Mar 12, 2009 - of Social Sciences, International Islamic University, ... Accepted by the Department of Education, Faculty of Social ... External Examiner 1: ...
RELATIONSHIP BETWEEN MATHEMATICAL THINKING AND ACHIEVEMENT IN MATHEMATICS AMONG SECONDARY SCHOOL STUDENTS OF NORTH WEST FRONTIER PROVINCE, PAKISTAN
AMIR ZAMAN 44 – FSS / Ph.D. (EDU) / F05
DEPARTMENT OF EDUCATION FACULTY OF SOCIAL SCIENCES INTERNATIONAL ISLAMIC UNIVERSITY ISLAMABAD 2011
RELATIONSHIP BETWEEN MATHEMATICALTHINKING AND ACHIEVEMENT IN MATHEMATICS AMONG SECONDARY SCHOOL STUDENTS OF NORTH WEST FRONTIER PROVINCE, PAKISTAN
By AMIR ZAMAN 44 – FSS / Ph. D. (EDU) / F05
Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in Education at the Faculty of Social Sciences, International Islamic University, Islamabad.
SUPERVISOR: Dr. R. A. FAROOQ
DEPARTMENT OF EDUCATION FACULTY OF SOCIAL SCIENCES INTERNATIONAL ISLAMIC UNIVERSITY ISLAMABAD 2011 i
ii
DEDICATED
To
MY PARENTS, WIFE AND CHILDREN
WHOSE PRAYERS, LOVE AND PATIENCE TOOK ME TO THE APEX OF GLORY AND TRANSFORM MY DREAM INTO REALITY
iii
APPROVAL SHEET RELATIONSHIP BETWEEN MATHEMATICAL THINKING AND ACHIEVEMENT IN MATHEMATICS AMONG SECONDARY SCHOOL STUDENTS OF NORTH WEST FRONTIER PROVINCE, PAKISTAN By AMIR ZAMAN 44- SS- (Ph.D. ( Edu.) / F05 Accepted by the Department of Education, Faculty of Social Sciences, International Islamic University, Islamabad as partial fulfillment of the requirements for the award of degree “DOCTOR OF PHILOSOPHY IN EDUCATION” Supervisor: ______________________________ ( Dr. R. A Farooq)
Internal Examiner: _________________________ (Dr. Nabi Bux Jumani)
External Examiner 1: __________________________ (Dr. Maqsud Alam Bukhari)
External Examiner 2: _________________________ (Dr. Javed Iqbal)
Head,
Dean,
Department of Education, International Islamic University, Islamabad.
Faculty of Social Sciences, International Islamic University, Islamabad. iv
SUPERVISOR’S CERTIFICATE
It is certified that the contents and form of thesis entitled” Relationship between Mathematical Thinking and Achievement in Mathematics among secondary school students of North West Frontier Province, Pakistan” submitted by Mr. Amir Zaman registration No. 44-SS/Ph.D. (Edu) /F05 have been found satisfactory for the requirement of degree.
Dated:
/
/ 2011
Supervisor: ______________________ (Dr. R. A. Farooq)
v
ACKNOWLEDGEMENTS I feel pleasure to honor Higher Education Commission of Pakistan for its award of scholarships under its indigenous scheme to pursue my Ph.D. studies at International Islamic University, Islamabad I also honor International Research Support Initiatives programs for Curtin University, Australia to complete my Ph.D. thesis successfully. I wish to express my gratitude to Secretary Education North West Frontier Province for granting me with study leave without which I would have faced a lot of troubles in accomplishment of this research work. I am highly grateful to my supervisor Dr R.A. Farooq and external supervisor, Dr Bill Atweh for their encouragement and patient guidance in completion of this thesis. I am highly thankful to my wife, parents and other family members whose patience and pray enabled me to complete this work. I can forget my friend Dr Ma’Mamoon Mubarak who provided me useful advices during tool development.
I
would like to thank all of my friends specially Mr. Alamgir, Mr. Atiq-ur-Rahman and Mr. Manzoor Ahmad who all have generously supported me, during my stay abroad, in the accomplishment of this research.
Amir Zaman March, 2011
vi
ABSTRACT This study aimed to investigate the relationships between mathematical thinking and achievement in mathematics in the 9th grade scientific stream students in North West Frontier Province, Pakistan which was renamed as Khyber Pakhtunkhwa by the parliament of Pakistan through 18th constitutional amendment in 2009. In addition, this study also investigated the gender, school sector, and school location differences in mathematical thinking and mathematics achievement. Besides, this study also investigated the interaction between gender and school location, gender and sector, sector and location, in relation to mathematical thinking and achievement. Further, this study examined conceptions of teachers about mathematical thinking and any inconsistencies between the teachers’ opinions about the aspects of mathematical thinking in terms of level of importance, level of difficulty and the results derived from the quantitative analyses of the student answers to the mathematical thinking test. In relation to the student interviews, the study also aimed to identify popular strategies used by students to arrive at solutions, question understanding, their perceptions about difficulty level of aspects of mathematical thinking and to differentiate between the thinking skills they used and to ascertain their attitude towards optimal solutions like checking their solutions. To achieve the above mentioned objectives, a model of mathematical thinking was first developed representing the best possible six aspects of mathematical thinking relevant to the curriculum of grade 9 in Pakistan with the help of experts and practising teachers. A test of mathematical thinking was developed based on this model to measure vii
mathematical thinking of the students. Besides a test of mathematics achievement with items taken from textbook of grade 9 and two interviews (one for teachers and one for students) were also conducted for data collection. Quantitative data was analyzed using t-test, Anova and regression analysis. Similarly qualitative data from interviews was analyzed using qualitative data analysis techniques. Main findings of the study showed that mathematical thinking was moderately related to mathematics achievement with proofs as the most correlated aspect.
The study further showed that male students
outperformed their female counterparts. In the same way, the students of the private schools scored significantly better than the students of public schools in all aspects of mathematical thinking and mathematics achievement. Teachers’ interview revealed that problem solving, induction, deduction, logical thinking, proofs and optimal solution were the most important aspects of mathematical thinking with proofs as the most difficult and generalization as the easiest aspects. Teachers expressed their opinions that student-centered teaching was the effective way in teaching of mathematics.
viii
TABLE OF CONTENTS No.
Contents
Page
Abstract…………………………………………………………. vii List of Tables……………………………………………………
xvi
List of Figures…………………………………………………... xx CHAPTER 1 Introduction 1.1
Background of the study……………………………………….
1
1.2
Statement of the Problem…………………………………….....
4
1.3
Objectives of the Study…………………………………………
5
1.4
Research Questions...…………………………………………...
6
1.5
Delimitation of the Study……………………………………….
7
1.6
Significance of the Study……………………………………….
7
1.7
Theoretical Framework…………………………………………
8
1.8
Research Paradigm and Research Design ……………………...
11
1.8.1
Population of the Study…………………………………………
11
1.8.2
Sample of the Study……………………………………………
12
1.8.3
Research Instruments………………………………………….... 11
1.8.4
Data Collection………………………………………………….
1.8.5
Data Analysis…………………………………………………… 12
1.9
Definition of Key Words…………………………………….... CHAPTER 2
2.1
12
13
Review of Literature
Mathematical Thinking and its Aspects………………………...
16 ix
2.2
Mathematics Curriculum in Pakistan and Mathematical Thinking………………………………………………………...
23
Description of Aspects of Mathematical Thinking in the Model.
26
2.3.1
Generalization…………………………………………………
26
2.3.2
Logical thinking………………………………………………… 29
2.3.3
Deduction ………………………………………………………
30
2.3.4
Induction………………………………………………………
32
2.3.5
Proofs………………………………………………………….
33
2.3.6
Problem Solving………………………………………………
36
2.3
2.4
Studies Regarding Mathematical Thinking with Gender Comparison……………………………………………………..
38
Gender and Mathematical Thinking……………………………
38
2.4.1.1
Gender Comparison in Generalization and Using Symbols……
39
2.4.1.2
Gender Comparison in Geometry Performance……..…………
41
2.4.1.3
Gender Comparison in Problem Solving Performance…………
44
2.4.1.4
Gender Comparison in Logical Thinking………………………
48
2.4.1.5
Gender Comparison in Induction and Deduction Performance.
52
Mathematics Achievement and Gender Comparison…………...
53
2.4.1
2.4.2 2.4.2.1
Studies that Report Male Superiority in Mathematics Achievement……………………………………………………
2.4.2.2
53
Studies that Report Female Superiority in Mathematics Achievement……………………………………………………. 56
x
2.4.2.3
Studies that Shows no Gender Difference in Mathematics Achievement……………………………………………………. 59
2.5
Mathematics Achievement and Location………………………
60
2.5.1
Mathematical Thinking and Location…………………………
61
2.5.2
Studies that Link Mathematics Achievement and Location……
62
2.5.2.1
Studies that Show Better Performance by Urban Students…….
63
2.5.2.2
Studies that Show Better Performance by Rural Students……
65
2.5.2.3
Studies that Show No Location Difference in Mathematics
2.6 2.6.1
2.7
Achievement…………………………………………………
68
Private schools in Pakistan……………………………………..
70
Studies about Public /Private Schools Comparison in Mathematics Achievement……………………………………
71
Conclusion of Review of Literature…………………………….
76
CHAPTER 3 Research Design 3.1
Research Paradigm……………………………………………..
81
3.2
Research Strategies……………………………………………
82
3.3
Research Methods………………………………………………
85
3.3.1
Scale Development…………………………………………….
86
3.3.2
Population of the Study…………………………………………
88
3.3.3
Sample of the Study…………………………………………….
89
3.3.4
Sub Sample……………………………………………………...
92
3.4
Development, Validity and Reliability of the Test of
xi
Mathematical Thinking
93
………………………………………………………. 3.4.1
Content Validity of the Test of Mathematical Thinking ………
95
3.4.2
Pilot Study………………………………………………………
96
3.4.3
Item Analysis…………………………………………………… 97
3.4.4
Construct Validity………………………………………………
99
3.4.5
Reliability of the Test of Mathematical Thinking ……………..
104
3.4.5.1
Scale Reliability of Generalization……………………………..
106
3.4.5.2
Scale Reliability of Induction………………………………….
107
3.4.5.3
Scale Reliability Problem Solving……………………………… 108
3.4.5.4
Scale Reliability of Deduction………………………………….
3.4.5.5
Scale Reliability of Proofs……………………………………… 109
3.4.5.6
Scale Reliability Logical Thinking……………………………..
110
3.5
Test of Mathematics Achievement……………………………..
111
3.6
Test Administration ……………………………………………
112
3.7
Scoring of the Tests……………………………………………
114
3.8
Interviews……………………………………………………….
116
3.9
Access Provision………………………………………………..
118
3.10
Ethical Consideration…………………………………………...
118
3.11
Data Analysis…………………………………………………… 119
109
CHAPTER 4 Analysis of Data 4.1
Analysis of Teachers’ Responses on the Model of
xii
Mathematical Thinking………………………………………… 4.2
122
Mathematics Achievement and Level of Mathematical Thinking of the Students of Grade 9……………………………
4.3
123
Relationship between Mathematical Thinking and Mathematics Achievement of the Students at Secondary Level………………
4.4
124
Comparison of Mathematical Thinking and Mathematics Achievement of the students Sector-wise, Locality-wise and Gender-wise……………………………….
125
4.4.1
Gender wise Comparison………………………………………
125
4.4.2
Location wise Comparison……………………………………..
127
4.4.3
Sector wise Comparison………………………………………..
128
4.5
Interaction Effect When Gender and Location Are Taken as Independent Variables…………………………………………
4.6
130
Interaction Effect When Sector and Gender Are Taken as Independent Variables………………………………………….
4.7
132
Interaction Effect When Location and Sector Are Taken into account……………………….....................................................
4.8
134
Regression Analysis of Mathematical Thinking with Mathematics Achievement……………………………………...
136
4.9
Summary of Quantitative Analysis……………………………..
140
4.10
Analysis of Qualitative Data……………………………………
143
Analysis of Teachers’ Interviews………………………………
143
4.10.1
xiii
4.10.1.1
Analysis of interviews from Teachers of Public Schools………
144
Summary of Teachers’ Interviews from Public schools………
149
4.10.2.1
Analysis of Interviews from Teachers of Private Schools……...
150
4.10.2.2
Summary of Teachers’ Interview from private schools………..
155
Analysis of Students’ Interviews……………………………….
156
4.11.1
Analysis of Students’ Interviews in School 1………………….
158
4.11.2
Analysis of Students’ Interviews in School 2………………….
160
4.11.3
Analysis of Students’ Interviews in School 3………………….
162
4.11.4
Analysis of Students’ Interviews in School 4………………….
165
4.11.5
Analysis of Students’ Interviews in School 5………………….
166
4.11.6
Analysis of Students’ Interviews in School 6………………….
168
4.11.7
Analysis of Students’ Interviews in School 7………………….
171
4.10.1.2
4.11
4.13
Summary of All Schools’ Interview……………………………. 173
4.13.1
Questions Understanding………………………………………
173
4.13.2
Solution Strategies……………………………………………..
174
4.13.3
Students’ Perception About Difficulty Level of Aspects……….
175
5.1
CHAPTER 5: Summary, Findings, Discussion Conclusion and Recommendations Summary……………………………………………………….
177
5.2
Findings ………………………………………………………..
178
5.3
Discussions …………………………………………………….
183
5.3.1
Aspects of Mathematical Thinking…………………………….
183
5.3.2
Students’ Performance in the Test of Mathematical Thinking
xiv
and Mathematics Achievement………………………………… 5.3.3
183
Relationship between Mathematical Thinking, Mathematics Achievement and Individual scales…………………………….
184
5.3.4
Gender Differences…………………………………………….
185
5.3.5
Location Differences……………………………………………
187
5.3.6
Private versus Public School Difference……………………….
188
5.3.7
Interaction between Gender and School Location…………….
190
5.3.8
The Interaction between Gender and School Sector…………..
191
5.3.9
Interaction between Location and School Sector…………….
192
Discussion of Teachers’ Interview Results…………………..
193
5.4.1
Teacher Understandings of Mathematical Thinking…………..
193
5.4.2
Aspects of Mathematical Thinking…………………………….
193
5.4.3
Teaching Strategies……………………………………………
196
5.4.4
Teachers’Perception of Difficulty Level of Aspects………....
196
5.4.5
Gender wise Comparison of the Student’s Performance from
5.4
Teachers’ Perspective…………………………………………. 5.4.6
Location wise Comparison of the Student’s Performance from Teachers’ Perspective………………………………………….
5.4.7
5.5.1
198
Sector wise Comparison of the Student’s Performance from Teachers’ Perspective………………………………………….
5.5
197
198
Discussion on Student’s Interviews…………………………….. 199 Understanding of Question…………………………………….
199
xv
5.5.2
Solution Process of the Students……………………………….
200
5.5.3
Difficulty Level………………………………………………..
201
5.6
Conclusions ……………………………………………………
202
5.7
Recommendations …………………………………………….
206
5.7.1
Recommendation for Authorities………………………………
206
5.7.2
Recommendation for Teacher Trainers…………………………
207
5.7.3
Recommendation for Teachers…………………………………
208
5.8
Suggestions for Further Research………………………………
208
5.9
Limitations of the Study ………………………………………..
210
5.10
Implication of the Study ……………………………………….
211
APPENDIX-A A-1 Information Letter for Experts……………………………………………
228
A-2 Information Letter for Teachers……………………………………………
230
A-3 Information Letter for School Principals and Headmasters………………..
236
A-4 Information Letter for Teachers for Interview……………………………..
232
A-5 Information Letter for Students…………………………………………….
236
A-6 Consent Form for School Principals and Headmasters…………………….
238
A-7 Consent Form for School Teachers for Interview………………………….
238
A-8 Consent Form for Students……………………………………………. ….
240
APPENDIX B B-1 Test of Mathematical Thinking in English…………………………………
241
xvi
B-3 Test of Mathematics Achievement in English……………………………... 252 B-6 Sample Solved Question for Student Guidance…………………………….
254
APPENDIX C C-1 Separate Factor Analysis for Scale……………………………………....... C-2 Interview Transcripts of Teachers…………………………………………
256 257
C-3 Transcripts of Students’ Interviews………………………………………..
279
C-3.1 Students Interviews in School 1………………………………………… C-3.2 Students Interviews in School 2………………………………………… C-3.3 Students Interviews in School 3………………………………………… C-3.4 Students Interviews in School 4………………………………………… C-3.5 Students Interviews in School 5…………………………………………
279 281 283 285 288
C-3.6 Students Interviews in School 6…………………………………………
290
C-3.7 Students Interviews in School 7………………………………………..
291
C-4 Scoring Rubrics Sample…………………………………………………... 294
xvii
LIST OF TABLES Table
Title
Page
2.1
Summaries of Aspects of Mathematical Thinking……………………….
21
2.2
Comparison of Students Grade in Private and Public Schools…………
72
2.3
Summary of Score in Test of Mathematics Achievement in Private and Public Schools……………………………………………………………
73
3.1
Details of the Districts in Different Zones…………………………………..
89
3.2
Percentage Composition of the Population………………………………
89
3.3
Percentage Composition of the Sample………………………………….
90
3.4
Item Analysis of Test of Mathematical Thinking……………………….
98
3.5
Actual and Average Eigen Value……………………………………….
101
3.6
Factor Analysis for Test of Mathematical Thinking …………………..
103
3.7
Reliability Statistics for Test of Mathematical Thinking …………….
104
3.8
Item wise Reliabilities Analysis for Test of Mathematical Thinking …
105
3.9
Item Statistics for Generalization Sub Scale…………………………….
107
3.10
Item Statistics for Induction Sub Scale…………………………………
107
3.11
Item Statistics for Problem Solving sub scale…………………………..
108
3.12
Item Statistics for Deduction sub scale………………………………..
109
3.13
Item Statistics for Proof sub scale…………………………………….
110
3.14
Item Statistics for Logical Thinking sub scale………………………….
111
xviii
3.15
Item Analysis for Multiple Choice Items of Test of Mathematics Achievement …………………………………………………………….
112
3.16
Scoring Rubric for Test of Mathematical Thinking ……………………..
115
3.17
Scoring Rubric for Test of Mathematics Achievement ………………….
116
4.1
Frequency of Distribution of Teachers Responses on Model of Mathematical Thinking
122
………………………………………………………………… 4.2
Description of Students Overall Achievement in All Variables of Mathematical Thinking and Mathematics Achievement…………………..
4.3
Pearson Correlation Coefficient of Aspects of Mathematical Thinking with Mathematical Thinking (total) and Mathematics Achievement……………
4.4
126
Location wise Analysis of Mathematical Thinking and Mathematics Achievement……………………………………………………………….
4.6
124
Gender wise Analysis of Mathematical Thinking and Mathematics Achievement………………………………………………………………..
4.5
123
128
Sector wise Analysis of Mathematical Thinking and Mathematics Achievement……………………………………………………………….
130
4.7
Interaction Effect between Gender and Location…………………………
131
4.8
Interaction Effect between Location and Sector…………………………..
133
4.9
Interaction Effect between Gender and Sector……………………………
135
4.10
Correlation, Standardized Coefficients with T-Value and Significance Level Shown for the Mathematical Thinking Aspects as Independent Variables and Mathematics Achievement as Dependant Variable……….. xix
138 4.11
Direct, Indirect and Total Effect in the Full Path Model…………………
139
4.12a
Interviews Summary of Public Rural Schools’ Teachers…………………
145
4.12b
Interviews Summary of Public Urban Schools’ Teachers……………….
146
4.13a
Interviews Summary of Private Rural Schools’ Teachers………………..
151
4.13b
Interviews Summary of Private Urban Schools’ Teachers……………….
152
xx
LIST OF FIGURES Figure
Page
3.1
Stages of the Research Studies……………………………………...
85
3.2
Development Paths for Mathematical Thinking Model……………
87
3.3
Sampling Distribution Map……………………………………….
91
3.4
Flowchart
of
Test
Development
Process
for
Mathematical
Thinking…………………………………………………………….
94
4.1
Interaction Graph of Gender and Location…………………………
132
4.2
Interaction Graphs of Gender and Sector……………………………
134
4.3
Interaction Graph of Sector and Location……………………………. 135
4.4
Full Path Model Including Both Direct and Indirect Effects on Mathematics Achievement of Each Variable………………………… 138
4.5
Student’s Sample Solution 1…………………………………………
167
4.6
Student’s Sample Solution 2………………………………………...
169
xxi
CHAPTER 1 Introduction 1.1 Background of the Study It is a societal expectation that academic achievement is the result of provision of education in schools, therefore society always looks for good schools and policy dictates what a good school is. Effectiveness of individual school is analogous to effectiveness of education system in a country. School effectiveness ultimately translates into students’ achievement. Therefore this study is also concerned with students’ achievement and is focused on theory of mathematical thinking and mathematics achievement. Central to this thesis is the relationship between mathematical thinking and mathematics achievement in relation to background variables i.e. gender, location and school sector. Mathematics is viewed as a dry subject at school level and therefore students achievement in mathematics has always been hot topic for discussion among teachers, parents and researchers. Low achievement in mathematics is most probably due to teaching of mathematics, as it is taught in schools by a way that is mostly outcome-based and teacher centered (Warick & Reimers, 1995) where teacher often solve questions on the blackboard and students note them in their note books ignoring the process based teaching with focus on understanding, reasoning, critical thinking and creativity (Warick & Reimers, 1995; Halai, 1998). Shoenfeld (1992) viewed mathematics as a living subject which is understanding the patterns that permeate both the world around us and mind within us. He further 1
stressed motivation for students to move beyond rules to be able to express things in the language of mathematics. This view leads to focus on seeking rule, not just memorizing procedures and exploring patterns, not just memorizing formula and formulating conjectures, not just doing exercise. Mathematics is an important part of cognition and thinking. Issues concerning mathematical thinking are one of the fundamental goals of mathematics curricula and instruction (Lutfiya, 98). Thinking involves manipulation of information during formation of concepts, solving problem, and reasoning. Mathematical thinking is fundamental to math.
It is hypothesis for high
achievement in mathematics where a high level of mathematical thinking is desired (Ma’moon, 2005). Moreover, thinking mathematically leads to students’ success in their life beyond schools. Mathematical thinking has diversity in definition e.g. Mason et al have defined mathematical thinking as “a dynamics process which, by enabling us to increase the complexity of ideas we can handle, expands our understanding” (p.158).
Similarly
Schoenfeld (1992) has defined it as “the development of a mathematical point of viewvaluing the process of mathematization and abstraction and having the predilection to apply them; and the development of competence with tools of the trade and using those tools in the service of the goal of understanding structure”(Schoenfeld, 1992, p.335). While Jim Ridgway (www.flaguide.org/cat/math/math/math6.php, last accessed on (6.12.2009) has described mathematical thinking as Thinking mathematically is about developing habits of mind that are always there when you need them-not in a book you can look up later. For me, a big part of education is about helping students develop uncommon common sense. I want students to develop ways of thinking that cross boundaries- between courses, and 2
between mathematics and daily life… people should be able to tackle new problems with some confidence…” To think mathematically does not necessarily mean that the individual is thinking in signs, symbols, and multiplication tables therefore can be done in any situation, the signs and symbols are merely the shorthand to free the situation from any superfluous complexities.
In other words to think mathematically is to free oneself from any
peculiarity of subject-matter and make inferences and deductions justified by the fundamental premises; to think mathematically is to develop and carefully weigh, one against the other, the various differential characteristics of the interrelations of objects of the physical or mental world and then to deduce from them the truths which they imply (Wren, pp.6-8). Petocz and Petocz (1994) has categorized mathematical thinking into two broad categories, inductive and deductive thinking. Inductive thinking involves the search for pattern which is a way of thinking that enables you to arrive at generalization while deductive thinking involve to arrive at valid conclusion from truth premises. Bruner (1960) distinguishes between the two types of complementary thinking-intuitive thinking and analytical thinking. Schielock, Chanceller and Childs (2000) have mentioned several aspect of mathematical thinking e.g. Symbolism, logical analysis, inference, optimization and abstraction. Focusing on students, mathematical thinking remains a power mechanism for bringing pedagogy, mathematics, and students understanding together (Franke & Kazemi, 2001). Mathematical thinking is the mental activity involved in the abstraction and generalization of mathematical ideas.
This definition draws on the research of 3
Krutestskii (1976) on level of mathematical knowledge and on the observable epistemic actions that result in abstraction and generalization as conceptualized by Dreyfus, Hershowitz, and Schwarz (2001). Mathematics is a core subject in school curriculum of Pakistan up to Secondary Education. The curriculum is organized around contents strands of number and number operations, measurement, geometry, data handling and algebra.
The importance of
mathematics has been embodied in mathematics curriculum’s document, 2006 by emphasizing development of ability of the students to extend and use their knowledge of mathematics in daily life and other fields as well. Moreover it has explicitly stressed in the objectives to enable students to think logically, reason systematically and make discerning conjectures, (MoE 2006, National Curriculum for Mathematics grades I – XII, p. 2-3)
1.2 Statement of the Problem The importance of mathematics at secondary level in Pakistan is reflected in its goals which stress to enable students to acquire understating of concepts of mathematics and its application, to reason consistently, to draw correct conclusions from given hypotheses, to inculcate in them a habit of examining any situation critically and analytically, to communicate their thoughts through symbolic expressions and graphs, to develop sense of distinction between relevant and irrelevant data and to give the students basic understanding and awareness of the power of mathematics in generalization and abstraction.
4
The objective clearly shows that thinking process like critical and analytical ability, the power of symbolic usage, generalization and abstraction has been highlighted which are different aspects of mathematical thinking have vital role in mathematics achievement (Ma’moon, 2005). So if it is desired to improve mathematics achievement, measures have to be taken to improve mathematical thinking. Mathematics achievement is viewed as a problem internationally (Young, 2003) and situation in Pakistan is almost the same showing low achievement among secondary school students of Pakistan. This fact is evident from Saadia (2010) investigation of mathematics achievement of middle grade students in Pakistan that students can only solve low-rigor items requiring simple mathematical skills and show poor performance in items that require reasoning. Research work on this particular aspect, investigating reasoning and thinking ability and its relationship with mathematics achievement is rare. Therefore to enquire about the problem of low achievement, relationship between mathematical thinking and mathematics achievement was focused in this study with further comparison of gender wise, location wise and school sector wise among students of secondary school in Pakistan.
1.3 Objectives of the Study The following were the objectives of the study: 1. To develop a model of mathematical thinking for students of grade 9 2. To assess the mathematics achievement and level of mathematical thinking of the students of grade 9
5
3. To find the relationship between mathematical thinking and mathematics achievement of the students at secondary level 4. To compare the mathematical thinking and mathematics achievement of the students private-public sector wise, location wise and gender wise 5. To assess conceptions of the teachers and the students about the importance and level of difficulty of different aspects of mathematical thinking.
1.4 Research Questions To achieve the above objectives the following research questions were investigated in this study. 1. What were most important aspects of mathematical thinking relevant to curriculum of mathematics for grade 9 students? 2. What was the level of achievement in the test of mathematics achievement and mathematical thinking for students of grade 9? 3. To what extent mathematical thinking and mathematics achievement were related? 4. Did scores in the tests of mathematical thinking and mathematics achievement differ for students, gender wise? 5.
Did significant difference exist between the score in the tests of mathematical thinking and mathematics achievement for students of private schools and public schools?
6. Did significant difference exist between the scores in tests of mathematical thinking and mathematics achievement for students of urban and rural schools? 6
7. Was there any interaction effect between gender and location? 8. Was there any interaction effect between sector and gender? 9. Was there any interaction effect between sector and location? 10. Were teachers and students consistent in their conceptions regarding difficulty level and importance of different aspects of mathematical thinking?
1.5 Delimitations of the study The study was delimited to 1. The science stream 9th grade students only. 2.
The schools imparting education using textbook designed by Khyber Pakhtunkhwa Text-Book Board.
3.
Two out of four zones only.
1.6 Significance of the Study The study which was mainly concerned with relationship of mathematics achievement with cognitive determinant of mathematical thinking in relation to non cognitive factor i.e. gender, location and school sector was significant for the following reasons. Firstly this study was likely to identify the cognitive component of mathematics achievement which ,on practical side, may help teachers to focus their instructional activities to enhance students’ achievement and teacher educators to train the teachers in context of developing their mathematical thinking which ultimately may have positive effect on students’ thinking. Secondly this study may inform the curriculum designer, 7
planner to focus on inclusion of mathematical thinking supporting materials in curriculum for better mathematics achievement which is backbone of science and technology, especially information technology that cannot be mastered without logical, creative thinking and problem solving which are important aspects of mathematical thinking. Thirdly this study may, on theoretical side, strengthen the idea of possible relationship between mathematical thinking and mathematics achievement pointed out by the earlier researcher e.g. Ma’moon (2005). Fourthly this study was likely to compare the variables under study across different strata of population like gender wise, locality and private/public sector wise which may unveil the causes of gap between students achievement among different strata of population particularly in private/public sectors that have wider gap shown by their results over the years. Lastly this study was to stimulate other areas of research in the focused area which ultimately may help the stakeholders to take better remedial actions to solve the problem. 1.7 Theoretical Framework The main investigation in this study concerned with the relationship between mathematics achievement and mathematical thinking.
Achievement is commonly
understood as the act of achieving or successful performance in other words accomplishment of assigned task. Mathematics achievement, in this study, is taken as the numerical score assigned to individual performance on a test of mathematics achievement based on the contents of the textbook of mathematics for grade 9. The achievement test was aimed to measure the conceptual and procedural knowledge of the students (Parks, 2003).Mathematical thinking is taken as the measures of thinking process in different 8
aspects of mathematical thinking discussed in details in chapter two of the thesis. Though, mathematical thinking is the term which is less defined in the history of mathematics education.
Different people have attempted to describe the nature of
mathematical thinking and list its aspects but none can be termed as exclusive. Barbara (2007) while addressing teachers described it as “I am going to ask you to do a mathematical task and while you are doing it to pay attention to any thinking you are doing that may be mathematical thinking”. Aspects of mathematical thinking often overlap that has been termed separate by many researchers e.g. Pattern recognition has been listed as separate aspects by Shatnawi (1971), Lutfiya (1998), and Ma’moon (2005) in their attempt to identify various aspect of mathematical thinking while during process of generalization the pattern recognition comes before generalization. Similarly creative and critical thinking is built in characteristic of problem solving.
This particular study operates on model of mathematical thinking having
relevancy to curriculum of grade 9 of Pakistan. The six aspects chosen in the model of mathematical thinking are generalization, induction, problem solving, deduction, proofs and logical thinking. Generalization is defined as the process of conjecturing from examples and making general rule.
Deduction is the applying of rule to identify
particular instance. Problem solving is the mathematical ability to solve a problem for which the immediate solution is not known. Logical thinking is the use of premises to reach and validate a conclusion. Mathematical proofs are using valid steps to prove a mathematical statement. However these aspect of mathematical thinking selected in the
9
model of mathematical thinking in this study are not totally independent and there is considerable overlap among the aspects. But it does not mean that we cannot measure them independently (Ma’moon, 2005). The learning and achievement of the students is not an isolated phenomenon and many factors are involved in it. To have a clear picture of the sample under study, major subsections of the population; gender, location and sector of the schools were taken as the background variable. Gender is a distinguished entity for this study and is taken with attributes rooted in social phenomena rather than biological one (Unger & Crawford, 1992). Although theories about biological reasons for sex differences in mathematics achievement have been advanced (e.g., Benbow & Stanley, 1980, 1982, 1983), socio cultural reasons are more widely accepted (Humphreys, Lin, & Fleishman, 1976; Meece et al., 1982; Sherman, 1978). The definition for location of the school was adopted from the ministry of welfare in Khyber Pakhtunkhwa where urban school was defined as a school which lies in jurisdiction of town or municipal committee and rural school as the one which is not in the area of town committee or municipal committee. Private schools in Pakistan are schools which are run by organizations or persons mostly for profit motives and self-employment (Shami et al, 2004). Private schools are making up a growing sector and attracting a large population mostly richer who can afford to admit their children in such school. The inclusion of private sector may help in understanding the effect of socio economic status on the achievement (Ma’moon, 2005, p.176) as well. There is a common view about inefficiency of public schools. On the other hand private schools are considered to be more efficient both in developed and developing countries of the world (Hanushek, 1995, 1997). 10
1.8 Research Paradigm and Research Design This researcher is following post positivist paradigm, consequently the researcher believes that reality cannot be measured exactly and that there are multiple realities within the world in contrast to positivist approach, which begins with the belief of existence of single reality. (Lincoln & Guba, 1985; Strauss & Corbin, 1990) Regarding research design, mixed method design was adopted for this study which relies on the combination of qualitative and quantitative research methods to obtain a clearer picture of the situation and those subjects involved in sample. This approach allows ‘triangulation of data’ which is fundamental in obtaining validity in research (Fraser & Tobin, 1991). If the researcher wants to build on the strengths of both types of data, mixed method research is a good alternative to traditional designs (Creswell, 2005, p.510). Teaching and Learning of mathematics is not a simple process and any problem dealing with it might be addressed appropriately using mixed methods research.
1.8.1 Population of the Study Population of the study was all 9th grade students in Khyber Pakhtunkhwa of Pakistan which had been divided into four zones by Public Service Commission of Khyber Pakhtunkeshwa.
11
1.8.2 Sample of the Study The sample selected for this study was 544 students from zone 2 and 4 comprising of 10 districts of the Khyber Pakhtunkhwa province.
Stratified random sampling
technique was used to take the sample for this study. Among these 544 students, 70% were rural and 30% was urban. Similarly the sample was bifurcated into private and public schools sector on equal basis i.e. 50% each from each sector. Also the sample was stratified into 70% of male students and 30% female ones.
1.8.3 Research Instruments Two tools were used in the survey to collect data from students, a test of mathematical thinking and a test of mathematics achievement to collect quantitative data. These instruments were supplemented by two interviews one from students and the other from teachers to collect qualitative data.
1.8.4 Data Collection The data was collected by the researcher himself by administering the instruments from randomly selected students in different schools. The data collection process took almost four months.
1.8.5 Data Analysis SPSS version 17 was used to analyze quantitative data with the help of statistical techniques like mean, standard deviation, t-test, ANOVA, correlation and regression analysis. Qualitative data from interview was analyzed using thematic technique to identify popular themes and patterns in responses. 12
1.9 Definition of Key Terms Mathematical thinking: Mathematical thinking in this thesis is defined as the abstraction and generalization of mathematical idea. It is dynamic process which, by enabling us to increase the complexity of ideas we can handle, expands our understanding. Mathematics achievement: In this study mathematics achievement is taken as the numerical score assigned to individual performance on test of mathematics achievement based on the contents of the textbook of mathematics for grade 9. Gender: Gender is a distinguished entity for this study and is taken with attributes rooted in social phenomena rather than biological one. Location: Location of the schools in this study has two components i.e. urban and rural. The distribution is based on the division made by ministry of social welfare where the schools falling in jurisdiction of town committee or municipal committee are termed as urban schools and those falling outside it are termed as rural schools. School Sector: Public schools in this study refer to those schools which are run by government while private schools are those which are run and owned by persons. Generalization: Generalization has been identified by Davydov (1990) as a process of abstracting from the particular to the general that involves the finding of similar qualities in all subjects of the same class and includes a searching for the invariants in an assortment of objects and their properties, and the use of those invariants to identify objects in a given (general) assortment. Deduction: Deduction is defined as the process that “yields valid conclusions, which must be true given that their premises are true” ( Johnson-Laird, P.110).
13
Problem solving: Problem solving is the mathematical ability to solve a problem for which the immediate solution is not known. Logical thinking: Macdonal (1986) described logical thinking as “the idea that there are certain basic rules of grammar with which we can organize our discussion in mathematics is what makes it possible to establish that certain things are “true” in mathematics. Induction: Inductive reasoning is the process of inferring unknown principles from information or observation. Stated more simply, it is the process of making general conclusions from specific information or observation. Proofs: A mathematical proof is a mathematical argument that begins with a truth and step by step arrives at valid conclusion by convincing demonstration within accepted standards of the field. It is also defined as using logical evidence to show the correctness of an expression that follows from the proof of previous expression.
14
CHAPTER 2 REVIEW OF LITERATURE The aim of literature review is to give an overview of the work of past researcher regarding the problem understudy. Literature review is necessary in many ways. It can document how this study will add to existing literature and to convince the educator why they need this study (Creswell, 2007, p.79). This chapter provides a review of the scholarly literature focusing on mathematical thinking, various aspects of mathematical thinking mentioned by different researcher, and curriculum of mathematics in Pakistan. Further, studies that have focused students’ comparison in mathematics achievement and mathematical thinking in relation to back ground variables i.e. gender, location and school sector are presented. As many of the existing studies do not make specific mentioning of particular aspects, therefore literature review is organized as follows. Generalization is discussed with studies related to use of Symbols and algebra. Studies about proofs are presented with studies related to geometry as most of the studies in this area involve geometrical proofs.
Problem solving, induction and deduction are
studied separately and finally logical thinking is related to the studies about mathematical reasoning. The literature as it pertains to achievement levels in mathematics in relation to gender alone, geographical location alone, school type alone and both gender and location, gender and school type is reviewed.
15
2.1 Mathematical Thinking and its Aspects Thinking is part of human nature that allows individual to grasp a clearer picture of the reality around him. Burton (1984, p.36) argued that "thinking is the means used by humans to improve their understanding of, and exert some control over, their environment” To promote thinking skills in the students a range of disciplines are used particularly mathematics.
Issues concerning mathematical thinking are one of the
fundamental goals of mathematics curricula and instruction (Lutfiya, 1998) and therefore have gained attention of researchers. Mathematical thinking has central focus in the issues regarding curriculum and instructions of school level mathematics. NCTM standards call for reasoning, critical thinking and proofs which is embodied in the following statement. A climate should be established in the classroom that places critical thinking at the heart of instruction.... To give students access to mathematics as a powerful way of making sense of the world, it is essential that an emphasis on reasoning pervade all mathematical activity Inductive and deductive reasoning are required individually and in concert in all areas of mathematics. ( NCTM standards, 1989) Mathematical thinking has a considerable diversity in definition. Discussion with several mathematicians and mathematics educators usually results in several different interpretations.
However several researchers have attempted to define it, e.g.
“A
dynamics process, which, by enabling us to increase the complexity of ideas, we can handle expands our understanding” (Mason et al p.158). While Schoenfeld (1992) has defined it as the development of a mathematical point of view- valuing the process of mathematization and abstraction and having the predilection to apply them; and the development of competence with tools of the trade, and using those tools in the service of 16
the goal of understanding structure. Similarly Dreyfus, Hershowitz, and Schwarz (2001) viewed mathematical thinking as the mental activity involved in the abstraction and generalization of mathematical ideas. Barbara (2007) while in an address to teachers described it as “I am going to ask you to do a mathematical task and while you are doing it to pay attention to any thinking you are doing that may be mathematical thinking”. Due to complex nature of mathematical thinking it is not a simple question to answer that what is mathematical thinking? There is, always, an expectation of subjectivity when seeking views about mathematical thinking from experts in different contexts; therefore it is not unexpected to hear different views from a mathematician, a psychologist, a school teacher, an expert or a person on the street. For this reason it is better (e.g. Lutfiya, 1998) to ask a question like what characterizes thinking of a student with high ability in mathematics instead what is mathematical thinking? Mathematical thinking can be done in any situation, the signs and symbols are merely the shorthand to free the situation from any superfluous complexities. In other words to think mathematically is to free oneself from any peculiarity of subject-matter and make inferences and deductions justified by the fundamental premises; to think mathematically is to develop and carefully weigh, one against the other, the various differential characteristics of the interrelations of objects of the physical or mental world and then to deduce from them the truths which they imply ( Wren, pp.6-8). Focusing on students’ mathematical thinking remains a power mechanism for bringing pedagogy, mathematics, and students understanding together (Franke & Kazemi 2001). Mathematical thinking is the mental activity involved in the abstraction and
17
generalization of mathematical ideas.
This definition draws on the research of
Krutestskii (1976) on level of mathematical knowledge and on the observable epistemic actions that result in abstraction and generalization, as conceptualized by Dreyfus, Hershowitz, and Schwarz (2001). Mathematical thinking is emphasized internationally. For example the following extract from National Research Council’s Publication (1989) “Everybody counts: A report to the nation on the future of mathematics education:” states More than ever before, Americans need to think for a living; more than ever before, they need to think mathematically…Wake up, American! Your children are at risk. Three of every four Americans stop studying mathematics before completing career or job prerequisites. (US Govt.1989, pp.1-2) The document further emphasized four major thinking processes i.e. observing, inferring, comparing and sequencing to help students strengthen their ability to think mathematically. Mathematical reasoning is considered a characteristic of Mathematical thinking. National Council of Mathematics Teachers have tried to explain mathematical reasoning with examples in its standards but these examples have not lead to a clear distinction of mathematical reasoning to logical reasoning, mathematical thinking or critical thinking. Mathematical thinking leads to understanding of ideas and discovery of relationship by using mathematically rich thinking skills. According to Burton (1984), Mathematical thinking is mathematical not because it is thinking about mathematics but because the operations on which it relies are mathematical operations. Furthermore, the key to recognizing and using mathematical thinking lies in creating an atmosphere that builds the confidence to question, challenge, and reflect. ( p.36, 48) These views by Burton (1984) have been supported by Skemp (1971) that by creating conducive atmosphere students can be driven towards advance mathematical 18
thinking by enabling them to formalize and systemize mathematics which is considered to be the final stage of mathematical thinking and should not be taken as the total activity. Williams (2000) created a framework to describe mathematical thinking of students that initially drew on the work of Krutetskii (1976) and used his empirical data on “mental activities” to systematically gratify cognitive complexity in order to develop a hierarchy of the cognitive activities students used when solving mathematical problems. The development of this hierarchy was further supported through consistency with Bloom’s (1956) hierarchy of learning objectives, which have since been used to describe cognitive activity.
The cognitive activities in the hierarchy, starting with the least
demanding, consist of the following: comprehending, applying, analyzing, syntheticanalyzing, evaluative-analyzing, and evaluating. These cognitive activities are assumed to be cumulative, with each activity of the system building on successful completion of the previous activity. As mentioned before Petocz and Petocz (1994) have mentioned two broad categories of mathematical thinking (1994), inductive and deductive thinking. Inductive thinking involves the search for pattern which is a way of thinking that enables you to arrive at generalization while deductive thinking involve to arrive at valid conclusion from truth premises. Bruner (1960) distinguishes between two types of complementary thinking intuitive thinking and analytical thinking. Schielock, Chanceller and Childs (2000) have mentioned several aspect of mathematical thinking e.g. symbolism, logical analysis, inference, optimization and abstraction.
Likewise Burton (1984) has also
identified four aspects of mathematical thinking which she describes as specializing,
19
generalizing, conjecturing, and convincing. These aspects have a central importance and general application. Litwin (2006) has described mathematical thinking as involving inducting reasoning, deductive and analogical reasoning. He further divided mathematical thinking into three areas.
The first one is more macro that includes
generalization, abstraction, modeling, conjecture. The second is logical thinking, such as classification, reverse thinking, counter example, specification, induction and deduction. The third area is specific mathematics skills, such as substitution, completion of square, determination of coefficients. Schielock, Chanceller and Childs (2000) have mentioned several aspect of mathematical thinking e.g. Symbolism, logical analysis, inference, optimization and abstraction. Wren (2006, p.8) has emphasized four aspect of mathematical thinking for elementary students which are observing and inferring, comparing, classifying and sequencing. He further described characteristics of mathematical thinking as: 1. The ability to set up clear cut premises and definitions: 2. The ability to reason coherently and critically and 3. The ability to draw implied conclusion. Lutfi, (1998) identified some aspects of mathematical thinking through teachers’ survey to develop a test for his study about performance on mathematical thinking of secondary school students in Nebraska. The aspects thus identified were generalization, use of symbols, logical reasoning, induction, deduction, and mathematical proof, problem solving, systematic thinking, abstraction, critical thinking and logical thinking.
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Table 2.1 Summaries of Aspects of Mathematical Thinking Ma’moon’s (2005) Study
Shatnawi’s (1971) Study
Lutfi’s (1998) Study
Generalization
Generalization,
Induction
Generalization and abstraction Induction
Deduction
Deduction
Deduction
Use of symbols
Using symbols
The Use of Symbols,
Mathematical proof
Mathematical proof
Proofs
Reasoning
Reasoning
Logical Reasoning,
Problem solving
Problem solving
Problem Solving,
Creative thinking
Creative thinking
Specialization
Systematic thinking
Systematic Thinking,
Using patterns
Modeling (pattern
Abstraction,
Induction
cognition) Ability to find optimal solution Inference from premises
Imaginative thinking Critical thinking
Critical Thinking,
Using mathematical expression or the ability to translate from words to equations
Building a concept
Logical Thinking,
Likewise Ma’moon, (2005, p.65) worked on the relationship of mathematical thinking with mathematics achievement for grade 11 students and identified several aspects i.e. generalization, induction, deduction, use of symbols, mathematical proof, reasoning, problem solving, creative thinking, specialization, using patterns, ability to find optimal solution, inference from premises and using mathematical expression or the 21
ability to translate from words to equations as important aspect of mathematical thinking. He further developed a model of mathematical thinking for his study with the help of teachers’ survey with six aspects out of his list. Six aspects included in his model were generalization, induction, and deduction, use of symbols, mathematical proof and logical thinking. In the same way, Shatnawi (cited in Ma’moon, 2005) worked out different aspects of mathematical thinking during his research study. He also used teachers’ survey to identify the various aspect of mathematical thinking including generalization, abstraction, induction, deduction, using symbols, mathematical proof, reasoning, problem solving, creative thinking, systematic thinking, modeling (pattern cognition), imaginative thinking, critical thinking and building a concept. Different aspects of mathematical mentioned by Ma’moon (2005), Shatnawi (1971) and Lutfiya (1998) are listed in Table 2.1 above.
Conclusion Most of the studies have not discussed aspect mathematical thinking in details e.g. Bruner (1960), Petocz and Petocz (1994) Schielock, Chanceller and Childs (2000), Williams (2000) and Schielock, Chanceller and Childs (2000) have mentioned few aspects of mathematical thinking. However three of the above studies Shatnawi (1971), Lutfiyya (1998) and Ma’ moon (2003) have comparatively given a detail description about aspects of mathematical thinking. There are considerable overlaps among these aspects figured out by different researcher leaving space for more work with more refined criteria of selection of the aspects using rigorous methods involving experts and teachers’ opinions. 22
2.2 Mathematics Curriculum in Pakistan and Mathematical Thinking The importance of mathematics has been embodied in mathematics curriculum document, 2006 as Mathematical structures, operations and processes provide students with a framework and tools for reasoning, justifying conclusions and expressing ideas clearly. As students identify relationships between mathematical concepts and everyday situations and make connections between mathematics and other subjects, they develop the ability to use mathematics to extend and apply their knowledge in other fields. (Moe, www.moe.edu.pk,) First national curriculum was developed in 1975-76 which was further reviewed in 1984-85 and then in 1994-95. The next review took place in the years 2000 and 2002 for science and social science subjects respectively. The new reforms in 2006 are aimed to make the curriculum more vibrant and more responsive to the modern, socio-economic, technical, and professional and labor market needs of the country. The most important feature of this curriculum is its continued focus on the content of the mathematics standards. The following themes permeate the national curriculum for mathematics. •
The curriculum is designed to help students build the solid conceptual foundation in mathematics that will enable them to apply their knowledge skillfully and further their learning successfully.
•
The curriculum emphasizes on the geometrical concepts that enable the students to think logically, reason systematically and conjecture astutely.
•
The curriculum stresses graphics that the students to visualize and interpret mathematical expressions correctly rather to manipulate them blindly.
23
•
The curriculum recognizes the benefits that current technologies can bring to the learning and doing mathematics. It, therefore, integrate the use of appropriate technologies to enhance learning in an ever increasingly information-rich world.
National curriculum for mathematics (2006) is organized in five standards which have been kept broad for flexible interpretations. The competencies are intentionally kept broad as to allow flexibility to the teachers in accordance with their students. These standards are: I) Numbers and Operations, II) Algebra, III) Measurements and geometry, IV, Information handling, V) Reasoning and logical thinking. Assessment in mathematics has also been addressed in the curriculum document stressing a variety of assessment techniques while assessing mathematical abilities of the students.
The document comments elaborates that a single type of assessment in
mathematics can frustrate students, diminish their self confidence and make them feel anxious about the subject because the understanding of mathematical concepts and mathematical abilities encompass a broad range of abilities.
It further states that
assessment practices must include by focusing on a student’s ability to: •
Communicate mathematically.
•
Reason and analyze, and to think and act in positive ways.
•
Comprehend the key concepts.
•
Evaluate the effectiveness of using different strategies to address the same problem.
•
Use a variety of strategies to problem solving and to make mathematical connections. 24
•
Discriminate between relevant and irrelevant attributes of a concept in selecting examples.
•
Integrate and to make sense of mathematical concept and procedure.
•
Examine real life situations by reasoning mathematically. (p.137)
The above points reveal that the curriculum does not say anything explicitly about mathematical thinking but have discussed themes, content standards, bench marks and assessment in mathematics which are associated with the concept of mathematical thinking e.g. reasoning, justifying conclusion identifying relationship and applying knowledge in other fields discussed in importance of mathematics as subject are certainly aspects of mathematical thinking.
Similarly the curriculum themes discussed in
Justification of the mathematics curriculum stress on skilful application of knowledge, to think logically, reason systematically and conjecture astutely.
In guidelines for
assessment in mathematics the documents lay stress by focusing on students ability to communicate mathematically, reason, analyze and to think in positive way, variety of strategies in problem solving, to screen relevant information in giving example for a concept, examine real life situation by reasoning mathematically.
this analysis of
curriculum give us a clear picture that mathematical thinking has been highly embodied and stressed but still the mathematics achievement in Pakistan is not satisfactory and this is because that the assessment practices in classroom and curriculum guidelines do not correlates.
Similarly classroom activity in teaching of mathematics
also typically
outcome based in the form of right answer and no attention is given to process based teaching with focus on understanding, reasoning, critical thinking and creativity (Warick 25
& Reimers, 1995; Halai, 1998 ; SPDC Survey, 2003). However efforts are made to change the situation which is reflected in national curriculum for mathematics by changing the role of teachers from dispensing information to planning investigative tasks, managing cooperative learning environment, and supporting students’ creativity in developing rational understanding of the concepts of mathematics. (MoE 2006, National Curriculum for Mathematics grades I – XII, p. 2-3)
2.3 Description of Important Aspects of Mathematical Thinking Important aspects of mathematical thinking conceptualized in theoretical framework of this study are discussed in this section. Further there theoretical position in literature is supported with examples of items taken from the actual test used in the main study.
2.3.1 Generalization Generalization is important aspect of mathematical thinking and is considered as life-blood of mathematics by Mason et al (91, p-8). Generalizing or making claims that extend beyond particular situations is a central mathematical practice and a focus of classroom mathematics instruction. It is defined as a statement presented as a general truth based on specific cases. Polya (1990, p.108) defines generalization as “leading from one observation to a remarkable general law”. According to Dorfler (1991) generalization is both “an object and a means of thinking and communicating”. Generalization has been identified by Davydov (1990) as process of abstracting from the particular to the general that involves
26
the finding of similar qualities in all subjects of the same class and includes a searching for the invariants in an assortment of objects and their properties, and the use of those invariants to identify objects in a given (general) assortment. Generalization requires symbols connecting the invariants with the objects and phenomena that have the common elements. Mason (1987, 1996) argues that generalization may well be a basic life skill. Generalization and using symbols have been studied as different aspect of mathematical thinking (e.g. Lutfiya, 98; Mamoon, 2003). In this particular study the researcher has attempted to use combined items for symbols and generalization due to higher correlation among them. The reason behind this nesting is that working with repeating pattern and symbols is conceptual stepping stone to algebra and a context for generalization (Threlfall, 1999). Algebraic thinking describes generalizations succinctly by being concerned with the structure of a mathematical statement (MacGregor, 1993). Whenever generalization is involved there is role of symbols. Algebra as it relates to generalization is defined as formal means to describe relationship among quantities (Kieren, 1992). A symbol may be a letter, relationship or abbreviation representing an expression, quantity, idea, concept or mathematical process. Expressing through symbols means the use of symbols to communicate mathematical idea or verbal problem. Shatnawi (1982, cited in Ma’moon, 2005) defined use of symbols as using symbols as a language to express ideas and mathematical information.
Mason (1996) describes
“expressing generality” as one of the roots of, and routes into, algebra. Patterning approach might be helpful in introduction of variables which makes students not only to observe and verbalize a general pattern but also provide them opportunity to expressed 27
them using symbols (English and Warren, 1998). Algebra being highly dependent on using symbols especially letters can provide opportunities for students in exploring, analyzing and representing mathematical concepts. Therefore using symbols and generalization are quite logical to correlate and can be measured using combined items. Algebra has the capability to generalize, describe and model relationships both from pure mathematical and real world situation, this was viewed by (Lee, 1996, p. 103) as well and stated that “algebra, and indeed all of mathematics is about generalizing patterns” Ability to see and describe patterns is a “central mathematical practice and it is widely agreed that students should learn to generalize from pre-K to the 12th grade” (National Council of Teachers of Mathematics, 2000, p.84). Identifying important attributes in mathematical instances and ignoring unimportant is important process.
Besides
communicating and selecting mathematical forms for this generalization is also vital like graphs, letters and other symbols (Goodwin, 1994; Latour, 1987). Thus generalization being a central mathematical practice is important for students to learn how it is described and modeled. Krutetskii (1976) has argued the ability to generalize is of two types, the first is to see a similar situation, i.e., where to apply it, and secondly the mastery of the generalized type of solution; the generalized scheme of proof.
This item demands from students to model a situation into a generalized form using symbols.
28
2.3.2 Logical Thinking Mathematics has been defined by Devlin (2001) as the science of patterns and then more fully as: "the science of order, patterns, structure, and logical relationships" (p. 73).
Enhancing students' ability to reason is one of the primary goals of mathematics
education and every discipline needs the capability to think logically mathematics in particular.
(Frances, 1995) Thinking logically and thinking in algorithm differs in
teaching mathematics, the former has an advantage over later that students learn formulae and memorize the situation where they can be used. Consequently a student can easily solve a similar task to that in which he has memorized the formula but has difficulty when confronted with unknown task, even if he has all the required knowledge for its solution. Learning of logical concepts is helpful in deeper understanding of other parts of mathematics. ( Bako, www.cimt.plymouth.ac.uk/journal/bakom.pdf, accessed on 12.9,09). Macdonal (1986, p.37) described logical thinking as “the idea that there are certain basic rules of grammar with which we can organize our discussion in mathematics is what makes it possible to establish that certain things are “true” in mathematics. Shatnawi (1982, p.6) defined logical thinking as the transition from the known to the unknown guided by objective rules and principles, which are the grammar of logic. Logic can be described as science of reasoning. Logical thinking seek conclusion on the basis of some premises and if the premises are mathematical in nature then it can be said that we are working in mathematical logic. NCTM standards (2000) have recognized that systematic reasoning, exploring, justifying, and using mathematical conjectures are common to all content areas though level of rigor may vary with different level. 29
Through the use of reasoning, students learn that mathematics makes sense. All habits of mind including reasoning mathematically can be developed with consistently using it in different contexts and from the earliest grades. Concept, judgment and conclusion are basic form of thinking used widely in logical thinking and can help in grasping reality.
Therefore mathematical thinking can support humans within their
logical thinking about realities which results in sensible actions (Wille, 2009).
There is a logical relationship between the set A, B and C and the same relationship has been translated in Vein diagram as well, students are required to identify the a correct option along with justification.
2.3.3 Deduction A deductive thinking or simply deduction can be considered as the concatenation of ideas, each one determined by the previous one. Deduction, in mathematics, is used to validate a conjecture and using general facts to justify less general or particular facts. 30
Deduction must be considered as part of mathematical reasoning whose function is to validate hypotheses. Deductive thinking is one of the most commonly used patterns of thinking (Harry, 2002). Mathematics is different from other discipline, at least at school level in the sense that it establishes truth by deductive thinking (Porteous, 1990). Deduction is defined as the process that “yields valid conclusions, which must be true given that their premises are true” ( Johnson-Laird, 1999, P.110). While Shatnawi (1982, p.10) defined it as “arriving at a particular results from unknown or assumed principle”. In deductive process known statements are used to draw a valid conclusion and thus it helps in interpreting and formulating instructions, plans of actions and general principles. Furthermore assessment of data, transfer of knowledge and choice among competing ideas can be made using deductive process due to its truth preservation nature. Validity of arguments does not change if new premises are added to a valid deductive argument. Deductive validity is an all-or-nothing matter; instead of occurring in degrees, an argument is totally valid or it is invalid. The current research has taken the deduction as the use of a known rule to identify the unknown the following example from the actual instrument can best illustrate the conception. Below an item is given for illustration from actual test for the study. Q.1 what is the value of angle A?
A. 70o
B. 100o
c. 110o
D. 120o 31
In this question the students is required to apply the known rule in their previous knowledge i.e. the sum of angles on a straight line is 180 degree and also the rule that vertical angles are equal.
2.3.4. Induction Inductive reasoning is the process of inferring unknown principles from information or observation. Stated more simply, it is the process of making general conclusions from specific information or observation. Polya (1964) defined induction as the process of discovering general laws by the observation and combination of particular instances. Induction occurs after checking whether the general rule or generalization is true for all cases rather than specific cases from which it is derived (Ma’moon, 2005). The researcher has differentiated induction from generalization on the basis of its process. Induction can be proved mathematically using the process of mathematical induction as the demonstrative procedure.
It is demonstrative reasoning instead of
plausible reasoning that completes the inductive reasoning (Polya, 1964).
Making
induction usually involves seeking specific information, connection or patterns and finally general conclusions or prediction.
Additionally it looks for any need in
conclusion or prediction when more information comes. Inductive approach usually comprised of (1) Experimentation. (2) Observation. (3) Forming of a hypothesis. (4) Further experimentation in order to test the hypothesis. Polya has linked mathematical induction with induction as it is the former which completes the later. An example of the problem showing induction is given below Q.1 if a series of number is as 1, 4,7,10, 13…. A 30
B 28 C 25
Then what is 10th number in the series?
D 33 32
The above question require from students to abstract a rule that each number in the series is in a sequence with common difference of 3, further they are required to exhibit their understanding of the rule by identifying the 10th number in the series.
2.3.5 Proofs Mathematical claims vow greatly to proofs as stated by Nyaumwe and Buzuzi, (2007, p.2) “Mathematics may be viewed as a subject, in which the viability of claims is not certain until claims are proved. This view presents mathematical knowledge as tentative and therefore not to be taken for granted as viable”. Without proof there would be no mathematical knowledge, in a strict sense. It is the form of mathematical proof which sets mathematics apart from the empirical sciences (Milton & Reeve, 2003). Proving has been termed as central activities to mathematics (Porteous, 1990). In NCTM (2000) standards the importance of proofs are recognized in the following words “Recognize reasoning and proof as fundamental aspects of mathematics; make and investigate mathematical conjectures; develop and evaluate mathematical arguments and proofs; select and use various types of reasoning and methods of proof.” (www.nctm.org/standards/content.aspx?id=23749, accessed on 12-06-2008) Geometry provides opportunities to students, while working with proofs, to enhance their reasoning and justification skills.
Proof of mathematical claims is viewed as a
process that involves active justification of the claims rather than a passive process of validating the viability of the claims (Nyaumwe & Buzuzi, 2007).
According to
Brumbaugh and Rock (2001) mathematical proofs are logically structured arguments that follow certain established sequences of steps to confirm or refute the viability of 33
mathematical conjectures. From a pedagogical point of view, Hanna (1996) views proofs as a transparent argument in which all the information used, and the rules of reasoning that are used to verify or falsify mathematical results are clearly shown and are open to criticism. Proofs show beyond reasonable doubt that a conjecture is indisputably viable or false, Proofs are not simply about justifying why mathematical results are true; they also include discourse that may lead to social construction of mathematical results. Such discourse enhances students understanding and construction of mathematical knowledge and characterizes learner centered classroom (Wheatley, 1992). A mathematical proof is a mathematical argument that begins with a truth and steps by steps arrive at valid conclusion by convincing demonstration within accepted standards of the field. Shatnawi (1982, p.6) defined mathematical proof as “using logical evidence to show the correctness of an expression that follows from the proof of previous expression”.
The goals of proof are not mere verification of truth but gaining of
understanding because it is efficient way of retaining information than memorization. (p.32) As noted by Herbst (2002), proof is an active process because it involves organizing previously learned concepts and presenting them as logical arguments that deduce or refute the viability of hypotheses, conjectures or theorems, The process of proving mathematical conjectures involves an understanding of the structure of concepts through selective use of existing knowledge, use of logical reasoning presented at appropriate stages in order to strengthen or weaken arguments for the viability of the mathematical claims, Foresight and appropriate use of axioms are important skills that enhance the presentation of arguments that establish the viability of mathematical claims, 34
Because justification of the viability of mathematical results is an important goal of proof, it is important to explore the pedagogical roles in relation to proof in learnercentred classrooms.
“Mathematical proof as human activity requires not only an
understanding of the concept definitions and the logical processes but also insight into how and why it works”. (Tall, 2007, p.506) Proof is concerned “not simply with the formal presentation of arguments, but with the student’s own activity of arriving at conviction, of making verification, and communicating convictions about results to others” ( Bell, 1978, p.78) Traditionally two-column proofs that involve the writing of two adjacent columns of statement and reason (Herbst, 2002) are popular that present logically structured sequence of steps and their justifications to establish the logic and correctness of argument for confirming or refuting a claim but the use of technology generated proofs and the growing recognition given to mathematical experimentation in student-centered classrooms has led to acceptance of other forms of validating the viability of mathematical conjectures such as rigorous and non-rigorous proofs (Hanna, 1996).
This item involves proof of two smaller angles which are the part of two different angles and are given to be equal in the statement. The students are required to split equal
35
angles in parts and then equate them. The common angles can be subtracted from both sides leading to the required proof.
2.3.6 Problem Solving Problem solving in mathematics has been described as complex cognitive activity. Some mathematical literature described mathematics problem solving as several separate activities such as doing word problems, creating patterns, interpreting figures, developing geometric constructions and proving theorems (Willson, Fernandez and Hadaway, 1993). Reitman (1965) described a problem solver as someone who received information and a goal without an immediate means to achieve the goal. Polya’s theory (Polya, in Willson, Fernandez and Hadaway, 1993) defined mathematical problem solving as a process that involved several dynamic activities: understanding the problem, making a plan, carrying out the plan and looking back. While Goldstein and Levin, (1987) defined it as “higherorder cognitive process that requires the modulation and control of more routine or fundamental skills”. To help students to become good problem solver is of central importance in mathematics so that they may not rely on rote learning which causes a lot of difficulties in mathematics achievement (Hiebert, 2003). In order to achieve the goal, the mathematical problem solver must develop a base of mathematics knowledge and organize it, create an algorithm and generalize it to a specific set of applications, and use heuristics (strategies, techniques, shortcuts) and manage them (Wilson, Fernandez and Hadaway, 1993). During the process, students might apply a number of general strategies such as a solution rubric, a logical 36
mathematical reasoning, a trial-and-error approach and an outright guess to derive answers on mathematical problem solving tests (Gallagher, DeLisi, Holst, McGillicuddyDeLisi, Morely and Cahalan, 2000).
Mayer (2003) divided mathematical problem
solving into four cognitive phases: translating, integrating, planning and execution. Royer and Garofoli (2005) classified them into two stages: representation of a problem and solving the problem. Both of them regarded representing the problem successfully as the basis for understanding the problem and making a plan to solve the problem. NCTM updated standards for mathematics (2000) has recognized that problem solving is an integral part of all mathematics learning. Solving problems is not only a goal of learning mathematics but also a major means of doing so. Problem solving should not be an isolated part of the curriculum but should involve all Content Standards. Good problems stimulate new learning and give students the chance to solidify and extend their knowledge. Below is a sample of problem that puts the students in a challenging
situation.
This item is typical problem solving item for the students in the sense that they have no prior knowledge in their textbook to solve them directly and they can use trial and error method or alternatively they can use equation to solve algebraically. 37
2.4 Studies Regarding Mathematical Thinking and Mathematics achievement with Gender Comparison Preceding section is related to review of studies regarding mathematical thinking and mathematics achievement in relation to their back ground variables i.e. gender, location and school sector.
2.4.1 Gender and Mathematical Thinking Gender and mathematics achievement has been focus of research since 1974, however few studies have discussed gender and mathematical thinking explicitly (Ma’moon, 2005). This section will focus on gender difference in mathematical thinking. The study will compare gender wise performance in different aspect of mathematical thinking. Ma’moon (2005) conducted a study in Jordan comparing mathematical thinking and mathematics achievement of year 11 students. The sample was 500 students and tools used were a test of mathematical thinking and achievement test along with interview to get greater insight.
He concluded that when mean scores for the male and female
students on total scores of mathematical thinking were compared, there was a significant gender difference for the total score of mathematical thinking. He further reported a significant interaction effect among gender and location in the total score of mathematical thinking. Another
study in this particular area was done by Lutfi (1998) in which he administered a test of 38
mathematical thinking to the student of grade 9, 10, 11 and grade 12.
Test was
comprised of 35 items which were aimed to measure induction, deduction, symbolism, proofs, logic and generalization. He found that across different grades i.e. from grade 9 to grade 12, students score was different in favor of senior students in all aspects of mathematical thinking except proofs. However this difference was not significant on any of the six characteristics. Similarly gender wise results were also not significant for any of the aspects in the model. When interaction of group and gender was taken into account he found class wise and gender wise interaction was also not significant.
2.4.1.1 Gender Comparison in Generalization, Using Symbols Generalization has been considered as life-blood of mathematics (Mason et al. 91) and has been recognized both as an object and a means of thinking and communicating (Dorfler, 1991). Algebra provides opportunities to observe and verbalize their generalization and record them symbolically. According to Wagner & Kieran (1989) algebra is conceived as branch of mathematics that deals with symbolizing and generalizing numerical relationships and structures and operating with those structures. This section will examine research studies comparing genders in generalization and studies of algebra and using symbols with relationship to generalization. Some studies have reported the superior performance by male than female e.g. Bessoondyal (2005) reported that boys outperformed girls in overall test and also in number, algebra in a separate strand of test in Mauritius during his studies investigating gender difference in mathematics. Ma (1995) conducted a study on a sample of 960 students from both senior and junior classes in four countries namely British Columbia, 39
Ontario, Hong Kong, and Japan. The aim of the study was investigate gender difference across different education system in domain of algebra and geometry. The study used data from the Second International Mathematics and Science Study (SIMSS, 1999) and reported that a significant gender differences among senior students where the males outperformed the females on the geometry subtest. In contrast several studies have reported that there is no significant gender difference in algebra, symbolism and generalization. Ma’moon (2005) conducted a study in Jordan comparing mathematical thinking and mathematics achievement of year 11 students. He administered a test of mathematical thinking with 30 items which was aimed to measured six different aspect using symbols, generalization, deduction, induction, geometrical proofs and logical thinking over a sample of 468 grade 11 students. He concluded that there was no significant difference between the mean scores for the male and female students in generalization and using symbols. In TIMSS (2003) results shows that in almost half of the countries there was no significant gender difference in the algebra subtest. Similarly Swafford (1980) also found that there is no significant gender difference in his studies comparing the performance of girls and boys with comparable math backgrounds in first-year algebra classes in high schools across the U.S. Also , in a survey in the United States conducted by Armstrong (1981) on year 7 and secondary schools students to examine any gender differences in various mathematical areas. He found that there was no significant gender difference in algebra subtest in year 7 and 12. Stites, Kennison, & Horton, (2004) investigated gender effect on solving algebraic word problems and found no significant difference across gender. El Hassan (2001) reported gender differences in a Mathematics test that covered five topics: 40
numbers, operations, geometry, measurement, and problem solving at the 9 grade, and three topics: operation, geometry, and algebra at the 13th achievement in mathematics were not statistically different for males and females in 9 and 13 grades. Generally, the studies regarding gender performance in generalization, symbols and algebra have concluded that there is no difference among them (Ma’moon, 2005; TIMSS, 2003; Swafford, 1980; Armstrong, 81; El Hassan, 2001) in contrast some have found that male are doing well (Bessoondyal , 2005; Ma’, 1995). These studies have given important information regarding gender difference in generalization but were either superficially conducted or failed to integrate the role of symbolization in generalization.
2.4.1.3 Gender Comparison in Geometry Performance Geometry has been recognized as the core component of curriculum in mathematics and is considered as important to success in mathematics (John, 2004). In their revised work Principles and Standards for School Mathematics (2000), the NCTM states that “Geometry has long been regarded as the place in the school mathematics curriculum where students learn to reason and to see the axiomatic structure of mathematics” (p. 40), and that “Geometry offers a means of describing, analyzing, and understanding the world and seeing beauty in its structures” (p. 309). Supporting their 1989 statement that geometry was useful for the learning of other branches of mathematics and problem solving, the NCTM further stated in 2000 “Geometric ideas can be useful in other areas of mathematics and in applied settings” (p. 309).This section will summarize the studies which are focused on gender comparison on mathematical
41
thinking with relationship of geometry, proofs and spatial reasoning. The goals of proof are not the mere verification of truth but the gaining of understanding. Most of the studies have reported the superior performance of boys over girls’ e.g. Bessoondyal (2005) studied the gender difference in mathematics in Mauritius during his PhD studies.
He administered a test consisting of multiple choice items and word
problems from strands, number, algebra, geometry and probability to secondary level students and reported the boys performed significantly better than girls in overall test, also in separate strands of the test boys outperformed girls in geometry. Battista (1990) conducted a study concerned with the spatial and geometrical thinking of students. A sample of 145 high school geometry students both male and female were tested in four areas; spatial visualization, logical reasoning, geometrical knowledge, and geometrical problem solving.
Battista (1990) found that males scored significantly higher than
females on geometrical knowledge and geometrical problem solving. However was his study was more concerned with measuring achievement in geometry in terms of comprehension of basic concepts, techniques, and principles and the ability to apply understanding to new situations instead of proofs. El-Hassan (2001) in Lebanon found that, at the 13 grade in operation and geometry topics, males performed better than females. Ma (1995) conducted a study on a sample of 960 students from both senior and junior classes in four countries namely British Columbia, Ontario, Hong Kong, and Japan. The aim of the study was investigate gender difference across different education system in domain of algebra and geometry. The study used data from the Second International Mathematics and Science Study (SIMSS) and reported a significant gender
42
differences among senior students where male students outperformed female on the geometry subtest. In contrast some studies have found that girls are performing well in geometry e.g. Ma’moon (2005) as discussed earlier found that females had significantly higher scores than males for subtest of Mathematical proof in his test of mathematical thinking. TIMSS (2007) study which was undertaken in 59 countries for grade 4 and 8 students in target content areas in mathematics for grade 4 were number, geometric shapes and measures, data display and for grade 8 number, algebra, geometry, data and chance. It is reported for both grades that girls had higher achievement on average in geometry. Girls had higher achievement in 15 countries and boys in 6 countries. Healy and Hoyles (2000) conducted a study on Proof conceptions in Algebra by surveying high-attaining 14 and 15-year-old students and concluded that gender of the students was significantly associated with achievement where girls obtained higher scores than boys in construction of proofs. In TIMSS (2003) Jordanian females had a significantly higher average score than males consistent with seven other countries in geometry. Senk and Usiskin (1983) conducted a study on large sample of 2699 in 99 different classes to investigate any gender differences in the understanding on geometrical proof for senior high students ranging from 7 grades to 12 grades. Three forms of a proof test were devised so that performance on a greater number of proofs could be analyzed. The students were tested on their knowledge of geometry at the beginning of the year and their understanding of three types of standard geometry proofs at the end of the year. They found that though boys had a slight higher score but no consistent pattern of statistically significant differences favoring either sex on any form 43
of proof tests was found. Huntley (1990), in his study regarding effect of diagram formats on performance on geometry items administered a 32 experimental, multiplechoice geometry items were administered in two pretest versions; one version did and one version did not provide a relevant diagram. Log linear analysis of the data shows that there were no significant differences between males and females students performance on these items. Literature review shows that there is no single direction regarding gender differences in the performance of students in geometry.
Some studies support the
superior performance of male over female in geometry and mathematical proofs (Bessoondyal, 2005; Battista, 1990; El-Hassan, 2001; Ma’, 1995) while some other report better performance in geometry by female students or no difference e.g. (Ma’moon, 2005; TIMSS, 2007; TIMSS, 2003; Senk and Usiskin, 1983; Huntely, 1990 ).
2.4.1.4 Gender Comparison in Problem Solving Performance Problem solving ability is important in enabling students to be independent thinkers and to find Solutions in all areas of life (Arora, 2003) therefore the National Council of Teachers of Mathematics has also recognized problem solving as one of the five
fundamental
mathematical
process
standards.([NCTM],
http://standards.nctm.org/document/chapter3/index.htm).
2000,
Problem solving is now
considered a great source of attention and foundation for all mathematical activities (Reys, Lindquist, Lambdin, Smith, & Suydam, 2001).
According to Burn (2000)
problem solving ability is necessary in a wider context to function in our complex and changing society. Achievement in problem solving across mathematics and science is strongly related to achievement in mathematics and to achievement in science as 44
measured and reported in TIMSS (2003). This section will provide an overview of gender difference in problem solving achievement. Johnson (1984) during a series of experiments on students in which all subject participated in one study only. The typical experimental session for all studies lasted for 90 minutes. Subjects were tested in mixed-sex groups numbering 20 to 50. No mention of sex differences was made on sign-up sheets or in introductory comments. Subjects were allowed exactly 3 minutes per problem. Looking ahead in the paper and returning to previous problems was not allowed. The series of studies reported here has attempted to discover the reason for the male superiority among college students in solving word problems. A major finding is that the male advantage, with a median value of 35% over nine experiments, The male advantage seems to extend rather broadly through the domain of word problems (Experiments 1 and 5) and to be independent of prior exposure to the particular problems used (Experiments 2 and 3). However, in formal deductive problem-solving tasks, the male advantage diminishes into insignificance (Experiments 4 and 6). He concluded that the male advantage may have something to do with translating a verbally expressed situation into a representation that can be attacked analytically or mathematically. Bessoondyal (2005) has reported significant gender differences in the performance of measurement and problem solving in favor of boys. However these differences were revealed at the knowledge and analysis levels only. Battista (1990) conducted a study to measure geometrical problem solving of 145 high school students and found that male significantly outperformed female in their geometrical problem 45
solving score.
A study was conducted by Manger and Gjestad (1997) in Norway
comparing males and female about mathematics achievement in which forty nine thirdgrade school classes in the city of Bergen were randomly selected for participation in the study in the spring term of 1992. The 49 classes included 924 of the 2,346 third-grade students in Bergen that term (there were 440 girls and 484 boys in the 49 classes). Students start school at the age of seven in Norway, and the modal age was ten years when data were collected in May 1992. The results suggest that there is a significant effect of gender favoring boys in mathematical achievement at the Norwegian third-grade primary-school level in multiplication and division, measurement problems, fraction problems, geometry problems, and verbal problems. In contrast, other studies show the superiority of female over male for example Fennema (1978) conducted Romberg-Wearne Problem Solving Test (R-W) (Wearne, 1976), on the 1320 students from 6th-8th grade enrolled in middle schools that were feeder schools for the four high schools studied the year before. The test of problem solving is in some respects unique and warrants description. This test was composed of twenty-three super items, each of which contains a stem and three questions related to the stem: a comprehension question, an application question, and a true problem-solving question.
Each comprehension question assesses understanding of the information
contained either implicitly or explicitly in the stem. The application questions assess mastery of a prerequisite concept or skill necessary for solving the problem.
The
problem-solving question poses a problem with no immediately available solution, that is, a situation that does not lend itself to an immediate application of some rule or algorithm. Although the application and problem-solving questions both refer to the 46
stem, the questions are independent to the extent that the response to one is not used to respond to the other. He found that girls were slightly doing well in problem solving. (i) Carr & Jessup (1997) found limited evidence on problem solving gender wise that there may be some gender differences in problem-solving strategies, girls tending to use counting while boys relied on mental strategies. Gallagher and DeLisi (1994) in his study of high-ability secondary school students found that females had the tendency to use more conventional (commonly taught) strategies, against males who tended to use more untaught strategies but overall there was no gender differences in achievement. Randhawa (1987) conducted a study in Canada for grades 4, 7 and 10 and found that in the Mathematics Problem Solving sub skills analysis; there was a trend from a non significant sex difference at Grade 4 to a significant sex difference at Grades 7 and l0. While some studies have reported no gender differences in problem solving construct e.g. Fennema et al (1998) conducted a three years longitudinal study with a sample of 44 boys and 38 girls. She conducted five interviews, besides considering other variables like race and those who received lunched and those who do not, gender was also taken as determinant. Results showed that difference was not significant when performance of boys and girls was compared during these three years for number facts, addition/subtraction, or non routine problems. In grade 3, boys solved significantly more extension problems than did the girls. However, for problem solving construct there were strong and consistent gender differences in terms of using strategies to solve problems. Girls were found to use more concrete strategies like modeling and counting while boys showed tendency to use more abstract strategies reflecting conceptual understanding. Moreover at the end of this longitudinal study strategies used by girls 47
were more standardized than boys used. On the other hand problems where extension procedure was required, boys were found to outperformed girls. As mentioned earlier El Hassan (2001) reported gender differences in Lebanon on a mathematics test covering five topics: numbers, operations, geometry, measurement, and problem solving using the Monitoring Learning Achievement (MLA). The Mathematics test covered five topics: numbers, operations, geometry, measurement, and problem solving. He found no gender difference in problem solving in 9 and 13 grades. Most of the studies have reported superior performance by male in problem solving e.g. (Johnson ,1984; Bessoondyal, 2005; Battista,1990; Terje & Rolf, 1997) while other reports better performance by female like (Fennema et al,1978;Carr & Jessup, 1997; and Gallagher & Delsi, 1994). In contrast two studies Fennema (1998) and El Hasan (2001) have found no gender difference in problem solving. Almost all of the above studies show that multiple choice tasks were used to examine the gender differences in solving routine word problem leaving the room for task based assessment, with justification of their solution, to examine the gender difference along with broader context of location, school sectors and other context.
2.4.1.5 Gender Comparison in Logical Thinking Logical thinking and reasoning abilities are important in mathematics (Dyke and Frances, 1995) and has been associated with mathematical achievement.
NCTM
standards (2000) have recognized systematic reasoning is a defining feature of mathematics. Exploration, identifying and justifying is not limited to a particular content in mathematics. This section will examine the studies gender performance with logical thinking in connection with reasoning abilities. 48
Bessoondyal, (2005) studied the gender difference in mathematics in Mauritius during his research for PhD studies. He administered a test consisting of multiple choice items and word problems from strands, number, algebra, geometry and probability to secondary level students and reported the boys performed significantly better than girls in overall test. The boy’s performance was better significantly in the questions with if-then statements. However the test was not validated properly and administered by the class teacher instead by the researcher which might have affected the results.
Also the
inclusion of high achiever, low achiever and average students in the sample was based on the assumption that students are comparable across schools. Five content areas were assessed in math at the 9th grade level: numbers, operations, geometry, measurement and problem solving.
Overall achievement in mathematics did not show statistically
significant differences, although the boys exhibited an advantage. Pearson and Stanley (1982) conducted a study aimed at comparing gender in mathematical reasoning test. He reported that a large sex difference favoring males on the SAT-M, a test of mathematical reasoning ability. This study followed up at high school graduation, the students studied by Benbow and Stanley (1980b) who, as 7th or 8th graders, had participated in any of the first three talent searches and had scored as well as a national sample of 1lth and 12th grade females do on the SAT. The objective was to investigate the development and consequences of this initial sex difference. Many consequences were found. This study and the Benbow and Stanley (l980b, 1981) paper demonstrate that (1) sex differences in mathematical reasoning ability are found at an early age among mathematically talented students favoring male students and they persist over several years and are related to subsequent differences in mathematics achievement. Benbow (1983) in his study about 49
talent searches conducted over period of eight years in the Mid-Atlantic States. Almost 10,000 students from grade 7 and 8 who were reckoned as gifted students according to their achievement in previous performance. He found that boys scored substantially better than their female counterparts on a difficult test of mathematical reasoning ability. However there were no sex differences on the equally difficult verbal reasoning test. Ginsburg, Cooke, Leinwand, Noell, & Pollock, (2005) analyzed TIMSS (2003) results in context of gender performance in three domains including reasoning and they found that boys have in the United States and Italy consistently outperform girls in reasoning domain though the difference was small i.e. less than a tenth of a standard deviation. Bitner-Corvin (1987) conducted a study aimed to determine the level of reasoning of all seventh through twelfth grade students in rural Arkansas. Group Assessment of Logical Thinking (GALT) was made with a 12-items abbreviated form of GALT. He concluded that significant male performed significantly better on the conservation mode, conservation of volume, and probabilistic reasoning while female had better score on combinatorial logic. Gonzales et al (2004) analyzed TIMSS (2003) data for reasoning domain and conclude that eighth-grade girls had significantly higher achievement than boys across all countries except Morocco and Tunisia where boys scored better than girls. Such pattern of gender difference was also observed at the fourth grade but was less pronounced in the sense that girls performed better in three countries and boys showed no superiority in any country. Ma’moon (2005) as discussed earlier found that females had significantly higher scores than males for subtest of logical thinking in his test of mathematical thinking. In Cox’s 50
(2000) study, while working on a project CAT i.e.
Common Assessment Task CAT1
was in investigative project, CAT2 was a challenging problem, CAT3 ,involved facts and skills and CAT4 was an analysis task, females scored significantly higher in four areas for CAT1 (extensions space and number, change and approximation, extensions change and approximation and reasoning and data), and three of the subjects for CAT2 (extensions space and number, extensions change and approximation and reasoning and data), whereas males were significantly higher in four of the subjects for CAT3 (space and number, change and approximation, extensions change and approximation and reasoning and data), and three of the subjects for CAT4 (space and number, extensions change and approximation and reasoning and data). This study emphasized that females did better than males in terms of reasoning on investigation and challenging problems, however, males did better than females in terms of reasoning on facts and skills and analysis of problems. Kiamanesh, (1999) analyzed data from TIMSS (1999) and TIMSS (2003) for Iranian students and concluded that the girls’ achievement increased in two performance categories, i.e., “solving routine problems” and “reasoning”. TIMSS (1999) data showed that the average percent correct for boys in all of the four performance expectations was higher than that of girls; however, in TIMSS (2003) girls had superiority over boys in “knowing facts and procedures” and “reasoning”. While some studies have found that there is no significant relationship between gender and reasoning in mathematics.
For example Battista (1990) used an
experimentally constructed test to measure logical reasoning and reported that, for reasoning, no gender differences were found.
51
In general, most of these studies comparing gender in reasoning have reported superior performance by male students (Bessoondyl, 2005; Stanley, 1982, Mamoon, 2005; Benbow and Stanley 1980b; Benbow and Stanley 1982;, Benbow 1983; Ginsburg, Cooke, Leinwand, Noell, & Pollock, 2005), however some studies have reported the opposite trends favoring female in their performance (Cox, 2000; Gonzales, Guzmán, Partelow,
Pahlke,
and Kastberg, 2004; Kiamanesh, 1999; Bitner-Corvin, 1987).
However, there is one study reported (Battista; 1990) which shows no gender difference in performance of reasoning ability.
2.4.1.6 Gender Comparison in Induction and Deduction Performance Inductive and deductive ways of thinking are two prime modes of thinking. Fundamental difference in these two types of thinking among boys and girls was investigated by Gurian, Henley and Trueman (2001). They concluded that boys tend to be deductive in their conceptualization, starting from their reasoning process frequently from a general principle and applying it. They also tend to do deductive thinking more quickly than girls. This is the reason that boys do well in multiple choice item in scholastic achievement test. Girls on other hand tend to favor inductive thinking adding more and more to their base of conceptualization. More often than boys, girls begin with specific examples and then build their general theory. Not many specific studies are available discussing students’ achievement in induction and deduction. The only major study found was that of Ma’moon (2005) in Jordan comparing mathematical thinking and mathematics achievement of year 11 students. He administered a test of mathematical thinking with 30 items which was aimed to measured six different aspect using symbols, generalization, deduction, 52
induction, geometrical proofs and logical thinking over a sample of 468 students at grade 11. He concluded that there was no significant difference between mean score for male and female students in induction and deduction sub scales. When gender and location were combined as independent variables with the six aspects of mathematical thinking and mathematics achievement as dependent variables, females outperformed males in induction while in deduction there was no significant difference. The scarcity of literature in this particular context shows that more study is needed to investigate induction and deduction aspects of mathematical thinking, particularly with a new setting e.g. different sample, curriculum and grades.
2.4.2 Mathematics Achievement and Gender Comparison Gender has been an important issue in the context of mathematics achievement and many studies have been conducted during past thirty years (Richard, 2002). Gender differences in mathematics learning continue to attract much attention from practitioners, administrators, school systems, government initiatives and researchers ( Leder & Forgasz, 1992), and have been studied intensely for about 20 years ( Mamoon 2004, Ai, 2002; Al khateeb, 2001; Dennis, 1993; Hanna, 1986; Low & Over 1993; Ma, 1995; Randhawa, 1988; Uekawa & Lange 1998; Young 1998). Equity in mathematics learning and achievement for male and female has been incentive for such studies in many countries as negative beliefs may hampers performance in girls.”(Fennema,1993, p.1). This section presents the studies focused on gender comparison in mathematics achievement.
2.4.2.1 Studies that Report Male Superiority in Mathematics Achievement
53
Many studies have reported that male students have performed well than female students in mathematics achievement.
Academy of Educational Planning and
Management (AEPAM, 1999) conducted a national level study to measure learning achievement at primary level in Pakistan. The sample comprised 2, 794 students from class V from 145 government schools (75 boys school & 70 girls schools) across the country. It looked at the learning achievements of the students at the primary level in Science, Mathematics and Language (Urdu) and to identify major factors associated with the students’ performance at this level. Gender wise analysis shows that girls performed significantly better than boys in both subjects (Moe, 2002, AEPAM Study No. 167). Similarly Shami et.al (2005) in their study on quality of education learning achievement at primary level have reported that male students’ performance was significantly better than their female counterparts.
Chudgar and Sankar (2008) conducted a survey on
relationship between teacher gender and student achievement in five Indian states in 300 schools, besides other reports they concluded that that while girls did poorly in mathematics overall compared with boys, that performance of boys was better than girls in overall in mathematics achievement. SPDC (2003) survey administered achievement tests in mathematics, physics, chemistry and biology in 20 colleges. Besides other subjects, score in mathematics was significantly higher for male students. Saeed, Bashir, and Bushra (2005) conducted a study to assess achievement level at primary grade in Pakistan. The sample was 1080 student from 36 schools in nine districts in province of Punjab. Regarding gender wise comparison they found that male (mean= 17.12) performance was slightly better than female (mean=15.78). Similarly World Bank’s Primary Education Project study in 1984 54
compared Science and Mathematics Achievement in a large scale assessment in Pakistan. A representative sample of 3,300 students of grades 4 and 5 project and non-project schools in Punjab, Sindh and the Khyber Pakhtunkhwa. It was summarized that boys scored higher in mathematics. In another study, results from TIMSS (2003) were analyzed by Alkhateeb (2004) for year 8 students for both Islamic and non Islamic countries, results indicated that in 20% of the countries students of male schools outperformed female counterparts, and in less than 20% of the countries female students outperformed males while a larger proportion of the countries there was no gender difference in mathematics achievement. In the same way a study was conducted by Manger and Gjestad (1997) in Norway comparing males and female about mathematics achievement. Forty nine third-grade school classes in the city of Bergen were randomly selected for participation in the study in the spring term of 1992. The 49 classes included 924 of the 2,346 third-grade students in Bergen that term (there were 440 girls and 484 boys in the 49 classes). Students start school at the age of seven in Norway, most of the students was aged around 10 years. The instrument used in this study was comprised of 100 items aimed to measure achievement in mathematics.
The results presented above suggest that there is a
significant effect of gender favoring boys in mathematical achievement at the Norwegian third-grade primary-school level. Hopkins (2004) conducted a study in Tennessee with a focus on mathematics achievement, locale and location. Findings shows that female have edge over their male counter parts in the middle school years while males are achieving higher at the high school level. A mid-term survey (NCERT, 2005-07) in state of Karnataka of India for students of grade 3 and 4 also provide evidences that boys are 55
superior in mathematics achievement than girls. Girls have scored better than boys in Mathematics. But there is no significant difference in the overall achievement of Boys & Girls of grade 3 however at grade 4 girls have scored better than Boys in Karnataka. Analysis of TIMSS (1995) by Maryellen et al (1997 on performance assessment task in mathematics for grade 4 and 8 shows that eighth grade boys in Romania did better than girls on the around the Bend task and in the United States boys had higher achievement on Shadows.
2.4.2.2 Studies that Report Female Superiority in Mathematics Achievement In contrast some studies have reported better performance from female than male in mathematics achievement. For example Christine (1996) did a meta analysis of 16 statistics achievement studies containing 318 subjects per sample on average 169, males and 149 females. Sample sizes ranged from 11 males and 34 females in Harvey, Plake, and Wise's (1985) Sample A to 1,010 males and 685 females in Mogull (1989). Of the 18 samples, 13 were from studies containing more than one sample or from studies by authors who contributed more than one article to this literature. Ten of the 13 articles either focused exclusively on gender differences or drew conclusions regarding gender as a major focus of the article. The median year of publication was 1987.Studies included in this meta-analysis examined, at least, a measure of statistics achievement from an applied statistics 56 gender differences course. Subjects of interest were under graduate or graduate students enrolled in statistics courses offered by psychology, education, and business departments (rather than by mathematics departments). Also, results of statistics performance had to be presented by gender. The general trend of the results was similar 56
to that found in studies of gender differences in mathematics: women outperform men when the outcome is grades, and men are favored when tests are used. Ismail (2009) analyzed the TIMSS (2003) data for Malaysia and Singapore and concluded that gender differences with girls scored higher than boys in both countries. Also they reported that in Singapore mean scores for girls are higher than Malaysian girls. A midterm national survey in India was conducted by National Council of Educational Research and Training on learning achievement of class-v children. The sample was 30 schools from each district of India and 15 students from each schools both urban and rural schools. There was significant gender difference overall but there was significant difference in achievement favouring urban girls than urban boys. Paul, Barbara and Ormond (1987) conducted a study on children's mathematics achievement in Hawaii. The data was taken from Hawaii State Department of Education (DOE) from its annual statewide administration of the Stanford Achievement Test series for two school years (1982-83 and 1983-84) on three mathematics subtests for grades 4, 6, and 8 and on one mathematics subtest for grade 10 were analyzed. Only data on Hawaii's four major ethnic groups (Caucasians, Filipinos, Hawaiians, and Japanese) were examined. They reported that girls’ achievement was higher than boys. Basey, Joshua and Asim (2008) conducted a survey type study in rural setting in Nigeria on a sample of 2000 students taken randomly. The sample had 50% proportion gender wise with a mean age 16.8 years. A forty five minutes 30-items test was administered to them and the result was that there is significant difference among the scores of boys and girls. The boys mean score was higher than the girls. AEPAM (2004) conducted a study aimed at comparing quality of schools in terms of students’ achievement on a sample of grade V students, 57
selected randomly from both private and public schools.
Result shows that in
mathematics achievement performance of girls was higher significantly than boys in both rural and urban. However this trend was reverse in private schools where boys score better than girls. The girls performed significantly better than the boys in both Science and Mathematics.
Girls performed significantly better than boys in both subjects,
however, this finding is in contradiction with the SPDC (2003) survey and earlier reports of gender gap in student achievement (Warick & Reimers, 1995). A possible reason could be that the NEAS and AEPAM reports are based on achievement in primary school while SPDC reports are based on achievements in secondary school where reportedly the gap in teacher quality in boys and girls school is wider. Khan et al (1999) conducted a study aimed to assess learning achievement in Science, Mathematics, and Language (Urdu) on a sample of 2794 from grade 4 students. They used test instrument which was directly comprised of items taken from their textbook. Results showed that male students outperformed their female counterparts in mathematics. In another study Awang and Halima (2009) used TIMSS (1999) data for gender analysis in mathematics for overall performance and in the main content areas of fractions, number sense, data representation, geometry and algebra as well.
They
concluded that there was a significant gender difference in favor of girls in overall performance and in the content area of number sense, algebra and data representation as well. Gender differences in various subjects, particularly in mathematics (mathematics concepts and computation) were investigated by Randhawa and Hunt (1987) in their study conducted in mid-western province in Canada on standardized tests. A random 58
sample of grades 4, 7, and 10 was taken. At Grades 4 and 7, the mathematics subtests are grouped into knowledge of mathematical concepts and competence in problem solving. At the Grade 10 level, the subtests are divided into knowledge of mathematical concepts and competence at mathematical computation. in the subtest involving concepts at grade 10 level, the males performed better and maintained their superiority; but, in the subtest involving computation, the females were superior In terms of grade 10, Randhawa and Hunt (1987) found that females scored better than males on mathematics computation subtests, whereas males scored better than females only on mathematics concepts. As discussed earlier, Maryellen, H. et al. (1997). Analysis of TIMSS (1995) data shows that girls had higher achievement than boys on the Solutions and Dice tasks in Australia while Swedish girls did better than boys on the Packaging task similarly significant gender differences on Plasticine task was found
in Australia, where girls had higher
achievement than boys.
2.4.2.3 Studies that Shows no Gender Difference in Mathematics Achievement Maryellen et al. (1997) analysis of the data from TIMSS on performance assessment tasks in mathematics for grade 4 and 8 shows that both genders had almost equal scores on both overall test and individual tasks. However for 8th grade some gender difference was reported on individual tasks.
For most countries, gender
differences on the written assessment were small or not significant in mathematics achievement overall. Abiam and Odok (2006), in another study in Nigeria, found that
there is no significant relationship among gender and achievement in number. Seventh Survey of Mathematics (2004) in Scotland shows that there is no significant 59
gender difference in attainment in mathematics, at any stage. A study was conducted by Academy of Educational planning and management in primary schools in Pakistan to assess student’s performance in different subjects. There was no significant difference between the scores of boys and girls in mathematics. Olson (2002) conducted a study on the affect of cooperative learning on mathematics achievement. In this four group experimental study he concluded that there is no significant gender difference in achievement though females were having slightly better course grades. In their study, Walsh, Hickey, and Duffy (cited in Olson, 2000) conducted experiments which Canadian students of two age groups to determine any gender difference on problem solving. A sample of 7th and 8th grade were given a modified version of the Canadian test of Basic skills, and sample of university students were given a modified version of a model standardized achievement test. But he found that male and female were both equal in giving correct answers. TIMSS (2003) was analyzed by Beaton and Robitaille (2006) for different categories of population as in population 1 all students in the two adjacent grades that contain the largest proportion of 9 years old at the time of testing. These students are in grade 3 and 4 in most countries. While in population 2 all students in the two adjacent grades that contain the largest proportion of 13 years old at the time of testing. These students are in grade 7 and 8 in most countries. For population 1 most of the countries, gender difference in mathematics achievement were small or essential nonexistent. However, the direction of the few gender difference that did exist favored boys over girls. Within the mathematics content areas, there were few differences in performance between boys and girls, except in
60
measurement, where the differences favored boys.
While for population 2 where
difference exists in case of algebra girls were found to performing well.
2.5 Mathematics Achievement and Location Mathematics achievement and location has been studied in the past and interest in such studies is understandable as an issue of equity in education in general and mathematics achievement in particular. There is a view among scholars that schooling in rural area suffers from deficiency particularly due to lack of facilities and environmental issues ( Herzog & Pittman, 1995). This view is further supported by Hopkins (2004, p.19) that media has promoted such perceptions that rural people have uninformed views about real world. These perceptions are further strengthened by research studies that had reported better performance of urban students than rural students (e.g. Hopkins, 2004; Khan et al 1999 and Young, 1998). Though, some researchers have reported that rural
students are doing well in mathematics (e.g. Saeed, Bashir and Bushra. 2005; Craig, Howley and Erik Gunn. 2003; Cox. 2000) while some studies tells that there is no difference in performance (e.g. Fan and Chell. 1999; Randhawa.1988)
2.5.2. Mathematical Thinking and Location Not many studies as stated before are available, addressing the issue of mathematical thinking and location explicitly except Ma’moon (2005), others have studied aspects of mathematical thinking like geometry, algebra, problem solving and logical thinking implicitly with comparison of mathematics achievement.
Ma’moon
(2005), in his study that compared the Jordanian year 11 students in mathematical thinking and mathematics achievement for three categories of regions, urban, sub urban 61
and rural. He found that there were significant performance differences for mathematical thinking (total) where the students of sub urban area were performing well than rural and urban areas similarly the students of rural areas were better than urban area in the score on test of mathematical thinking. Furthermore he found that in subscale of generalization and logical thinking, suburban students were found scoring better than urban students. While for sub scale induction also rural student’s outperformed urban students. For sub scale use of symbols o linear relationship was found among students achievement and population density where sub urban outperformed urban and rural students.
2.5.2 Studies that Link Mathematics Achievement and Location Mathematics achievement in the context of location as determinant has been the focus of research since long. Rural Education has been viewed as the deficient model in general whenever students’ achievement is discussed and mathematics achievement is not the exception.
However there are contrasting reports about location as a factor in
mathematics achievement.
2.4.2.1 Studies that shows better Performance by Urban Students Hopkins (2004) conducted a study in Tennessee with a focus on mathematics achievement and enrollment in context of location, and reported that urban students were found better than rural students in mathematics achievement.
Khan et al (1999) as
discussed earlier has reported that urban students scored higher on all three sections than rural students. However the sample in this study was not random.Young (1998) conducted a study
in Western Australia comparing student’s achievement in science and mathematics in context of urban/rural comparison. An 18 items test was used for this study which was already used by TIMSS and was validated. Only 18 multiple choice items were used in 62
test that had to be completed in 45 minutes. Sample for this study includes schools from a variety of location in Western Australian. A sample of 3397 students from 28 schools participated in the study.
This analysis showed that the location of the school had a
strong effect on mathematics achievement after background variables were ignored. The location of the school was categorized from 1 (urban) to 5 (remote), and the strong negative effect demonstrated that the more rural and remote schools had significantly lower achievement in mathematics achievement. School level differences in mathematics achievement showed that location of School appeared to account for 21.5% of the variation in achievement. The national survey conducted by the Multi-Donor Support Unit for the Social Action Program (SAP, 1995, cited in Das et al 2006) was conducted with main objective to assess student achievement in different subject including mathematics at primary level and associated variables .A sample of over 1100 students from grade 5 participated in the study.
Students from populous and urban set up like
province of Punjab outperformed their counterparts in rural set up like Federally Administered Northern Area (FANA). Executive summary of survey conducted by National Council for Research and training (NCERT, 1994) in India on student’s achievement in different subjects reported that in Mathematics the performance of urban students, both boys and girls was significantly better than their counterparts in rural areas. The achievement of boys was better than girls both in urban and rural areas. This result was confirmed by midterm survey by NCERT (2001) showing that in Mathematics in ST (Scheduled tribe) category, the urban students performed significantly better than rural students.
In a study
conducted by Academy of Educational Planning and Management (AEPAM, 1999) with 63
a sample of about three thousands students from 145 schools at grade V with equal distribution of students from both genders. The focus of the survey was to measure learning achievement in core subjects of science, mathematics and National language Urdu and important factors associated with students’ achievement. Urban students have performed significantly better than the rural students in both Mathematics and Science. AEPAM (2004) conducted a study to compare school performance to understand which schools are doing better by assessing and comparing quality. This study also looked at student achievement of class V students in the country’s government and private schools. Achievement tests were administered in mathematics, science and Urdu to randomly selected students. Half of the students both in private and public did not qualify the test and they got grade ‘D’ which indicates that the majority of the students lack the basic competency in Mathematics. The performance of urban and rural girl students from public schools was better than that of urban and rural boys and the difference of the mean score was significant. Similarly, the performance of rural boys in the private schools was better than that of the rural girls and the difference of the mean score was also significant. The study further reports that average percentage score in mathematics for urban boys was 62% and it was 56% for rural boys, which indicates better performance of urban boys over the rural boys. The average percentage score for urban girls was 62% and it was 52% for rural girls, which shows that urban girls outperformed rural girls. The total score of urban students was 62% whereas it was 54% for rural students in mathematics. A significant difference was observed between the students of urban and rural students. It was inferred that urban students outperformed their rural counter parts in mathematics. 64
Williams (2005) compared students from 24 countries from rural and non rural areas. He used data from Program of International Student’s Assessment (PISA, 2000) collected by the Organisation for Economic Co-operation and Development (OECD). The report shows that students’ scores in 14 of 24 countries were higher for urban students than their rural counterparts.
These results show that rural urban gap is
common but not a universal phenomena and only Australian students follow a linear pattern between community size and average math score. In other countries, such as Japan, medium-size communities scored highest, followed by students in urban locales, then rural.
2.5.2.2 Studies that Shows better Performance by Rural Students In contrast some studies have reported that rural students are doing better than their urban counter parts. For example Saeed, Bashir and Bushra (2005) conducted a study to assess achievement level at primary grade in Pakistan .A sample of 1080 student from 36 schools in nine districts in province of Punjab was administered achievement test. Comparing location wise students’ achievement, they found that performance of rural student was better than urban counterparts. This was termed surprising despite lacking of physical facilities in rural schools. Similarly Shami and Sabir (2007) collected both quantitative and qualitative data at primary level in Pakistan with sample coming from both private and public schools. Their results show that locality was not material in private schools where no difference was found between rural and urban students score areas, whereas performance of rural students was better than that of urban students in public schools and difference of mean score was significant. Also Howley and Erik (2003) in their findings reported that in rural areas, boys performed significantly better 65
than girls in mathematics whereas in urban areas, girls performed significantly better than boys. At the state level a rural /non rural achievement gap exist in just 40% of the states favoring non rural students in 20% of states and favoring rural students in the other 20% of state.
These conclusions falsify common assumptions about rural deficiency in
mathematics achievement at least in comparison to national averages.As discussed in previous section in the study by James (2005) comparing students from 24 countries for rural and non rural comparison. Students in rural settings of United Kingdom obtained highest average achievement followed by Finland, New Zealand, Japan, Belgium, and Australia. Relative size of the gap was also estimated to get a clearer picture, which resulted into a more pronounced gap between these two competing communities in eleven countries out of twenty four where score was significant.
While scores in
Belgium, U.S., and Ireland was not significant but rural scores were higher, on average, than urban scores particularly in Belgium. In the U.S., rural scores were more than one third of a standard deviation higher, though, again, the differences were only marginally significant. Regression analyses with socioeconomic status and location as dependant variables and mathematics achievement as independent variables was also carried out. When SES was controlled location was significant predictor of mathematics achievement for four countries only while for three countries i.e. Sweden, Germany, and New Zealand out of these four rural locations were positively associated with mathematics achievement. Cox (2000) in his study found that students in metropolitan schools did better than students in the country schools in three of subjects in (CATs), CAT1, CAT2 and CAT3, and one subject in CAT4 (CAT1 was in investigative project, CAT2 was a 66
challenging problem, CAT3 involved facts and skills, and CAT4 was an analysis task). Each CAT contained six distinct sections, i.e. space and number, extensions space and number, change and approximation, extensions change and approximation, reasoning and data, and extensions reasoning and data).Cox (2000) found also that students in the country schools did better than students in the metropolitan areas in two of subjects in CAT4.
The interaction between gender and location indicated that urban males
outperformed rural males on half the subjects in CAT1 to CAT3, whereas, rural males outperformed urban males on only two subjects in CAT4. However, urban females outperformed rural females on, five subjects in CAT1 to CAT3 but, rural females outperformed urban females on only one subject in CAT4. In the same way Academy of Educational Planning and Management (2001) compared students from both private and public sectors, the reports of this comparison shows that there was a significant difference in the performance of students in favour of female students in both urban and rural areas. However in case of private schools male students in rural areas were found better than their female counterparts. There was an interesting results regarding interaction between school location and disadvantaged students where rural schools was found advantageous in case of low achiever in mathematics achievement. However the researcher could not answer why this pattern occurs. A greater sense of community may possibly be the reason for better achievement in rural area along with smaller schools and smaller class size where environment is often more nurturing. Uekawa K. and Lange R. (1998) used data from third international mathematics and science study in 1995 by studying rural, urban and suburban schools in an international context relating social capitals and technological factors with mathematics achievement in Korea and United States. They 67
found that in both countries eighth grade students in urban schools perform better than suburban and sub urban were found significantly better than rural schools with d= 15.2 at p
