# Mathematics Teachers' Beliefs and their Practices towards Collaborative Learning in Public and Private Schools: A Comparative Case Study.

Byline: Maria Shiraz and Shahzada QaisarAbstract

A multiple case study was designed to find the consistency of mathematics teachers' beliefs towards collaborative learning with their practices. Moreover, it explored the similarities\differences of public and private school teachers' practices who highly believe in collaborative learning. Careful analysis of the transcribed classroom observations into four categories; classroom tasks, classroom discourse, classroom environment and types of evaluation, revealed that the collaborative pedagogical beliefs and practices of both teachers are not consistent. However, the public-school teacher used collaborative layout in some of her classes but could not implement the social pedagogy in true spirit. On the other hand, the private school teacher did not implement the collaborative learning in her classroom.

Keywords: Collaborative learning, Beliefs, Mathematics Teachers' practices

Introduction

A plethera of researches have investigated the relationship between beliefs of teachers and their instructional practices and have reported that the beliefs are in line with their practices (Beswick, 2007; Ertmer, 2005). The studies conducted on teachers' beliefs have projected the connection between instructors' beliefs and didactic practices (Cohen, Moffitt, and Goldin, 2007). On the contrary, there are researches that report the connection between the beliefs and practices of the teachers to be inconsistent e.g. (Ertmer, Ottenbreit-Leftwich, Sadik, Sendurur, and Sendurur, 2012; Phipps and Borg, 2009). Hence, the connection between teachers' beliefs and classroom practices as (Pehkonen, 2009) argues is exceptionally complicated and is often found to be a weak one.

In Pakistan, mathematics research is at the initial phase of development (Alia and Sadia, 2013) and has a lot of unanswered questions. Though collaborative learning (CL) and beliefs of mathematics teachers have been separately investigated in Pakistan (Qaisar and Butt, 2015), yet no study has been carried out to explore the relationship among the beliefs and practices towards a particular teaching approach. In the present study we will illuminate the similarities\differences of the practices of teachers from both public and private schools regarding collaborative learning.

The association between beliefs of mathematics teachers and their practices is highly uncertain and the aim of this research is to seek the consistency between the two constructs. For the purpose of this research we are initially defining mathematics teachers' beliefs and collaborative learning.

Mathematics Teachers' Beliefs

Some researchers have said that beliefs act as a "regulating system" that drive actions and that they are the determinants of actions (Pehkonen, 2009).

Collaborative Learning

Research reveals that in Pakistan collaborative learning in mathematics classroom is limited whereby teachers mostly use a teacher-centered approach (Barwell, 2007; Alia and Sadia, 2013). In collaborative learning students cooperate in groups with the teacher in an effort to develop knowledge (Davidson and Major, 2014, p. 3).

Research Design and Research Questions

A multiple case study design was selected for this study. This qualitative approach is often the preferred choice for empirical inquiry that explores a phenomenon within its real-world context (Robson, 2016; Yin 2014). In this study, the meaning of 'cases' refer to the individual mathematics teachers. Hence two teachers make the two cases of the present study. Guiding this study are the following research questions:

Q.1. To what extent are mathematics teachers' beliefs aligned with their practices regarding collaborative learning?

Q.2. What are the similarities and differences between the practices of public and private school teachers?

Participants and Context

This article reports on data gathered through questionnaires from fifty elementary mathematics school teachers. The analysis shows that only twelve out of fifty questionnaires showed that the teachers believed in the use of collaborative learning. We selected four teachers out of twelve. The grounds for selecting those teachers from the wider study were twofold: (1) their beliefs regarding collaborative learning were higher than the rest; (2) they all had their bachelors in Mathematics and were having an average of three years teaching experience. We are reporting two cases for the present study; "Noureen", who is a public-school teacher and "Sana", who is a private school teacher.

Data Collection

The data was collected over a period of five months, which took place in 2014. The study is divided into the following two stages:

Stage 1: Teachers' beliefs Questionnaire

Mathematics researchers have classified teachers' beliefs into three types: beliefs about the nature of mathematics, beliefs about mathematics teaching, and beliefs about students' learning (Loucks-Horsley, Stiles, Mundry, Love, and Hewson, 2009). The results of the questionnaire helped us to find teachers whose beliefs regarding CL in mathematics were the highest.

Stage 2: Observation (Video Recordings)

Two teachers (cases) were selected; one from the private school and the other from the public school. The researchers observed ten classes of each case. Hence 20 formal observations (video recordings) were done altogether at various intervals during the term. The purpose of recording the classroom practices of these cases was to see the alignment of beliefs of teachers and their practices.

Data Analysis

We used the analytical framework of Raymond (1997) for the analysis of the data of the video recordings which comprises of four themes (table 1).

The Analytical Framework

There are different indicators given with their respective codes for each theme.

Table 1 Criteria for the Categorization of Teachers' Mathematics Teaching Practice

Themes###Indicators###Categories

Tasks###The teacher instructs solely from the textbook###T1

###The teacher instructs primarily from the textbook with occasional###T2

###diversions from the text

###The teacher teaches equally from textbook and problem-solving###T3

###activities

###The teacher solely provides problem-solving tasks###T4

###The teacher selects tasks based on students' interest and###T5

###experience

###The teacher selects tasks that stimulate students to make###T6

###connections

###The teacher selects tasks that promote communication about###T7

###mathematics

Discourse###The teacher approaches mathematics topics in isolation###D1

###The teacher approaches mathematics instruction in the same###D2

###pattern daily

###The teacher primarily encourages teacher-directed discourse,###D3

###only occasionally allowing for student-directed interactions

###The teacher encourages teacher-directed and student-directed###D4

###discourse

###The teacher encourages mostly student- directed discourse###D5

###The teacher poses questions that engage and challenge students'###D6

###thinking

###The teacher has students clarify and justify their ideas orally and###D7

###in writing

Environment###The teacher creates an environment in which students are passive###E1

###learners

###The teacher creates an environment in which students are passive###E2

###learners, occasionally calling on them to play a more active role

###The teacher creates a learning environment that at times allows###E3

###students to be passive learners and at times active explorers

###The teacher presents an environment in which students are to be###E4

###active learners, occasionally having them play a more passive

###role

###The teacher creates an environment that reflects respect for###E5

###students' ideas and structures the time necessary to grapple with

###ideas and problems

###The teacher has students work cooperatively, encouraging###E6

###communication

Evaluation###The teacher poses questions in search of specific, predetermined###Ev1

###responses

###The teacher evaluates students solely via questions seeking "right###Ev2

###answers"

###The teacher primarily evaluates students through set questions###Ev3

###from the textbook, only occasionally using other means

###The teacher evaluates students' learning equally through set###Ev4

###questions from the textbook and alternative means, such as

###observations and writing

###The teacher primarily evaluates students using means beyond the###Ev5

###textbook

###The teacher observes and listens to students to assess learning###Ev6

The Interpretation of Scores

For the sake of drawing out the meaning from the data, we used the following criteria for qualification such as categories of analytical framework to quantification in the form of percentage of collaborative work in the classroom. We admit that there is no statistical ground for these boundaries. It is a kind of arbitrary common-sense scale that will help us to know the extent/nature of collaborative learning of these teachers.

Table 2 Categorization of Collaborative Learning against their respective percentages

Categories###Percentage of Collaborative Learning

T1, D1, D2, E1, Ev1, Ev2###0%

T2, D3, E2, Ev3###25%

T3, D4, E3, Ev4###50%

T4, D5, E4, Ev5###75%

T5, T6, T7, D6, D7, E5, E6, Ev6###100%

When we were arranging the categories from stage 2 (observations) data, we were noting the regularities as well as the patterns within emerging data. The data having similar features were placed within a category in order to accommodate emerging information (table 1). Teachers' teachings were observed and categorized accordingly (figures 1 to 4).

Background and setting

"Noureen" (Pseudonym) was an elementary mathematics teacher in a Government Girls High School. She had a post graduate degree in mathematics. When we explained about our study she was a bit reluctant to video record her lessons. But on the assurance to maintain the confidentiality she agreed to record her classes. The classroom of Ms. Noureen was very comfortable and spacious.

"Sana" (pseudonym) has been teaching in a private school at elementary level. She had a graduate degree in mathematics and a post graduate degree in communication. She was not enthusiastic in teaching and wanted to leave this profession. Ms Sana had no issue in recording her teachings since she had already been filmed by some other researcher. The physical conditions of Ms Sana's classroom were miserable. During electricity failure it became very difficult to continue teaching because of poor lightening.

Teaching Practice

The teaching practices of both the teachers were grouped into four categories (Table 1).

Case-1: Noureen's Classroom Task1

The table 3 shows that the tasks used by Noureen fall under the category T2

Table 3 Analysis of Classroom Tasks (Noureen's Teaching)

Episode Number###Category

Epi-1, Epi-2, Epi-3, Epi-4, Epi-5, Epi-6, Epi-7, Epi-8, Epi-9 and Epi-###T-2

10

Evidence from the Context

The teacher used tasks from the textbook in all the episodes and adopted the same procedure for teaching so the students were familiar with the teacher's source of choosing the tasks which helped them to give the answers (given at the end of book). Following extract is an example:

12.1 Teacher (After exchanging the greetings asks them to) Open 69 Ex 4.11

12.2 Teacher (confirms) Did you complete Part 1 in your homework?

12.3 Students (Altogether) Yes teacher.

12.4 Teacher (smiles) Ok! Let's see how much you remember (pauses, looks into the book)

In episode 3, for the first time Noureen gave a task to the students other than the textbook. The students became very excited and started discussing in their respective groups. They gave their answers however; they were not very much sure about their answer, because they could not check them since were given at the end of the textbook.

Case II - Sana's Classroom Task

Throughout the episodes that were recorded the teacher instructed solely from the textbook. This indicated that the tasks used in the class were non-collaborative.

Table 4 Analysis of Classroom Tasks (Sana's Teaching)

Episode Number###Category

Epi-1 Epi-2, Epi-3, Epi-4, Epi-5, Epi-6, Epi-7, Epi-8, Epi-9###T-1

and Epi-10

Evidence from the Context

The following excerpt shows how the teacher uses the traditional way to solve the tasks on the board.

31.4 Teacher (Asks) Area of trapezium (draws as well)

31.5 Students = A1/2 x h x (sum of 11 side) (all of them)

Such tasks used by the teacher did not stimulate the students to make connections as well as did not encourage them to think of alternative ways of doing mathematics. The teacher follows the same pattern and uses the textbook tasks. She kept walking between desks as long as students noted down the solutions from the board.

Case-1: Noreen's Classroom Discourse

Noureen's classroom discourse was not collaborative.

Table 5 Analysis of Discourse (Noureen's Teaching)

###Episode Number###Category

###Epi-1, Epi-3, Epi-4, and Epi-8###D-3

###Epi-2, Epi-5, Epi-6, Epi-7, Epi-9 and Epi-10###D-4

Evidence from the context

The following extract from episode 4 shows the case

14.9 Teacher (Smiles and asks from the rest of the class) Where have you seen it?

14.10 (All students start discussing)

14.11 Student (One of the student stands up and answers) In our result reports.

It was the routine of the teacher that appeared in almost all the episodes to direct the students about the page number of the textbook. The purpose of this announcement was to keep the students focused without wasting time. However, the discourse that proceeded was teacher led, where the teacher was asking questions and the students are replying. Ethno mathematics activities help the students to understand those concepts of mathematics by linking them with the students' cultural and daily experiences. Such kind of activities used by the teacher while introducing the concept of percentages facilitates the students in developing critical thinking in mathematics.

In the D4 category the teacher encouraged mostly student-directed discourse. Students discussed their confusions with each other as the teacher 'created situations that let' the students to discuss together of how to solve questions first before starting the exercise.

Case II - Sana's Classroom Discourse

The classroom discourse of Sana was not collaborative.

Table 6 Analysis of Classroom Discourse (Sana's Teaching)

Episode Number###Category

Epi-1, Epi-2 and Epi-3###D-2

Epi-5, Epi-6 and Epi-7###D-3

Epi-4, Epi-8, Epi-9 and Epi-10###D-4

Evidence from the Context

Sana's classroom discourse fall under the category D2, whenever the teacher started teaching a new concept. Following is an example:

33.5 Teacher So, Volume = base area x height (tells them and writes)

33.10 In Ex 9c lets see Q 1

The students are listening to the teacher's instruction quietly (lines 33.5-33.9). The teacher did not provide any opportunity to the students to share their ideas with each other. Dialogue-rich mathematics classrooms on the other hand are only possible if students are encouraged to discuss the new concept.

Case-1: Noureen's Classroom Environment

Noureen's recordings show that her classroom environment was collaborative.

Table 7 Analysis of Classroom Environment (Noureen's Teaching)

Episode Number###Category

Epi-1, Epi-3, Epi-5, Epi-7 and Epi-9###E-3

Epi-2, Epi-4, Epi-6, Epi-8 and Epi-10###E-4

Evidence from the Context

In the E3 category the teacher creates a learning environment that sometimes allows students to be inactive learners and at times active explorers. Following segment is an example of this.

16.28 Teacher (Writes) 16A1/2 %

16.29 = 33/2 % (Students are actively participating by answering the question)

The students would have learnt the above taught concept more effectively through investigation and discovery. Although the teacher encouraged learner participation, there is still a step-by-step direction by the teacher.

Episode 1 is an example of E4 category where Noureen gives the students the autonomy to discuss and share their findings within the group. Students were seen to 'enjoy' a close relationship with peers.

Case II - Sana's Classroom Environment

The classroom environment provided by Sana was not collaborative.

Table 8 Analysis of Classroom Environment (Sana's Teaching)

Episode Number###Category

Epi-1, Epi-2 and Epi-3###E-2

Epi-4, Epi-5, Epi-6, Epi-7, Epi-8, Epi-9 and Epi-10###E-3

Evidence from the Context

In the E2 category the teacher creates an atmosphere in which students are not active, occasionally asking them to play a more active role. The following teaching scenario portrays the E2 category.

32.12 Teacher (Writes and Explains) Q 1 (a) M.P = $ 100, S.P - and 88

32.13 Teacher [So] discount = (100 - 88) = $ 12 (students are attentively looking at the board)

32.14 Teacher [Now] percentage Discount = Discount/M.P x 100

As in the above example it is noted that the teacher is doing all the work. She is explaining the process of finding discount while on the other hand students are passively sitting and listening to the teacher. However, in a constructivist classroom it is the teacher who is responsible for establishing a conducive learning environment that develops the interest of children towards mathematics.

In episode 7, the students' role is passive in the beginning since they are listening to the teacher's explanation. However, it is seen that after Sana's explanation when she asks the students to "do the next questions together"; the students become "very active" and start the discussion. Consequently, the classroom culture becomes positive and lively.

Case-1: Noureen's Classroom Evaluation

The assessment procedures used by Noureen were non-collaborative.

Table 9 Analysis of Classroom Evaluation (Noureen's Teaching)

Episode Number###Category

Epi-1, Epi-2, Epi-3, Epi-4, Epi-5, Epi-6, Epi-9 and Epi-10###Ev-3

Epi-7 and Epi-8###Ev-4

Evidence from the Context

According to the Ev 3 category, the teacher primarily evaluates students through a set of questions from the textbook, only occasionally using other means. Students were seen to be mostly evaluated from the previous day's lesson at the beginning of the class. This extract justifies the case.

15.2 Teacher What did we do yesterday Madiha? (the teacher goes near her)

15.3 Madiha (Confidently answers) Percentage; Ex 4.14

In the Ev4 category the teacher evaluates students' learning equally through set questions from the textbook and other means, such as observations and writing. The teacher is seen using her observations when one of the students who is a bit confused about the working of the question posed by the teacher. When some of the students are unable to solve the sum in their notebooks, the teacher facilitated them by asking if anyone could solve the question on the board.

Case II - Sana's Classroom Evaluation

The evaluation procedure adopted is non-collaborative.

Table 10 Analysis of Classroom Evaluation (Sana's Teaching)

Episode Number###Category

Epi-1 Epi-2, Epi-3, Epi-4, Epi-5, Epi-6, Epi-7, Epi-8,###Ev-1, Ev-2

Epi-9 and Epi-10

Evidence from the Context

This example illustrates the assessment strategies the teacher used in her class time

38.4 Teacher (writes)Q1 (a) 2/7 x 90Adeg = 180Adeg/7 = 25.7

38.5 (asks) Is it acute, obtuse or right angle?

38.6 Students It is an acute angle (all)

Teachers' questioning strategies also play a vital role in the quality of instruction students receive and in the evaluation process. Teachers can foster students' reasoning ability by asking questions that promote student thinking. However, our interpretation both at the time of this episode in the classroom and upon later analysis was that Sana was just evaluating them by asking very basic questions that were based on recall and did not provide opportunities to the students to think creatively and collaboratively.

Discussion

Sewornoo (2016) states that the beliefs of teachers have a strong impact on teachers' instructional practices. However, the results of current study show that both the teachers' beliefs are not completely aligned with their instructional practices towards collaborative learning. Such types of discrepancies are also reported by many other researchers (Thompson, 1992). Classroom task, classroom discourse, classroom environment, and classroom evaluation are used as analytical framework's categories to identify whether the collaborative practices are being followed by the teachers. These categories are discussed in detail as follows:

Classroom Tasks

The results show that project teachers did not practice collaborative tasks. The tasks used by both the teachers are mostly close ended practicing tasks that did not provide the opportunity to the student for thinking actively. Both teachers frequently selected tasks from the textbooks. We are not challenging whether the nature of tasks given in the textbooks are collaborative or non-collaborative, however; one of the indicators of the analytical framework informs that tasks chosen from the textbook are considered non-collaborative. Hence, it was very rarely observed that when both teachers used the tasks from another source or developed them on their own. They did not use any tasks that stimulated students to make connections through problem solving activities. In collaborative learning tasks students are given the opportunity to explore new ideas through discussing, questioning, and organizing processes.

This then helped to develop students' comprehension and internalization of critical concepts (Palincsar and Herrenkohl, 2002). It is also apparent from different researches (Kabooha, 2016) that the textbook is not the only source that the teachers should rely on for their task selection. It is expected from the teachers that they should develop and use constructivist tasks in their classrooms for effective learning of mathematics.

Classroom Discourse

Both public and private teachers controlled the behaviour of the students and stressed that they must memorize all definitions. And the teachers indicator "these definitions are very important from the examination point of view" also encourage to do so. The teachers explained all the concepts without involving the students. The conceptual explanation is not done by the teachers. According to Gilbert (2006), a mathematical explanation consists of two parts. The first one is the calculation explanation that describes the process of reaching to an answer and second one is the conceptual explanation that rationalizes as to why a process was selected. Hence, students will not only be able to solve a mathematics question but will also be able to justify as to why they choose a particular method to solve the given question.

On the contrary, in collaborative learning teachers focus on such context where definitions are socially constructed within a discourse community. So, both teachers have transmission absolutist view in which the teacher is the dispenser of knowledge. Although, Noureen had a professional training before joining the schools but her classroom discourse was not different from Sana.

It was observed in both cases that the classroom discourse was more teacher-led. This may often be because of the expected classroom management issues that accompany as well as a perceived urgency to complete a syllabus (Ingram, 2012).

Classroom Environment

Noureen's classroom environment was very appropriate. It had all the basic facilities (space, proper lighting). Students were given tasks to solve in their respective groups. Noureen's teaching practices with respect to environment (as far as the physical layout of the classrooms is concerned) seemed to align with her beliefs. In Noureen's class the students were given the opportunity to discuss the mathematics problems in groups but the nature of discussions was controlled by the teacher that did not motivate students to evaluate their understanding and explore their errors on their own. Teachers' intervention rate was very high during collaborative work since she did not want to lose her authority. On the contrary, it is essential in a collaborative classroom environment that students have the opportunity to discuss mathematics with one another, refining and critiquing each other's ideas.

It seemed to us that Noureen knows how to make a layout of collaborative work but could not implement it in 'true sense' where students construct their knowledge on their own and role of teacher is very passive and he provides process help if needed.

Sana's classroom environment did not indicate any signs of collaborative learning; although she believed in collaborative learning. It may be assumed that Sana was trying to avoid management problems. Menzies and Bruhn (2010) argue that environmental factors, such as size of the classroom and basic facilities may affect the motivation of students to engage verbally in the classroom and perform specific activity patterns. In Sana's case she did not put her effort in developing collaborative learning environment. Classroom space seems to be an issue nowadays for private schools which are merely money-makers who keep on having admissions to raise funds. Sana made same textbook diagram on the board even she may have referred towards book but it seems she was just killing her time. In Pakistan, just like Sana mostly teachers join the teaching profession as their last choice and do not have professional training.

They get jobs on the basis of their highest qualifications and English proficiency. Females are seen in majority in this profession since according to them and their families it is the 'most respectable profession and they feel more secure in same gender school.

Classroom Evaluation

Noureen's patterns of evaluation were not collaborative. She called only those students who knew the answers. One of the reasons might be that she does not want to waste time on students' wrong answers. However, in some collaborative learning classroom students when engaged in questioning should have equal participation of all the students. Similarly, Sana did not bother to ask any question from the previous day's lesson. This may be because she thought that assessing them at the beginning of the lesson may be a burden to her teaching. Classroom assessment should not be viewed as an added requirement but should be thought as a means to improve teachers' teaching. Although Noureen had gone through a teachers' training program however, her evaluation methods were not better than Sana who had not gone through any professional training.

It seems that one of the factors of having no difference in assessments process between trained and untrained teachers was what Christenson et al. (2008) explains that most teacher education programs skim over classroom assessment, leaving teachers to assess in the way they were assessed when they were in school. The training programs that Noureen had gone through were not rigorous in terms of collaborative assessment techniques. As Crane et al. (2012) suggest that specific training is necessary for teachers to learn how to foster collaborative assessment in their classrooms.

Conclusion

The results of these video recordings revealed that although both the teachers highly believed in collaborative learning but their beliefs were not practically seen in their classrooms. The classroom practices of both the teachers were similar in terms of classroom task, classroom discourse and classroom evaluation. However, they appeared different in terms of classroom environment.

We admit that the relationship between beliefs and practices of mathematics teachers is a complex phenomenon and teachers' beliefs were seemed to play an important role in 'modulating' teachers' practices but the result of this study show that it is not always true that the teachers' practices are formed by their beliefs. Hence in order to bring any reform in the teaching practices of mathematics teachers in Pakistan it is necessary to find the influence of other factors other than teachers' beliefs that influence teaching learning process. The findings in this article therefore, indicate a need for further research. Specifically, a study needs to be done in finding the constraints and the extent to which they directly influence the instructional practices decisions teachers make against their beliefs. The present study is inevitably limited because of the methodology that has been adopted to conduct the study.

Although the data provides us with in depth understanding about the phenomena; still it is gathered from a very small number of cases. Thus, we are not in a position to make generalizations. It must be borne in mind that more research is needed to identify the constraints\opportunities of teachers who believe in practicing collaborative learning in mathematics classrooms.

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Author: | Shiraz, Maria; Qaisar, Shahzada |
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Publication: | Journal of Educational Research |

Article Type: | Case study |

Date: | Dec 31, 2017 |

Words: | 5646 |

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