As in all over the world, developments and changes in science and technology are also continuing in an increa-sing way in our country, which naturally affects education programmes. As a result of the new primary school pro-gramme implemented in 2005, new applications in mathematics teaching programme were introduced. Changing duties of teachers and students, changes in educational contexts, variations in the approaches to assessment and evaluation are some of them. In the new programme one of the most important things that draws attention is the fact that for the first time that the terms mathematical model and modelling have been handled so comprehensively (MEB, 2005). In the new programme modelling has become one of the basic elements since mathematical modelling finds its way in teaching programmes of many countries as a result of the improvements in mathematics teaching (Güzel and U?urel, 2010).

In recent years, the importance of mathematical modelling has been emphasized by NCTM (2001) and many mathematics teachers (Kertil, 2008). Nowadays mathematical modelling is not only used in the area of mathematics but also in technology, architecture, economy, engineering, medicine and many others. In order to keep up with the rapid changes in society individuals who are at peace with technology, have creative thinking ability and can make mathematical model are needed (Lingefjard, 2002). Mathematical modelling has become a necessity for those who have their education in these areas. The fact that mathematical modelling is used in different areas accounts for its importance.

Studies on mathematical model and modelling receive increasing attention (Blum and Ferri, 2009). There are limited number of studies on the terms model and modelling in our country (Erarslan, 2011). In his study, Keskin (2008) made use of mathematical modelling questionnaire and aptitude tests in order to have knowledge of opinions and skills of third grade preservice mathematics teachers. Ayd?n (2008) carried out a qualitative study as to the opinions of teachers and students in England about the use of modelling of movable objects in their courses. In another qualitative study it was researched on how problem solving skills of mathematics preservice teachers emerged in the process of mathematical modelling (Kertil, 2008). Güzel and U?urel (2010) observed the relationship between mathematics preservice teachers’ success in Analysis I and their opinions about mathematical modelling. Finally, Erarslan (2011) observed opinions of primary school mathematics preservice teachers about teachers’ efficiency in preparing models and contribution of these models into mathematic education.

Problem status

In this study, literature was scanned in detail and research question was identified as ‘what are the opinions of secondary school mathematics teachers about mathematical modelling?’ Sub-problems in the process of data collection and analysis are as follows:

1. What is knowledge level of secondary school mathematics teachers as to mathematical modelling?

2. What are the opinions of secondary school mathematics teachers as to the appearance of mathematical modelling in secondary school mathematics programme

3. How do attitudes of students towards the course in which mathematical modelling is used change?

4. What are the opinions of secondary school mathematics teachers about their mathematical modelling skills?

5. What are the opinions of secondary school mathematics teachers about the contribution of departmental courses to mathematical modelling.

In this study opinions of secondary school mathematics teachers as to the mathematical modelling were analyzed. In the light of the related literature it was seen that there were not many studies in this area which accounts for the importance of this study. It is thought that this study will present significant data for prospective studies and make important contributions to mathematics education. 

Model of the study

In educational research there exists qualitative and quantitative methods in terms of data collection and analysis and which one to use is the most important element in identifying the nature of the study.

In this study interview technique as a method of qualitative research was used in order to identify the opinions of secondary school mathematics teachers about mathematical modelling.

Study group

The participants of the study were 40 secondary school mathematics teachers working in the Bingöl Province in Turkey 2012-2013 education year. Statistical data about the participants are presented in Table 1. 

When Table 1 is observed, it is seen that 40% of the participants are females and 60% of them are males. All of the participants are graduates of education faculty. 65% of the participants have 1-5 years of working experience, 27.5% have 6-10 years, 5% have 11-15 years and 2.5% have more than 16 years of work experience.

Data Collection Tools

In this study semi structured interview form was used for data collection. Before the preparation of the interview form a comprehensive literature scanning about modelling and mathematical modelling was carried out by the researcher. In the light of the related literature firstly an interview form consisting of 12 questions was prepared by the researcher. Before the implementation of the form pilot study was carried out. In the pilot study 12 secondary school mathematics teachers who worked in 3 different schools in Bingöl were selected and the interview form was implemented. The form was made to be examined by 4 expert educators, as a result of which questions were taken out of the form in order to provide its content validity. As it is known in qualitative studies internal validity is related to whether the researcher can really measure the data with the tool or method that he uses in order to measure it (Y?ld?r?m and ?im?ek, 2004).

Data analysis

In this study data analysis consists of two phases in the first of which descriptive analysis was carried out in order to evaluate the data in terms of problem status (Y?ld?r?m and ?im?ek, 2011). In the second phase content analysis method was utilized. Content analysis is described as a systematic and replicable technique in which some words of a text are summarized with smaller content categories by means of particular codifications (Büyüköztürk, 2008). Categories were specified by making codification of the raw data categorized under these categories in order to make it meaningful for the reader. The processes of codification and categrorization were carried out repeatedly by the researcher. Therefore un-necessary codifications were removed and new codifications were added when necessary. As a result tables in which ideas of each participant could be seen separately were attained. 

In this section findings are presented in tables and column charts were used in order to show numerical value of each answer and to facilitate visual elements. In addition some of the answers of the teachers were presented after scanning.

Question 1 and its findings are as follows.

The answers of the participants are presented in Table 2.

The answers of the participants are presented in Table 3. When Table 3 is observed it can be seen that the answers of the participants have been categorized under 4 main titles which are easy, time consuming, depends on the subject and difficult. According to Table 34 30% of teachers think that creating mathematical modelling is easy, 20% of them think it is time consuming, 32.5% of them think it depends on the subject and finally 17.5% of them think it is difficult.

Question 4 and its findings are as follows:

The answers of the participants are presented in Table 4. 

When Table 4 is observed it can be seen that the answers have been categorized under 5 main titles. These are fractions, counters, algebraic statements, identities, pattern and decoration According to Table 4, 37.5% of teachers use mathematical modelling for fractions, 20% of them for counters, 12.5% of them for algebraic statements and 10% of the teachers use mathematical modelling for pattern and decoration.

Question 5 and its findings are as follows:

The answers of the participants are presented in Table 5.

When Table 5 is observed it can be seen that the answers have been categorized under 3 main titles which are change negatively, not change, change positively. According to Table 5, 75% of teachers believe that there will be positive changes in attitudes of students towards the course in which mathematical modelling is used. While 15% of teachers believe that there is no change regarding the attitudes of students, 10% of teachers think the change will be negative.   

Question 6 and its findings are as follows:

The answers of the participants are presented in Table 6.

When Table 6 is observed it can be seen that the answers have been categorized under 3 main titles. These are 30-40 min; 50-60 min and depend on the subject. According to Table 6, 12.5% of teachers spend 30-40 min for mathematical modelling, 47.5% of teachers spend 50-60 min and according to 40% of teachers time spent for mathematical modelling depends on the subject. In addition the results are presented graphically below. The following graphic shows the numerical value of the answers of teachers about time necessary for mathematical modelling. 

Question 7 and its findings are as follows:

The answers of the participants are presented in Table 7.

When Table 7 is observed it can be seen that the answers have been categorized under 3 main titles which are not beneficial, slightly beneficial and quite beneficial. According to Table 7, 35% of teachers think the courses they took at university are not beneficial for mathematical modelling. 10% of teachers think they provide slight benefit for mathematical modelling and 55% of teachers think they are quite beneficial for mathematical modelling.

Question 8 and its findings are as follows:

The answers of the participants are presented in Table 8.

When Table 8 is observed it can be seen that the answers have been categorized under 3 main titles. These are should be included, should be included for particular subjects, should definitely be included. According to Table 8 35% of teachers think mathematical modelling should be included in the program, 15% of teachers think it should be included for only particular subjects while 50% of teachers think it should definitely be included in the program.

In this section the results have been presented under titles and these results have been compared with the related literature. The titles have been selected based on the problem situations of the study.

Conclusions as to the knowledge level of mathematics teachers about mathematical modelling

This study in addition to presenting the ideas of secondary school mathematics teachers’ opinions on mathematical modelling, aims at showing how often mathematical modelling is used by teachers. For this purpose interviews with 40 teachers in Bingöl were carried out. Findings were analyzed in accordance with the sub problems of the study.

When opinions of secondary school mathematics teachers about the definition of mathematical modelling were observed, it was seen that the terms concretization of mathematical statements, attempts to use materials, visualization of mathematical statements, exemplification with figures and diagrams were frequently used. While Niss (1999) defines mathematical modelling as the integration of one or more than one mathematical formation selected for the aim of representing real life situations and the relationship between them, Galbraith and Catworthy (1990) define it as the application of mathematics into the problems not structured in real life. Verschaffel et al. (2002) define mathematical modelling in the most general sense as the process of interpreting events and relationship between these events mathematically and to create mathematical patterns among these events. When the results are compared with the related literature, it can be said that the participants could not make a complete correct definition of the term ‘mathematical modelling’. Keskin (2008), in his study on the improvement of preservice mathematics teachers’ skills of making mathematical model, concluded that preservice teachers could not make the complete definition of mathematical modelling, which is in parallel with the result of this study. It was seen that the examples preservice teachers gave for mathematical modelling generally consisted of the subjects of the course and there were few examples from daily life.

Results of the opinions of teachers about the involvement of mathematical modelling in the programme

When teachers’ opinions about the involvement of mathematical modelling in secondary school mathematics programme were observed, it was concluded that it should be included in the programme. Teachers’ providing students with variety of presentations will help students learn algebra and to understand mathematics by making associations with real life. In teaching contexts suitable methods and techniques should be used based on interests, needs and individual differences of students.

For that it will be useful to know learning styles of students and to use different materials in the learning process (MEB, 2005). Thus it will help to improve problem solving skills of students. In Messmer’s (1989), Hermann and Hirsberg’s (1989) studies they expressed that mathematical modelling should be included in teaching programme. Maab (2007) stated mathematical modelling should be included in teaching programme of secondary schools and Anker (1989) stated that it should be included in primary school mathematics teaching programme.

These results of the literature overlap with the result of this study. In addition some of the teachers expressed that mathematical applications course at 5^th grade could include mathematical modelling applications.

Teachers’ opinions about attitudes of students towards the course in which mathematical modelling is used

When opinions of teachers were observed about the attitudes of students, it was concluded that attitudes of students towards the course changed positively, which also meant that their interest in the course increased. Yu and Chang (2009) stated that students’ listening to teacher lecture for a long time and doing only what was wanted could hinder their creative thinking ability. In mathematics courses mostly students memorize formulas, about which they do not have any idea, and make calculations. Therefore teaching methods should be revised so that students can improve their abilities of thinking, expressing and interpreting. In that context model setting activities which enable high level thinking can be helpful (Eraslan, 2011). Lesh and Doerr (2003) stated that students could define, explain, interpret real life problems, create different solutions, and produce designs by means of model making activities. The importance of making use of different representational systems and making switches between these systems in the process of problem solving and teaching and learning of mathematical terms was emphasized by many researchers like Kaput (1987) and Goldin (1998). The result of this study is in parallel with the results in the literature.

Results of mathematics teachers’ making mathematical model

As a result of the interviews with the 40 participants of the study, it was concluded that difficulty level of making a mathematical model for mathematics teachers was subject dependant, changing from topic to topic. As Mayer (1998) states what preservice teachers should be careful with while making models is to know when and how to use appropriate terms and procedures. In their studies Bozkurt and Polat (2011) carried out interviews with teachers about the effect of modelling with counters to the learning of whole numbers and concluded that teachers did not lean on modelling with counters. Harman and Ak?n (2008) investigated the effect of teaching Pascal’s triangle and some identities by using perfect cube model and concluded that teaching by making use of perfect cube model had a positive effect on students’ success. In his study Eraslan (2011) concluded that when model making activities were planned with particular limits, they could be applied at all levels and would contribute to mathematical development of children. When secondary school mathematics teacher’s preferences for the subjects to use mathematical modelling are observed it can be seen that they mostly tend to use it in fractions and counters. When Primary School Mathematics Teaching Programme (2008) is observed it is seen that there exists a number of modelling examples for fractions, based on which it can be said that teachers make use of examples that overlap with the ones in the curriculum. As for the opinions of teachers about the duration of course for mathematical modelling, it is stated that it depends on the subject. Teachers expressed the main reason of not being able to use mathematical modelling in their courses as time limitations.

Teachers’ ideas on the contribution of the courses they took at university to the process of making mathematical model

When secondary school mathematics teachers’ ideas on the contribution of the courses they took at university to the process of making mathematical model are observed it is seen that there is a high consensus among teachers about courses like instructional technologies and material development, computer programming, computer assisted mathematics teaching, teaching of dynamic geometry, special teaching methods contributed a lot to the making of mathematical model. Zambujo (1989) and Rose (1974) stated that most of the courses taken at university were necessary for mathematical modelling. Hermann and Hirsberg (1989) stated that examples of mathematical modelling on differential equations should be included in the curriculum. Moreover some of secondary school mathematics teachers expressed that the courses they took at university had no contribution to mathematical modelling. Macgregory and Stacey (1997) stated that the misconceptions of students about algebraic symbols were not only related to cognitive development but also to several environmental factors and teaching methods and techniques. (Hubbard, 2003, p:1). Kertil (2008) as a result of his study emphasized that university curriculum should include courses about mathematical modelling activities. The literature results presented above supports the result of this study. 

The authors have not declared any conflict of interests.