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The influence of the teacher's subject matter knowledge and beliefs on teaching practices: A case study of an Indonesian teacher teaching graph theory in Indonesia

Theresia Tirta Seputro
National Key Center of School Science and Mathematics
Curtin University of Technology
Networks or Graph Theory is a new optional mathematics topic in Indonesian secondary schools. This paper describes the influence of the teacher's subject matter knowledge and beliefs and the effect of constraints and opportunities provided by the social context of teaching on his instructional practices. Issues arising from this study for the mathematics teacher education program are discussed along with implications for the teaching and learning of the Graph Theory topic.


Graph theory provides an excellent link with a wide range of topics in mathematics and computer science. Since 1996, graph theory has been taught as one of the optional mathematics topics in the Year 11 of the Indonesian secondary school system (Depdikbud, 1994). It is now clear that, with the inclusion of this topic in the school mathematics curriculum, there is a need for secondary school teachers to teach the topic well.

Recently, concerns were expressed about the teaching of the Graph Theory topic in Indonesian schools. There were indications of a lack of textbooks and inadequacies in teachers' mastery of the topic. In spite of the shortcomings, there were also indications of teachers' efforts to overcome these in their preparation for teaching the topic.

Canon & Sri Widodo (1994) indicated that there are problems in teaching and learning in Indonesian universities. Among these were an emphasis on oral methods, low quality libraries, and inadequate levels of equipment and materials for teaching. It appears that Indonesian secondary schools share similar problems to the universities. Additionally, Drost (1997), Top (1997), Wis (1996a), Wis (1996b), and Wis & Ken (1996) expressed concerns over the overcrowded and overloaded recent mathematics syllabus, the Kurikulum Matematika SMU 1994. These problems in turn led to the dominance of traditional approaches to teaching in both kinds of institutions.

A visit to an Indonesian secondary school, and the observation of a senior mathematics teacher carrying out his teaching practices, provided a window through which to see the underlying frameworks to his practices. Understanding a senior teacher's practices could contribute to improving the teaching and learning of mathematics in Indonesia. This was the goal of the study reported in this paper.

Theoretical framework

Research on exemplary practitioners

Several authors have reported research conducted on exemplary teachers (Lloyd, 1998; Tobin & Fraser, 1987; and Berliner, 1986). Furthermore, Driscoll (1985) and Kennedy (1986) note that there has been much support for the idea that teachers and curriculum developers have much to learn from exemplary practitioners. It was even suggested by Berliner (1986) and Shulman (1986) that case studies of expert teachers should form a part of teacher education programs. Studying advanced practices could motivate and guide preservice and inservice teachers in their practice.

The link between teachers' subject matter knowledge and instructional practices Shulman (1986, p.9) defines subject matter content knowledge as "the amount and organisation of knowledge per se in the mind of the teacher", and pedagogical content knowledge as "the ways of representing and formulating the subject that make it comprehensible to others". Fennema and Franke (1992) indicate that pedagogical content knowledge also includes an understanding of what makes the learning of specific togpics easy or difficult, and the conceptions and preconceptions that students of different ages and backgrounds bring with them to the learning of the most frequently taught topics and lessons. In this study, a teacher's subject matter knowledge refers to the subject matter content knowledge, together with the pedagogical content knowledge.

Brophy (1986), Shulman (1986) and the National Council of Teachers of Mathematics (NCTM) (1989) propose that the study of teachers' subject matter knowledge, as it impacts on mathematics instruction, is an important and timely topic for both research on teaching and on curriculum reforms. Despite questions dealing with teacher's knowledge being complex, ill defined, and often poorly studied, Fennema and Franke (1992) state that increasing attention has been given to this area of research.

The connections between teachers' subject matter knowledge and instructional practices have been recognised by studies focusing on a variety of subject areas. Lloyd (1998), Haimes (1996), Gamoran (1994), Stein, Baxter, & Leinhardt (1990), McGraw (1987), and Gudmundsdottir & Shulman (1987) have provided evidence that teachers' knowledge affects both the content and the processes of their instruction, influencing both what they teach and how they teach.

Ernest (1989) suggests that one of the key elements that determines the practice of teaching mathematics is a teacher's mental schema. He explains that teachers' mental schemas include knowledge of mathematics, beliefs concerning mathematics and its teaching and learning, as well as several other factors. Koehler and Grouws (1992) too are of the view that a teacher's behaviour is influenced by his or her subject matter knowledge, attitudes, and beliefs about mathematics and its teaching and learning.

The links of teachers' beliefs to instructional practices

Green (1971) and Rokeach (1960), as quoted by Thompson (1992), introduced the concept of a beliefs system and use this notion as a metaphor for examining and describing how an individual's beliefs are organised. However, Ernest (1989) suggests the definition of beliefs as views regarding the nature of teaching that topic (the purpose, the self-perceived role of the teacher and the relationship between students and teachers), the process of students' learning of that topic, and the nature of that topic. The categorisations of teachers' beliefs in this study, are based on Ernest's (1989) views.

Teachers' beliefs have always been considered important in devising approaches to the teaching of the subject matter content. In adopting certain models of learning, teachers' beliefs do have an important role. Brophy and Good (1974) indicated that teachers' beliefs system are especially important in guiding their instructional decisions. Ernest (1989) suggests that teachers' beliefs about mathematics teaching encompasses three components: the teacher's views of the nature of mathematics, the nature of mathematics teaching, and the process of learning mathematics. However, research carried out by Kesler (1985), Parmelee (1992), and Van Zoest, Jones, & Thornton (1994) found that actions are not always consistent with stated beliefs. It is sugges ted by Pajares (1992) that beliefs about teaching are well established by the time students go to college, and as a result, such beliefs act as a filter through which new information on teaching is sifted. Meanwhile, Prawat (1989) asserts that teachers with the same level of conceptual understanding may teach differently depending upon their educational beliefs (ie, their beliefs about teaching and learning).

Distinctions between knowledge and beliefs

Although it is difficult to distinguish between knowledge and beliefs, Abelson (1979) (quoted in Thompson, 1992, p. 129), indicated two distinctive features of beliefs and knowledge. The first feature is that beliefs can be held with varying degrees of conviction. The second feature is that beliefs are not consensual. Disputability is associated with beliefs, but truth or certainty is associated with knowledge. Scheffler (1965, quoted by Thompson, 1992, p. 129) explained the second distinction between beliefs and knowledge by suggesting that a claim to knowledge must satisfy a truth condition, whereas beliefs are independent of their validity. Nespor (1987) strengthened Scheffler's explanation by indicating that knowledge must meet criteria involving canons of evidence, while beliefs on the other hand are often held or justified for reasons that do not meet those criteria of evidence. Therefore beliefs are distinguished from knowledge by a lack of agreement over the evaluation or judgement. This study is following the line of making beliefs and knowledge distinctive entities.

The role of the concept of graphs in mathematical modelling Graph theory is a branch of mathematics that deals with problems involving arrangement of certain objects and the relationships between the objects. Wilson and Watkins (1990) suggest that much of the impetus to the Graph Theory topic in the last decade has arisen out of the need to solve particular problems in industry, where substantial saving in time or money has been made possible by applying techniques involving networks analysis.

The advantages of using graphs as models to solve practical problems in various fields can be enhanced if one is able to formulate a problem in such a way that it can be attacked by techniques of graph theory. The way in which the modelling process is carried out, and the degree to which the mathematical model accurately represents the original problem, varies from problem to problem. Familiarity in working with graphs and using graph theory techniques surely helps to recognise objects and relationships between them in many different real world situations.

It is expected that students will appreciate the opportunity to see real mathematics in action solving real world problems. It is also expected that they will see the importance and usefulness of developing a mathematical model or framework, which can be used to interrelate different problems and provide means to solving them. Part of mathematics teachers' duties is to expose their students to this opportunity.

Focus of the study

The main focus of this study was on the teacher's teaching practices in relation to his subject matter knowledge and beliefs within the constraints and opportunities provided by the social context of teaching. The following are the questions, which helped guide the collection of data:
  1. What is the extent of the teacher's subject matter knowledge of the Graph Theory topic?

  2. What kind of beliefs regarding the nature of mathematics, and the teaching and learning of mathematics, does the teacher have?

  3. How do the teacher's subject matter knowledge and beliefs influence his teaching practices when teaching the Graph Theory topic?


A qualitative case study approach was adopted (Merriam, 1988), using an ethnographic framework. The interpretive research methodology adopted (Ericson, 1986) utilised both quantitative and qualitative research techniques. The study involved a senior Indonesian mathematics teacher, Mr Sap, who expressed a willingness to be the subject of the research, along with his 48 Year 11 students. The study was conducted in an Indonesian senior high school, in the northern part of Surabaya, the capital city of Eastern Java Province. The author proposed to make assertions regarding Mr Sap's practices and then, test these assertions by looking for conforming and disconforming evidence.

Data collection

Beside the researcher herself as the research instrument (Merriam, 1988), the study utilised an interview guide and a sequence of observations. The interview guide was developed by the researcher and consisted of questions devised to ascertain the interviewee's views of the nature of mathematics, and of mathematics teaching and learning. It also contained questions with regard to the teachers' subject matter knowledge.

Information about the teacher's subject matter knowledge and beliefs obtained by the interview guide were complemented by a series of observations of the teacher's practice in the natural setting of his classroom. Interviews with the teacher were conducted in between observations. The six hours of observations on the teaching of Graph Theory lasted over a two weeks period. As well as teaching Graph Theory during this period, Mr Sap also conducted revisions of previous topics. With this amount of time left for teaching the Graph Theory topic, only part of the material originally selected by Mr Sap could be taught. Observations focused on the pattern of Mr Sap's teaching and his ways of representing and formulating the topic to the students, in order to establish the impact of Mr Sap's subject matter knowledge and beliefs on his teaching practices.

The interviews with Mr Sap were audiotaped and summaries of the interviews were shown to him. Mr Sap's lessons were also videorecorded. The analysis and interpretation of his teaching which resulted were also shown to him. In addition, the researcher discussed various aspects of the teaching practices observed with Mr Sap.

Interviews were also conducted with three top students, three middle ability students, and three low ability students. This was done in order to obtain information from three groups of students of differing ability. The interviews sought students' opinions on Mr Sap's teaching, on the teaching materials, and on their learning.


Analysis and data collection occurred simultaneously and evolved as the interpretation of each phase of data collection influenced subsequent phases. Each interview conducted was based on the themes of previous observation and interviews. The processes of data collection and analysis were recursive and dynamic (Merriam, 1988, p. 123). All the data for the study were organised into a case study database (Yin, 1994) which was constructed by listening to tapes, watching the videos, reading the transcripts, and taking fieldnotes, and examining classroom artifacts. The videos helped to create a chronological account of lessons and a base for interview themes. Retracing the phases of data collection helped in identifying the emerging patterns and providing a basis for coding them thematically. The final stage of analysis categorised and synthesised the emergent themes within and across the data sources to lend coherence and richness to the written case study report.

Limitations of the study

The fact that a case study does not seek generalisations requires care to be taken in generalising from the results. However, when several case studies give similar results, then some generalisations can be drawn. Because no single reality can be claimed in a particular setting, the best interpretation of a solution that can be achieved is an interpretation filtered by the researcher's own point of view and social theory. Many different methods are used in this study to focus on the effects of teachers' content knowledge and perceived roles on their teaching practices, so avoiding highly idiosyncratic interpretations. Also, these multiple inquiries are utilised to ensure the validity of the research findings. The teacher who pa rticipated in this study might not have been the best in teaching the mathematical topic of Graph Theory. Although other teachers and the principal of the school identified this senior teacher, Mr Sap, the selection was also based on the teacher's availability and willingness to be involved. The definition of the term 'senior' mathematics teacher is possibly a further limitation. In this study, the term "senior mathematics teacher" tends to mean a successful, experienced mathematics teacher. A final limitation involves the instrumentation used, which may have contained some weaknesses. Although the instrument was trialed before the research was conducted, possible personal subjectivity as well as language deficiencies in devising the additional statements in English and then translating them into Indonesian, might have resulted in biased understandings.

The social context

The study was conducted in Surabaya, the capital city of Eastern Java province, during a period of social and economical crisis in Indonesia. Social unrest caused by a monetary crisis sharpened the religious, ethnic, social-politic, and economic differences in many places in Indonesia. Many businesses had to be closed down and unemployment rates escalated. Being sensitive to people's suffering and having endured continued disappointment with perceived practices of nepotism, collusion, and corruption by the government, students demonstrated to demand political reform.

The school selected for the study was located in the northern part of Surabaya. Mass demonstrations in, and surrounding the Province Parliament Building not far from the school blocked the traffic, thus affecting the school's teaching and learning activities. Parents and teachers feared for the students' safety. During the height of the crisis, schools had to be closed, and the national primary examinations had to be delayed. Teachers complained that students could not concentrate on the lessons. In effect, students were traumatised. As with other schools, the teaching programs of this particular school had to be modified and reduced once some order was restored.

The biggest riot occurred in the capital city of the country, Jakarta, and caused million of dollars material loss and deep scars in the hearts of people being looted, having family and friends killed, burnt or raped. The burning down of business centres, and similar riots in other cities and towns including Surabaya, contributed to the already crippled economical situation. After a new President had been elected, the schools were re-opened and the situation seemed to be back to normal. However, the loss of two weeks effective study time could not be recovered. The 14 hours planned for teaching the Graph Theory topic as suggested in the syllabus, could not be carried out. After cancelling some of the mathematics revision program, the school managed to re-schedule just six hours for teaching the topic.

The Indonesian curriculum

Educational philosophy

In the 1920s, while still under Dutch colonisation, Ki Hajar Dewantara inspired Indonesians with the establishment of his Taman Siswa school and by suggesting his famous teaching philosophy: Ing Ngarso Sung Tulodo, Ing Madyo Mangun Karso, Tut Wuri Handayani. This meant that being in front, a teacher is the model of the class; being in the middle, a teacher should generate students' spirit to learn; being at the back, a teacher should facilitate and encourage students' learning. In a loose way, a teacher's roles in those three different positions in the class can be translated as the roles of a model, a friend, and a parent. Guru in Javanese, the dialect spoken by people who live in Central and Eastern Java Island, meant 'digugu' - to be listened and 'ditiru' to be modelled and followed. After Indonesia gained its independence, the Indonesian Ministry of Education and Culture adopted Tut Wuri Handayani as its educational philosophy.

Formal education structure and program

Indonesian Basic Education (Pendidikan Dasar - PD) ranges from Year 1 to Year 9, and it is compulsory for all Indonesians. After Year 9, students can pursue further study in either the General or the Vocational Secondary School or the informal education courses. The General Secondary School (Sekolah Menengah Umum - SMU) consists of Year 10 to Year 12. The syllabi for Year 1 PD to Year 11 SMU are the same throughout Indonesia. There are two strands of Year 12 SMU: the Natural Sciences (Ilmu Pengetahuan Alam - IPA) and the Social Sciences (Ilmu Pengetahuan Sosial - IPS).

The teaching year commences in the middle of July and ends in the middle of June. One year's study in SMU is divided into quarters, therefore each quarter consists of three months of school programs. All of the subjects and topics are offered as a package for all students. This means all students should learn all subjects and all topics to the same extent, without any regard to their abilities. Only the Year 12 students are divided into two different strands. But again, each strand offers the subjects and topics as a whole package, and no students are allowed to take only part of the package.

The school syllabi were designed by a group of subject educators, scientists, teachers, people from industry, consumers and lay persons, together with expert staff of the Curriculum Unit of The Central Ministry of Education and Culture. The school syllabi issued are the same across all of Indonesia. Draft syllabi were piloted for at least three monthly periods. One of the main factors influencing the shape of the curriculum was the intention to design SMU as the vehicle for students to enter university study. Students who intend to go straight to work were to study the Vocational Secondary School (Sekolah Menengah Kejuruan - SMK) syllabus instead. Apparently, this intention is not well received, since students graduating from Year 9 will likely prefer to continue their study in SMU. Only a very small percentage of students choose to study in the formal SMK.

The Graph Theory topic in Indonesian secondary mathematics

Mathematics is one of the compulsory subjects for Year 1 to Year 12. Graph Theory was included in the 1994 secondary mathematics syllabus (Kurikulum 1994) as an optional mathematics topic in Year 11, allocated in Quarter 3, of the Indonesian secondary school system (Depdikbud, 1994). According to the 1994 Mathematics Syllabus, the Graph Theory topic is mainly a theoretical introduction for the same topic in tertiary institutions. it consists of six subtopics as follows: Simple graphs and multiple graphs, Connectivity, Labelled graphs, Graphs and matrices, Planar graphs, and Graph colouring. The introductory role for the teaching of this subject is reflected by the allocation of only 8% of the whole time provided for mathematics in Year 11.

The teacher

Participant and his background

Mr Sap was a full time teacher at his school and his involvement with the study resulted from the recommendation of his school principal. One of the reasons for recommending Mr Sap was that he had been a successful teacher in preparing students for the Mathematics National Examination and the Tertiary Entrance Examinations. Besides nine years of teaching experience in that school, he was regarded as a teacher with a high standard of moral integrity and honesty. Subsequently, Mr Sap expressed a willingness to allow the researcher to conduct the study in his classroom. He was happy to make some adjustment to his timetable in order that the period of observation could be accommodated. Although Mr Sap was very busy with his teaching, he made himself available to the researcher and was very helpful in providing data.

Mr Sap was a graduate from the mathematics department of one of the Teacher Training and Education Institutions in a city in Central Java. Immediately upon graduation following four years study, he did some mathematics teaching in a public school for one semester. This was his first teaching experience in a school. Following this, he was accepted to teach mathematics in his present school . The initial guidance and advice of one of his senior colleagues proved very helpful to his teaching performance and in adjusting to the school's environment and students. While Mr Sap came from a village in Central Java, about 90% of the students in his school were ethnic Chinese from big cities in Eastern Java and other provinces of Indonesia.

Mr Sap's school

The school was established in 1967 and, at the time of the study, had a total of 1174 students including 454 students of Year 10, 394 students of Year 11, and 326 students of Year 12 (129 Natural Science (IPA) students and 197 Social Science (IPS) students). Classes were conducted in two shifts with morning and afternoon classes. Among the 53 teachers working in the school, there were 17 full time teachers including Mr Sap. A full time teacher of the school was required to conduct a minimum of 24 teaching hours. The students came mostly from middle class families. Practicing Catholicism, the school did not allow students to be excluded from being educated because of financial hardship. At the school, the wealthy families assisted the poor. As well, the financially healthy schools in the cities supported the financing of poorer schools in the countryside.

Mr Sap's subject matter knowledge

Mr Sap had tried hard to equip himself with knowledge of the Graph Theory topic by attending lectures and a seminar provided. He spent much time studying the topic by himself, which he found useful. The lectures and seminar he attended did not provide that much help, since his mind set was not really ready at that time. Mr Sap honestly admitted that his 'inadequate' knowledge of Graph Theory was a barrier to giving clear views of the topic to his students. He mentioned, "If I could teach the Graph Theory topic more confidently, students would be more interested to learn it".

Mr Sap found that his own study of this topic had a profound effect on him. He now had a broader view of the subject and of mathematics in general. But, as he indicated, inadequacies in his subject matter content knowledge did not give him much confidence in helping his students to construct a thorough understanding of the subject. He admitted that sometimes he was uncertain when explaining particular understandings or concepts, or in answering students' questions.

Mr Sap's beliefs

In spite of the problems envisaged by Mr Sap in teaching the Graph Theory topic, he admitted that he was amazed to see the uses of this topic in providing solutions to real world problems in many different fields. He believed that this topic offered a lot to students in seeing mathematics in a different way than he experienced it himself. He felt that the real world applications of the Graph Theory topic would be especially attractive to students and would motivate them to learn more mathematics.

Before studying the Graph Theory topic, Mr Sap's view on the nature of mathematics was as a mixture of a static but unified body of knowledge and an accumulation of facts, rules and skills to be used in pursuance of some external end. "Mathematics has absolute truths", he often said. But his experience in studying the Graph Theory topic made him develop a view of mathematics as a dynamic, continuing, expanding field. He believed, "Mathematics is expanding and previously existing theories continue to be refuted".

In order to have the 48 students in his mathematics class behave themselves, Mr Sap saw himself as captain of the team. He believed that he had to be firm in deciding which was right and wrong when students asked about the solution of problems or definition of concepts. Mr Sap believed also that he had to be firm in controlling the class, so that all students had an equal opportunity to learn. A silent classroom, except when he used a student-tutoring strategy, encouraged teaching and learning to occur.

Mr Sap believed that it was important to cover the whole syllabus content. Consequently, there were times that he felt the need to go ahead with a lesson regardless of the less able students' questions and confusion with a previous lesson. Mr Sap's goal was to cover all the material listed in the syllabus in the space of time provided. He believed that this pressure would somehow push the less able students to do more study in mathematics by themselves. The risk of being not eligible to move on to Year 12 was of deep concern to most Year 11 students.

Mr Sap viewed the learning process as an active search of objects one had not known until one knew them well. Students should realise that they should not depend on the teacher, that the effort to learn should come from them. "A teacher cannot change the students. The students have to work seriously on their own learning". The fact that Mr Sap's effort to facilitate students' active construction of knowledge was limited by time and his teaching duties, could lead to a rudimentary development of students' autonomy and interests in mathematics. However, it enhanced Mr Sap's acceptance of a view of learners as submissive and compliant. He mentioned, "I expect the students to be always ready with their assignments, be prepared to study new topics. I hope that they own the habit of studying hard in their school work". The emerging dominant model of learning in Mr Sap's class was the students' compliant behaviour and a mastery skills model. It seemed that one way communication was dominant in Mr Sap's classroom. In spite of his domination in the classroom, he believed that he provided more help to the students by being available and open to their questions during recess time.

Mr Sap viewed his responsibility as a teacher as (a) representing the profession as best as he can - by conducting his teaching sincerely; (b) applying Dewantara's educational philosophy - a teacher is a model; and (c) encouraging students' learning - helping the students in their learning sincerely. For him, an ideal teacher should be (a) an expert in his or her field - have knowledge mastery; (b) sensitive - able to interpret class' situations; and (c) able to educate students - planting and developing good values in students.

Problems encountered by Mr Sap in teaching Graph Theory

Mr Sap admitted that he had a lot of problems in teaching the Graph Theory topic. Firstly, the topic was not provided in his tertiary education. He was aware of the topic from the 1994 Mathematics Syllabus, which was available in schools only in 1996. At that time many teachers raised concerns with their principals of not being knowledgeable about the topic. Subsequently, the School Education Board invited an expert on Graph Theory from the public Teacher Training and Education Institution to give lectures on the topic. A seminar was provided by the Local Ministry of Education some time after that. A second problem for Mr Sap was that he had been confused about several of the terms used in Graph Theory. Teachers, including Mr Sap, considered that the language used in the chapter of the textbook available for the Graph Theory topic was difficult to understand and that its scope was beyond the syllabus. The reference textbooks available sometimes used different terms for the same concepts. Thirdly, Mr Sap encountered the enduring problem of time constraints. There was difficulty in covering all nine topics in Year 11 mathematics. The previous year and the year before, there was no time to teach Graph Theory. The present year was the first time the school had provided the topic. Fourthly, the Graph Theory topic consisted of too much material. Mr Sap mentioned, "Probably because this is a new topic for me, I myself find it difficult to remember different kinds of graphs. I think the 1994 Mathematics Syllabus has too much weight". Finally, schools and teachers did not think that the Graph Theory topic would be tested in either the National Mathematics Examination or the Tertiary Entrance Examinations. So, most of the schools and teachers did not support the teaching of this topic in their schools. Furthermore, the students did not believe that Graph Theory would contribute marks to th eir final score in mathematics.

The general pattern of Mr Sap's lessons

Six of Mr Sap's lessons on teaching Graph Theory were observed and certain patterns became visible during the observations. In starting the lessons, Mr Sap reviewed the materials taught in the previous lesson and raised several review questions. These questions were usually answered by students and sometimes short discussion followed. Mr Sap then moved on to discuss the assigned homework. He explained difficult problems to the whole class, or one or two students shared their work with the rest of the class by working the problems on the board. Following this, Mr Sap introduced the topic he intended for that day. After directing several questions to students to get input about the new topic, he continued by giving the day's homework. It was usual for each lesson segment to flow smoothly to the next.

In general, Mr Sap developed ideas on the board using an Over Head Projector (OHP) and invited input from students. His teaching style was whole-class interaction. He tried to involve all of the students by sometimes directing questions about ideas being developed to individual students. Each lesson segment was conducted quickly without much time provided for students to interrupt or ask questions, but it seemed that they did not want to ask too many questions.

Mr Sap asked the students to be quiet and listen to his lesson presentation. He was the leader in almost every lesson segment and had control of the whole class. The fact that his explanation dominated most of the lessons clearly demonstrated his emphasis on the use of oral methods. Although some of the students responded actively to the teacher's questions, most of them remained quiet. Subsequent interviews with students indicated that middle and low ability students found Mr Sap's teaching was paced too quickly. For these groups of students, he did not have the time to give more attention to helping them. They tended to disguise their confusion because they did not want to disappoint Mr Sap and were ashamed to expose their lack of understanding to the more able students. Actually Mr Sap, in his role of a model, demanded these groups of students to listen to his advice to be more active in their own learning, but the gap constructed unconsciously between himself (as the model) and these students (as followers), discouraged communication. Relating to Dewantara's philosophy, Mr Sap seemed to place emphasis more on his role as a good model for his students, partly to the exclusion of roles as a friend and a parent as suggested in the philosophy.

Mr Sap listened to the students and welcomed their opinions or questions. He showed some patience in repeating his explanations or answering students' opinions and questions. "I do not want to discourage their learning", he mentioned during an interview. He had been trying to explain the mathematics concepts as clearly as he could, but he had to move to the next topic quickly in order to cover the material on time according to the teaching plan. Some students looked quiet and did not respond to Mr Sap's questions and explanations. These students did not express their feelings and there was no other choice for them except listening to Mr Sap's explanation about the next topic.

In general Mr Sap followed the definitions and terms in the textbook used by the school. The use of the same term by another book to refer a different concept confused him, especially in making a decision on the right one to use. His confusion led to students' confusion. A technique he employed was to ask the students to read about the Graph Theory topic in other books, especially with regard to confusing concepts or terms, and to let him know of their findings. Mr Sap considered this as the best way for both the students and himself to construct better understandings.

Mr Sap appreciated the use of graphs to model real world problems in several different fields. However, in his explanation about graphs, he did not emphasise their use. He only touched on it and focused his teaching on defining the concepts discussed, covering the content subject matter, and the acquisition of techniques in solving the problems.

Mr Sap used different teaching approaches in spite of the tight timetable and the amount of material to cover. Sometimes he used a student-tutoring strategy. For this, he asked the students to form groups of five or six. The more able students in the group then helped the less able by explaining how to solve difficult homework problems. At other times, he conducted worksheet activities. Mr Sap devised the worksheet and the student chief of the class made copies of it for all of the class. Then, Mr Sap guided the students in completing the worksheet and worked together with them in solving problems provided. By leaving part of the problems' solutions blank, Mr Sap encouraged his students to solve them for themselves.

There were not many opportunities to develop teacher-student interactions. There was also little time provided for student-student interactions. In the time space provided, each student could interact mostly with his or her desk mate.

In assessing student achievement as correct solutions of problems in paper and pencil tests, Mr Sap demonstrated that his intended teaching outcomes were mostly skills mastery with correct performance and conceptual understanding with unified knowledge. Mr Sap had wanted in his teaching to give students some problem solving skills, but he found there was no time to provide problems for mathematical investigations designed to enhance these.

Assertions about Mr Sap's teaching

  1. Mr Sap adopted the roles of teacher as model, partly as the authoritative source of knowledge, and partly as a learning facilitator, with more emphasis on the former.

    When interviewed, Mr Sap stated that, "A teacher should make himself entitled to be listened and modelled by the students". He tried hard to project the model image to his students, by being fluent in all topics of mathematics. But this new topic encouraged him to accept that he was not the only authoritative model and source of knowledge. He said, "A teacher is part of the knowledge resource for students". He continued with, "Nowadays there are books and electronic or computer software as knowledge resources as well". He viewed himself learning along with his students while teaching. He admitted, "It is not because I am a teacher that I should know everything. I do not always know everything, even mathematics". As a knowledge resource, Mr Sap viewed himself as an instructor to be listened and followed, and as an explainer. However, in restructuring his knowledge when conducting his learning together with students he viewed himself as a learning facilitator.

  2. Mr Sap viewed mathematics more as a combination of Platonist and instrumentalist views than from a problem solving view.

    Mr Sap's respect for his students made him always try to maximise student learning by pacing his teaching to his students' learning. But the constraints of time limited his efforts to help students. In line with Platonist and instrumentalist views of mathematics, Mr Sap emphasised covering the content, mostly by oral methods, and fostered students' acquisitions of techniques in solving problems, with correct performance and procedures, rather than focusing on the values and affective aspects of mathematics. He had to avoid problem solving because of the time constraints and the amount of material to be covered.

  3. Mr Sap's action was not always consistent with his stated beliefs.

    Although in interviews Mr Sap stated that the learning process should be active, in his classroom, his actions reinforced that learning was submissive and compliant. School mathematics was considered as more a mind-training medium and independent of any intrinsic value it may have. It needed to be difficult and the difficulty in learning mathematics was considered as a normal thing to happen. Although Mr Sap stated that "Students' minds should be active," his teaching did not show many opportunities for students to express themselves.

  4. Mr Sap's teaching practice was a reflection of his subject matter knowledge and beliefs with some distortion caused by the social constraints and opportunities.

    Mr Sap's intended teaching outcomes were mostly skills mastery with correct performance and conceptual understanding with unified knowledge, although he actually desired that problem solving skills be among his teaching outcomes. The time constraints did not allow the development of students' confident problem solving activities. Mr Sap's pattern of use of curricular materials was as a strict follower of a text or scheme (certain textbook or teaching scheme). He combined problems and activities, sometimes designed by himself, but not necessarily in the textbooks used.

  5. The single syllabus, the amount of material to be taught and the class size encouraged Mr Sap's adoption of traditional teaching practice.

    The large number of students in Mr Sap's class, the time constraints and the single syllabus, encouraged him to restrict the students' autonomy and interest in learning mathematics. "Each student should be able to learn in the class, so peace and silence should be promoted in the class", Mr Sap mentioned. "Too much material needs to be covered", he added. "There is only one syllabus for Year 11 students", he said in frustration. All students, with their different mathematical abilities, had to learn the same curriculum. All of these fostered the conduct of traditional teaching of mathematics. Unconsciously, the acceptance of the myths of cold reason and hard control in the mathematics classrooms were encouraged.

  6. Mr Sap's traditional approach to teaching was unconsciously adopted and practiced despite his effort to change.

    Mr Sap's instrumentalist view and Platonist view of mathematics was due to the learning process he himself experienced in studying mathematics in his secondary and tertiary institutions. His experience in teaching the Graph Theory topic, a new topic he himself did not encounter in his formal education, led him to a problem solving view of mathematics, although this latest view had not been rooted nor well established. However, these views provided a basis for his mental models of teaching and learning of mathematics. His dominant role as an instructor, a model, and an explainer came from his previous views of the nature of mathematics. The Platonist view of seeing mathematics as a static unified body of knowledge led to Mr Sap's adopted role as explainer to help students' conceptual understanding and unified knowledge of mathematics. The instrumentalist view of seeing mathematics as an accumulation of facts, rules and skills led to Mr Sap's role as an instructor to help students develop skills mastery with correct performance. This practice had been accepted as normal in teaching mathematics in Indonesian classrooms from generations to generations. The teacher transmitted or transferred the information to students and the students absorbed it. The emphasis on tests and examinations as success in learning encouraged the establishment of these views of mathematics.

    Mr Sap's recently added problem solving view of mathematics led him to a role as a facilitator of students learning, which unfortunately could not develop well because of the social and time constraints. Essentially, Mr Sap was a strict follower of a certain textbook. But, his problems in understanding the concepts of the new Graph Theory topic made him to look at other textbooks, to compare the development of concepts, and led him to sometimes devising his own curriculum. Although he was a supporter of the active construction of understanding, he had to often defer to the passive reception of understanding and students' compliant behaviour and mastery of skills. because of the social context of the school, in a punishment/reward situation through the awarding of marks the students had no option other than to try.

  7. Mr Sap's lack of familiarity with graph theory led him to place emphasis on memorising definitions, content coverage and the acquisitions of techniques for solving certain problems.

    Mr Sap did not place much emphasis on showing how graph theory deals with various problems involving the arrangements of certain objects and relationships between them. His focus on memorising the definitions, covering the content and mastering the skills was a result of his lack of familiarity with the Graph Theory topic. In interviews he stated that he wished to know more about the topic so that he could better carry out teaching.

  8. Positive teacher-student interactions occurred only with the high ability and some of the middle ability students.

    The rest of the class, especially the low ability students, did not get as much help as they felt they needed. Time constraints did not allow Mr Sap to provide much help for this group of students. However, he maintained an expectation that they would be able to keep up with the lessons. Interview with the students indicated that some of the low and middle ability students found that Mr Sap had tried hard to help them, especially when he was asked.


The findings of this study lend strength to previous indications of problems in teaching and learning of the Graph Theory topic and mathematics in general. These include an emphasis on oral methods and lack of suitable textbooks, (as suggested by Canon & Sri Widodo, 1994), inadequacies in teacher's subject matter knowledge, and (as Drost (1997) suggested) a crowded and heavy-loaded syllabus. In line with Lloyd (1998); Haimes (1996); Gamoran (1994); Stein, Baxter, & Leinhardt (1990); McGraw (1987); Gudmmundsdottir & Shulman (1987), these findings support the strong interdependence of a teacher's subject matter knowledge and beliefs and instructional practices.

The problems Mr Sap experienced in teaching a new topic in the syllabus, in spite of his experience as a senior teacher of mathematics at his school, gives much for teachers and curriculum developers to learn as suggested by Driscoll (1985) and Kennedy (1986). In line with Berliner (1986) and Shulman (1986) it is also useful for teacher education institutions to develop programs such as professional development for inservice teachers.

Mr Sap's strong beliefs on Platonist mathematics led him to teach the Graph Theory topic by emphasising the content only and by demanding students to perform correct procedures and skills mastery in achieving solutions. Mr Sap's inadequacy in his subject matter content knowledge led him to structuring the knowledge of the Graph Theory topic in a different way than he wanted to. His pedagogical content knowledge for comprehensively transferring knowledge to students, was undeveloped. His mental schema of the Graph Theory topic had not been rigorous enough for delivering clear and simple explanations and understandings. As Pajares (1992) suggested, not only were Mr Sap's beliefs about teaching well established by the time he went to college, but also his beliefs about the nature of mathematics and learning. Mr Sap's teaching practice enforced his role as a model, instructor, and explainer, and also his Platonist and instrumentalist views of mathematics. These beliefs guided Mr Sap in making instructional decisions and acted as lenses in filtering new information. Mr Sap's methods and approaches to teaching, the created gap between himself and his students, was very similar to the traditional teaching he experienced himself.

The findings also support the inconsistency between Mr Sap's stated beliefs and his actions as Kesler (1985), Parmelee (1992), and Van Zoest, Jones, & Thornton (1994) suggested. Although in interviews Mr Sap stated that he promoted active learning in his teaching, his practice encouraged the performance of learning as submissive and compliance. While desiring that his students be active in the construction of mathematical understandings, he was not able to allow the students to express themselves due to time constraints. Among the possible causes of the inconsistency between Mr Sap's stated beliefs and his actions were the social constraints and opportunities provided in the setting. One of the school's criteria in determining teachers' standard of ability was success in leading students to pass the National Mathematics Examination and Tertiary Mathematics Entrance Examinations. This emphasis on exams undoubtedly added credence to the acceptance of traditional teaching methodologies for mathematics.


Being a teacher herself, the researcher recognised the difficult position that a teacher has. Mr Sap's struggle to fulfil his teaching tasks despite of the social context and opportunity reflected his commitments to teaching profession. Therefore, teachers should not be the only scapegoat of education reform and be blamed for all the ills or problems in education. Functioning collaborations between the Ministry of Education and Culture, Teacher Training and Education Institutions, and inservice or preservice teachers should be strengthened and maintained.

The fact that the Graph Theory topic was optional allowed a choice for most of the schools in whether to not provide it. Factors such as a lack of teachers who had mastered the topic and the extra time needed, were the main problems for schools. In addition to those two factors, the non inclusion of Graph Theory in the Mathematics National Examination and the Tertiary Entrance Test did not encourage schools to provide this topic. In fact, only a few of the top schools decided to provide the topic in their mathematics program.

The teaching of the Graph Theory topic needs integrated action

To promote the teaching of the Graph Theory topic in schools, several factors need to be considered:
  1. Changes in syllabi. Schools need more encouragement and preparation to provide the Graph Theory topic. Originally, the optional preference for schools in providing the Graph Theory topic was to give the opportunity for schools to prepare their teachers to teach it. Changes to the syllabus by reducing the mathematics content and having at least two strands of mathematics for Year 11, would give some space and time for teaching the Graph Theory topic. Looking at the many applications of the Graph Theory topic in mathematical modelling and a change from an optional preference topic to a compulsory topic after several years, might be good for Indonesian school mathematics.

  2. Increasing professional development for teachers. Teachers, especially the ones without any background in the Graph Theory topic, need to have some training before teaching the topic. More opportunities for teachers to attend professional development sessions would encourage them to do further study in their special interest topics and teaching methodologies.

  3. Revising the textbooks. The chapter on the Graph Theory topic provided in the mathematics textbook published by the government needed to be revised. Teachers found that its language was not easy to comprehend and its scope was beyond the 1994 Mathematics Syllabus. This encouraged teachers to use other books published privately as the main source for teaching. But, some confusion developed due to the terms used to refer to certain concepts. Collaboration between scientists, educators, and teachers to formulate agreement on Graph Theory terms, topics, scope, and emphases, as well as teaching methods, would alleviate problems and prevent confusion.


The author would like to thank Dr David Haimes and Prof. John Malone who spent much time in editing earlier versions of this paper.

This study is part of a larger case study of senior mathematics teachers in Australia and Indonesia teaching the topic of Networks or Graph Theory.


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Please cite as: Seputro, T. T. (1998). The influence of the teacher's subject matter knowledge and beliefs on teaching practices: A case study of an Indonesian teacher teaching graph theory in Indonesia. Proceedings Western Australian Institute for Educational Research Forum 1998. http://www.waier.org.au/forums/1998/seputro.html

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