INDONESIAN MATHEMATICS TEACHERS' KNOWLEDGE OF

ISSN 2087-8885

E-ISSN 2407-0610

Journal on Mathematics Education

Volume 12, No. 2, May 2021, pp. 223-238

223

INDONESIAN MATHEMATICS TEACHERS’ KNOWLEDGE OF

CONTENT AND STUDENTS OF AREA AND PERIMETER OF

RECTANGLE

Wahid Yunianto

, Rully Charitas Indra Prahmana

, Cosette Crisan

SEAMEO QITEP in Mathematics, Jl. Kaliurang, Condongcatur, Depok, Sleman, Yogyakarta 55281, Indonesia

Universitas Ahmad Dahlan, Jl. Pramuka 42, Pandeyan, Umbulharjo, Yogyakarta, Indonesia

University College London, Gower St, Bloomsbury, London WC1E 6BT, United Kingdom

Email: [email protected]

Abstract

Measuring teachers' skills and competencies is necessary to ensure teacher quality and contribute to education

quality. Research has shown teachers competencies and skills influence students’ performances. Previous studies

explored teachers’ knowledge through testing. Teachers' knowledge of the topic of area-perimeter and teaching

strategies has been assessed through testing. In general, items or tasks to assess mathematics teacher knowledge

in the previous studies were dominated by subject matter knowledge problems. Thus, it seems that the assessment

has not fully covered the full range of teacher knowledge and competencies. In this study, the researchers

investigated mathematics teachers’ Knowledge of Content and Students (KCS) through lesson plans developed

by the teachers. To accommodate the gap in the previous studies, this study focuses on KCS on the topic of area-

perimeter through their designed lesson plans. Twenty-nine mathematics teachers attended a professional

development activity voluntarily participated in this study. Two teachers were selected to be the focus of this

case study. Content analysis of the lesson plan and semi-structured interviews were conducted, and then data

were analyzed. It revealed that the participating teachers were challenged when making predictions of students'

possible responses. They seemed unaware of the ordinary students' strategies used to solve maximizing area from

a given perimeter. With limited knowledge of students' possible strategies and mistakes, these teachers were

poorly prepared to support student learning.

Keywords: Knowledge of Content and Students, Mathematics Teacher, Area and Perimeter, Teachers’ Skills

and Competencies

Abstrak

Mengukur keterampilan dan kompetensi guru diperlukan untuk memastikan kualitas guru dan berkontribusi pada

kualitas pendidikan. Penelitian ini menunjukkan bahwa komptensi dan keterampilan guru mempengaruhi

performa siswa. Penelitian sebelumnya telah mengkaji pengetahuan gru melalui tes. Pengetahuan guru pada topik

keliling-luas dan strategi pembelajaran juga telah dikaji melalui tes. Pada umumnya, banyaknya soal pada tes

didominasi oleh soal-soal tentang pengetahuan subjek yang diajarkan. Oleh karena itu, asesmen seperti ini belum

mencakup kesuluruhan pengetahuan dan kompetensi guru. Pada studi ini, peneliti menginvestigasi pengetahuan

guru matematika tentang KCS pada rencana pelaksanaan pembelajaran yang mereka kembangkan. Untuk

mengakomodasi kesenjangan pada penelitian sebelumnya, penelitian kali ini berfokus pada pengetahuan tentang

konten dan siswa (KCS) pada topik keliling-luas pada rencana pelaksanaan pembelajaran. Dua puluh Sembilan

guru matematika yang sedang mengikuti pelatihan peningkatan kompetensi secara suka rela mengikuti penelitian

ini. Dua guru matematika menjadi fokus penelitian studi kasus ini. Konten analisis dan interview semi terstruktur

dilakukan dan datanya dianalisis. Terungkap bahwa peserta ini mengalami tantangan dalam memprediksi

kemungkinan respon yang diberikan siswa. Mereka belum menyadari strategi siswa yang biasanya digunakan

untuk menyelesaikan persoalan memaksimalkan luas dari keliling yang ditentukan. Dengan pengetahuan yang

terbatas pada kemungkinan strategi siswa dan kesalahan siswa, guru ini kurang siap dalam mendukung siswanya

Kata kunci: Pengetahuan tentang Materi dan Siswa, Guru Matematika, Luas dan Keliling, Keterampilan dan

Kompetensi Guru

How to Cite: Yunianto, W., Prahmana, R.C.I., & Crisan, C. (2021). Indonesian Mathematics Teachers’

Knowledge of Content and Students of Area and Perimeter of Rectangle. Journal on Mathematics Education,

12(2), 223-238. http://doi.org/10.22342/jme.12.2.13537.223-238

224 Journal on Mathematics Education, Volume 12, No. 2, May 2021, pp. 223-238

Shulman (1986) refers to Pedagogical Content Knowledge (PCK) as the ways of representing and

formulating the subject that is understandable to others. Research have shown that student achievements

are more affected by PCK than Subject Matter Knowledge (SMK) as the quality of instruction is related

to PCK (Baumert et al., 2010; Hill, Rowan, & Ball, 2005; Hill, Ball, & Schilling, 2008). As the use of

SMK terminology varies, SMK in this paper refers to common content knowledge (CCK) which is part

of SMK (see Figure 1).

Hill, Ball and Shilling (2008), in seeking to conceptualize the domain of effective teachers'

unique knowledge of students' mathematical ideas and thinking, proposed the following

domain map for mathematical knowledge for teaching (see Figure 1) (White, et al., 2012,

p.394).

One specific aspect of PCK is the Knowledge of Content and Students (KCS). KCS is ‘knowledge

that combines knowing about students and knowing about mathematics (Ball, Thames, & Phelps, 2008,

p. 401). It consists of anticipating what students are likely to think about, what they could find confusing

or complicated, and what students are expected to do mathematically to complete the chosen task.

Figure 1. Domain map for mathematical knowledge for teaching (Hill, Ball, & Schilling, 2008, p. 377)

There are some teacher assessment models which measure knowledge for teaching. The Teacher

Education and Development Study in Mathematics (TEDS-M) is one of the international assessments

intended for pre-service mathematics teachers (Tatto et al., 2012). Some researchers assert that the

Assessment of Teachers’ PCK could be done through micro-teaching (Setyaningrum, Mahmudi, &

Murdanu, 2018; Ünver, Özgür, & Güzel, 2020). In the case of pre-service teachers, they have challenges

with student’s thinking, mistakes and responding (Korkmaz & Şahin, 2019; Setyaningrum et al., 2018;

Ünver et al., 2020). It makes sense as they have limited teaching experiences or even have not taught

Yunianto, Prahmana, & Crisan, Indonesian Mathematics Teachers’ Knowledge of Content and Students … 225

yet. For in-service teachers, Baumert and Kunter (2013) developed instruments to measure teacher’s

professional competence (COACTIV). The COACTIV adopted the three main core knowledge CK,

PCK and PK from Shulman’s work and extended it.

As one of the ways, testing is used to assess teachers. The Ministry of Education and Culture

(MoEC) of the Republic of Indonesia has also implemented Teacher Competency Tests (TCT) to

evaluate teachers’ knowledge. The result of this assessment is both to evaluate teachers and to provide

support for them (Widodo & Tamimudin, 2014). However, the content of this assessment is commonly

dominated by SMK, in this case within the mathematical problems. It seems that the PCK has not been

measured fully through this wide assessment. Another study using testing faced challenges in measuring

teachers’ knowledge (Fauskanger, 2015). An interesting finding of a study of pre-service teachers is

that they possessed higher PCK scores than SMK from limited test items (Kristanto, Panuluh, &

Atmajati, 2020). A case study in South Korea revealed that teachers with sufficient SMK of a certain

competence/ topic faced challenges in incorporating KCS and KCT of that topic (Lee, Capraro, &

Capraro, 2018). Therefore, testing to measure teachers’ knowledge still face challenges.

Lesson plans are considered to play an important role in teaching and learning. Having a good

lesson plan is important in ensuring that learning would take place during the lesson (Jones & Edwards,

2010). Academics argue that the key determinant of success in teaching is the effectiveness of planning

and how well a plan is carried out in the classroom. Effective lesson planning considers possible

classroom problems and how to tackle them adequately (Jones & Edwards, 2010). In the common

Japanese lesson plan, it contains detailed instruction so that teachers can easily understand it when

reading it (Nakahara & Koyama, 2000). Japanese lesson plans also include possible student solutions

and errors. The blackboard is also carefully planned. Called, ‘Bansho’, which anticipates and tries to

elicit student mathematical thinking and student thinking schema for solving the given problems.

In developing lesson plans, teachers integrate their knowledge, such as subject matter knowledge

and pedagogical content knowledge (An, Kulm, & Wu, 2004; Burns & Lash, 1988; Simon, 1995). A

study in Australia revealed that the teacher, in planning a lesson, gave attention to students’ engagement

(Clarke, Clarke, Roche, & Chan, 2015). The students’ engagement involves a choice from many

pedagogical strategies, all designed to motivate the students to engage with the topic. It has been shown

by several studies that novice teachers improved their PCK by teaching and preparing to teach

(Turnuklu & Yesildere, 2007). There is a reciprocal relationship between teacher thought process

(including planning) and teachers actions, the latter much influenced by the former (Clark & Peterson,

1986; Superfine, 2008). In other words, teacher classroom practices are influenced by a complex mix

of teacher beliefs, attitudes knowledge and intentions Therefore, arguably it is possible to look at teacher

lesson plans to investigate their knowledge. The illustration of a model of teacher knowledge and

planning can be seen in Figure 2.

226 Journal on Mathematics Education, Volume 12, No. 2, May 2021, pp. 223-238

Figure 2. Model of teacher knowledge and planning (Burns & Lash, 1988, p. 382)

Carle (1993) has investigated several student misconceptions related to the area-perimeter topic.

A meta-analysis of research has shown some student misconceptions on area measurement was due to

area being taught together with perimeter causing many students to confuse area and perimeter (Watson,

Jones, & Pratt, 2013; Cavanagh, 2007). Cavanagh (2007) studied Australian Year 7 secondary students

and reported students experienced difficulties dealing with area concepts because of the above

confusion with perimeter. As a consequence, students used slant and perpendicular height

interchangeably. Zacahros & Chassapis, (2012) reported Greek Year 6 elementary students added the

base plus the height instead of multiplying base with height to find the area of a rectangle. Özerem

(2012) reported that seventh year secondary school students in Cyprus had a number of misconceptions

due to a lack of knowledge related to geometry, resulting in them using the wrong formula. This lack

of understanding of the concept of area resulted in students memorizing the formulas. Students who

learn through manipulating area seem likely to avoid misconceptions on area measurement (Watson et

al., 2013). It seems to make sense as they could manipulate and observe what changes happen by

reshaping a figure (Yunianto, 2015).

It has been shown that SMK and PCK of mathematics teachers influenced student performance

(Baumert et al., 2010). Thus, we should not expect teachers to deliver mathematics well if they do not

have mastered it and do not understand how to teach it. Kow and Yeo (2008) explored the importance

of SMK and PCK in the topic of area-perimeter from the planning of the lesson to its delivery. It was

found that teachers with strong SMK and PCK provided more freedom to students to approach the task.

Baturo and Nason (1996) evaluated first-year teacher education student understanding of subject matter

Yunianto, Prahmana, & Crisan, Indonesian Mathematics Teachers’ Knowledge of Content and Students … 227

knowledge in the domain of area measurement and uncovered many misconceptions. Success was

related to their experience of learning the topic. John (2006) argued that novice teachers have difficulty

making predictions about student responses and how to respond to unpredicted situations they

encountered. In line with this, lack of mathematics pedagogical content knowledge of the teacher

potentially lead to students having misconceptions (Kow & Yeo, 2008).

This study intends to focus on a part of PCK, the KCS within lesson plans on the topic of area-

perimeter of a rectangle. It is necessary to obtain a fuller insight into teacher knowledge as it influence

students’ performance. Beside testing, there might be alternative way such as lesson plans to investigate

teachers’ knowledge. How are mathematics teachers prepare their lesson plans and how is PCK

integrated in their lesson plans? How are the KCS integrated in the lesson plans? In the next section,

the ways of gaining this insight will be discussed and the strategies used in collecting and analyzing the

data. Furthermore, the results and discussion sections will describe the KCS evident in the lesson plans

and the interviews with the respondents.

METHOD

This research involved humans and has been approved by IOE research ethics of University

College London (IOE.researchethics@ucl.ac.uk) as this is a part of completion of the first author’s

dissertation. This study administrated a case study approach. This approach suits this study as it does

not seek to generalize the findings but to gain deeper insight into the issue (Denscombe, 2010; Yin,

2014). The research subjects were the mathematics teachers in Yogyakarta and its surrounding

registered themselves to participate on PD organized by SEAMEO QITEP in Mathematics. Some

teachers teach across multi-grades. The first researcher who was facilitating one of the sessions asked

the participants to develop a lesson plan as part of the whole PD. It was done somewhere in the middle

of all complete sessions. As it is a case study, the researchers examined two selected lesson plans of

two mathematics teachers. The remaining lesson plans have not been analyzed due to time limitation.

The sample was chosen from twenty-nine teachers who attended a professional development (PD)

session, and two teachers were selected for the lesson plan analysis and interview. Additionally. these

teachers were selected based on their teaching experience; at least five years. The interview scenario

was a semi-structured interview, and the two teachers were interviewed together. The two teachers who

had been interviewed were a female teacher and a male teacher. They have different years of teaching

experience. The female teacher teaches in a city while the male teacher teachers in a rural area.

Participation in this study was voluntarily. The Indonesian mathematics teachers attending this PD were

teaching grade 7 to grade 9. The topic that would be taught was area and perimeter for grade 7. The

“Gold Rush/Mining” task was selected. This task was chosen because it is a problem-solving task and

has several ways to be solved on area-perimeter of a rectangle (see Figure 3). Additionally, the complete

Gold Rush activity showed the mistakes that students might do. Thus, it is considered as a good activity

to be explored to understand how teachers prepare this activity.

228 Journal on Mathematics Education, Volume 12, No. 2, May 2021, pp. 223-238

Figure 3. The Gold Rush problem (https://www.map.mathshell.org/download.php?fileid=1637)

To analyze the lesson plans, the researchers used content analysis. This method has the ‘potential

to disclose many hidden aspects of what is being communicated through the written text’ (Denscombe,

2010, p. 282). From the lesson plan, the researcher would investigate to what extent the teachers’

knowledge of students’ conceptions and misconceptions is reflected in their written lesson plans (Table

1). The two lesson plans were coded to find themes by classifying instructions and KCS integrated in

the lesson plans.

Table 1. Knowledge of Content and Student (KCS) (Ball et al., 2008, p. 401)

No.

Knowledge of Content and Student

The ability to anticipate what students are likely to think and what they will find confusing

The ability to predict what students will find interesting and motivating when choosing a task

The ability to anticipate how students are likely to solve a given task and whether they will find

it easy or difficult

The ability to hear and interpret students’ emerging and incomplete thinking

By using Table 2, it is easy to differentiate instructions’ categories. These themes were useful in

providing information on what the lesson plans contained. It focused on whether or not, the teachers

Yunianto, Prahmana, & Crisan, Indonesian Mathematics Teachers’ Knowledge of Content and Students … 229

included information about what students would do to the task (KCS). The data were presented

descriptively.

The two lesson plans were coded and analyzed. There were three types of instructions to refer to

with the codes. First, general instruction (GI) is where the teacher gives students instructions in a general

way. This type of instruction is relatively simple, short and contains the doer(s) and their actions (verb)

but leads to some mysteriousness (unclear). The second type of instruction is specific instruction with

no detail (SIND). This refers to specific action, which has more information than GI but lacks detail in

necessary aspects. The last type of instruction is specific instruction with detail information (SID). This

instruction provides more detail and clearer information. Some forms of SID are short and require no

detail, as it can be found easily or understood easily in other parts of the text. Looking through the

instruction types, the researcher seeks evidence of KCS on the lesson plans (Table 2).

Table 2. Coding for instructions

Code

Example 1

Example 2

Teacher asks a question to students

Teacher asks students to present their work

SIND

Teacher asks a question to students about their

strategy.

Teacher asks two groups to present their

work

SID

Teacher asks a question to students about their

strategy. “what did you do and How did you

do it? How are you convinced with your

strategies?

Teacher asks two groups with different

strategies to present their work starting with

the group with less sophisticated strategy.

The two teachers were also interviewed to gain more insight. They were interviewed together

(focus-group interview). The researcher wanted to clarify what was written on the lesson plans and why.

Through a semi-formal interview style, data were collected through voice recording as well as video

recording. From the records, data were transcribed and analyzed.

RESULTS AND DISCUSSION

Using the codes, the lesson plans revealed some interesting findings. Teachers 1 (T1) and

Teachers (T2) have different proportions of the use of the instructions (Table 3). The percentage is from

type of instruction per total instructions written on the lesson plans.

Indonesian teachers follow the prescribed template of a lesson plan by MoEC. The template consists

of three main parts namely; introduction, main and closure. It also consists learning goals and how teachers

and students would do in the classroom.

230 Journal on Mathematics Education, Volume 12, No. 2, May 2021, pp. 223-238

Table 3. Proportions of the instructions

Instruction

8 (35%)

6 (31.6%)

SIND

6 (26%)

7 (36.8%)

SID

9 (39%)

6 (31.6%)

Total

23 (100%)

19 (100%)

Based on the partition T1 used more instruction in the introduction and has less instruction in the main

body. Interestingly, T2 has more instructions in the Main body with detailed information. Compared to T1,

T2 had fewer total instructions, and detailed instructions (SID). From T2’s SID, there were several

instructions that provided information relating to PCK (Table 4).

Table 4. Comparison of Instructions

Code

Introduction

Main

Closure

SIND

SID

Total

T1 put more details of what students would ask to her on her lesson plan. For instance: ‘Can I solve

it freely?’ has been put on her lesson plan. This is a proof of PCK in the lesson plan, but not specific to KCS.

Figure 4. Teacher 1 Lesson Plan

Yunianto, Prahmana, & Crisan, Indonesian Mathematics Teachers’ Knowledge of Content and Students … 231

In addition, the way she would organize the discussion are provided in detail. This would provide

information to other readers/ teachers how the classroom discourse was managed (Figure 4). On the phase

of guiding the individual and group investigation which be rich of KCS. In this lesson plan, detail ways of

students might solve it or make mistakes and how to facilitate it have not been depicted.

The T2 lesson plan of rectangle using Gold Rush task depicted detailed information about a

possible student strategy (KCS). Figure 5 shows that T2 considered one strategy that students would

utilize by asking students to make a table. T2 prompted students to make a table and gave an example

to start with simple numbers. Within that table students would investigate the largest area by filling the

lengths and widths that added to 100. More interestingly, two examples with easy numbers were

provided to support students. Therefore, T2’s instruction can be understood as providing a method to

solve the task, with much support given to students.

Figure 5. Teacher 2 Lesson Plan of Gold Rush

After finding the largest area of the rectangle, students had to find the largest area by joining two

miners’ ropes and how would they join it. T2 also offered questions for students, revealing the

organization on their lesson plan. T2 has also provided students actions in Figure 6.

Figure 6. T2's lesson plan on organizing the classroom discussion

Students were expected to evaluate and generalize during discussion. Although it was unclear

what kind of evaluations and generalizations would be made. It would be clear if he put, for instance,

that the generalization would be that ‘the largest area would always be a square’. This generalization

might come out from students. In addition, it was not clear how T2 would organize the presentation, or

232 Journal on Mathematics Education, Volume 12, No. 2, May 2021, pp. 223-238

which group would present first. If there were two groups with different strategies or different

conclusions, it is not clear how it would be organized.

Teachers T1 and T2 have more than five-years teaching experience each. Based on the

questionnaire and interview, their schools are different in terms of location and students’ background.

These teachers themselves employed different abilities in solving the Gold Mining problem (Figure 3).

From the conversation below, it seems that they have three correct strategies or less to solve it: T1-Ms.

Excel integration and T2 -table, quadratic function and graph. However, there is a significant difference

between the two teachers. T1 allowed the students to solve the task freely (students’ own ways).

The interview with Teacher 1 showed that she has the ability to solve the problem.

Are there other ways T1?

Yesterday, I just did that one.

…just let students find the ways to solve it …. Then, I will let them know that there are some

ways to solve it. I give that opportunity to students

This teacher (T1) would allow her students to approach the task in their own ways. However, T2 had a

different way of letting students approach the task, providing only one strategy.

To me, I could do it directly because I already knew it but to students if I want to students to

learn it, I make a table for them. If the table is not made, students will find it difficult to

solve it for students in my school.

So, you (T2), induce them by using the table?

Yes, by the table.

What do you think, how many ways to solve it?

To me, I did one way I know it directly it would be a square. I knew it already. But for

students, with table, students will measure the perimeter, area, so if the length is 5, how

long is the width, if the length is 10, how long is the width, and.., they will list it, this is how

I let them learn. If I do not do it they will have no clue to solve it.

From the transcript of T2, he seemed to only allow his students to use one strategy. He believed

that his students would not be able to approach the task without inducing the table. He has had previous

experiences where students were unable to complete a similar task.

I have tried several times an easier task, for instance, given the perimeter of a rectangle and

how big is the area, changing from the perimeter to area, I let them do it and facilitated

them, but students were not able. For the story problem, the reading comprehension, the task

asks to go to the East, most of my students go to the West (metaphor).

However, I have thought only one strategy, which is global to solve a task. … I, I... know at

least I understand my students' characteristic so that it will be difficult for my students. … It

is not possible to come up if I let them to do it freely. … I am so careful to give it the various

strategies because students would get confuse

To know how to solve the mathematical task, these teachers tried the problem themselves. During the

interview, T2 seemed to be familiar with the task and had three ways of finding the answer. Meanwhile,

T1 only thought of one strategy.

Yunianto, Prahmana, & Crisan, Indonesian Mathematics Teachers’ Knowledge of Content and Students … 233

By using the strategy of making rectangles with certain sizes and order them and estimate

the biggest area.

To me, I did one way I know it directly it would be a square. I knew it already

…instead of table, we can make the variable x, then I will be a quadratic function,

Are there other ways to solve it?

For the time being, not yet, making rectangles and to the square

Do you think there are still other ways to solve that problem?

I could use the graph …

To some extent, from the lesson plan, T2 gave students a global strategy (table) to solve the task

based on his previous experiences, although there is no guarantee that students would continue to have

the same issues with the task (Figure 5). However, by giving the students the strategy, he inadvertently

is making the students dependent on him. Whereas, from the lesson plan, T1 is helping the students to

make decisions themselves (Figure 4). From the interview evidence, the two teachers have different

abilities in solving the task and differ on the approaches they offer to their students.

In relation to students’ possible mistakes and misconceptions, it seems that these teachers had

some ideas as to what their students would find difficult.

The task has missing information, it should be more, and some students would think that. So

that they have not thought yet the possible ways to solve it. In average, students can directly

solve it with possible ways to do. They can find it directly.

100. Maybe they thought that that’s the only think they know.

… So, they would answer it 100, possibly

Yeah, possibly

… for those who did not understand, they would not know what 100 m rope is to with the

perimeter. So that the concept of perimeter, for those who understood, they already make it

but later they would not think the rectangles can be varied.

Students would confuse the meaning of maximum, which is the largest, they have not

thought about it. So that students' thinking is not yet there. Their thinking is still circulated

on the perimeter not yet the perimeter to area and from area to find maximum area.

Teachers also have ways of responding to students’ mistakes, prompted by the researcher (Figure 7).

The researcher proposed a possible mistake by a student of which the shape looks like a rectangle 25 x

26,5.

Figure 7. A student's possible mistake proposed by the researcher

234 Journal on Mathematics Education, Volume 12, No. 2, May 2021, pp. 223-238

If faced with a student mistake that they have not thought of before, both teachers seemed to

engage thoughtfully with the scenario presented and sought ways of supporting students in addressing

the mistake. Rather than telling a student their answer was incorrect, they asked what the task wants,

and told them to check whether the shape is a rectangle or not.

If it happens if you see this (showing)

I would ask students back to try it then you calculate it as what being asked to you

They have not yet known the result!

Try, try it, by trialing they would know that it is different, this one is more, and that one is

like that, ....

T2, what if your students did this? what would you do?

I would check it first, is it correct or not, the shape is a rectangle or not, they said that it is

not, so I asked whether the perimeter is 100 cm or not. So, by knowing that it is a rectangle,

the length would be equal, and the width would be equal (opposite sides), so that the

perimeter would be 100 cm...

In this study, the lesson plans facilitated an insight into teachers’ knowledge. In this case, it

showed teacher’s pedagogical knowledge as well as PCK. Lesson plans can contain rich information

on how the lesson is expected to be carried out. This is potential data to be used for assessing teachers’

knowledge. How the teachers organize and manages the classroom, task, and the discussion would be

depicted in the lesson plans. This resonates with Burns and Lash (1988) and Simon (1995) who argue

that in developing lesson plans, teachers integrate their knowledge, such as SMK and PCK. On the other

hand, experienced teachers may not use paper planning (written lesson plan) or just outlines as they

have knowledge of what will work best (Butt, 2008; Jones & Edwards, 2010). In addition teachers also

do mental planning for the lesson plans and the lesson plans are not written (Borko, Livingston, &

Shavelson, 1990). The dynamics of a classroom are very fluid, and a teacher must adjust to that fluidity

while following the plan. It is rare for a lesson to go exactly to plan. Yet, the execution of the lesson

plan determines the effectiveness of the lesson (Kow & Yeo, 2008). In Japanese lesson plans, they

contain more detailed instructions (Nakahara & Koyama, 2000) which shows more information about

teachers knowledge. In contrast, the two case of teachers in this study, have not yet shown detailed

instructions but more in general instruction.

Teachers have different ways of supporting students to solve tasks (Yeo, 2008). Students’

performance is more affected from teachers’ PCK (Baumert et al., 2010). However, SMK is basis

knowledge for teachers (Shulman, 1986; Turnuklu & Yesildere, 2007). It is not usual that teachers teach

‘something’ before mastering the subject matter thus reducing the possibility of teaching effectively

(Turnuklu & Yesildere, 2007). The teachers in this study were able to solve the task and had some ways

to respond to students when they made mistakes in solving the given task (possessing SMK and PCK).

However, these results are not generalizable. The limited sample was not chosen randomly and as these

teachers came from relatively developed areas in Java and have at least five years teaching experiences

they are not representatives of the wider Indonesian teaching population. Mathematics teachers in this

Yunianto, Prahmana, & Crisan, Indonesian Mathematics Teachers’ Knowledge of Content and Students … 235

study might not show detail information on their lesson plans and have not fully been aware of

integrating PCK on developing their lesson plans. This study might not cover all mathematics teachers’

PCK profile in Yogyakarta or broadly in Indonesia. However, this study has provided an interesting

glimpse into one part of the very complex decision and knowledge processes that are involved in teacher

pedagogical knowledge.

CONCLUSION

This study indicates that it is possible to assess teachers' KCS of a specific topic through analysis

of the lesson plans when supported by interviews. There is evidence that these teachers had some

knowledge about student strategies and misconceptions about the area-perimeter of rectangle topic, and

that this knowledge was not necessarily fully integrated into their lesson plans. When prompted to think

about possible misconception, the teachers found that it was challenging. Understanding possible

misconceptions, making predictions and the anticipation of student responses would help teachers to be

better prepared in facing the situations during teaching. Developing problem solving skills and

autonomy among students requires teachers to stop providing a particular way (limiting students'

strategies) but rather provide an environment where students are able to choose strategies, to make

mistakes and to explore. Training for teachers could be more supportive in providing pedagogy that

promotes such an environment. Additionally, this study explored a rectangle topic, the result might vary

in different topics. Therefore, further investigation on different topic could be conducted. This study is

not generalizable as it used limited research subjects.

ACKNOWLEDGMENTS

The authors wish to thank to Ministry of Education of Republic of Indonesia; Planning and

Cooperation of Foreign Affairs for the scholarship. The authors also would like to thank SEAMEO

QITEP in Mathematics for its endless support.

REFERENCES

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