling course. The sub-competencies of mathematical modelling of the student mathematics teachers who participated in the mathematical modelling course were examined using an analytic rubric that was designed according to the holistic approach applied within the context of the study. In order to determine whether the observed changes were related to the experiences of mathematical modelling, these results were compared to the sub-competencies of mathematical modelling of the control group. The findings showed that experience within a learning environment based on mathematical modelling, as well as affective factors, supported the development of modelling competencies. On the other hand, it was observed that certain sub-competencies were difficult to develop; while many sub-competencies were enhanced by the modelling experiences, others were adversely affected. Furthermore, a certain group of sub-competencies was found not to be directly linked to modelling experiences. 1. Introduction Mathematical modelling plays a major role in solutions to daily life with mathematical view. In general, ‘mathematical modelling’ refers to a cyclic process of translating a real-life problem to a mathematical one with a view to developing mathematical solutions and implementing secondary processes to interpret these mathematical solutions in applied environments (Borromeo-Ferri, 2006; Lesh & Doerr, 2003). This process is crucial for solving real-life problems and requires individuals to understand the relationship between the mathematics learned at school and the real-life experiences. Mathematical modelling began to gain prominence in early 1980s, followed by the inclusion of mathematical modelling in school mathematics curriculum (Blomhøj & Kjeldsen, 2006) and emphasis on the issue among the skills to be developed within the framework of school mathematics teaching in various countries (National Council of Teachers of Mathematics [NCTM], 2000). There are also studies which emphasize the applicability of mathematical modelling from preschool years till the very last year of high school (Borromeo-Ferri, 2006) and the necessity to include mathematical modelling in school mathematics (NCTM, 2000). In Turkey, mathematics curriculum in elementary and secondary school levels also focuses on the raising of individual’s abilities of using mathematical concepts, developing models and associating such models with verbal and mathematical statements, as important objectives (Ministry of National Education [MNE], 2013). 1.1 Theoretical framework Examination of the studies concerning mathematical modelling reveals that the concept is interpreted in two distinct ways in educational programmes and applications. One of these interpretations considers mathematical modelling as a means to reveal and develop the relationships among certain mathematical contexts and to motivate students to work on real-life problems (Chinnappan, 2010). In the second interpretation, mathematical modelling is considered as a goal in and of itself, in terms of the achievement of the objectives of education rather than as a means to improve certain types of mathematical learning through educational processes. Both interpretations, however, consider mathematical modelling as a wider process covering certain elements such as formulation, mathematization, solution, interpretation and assessment (Stillman, 2012). In this context, as both perspectives emphasize the importance of the process of mathematical modelling, one should first try to define the sub-processes the students would be required to experience in the process of mathematical modelling, the competencies for the completion of such sub-processes and the sub-competencies they involve. In this study mathematical modelling competencies and sub-competencies approaches which are defined by Blum & Kaiser (1997, quoted in the study by Maaß, 2006) are adopted. 1.2 Studies on the development of mathematical modelling competencies Related literature indicates that it is possible to come up with rough categorizations of the studies with reference to certain characteristics, even though a widely accepted shared perspective regarding the development and evaluation of mathematical modelling competencies is lacking. A review of the studies considering mathematical modelling as an educational objective and as a learning environment to develop mathematical modelling competencies contains within itself two distinct approaches. These approaches are ‘micro-level approach’ and ‘holistic approach’ (Grünewald, 2012). Furthermore, there are also some studies which adopt a ‘mixed’ perspective, underlining the need to establish a balance of the micro-level and holistic approaches both (Blomhøj, 2007). The aim of learning environments which embrace a holistic approach is to ensure that the individuals go through the whole mathematical modelling process defined through a model-development activity. On the other hand, the learning environments based on the micro-level approach intend to enable individuals experience a specific sub-process of the mathematical modelling process. Therefore, in the holistic approach all competencies and sub-competencies expected through the mathematical modelling process are implemented through the activity (Maaß, 2006), while micro-level approach implements various sub-competencies within the framework of distinct activities (Crouch & Haines, 2004). It is possible to categorize the studies aiming to develop and evaluate mathematical modelling competencies with a perspective of mathematical modelling competency as the objective of mathematics education, with reference to their approaches to integration to the learning environment. The categorization is summarized in Table 1. Table 1 Learning environment approaches of mathematical modelling Approaches Learning environments approaches Examples of literature 1. Micro-level approach Sub-competency-oriented Crouch & Haines (2004) 2. Holistic approach Theoretical knowledge-based Kaiser (2007) Open Model Eliciting Activity [MEA]-oriented Focusing on the procedure to follow the steps of mathematical modelling Blum & Borromeo-Ferri (2009) 3. Mixed approach Contains both micro-level and holistic approaches Blomhøj (2007) (Balance of micro-level approach and holistic approach) Approaches Learning environments approaches Examples of literature 1. Micro-level approach Sub-competency-oriented Crouch & Haines (2004) 2. Holistic approach Theoretical knowledge-based Kaiser (2007) Open Model Eliciting Activity [MEA]-oriented Focusing on the procedure to follow the steps of mathematical modelling Blum & Borromeo-Ferri (2009) 3. Mixed approach Contains both micro-level and holistic approaches Blomhøj (2007) (Balance of micro-level approach and holistic approach) View Large Table 1 Learning environment approaches of mathematical modelling Approaches Learning environments approaches Examples of literature 1. Micro-level approach Sub-competency-oriented Crouch & Haines (2004) 2. Holistic approach Theoretical knowledge-based Kaiser (2007) Open Model Eliciting Activity [MEA]-oriented Focusing on the procedure to follow the steps of mathematical modelling Blum & Borromeo-Ferri (2009) 3. Mixed approach Contains both micro-level and holistic approaches Blomhøj (2007) (Balance of micro-level approach and holistic approach) Approaches Learning environments approaches Examples of literature 1. Micro-level approach Sub-competency-oriented Crouch & Haines (2004) 2. Holistic approach Theoretical knowledge-based Kaiser (2007) Open Model Eliciting Activity [MEA]-oriented Focusing on the procedure to follow the steps of mathematical modelling Blum & Borromeo-Ferri (2009) 3. Mixed approach Contains both micro-level and holistic approaches Blomhøj (2007) (Balance of micro-level approach and holistic approach) View Large As these studies focus on competencies, the sub-competencies leading to the failure in terms of a given competency may not always be clear. Taking into account the fact that the acquisition of certain competencies may be slower compared to others (Blum & Niss, 1991), the need to design learning environments to review individual sub-competencies leading to a failure in the development of a given competency deemed necessary for a successful mathematical modelling becomes evident. 1.3 Studies on the assessment of mathematical modelling competencies The consensus on the importance of mathematical modelling in education led to research studies aimed to assess mathematical modelling competencies. As a result of this trend, many studies recently provided means to assess mathematical modelling competencies (Borromeo-Ferri, 2006; Zöttl et al., 2011). The studies so far discuss mathematical modelling competence as a variable lending to assessment. Even a comprehensive agreement about the need of assessment in the literature, the actual means to the assessment of mathematical modelling competencies are still an issue of debate. As yet, the literature offers no categorization of approaches to assessment. However, a review of the studies on the assessment of mathematical modelling competencies clarifies that the assessment approaches can also be categorized with reference to ‘micro-level’ and ‘holistic’ approaches, in parallel to the categorization of learning environments. The approaches to the assessment of mathematical modelling competencies can be categorized as shown in Table 2. Table 2 Assessment approaches of mathematical modelling competencies Approaches Assessment approaches Examples of literature 1. Micro-level approach One or more sub-competence to deal separately Haines et al. (2001, quoted in Kaiser, 2007) 2. Holistic approach A. Competency evaluation a. Identification of existing competencies (identification of the steps implemented) Schwarz & Kaiser (2007) b. Identification of the actual level of implementation of individual competencies (steps) Borromeo-Ferri (2006) c. Establishment of the level of progress regarding the efforts so far (with reference to individual steps) Ludwig & Xu (2010) (Some studies have dealt with the process steps) d. Identification of individual sub-competencies implemented within the framework of competencies Tekin-Dede & Yilmaz (2013) B. Level determination Henning & Keune (2007) C. Multidimensional evaluation Niss & Jensen (2006, quoted in Jensen, 2007) 3. Integrated approach Balance of micro-level approach and holistic approach Zöttl et al. (2011) Approaches Assessment approaches Examples of literature 1. Micro-level approach One or more sub-competence to deal separately Haines et al. (2001, quoted in Kaiser, 2007) 2. Holistic approach A. Competency evaluation a. Identification of existing competencies (identification of the steps implemented) Schwarz & Kaiser (2007) b. Identification of the actual level of implementation of individual competencies (steps) Borromeo-Ferri (2006) c. Establishment of the level of progress regarding the efforts so far (with reference to individual steps) Ludwig & Xu (2010) (Some studies have dealt with the process steps) d. Identification of individual sub-competencies implemented within the framework of competencies Tekin-Dede & Yilmaz (2013) B. Level determination Henning & Keune (2007) C. Multidimensional evaluation Niss & Jensen (2006, quoted in Jensen, 2007) 3. Integrated approach Balance of micro-level approach and holistic approach Zöttl et al. (2011) View Large Table 2 Assessment approaches of mathematical modelling competencies Approaches Assessment approaches Examples of literature 1. Micro-level approach One or more sub-competence to deal separately Haines et al. (2001, quoted in Kaiser, 2007) 2. Holistic approach A. Competency evaluation a. Identification of existing competencies (identification of the steps implemented) Schwarz & Kaiser (2007) b. Identification of the actual level of implementation of individual competencies (steps) Borromeo-Ferri (2006) c. Establishment of the level of progress regarding the efforts so far (with reference to individual steps) Ludwig & Xu (2010) (Some studies have dealt with the process steps) d. Identification of individual sub-competencies implemented within the framework of competencies Tekin-Dede & Yilmaz (2013) B. Level determination Henning & Keune (2007) C. Multidimensional evaluation Niss & Jensen (2006, quoted in Jensen, 2007) 3. Integrated approach Balance of micro-level approach and holistic approach Zöttl et al. (2011) Approaches Assessment approaches Examples of literature 1. Micro-level approach One or more sub-competence to deal separately Haines et al. (2001, quoted in Kaiser, 2007) 2. Holistic approach A. Competency evaluation a. Identification of existing competencies (identification of the steps implemented) Schwarz & Kaiser (2007) b. Identification of the actual level of implementation of individual competencies (steps) Borromeo-Ferri (2006) c. Establishment of the level of progress regarding the efforts so far (with reference to individual steps) Ludwig & Xu (2010) (Some studies have dealt with the process steps) d. Identification of individual sub-competencies implemented within the framework of competencies Tekin-Dede & Yilmaz (2013) B. Level determination Henning & Keune (2007) C. Multidimensional evaluation Niss & Jensen (2006, quoted in Jensen, 2007) 3. Integrated approach Balance of micro-level approach and holistic approach Zöttl et al. (2011) View Large Against this background, the need to assess the students’ mathematical modelling competencies with special emphasis on sub-competencies is obvious. Such assessments would provide insights into specific sub-competencies causing a failure to achieve, at least on a sufficient level, individual competencies in learning environments designed to foster the development of mathematical modelling competencies. The awareness of specific sub-competencies blocking the development of mathematical modelling competencies, on the other hand, is crucial for it sheds light on the sub-competencies requiring micro-level attention in learning environments designed with respect to mathematical modelling competencies. Although there are studies (Galbraith & Stillman, 2006; Galbraith et al., 2006, 2007) that reveal the blockages—although not focusing in sub-competencies— that are experienced during the transition from one stage of the modelling cycle to the next passage, it is not known what kind of picture emerges when working to remove the blockages. Such an approach would provide insights to mathematics educators intending to teach mathematical modelling competencies to students, about which specific sub-competencies incline to resistance to development, and hence give them clues to revise the learning environment design. 1.4 Objective of the study The related literature highlights that students at various level of education have been suffering difficulties in the process of mathematical modelling and were failing to achieve expected level of mathematical modelling competencies (Bukova-Güzel, 2011; Maaß, 2006). However, the direction of the development of sub-competencies and the specific sub-competencies leading to a failure of the development of development-resistant competencies are still matters of debate. Against this background, the need to do away with the shortcomings identified in the literature and to design and assess learning environments to enable the development of mathematical modelling sub-competencies is obvious. From this aspect this research is a next step to deepening our understanding of the students’ development of mathematical modelling sub-competencies and on how to support and assess such development. On this background, the objective of this study is to establish how exactly the sub-competency-assisted learning environment process designed to develop mathematical modelling competencies through a holistic approach would contribute to the development of mathematical modelling competencies of student mathematics teachers. The review of mathematical modelling competencies in this study was based on the theoretical framework identified by Blum & Kaiser (1997, quoted in Maaß, 2006). Thus, the main research problem in this study can be stated as follows: How do mathematical modelling competencies of student mathematics teachers develop through the course providing mathematical modelling processes based on a holistic approach? The sub-problems discussed in this context are as follows: 1. How does the competency to understand the real problem and to develop a model based on the reality develop with student mathematics teachers? 2. How does the competency to create a mathematical model out of the model based on the reality develop with student mathematics teachers? 3. How does the mathematical model thus developed help improve student mathematics teachers’ competency to solve mathematical problems? 4. How does the student mathematics teachers’ competency to interpret mathematical results against real cases develop? 5. How does the student mathematics teachers’ competency to validate the solution develop? 2. Methodology Within the framework of the present study, a learning environment was designed by taking into account the shortcomings discussed in the literature and implemented as an undergraduate course in the Department of Mathematics Education in the Faculty of Education of Giresun University. In this context, it is possible to call the study an example of action research carried out as researcher and as teacher. Furthermore, it would not be far-fetched to call the study a qualitative one, given the qualitative approach to the processes experienced in the learning environment designed within the framework of the study. 2.1 Participants The study was carried out by two distinct groups of participants from two different universities, with reference to the research questions at hand. The first group consists of 40 sophomore-year student mathematics teachers who took the mathematical modelling course in the academic year 2013–2014 at the Department of Mathematics Education in Giresun University. The second group consists of three sophomore-year student mathematics teachers at the Department of Mathematics Education in the Education Faculty of Karadeniz Technical University. The second group did not have any experience of mathematical modelling and would be unable to contact the first group. These students comprise the second tier of the study group, providing a means of control. In the first group, 40 student mathematics teachers were asked to form groups of four to six as they wished. In response seven groups were formed. As the study was intended to be based on a detailed process analysis, two of such groups were selected on a voluntary basis. The initial participants of the study were nine student mathematics teachers who formed those groups. The ensuing sections of the study shall refer to these student mathematics teachers as K1, K2, K3,…, K9. No assessment to investigate existing levels of mathematical modelling competencies of the student mathematics teachers to take part in the learning environment was carried out in the beginning. Therefore, the prior mathematical modelling competency levels of the student mathematics teachers involved in the learning environment are not known. As a different MEA design was required for each student teacher who was not included in the learning environment, not to mention the need to carry out individual clinical interviews with each and every one of them, the participant count of the control group was limited to three. Throughout their secondary and higher education tenures, the participants had taken courses concerning mathematical concepts required to complete the MEAs discussed in this study. Not all participants of the study had formal experience regarding mathematical modelling. 2.2 Learning environment design Once the existing problem was identified within an action research framework, the relevant literature was reviewed to see potential solutions for the problem and a decision on how an environment reached the requirements. The literature survey revealed that experiences to help students go through all steps of the mathematical modelling process were required to help enhance their mathematical modelling competencies and sub-mathematical modelling competencies. On the bases of this background, the holistic approach was chosen to provide such an environment to support the development of mathematical modelling competencies and thus used as the basis of the learning environment design. The design was implemented in observation of the following principles: • The directives regarding sub-competencies should enable the development in individuals of the cognitive structure for the mathematical modelling process. • In the case of individuals who lack formal experience required for the development of models, the activities should be ranked from simple to complex and from structured to unstructured. • The activities should be designed to enable gradual transitions between the world of mathematics and the real world, after a start in the former. • Activities requiring different processes applied on different mathematical contents against different backgrounds should be employed. • A holistic structure should be provided through activities with associations in contextual and mathematical frameworks the students are experienced with. • With a view to increasing the level of interest and motivation on part of the students, materials about daily life such as videos, news, posters and so on should be used to enrich the learning environment with respect to the contexts of activity. • Opportunity to enable group and individual work inside and outside the classroom, presentation of such work and a discussion of modelling processes and models per se should be provided. The guidelines regarding the activities in the learning environment designed in homage to these principles were drawn up to offer students experiences regarding such sub-competencies and to regulate their studies. Examples of the guidelines intending to provide an experience concerning sub-competencies are provided in Table 3. Table 3 Examples of guidelines that are related to mathematical modelling competencies and sub-competencies A. Competencies to understand the real problem and to set up a model based on reality A1. Competency to make assumptions for the problem and simplify the situation What do you know? What do you want to learn? What could be the assumptions? A2. Competency to recognize quantities that influence the situation, to name them and to identify key variables Which variables are most important? Which variables you can ignore? A3. Competency to construct relations between the variables Can you create an initial model to elaborate later? A4. Competency to look at available information and to differentiate between relevant and irrelevant information — A. Competencies to understand the real problem and to set up a model based on reality A1. Competency to make assumptions for the problem and simplify the situation What do you know? What do you want to learn? What could be the assumptions? A2. Competency to recognize quantities that influence the situation, to name them and to identify key variables Which variables are most important? Which variables you can ignore? A3. Competency to construct relations between the variables Can you create an initial model to elaborate later? A4. Competency to look at available information and to differentiate between relevant and irrelevant information — Table 3 Examples of guidelines that are related to mathematical modelling competencies and sub-competencies A. Competencies to understand the real problem and to set up a model based on reality A1. Competency to make assumptions for the problem and simplify the situation What do you know? What do you want to learn? What could be the assumptions? A2. Competency to recognize quantities that influence the situation, to name them and to identify key variables Which variables are most important? Which variables you can ignore? A3. Competency to construct relations between the variables Can you create an initial model to elaborate later? A4. Competency to look at available information and to differentiate between relevant and irrelevant information — A. Competencies to understand the real problem and to set up a model based on reality A1. Competency to make assumptions for the problem and simplify the situation What do you know? What do you want to learn? What could be the assumptions? A2. Competency to recognize quantities that influence the situation, to name them and to identify key variables Which variables are most important? Which variables you can ignore? A3. Competency to construct relations between the variables Can you create an initial model to elaborate later? A4. Competency to look at available information and to differentiate between relevant and irrelevant information — However, no guidelines were developed with respect to certain sub-competencies. The lack on this front is perhaps associated with the perceived need to develop these sub-competencies with other guidelines and interactions through the process or the need to do away with instructions regarding such competencies. In the learning environment thus designed, the students worked on MEAs, as groups they themselves formed. In conclusion of the group work stage, each group presented the model it had developed, paving the way for discussion in the class. Furthermore, the students were given group- and individual-based project assignments. Such project assignments were again presented in the class, followed by discussions on the models developed. 2.3 Activities and design process In line with the objective of developing a ‘Learning Environment to Support Mathematical Modelling Sub-Competencies (LES-MMC)’, MEAs with embedded guidelines encouraging the implementation of sub-competencies expected to arise during the mathematical modelling were developed. In the development of MEAs, the principles of ‘model construction, reality, self-assessment, model documentation, effective prototype, model share-ability and reusability’ identified by Lesh et al. (2000) during model-eliciting activities as principles representing real-life problems were adopted. When designing the MEAs, special attention was paid to make them contain contexts student mathematics teachers are familiar but not extremely involved with. The objectives of MEAs are presented in Table 4. Table 4 Some MEAs implemented in the designed learning environment and objectives thereof Number* MEA Objective 1 Big Food (BF) The goal is to determine student teachers’ sub-competencies without any guidelines before any experience on mathematical modelling 5 Traffic Accident Statistics (TAS) A similar situation to the previous activity, however, ensuring that students decide which data and which model will be more meaningful. Different linear models may emerge. Now the goal is to reduce the guidelines and allow the use of the sub-competencies that were targeted in the previous lessons. 8 Individual Project Development (IP) Making student mathematics teachers realize real-life problems which can be solved through mathematical knowledge. Enabling solutions of the real-life problems they had identified, through the implementation of mathematical modelling sub-competencies. Number* MEA Objective 1 Big Food (BF) The goal is to determine student teachers’ sub-competencies without any guidelines before any experience on mathematical modelling 5 Traffic Accident Statistics (TAS) A similar situation to the previous activity, however, ensuring that students decide which data and which model will be more meaningful. Different linear models may emerge. Now the goal is to reduce the guidelines and allow the use of the sub-competencies that were targeted in the previous lessons. 8 Individual Project Development (IP) Making student mathematics teachers realize real-life problems which can be solved through mathematical knowledge. Enabling solutions of the real-life problems they had identified, through the implementation of mathematical modelling sub-competencies. *Numbers indicate the order of implementation. Table 4 Some MEAs implemented in the designed learning environment and objectives thereof Number* MEA Objective 1 Big Food (BF) The goal is to determine student teachers’ sub-competencies without any guidelines before any experience on mathematical modelling 5 Traffic Accident Statistics (TAS) A similar situation to the previous activity, however, ensuring that students decide which data and which model will be more meaningful. Different linear models may emerge. Now the goal is to reduce the guidelines and allow the use of the sub-competencies that were targeted in the previous lessons. 8 Individual Project Development (IP) Making student mathematics teachers realize real-life problems which can be solved through mathematical knowledge. Enabling solutions of the real-life problems they had identified, through the implementation of mathematical modelling sub-competencies. Number* MEA Objective 1 Big Food (BF) The goal is to determine student teachers’ sub-competencies without any guidelines before any experience on mathematical modelling 5 Traffic Accident Statistics (TAS) A similar situation to the previous activity, however, ensuring that students decide which data and which model will be more meaningful. Different linear models may emerge. Now the goal is to reduce the guidelines and allow the use of the sub-competencies that were targeted in the previous lessons. 8 Individual Project Development (IP) Making student mathematics teachers realize real-life problems which can be solved through mathematical knowledge. Enabling solutions of the real-life problems they had identified, through the implementation of mathematical modelling sub-competencies. *Numbers indicate the order of implementation. With a view to comply with the learning environment principles established through the literature survey, MEAs were ranked in the order of complexity, as well as from the guidelines concerning sub-competencies to guidelines concerning primary competencies, as well as activities without guidelines. In first MEA, there are many guidelines and it includes all mathematical modelling process. The other MEAs are sorted correctly from focusing some sub-competencies at the same time including all competencies to focusing all competencies, from use of known mathematical constructions to require the construction of new structures. In the Table 4 there are examples of objectives of some MEAs. Full information of all MEAs and their objectives can be found in the PhD thesis written by Aydin-Güç (2015). In order to prevent any influence student mathematics teachers who are not included in the designed learning environment may have on each other while working on MEAs, the decision to issue a different MEA to each one was taken and hence three distinct MEAs were designed. These MEAs are compliant with the criteria established through the research and the six principles representing real-life problems as defined by Lesh et al. (2000). In the first hour of the lesson, the student mathematics teachers were provided an introduction on the execution of the course and groups were formed. The group lessons carried out until individual works were as follows: groups worked on the related MEA, shared their works with all students, discussed insights and identified applicable and optimal mathematical models before starting the next MEA. During the last two weeks IP presentations and discussions were conducted. 2.4 Data gathering and analysis process Nine student mathematics teachers in the first group in the study have completed seven MEAs with their group. The process was concluded with the completion of an individual modelling cycle to develop a solution for real-life problems the student mathematics teachers chose individually. Both group work and the classroom debates were recorded on video. The competencies of the student mathematics teachers who were not included in the learning environment were discussed with reference to individual clinical interviews on the MEAs assigned. Each student teacher was interviewed for a total of three weeks, with one interview per week, discussing the solutions regarding the activities. Understanding the depth and scale of the students’ preparedness in terms of coping with the issues observed with each sub-competency is crucial in terms of the assessment of modelling competencies. Therefore, in this study, an analytical rubric to allow evaluation with reference to performance criteria was developed to ensure the assessment of mathematical modelling sub-competencies. The study opted to go with the monitoring of sub-competencies pertaining to competencies described in detail by Blum & Kaiser (1997) and quoted by Maaß (2006). Thereafter, performance indicators were defined. A pilot study with sophomore-year student mathematics teachers enrolled in Primary School Mathematics Teacher programmes comparable to the student mathematics teachers to take part in the actual implementation was carried out with a view to fine-tuning sub-competencies. The sub-competencies discussed in the analytical rubric are listed in Table 5. Table 5 Revised mathematical modelling competencies in consequence of pilot study Mathematical modelling competencies and sub-competencies A. Competencies to understand the real problem and to set up a model based on reality   A1. Competency to make assumptions for the problem   A2. Competency to simplify the situation   A3. Competency to recognize quantities that influence the situation, to name them   A4. Competency to identify key variables that influence the situation   A5. Competency to construct relations between the variables   A6. Competency to look for available/not available information to solve the problem   A7. Competency to differentiate the related/unrelated information to solve the problem B. Competencies to set up a mathematical model from real model   B1. Competency to mathematize relevant qualities/quantities   B2. Competency to mathematize the relations between the related qualities/quantities   B3. Competency to reduce the number of relevant quantities and simplify their quantities when necessary   B4. Competency to reduce the number of relations between related quantities and simplify the relations when necessary   B5. Competency to choose appropriate mathematical notations and represent situations graphics if necessary C. Competencies to solve mathematical questions within this mathematical model   C1. Competency to use heuristic strategies (division of the problem into part problems, establishing relations to similar or analogue problems, viewing the problem in a different form, varying the quantities or the available data, etc.)   C2. Competency to use mathematical knowledge to solve the problem D. Competencies to interpret mathematical results in a real situation   D1. Competency to interpret mathematical results in extra-mathematical contexts   D2. Competency to generalize solutions that were developed for a special situation E. Competencies to validate the solution   E1. Competency to reflect on found solutions   E2. Competency to critically check on found solutions   E3. Competency to review some parts of the model or again go through the modelling process if the solutions do not fit the situation   E4. Competency to reflect on other ways of solving the problem or if solutions can be developed differently Mathematical modelling competencies and sub-competencies A. Competencies to understand the real problem and to set up a model based on reality   A1. Competency to make assumptions for the problem   A2. Competency to simplify the situation   A3. Competency to recognize quantities that influence the situation, to name them   A4. Competency to identify key variables that influence the situation   A5. Competency to construct relations between the variables   A6. Competency to look for available/not available information to solve the problem   A7. Competency to differentiate the related/unrelated information to solve the problem B. Competencies to set up a mathematical model from real model   B1. Competency to mathematize relevant qualities/quantities   B2. Competency to mathematize the relations between the related qualities/quantities   B3. Competency to reduce the number of relevant quantities and simplify their quantities when necessary   B4. Competency to reduce the number of relations between related quantities and simplify the relations when necessary   B5. Competency to choose appropriate mathematical notations and represent situations graphics if necessary C. Competencies to solve mathematical questions within this mathematical model   C1. Competency to use heuristic strategies (division of the problem into part problems, establishing relations to similar or analogue problems, viewing the problem in a different form, varying the quantities or the available data, etc.)   C2. Competency to use mathematical knowledge to solve the problem D. Competencies to interpret mathematical results in a real situation   D1. Competency to interpret mathematical results in extra-mathematical contexts   D2. Competency to generalize solutions that were developed for a special situation E. Competencies to validate the solution   E1. Competency to reflect on found solutions   E2. Competency to critically check on found solutions   E3. Competency to review some parts of the model or again go through the modelling process if the solutions do not fit the situation   E4. Competency to reflect on other ways of solving the problem or if solutions can be developed differently Table 5 Revised mathematical modelling competencies in consequence of pilot study Mathematical modelling competencies and sub-competencies A. Competencies to understand the real problem and to set up a model based on reality   A1. Competency to make assumptions for the problem   A2. Competency to simplify the situation   A3. Competency to recognize quantities that influence the situation, to name them   A4. Competency to identify key variables that influence the situation   A5. Competency to construct relations between the variables   A6. Competency to look for available/not available information to solve the problem   A7. Competency to differentiate the related/unrelated information to solve the problem B. Competencies to set up a mathematical model from real model   B1. Competency to mathematize relevant qualities/quantities   B2. Competency to mathematize the relations between the related qualities/quantities   B3. Competency to reduce the number of relevant quantities and simplify their quantities when necessary   B4. Competency to reduce the number of relations between related quantities and simplify the relations when necessary   B5. Competency to choose appropriate mathematical notations and represent situations graphics if necessary C. Competencies to solve mathematical questions within this mathematical model   C1. Competency to use heuristic strategies (division of the problem into part problems, establishing relations to similar or analogue problems, viewing the problem in a different form, varying the quantities or the available data, etc.)   C2. Competency to use mathematical knowledge to solve the problem D. Competencies to interpret mathematical results in a real situation   D1. Competency to interpret mathematical results in extra-mathematical contexts   D2. Competency to generalize solutions that were developed for a special situation E. Competencies to validate the solution   E1. Competency to reflect on found solutions   E2. Competency to critically check on found solutions   E3. Competency to review some parts of the model or again go through the modelling process if the solutions do not fit the situation   E4. Competency to reflect on other ways of solving the problem or if solutions can be developed differently Mathematical modelling competencies and sub-competencies A. Competencies to understand the real problem and to set up a model based on reality   A1. Competency to make assumptions for the problem   A2. Competency to simplify the situation   A3. Competency to recognize quantities that influence the situation, to name them   A4. Competency to identify key variables that influence the situation   A5. Competency to construct relations between the variables   A6. Competency to look for available/not available information to solve the problem   A7. Competency to differentiate the related/unrelated information to solve the problem B. Competencies to set up a mathematical model from real model   B1. Competency to mathematize relevant qualities/quantities   B2. Competency to mathematize the relations between the related qualities/quantities   B3. Competency to reduce the number of relevant quantities and simplify their quantities when necessary   B4. Competency to reduce the number of relations between related quantities and simplify the relations when necessary   B5. Competency to choose appropriate mathematical notations and represent situations graphics if necessary C. Competencies to solve mathematical questions within this mathematical model   C1. Competency to use heuristic strategies (division of the problem into part problems, establishing relations to similar or analogue problems, viewing the problem in a different form, varying the quantities or the available data, etc.)   C2. Competency to use mathematical knowledge to solve the problem D. Competencies to interpret mathematical results in a real situation   D1. Competency to interpret mathematical results in extra-mathematical contexts   D2. Competency to generalize solutions that were developed for a special situation E. Competencies to validate the solution   E1. Competency to reflect on found solutions   E2. Competency to critically check on found solutions   E3. Competency to review some parts of the model or again go through the modelling process if the solutions do not fit the situation   E4. Competency to reflect on other ways of solving the problem or if solutions can be developed differently The performance indicators of the analytical rubric developed with reference to the sub-competencies listed in Table 5 are presented in Table 11, with reference to a specific MEA to serve as an example. When analyzing the data, it was seen that between each stage the student mathematics teachers move forth and back many times during the task solving. In this case, the sub-competencies of the student teachers were evaluated using analytical rubric. Although this assessment appears to be quantitative, it was originally made to see qualitative change. So the criteria were not taken as scoring; students’ full scores were not calculated. When a student teacher studies at this sub-competency again, the criteria of the study were redetermined. In a MEA, student teachers’ competencies had encoded considering their last study on a sub-competency. With a view to ensuring the reliability of data analysis, a part of the transcript concerning a group study was analyzed by two distinct researchers with experience on mathematical modelling competencies studies, using the analytical rubric developed previously. The consistency rate proposed by Miles & Huberman (1994) was found to be 0.89. Therefore, one can argue that the data analysis is reliable. Table 6 The studies on competency A and expected analyses Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) A. Competencies to understand the real problem and to set up a model based on reality A1 K8: [Reading the guidelines] Asks what are your assumptions? What are our assumptions? K6: What did we do? We assumed a consistent rate of fall each year: Other than that... What else did we do? K8: What can we assume? We assumed the gender to be the same. K6: That would not do. K8: OK whatever. Coming up with fairly robust assumptions to make the problem simpler; to present the case with justification. For example: Let’s assume that the deaths in traffic accidents occur only due to the physical impact. There may be other cases of death to take place after the accident, due to the fear factor or another condition triggered. A2 K4: Three thousand. 3835. For instance, assuming a given number of deaths in a specific number of accidents… K3: We could calculate the probability of increase in accident count. That would give us the number of dead. Something akin to an equation with two unknowns. K1: I believe it is not about the accident, but rather about the number of people involved. K3: Driver’s licence, etc. Do you suggest we take a look on these? K4: Then we will check the ones with driver’s licence. For, looking at the number of people alone does not help, as all are not drivers. In other words, if you say that this number of people died, the dead who died on their own may have died due to normal causes, rather than due to the accident. K3: But now there is the number of people who do not have a driver’s licence. And then there are who do not have a driver’s licence, but who still take the wheel. K1: But the number of dead here… K3: Then we should correlate it with the whole population. For, there are those who do not have a driver’s licence, but still cause an accident. Applying robust simplifications to be able to develop a model. For example: Deaths in traffic accidents can be associated with numerous variables. For example, there could be deaths caused by accidents involving drivers who do not have a licence. We need to know the exact numbers of such cases. This is an extreme case we cannot control for. A3 K3: I mean, the population is growing. Say, 3 out of every 5 person get a driver’s licence. The licences are obtained without thinking about the consequences. That’s why the accident rates are rising. Identifying, naming and providing justification for the quantities which affect the case. For example: the number of deaths in traffic accidents is affected by a number of variables including the measures taken, the state of the roads, the traffic volume, the requirements for a driver’s licence, traffic assistance and first-aid capabilities and the state of affairs regarding these variables through a number of years. A4 K4: I said, an increase was observed with respect to driver’s licence numbers, due to the increase in population. The increase in the number of driver’s licences, in turn, led to an increase in the overall number of accidents. However, given the fall in the number of deaths and the rise in the number of injured in the accidents with deaths, one can talk about an increase in the measures taken through the years. Researcher: What was the thing you were trying to estimate? [Group discussion is taking place.] K4: The number of dead. Researcher: How will you associate these variables in order to estimate the dead count? K4: With traffic accidents. Researcher: I mean, will you be estimating the change in dead count with reference to the number of traffic accidents? [Group discussion is taking place.] K4: Varies by the year, I mean, falls in parallel. Having a robust identification of the key variables and explaining them with proper basis. For example: We are asked to provide the dead count caused by traffic accidents in the next year. We should forecast the next year by going over the change occurring in dead count through years. A5 K7: The population had fallen. K8: The population is rising, so is driver’s licence count. K7: It is rising and falling; see.... K8: It fell just once. On other occasions it had been rising. K7: Driver’s licences are increasing. K8: Number of accidents is increasing. K7: It rose substantially. K9: The higher the number of people, the higher the accident count. K7: The number of deaths is falling. So perhaps the safety measures are increasing. K8: Yes. K7: The number of injuries grew. That means, the number of survivors is increasing. Having a robust identification of the relationships between the variables including the key variables and explaining them with proper basis. For example: If the requirements for getting a driver’s licence were made stricter and if the trainings are increased, the number of deaths caused by traffic accidents would fall. Therefore, in cases where the number of driver’s licences is increasing, the number of dead would fall even if the number of accidents rises. If the first-aid mechanisms were improved year over year, the number of deaths would again fall despite a possible increase in the number of accidents. A6 K7: The number of deaths is falling. So perhaps the safety measures are increasing. Focusing on fairly applicable and accessible information in order to solve the problem. For example: Embracing the idea that we cannot have the exact figure of those who took the wheel without a driver’s licence. Reaching across to the data for specified variables, if it had no such data been provided in this activity. A7 K8: 3835. For instance, assuming a given number of deaths in a specific number of accidents… K7: We could calculate the probability of increase in accident count. That would give us the number of dead. Something akin to an equation with two unknowns. K6: I believe it is not about the accident, but rather about the number of people involved. K8: Then we will check the ones with driver’s licence. For, looking at the number of people alone does not help, as all are not drivers. In other words, if you say that this number of people died, the dead who died due to natural causes, rather than due to the accident may be included. Having clear view of what is related and what is not for solving the problem. For example: The population of the country, the number of driver’s licences, total number of accidents, number of accidents with death or injury, the number of injured individuals are unrelated data associated with our assumptions. What is crucial for us is the number of dead varying through the years. Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) A. Competencies to understand the real problem and to set up a model based on reality A1 K8: [Reading the guidelines] Asks what are your assumptions? What are our assumptions? K6: What did we do? We assumed a consistent rate of fall each year: Other than that... What else did we do? K8: What can we assume? We assumed the gender to be the same. K6: That would not do. K8: OK whatever. Coming up with fairly robust assumptions to make the problem simpler; to present the case with justification. For example: Let’s assume that the deaths in traffic accidents occur only due to the physical impact. There may be other cases of death to take place after the accident, due to the fear factor or another condition triggered. A2 K4: Three thousand. 3835. For instance, assuming a given number of deaths in a specific number of accidents… K3: We could calculate the probability of increase in accident count. That would give us the number of dead. Something akin to an equation with two unknowns. K1: I believe it is not about the accident, but rather about the number of people involved. K3: Driver’s licence, etc. Do you suggest we take a look on these? K4: Then we will check the ones with driver’s licence. For, looking at the number of people alone does not help, as all are not drivers. In other words, if you say that this number of people died, the dead who died on their own may have died due to normal causes, rather than due to the accident. K3: But now there is the number of people who do not have a driver’s licence. And then there are who do not have a driver’s licence, but who still take the wheel. K1: But the number of dead here… K3: Then we should correlate it with the whole population. For, there are those who do not have a driver’s licence, but still cause an accident. Applying robust simplifications to be able to develop a model. For example: Deaths in traffic accidents can be associated with numerous variables. For example, there could be deaths caused by accidents involving drivers who do not have a licence. We need to know the exact numbers of such cases. This is an extreme case we cannot control for. A3 K3: I mean, the population is growing. Say, 3 out of every 5 person get a driver’s licence. The licences are obtained without thinking about the consequences. That’s why the accident rates are rising. Identifying, naming and providing justification for the quantities which affect the case. For example: the number of deaths in traffic accidents is affected by a number of variables including the measures taken, the state of the roads, the traffic volume, the requirements for a driver’s licence, traffic assistance and first-aid capabilities and the state of affairs regarding these variables through a number of years. A4 K4: I said, an increase was observed with respect to driver’s licence numbers, due to the increase in population. The increase in the number of driver’s licences, in turn, led to an increase in the overall number of accidents. However, given the fall in the number of deaths and the rise in the number of injured in the accidents with deaths, one can talk about an increase in the measures taken through the years. Researcher: What was the thing you were trying to estimate? [Group discussion is taking place.] K4: The number of dead. Researcher: How will you associate these variables in order to estimate the dead count? K4: With traffic accidents. Researcher: I mean, will you be estimating the change in dead count with reference to the number of traffic accidents? [Group discussion is taking place.] K4: Varies by the year, I mean, falls in parallel. Having a robust identification of the key variables and explaining them with proper basis. For example: We are asked to provide the dead count caused by traffic accidents in the next year. We should forecast the next year by going over the change occurring in dead count through years. A5 K7: The population had fallen. K8: The population is rising, so is driver’s licence count. K7: It is rising and falling; see.... K8: It fell just once. On other occasions it had been rising. K7: Driver’s licences are increasing. K8: Number of accidents is increasing. K7: It rose substantially. K9: The higher the number of people, the higher the accident count. K7: The number of deaths is falling. So perhaps the safety measures are increasing. K8: Yes. K7: The number of injuries grew. That means, the number of survivors is increasing. Having a robust identification of the relationships between the variables including the key variables and explaining them with proper basis. For example: If the requirements for getting a driver’s licence were made stricter and if the trainings are increased, the number of deaths caused by traffic accidents would fall. Therefore, in cases where the number of driver’s licences is increasing, the number of dead would fall even if the number of accidents rises. If the first-aid mechanisms were improved year over year, the number of deaths would again fall despite a possible increase in the number of accidents. A6 K7: The number of deaths is falling. So perhaps the safety measures are increasing. Focusing on fairly applicable and accessible information in order to solve the problem. For example: Embracing the idea that we cannot have the exact figure of those who took the wheel without a driver’s licence. Reaching across to the data for specified variables, if it had no such data been provided in this activity. A7 K8: 3835. For instance, assuming a given number of deaths in a specific number of accidents… K7: We could calculate the probability of increase in accident count. That would give us the number of dead. Something akin to an equation with two unknowns. K6: I believe it is not about the accident, but rather about the number of people involved. K8: Then we will check the ones with driver’s licence. For, looking at the number of people alone does not help, as all are not drivers. In other words, if you say that this number of people died, the dead who died due to natural causes, rather than due to the accident may be included. Having clear view of what is related and what is not for solving the problem. For example: The population of the country, the number of driver’s licences, total number of accidents, number of accidents with death or injury, the number of injured individuals are unrelated data associated with our assumptions. What is crucial for us is the number of dead varying through the years. Table 6 The studies on competency A and expected analyses Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) A. Competencies to understand the real problem and to set up a model based on reality A1 K8: [Reading the guidelines] Asks what are your assumptions? What are our assumptions? K6: What did we do? We assumed a consistent rate of fall each year: Other than that... What else did we do? K8: What can we assume? We assumed the gender to be the same. K6: That would not do. K8: OK whatever. Coming up with fairly robust assumptions to make the problem simpler; to present the case with justification. For example: Let’s assume that the deaths in traffic accidents occur only due to the physical impact. There may be other cases of death to take place after the accident, due to the fear factor or another condition triggered. A2 K4: Three thousand. 3835. For instance, assuming a given number of deaths in a specific number of accidents… K3: We could calculate the probability of increase in accident count. That would give us the number of dead. Something akin to an equation with two unknowns. K1: I believe it is not about the accident, but rather about the number of people involved. K3: Driver’s licence, etc. Do you suggest we take a look on these? K4: Then we will check the ones with driver’s licence. For, looking at the number of people alone does not help, as all are not drivers. In other words, if you say that this number of people died, the dead who died on their own may have died due to normal causes, rather than due to the accident. K3: But now there is the number of people who do not have a driver’s licence. And then there are who do not have a driver’s licence, but who still take the wheel. K1: But the number of dead here… K3: Then we should correlate it with the whole population. For, there are those who do not have a driver’s licence, but still cause an accident. Applying robust simplifications to be able to develop a model. For example: Deaths in traffic accidents can be associated with numerous variables. For example, there could be deaths caused by accidents involving drivers who do not have a licence. We need to know the exact numbers of such cases. This is an extreme case we cannot control for. A3 K3: I mean, the population is growing. Say, 3 out of every 5 person get a driver’s licence. The licences are obtained without thinking about the consequences. That’s why the accident rates are rising. Identifying, naming and providing justification for the quantities which affect the case. For example: the number of deaths in traffic accidents is affected by a number of variables including the measures taken, the state of the roads, the traffic volume, the requirements for a driver’s licence, traffic assistance and first-aid capabilities and the state of affairs regarding these variables through a number of years. A4 K4: I said, an increase was observed with respect to driver’s licence numbers, due to the increase in population. The increase in the number of driver’s licences, in turn, led to an increase in the overall number of accidents. However, given the fall in the number of deaths and the rise in the number of injured in the accidents with deaths, one can talk about an increase in the measures taken through the years. Researcher: What was the thing you were trying to estimate? [Group discussion is taking place.] K4: The number of dead. Researcher: How will you associate these variables in order to estimate the dead count? K4: With traffic accidents. Researcher: I mean, will you be estimating the change in dead count with reference to the number of traffic accidents? [Group discussion is taking place.] K4: Varies by the year, I mean, falls in parallel. Having a robust identification of the key variables and explaining them with proper basis. For example: We are asked to provide the dead count caused by traffic accidents in the next year. We should forecast the next year by going over the change occurring in dead count through years. A5 K7: The population had fallen. K8: The population is rising, so is driver’s licence count. K7: It is rising and falling; see.... K8: It fell just once. On other occasions it had been rising. K7: Driver’s licences are increasing. K8: Number of accidents is increasing. K7: It rose substantially. K9: The higher the number of people, the higher the accident count. K7: The number of deaths is falling. So perhaps the safety measures are increasing. K8: Yes. K7: The number of injuries grew. That means, the number of survivors is increasing. Having a robust identification of the relationships between the variables including the key variables and explaining them with proper basis. For example: If the requirements for getting a driver’s licence were made stricter and if the trainings are increased, the number of deaths caused by traffic accidents would fall. Therefore, in cases where the number of driver’s licences is increasing, the number of dead would fall even if the number of accidents rises. If the first-aid mechanisms were improved year over year, the number of deaths would again fall despite a possible increase in the number of accidents. A6 K7: The number of deaths is falling. So perhaps the safety measures are increasing. Focusing on fairly applicable and accessible information in order to solve the problem. For example: Embracing the idea that we cannot have the exact figure of those who took the wheel without a driver’s licence. Reaching across to the data for specified variables, if it had no such data been provided in this activity. A7 K8: 3835. For instance, assuming a given number of deaths in a specific number of accidents… K7: We could calculate the probability of increase in accident count. That would give us the number of dead. Something akin to an equation with two unknowns. K6: I believe it is not about the accident, but rather about the number of people involved. K8: Then we will check the ones with driver’s licence. For, looking at the number of people alone does not help, as all are not drivers. In other words, if you say that this number of people died, the dead who died due to natural causes, rather than due to the accident may be included. Having clear view of what is related and what is not for solving the problem. For example: The population of the country, the number of driver’s licences, total number of accidents, number of accidents with death or injury, the number of injured individuals are unrelated data associated with our assumptions. What is crucial for us is the number of dead varying through the years. Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) A. Competencies to understand the real problem and to set up a model based on reality A1 K8: [Reading the guidelines] Asks what are your assumptions? What are our assumptions? K6: What did we do? We assumed a consistent rate of fall each year: Other than that... What else did we do? K8: What can we assume? We assumed the gender to be the same. K6: That would not do. K8: OK whatever. Coming up with fairly robust assumptions to make the problem simpler; to present the case with justification. For example: Let’s assume that the deaths in traffic accidents occur only due to the physical impact. There may be other cases of death to take place after the accident, due to the fear factor or another condition triggered. A2 K4: Three thousand. 3835. For instance, assuming a given number of deaths in a specific number of accidents… K3: We could calculate the probability of increase in accident count. That would give us the number of dead. Something akin to an equation with two unknowns. K1: I believe it is not about the accident, but rather about the number of people involved. K3: Driver’s licence, etc. Do you suggest we take a look on these? K4: Then we will check the ones with driver’s licence. For, looking at the number of people alone does not help, as all are not drivers. In other words, if you say that this number of people died, the dead who died on their own may have died due to normal causes, rather than due to the accident. K3: But now there is the number of people who do not have a driver’s licence. And then there are who do not have a driver’s licence, but who still take the wheel. K1: But the number of dead here… K3: Then we should correlate it with the whole population. For, there are those who do not have a driver’s licence, but still cause an accident. Applying robust simplifications to be able to develop a model. For example: Deaths in traffic accidents can be associated with numerous variables. For example, there could be deaths caused by accidents involving drivers who do not have a licence. We need to know the exact numbers of such cases. This is an extreme case we cannot control for. A3 K3: I mean, the population is growing. Say, 3 out of every 5 person get a driver’s licence. The licences are obtained without thinking about the consequences. That’s why the accident rates are rising. Identifying, naming and providing justification for the quantities which affect the case. For example: the number of deaths in traffic accidents is affected by a number of variables including the measures taken, the state of the roads, the traffic volume, the requirements for a driver’s licence, traffic assistance and first-aid capabilities and the state of affairs regarding these variables through a number of years. A4 K4: I said, an increase was observed with respect to driver’s licence numbers, due to the increase in population. The increase in the number of driver’s licences, in turn, led to an increase in the overall number of accidents. However, given the fall in the number of deaths and the rise in the number of injured in the accidents with deaths, one can talk about an increase in the measures taken through the years. Researcher: What was the thing you were trying to estimate? [Group discussion is taking place.] K4: The number of dead. Researcher: How will you associate these variables in order to estimate the dead count? K4: With traffic accidents. Researcher: I mean, will you be estimating the change in dead count with reference to the number of traffic accidents? [Group discussion is taking place.] K4: Varies by the year, I mean, falls in parallel. Having a robust identification of the key variables and explaining them with proper basis. For example: We are asked to provide the dead count caused by traffic accidents in the next year. We should forecast the next year by going over the change occurring in dead count through years. A5 K7: The population had fallen. K8: The population is rising, so is driver’s licence count. K7: It is rising and falling; see.... K8: It fell just once. On other occasions it had been rising. K7: Driver’s licences are increasing. K8: Number of accidents is increasing. K7: It rose substantially. K9: The higher the number of people, the higher the accident count. K7: The number of deaths is falling. So perhaps the safety measures are increasing. K8: Yes. K7: The number of injuries grew. That means, the number of survivors is increasing. Having a robust identification of the relationships between the variables including the key variables and explaining them with proper basis. For example: If the requirements for getting a driver’s licence were made stricter and if the trainings are increased, the number of deaths caused by traffic accidents would fall. Therefore, in cases where the number of driver’s licences is increasing, the number of dead would fall even if the number of accidents rises. If the first-aid mechanisms were improved year over year, the number of deaths would again fall despite a possible increase in the number of accidents. A6 K7: The number of deaths is falling. So perhaps the safety measures are increasing. Focusing on fairly applicable and accessible information in order to solve the problem. For example: Embracing the idea that we cannot have the exact figure of those who took the wheel without a driver’s licence. Reaching across to the data for specified variables, if it had no such data been provided in this activity. A7 K8: 3835. For instance, assuming a given number of deaths in a specific number of accidents… K7: We could calculate the probability of increase in accident count. That would give us the number of dead. Something akin to an equation with two unknowns. K6: I believe it is not about the accident, but rather about the number of people involved. K8: Then we will check the ones with driver’s licence. For, looking at the number of people alone does not help, as all are not drivers. In other words, if you say that this number of people died, the dead who died due to natural causes, rather than due to the accident may be included. Having clear view of what is related and what is not for solving the problem. For example: The population of the country, the number of driver’s licences, total number of accidents, number of accidents with death or injury, the number of injured individuals are unrelated data associated with our assumptions. What is crucial for us is the number of dead varying through the years. Below, a model-eliciting activity is described, along with specific experiences concerning certain sub-processes throughout the modelling cycle during the execution of the activity, in the light of the group study and classroom debate. 2.5 An MEA example: TAS MEA’s characteristics: This MEA asks student teachers to determine whether the measures taken for traffic accidents are sufficient. This activity to be implemented through group work will allow estimations for future years, based on the data for past years. The data for past years, concerning the context, were available on Turkish Statistics Agency (TurkStat) website. Thereafter, the real-life problem to be dealt with within the framework of the activity was described alongside justification. In this activity, all data received from TurkStat were presented directly, without identification of key variables. The student mathematics teachers were asked to work with data for years 2006 on. The key variables in this activity were the year and the death count. A glance at the data for key variables revealed a linear direction. This data set is conducive to representation through a linear model. Student mathematics teachers’ experiences with the implementation of the TAS activity were analyzed through taking into account the rubric presented in the following tables. The studies on the competency A were analyzed through taking into account the rubric presented in Table 6. As illustrated in the Table 6, the work on A1 sub-competency reveals that K6 developed valid assumptions required to develop the model which could provide a solution for the problem, even if she did not provide justification, while K8 engaged in an unrelated assumption. K8 did not inquire K6’s critical view’s justification and duly abandoned her idea. That is why K6 is deemed to have sub-competency A1 at the level of criteria 1, while K8’s competency A1 is deemed to be at the level of criteria 0. Work on sub-competency A2, in turn, reveals that certain student mathematics teachers have reviewed the variables affecting the state of the problem on a wide perspective, while some others had a more realistic outlook, seeking to simplify the variables in turn. Yet, as the simplifications focused on incorrect key variables, student mathematics teachers K1, K3 and K4 were considered to meet only criteria 2 for these sub-competencies. A look at sub-competency A3 reveals that K3 voiced variables concerning population increase, the number of driver’s licences and the rate of accidents among all the variables to affect the picture. In this context, an increase in the number of deaths should have arisen. However, the measures taken and other variables such as first-aid measures contribute to a fall in the number of deaths. Against this background, K3 was observed to identify various variables but fell short of being able to interpret the state of affairs concerning those variables. Hence, K3 is assessed to meet criteria 2 for this sub-competency. With reference to sub-competency A4, K4 was observed to identify the linkages between various variables affecting the state of affairs but remained unable to take note of key variables. Hence, K4 is assessed to meet criteria 0 for sub-competency A4. The in-group debates executed with the counsel of the researcher paved the way for K4 to accurately identify the key variables. The discussions with the researcher and the group have led K4 to criteria 3 level for sub-competency A4. With reference to sub-competency A5, K7, K8 and K9 were observed to focus on the variables specified in the activity and try to establish the linkages between such variables. Yet, the relationships between other variables which would have an impact on the state of affairs but which were not covered in the activity, such as the measures taken, and the desired state of affairs had not been covered. Only K7 was observed to establish a link between the measures taken —the key variable— and other variables. Nonetheless, K7 established the said relationship not with the number of deaths with reference to the year —the key variable— but with the fall in the number of injuries. Taking all these into account, K8 and K9 exhibited the ability to establish connections between the variables, to the tune of criteria 1, while K7’s skills were assessed to meet criteria 2. In terms of the work on sub-competency A6, K7 was observed to focus on the causes of the fall in the number of deaths, without any attempt to check the availability of such data, instead focusing on the information provided. Hence, her competence level for A6 is assessed to meet criteria 0. A glance at the work on sub-competency A7 reveals that the student mathematics teachers did not associate the key variable they identified when working on the variables to affect the state of affairs, with the year variable entailed in the problem case, but instead tried their luck with other variables which are not related with the desired result. Therefore, these student mathematics teachers meet criteria 0 for sub-competency A7. The studies on the competency B were analyzed through taking into account the rubric presented in Table 7. Table 7 The studies on competency B and expected analyses Table 7 The studies on competency B and expected analyses The work on sub-competency B1 suggests that K5 is aware of the need for a mathematical expression of the qualities identified in accordance with the state of the problem, through an interpretation of data. However, a mathematical expression still eludes them. Therefore, K5 meets criteria 1 for sub-competency B1. The work on sub-competency B2 reveals that student mathematics teachers have expressed the equation to lead to the number of deaths with reference to the average rate of change. Hence, K6, who developed this idea, is assessed to meet criteria 3 for sub-competency B2. On sub-competency B3, K6 was observed to offer some means to simplify the quantities and to provide some remarks on the cases where the model would apply after simplifications. K7 too provides accurate remarks on why the approach could be the right one and confirms the case with real-life data. Therefore, the student mathematics teacher meets criteria 3 for sub-competency B3. With reference to sub-competency B4, the simplifications required to render the mathematical model usable can be observed to be effected. The simplifications render the model more usable, while remaining mathematically accurate. Therefore, student mathematics teacher K6 who effected such simplifications exhibits the competence to reduce the number of relationships between relevant quantities and to simplify the relationships, where necessary, to the tune of criteria 3. A glance at the work for sub-competency B5 reveals that student mathematics teachers had identified the relationships explaining the case and were able to come up with a mathematical model by choosing applicable mathematical expressions to match the mathematical operations applied for this purpose. Furthermore, it is evident that the student mathematics teachers fell short of using regression or least squares method which would provide a better representation of the data set, when developing the mathematical model to represent the data for the case. Therefore, K6, who chose the mathematical expressions used for the development of the model, exhibits the competence to choose applicable mathematical expressions and represent them graphically where necessary, to the tune of criteria 2. The studies on the competency C were analyzed through taking into account the rubric presented in Table 8. Table 8 The studies on competency C and expected analyses Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) C. Competencies to solve mathematical questions within this mathematical model C1 No work was carried out. Making robust use of the heuristic strategies to assist the solution of the problem; providing explanations with justification. For example: Developing solutions to the problem by taking part of the data into account; or using the model by establishing links with the previous activity which is similar in nature. C2 Making very good use of the mathematical information required to solve the problem. For example: Reaching to the required information in a mathematically correct form, by accurately noting the known data on the linear model developed for this purpose. Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) C. Competencies to solve mathematical questions within this mathematical model C1 No work was carried out. Making robust use of the heuristic strategies to assist the solution of the problem; providing explanations with justification. For example: Developing solutions to the problem by taking part of the data into account; or using the model by establishing links with the previous activity which is similar in nature. C2 Making very good use of the mathematical information required to solve the problem. For example: Reaching to the required information in a mathematically correct form, by accurately noting the known data on the linear model developed for this purpose. Table 8 The studies on competency C and expected analyses Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) C. Competencies to solve mathematical questions within this mathematical model C1 No work was carried out. Making robust use of the heuristic strategies to assist the solution of the problem; providing explanations with justification. For example: Developing solutions to the problem by taking part of the data into account; or using the model by establishing links with the previous activity which is similar in nature. C2 Making very good use of the mathematical information required to solve the problem. For example: Reaching to the required information in a mathematically correct form, by accurately noting the known data on the linear model developed for this purpose. Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) C. Competencies to solve mathematical questions within this mathematical model C1 No work was carried out. Making robust use of the heuristic strategies to assist the solution of the problem; providing explanations with justification. For example: Developing solutions to the problem by taking part of the data into account; or using the model by establishing links with the previous activity which is similar in nature. C2 Making very good use of the mathematical information required to solve the problem. For example: Reaching to the required information in a mathematically correct form, by accurately noting the known data on the linear model developed for this purpose. All student mathematics teachers were found to meet criteria 0 for sub-competency C1 regarding TAS activities. This is caused by the lack of any heuristic strategy, rather than by the use of inapplicable heuristic strategies. A glance at work on sub-competency C2 reveals that student mathematics teachers had made very effective use of their mathematical knowledge when solving the problem through the mathematical model they had developed and were able to reach to the result through the model. Therefore, K6 and K7 which carried out these procedures exhibit competence to use mathematical information to solve the problem, to the tune of criteria 3. The studies on the competency D were analyzed through taking into account the rubric presented in Table 9. Table 9 The studies on competency D and expected analyses Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) D. Competencies to interpret mathematical results in a real situation D1 K9 : But after some years, we will reach to a state... where deaths will be no more. Offering very robust interpretations of mathematical results in non-mathematical contexts and providing justifications. For example: If the result is negative, the case would refer to a negative dead count, which should not be interpreted to suggest unexpected births but rather as a reflection of our linear reduction perspective. D2 K8 : Asking is it generalizable? K6 : It can be generalized in cases of continuous fall. K7 : It can be generalized in cases where no huge increase occurs. K5 : It cannot be generalized in cases of excessive fall or increase. OK. Providing very effective generalizations of the solutions developed for special cases; providing justifications for doing so. For example: Realizing that the mathematical model thus developed can be generalized only in linear cases with a fall in the rate of change and developing a general mathematical model in this direction (for a linear change; estimated number of deaths = rate of change reflecting the data x time frame from the base year to the focus year + the number of deaths in the base year). Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) D. Competencies to interpret mathematical results in a real situation D1 K9 : But after some years, we will reach to a state... where deaths will be no more. Offering very robust interpretations of mathematical results in non-mathematical contexts and providing justifications. For example: If the result is negative, the case would refer to a negative dead count, which should not be interpreted to suggest unexpected births but rather as a reflection of our linear reduction perspective. D2 K8 : Asking is it generalizable? K6 : It can be generalized in cases of continuous fall. K7 : It can be generalized in cases where no huge increase occurs. K5 : It cannot be generalized in cases of excessive fall or increase. OK. Providing very effective generalizations of the solutions developed for special cases; providing justifications for doing so. For example: Realizing that the mathematical model thus developed can be generalized only in linear cases with a fall in the rate of change and developing a general mathematical model in this direction (for a linear change; estimated number of deaths = rate of change reflecting the data x time frame from the base year to the focus year + the number of deaths in the base year). Table 9 The studies on competency D and expected analyses Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) D. Competencies to interpret mathematical results in a real situation D1 K9 : But after some years, we will reach to a state... where deaths will be no more. Offering very robust interpretations of mathematical results in non-mathematical contexts and providing justifications. For example: If the result is negative, the case would refer to a negative dead count, which should not be interpreted to suggest unexpected births but rather as a reflection of our linear reduction perspective. D2 K8 : Asking is it generalizable? K6 : It can be generalized in cases of continuous fall. K7 : It can be generalized in cases where no huge increase occurs. K5 : It cannot be generalized in cases of excessive fall or increase. OK. Providing very effective generalizations of the solutions developed for special cases; providing justifications for doing so. For example: Realizing that the mathematical model thus developed can be generalized only in linear cases with a fall in the rate of change and developing a general mathematical model in this direction (for a linear change; estimated number of deaths = rate of change reflecting the data x time frame from the base year to the focus year + the number of deaths in the base year). Student mathematics teachers’ works on sub-competencies and coding Expected perfect works (criteria 3) D. Competencies to interpret mathematical results in a real situation D1 K9 : But after some years, we will reach to a state... where deaths will be no more. Offering very robust interpretations of mathematical results in non-mathematical contexts and providing justifications. For example: If the result is negative, the case would refer to a negative dead count, which should not be interpreted to suggest unexpected births but rather as a reflection of our linear reduction perspective. D2 K8 : Asking is it generalizable? K6 : It can be generalized in cases of continuous fall. K7 : It can be generalized in cases where no huge increase occurs. K5 : It cannot be generalized in cases of excessive fall or increase. OK. Providing very effective generalizations of the solutions developed for special cases; providing justifications for doing so. For example: Realizing that the mathematical model thus developed can be generalized only in linear cases with a fall in the rate of change and developing a general mathematical model in this direction (for a linear change; estimated number of deaths = rate of change reflecting the data x time frame from the base year to the focus year + the number of deaths in the base year). A glance at work on sub-competency D1 reveals that K9 was able to interpret the results reached through the mathematical model and to reach to inferences regarding the actual context. However, no remarks were made on whether or not the lack of any deaths in traffic accidents was a realistic case against the actual state of affairs. That is why K9 is assessed to exhibit the ability to interpret mathematical results in non-mathematical contexts, to the tune of criteria 1. A glance at work on sub-competency D2 reveals that K6 and K7 provided remarks on the justifications on the cases where the mathematical method developed could be generalized. However, as various simplifications were applied to render the model practical, the mathematical model can be used to estimate the time frame 2000–2099. Furthermore, as no constant increase or fall took place in the actual state of affairs, in its current form, the model cannot be generalized. The student mathematics teachers are aware of the conditions required for the generalization of the model yet fall short of the generalization work by assigning variables to stand for actual data, so as to be able to compensate for such shortcomings. Taking all these into account, K6 and K7 exhibit the competence to generalize the solutions developed for a special case, to the tune of criteria 1. The studies on the competency E were analyzed like the other competencies. The sub-competency levels of each student teacher involved in group work were assessed separately. Individual assessments allowed the observation of the development of competencies on part of each student teacher through the process, as well as the combined interpretation of competencies to arise through the activities carried out as part of the group work, as well as through the individual activities. With a view to providing an example of how the analysis process took place, the rubric developed to assess the student mathematics teachers’ mathematical modelling sub-competencies and the examples of top performance indicators for the MEA at hand are presented in Table 11. The processes in MEAs were reviewed using the analytical rubric exemplified in Table 10, to identify the student mathematics teachers’ sub-competencies regarding each MEA. The number of student mathematics teachers with specific sub-competencies was established with respect to each MEA. As an example, the data set generated for competency B regarding activity BA is provided in Table 12. Table 10 Mathematical modelling competencies analysis rubric and examples of student work regarding TAS activity, at criteria 3 level No attempt or unacceptable performance (criteria 0) Insufficient performance (criteria 1) Imperfect yet acceptable performance (criteria 2) Perfect performance (criteria 3) B2 The relationships between the applicable quantity/quantities were not mathematized, or the mathematization implemented is not applicable or is irrelevant. Attempts were made to mathematize the relationships between the applicable quantity/quantities, but they are not sufficient for the development of a model. The relationships between the applicable quantity/quantities were mathematized; but some small insignificant errors exist or no justifications were provided. The relationships between the applicable quantity/quantities were mathematized quite well; justifications were provided to explain. For example: There’s a general fall. The data exhibit a linear fall. Stating the most applicable correct equation to represent the data such as [(y-y0)=m(x-x0)]. B3 The number of applicable quantities was not reduced even though such a reduction was in order; no simplification of quantities was applied, or the applied simplification is invalid. Some remarks were made about the simplification of applicable quantities were voiced; however the implementation of such remarks is not sufficient, or the simplifications are not suitable in terms of usability. Necessary simplifications were effected with respect to applicable quantities; however certain small and insignificant errors persist; or no investigation of the impact of simplifications on the model was applied. The simplifications concerning applicable quantities were carried out quite well; a review of their impact on the model was carried out, along with a discussion of justifications. For example: The average change per year is 159.6. Since we are talking about the number of individuals, rounding it up to 160 would have only a minimal impact on the error margins. For our estimations will certainly be in four digits. We can apply such a simplification. No attempt or unacceptable performance (criteria 0) Insufficient performance (criteria 1) Imperfect yet acceptable performance (criteria 2) Perfect performance (criteria 3) B2 The relationships between the applicable quantity/quantities were not mathematized, or the mathematization implemented is not applicable or is irrelevant. Attempts were made to mathematize the relationships between the applicable quantity/quantities, but they are not sufficient for the development of a model. The relationships between the applicable quantity/quantities were mathematized; but some small insignificant errors exist or no justifications were provided. The relationships between the applicable quantity/quantities were mathematized quite well; justifications were provided to explain. For example: There’s a general fall. The data exhibit a linear fall. Stating the most applicable correct equation to represent the data such as [(y-y0)=m(x-x0)]. B3 The number of applicable quantities was not reduced even though such a reduction was in order; no simplification of quantities was applied, or the applied simplification is invalid. Some remarks were made about the simplification of applicable quantities were voiced; however the implementation of such remarks is not sufficient, or the simplifications are not suitable in terms of usability. Necessary simplifications were effected with respect to applicable quantities; however certain small and insignificant errors persist; or no investigation of the impact of simplifications on the model was applied. The simplifications concerning applicable quantities were carried out quite well; a review of their impact on the model was carried out, along with a discussion of justifications. For example: The average change per year is 159.6. Since we are talking about the number of individuals, rounding it up to 160 would have only a minimal impact on the error margins. For our estimations will certainly be in four digits. We can apply such a simplification. Table 10 Mathematical modelling competencies analysis rubric and examples of student work regarding TAS activity, at criteria 3 level No attempt or unacceptable performance (criteria 0) Insufficient performance (criteria 1) Imperfect yet acceptable performance (criteria 2) Perfect performance (criteria 3) B2 The relationships between the applicable quantity/quantities were not mathematized, or the mathematization implemented is not applicable or is irrelevant. Attempts were made to mathematize the relationships between the applicable quantity/quantities, but they are not sufficient for the development of a model. The relationships between the applicable quantity/quantities were mathematized; but some small insignificant errors exist or no justifications were provided. The relationships between the applicable quantity/quantities were mathematized quite well; justifications were provided to explain. For example: There’s a general fall. The data exhibit a linear fall. Stating the most applicable correct equation to represent the data such as [(y-y0)=m(x-x0)]. B3 The number of applicable quantities was not reduced even though such a reduction was in order; no simplification of quantities was applied, or the applied simplification is invalid. Some remarks were made about the simplification of applicable quantities were voiced; however the implementation of such remarks is not sufficient, or the simplifications are not suitable in terms of usability. Necessary simplifications were effected with respect to applicable quantities; however certain small and insignificant errors persist; or no investigation of the impact of simplifications on the model was applied. The simplifications concerning applicable quantities were carried out quite well; a review of their impact on the model was carried out, along with a discussion of justifications. For example: The average change per year is 159.6. Since we are talking about the number of individuals, rounding it up to 160 would have only a minimal impact on the error margins. For our estimations will certainly be in four digits. We can apply such a simplification. No attempt or unacceptable performance (criteria 0) Insufficient performance (criteria 1) Imperfect yet acceptable performance (criteria 2) Perfect performance (criteria 3) B2 The relationships between the applicable quantity/quantities were not mathematized, or the mathematization implemented is not applicable or is irrelevant. Attempts were made to mathematize the relationships between the applicable quantity/quantities, but they are not sufficient for the development of a model. The relationships between the applicable quantity/quantities were mathematized; but some small insignificant errors exist or no justifications were provided. The relationships between the applicable quantity/quantities were mathematized quite well; justifications were provided to explain. For example: There’s a general fall. The data exhibit a linear fall. Stating the most applicable correct equation to represent the data such as [(y-y0)=m(x-x0)]. B3 The number of applicable quantities was not reduced even though such a reduction was in order; no simplification of quantities was applied, or the applied simplification is invalid. Some remarks were made about the simplification of applicable quantities were voiced; however the implementation of such remarks is not sufficient, or the simplifications are not suitable in terms of usability. Necessary simplifications were effected with respect to applicable quantities; however certain small and insignificant errors persist; or no investigation of the impact of simplifications on the model was applied. The simplifications concerning applicable quantities were carried out quite well; a review of their impact on the model was carried out, along with a discussion of justifications. For example: The average change per year is 159.6. Since we are talking about the number of individuals, rounding it up to 160 would have only a minimal impact on the error margins. For our estimations will certainly be in four digits. We can apply such a simplification. Table 11 The number of student mathematics teachers to meet the established competency levels, as observed through the evaluation of the work carried out in first activity (BF), using the analytical rubric Number of student teacher Sub-competencies Criteria 0 Criteria 1 Criteria 2 Criteria 3 Total B2 8 1 0 0 9 B3 9 0 0 0 9 Number of student teacher Sub-competencies Criteria 0 Criteria 1 Criteria 2 Criteria 3 Total B2 8 1 0 0 9 B3 9 0 0 0 9 Table 11 The number of student mathematics teachers to meet the established competency levels, as observed through the evaluation of the work carried out in first activity (BF), using the analytical rubric Number of student teacher Sub-competencies Criteria 0 Criteria 1 Criteria 2 Criteria 3 Total B2 8 1 0 0 9 B3 9 0 0 0 9 Number of student teacher Sub-competencies Criteria 0 Criteria 1 Criteria 2 Criteria 3 Total B2 8 1 0 0 9 B3 9 0 0 0 9 Once the data presented in Table 11 were established for each activity, tables pertaining to the criteria observed in activities for competencies were drawn up with a view to reviewing the development of each sub-competency within the framework of the process. For instance, one can review Table 13 indicating the distribution regarding sub-competency B3 to arise in all MEAs. Table 12 Distribution of B3 sub-competency levels regarding assumptions about the all MEAs Number of student teacher MEA Criteria 0 Criteria 1 Criteria 2 Criteria 3 Total 1 9 0 0 0 9 2 7 2 0 0 9 3 5 1 2 1 9 4 6 3 0 0 9 5 6 0 1 2 9 6 9 0 0 0 9 7 6 0 0 3 9 8 1 0 1 7 9 Number of student teacher MEA Criteria 0 Criteria 1 Criteria 2 Criteria 3 Total 1 9 0 0 0 9 2 7 2 0 0 9 3 5 1 2 1 9 4 6 3 0 0 9 5 6 0 1 2 9 6 9 0 0 0 9 7 6 0 0 3 9 8 1 0 1 7 9 Table 12 Distribution of B3 sub-competency levels regarding assumptions about the all MEAs Number of student teacher MEA Criteria 0 Criteria 1 Criteria 2 Criteria 3 Total 1 9 0 0 0 9 2 7 2 0 0 9 3 5 1 2 1 9 4 6 3 0 0 9 5 6 0 1 2 9 6 9 0 0 0 9 7 6 0 0 3 9 8 1 0 1 7 9 Number of student teacher MEA Criteria 0 Criteria 1 Criteria 2 Criteria 3 Total 1 9 0 0 0 9 2 7 2 0 0 9 3 5 1 2 1 9 4 6 3 0 0 9 5 6 0 1 2 9 6 9 0 0 0 9 7 6 0 0 3 9 8 1 0 1 7 9 Table 13 An example for the evaluation of development within the framework of the process Process Example for B2 sub-competency Development for participants (both group work and individual work) Sub-competencies for each MEA were evaluated on an individual basis, followed by the review of general change. Individual development (both group work and individual work) Sub-competencies for each MEA were evaluated on an individual basis, followed by the review of individual change. Without intervention The sub-competencies of student mathematics teachers who are not involved with learning environment. A general review of the work three student mathematics teachers who were not involved with the learning environment carried out with MEAs provided reveals that just one of the student mathematics teachers reached criteria 3 levels regarding sub-competency B3, while others remained at criteria 0. The student mathematics teacher who had reached criteria 3 levels is the same one who scored criteria 3 levels for sub-competencies B1 and B2. Student mathematics teachers with low levels of sub-competency for B1 and B2 failed to provide mathematical expression of the relationships between the variables, and hence did not get the chance to work on simplifications regarding quantities. This finding suggests that sub-competency B3 is affected by other sub-competencies. Process Example for B2 sub-competency Development for participants (both group work and individual work) Sub-competencies for each MEA were evaluated on an individual basis, followed by the review of general change. Individual development (both group work and individual work) Sub-competencies for each MEA were evaluated on an individual basis, followed by the review of individual change. Without intervention The sub-competencies of student mathematics teachers who are not involved with learning environment. A general review of the work three student mathematics teachers who were not involved with the learning environment carried out with MEAs provided reveals that just one of the student mathematics teachers reached criteria 3 levels regarding sub-competency B3, while others remained at criteria 0. The student mathematics teacher who had reached criteria 3 levels is the same one who scored criteria 3 levels for sub-competencies B1 and B2. Student mathematics teachers with low levels of sub-competency for B1 and B2 failed to provide mathematical expression of the relationships between the variables, and hence did not get the chance to work on simplifications regarding quantities. This finding suggests that sub-competency B3 is affected by other sub-competencies. Table 13 An example for the evaluation of development within the framework of the process Process Example for B2 sub-competency Development for participants (both group work and individual work) Sub-competencies for each MEA were evaluated on an individual basis, followed by the review of general change. Individual development (both group work and individual work) Sub-competencies for each MEA were evaluated on an individual basis, followed by the review of individual change. Without intervention The sub-competencies of student mathematics teachers who are not involved with learning environment. A general review of the work three student mathematics teachers who were not involved with the learning environment carried out with MEAs provided reveals that just one of the student mathematics teachers reached criteria 3 levels regarding sub-competency B3, while others remained at criteria 0. The student mathematics teacher who had reached criteria 3 levels is the same one who scored criteria 3 levels for sub-competencies B1 and B2. Student mathematics teachers with low levels of sub-competency for B1 and B2 failed to provide mathematical expression of the relationships between the variables, and hence did not get the chance to work on simplifications regarding quantities. This finding suggests that sub-competency B3 is affected by other sub-competencies. Process Example for B2 sub-competency Development for participants (both group work and individual work) Sub-competencies for each MEA were evaluated on an individual basis, followed by the review of general change. Individual development (both group work and individual work) Sub-competencies for each MEA were evaluated on an individual basis, followed by the review of individual change. Without intervention The sub-competencies of student mathematics teachers who are not involved with learning environment. A general review of the work three student mathematics teachers who were not involved with the learning environment carried out with MEAs provided reveals that just one of the student mathematics teachers reached criteria 3 levels regarding sub-competency B3, while others remained at criteria 0. The student mathematics teacher who had reached criteria 3 levels is the same one who scored criteria 3 levels for sub-competencies B1 and B2. Student mathematics teachers with low levels of sub-competency for B1 and B2 failed to provide mathematical expression of the relationships between the variables, and hence did not get the chance to work on simplifications regarding quantities. This finding suggests that sub-competency B3 is affected by other sub-competencies. Table 13 presents an example on the evaluation of the sub-competency development in line with these data. The sub-competence levels of the mathematics teachers who were involved and did not participate in the learning environment were compared. In this way the contribution to the development of the learning environment was perceived. A general review of the graph presenting the change student mathematics teachers, who were involved in the learning environment, went through in terms of their qualification levels for sub-competency B3 reveals that the student mathematics teachers did not get sub-competency B3 without receiving a training on mathematical modelling; that the level of qualification for this sub-competency developed, even if at a limited rate, compared to the models developed through the process; and that at the end of the process, the majority of the student mathematics teachers exhibited criteria 3 levels of competence in this context. When individual opportunities were extended (with IPs), the student mathematics teachers who did not engage in work on sub-competency B3 through the process were, strikingly, observed to exhibit the sub-competency B3 in their own projects. The sub-competency B3 scores in IPs were higher then group work scores. Because the student mathematics teachers needed to render their mathematical model practical so as to become able to generate solutions for their individual problems. They work together in the groups' mathematical models, so they do not need practical models. They overcome the non-practical mathematical models in the group works. (We can say that the learning environment is ‘Not Directly Influential (NDI)’ on B3 sub-competency and it is activated in individual work process.) A comparison of the student mathematics teachers’ competence levels for sub-competency B3 with the case of student mathematics teachers who were not involved with the learning environment leads to the conclusion that those who were involved with the learning environment were found to reach better levels at the end of the process. However, considering the view that the sub-competency arises with reference to the structure of the mathematical model thus developed and the existence of preceding sub-competencies, one can forcefully argue that sub-competency B3 is not directly associated with the mathematical modelling experience. (We can say that B3 sub-competency’s development is ‘Not Directly Dependent on Experience (NDDE)’ of mathematical modelling.) Only an example of analysis for B3 sub-competency is given in this section. Similar processes have been carried out for other sub-competencies. The development of sub-competencies of student mathematics teachers involved with the learning environment, to arise within the group, for each sub-competency, as well as their sub-competencies to arise in individual projects and the sub-competencies of the student mathematics teachers who were not involved with the learning environment were compared to establish the contribution of the learning environment to the development of sub-competencies. If a sub-competency has not developed to the expected criteria for more student mathematics teachers, we can say that the learning environment is ‘Influential (I)’ on this sub-competency. If a sub-competency is developed in both general and individual in high criteria, we can say that the learning environment is ‘Extremely Influential (EI)’ on this sub-competency and if student mathematics teachers who are not involved with learning environment have not this sub-competency, we can say that this sub-competency’s development is ‘Depends on Experience (DE)’ or if student mathematics teachers who are not involved with learning environment have this sub-competency even if low criteria, we can say that this sub-competency’s development is ‘Somewhat Depends on Experience (SDE)’. If many student mathematics teachers have a sub-competency in high criteria and this sub-competency appears in low criteria late in the process, we can say that the learning environment is ‘Negative Influence (NI)’ on this sub-competency and if student mathematics teachers who are not involved with learning environment have engaged in better studies regarding this sub-competency, we can say that this sub-competency’s regression is ‘Depends on Experience (DE)’ on mathematical modelling. 3. Results The objective of this study is to reveal how LES-MMC process contributes to the development of mathematical modelling sub-competencies. In line with this objective, the development of sub-competencies of participants involved with the LES-MMC process was analyzed for individual participants, as well as for the whole group. The findings were compared against the sub-competencies of the student mathematics teachers who were not involved with the learning environment, leading to the analysis of the impact of the designed environment on the development of mathematical modelling sub-competencies. Given the limitations of space, it is not possible to cover the sub-competency related findings for all participants as well as the student mathematics teachers who are not involved in the learning environment and is not within the realm of possibilities. Therefore, this section will compare the three data sets developed for each sub-competency to discuss the conclusion reached with respect to the development of sub-competencies in the light of the literature. Also only the interesting findings will be discussed in this section. The full PhD thesis written by Aydin-Güç (2015) can be reviewed for further results and discussion. The summaries regarding the findings concerning the competency to comprehend the real problem and to develop a model based on the reality (A) and the conclusions reached with respect to associated sub-competencies are presented in Table 14. Table 14 Development of sub-competencies associated with competency A, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature A1. Competency to make assumptions for the problem EI DE Bukova-Güzel (2011) A2. Competency to simplify the situation I SDE Blum & Borromeo-Ferri (2009); Şen-Zeytun (2013); A3. Competency to recognize quantities that influence the situation, to name them NI DE - A4. Competency to identify key variables that influence the situation I SDE - A5. Competency to construct relations between the variables I SDE - A6. Competency to look for available/not available information to solve the problem I SDE - A7. Competency to differentiate the related/unrelated information to solve the problem I SDE - Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature A1. Competency to make assumptions for the problem EI DE Bukova-Güzel (2011) A2. Competency to simplify the situation I SDE Blum & Borromeo-Ferri (2009); Şen-Zeytun (2013); A3. Competency to recognize quantities that influence the situation, to name them NI DE - A4. Competency to identify key variables that influence the situation I SDE - A5. Competency to construct relations between the variables I SDE - A6. Competency to look for available/not available information to solve the problem I SDE - A7. Competency to differentiate the related/unrelated information to solve the problem I SDE - Table 14 Development of sub-competencies associated with competency A, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature A1. Competency to make assumptions for the problem EI DE Bukova-Güzel (2011) A2. Competency to simplify the situation I SDE Blum & Borromeo-Ferri (2009); Şen-Zeytun (2013); A3. Competency to recognize quantities that influence the situation, to name them NI DE - A4. Competency to identify key variables that influence the situation I SDE - A5. Competency to construct relations between the variables I SDE - A6. Competency to look for available/not available information to solve the problem I SDE - A7. Competency to differentiate the related/unrelated information to solve the problem I SDE - Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature A1. Competency to make assumptions for the problem EI DE Bukova-Güzel (2011) A2. Competency to simplify the situation I SDE Blum & Borromeo-Ferri (2009); Şen-Zeytun (2013); A3. Competency to recognize quantities that influence the situation, to name them NI DE - A4. Competency to identify key variables that influence the situation I SDE - A5. Competency to construct relations between the variables I SDE - A6. Competency to look for available/not available information to solve the problem I SDE - A7. Competency to differentiate the related/unrelated information to solve the problem I SDE - It is clear that the designed learning environment is very influential in the development of A1 sub-competency. A comparison of A1 sub-competencies of student mathematics teachers who are involved with the learning environment against those of the student mathematics teachers not involved suggests that the development of this sub-competency depends on the learning environment. The literature is not bereft of other studies with similar results on the experience-based development of that competency (Bukova-Güzel, 2011). The development of the A2 sub-competency based on experience concerning mathematical modelling, albeit not at the desired level, suggests that the learning environment design influences the development of A2 sub-competency. Related literature suggests that the simplification competencies of the students leave some to be desired, given the fact that they engage only in crude simplifications, even though the studies do not investigate sub-competencies of mathematical modelling individually. Furthermore, these studies emphasize the fact that the competencies to make assumptions regarding the context are less than stellar (Blum & Borromeo-Ferri, 2009; Şen-Zeytun, 2013). The A4, A5, A6 and A7 sub-competencies of student mathematics teachers who were not involved with the learning environment, compared against the development registered by the student mathematics teachers who were involved, revealed that the latter group achieved higher competency levels in practices concerning such sub-competencies. This state of affairs suggests that the learning environment design influences the development of A4, A5, A6 and A7 sub-competencies and that such sub-competencies develop in parallel to mathematical modelling, albeit not at the desired level. Given the fact that the studies and the results of which are discussed in the literature do not discuss the sub-competencies separately, the findings concerning sub-competencies A3, A4, A5, A6 and A7 would certainly shed light on sub-competencies. The summaries regarding the findings concerning the competency to develop a mathematical model based on the real one (B) and the conclusions reached with respect to associated sub-competencies are presented in Table 15. Table 15 Development of sub-competencies associated with competency B, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature B1. Competency to mathematize relevant qualities/quantities I SDE - B2. Competency to mathematize the relations between the related qualities/quantities I SDE - B3. Competency to reduce the number of relevant quantities and simplify their quantities when necessary NDI- individual use NDDE - B4. Competency to reduce the number of relations between related quantities and simplify the relations when necessary NDI-individual use NDDE - B5. Competency to choose appropriate mathematical notations and represent situations graphics if necessary I SDE - Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature B1. Competency to mathematize relevant qualities/quantities I SDE - B2. Competency to mathematize the relations between the related qualities/quantities I SDE - B3. Competency to reduce the number of relevant quantities and simplify their quantities when necessary NDI- individual use NDDE - B4. Competency to reduce the number of relations between related quantities and simplify the relations when necessary NDI-individual use NDDE - B5. Competency to choose appropriate mathematical notations and represent situations graphics if necessary I SDE - Table 15 Development of sub-competencies associated with competency B, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature B1. Competency to mathematize relevant qualities/quantities I SDE - B2. Competency to mathematize the relations between the related qualities/quantities I SDE - B3. Competency to reduce the number of relevant quantities and simplify their quantities when necessary NDI- individual use NDDE - B4. Competency to reduce the number of relations between related quantities and simplify the relations when necessary NDI-individual use NDDE - B5. Competency to choose appropriate mathematical notations and represent situations graphics if necessary I SDE - Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature B1. Competency to mathematize relevant qualities/quantities I SDE - B2. Competency to mathematize the relations between the related qualities/quantities I SDE - B3. Competency to reduce the number of relevant quantities and simplify their quantities when necessary NDI- individual use NDDE - B4. Competency to reduce the number of relations between related quantities and simplify the relations when necessary NDI-individual use NDDE - B5. Competency to choose appropriate mathematical notations and represent situations graphics if necessary I SDE - Sub-competencies B1, B2 and B5 are found to have developed through the process. Yet another telling result is that upon the completion of the process the student mathematics teachers fall short of the maximum level of achievement concerning these sub-competencies. Furthermore, some student mathematics teachers who are not involved with the learning environment were found to engage in activities regarding such sub-competencies, even though at a limited scale. Still, a comparison of these sub-competencies in the case of student mathematics teachers involved with the learning environment against those who were not involved reveals that the learning environment had affected the development of these sub-competencies, and hence such sub-competencies had been somewhat dependent on the experience. Even though the studies on mathematical modelling had not yet discussed mathematical modelling competencies within the framework of individual sub-competencies, they often note the difficulties students experience during the transition from the real world to the mathematical one (Şen-Zeytun, 2013). In that transition, B1 and B2 sub-competencies are extremely important. In this context, one can possibly argue that this learning environment designed with a holistic approach focusing on sub-competencies is influential in the elimination of the difficulties faced during the transition from the real world to the mathematical one, as discussed in the literature. The B3 and B4 sub-competencies of student mathematics teachers who were not involved with the learning environment, compared against the development registered by the student mathematics teachers who were involved, revealed no significant differences in their sub-competency levels in this context. It is evident that the student mathematics teachers involved with the learning environment did not engage in activities concerning B3 and B4 sub-competencies with a view to making the model developed through group work more practical. However, the student mathematics teachers individually apply B3 and B4 sub-competencies in the model developed in line with the problem they had identified. Taking into account the importance of the practical applicability of the model, one can reach to the conclusion that the learning environments designed thus do not support B3 and B4 sub-competencies, which are crucial in terms of practical application. The findings of this study, on the other hand, reveal that experience is influential in the development of sub-competencies B1, B2 and B5, while sub-competencies B3 and B4 were not associated with experience. The summaries regarding the findings concerning the competency to solve mathematical problems through the mathematical model thus developed (C) and the conclusions reached with respect to associated sub-competencies are presented in Table 16. Table 16 Development of sub-competencies associated with competency C, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature C1. Competency to use heuristic strategies I SDE - C2. Competency to use mathematical knowledge to solve the problem I SDE - Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature C1. Competency to use heuristic strategies I SDE - C2. Competency to use mathematical knowledge to solve the problem I SDE - Table 16 Development of sub-competencies associated with competency C, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature C1. Competency to use heuristic strategies I SDE - C2. Competency to use mathematical knowledge to solve the problem I SDE - Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature C1. Competency to use heuristic strategies I SDE - C2. Competency to use mathematical knowledge to solve the problem I SDE - The C1 and C2 sub-competencies of student mathematics teachers who were not involved with the learning environment, compared against the development registered by the student mathematics teachers who were involved, revealed that the latter group achieved higher competency levels in practices concerning such sub-competencies. This leads to the argument that C1 and C2 sub-competencies have been developed based on experience concerning mathematical modelling, while the learning environment design influences the development of C1 and C2 sub-competencies albeit not at the desired level. The studies in the literature emphasize, in parallel to this finding, that mathematical knowledge influences mathematical modelling competencies, even if the sub-competencies are not discussed specifically (Eraslan, 2012; Grünewald, 2012; Maaß, 2006; Şen-Zeytun, 2013). The summaries regarding the findings concerning the competency to interpret mathematical results against real-life cases (D) and the conclusions reached with respect to associated sub-competencies are presented in Table 17. Table 17 Development of sub-competencies associated with competency D, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature D1. Competency to interpret mathematical results in extra-mathematical contexts I SDE Tekin-Dede & Yilmaz (2013); Blum (2011) D2. Competency to generalize solutions that were developed for a special situation I SDE - Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature D1. Competency to interpret mathematical results in extra-mathematical contexts I SDE Tekin-Dede & Yilmaz (2013); Blum (2011) D2. Competency to generalize solutions that were developed for a special situation I SDE - Table 17 Development of sub-competencies associated with competency D, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature D1. Competency to interpret mathematical results in extra-mathematical contexts I SDE Tekin-Dede & Yilmaz (2013); Blum (2011) D2. Competency to generalize solutions that were developed for a special situation I SDE - Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature D1. Competency to interpret mathematical results in extra-mathematical contexts I SDE Tekin-Dede & Yilmaz (2013); Blum (2011) D2. Competency to generalize solutions that were developed for a special situation I SDE - The D1 sub-competency of student mathematics teachers who were not involved with the learning environment, compared against the development registered by the student mathematics teachers who were involved, revealed that the latter group achieved higher competency levels in practices concerning that sub-competency. Therefore, the development of the D1 sub-competency based on experience concerning mathematical modelling, albeit not at the desired level, suggests that the learning environment design influences the development of D1 sub-competency. In parallel to this finding, there are also studies leading to the conclusion that the experience regarding mathematical modelling is influential in terms of understanding the real-life impact of the results (Blum, 2011). Yet, there are findings to the contrary as well. There are indeed studies which note the difficulties students may have in the interpretation of mathematical results against the real world, despite experience regarding mathematical modelling (Bukova-Güzel, 2011; Blum, 2011; Maaß, 2006; Şen-Zeytun, 2013). Zbiek & Conner (2006) regard low levels of interpretation competency as an act of subconscious and may be caused by lack of clear statements at all times. Tekin-Dede & Yilmaz (2013), on the other hand, argue that making student mathematics teacher’s work on MEAs by following mathematical modelling cycle through would lead to the validation competency. This study, however, reveals that experience gained in the order of hierarchy from structured to non-structured through guidelines taking sub-competencies concerning mathematical modelling process into account, without actually providing detailed theoretical insights into mathematical modelling, would contribute to the improvements in D1 sub-competency, even if not at the desired scale. Table 18 Development of sub-competencies associated with competency E, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature E1. Competency to reflect on found solutions I SDE - E2. Competency to critically check on found solutions I SDE Tekin-Dede & Yilmaz (2013) E3. Competency to review some parts of the model or again go through the modelling process if the solutions do not fit the situation NDI resistant to development NDDE - E4. Competency to reflect on other ways of solving the problem or if solutions can be developed differently NDI resistant to development NDDE Şen-Zeytun (2013) Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature E1. Competency to reflect on found solutions I SDE - E2. Competency to critically check on found solutions I SDE Tekin-Dede & Yilmaz (2013) E3. Competency to review some parts of the model or again go through the modelling process if the solutions do not fit the situation NDI resistant to development NDDE - E4. Competency to reflect on other ways of solving the problem or if solutions can be developed differently NDI resistant to development NDDE Şen-Zeytun (2013) Table 18 Development of sub-competencies associated with competency E, in the LES-MMC process Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature E1. Competency to reflect on found solutions I SDE - E2. Competency to critically check on found solutions I SDE Tekin-Dede & Yilmaz (2013) E3. Competency to review some parts of the model or again go through the modelling process if the solutions do not fit the situation NDI resistant to development NDDE - E4. Competency to reflect on other ways of solving the problem or if solutions can be developed differently NDI resistant to development NDDE Şen-Zeytun (2013) Sub-competency Change through the process Comparison of the participants in the learning environment and those who are not involved Comparable results in the literature E1. Competency to reflect on found solutions I SDE - E2. Competency to critically check on found solutions I SDE Tekin-Dede & Yilmaz (2013) E3. Competency to review some parts of the model or again go through the modelling process if the solutions do not fit the situation NDI resistant to development NDDE - E4. Competency to reflect on other ways of solving the problem or if solutions can be developed differently NDI resistant to development NDDE Şen-Zeytun (2013) The summaries regarding the findings concerning the competency to verify the solution (E) and the conclusions reached with respect to associated sub-competencies are presented in Table 18. A comparison of E1 sub-competencies of student mathematics teachers who are involved with the learning environment against those of the student mathematics teachers not involved suggests that the first group generally have more competencies in their endeavours in this front. This suggests that the learning environment thus designed is influential in the development of the E1 sub-competency. In the same vein, the development of the E2 sub-competency based on experience concerning mathematical modelling, albeit not at the desired level, suggests that the learning environment design influences the development of E2 sub-competency. Even though sub-competencies of the validate the solutioncompetency are not discussed separately, the literature is rich in studies suggesting that the students do not see a reason to verify their solutions confident in their accuracy and that they do not check for calculation errors and the accuracy of solutions regarding the state of the problem (e.g., Blum & Borromeo-Ferri, 2009; Maaß, 2006; Şen-Zeytun, 2013). Şen-Zeytun (2013) and Blum & Borromeo-Ferri (2009) note that this is caused by the students’ conviction that the validation of the solution is actually the job of the instructor. Even though emphasis on the review of the modelling process and the development of different solutions has not been lacking where the solutions are not consistent with the case in hand in the learning environment, the student mathematics teachers’ E3 and E4 sub-competencies were found to be extremely resistant to development. The comparison of the E3 and E4 sub-competencies of student mathematics teachers who were involved with the learning environment against those of the control group revealed that these sub-competencies were either non-existent or were weak and far in between in both groups. Therefore, one can argue that E3 and E4 sub-competencies are not directly associated with the experience for mathematical modelling and that the learning environment design does not contribute directly to the development of these sub-competencies. The literature, on the other hand, contains some studies suggesting that student mathematics teachers with mathematical modelling experience question the model and go through it once again where necessary (Tekin-Dede & Yilmaz, 2013). As the modelling cycle contains the step of ‘validating’ as well, going through the modelling cycle deemed a guideline may have led student mathematics teachers to carrying out studies concerning E3 sub-competency. Therefore, the E3 sub-competency applied by going through the guiding cycle does not necessitate the comparable application when working with free MEAs. Even though certain activities entail guidelines directing student mathematics teachers to envisaging different solutions, the student mathematics teachers chose to skip the guideline altogether, rather than responding to it. The student mathematics teachers consider a single route to solution sufficient and do not bother to seek different routes. This fact, on the other hand, indicates that the importance attached to mathematical modelling process influences the E4 competency. The literature is not devoid of studies which have reached to similar conclusions that different routes to solutions were not sought after when working on MEAs (Şen-Zeytun, 2013). 4. Conclusion and implications While some of the literature emphasizes supra-cognitive and sensory mathematical modelling competencies, we focused solely on cognitive mathematical modelling competencies as evidenced in the actions, speech and work of our participants. Our findings depend on comparing groups whose educational background was similar, rather than assessing competences prior to the LES-MMC. We have used qualitative methods that have enabled us to identify details and differences in performance between those who have and have not undergone the LES-MMC. The LES-MMC process appears to have influenced positively the development of sub-competency A1, that is the making of assumptions about the problem situation. By and large the student mathematics teachers had made progress with respect to most sub-competencies; they were more able to simplify the problem, identify variables and relate them, decide on appropriate information for the problem, mathematize quantities and relations between them, use appropriate representations, apply heuristics and mathematical knowledge and interpret and check findings. However, context appeared to sometimes lead to the identification of inappropriate variables that did not affect the situation mathematically, arising from mental imagery about similar situations, thus blocking the development of accurate representations. Furthermore, those students who were not involved with the learning environment were found to have done better regarding the associated sub-competency A3, identification of influential quantities, possibly because they had not been immersed in situations they could imagine from general knowledge. Sub-competencies associated with reduction and simplification of variables and relations, B3 and B4, and those associated with reviewing and developing the model, E3 and E4, did not develop in the study, possibly because the student mathematics teachers did not perceive a need to do this, since their aim was to produce a model, not to use it in reality. Our assessment of mathematical modelling through this holistic approach revealed that the development of the competencies is not linear. Therefore, a holistic approach is more advisable than written tests to assess such competencies. Assessment based on written documents alone would not suffice to show a true picture of mathematical modelling competency. It would be advisable for future studies to develop assessment approaches that make it easier to assess mathematical modelling competencies throughout the process. As the present study discussed sub-competencies on an individual basis, it has been possible to identify some resistant ones. It would be advisable for future studies to design learning environments at a similar micro level, providing more experience regarding those sub-competencies which had been influenced negatively in this study and those which had developed at a level less than expected or which appeared resistant to development. Acknowledgements This study was produced from PhD thesis titled ‘Examining mathematical modelling competencies of teacher candidates in learning environments designed to improve mathematical modelling competencies’. Funda Aydin-Güç is an Assistant Professor at the Mathematics and Science Education Department in Giresun University, Giresun, Turkey. She received a PhD in Mathematics Education from Institute of Education of Karadeniz Technical University in 2015. Her research interests are teaching and learning mathematics. Adnan Baki is a Professor at the Mathematics and Science Education Department in Karadeniz Technical University, Trabzon, Turkey. He was awarded the degree of MD in Mathematics Education in 1989 from the University of New Brunswick in Canada. He received a PhD from the Institute of Education of the University of London in 1994. He publishes papers on teaching and learning mathematics. References Aydin-Güç , F. ( 2015 ) Examining mathematical modelling competencies of teacher candidates in learning environments designed to improve mathematical modelling competencies . Ph.D. Thesis , Karadeniz Technical University , Trabzon . Blomhøj , M. ( 2007 ) Developing mathematical modelling competency through problem based project work - experiences from Roskilde University . Philosophy and Science Teaching Conference . [Avaible online at: http://www.ucalgary.ca/ihpst07/proceedings/ihpst07%20papers/125%20blomhoj.pdf], retrieved on October, 05, 2011, pp. 1–6. Blomhøj , M. & Kjeldsen , T. H. 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For permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com

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Teaching Mathematics and Its Applications: International Journal of the IMAOxford University Press

Published: Mar 26, 2018

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