**Prospective Teachers’ Beliefs of an ideal Mathematics Classroom: Changes Over Time **

**By Dhitta Puti (Sampoerna University)**

The term 'beliefs' is usually used in many ways.

Opinions, perceptions, values, judgments, are examples of terms used to define beliefs. Despite that beliefs have a variety of meanings; beliefs can be differentiated from knowledge.

The foundation of someone's belief is their evaluation or judgment.

Teachers' beliefs are important because it affects how they act, including when they teach in the classroom. They also influence how teachers interpret and perceive teaching situations and how they respond to those situations (König, 2012). This research focuses on the prospective teachers' beliefs of an ideal mathematics classroom - the beliefs that teachers have when they before they really become teachers.

Throughout their time in university, prospective teachers' process of learning, both inside and outside of the classroom might affect the beliefs of their ideal mathematics classroom. As part of the curriculum, all prospective teachers, who are students at the Mathematics Education (MEdu) Study Program at the Faculty of Education (FoE), Sampoerna University (SU) must take courses related to mathematics and mathematics teaching. They must also join the School Experience Program (SEP).

This research aims to qualitatively portrait the changes of beliefs that the prospective teachers have since the middle of their second year until the end of their third year in university. The data from this research is taken from the reflection two prospective teachers made in February 2017 and an interview done on the same prospective teachers in May 2019. Specifically, this research addresses the following questions: (1) What kind of changes do the prospective teachers have regarding their belief of an ideal mathematics classroom? (2) What aspects did affect those changes? By answering these two questions, we might gain insights to understand what teacher education institutions can do to support prospective teachers in analyzing their beliefs about teaching and learning and creating new visions about teaching and learning.

To portrait prospective mathematics teachers' changes of beliefs of an ideal mathematics classroom, a qualitative methodology is used. Qualitative research is chosen because it ‘places more emphasis on the study of phenomena from the perspective of insiders’ (Lapan et al., 2012). In this research, the ‘insiders’ are the prospective mathematics teachers who participated in this

Two participants, P1M and P2F took part in this research. They both are prospective teachers in the MEdu, FoE, SU. P1M is male while P2F is female. The data for this research is taken from their written reflections, about their ideal mathematics classroom, which they made in February 2017. The reflection was part of their assignment in a course they took in the second year of the university named ‘Teaching and Learning Mathematics (TLM) 1’. It should be noted that before PIM and P2F wrote their reflection, the author, as the lecturer gave an example of her own ideal mathematics classroom. They were also asked to read the book chapter “Introduction: Understanding the Urgency” from “What’s Math Got to Do With It?: How Parents and Teachers Can Help Children Learn to Love Their Least Favorite Subject” written by Boaler (2015). Both, the example and the reading might have influenced their writings.

In May 2019, both participants were already in their third year of university. They were interviewed to find out whether have they changed their belief in an ideal mathematics classroom. Firstly, they were asked to reread their own writing, about the ideal mathematics classroom, which they made in 2017. Then, they were asked to describe the kinds of changes in their belief, if any. They were also asked to explain the reasons why their beliefs changed.

Based on the findings, both participants have changed some of their beliefs about an ideal mathematics classroom. P1M explained that his perspective on how mathematics should be taught has changed. On the other hand, P2F's belief in an ideal classroom did not change much.

P1M used to think that students’ learning happens when teachers explain a material and students listen. Although his view is not completely wrong, in 2019 he noticed that there are other ways of learning mathematics. After reading his previous writing, P1M said he had changed most of his ideas about an ideal mathematics classroom:

“Before I believed that teaching mathematics meant the teacher explains then the students listen. Now my view has changed. I now believed that mathematics can be learned in many other ways, for example through games or through creating projects.”

P1M mentioned two main events that changed his idea of an ideal mathematics classroom. First, his experience during SEP. Second, the Problem and Project-Based Learning Course he took during his second year at university.

P2F has always believed that her ideal mathematics classroom would be a place where her students felt comfortable. She talks about an ideal mathematics classroom through a picture.

The picture means that her ideal mathematics classroom is where her students feel safe and comfortable just like at home. She wanted to coordinate with each other to solve problems, this will resulted in faster problems solving. She wanted to create a situation where her students are interested in exploring everything, but still in her control. In other words, students have a great sense of curiosity. Lastly is communication, which is very important in the classroom. When there is good communication, students and teachers are become closer and understand each other and prevent misunderstandings. Therefore, serious but relax is her principle of teaching.

After reading her previous writing, P2F said that there are still values, which are reflected in her ideal mathematics classroom that she still believes in. However, P2F’s belief in ideal mathematics has changed slightly. She started to realize that learning mathematics can be done by integrating technology. This view was influenced by an “Educational Technology” which she took. Besides that, she realizes that most of her lecturers use Learning Management Systems (LMS) for teaching. Seeing that, she imagines that her ideal mathematics classroom could happen not only through face-to-face sessions but also online.

Furthermore, P2F also was affected by a workshop she attended on Hypothetical Learning Trajectory (HLT). The facilitator of the workshop exhibited various ways of using GeoGebra for teaching mathematics. P2F also explained that now she considers play as an important part of learning mathematics. This was affected by two of the courses she took which were Social Emotional Learning and Number Theory. P2F also took two courses which were TLM 1 and TLM 2. Both encouraged her to read more references on teaching and learning mathematics. Besides the courses she took, P2F's belief of an ideal mathematics classroom was also influenced by SEP.

This research reveals that prospective teachers’ belief in an ideal mathematics classroom is not static. From February 2017 until May 2019, the prospective teachers’ beliefs have changed. Both P1M and P2F are affected by the courses they take in university. Although both take the same courses, what influences each prospective teacher are not the same. P1M was influenced by the Problem and Project-Based Learning Course, while P2F was influenced by many other courses such as Educational Technology, Social-Emotional Learning, Teaching and Learning Mathematics 1 & 2, and Number Theory. P2F's change of belief in an ideal mathematics classroom was also affected by her readings about teaching and learning mathematics. P1M and P2F were both influenced by their clinical practices in schools.

The finding shows that the education that the prospective teacher gets does affect their beliefs. Consequently, teacher education institution must design their curriculum carefully, by considering the beliefs the prospective teachers have and might have in the future. Further research regarding this issue can be done by analyzing the changes of prospective teacher beliefs about their ideal mathematics classroom in teacher education institutes programs. This may give a broader insight to understand how change can happen in different teacher education contexts.

**Dhitta Puti (Sampoerna University)**

Dhitta Puti has 19 years of experience in education, with 10 years of experience in teacher educatin (pre-service & in-service teachers. She holds a bachelor’s degree in Engineering from Institut Teknologi Bandung (ITB) and a master’s degree in Education from University of Bristol. She joined Sampoerna University in 2011 as the lecturer at the Mathematics Education, Faculty of Education. She has taught various subjects such as Foundation of Teaching & Learning, Precalculus Algebra & Trigonometry, Educational Research Methods or Foundations. Besides teaching, she also actively participates as the content coordinator of *Gerakan Nasional Pemberantasan Buta Matematika* (Gernas Tastaka).

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