The Compatibility of Model Eliciting Activities of Secondary School Teacher Candidates with Design Principles

The purpose of this study is the investigation of the compatibility of model eliciting activities of secondary school teacher candidates with design principles. This study was conducted in the scope of Mathematical Modelling course with the students who were the secondary school mathematics teacher candidates. The participants of this case study were thirty-nine mathematics teacher candidates who worked in eight groups. The data of this study consisted of eight model eliciting activities which were created within the eight groups and their analysis. The activities created by the groups were analyzed by document analysis method in terms of design principles that were defined for model eliciting activities. It was concluded that the created model eliciting activities satisfied the construct share ability and reusability principle at minimum while they satisfied the reality principle at maximum. The effective prototype principle could not be determined. It can be ensured that the secondary school mathematics teacher candidates gain more experience by making more implementations related to model eliciting activities. The implementation of model eliciting activities in class can be effective in reducing the modelling deficiencies of secondary school mathematics teacher candidates.

interdisciplinary relationships and denoted that students develop skills such as reading comprehension, communicating with peers and explaining their solutions and writing them. Lesh, Hoover, Hole, Kelly and Post (2000) state that the use of MEAs is effective in teaching and evaluation process as well as their use to reveal students' thoughts for research purposes. It is expressed that when teachers observe their students while working on MEAs and examine the solutions they produce, they can have an idea about their students' conceptual strengths and weaknesses, and they can make their teaching more effective (Lesh et al., 2000). Mousoulides, Christou and Sriraman (2008) state that MEA implementations contribute to students' mathematical literacy, conceptual understanding, social development and metacognition and to teachers' development of pedagogical approaches and teaching practices. From this point of view, MEAs are important tools that can be used for successful mathematics teaching (Tekin-Dede & Bukova-Güzel, 2014). It is believed that knowing this important tool, which can be used in mathematics teaching, by both mathematics teachers and mathematics teacher candidates, is very significant to find their ways of integrating it into the teaching process, and to develop themselves in designing different MEAs to use in their lessons.
The studies that were done related with MEAs can be listed as follow;  (2012), it was stated that the teachers cared for the reality principle at maximum while they paid the least attention to the prototype principle in the MEAs they designed. Tekin-Dede and Bukova-Güzel (2013) studied the MEA design process named "Obesity Problem" which was created by 4 mathematics teachers and its compatibility with MEA design principles. These MEAs were found completely appropriate for the reality, model reconstruction, and the construct documentation and construct share ability and reusability principle, and were only in compliance with self-assessment principle at some extent. It was determined that the MEAs did not satisfy the prototype principle. Deniz and Akgün (2016) investigated whether the secondary school mathematics teachers created activities compatible with model eliciting principles. It was concluded that all MEAs were totally appropriate in the reality and the construct share ability and reusability principles while they were only partially appropriate in self-assessment principle and the compatibility with the effective prototype principle was not investigated. Tekin-Dede, Hıdıroğlu and Bukova-Güzel (2017) analyzed the MEAs created by mathematics teachers in terms of MEA principles. Accordingly, the purpose of this study is the investigation of MEAs prepared by secondary mathematics teacher candidates with regards to MEA design principles.

Theoretical Framework
Model eliciting activities (MEA) which was first defined by Lesh et al. (2000) are stated as problem solving activities from real life that require to form a mathematical model (Lesh & Yoon, 2004). Beyond the representation of problem scenarios from real life in MEAs, it is required to develop a model that can be generalized by students in different contexts (Lesh & Harel, 2003). In addition, MEAs are used as research tools that aim to reveal the thoughts of teachers and students for the solution during implementation (Lesh et al., 2000).
At this point, it is necessary to seek for an answer to the questions of how to decide whether an activity is an MEA or what a teacher should consider if he/she wants to design his/her own activity. According to Lesh et al. (2000), teachers or researchers should take into account of the six principles that are shown in Figure 1 while they are creating their own activities or understand whether an activity is an MEA: The reality principle, which is the first of MEA design principles, the problem situation needs to be a situation that students might encounter in real life (Bukova-Güzel et al., 2016). The most precise way to determine whether an MEA satisfy this principle is to try to reply the question "Can a student come across with such a situation in his/her real life?" (Lesh et al., 2000). In MEAs, students make to develop a model by asking them to help a client or a customer, thus, they are expected to intuit that they are occupied with a real problem.
Model construction principle is that the problem situation requires to construct a model (Bukova-Güzel et al., 2016). Because of this principle, students are expected to create a model in order to reach a solution for a problem (Chamberlin & Moon, 2005). For the investigation of the presence of this principle, the questions of "Does the given situation require the students to create a model?" or is just answering a situation developed by others enough?" have been asked (Lesh et al., 2000).
In the self-assessment principle, the students are expected to evaluate the suitability and practicality of solutions by themselves without the support or the consent of their teacher (Bukova-Güzel et al., 2016). Therefore, the purpose of a problem that satisfies the self-assessment principle has to be clear and suitable for students' level (Chamberlin & Moon, 2005). In order to reveal whether an MEA satisfies the self-assessment principle or not, the questions of "Can students evaluate themselves when the answers are needed to be improved?", "Will the students realize that they finalize the solution of the problem or will they ask to their teachers if they need to continue to the solution?" has to be answered (Lesh et al., 2000).
In the construct documentation principle, students are required to use as much clear expressions as possible and explain their thoughts with details because they will create model/models for the purpose of helping a client or customer (Bukova-Güzel et al., 2016).
While students present their thoughts and solutions, they should document them as the people who encounter the problem can understand (Chamberlin & Moon, 2005). The survey on the presence of construct documentation principle is provided by answering the question of "Does students' answers given to the problem situation display how they think about this situation clearly?" (Lesh et al., 2000).
One of the MEA design principles is the construct share ability and reusability principle. In this principle, the purpose is not only using models created by the students for a specific situation and purpose but also using them for different situations and purposes at

Method
This study was designed in the case study, which is one of the qualitative research methods. Yin (1984) defines case study as a research method that is used when: 1) the research is focused on the "how" and "why" questions, 2) the researcher has little or no control over events, 3) the event or phenomenon is studied within its own natural environment, and 4) the connection between the event and real life is not clear enough.

Participants
The participants were selected with respect to convenience sampling method, which is one of the purposive sampling methods. Convenience sampling is to select the close and easily accessible group of participants in accordance with the purpose of the research.
Convenience sampling comparatively costs less and can be perceived as practical and easy.
This study was conducted with the 4th grade secondary school mathematics teacher candidates within the elementary school mathematics teacher department in a state university in Agean Region, Turkey. The participants were 28 females and 10 males who were registered to Mathematical Modelling course. The teacher candidates were divided into 8 groups with 4-6 people.

Data Collection
The data of the research consist of MEAs developed by secondary school mathematics teacher candidates. The teacher candidates designed an MEA as a group in the last three weeks of the Mathematical Modelling course, performed the solution and reported it. Accordingly, the data of the research were created from documents containing eight

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MEAs designed by the groups and their solution reports. The 14-weeks mathematical modelling lesson where the application took place was planned as indicated in Figure 2. Tekin-Dede, 2015) were studied within the groups. The obtained results within groups were discussed in classroom with other groups. In the 12th week, the participants were asked to create a MEA that could be a solution to a problem in the environment in order to establish a relationship with daily life and create awareness. The feedback was given by the researcher about the activity they created within their groups and the creation process of the MEAs was completed. In the 13 th week, the groups were asked to solve the MEAs they created. In the 14 th week, the designed MEAs and their solutions were reported.

Data Analysis
In the light of the theoretical framework, the MEA, which is the data collection tool of the research, has been tried to be revealed by document analysis in which extent it satisfies the modelling design principles. Çepni (2007) Yıldırım & Şimşek, 2008). In these evaluations, the three categories, which were created by Tekin-Dede et al. (2017), were taken as reference to examine the MEA design principles (See Table 1). In this context, the compatibility of aforementioned principle with each principle was examined in the categories of "totally appropriate", "partially appropriate" and "inappropriate". Including statements about the students' remembering of the problem statement and constructed models Including statements about the students' remembering of the problem statement and constructed models to some extent Not including statements about the students' remembering of the problem statement and constructed models In order to ensure the reliability of the analysis of the data, the data was analyzed separately by the researcher and an academic working as a faculty member in mathematics education. The MEAs compatible and incompatible aspects with design parameters were determined by the analysis of MEAs compliance to which categories and to what extent were stated. After this statement, the reliability calculation method of Miles and Huberman (1994) was utilized. The reliability of this study was calculated with the help of reliability formula given below, and determined to be 80%. When the reliability calculations reach over 70%, the study is accepted as reliable by Miles and Huberman (1994).