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Looking Through Different Points of Views: Characteristics of a Good Mathematics Teacher

Mathematics is essentially used in many fields. Due to its importance, nurturing mathematical knowledge in students is deemed crucial in education now. Providing a good mathematics teacher is an important factor in developing students' mathematical skills and knowledge. This study aims to explore the characteristics of a good mathematics teacher from the points of view of teachers and students. In order to investigate the characteristics, this study employed a qualitative method using semi-structured interviews and focus group sessions involving twenty-three informants consisting of mathematics teachers and students. The informants were purposively selected from several schools in the southern region of Malaysia. Once the data were collected, they were qualitatively analysed using thematic analysis. Based on the findings, there are two main perspectives on the characteristics of a good mathematics teacher based on the teacher's and student's perspectives. Teachers define the characteristic of a good mathematics teacher from the aspects of the teacher's knowledge and experience. In comparison, students viewed it from the aspects of teachers' personalities, teaching styles, and delivery instructions. So, in conclusion, mathematics teachers and students have different points of view when describing the characteristics of a good mathematics teacher.

abilities, and social-relational skills (Bullock, 2015). With the specific roles of teachers, these factors should be related to the teachers' essence to describe the characteristics. What kind of particular personalities are required to make a good teacher, and what behaviours should good teacher displays.
The teacher's teaching abilities should also be taken into account, and how a good teacher displays a good teacherstudents relation to show that the teacher possesses interpersonal skills. According to the pedagogies' requirement changes, students' needs and expectations have continuously changed throughout the decades. Since the radical shift in pedagogical engagement, such as studentcentered inquiry and problem-solving, had been widely advocated by educators (Haider & Jalal, 2018), teacher's roles also have been evolved; from the traditional position to that of a coach in students learning process (Blackie, Case & Jawitz, 2010).
Students' ideas should be heard, listened to, and considered because they are the ones who undergo the classroom experience with their respective teachers, and they are the ones who are directly affected by good and bad teaching. Students in action in the teaching and learning environment know better about what will suit their learning preferences to make the process much more effective (Follman, 1995). Understanding how students learn will help teachers find the best-suited way to teach a subject (Abdurrahman, Abdullah & Osman, 2020 According to Anthony and Walshaw (2009), ten prime concepts of an efficient mathematics pedagogy should not be neglected (refer to Figure 1). But to be clarified as part of a complex web of aspects affecting students' learning process.
These concepts include elements of methods related to the classroom community, a discourse, tasks that enhance students' thinking, and the important role of the teacher's knowledge.

A. Objectives
This study explores teachers' and students' perspectives on the characteristics of a good mathematics teacher. Based on the literature review, various studies have been conducted on the characteristics of good teachers from teachers' (Bayar, 2014) or students' (Martinez-Sierra, 2014;Ifeanyi, 2016) perspectives. Only a few studies specifically study the characteristics of good mathematics teachers and compare both parties' perspectives to find the overlapping characteristics. So, in this study, we focus specifically on teachers that teach mathematics subjects and take into account both mathematics teachers' and students' perspectives and find the common agreement in both.
• To explore the characteristics of a good mathematics teacher from the teachers' point of view.
• To explore the characteristics of a good mathematics teacher from the students' point of view.

B. Samples
A purposive sampling technique was used to select the participants. Purposive sampling is a not random technique that does not need any base theory or an exact number of informants. The researcher decides what must be known and built out to find a voluntary individual to provide the information by the integrity of education or experience (Bernard, 2002).
For this qualitative research, the participants selected are five secondary school teachers who teach mathematics and sixteen secondary students aged 13 years old to 17 years old from schools in the southern region of Johor. Students of this age range were chosen because pupils of lower age will be less suitable as the research involved a focus group session. This age range involved students from form 1 to form 5 of secondary school that learn mathematics as a compulsory subject. According to the interview protocol, the informants who anticipated may have given sufficient information to answer the questions.

C. Instrument
Data were collected through a semi-structured interview and focus group session. The interview was conducted using an interview protocol as a guide. The questions asked during the interview are as below: 1. How should a good mathematics teacher be? 2. What qualities do you like about (your) mathematics teacher?
3. In your opinion, (if you are) what makes you or will make you a good mathematics teacher?

D. Data Analysis
Data collected were analysed using thematic analysis, following Braun, Virginia and Clarke, Victoria (2006) recommendation. The interviews and focus group sessions were recorded using an audio recorder and transcribed by the researcher. The transcriptions were re-read several times before the coding process, and the emerging subthemes were conceptualised and categorised into two major themes. An expert for its consistency then cross-checked the themes.

A. The Characteristics of a Good Mathematics Teacher from the Teachers' Point of View
The characteristics of a good mathematics teacher were described using five main criteria: awareness in mathematics education, teacher's knowledge, teaching styles, personality, and the teacher-student relationship.
Each criterion is composed of a set of acceptable characteristics to the students.

Awareness in Mathematics Education
The first main criteria that lead to the characteristics of a good mathematics teacher from a mathematics teacher's point of view is the mathematics teacher's awareness in the field of the mathematics education field.
Mathematics awareness is crucial for a teacher to notice, articulate, and interprets aspects of classroom practice.
Awareness in this context refers to the teacher's knowledge and is rooted in the context of the actual classroom situations (Potari, 2013).
Quotation from Teacher 1 directly described the importance of high motivation in molding the characteristics of a good mathematics teacher.

"…A mathematics teacher who enters her class with spirit
and looks enthusiastic is the best, bringing all the motivation into her class…" (Teacher 1). Teacher 3 and Teacher 4 mentioned the importance of mathematics teachers acknowledging and being aware of mathematics anxiety symptoms among students.
"…Mathematics teachers should be observant in monitoring classes; mathematics anxiety among students is something that mathematics teachers should be cautious about…" (Teacher 3). " The assumption that mathematics anxiety started in school is partially false. Parents, too, play vital roles in these situations, and home is the start of everything. But of course, teachers too play a very crucial part in this, and being aware is the start." (Teacher 4).
Yanuarto also supports these opinions of mathematics teachers participants in his paper; mathematics anxiety has its roots in teaching and teachers. It has been tied to the poor academic performance of students and the effectiveness of elementary mathematics teachers.

Mathematics Teacher's Knowledge
The second criterion mathematics teachers mention in becoming a good mathematics teacher is the teacher's knowledge of the subject. According to them, general knowledge and syllabus/content knowledge are the two types of knowledge specifically essential for a good ensure the exercise be used as a reference later on and not just as a practice.
"…When the teacher gives exercise, it will be beneficial if the activity was arranged systematically from easy questions to harder questions…" (Student from Focus Group 2).

Teacher's Instruction Delivery Styles
Students also believe that a good mathematics teacher should have a specific instruction delivery style different from other teachers (Adler & Davis, 2006).  styles, their ability in teaching, and their management of using teaching aids. But both mathematics teachers and students share the same point of view on the characteristics of a good mathematics teacher. They agree that a mathematics teacher's knowledge level is essential in determining whether the mathematics teacher is good or vice versa. Both teachers and students share similar opinions on the importance of a good mathematics teacher's teaching style and personality. While both parties agree on