How do Iranian Graduate Students Learn to Teach Collegiate Mathematics as Future Mathematics Professors?

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1 12 th International Congress on Mathematical Education Program Name XX-YY-zz (pp. abcde-fghij) 8 July 15 July, 2012, COEX, Seoul, Korea (This part is for LOC use only. Please do not change this part.) How do Iranian Graduate Students Learn to Teach Collegiate Mathematics as Future Mathematics Professors? Azimeh Sadat Khakbaz Shahid Beheshti University. Iran Azimeh_khakbaz@yahoo.ie Research about education of professors in the field of mathematics education is a topic, largely neglected in the current debate. The field of teacher education is more focused on school teacher education and there is not much known about how do professors learn to teach. This research aims to study how do Iranian graduate students learn to teach collegiate mathematics as they would be future mathematics professors in Iranian educational system. The theoretical framework is based on the approach of pedagogical content knowledge (PCK), the concept which is introduced about specific teaching knowledge. The research is done using qualitative research methods, the data are gathered through semi-structured interviews and other kinds of qualitative data recourses and analysed through Creswell s approach (2008) in making themes and layering them. Analysing data revealed the understanding of what is PCK in Iranian university mathematics education and how this can be developed through graduate (PhD) mathematics curriculum which is mainly responsible about education of mathematics professors. Key words: Education of mathematics professors, Pedagogical Content Knowledge, Belief Systems in Teaching, Professional Development of mathematics teachers, Graduate education INTRODUCTION Higher education is responsible for training future humans as educated citizens, and curriculum is the heart of higher education in performing this responsibility. Professors have the most crucial role as they are the curriculum implementers in higher education to achieve this aim. However, how do professors learn to be educators and do this important role? Higher education curriculum, which is training the future human, is also responsible for educating future professors too. Although, education of educators is one of the influential missions of higher education curriculum, it is most often ignored (Short, 2002). Teaching is a neglected subject in higher education as instructors at the university level are hired based on their expertise as researchers, not on their teaching credentials (Brogt, 2009). Ordinarily this part is supported through the graduate curricula, but they largely include research knowledge in a special field, and knowledge and skills of how to teach people are excluded (Short, 2002). They often do not have formal preparation in teaching and usually go abcde

2 Khakbaz from their graduate work via a doctoral position into a faculty job. Instructors in higher education typically have content knowledge of the teaching subject at the Ph.D. level. They are experts and as such are likely to have an organization of knowledge that is very different than the organization of knowledge of a novice in the field, i.e. an undergraduate or even a non-major student (Brogt, 2009). This research has done in Iranian educational context. In this context graduate mathematics students in PhD level normally become mathematics teachers at university level after their graduation and they generally start teaching at undergraduate level, especially in non-mathematics major fields; for example engineering or humanities students who understand mathematics in different ways rather than a mathematics student. This research would explore what is going on in this debate in Iranian higher education context in mathematics field. Theoretical framework for this study was based on pedagogical content knowledge (PCK). Studies about education of educators would explore interpretation of professors teaching knowledge. In literature about teaching knowledge, one of the most prominent issues is what Shulman (1986, 1987) introduced as pedagogical content knowledge (PCK), the main part of teaching knowledge. PCK is described as the category most likely to distinguish the understanding of content specialist from that of pedagogue and does it by transforming their understanding of it into instruction that their students can comprehend. Although PCK is a suitable theoretical base to study about teaching knowledge, little is known about that in higher education field as Shulman introduced PCK in school curriculum and it is developed with school curriculum scholars (Lenze, 1994; Counts, 1999; Brogt, 2009). Therefore, it is needed to conceptualize PCK at first. This research would try to respond some 2 questions in this context: What is the concept of PCK in teaching mathematics in higher education? How do Iranian graduate students (PhD) develop PCK as the future mathematics teachers? THEORETICAL FRAMEWORK Teaching in higher education was under the shade of research because of the expectation of university is developing knowledge and it means doing research. But in 1990 with introduction of a concept, scholarship, by Ernest Boyer, teaching become more considerable. Boyer s main thesis does not focus on teaching in isolation, but on teaching as a part of the larger whole of academic work. Boyer argues that we should let go of the old research vs. teaching argument and focus on the idea that scholarship exists in all aspects of our academic work. He suggested there are four separate, but overlapping areas of scholarship. These are: (a) The scholarship of discovery: close to the old idea of research; (b) The scholarship of integration: this involves making connections across the disciplines and placing the specialties in larger context. (c) The scholarship of application: this goes beyond the application of research and develops a vital interaction and so informs the other; Abcde+3 ICME-12, 2012

3 Last names of authors in order as on the paper (d) The scholarship of teaching and learning: this educates future scholars by communicating the beauty and enlightenment at the heart of significant knowledge (Trigwell et.al, 2000). Rice (1992) proposed that the scholarship of teaching and learning (SoTL) is based on the development and application of its several distinct elements: synoptic capacity, what we know about learning and pedagogical content knowledge(pck). Synoptic capacity is similar to what Shulman (1987) has called content knowledge, scholarship in the content disciplines. The expression what we know about learning refers to an element of scholarship that is similar to what Shulman has called formal educational scholarship, especially about teaching and learning, and is one of the distinguishing characteristics of what Kreber and Cranton (1997) refer to as pedagogical knowledge, knowledge about how to best facilitate student learning. And finally, pedagogical content knowledge represented more succinctly, as knowledge about the interaction between learning process and academic content (Paulsen, 2001). Paulsen said: Pedagogical content knowledge can be viewed as the knowledge that stands at the nexus between content and pedagogical knowledge. Not merely the summation of the two parts, pedagogical content knowledge represents a special synthesis and unique relation of the two. Serving as the substantive and syntactical foundation for the scholarship of teaching, it provides the symbols, language, ideas, concepts, theories, metaphors, analogies, and other forms of knowledge representation, as well as the modes of inquiry, that constitute the knowledge base for effective teaching and learning. Therefore, in this view, PCK is studied as the main framework of SoTL which Garritz (2010) emphasized too. Although the importance of PCK in higher education teaching studies is considered, there are few studies in this field(fernandez Balboa & Stiehl, 1992; Lenze, 1995; Counts, 1999; Brogt, 2009); More literature is came from school education. PCK could be developed in higher education either, because of its focus on content which is relevant to higher education studies that are inevitably connected to the discipline as a crucial commonplace (Lenze, 1995). Therefore, studies about the nature of PCK development in higher education are necessary. Few studies which has done in this debate, asserted that Grossman s model is useful in studying PCK in higher education (Fernandez Balboa & Stiehl, 1992; Lenze, 1995; Counts, 1999; Brogt, 2009). Therefore we started by Grossman s model as a theoretical background. Grossman (1998) introduced two overarching categories: General and Specific PCK. These categories were later modified by Fernandez-Balboa & Stiehl (1995) to generic and specific. Specific PCK is particular to instruction of a specific subject or content area; and Generic PCK is common to instruction all content areas or subjects. Generic PCK refers to the fact that every discipline in higher education could have a concept of PCK. It means mathematics as a discipline has its PCK and surely this Generic PCK consist of Specific PCK, emerged from different subjects and content area. Grossman (1990) also described four categories embodied in PCK: ICME-12, 2012 abcde+2

4 Khakbaz 1- Knowledge about the purposes for teaching a given subject matter (subject matter issue); 2- Knowledge about the students understandings about the content (students issue); 3- Knowledge about the order in which subject matter should be presented (curricular issue); 4- Knowledge about the instructional strategies useful for teaching content (pedagogical issue). The approach of generic PCK and Grossman s model are the basis of the theoretical framework in this study. On the other hand as PCK is an academic concept for explanation of teaching as a practical experience, it has some limitations. Studies about PCK revealed that PCK does not consider about beliefs and values and it seems values are missing part of the missing paradigm (Gudmundsdottir, 1990). Shulman(1992) asserted that he forgot that important part and he referred to Fenstermacher(1992) who believed that when we talk about teaching, we should consider both manner and method. Therefore, in this paper PCK model of Grossman is based but with consideration that behind action part of PCK is an epistemological belief system. RESRACH METHOD This research has done through qualitative approach. Data were gathered through semi-structured interviews and participant observations mainly. Field notes and autobiographical writing and other documentations were used either. Data analysed through coding, finding themes and interpretation (Creswell, 2008). FINDINGS Answer to the first question in this research is the base for answering the second question which is the main question in this paper. Therefore it is presented first. For conceptualization of PCK in teaching mathematics at higher education, we found out three main themes: 1. Components 2. Basis Knowledge Behind 3. Development Components of PCK were those variables which respondents explained when they were talking about good teaching. This theme consists of four sub theme: Encouraging students interest Making connection between students knowledge and content Representing an integrated content Developing teaching and learning classroom environment Abcde+3 ICME-12, 2012

5 Last names of authors in order as on the paper One thing that every respondent explained was that a mathematics professor should encourage students to study the subject and he could do that through engaging with historical gap in mathematics or mathematics applications. Some respondents told that when students know why they are learning the mathematical subjects, they could do that with much more interests. They said that in mathematics classroom for mathematics students, they usually apply the first approach and when they teach mathematics as the service lesson, they usually use the second approach and try to focus on mathematical applications related to their major, for example engineering, Second variable was explained by respondents through two approaches too. They tried to make connections between students knowledge and mathematical content through giving intuition or focusing on the main idea behind a mathematics problem. The other thing was that they tried to represent an integrated content and for that, they reflect on the selection of content and making connections between different parts of content. The last variable has different sub-variables which I do not want to go to them in details: students engagement in activities, educational technology, body language, classroom management and writing, speaking, hearing and thinking in the class. These components explained the PCK for teaching mathematics in university from their point of view. On the other hand, PCK came from the integration of pedagogical knowledge and content knowledge. Therefore the basic knowledge for that, rooted in those two types of knowledge. For this reason we tried to find out the origins of those emerged components in pedagogical and mathematical knowledge. For mathematics knowledge, we found out that a mathematics professor should have a deep knowledge in the structure of mathematics as a discipline in two aspects: Substantial and Syntactical. Mathematics as a discipline has two structures: substantive structure in mathematics which refers to concepts, theorem, and formula and... in mathematics and syntactic structure in mathematics which refers to the methods of how the concepts of mathematics would be develops (Schwab, 1964). Substantial knowledge usually developed through mathematics curriculum, but syntactical knowledge is usually ignored. Knowledge about historical gap in mathematics, applications, intuition or big idea behind mathematical concepts usually is based on syntactical knowledge. For pedagogical knowledge, we found out that a mathematics professor should have a deep knowledge of pedagogy in two aspects: Theoretical and Practical. By theoretical it means theories from social sciences and psychology about education. By practical, it means practical arts which have two parts, practical which related to practical problem solving and eclecticism which is the art of working with theories (Schwab, 1969). Theories related to education usually are developed through training and development programs, but we could not make judgement about practical pars as easily as the theoretical part. ICME-12, 2012 abcde+2

6 Khakbaz The key theme in analyzing this part was development of PCK which is possible through deliberation. By deliberation here we mean that professors have different resources for developing PCK(like the exam sheets of students, the evaluation sheets of their teaching, feedback of students in class, talking with colleagues and.) but they themselves would reflect on their action, find a problem in action and try to find a solution for that and apply the solution and reflect on that. Just in this situation, external resources, like professional development opportunities would help them to develop PCK. This is a kind of problem solving which the problem is not a mathematical problem, but a mathematics education problem. I would like to go in details about deliberation in another place and now try to explain the answer of the second question: How do Iranian graduate students (PhD) develop PCK as the future mathematics teachers? Based on the answer of first research question, it means how do they learn deliberation? To answer this question, we studied PhD mathematics curriculum at two levels: Explicit, Implicit. At explicit level, graduate mathematics formal curriculum were studied and this study showed that there are 3 influential points in the history of mathematics graduate curriculum in Iran which were related to educating future mathematics professors about teaching knowledge, which can be seen at the following indicators: Establishment of Dr.Mosahab s Mathematics Institute. Establishment of Tarbiat Modares University which aimed to educate professors for every discipline. Teaching assistant (TA) program which mainly considered in Sharif University and Shahid Beheshti University. Dr.Mosahab s curriculum started 50 years ago in Iran. This curriculum was a master degree of mathematics but the aim of that was education of future mathematics university teachers. Dr.Mosahab selected the students after that they took part in a summer course and succeed in an exam. Those days Iran had few mathematicians and therefore Dr.Mosahab hired some professors from other countries to teach students mathematics. And in his own class, he trained students about how to teach mathematics. His approach was that he asked students to explain, prove a theorem or solve a problem and he noticed them about using the board, their voice, using Persian words and other things while they were engaging in explaining mathematics or solving problem. We named this approach: practical note. And the graduated students who are now working as mathematics professors in Iran told that they learned a lot from Dr. Mosahab s behaviour and personality about how to become a mathematics professor. He was so organized and had disciplines in work and now the professors try to follow his approach. We called this approach: practical modeling. Now this curriculum does not exist, because after Islamic revolution in Iran, the system of higher education became centralized and all the universities should follow a unique curriculum. Therefore, Mosahab s institution, now is a mathematics department in one of the universities, called Tarbiat Moallem University, has a curriculum in graduate levels(master and PhD) focused on mathematics and the aim of this institution not longer remain in education of future mathematics university teachers. Abcde+3 ICME-12, 2012

7 Last names of authors in order as on the paper After Islamic revolution, Tarbiat Modares University established with the aim of education of future university teachers. This university established because of shortage of professors in Iranian universities at that time. It had master degree at first and the difference with Dr.Mosahab s curriculum was that the master had thesis and this university educated people not only in mathematics but also in other majors. The curriculum was some general pedagogical courses which students should have passed parallel to their courses in their specific major and the thesis. After some years this university started PhD degrees. Now with development of higher education in Iran, the aim of this university is not focused on education of future professors as there are many graduated students in other universities who could be hired as faculty members without passing the pedagogical courses. Now these courses are not obligatory and if students in this university or even other universities pass these courses, they could obtain a teaching certification. The curriculum now is consist of general pedagogical knowledge which would be taught by professors in education department and a session of oral exam which would be done in the specific department considering the major of students. But interview with students who passed these courses showed that they do not really do this oral exam and their supervisor evaluate them in the defence session of their thesis which students are not satisfied with because they think their supervisors are not the right persons in teaching knowledge and they are not really evaluate their teaching knowledge. Therefore this curriculum is focused on general pedagogical knowledge and it is not focused on mathematics. Third influential point was Teaching Assistant (TA) program which is now considering in many universities in Iran. It started from Sharif University less than 10 years ago. In this program PhD students would become teaching assistant of a professor and they would teach in exercise classes or even as an independent teacher in other classes. They usually teach calculus to students in other majors in bachelor degrees. This curriculum also seems develop PCK in a practical field. Through this review in explicit curriculum, it seems PCK development is most probable in the first and last one as second one is focused on development of general pedagogical knowledge. In the next phase we focused on Dr.Mosahab s institution which is now mathematics department in Tarbiat Moallem University (TMU) and also in mathematics department of Shahid Beheshti University (SBU) as one of the universities that is considering TA program (and that was the university which I am studying). Interviews were done to know what is going on in those 2 universities about PCK development. Although we knew that Mosahab s curriculum is not exist anymore, we did this research because we found some symptoms of a culture which exists now and it is like a heritage of Dr.Mosahab s attitudes and analysing data revealed that too. When we asked mathematics professors to know what they do about educating their students to teach mathematics, they replied we try to do the best in our teaching and students would know through our teaching. This implies that as they learned through the behaviours of Dr.Mosahab, they try to be like him for their students. They said we try to behave in disciplines and organized and teach step by step in an organized way. They also said that they try to notice students about the way of presentation, writing, talking and other aspects of ICME-12, 2012 abcde+2

8 Khakbaz a mathematics teacher when the students are solving a problem on the board or presenting a paper for their classmate. They try to teach their students about writing in a meaningful way by correction the exercises and exam sheets through this lens. Therefore they are using practical notes and practical modelling in their teaching mathematics. When we asked PhD students that how they learn to teach mathematics, they exactly said their professors are like models from them and they try to learn their way. And they also referred to notices about presentation and writing and these aspects. So it seems that Dr.Mosahab s approach is still remains and although there is not an explicit curriculum about that, it exists as a cultural heritage. On the other hand, we knew that the special program does exist in SBU, TA program which is famous in higher education, especially in western countries and USA. Therefore we interviewed with mathematics professors and PhD students focused on this theme. Mathematics professors said that although there is a special program, but they did not think it would help students to learn how to teach because there is not a program to supervise them. It is like an opportunity to teach and they could learn just by experience; practical experience. There is no opportunity to reflect on their experience. They also said that PhD students usually applied to teach calculus to students in other majors, especially human sciences. Although the content which is teaching in these classes is easier, teaching to them is really complicated and needs experience. Through this program, the quality of teaching mathematics in higher education decreased. When PhD students applied in mathematics department, they usually are responsible with exercise classes. It is a good opportunity, but they are not in connection with professors. Therefore they are lonely in teaching. PhD students said that they are not connected to their professors and they usually prefer to communicate with their colleagues who are in the same situation. They think that professors may not pay attention to their problems in teaching because they might think these are not problems! So we focused on the relationship between PhD students and their professors in TA program. In SBU we found out a spectrum from close relationship to null relationship. In close relationship, we found out a supervisor who had a team of their PhD students and they had a weekly discussion. I cooperated in their weekly sessions and I found out that they engaged in a professional development community which directly influenced on PCK development. But there are also supervisors who would not like that their PhD students teach. We found some students who said I hide my teaching from my supervisor, how could I communicate with him about teaching? He does not like me to teach and he thinks PhD student should just do research. Through these findings, we found out although there is an explicit curriculum in SBU with this aim, but this curriculum does not receive to its goal in most times. It should be asserted that practical experience would be an opportunity to develop PCK but in a professional development community, it would be refined much more. Abcde+3 ICME-12, 2012

9 Last names of authors in order as on the paper CONCLUSION From the comparison of what is going on in these 2 universities we found out that practical modelling, practical notes and practical experience of teaching, would be opportunities for Iranian mathematics PhD students to develop PCK. But why these approaches would work? The reply is connected to what we said about PCK, which is really connected with belief system and could be develop through deliberation. These approaches, all have 2 important virtues: 1- They are all practical and not theoretical. 2- They are all make opportunities for participants to reflect on the real actions. Another result is the main role of professors, especially supervisors, on PCK development of future mathematics professors, because Iranian mathematics PhD students do look at their supervisors as practical models in teaching mathematics and they do expect their supervisors to supervise them about teaching knowledge through a learning community. Although at the beginning of this study we wanted to design a curriculum for PCK development in PhD mathematics professors, we found out the key of educating future mathematics professors in teaching knowledge is focusing on existing mathematics professors and show them how their role is really influential on the next generation. Because students would develop their beliefs in connection with their professors, especially their supervisors. It was interesting that there was a highly closed relation between the beliefs of PhD students as future mathematics professors, with their supervisors. Furthermore, it is essential to develop opportunities for communicating between professors and PhD students about teaching. It is not enough to satisfy just by providing opportunity to teach as TA program do, PhD students needs an opportunity to reflect on the teaching; this help them to learn deliberation as the syntactic structure of PCK and they would develop the substantial structure through changing beliefs. But what we have now is focusing on training and development of professors as machines without any beliefs and just convey different kinds of interesting methods of teaching. It means that, now we are walking in the surface and it is better to go in deep! REFERENCES Boyer, E. L. (1990). Scholarship reconsidered: Priorities of the professoriate. Princeton, NJ: The Carnegie Foundation for the Advancement of Teaching. Brogt, E. (2009). Pedagogical and curricular thinking of professional astronomers teaching the Hertzsprung- Russell diagram in introductory astronomy courses for non-science major. Unpublished PhD dissertation of the University of Arizona. Counts, M. C. A case study of a college physics professor's pedagogical content knowledge. PhD dissertation of the Gorgia State University. Creswell, J. W. (2008). Educational research. Pearson Education International. ICME-12, 2012 abcde+2

10 Khakbaz Fenstermacher, G. D. (1992). The concept of method and manner in teaching. In Oser, F. K., Dick, A. & Patry, J. Effective and responsible teaching. Jossey-Bass Publishers. Pp Fernandez-Balboa, J. & Stiehl, J. (1995). The generic of pedagogical content knowledge among college professors. Teaching and Teacher Education. 11(3) Garritz, A. (2010). Pedagogical content knowledge and the affective domain of Scholarship of Teaching and Learning. International Journal for the Scholarship of Teaching and Learning. 4(2). Grossman, P.L. (1998). A study in contrast: Sources of pedagogical content knowledge for secondary English. Unpublished PhD dissertation. Stanford University. Grossman, P.L. (1990).The making of a teacher: teacher knowledge & teacher education. New York: Teachers College Press. Gudmundsdottir, S. (1990). Values in pedagogical content knowledge. Journal of teacher education. 41(3). Pp Kreber, C., and Cranton, P. A. Teaching as scholarship: A model for instructional development. Issues and Inquiry in College Learning and Teaching, 1997, 19(2), Lenze, L.(1995). The pedagogical content knowledge of faculty relatively new to college teaching. PhD dissertation of the Graduate School, Illinois. Paulsen, M. B. (2001). The relationship between research and scholarship of teaching. New Directions for Teaching and Learning Rice, E. A. (1992) Toward a broader conception of scholarship: The American context. In T. G. Whiston, and R. L. Geiger (eds.), Research and Higher Education in the United Kingdom and the United States. Lancaster, England: Society for Research into Higher Education. Schwab, J.J. (1964). Structure of the disciplines: meanings and significances. In Ford, G. W. & Pugno, L. The structure of knowledge and the curriculum. Pp Schwab, J.J. (1969).The practical: A language for curriculum. School Review, Vol 76, No 1. Pp Shulman, L.S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2) Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, Shulman, L. S. (1992). Research on teaching: A historical and personal perspective. In Oser, F. K., Dick, A. & Patry, J. Effective and responsible teaching. Jossey-Bass Publishers. Pp Trigwell, K., ET. Al (2000). Scholarship of teaching: a model. Higher Education Research and Development. 19(2) Abcde+3 ICME-12, 2012