Effective teaching strategies which improve learning of geometry

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This chapter describes the overall design of this research work. The main aims of this study are to investigate the effective teaching strategies which can improve the learning of 2D geometry at the upper primary level in Mauritius and to explore other crucial factors related to the pupils' general abilities, their linguistic abilities, their socio-economic status and the educational facilities available at their homes which contribute in the improvement. In fact, a pretest-posttest-retention test design with three experimental groups and one control group is used as the overall research design and is described in details in the following sections.

3.2 Rationale of the Research Design

In this research study, a combination of several research methodologies (triangulation) is used in the investigation of the research questions in order to enhance confidence in the ensuing findings. The triangulation process aims at enhancing the potential insights available from existing data. The idea is to use different perspectives leading to the same results so that one can be more confident about the outcomes and thus, enhance the validity and reliability of the results. Triangulation increases the credibility of the results and is an application for generalizability in findings. It is worth noting that interpretation of the data collected from the subjects of the inquiry is open to bias when not triangulated.

Since for this study the aim is to explore the crucial factors that can influence learning of 2D geometry, multiple-triangulation is used. Multiple-triangulation refers to the use of two or more triangulation techniques in one study. The researcher combines multiple theoretical perspectives, sources of data and methodologies. In fact, theory, methods and data triangulation techniques are used (Cohen & Manion, 2000).

Theory triangulation involves using more than one theoretical scheme in the interpretation of the phenomenon. In this study both a structuralist and interactionist perspectives are considered. Thus, the performance in geometry is analyzed in terms of child-related factors like his/her geometrical skills and overall mathematical ability and in terms of home-related factors like socio-economic status, mother tongue, parent attitude towards education, parent educational background and availability of educational resources and facilities at home.

Methods triangulation is also used at both levels of design and data collection. Multiple measures of the same phenomena are more likely to show various aspects of it. Methods triangulation at the quasi-experimental and observational design levels, also called as between-method triangulation, involve contrasting research methods such as research questionnaires (multiple-choice and open-ended questions) and observation (through field notes and videofilming). Both methods are sequentially implemented in this research work. The researcher blends these approaches at the level of interpretation, merging findings from each technique to derive a consistent outcome. Methods triangulation at the level of data collection, also called as within-method triangulation, involves two different techniques of data collection but each technique is within the same research tradition. It provides a more holistic and better understanding of the phenomenon under study. In this study, two different types of questionnaires (one involving open-ended questions only and one involving multiple-choice questions only) are used to collect data concerning the pupils' performances in 2D geometry. This boost the researcher's confidence in a more reliable measure of the pupil's learning. Beside, any differences in results for the measures become interesting and informative.

Data triangulation is done to gather data through several sampling strategies, so that slices of data on a variety of pupils are gathered. Three types of data triangulation: time, space and person are used. The researcher uses triangulation to collect data. First data is collected just after the experimentation in the form of two questionnaires (one involving multiple choice questions only and the other involving open-ended questions only) and after six weeks in terms of the same two questionnaires as retention tests. Space triangulation consists of collecting data at more than one site. In this survey, data is collected from 4 different primary schools. By collecting data at different points of time and in different spaces, the researcher gains a clearer and more complete sampling space. Using person triangulation the researcher collects data from more than one group of persons. In fact, from each of the four schools, pupils of low, average and high abilities are selected for the experimentation process.

3.3 Sampling Procedure

3.3.1 Selection of Schools

The best sampling method is the one which meets the particular goals of the study in question. It was important to ensure that the samples were representative of students in Grades 4 and 5 in primary schools in Mauritius. It is also crucial in a quasi-experimantal design to control for variables that were not being considered in the study. A sampling strategy was adopted with two particular stages. At the first stage, all the primary schools are classified into three mutually exclusive and collectively exhaustive strata: high-performing, average-performing and low-performing. Then schools are randomly chosen for each of the three categories of schools. To stratify the population of the primary school into three categories whose elements are internally homogeneous, the average pass rate of pupils in Mathematics in the Certificate for Primary Education examinations for the recent years 2003, 2004 and 2005were analyzed.

Table 3.3.1: Number of Schools with their Average Pass Rate of Pupils in Mathematics

Pass rate of Pupils ( %)

0≤x<40

40≤x<50

50≤x<60

60≤x<70

70≤x<80

80≤x<90

90≤x<100

Number of school

2003

7

13

32

55

74

70

30

2004

7

8

42

48

80

70

28

2005

11

10

28

49

77

76

31

The pass rates of pupils in the schools for all the three years approximately follows a Normal distribution with lower quartile range 60≤ x <70 and median range 70≤ x <80. Based on these facts, three groups are categorized and presented in Table 3.3.2.

Table 3.3.2: Percentages of Schools with Corresponding Percentages of Passes

Year

Below 60% Passes

Between 60% to 80% Passes

Above 80% Passes

2003

18.5

44.7

35.6

2004

20.2

45.2

34.6

2005

17.4

44.7

37.9

Thus, all the schools with percentage of passes below 60% are classified low-performing; all those with 60% to 80% passes are classified average-performing and all those with above 80% of passes are classified high-performing.

To involve schools of different categories in the study, one high-performing school, two average-performing schools and one low-performing school were randomly selected. Hence, in total, 4 schools are involved in this study. The schools were internally heterogeneous because each class of the schools contained mixed-ability pupils although in high-performing schools, there were more above-average performing pupils and in low-performing schools, there were more below-average performing pupils. In this way, the sampling frame is a good representation of the different categories of schools in Mauritius.

Table 3.3.3: Schools involved in the Survey

School category

School

Pass rate (in %) Position in the Category

Low-performing

School 4

47.1 Rural region and mostly

low SES background

Average-performing

School 2

School 3

73.0 Urban region and mostly

70.4 middle SES backgroud

High-performing

School 1

85.3 City school and mostly

high SES backgroud

Pretest for Selection of Participants

A pretest involving geometrical items was carried out in each of the four selected schools. The basic aim of the pretest was to have an idea of the prior geometrical knowledge of the participants and to classify their geometrical ability as the pupils were to be assigned to matching groups. In fact, the criteria for the classification of the pupils was as follows: low performer if total marks earned was less than or equal to 50%, average performer if the total marks earned was in the range of 50% to 70% and high performer if the total marks earned was 70% or more. It must be noted that the pretest scores are not used for analysis purposes as the survey involves fourth graders and the latter had not done enough geometry at third grade level. In addition, the classification was validated based on their teacher's view of the overall mathematical ability of the pupils. Generally, it was found that their ability in geometry coincide with their overall mathematical ability.

The pretest questionnaire for the grade four pupils involves 10 multiple choice questions based on their knowledge of the geometrical curriculum up to fouth grade level (See the detailed pretest questionnaire in Appendix 1.1)

The pretest questionnaire for the grade five pupils involves 19 multiple choice questions based on their knowledge of the geometrical curriculum up to fifth grade level (See the detailed pretest questionnaire in Appendix 1.2)

Table 3.2.2.1: Design of the Pretest Questionnaire

No. of Grade 3 No. of Grade 4 No. of Grade 5

topic Questions topic Questions topic Questions

Direct VH1 Direct VH1 Harder VH2 Direct VH1 Harder VH2

Fourth Graders' Pretest Questionnaire requisites

5

2 3

_ _

Fifth Graders' Pretest Questionnaire requisites

5

8 5

1 _

VH1: Van Hiele level 1 VH2: Van Hiele level 2

Table 3.2.2.2: Participants of the Pretest

Number of Pupils

Grade 4

Grade 5

School 1 (High)

91

112

School 2 (Average)

96

101

School 3 (Average)

83

87

School 4 (Low)

88

77

Based on the pretest results and teachers suggestions, the pupils from every class were distributed to the three clusters: high-performers, average-performers and low-performers. Then pupils from each of the three clusters were randomly selected from the class and assigned to different groups for the experimental process.

Quasi-Experimental Design

The quasi-experimental design allows the researcher to alter systematically the variables of interest and observe what changes follow. The concept of causality cannot be proved with certainty but the probability of one variable being linked to another can be established convincingly. The foremost advantage is the researcher's ability to manipulate the independent variable. Consequently, the probability that changes in the dependent variables are a function of that manipulation increases. Also, the control group serves as a comparison to assess the existence and potency of the manipulation (Cooper & Schundler, 2003)

The survey involved both fourth and fifth graders from each of the four selected schools and were assigned to four groups. There were three experimental groups which received three different treatments and one control group with the usual teaching method in the class. The control serves the purpose whether the treatments had significant effects in learning. The same teacher taught all the groups.

The quasi-experimental design involved two independent variables: language of teaching and type of teaching strategies. The two categories for language of teaching were English and Creole and the two categories for teaching strategies were investigative and standard classroom teaching. Thus, it is a 2 Ã- 2 matrix design illustrated in Table 3.4.1.

Table 3.4.1: Two by Two Matrix Design

English

Creole

Standard Teaching Strategies

Group 1

Group 2

Investigative Teaching Strategies

Group 3

Group 4

Further details of the teaching procedures are provided in Table 3.4.2.

Table 3.4.2: Details of the Four Assigned Groups

Teaching Strategies

Description of Classroom Teaching

Group 1

Control

Regular Classroom Teaching

Resource: School Textbook

Language written and spoken: English

Procedure: Teacher demonstrates the examples from the textbook and pupils do the activities. Teacher and pupils interact in English only.

Group 2

Treatment 1

Resource: School Textbook

Language written: English

Language spoken: Creole

Procedure: Teacher demonstrates the examples from the textbook and pupils do the activities. Teacher uses Creole to interact with the pupils.

Group 3

Treament 2

Resources: Concrete materials & manipulatives

Language spoken and written: English

Procedure: Use of investigation and inquiry to teach the lessons. Make use of concrete materials and manipulatives. Teacher and pupils interact in English only.

Group 4

Treatment 4

Resources: Concrete materials & manipulatives

Language written: English

Language spoken: Creole

Procedure: Use of investigation and inquiry to teach the lessons. Make use of concrete materials and manipulatives. Teacher interacts in Creole with pupils.

For every school, the fourth and fifth graders were assigned to the four different groups. Validation considerations require that the groups be equivalent in every respect. In an attempt to increase the likelihood that the groups of pupils in the quasi-experimental design were heterogeneous and equivalent in ability, pupils from every class were assigned to the following three clusters: high-performers, average-performers and low-performers. Every group was formed from at most two classes in such a way that pupils receiving one treatment under a group were matched in ability with pupils receiving the other treatments under the other groups.

Table 3.4.3: Details of the Groups

Number of

High-performers

Number of

Average-performers

Number of

Low-performers

Group 1

7

8

7

Group 2

7

8

7

Group 3

7

8

7

Group 4

7

8

7

Each group had 22 students and in order to better control the effect of gender differences, boys and girls were taken in the combinations of 4 & 3 or 4 & 4.

The administration sector of the four schools provided the researcher with a classroom including a teacher desk, a whiteboard, chairs and tables for pupils. It was possible for the researcher to organize the classroom in a way to suit the survey. All the groups were taught by the researcher and videotaped.

Videotaping for Analysis of Classroom Interactions

For this study, a profession videofilmer was engaged by the researcher. All the lessons taught by the researcher were videofilmed on DVDs. In fact, the researcher explained to the videofilmer the aims of the videos, which was to mainly capture the reactions, gestures and attitudes of the pupils to the different treatments applied in the experimental teachings. Besides, the researcher was always giving the videofilmer signals with hand when he had to film some important episodes of the lessons. After 2 or 3 lessons, the videofilmer was able to do the task more accurately but was still given signals just to ensure that he did not miss any important part of the lessons.

This research work involved video-based classroom observation to explore the use and role of different teaching and learning strategies in 2D geometry. Videotaping enabled scrutinized examination of the different teaching strategies and materials from different points of view, making possible detailed description of many classroom lessons. It increased the opportunity to develop powerful theoretical observational systems to examine the use of resources in the classroom and learning situations.

Teaching Strategies

For every group, the teaching strategies were different but the main concepts of two-dimensional geometry were same. Both fourth and fifth graders of every corresponding group were taught the same lessons as it was also aimed to explore if age and maturation or instruction and learning experiences could lead to significant differences in learning of 2-D geometry. The researcher taught all the lessons to the pupils so that there was no disparity caused by different teaching styles of different teachers. The textbooks used for group 1 and group 2 were the standard school textbooks for grade 4 and grade 5 provided freely by the Ministry of Education and Human Resources to every pupil. As the lessons included both sections of 2D geometry from grade 4 and grade 5 textbooks, the fourth graders were provided with photocopies of the relevant pages from the grade 5 textbook and fifth graders with that from grade 4 textbook. The lessons also involved essential concepts of 2D geometry not found in the textbooks like slant lines, scalene triangles and trapeziums.

The detailed lesson plans are described in Appendices 5.1 and 5.2. The lessons for Group 1 and Group 2 are identical except that Creole is used for explanation in Group 2. Similarly the lessons for Group 3 and 4 are identical with the exception that Creole is used for explanation in Group 4. For convenience, only 2 categories of lesson plans are presented; one for Groups 1 & 2 and one for Groups 3 & 4 and the sections where translation in Creole are required, they are shown in italics within brackets.

The Design of the Classroom

In order to allow a better coordination between the pupils in the learning process, the seating plan was carefully and thoughtfully planned. The pupils had to work in pairs so that interaction could take place. The pupils were all facing the researcher and could see the whiteboard. They can also easily see the instructional aids that could be used. Pupils could do activities and learn cooperatively. Besides, it was easier for the researcher to monitor and help all the pupils (Ramsden, 1999). However, the problem with this type of arrangement is that some pupils were going to have to sit in the corners and at the back of the room. In these locations, pupils generally participate and interact less and more behavioral problems occur (Ramsden, 1999). To avoid this situation, the researcher included a maximum of 22 pupils only in the classroom which was normally met for 35 to 40 pupils. Thus, the pupils occupied about two -third of the classroom and were situated rather close to the researcher. Furthermore, the pupils with high abilities were kept farest from the researcher corner.

Figure 3.4.3.1 : Seating Plan of the Pupils in the Classroom

H

H

H

H

H

H

L

M

L

M

M

H

L

M

L

M

L

M

L

M

L

M

Whiteboard

Researcher Desk

L: Low Performer M: Average (Middle) Performer H: High Performer

It must be noted that troublesome pupils even though very bright and pupils who were very weak especially with writing problems were not involved in the study.

The combination L&H was not allowed as the high-performer would be too dominant.

Generally, pupils from the same class were involved in a particular group. But in School 1 and School 4, in some groups pupils from two different classes were assigned. For such groups, generally the pupils from the same class sat together because they knew each other well and would interact more efficiently.

The combination L&L was not allowed because if both pupils were weak, development and improvement might be a problem.

In some groups where pupils were absent, combination of M&M was allocated.

Teaching Lesson Schedule

The lessons were spread over a period of about four months from March 2008 to August 2008 and they were conducted during school hours. The schools' head masters and the teachers were very cooperative and allowed the researcher to take the classes at his convenience. During these months, there were no other lessons on 2-D geometry for the participating pupils by their usual teachers and the researcher taught the topic based the curriculum and not the schools' teachers work.

Initially, it was planned to have lessons of 75 to 90 minutes for each group where participants for groups 1 and 2 were selected from one particular class and those for groups 3 and 4 were selected from another class. This teaching time schedule was applied at School 1 for grade 4. Although the teachers were not officially complaining, it was observed that they were not able to work for at least half a day during the 5 days of experimental teaching.

Consequently, the researcher changed his teaching timetable for grade 5. Pupils from one class were only selected to represent one group in order to not disturb the same class twice daily during a whole week. Besides, the researcher decided to conduct longer lesson sessions with a break of 15 minutes in-between for both grades 4 and 5.

Table 3.4.4.1: Teaching Timetable

Time

Group

Remark

Day 1

9.00 - 11.45

1

Break of 15 minutes in- between and participants from one class only.

12.15 - 15.00

2

Break of 15 minutes in-between and participants from one class only.

Day 2

9.00 - 11.45

3

Break of 15 minutes in-between and participants from one class only.

12.15 - 15.00

4

Break of 15 minutes in-between and participants from one class only.

Lunch Break: 11. 45 - 12.15

The new teaching timetable was positively welcomed by the teachers as they were disturbed for half a day during two days per week only. Even the pupils were able to cope with the longer lesson sessions.

In fact the longer sessions were beneficial in the teaching of long lessons like types of triangles where the researcher could present the concepts continuously in one lesson. This was particularly helpful for groups 3 and 4 where more time was required to allow the pupils to inquire or investigate.

The new timetable worked perfectly with schools 1, 2 and 3 as they were large schools with more than four grade 4 and grade 5 classes. But School 4 was a small school with only three grade 4 and two grade 5 classes. Thus, the new timetable could not be applied. Luckily, the survey was conducted in the last two weeks of the second term in that particular school. By that time, teachers had already taught the majority of their planned lessons and conducted the assignments (excluding geometry as the researcher met them before the survey and convinced them to allow the researcher to conduct it). Consequently, the researcher was able to take the classes for the whole day and mixed pupils from at most 2 different classes to form a group.

Table 3.4.4.2: Lesson-Teaching Schedule

Grade

Lesson Date

Retention

Date

School 1

4

5, 6, 8, 9 and 13 May 2008

27 June 2008

5

15, 16, 19, 21 and 23 May 2008

27 June 2008

School 2

4

3, 5, 6, 9 and 10 June 2008

18 July 2008

5

26, 27, 29 and 30 May 2008 & 2 June 2008

18 July 2008

School 3

4

11,12,13,16 and 17 June 2008

31 July 2008

5

19, 20, 23, 24 and 25 June 2008

31 July 2008

School 4

4

3, 4, 7, 8 and 9 July 2008

26 August 2008

5

10, 11, 14, 15 and 16 July 2008

26 August 2008

Notes

The questionnaires for teacher, pupil and parent were distributed during the initial lessons and collected before the end of lessons. For the parent questionnaire, the pupil had to take the questionnaire home, made his/her parents filled it and then returned it to the researcher. Even the pupil questionnaires were asked to be filled at home with help of parents

Attendance of the pupils was taken in order to monitor their participation rate and eligibility to do the posttests and retention tests.

Instrumental Procedure

After the intervention, two evaluation instruments were collectively administered in all the groups for both fourth and fifth grades. In fact 2 posttests in the form of questionnaires were conducted in the last day just after the completion of the lessons or one day after the completion of the lessons. The first posttest contained multiple choice questions (MCQ) and the second one contained open-ended questions (OEQ).

Posttest Multiple-Choice Questionnaire (MCQ)

To analyze the level of learning 2-D geometry at the end of the experiment, a posttest containing 31 multiple-choice items was administered. There were 4 answer-options for every item and only one response was correct. It involved item measuring geometrical thoughts and understanding up to Van Hiele level 2 (see the detailed posttest MCQ questionnaire in Appendix 1.3).

Table 3.5.1.1: Design of the Posttest MCQ Questionnaire

Geometrical Skill Required

Visual

Verbal

Drawing

Logical

Applied

VH1

VH2

VH1

VH2

VH1

VH2

VH1

VH2

VH1

VH2

Q11

Q9

Q24

Q21

Q1

Q4

Q6

Q16

Q10

Q25

Q28*

Q2

Q8

Q17

Q14

Q27

Q3

Q15

Q28*

Q5

Q18

Q29

Q7

Q20

Q30

Q12

Q22

Q31*

Q13

Q23

Q19

Q26

Q31*

VH1: Van Hiele level 1 VH2: Van Hiele level 2 * : Require two geometrical skills

The questions set in the posttest MCQ questionnaire tested the skills of geometry lessons from grades 3, 4 and 5. The questions also demanded visual, verbal, drawing, logical and applied geometrical skills. The questions were set in simple plain English for the four groups of pupils. Since there were some word problems in the questionnaire, the time allocated to finish the paper was extended to 40 minutes.

Table 3.5.1.2: Participants of the Posttest MCQ Questionnaire

Number of Pupils

Grade 4

Grade 5

Group 1

Group 2

Group 3

Group 4

Group 1

Group 2

Group 3

Group 4

School 1

21

22

14

15

15

14

21

11

School 2

21

18

13

14

13

15

21

10

School 3

17

15

16

13

22

16

22

9

School 4

11

18

16

12

17

18

21

17

Total

70

73

59

54

67

63

85

47

Posttest Open-Ended (OEQ) Questionnaire

The Posttest MCQ questionnaire was based on paper and pencil multiple-choice items where they had restricted close-ended answers and tested single level of Van Hiele geometry thinking. But in practice, it is also possible that the correct geometrical reasoning of pupils take into account their capacity to use each one of the Van Hiele levels rather than assign a single level. That is, it is possible to identify several distinct ways of reasoning during the acquisition of a Van Hiele level (Jaime and Gutiémez, 1995). Consequently, paper and pencil open-ended items, where the students can freely explain the reasons for their answer are very reliable for assessing Van Hiele levels of reasoning. Such questions can be answered according to several level of thinking.

The posttest OEQ questionnaire was a one-hour paper with five questions divided into several parts measuring geometrical concepts. As decided during the piloting of the questionnaire, the researcher took an additional 10 minutes to explain the instructions given for each question prior to the administration to all the four groups. The questionnaire was completely in English but for groups 3 and 4, the researcher read the instructions in English and explained them in Creole. The test design was based on specific 2D geometry descriptors for the Van Hiele levels 1 to 3.

Generally, the Posttest OEQ Questionnaire was designed to test the following skills: shapes were judged by their appearance and were named, properties of shapes were defined, properties of 2D shapes were discovered by experimentation, logical classification of families of 2D shapes were made, properties of different 2D shapes were compared and simple inferences were carried out on their own

Table 3.5.2.1: Design of the Posttest OEQ Questionnaire

Question

Geometrical Skill Required

van Hiele Level of Reasoning Required

Q1.1(i), Q1.2(i), Q1.3(i), Q2(i)

Drawing

1 or 2

Q1.1(ii), Q1.2(ii), Q1.3(ii), Q2(ii)

Verbal

2

Q3(i), Q3(ii), Q3(iii)

Drawing

2

Q4(i)

Drawing

1 or 2

Q4(ii), Q4(iii)

Verbal

3

Q5(i), Q5(ii), Q5(iii)

Logical

1 or 2

(see the detailed posttest OEQ questionnaire in Appendix 1.4)

As this questionnaire was mostly targeted as a qualitative measure so that one can differentiate between pupils' geometrical abilities, the test was conducted to only a limited number of pupils from each of the 4 groups. Besides, it would have been too costly in terms of administration and analysis. In fact, pupils (both fourth and fifth graders) for groups 1 and 2 were taken from schools 1 & 2 only and pupils (both fourth and fifth graders) for groups 3 and 4 were taken from schools 3 & 4 only. It must also be noted that this questionnaire is analysed separately from the MCQ questionnaire and do not need to match its sample size.

Table 3.5.2.2: Participants of the Posttest OEQ Questionnaire

Number of Pupils

Grade 4

Grade 5

Group 1

Group 2

Group 3

Group 4

Group 1

Group 2

Group 3

Group 4

School 1

11

12

-

-

20

22

-

-

School 2

13

15

-

-

19

15

-

-

School 3

-

-

16

13

-

-

22

16

School 4

-

-

16

12

-

-

21

17

Because of the demanding nature, many questionnaires were left blank or improperly filled especially by the fourth graders. But, still reasonable and sufficient number of scripts was obtained to carry out the targeted analysis.

Retention Tests

The same two posttests were again conducted after 6 or 7 weeks as retention tests. The aim was to measure how well the pupils from different groups were able to remember details of the taught lessons and how significantly their performances were affected with passage of time.

The researcher went to each of the 4 schools again and the MCQ Questionnaires were administered in the regular classrooms. The class teachers invigilated and then submitted the scripts to the researcher. For the open-ended items questionnaire, the same batches of pupils (as in posttest) from each group were called upon and sat for the test in a classroom where the researcher invigilated. The researcher read and explained the questions for the pupils as done in the posttest.

Table 3.5.3.1: Participants of the Retention MCQ Questionnaire

Number of Pupils

Grade 4

Grade 5

Group 1

Group 2

Group 3

Group 4

Group 1

Group 2

Group 3

Group 4

School 1

20

19

17

9

15

12

22

13

School 2

19

13

13

16

15

13

20

13

School 3

18

20

14

14

20

16

18

15

School 4

11

19

14

17

16

17

22

16

Total

68

71

58

56

66

58

82

57

Table 3.5.3.2: Participants of the Retention OEQ Questionnaire

Number of Pupils

Grade 4

Grade 5

Group 1

Group 2

Group 3

Group 4

Group 1

Group 2

Group 3

Group 4

School 1

11

10

-

-

19

21

-

-

School 2

12

13

-

-

19

12

-

-

School 3

-

-

15

13

-

-

22

16

School 4

-

-

14

10

-

-

20

14

Impact of Contextual Factors on Geometry Performance

In this section the instrumentation for exploring the impact of home-related and pupil-related factors are discussed. In fact, two questionnaires were filled by the surveyed pupils and their respective parents and based on the two questionnaires; indexes were created to measure the impact of other contextual factors on learning outcome. A teacher questionnaire was also administered with the school teachers to elicit their view of the geometry curriculum in Mauritius.

Parent Questionnaire

The interaction between the school and its environment is not always harmonious. Several factors like the living conditions of pupils, what the parents expect from the school, the socioeconomic status of pupils and the availability of learning resources at home are of paramount importance in the process of acquiring education especially at the level of basic education (Chinapah, 1983; Carron & Chau, 1996; Mancebon & Mar Molinero, 2000; Griffith, 2000; Sukon & Jawahir, 2005). Consequently, the parent questionnaire was specifically designed to gather information regarding the home environment of the pupils that could affect their learning achievement. The questionnaire requested personal information such as education background and occupation of parents. Information was required on the demography of the home, such as the total number of people living in their houses including the number of relatives, siblings and other resident members of the household. In order to determine the pupils' socio-economic background, information was requested regarding income sources, type of dwelling, availability and quality of amenities, access to transport, possession of information & communication technology and food & nutritional status at home. Items also focused on attitudes of parents regarding the value of financial investment in schooling, their commitment to assist their children in their homework, the degree to which they were concerned with, and participated in school activities, as well as their aspirations for the education of their children. The detailed parent questionnaire is presented in Appendix 1.5.

Pupil Questionnaire

The pupil questionnaire aimed to elicit information on a range of situational, attitudinal and motivational characteristics of learners. It dealt with the following items: pupils' personal characteristics such as sex, language, the extent to which parents assist and encourage them; their attitudes towards school, towards further study, and towards homework given by the teacher as well as their ability regarding the homework's completion. It also measured their access to learning materials, libraries and information & communication technology. A detailed pupil questionnaire is presented in Appendix 1.6.

3.5.4.3 Teacher Questionnaire

The teacher questionnaire was structured so as to generate information on teacher that could assist in explaining variations in the profiles of learner achievement scores in geometry. It paid attention to basic personal and professional characteristics such as academic qualification, training and teaching experiences. It also involved questions related to the teachers' opinion about mathematics and geometry, their teaching strategies, use of teaching resources, importance of geometry in mathematics and language(s) used in teaching. A detailed teacher questionnaire is presented in Appendix 1.7.

Coding Procedure

For data analysis purpose, the questions from the questionnaires which were of ordinal or nominal forms were coded. The methods that were used to code the questions are described in this section.

Coding of questions from parent questionnaire is presented in Appendix 2.1.

Coding of questions from pupil questionnaire is presented in Appendix 2.2.

Coding of questions from teacher questionnaire is presented in Appendix 2.3.

Coding of questions from posttest OEQ questionnaire is presented in Appendix 2.4.

Construction of Indexes

In order to measure the impact of home-related and pupil-related factors in their learning process, several indexes are created from the pupil and parent questionnaires.

Parent Questionnaire

The fact that parents keep faith in the value of schooling and that they are on the whole rather positive about the education provided by schools to their children does not imply that they can shrug their shoulders. The parents need to provide support and reading materials at home to encourage their children in their studies. Based on the parent questionnaire, nine indexes are constructed to measure home-related characteristics. The nine indexes are described in Appendix 3.1.

Pupil Questionnaire

The home-related and school-related variables can be meaningless in the learning process of a child if he/she does not show keen interest in his/her studies. The pupils' personal characteristics contribute enormously towards making him/her successful in his/her learning process. Based on the pupil questionnaire, nine indexes are constructed to measure pupil-related characteristics. The nine indexes are described in Appendix 3.2.

Pilot Study

Although the questionnaires were meticulously prepared by the researcher based on his own experiences, comments from resource persons in the fields & his supervisors and the extensive literature available, they needed to be pilot tested with genuine respondents in order to identify miskeyed and malfunctioning questions. During the month of February 2008, the pilot study was conducted at Rajcoomar Gujadhur Government School to test the following questionnaires: Pre-test for grade 4, Pre-test for grade 5, Posttest MCQ, Posttest OEQ, teacher, and pupil.

To test the 2 pretest questionnaires, 10 pupils (5 boys & 5 girls) from one class of grade 4 and another group of 10 pupils from one class of grade 5 were selected. Based on the recommendation of the class teachers, 3 high, 4 average and 3 low performers in mathematics were selected. First of all, the ten fourth graders were pilot tested in a separate classroom. They were asked to query with the researcher if they encountered any difficulty with the questions. There were no major queries and the researcher was moving around the classroom to see how the pupils were performing. They had no problem with the questionnaire and completed it in less than 10 minutes. The same procedure was adapted for the pilot test of the fifth graders' questionnaire. All the ten pupils (5 boys & 5 girls) completed the 15 multiple choice questions in less than or equal to 15 minutes without any query. Consequently, there was no amendment required in the pretest questionnaires.

As the posttest could only be administered to pupils who had already studied geometry up to grade 5, the two posttests were pilot tested with 10 pupils who had just been promoted to grade 6. In fact a group of 10 pupils (5 boys & 5 girls) with 3 high, 4 average and 3 low performers were selected from a class for the test. They had to work out the 31 multiple choice questions from the posttest MCQ questionnaire in the presence of the researcher and to report any difficulty encountered. Discarding some questions on minor issues, the pupils got no problem and were able to finish the questions in less than 30 minutes.

For the posttest OEQ questionnaire, 2 high, 2 average and 2 low performers (3 boys & 3 girls) were selected from a class of six graders. The initial posttest OEQ questionnaire contained 10 long structured questions to be completed in 1 hour. Because of the demanding nature of the questionnaire, the researcher pilot tested the respondents two by two. The researcher took a seat close to the 2 respondents and observed their ways of proceeding with the questionnaire. There were many questions and queries from the respondents even from the bright ones. In fact, it was observed that pupils were struggling with the long and time-consuming questions. Many pupils, especially low and average ability, left many questions unanswered. The researcher took notes of all the problems in order to better amend the questionnaire. He made considerate modification to the paper especially to the language used and instructions given. In fact, a shorter but concise open-ended questionnaire was re-designed.

The time allocated was increased from 50 minutes to 1 hour. Besides, the researcher decided to take an additional 5 to 10 minutes to explain each question to the pupils before they would start answering them because the real experimentation would be carried out with pupils below grade six.

For the pupil questionnaire also, 2 high, 2 average and 2 low performers (3 boys & 3girls) were selected from a class of grade 4. They had to fill the questionnaires in the presence of the researcher. Except asking for the meaning of a few words, they encountered no problem in filling the questionnaires. Besides, during the survey, the respondents would have to fill them at home with the help of their parents. Thus, no alternation was made in the questionnaire.

Concerning the parent questionnaire, six parents from the researcher's vicinity, whose children were in upper primary grades, were asked to complete the questionnaire to pilot test it. As the questions were written in a very simple and straight forward language, the parents got no problem in answering it. There was no need to amend any question from the questionnaire.

The teacher questionnaire was prepared based on the concepts of geometry. It was cross-examined by a lecturer in mathematics education at primary level and pilot tested with three primary teachers following Master of Education course at the Mauritius Institute of Education, before finalizing its form.

Ethical Issues

In any research work it is always important to give due consideration to ethical issues so as to add value to it and therefore expect a better response from the participants. Prior to conducting the survey, the researcher took care to ensure that the relevant persons, committees and authorities were consulted, informed and that the necessary permission and approval had been obtained.

Firstly, an official written permission letter to carry out the survey in the 4 primary schools was sought from the supervising office of the Ministry of Education, Culture and Human Resources through the head of the school of University of Technology, Mauritius. The letter clearly explained the nature, purpose and procedures of the study. In fact, a detailed survey plan was also attached which showed the step by step procedures of the survey during the 4 months of experimentation at the chosen schools. (See Appendix 3.1for the detailed letter)

The Ministry of Education and Human Resources responded positively to the survey request and granted permission to conduct the survey (See Appendix 3.2 for detailed letter).

The researcher went to informally meet the heads of the four chosen schools and the subject teachers with a written request letter from University of Technology, Mauritius (See Appendix 3.3 for the detailed letters) and a copy of the letter from the Ministry of Education and Human Resources. Firstly, the intention was to make them aware of the Ministry's decision to allow the researcher to conduct the survey and secondly, to take cognition of any problem that it would caused. All the four heads of schools willingly agreed to allow the researcher to conduct the survey in their respective schools. They also made the provision of a classroom for the experimental lessons and showed no disagreement on the researcher's decision to videofilm his lessons.

The researcher also selected 4 grade four classes and 4 grade five classes. He explained to the class teachers the importance of the survey, its significance to the education system of Mauritius and its duration. They were also requested not to teach 2-D geometry to the pupils.

Concerning the teacher, pupil and parent questionnaires, they were anonymous where the participants cannot be personally identified; no formal consent forms of participation were required. However, on the parent questionnaire itself, the identity of the researcher, the aim of the survey and the anonymous nature of information were mentioned.

Statistical Procedure

Statistical methods are used to analyse the multiple-choice questions (MCQ) questionnaire and open-ended questions (OEQ) questionnaire.

The statistical analysis of the MCQ questionnaire is conducted using Quest Software (Adams & Khoo, 1992), a computer program developed from Rasch principles. The Rasch analysis is used to investigate the relationship between the pupils' ability estimates and the item difficulty estimates for both posttest and retention test. A person-item map is constructed for this purpose and the goodness of the fit of each item is computed in terms of 'infit' and 'outfit' statistics. The Rasch analysis provides the hierarchy of task-categories and discriminates ability between estimated level of task difficulty and between pupils with different ability levels.

Besides, the multivariate analysis of covariance (MANCOVA) is used to analyse the impact of the different learning environment on the pupil's achievement in the MCQ questionnaire in both posttest and retention test (in SPSS version 18.0). It shows Box's M test which tests the assumption that the covariance matrices are equal across the groups and the Levene's test which tests the assumption of homogeneity of error variance is met. It also involves partial eta squared as effect size to measure the degree to which the statistically significant results are tenable. In the MANCOVA, pupil's grade and ability are used as covariates, group (treatment) and gender as independent factors and posttest score and retention test score as dependent measures. Roy's largest root test is used to test the effect of each factor on the dependent groups.

The Discriminant analysis (involving eigenvalues, canonical correlation, Wilk's lambda and standard canonical discriminant function) is conducted to investigate the nature of the relationship between the two dependent variables: posttest score and retention test score with respect to the 4 groups.

Descriptive statistics and line-graphs are also presented to illustrate the impact of grade level, gender and pupil's ability on performance of MCQ questionnaire.

A one-way ANOVA test together with Games-Howell test and descriptive statistics are conducted to analyse the five questions set in the open-ended questionnaire.

To analyse the contextual factors influencing performance in Geometry at the upper primary level, three tests are conducted. First, Spearman correlation analysis is used to explore the relationship between pupil-related and school-related variables. Then a factor analysis (using principal component analysis with varimax rotation) is conducted to cluster the variables explaining some common dimensions. Finally, Analysis of Moment structures (AMOS 16.0) is used to analyse the impact of different contextual variables on geometry performance.

Limitations of the Study

Generally, this research work is a well-planned study analyzing the teaching of 2D geometry at upper primary level from several possible perspectives. However, some restrictions had taken place in the research design of the study.

Although this study are meant for upper primary pupils, sixth graders (age 12-13 years) are not involved in the survey. The reason for their exclusion is that they were preparing for the Certificate for Primary Education (CPE) examinations and the Ministry of Education, Culture and Human Resources would not have granted permission to conduct experimental teaching with them. Nevertheless, the fourth and fifth graders (age 9-12 years) are making a reliable sampling frame of upper primary pupils.

Concerning the sampling procedure, initially it was planned to involve equal number of high, average and low performing schools (that is 2 from each categories). But then the sample size would have been too large, about 900 pupils to administer and too costly in terms of preparation of teaching materials.

The OEQ questionnaire was not administered with all the pupils under the study for both posttest and retention test. It would have been too time-consuming to administer and analyze. As it was rather planned to be used as a qualitative tool, only limited groups of pupils were considered.

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