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UNIVERSITY OF EDUCATION, WINNEBA
EFFECTS OF THE CONSTRUCTIVIST TEACHING APPROACH ON SENIOR HIGH SCHOOL STUDENTS’ PERFORMANCE IN ALGEBRA IN THE GOMOA EAST DISTRICT OF GHANA
SAMUEL ADDO OSEI
MASTER OF PHILOSOPHY
2019 University of Education, Winneba http://ir.uew.edu.gh
UNIVERSITY OF EDUCATION, WINNEBA
EFFECTS OF THE CONSTRUCTIVIST TEACHING APPROACH ON SENIOR HIGH SCHOOL STUDENTS’ PERFORMANCE IN ALGEBRA IN THE GOMOA EAST DISTRICT OF GHANA
SAMUEL ADDO OSEI 8160110016
A thesis in the Department of Mathematics Education, Faculty of Science Education, submitted to the School of Graduate Studies, in partial fulfilment
of the requirements for the award of the degree of Master of Philosophy (Mathematics Education) in the University of Education, Winneba
SEPTEMBER, 2019 University of Education, Winneba http://ir.uew.edu.gh
DECLARATION
STUDENT’S DECLARATION
I, SAMUEL ADDO OSEI, hereby declare that this thesis, with the exception of quotations and references contained in the published works which have all been identified and duly acknowledged, is entirely my own original work, and it has not been submitted, either in part or whole, for another degree elsewhere.
SIGNATURE: …………………………………………
DATE: ………….…….………………………………..
SUPERVISOR’S DECLARATION
I hereby declare that the preparation and presentation of this work was supervised in accordance with the guidelines for supervision of thesis as laid down by the University of Education, Winneba.
NAME OF SUPERVISOR: DR. NYALA JOSEPH ISSAH
SIGNATURE: ……………………………………………
DATE: ……………………………………………………
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DEDICATION
To the Almighty God for His faithfulness in fulfilling His promise to me. To my dear wife, Dorcas Osei and my children – Kennedy Asare Osei, Franklin Gyamerah Osei and Christiana Nyarko Osei – for their sacrifices in helping me come this far.
I am most grateful.
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ACKNOWLEDGEMENT
First of all, my sincere thanks go to the almighty God whose love, protection, mercies and guidance has brought me this far. May your Holy name be praised forever and ever.
The immeasurable useful suggestions, corrections and excellent supervisory role played by Dr. Nyala Joseph Issah are appreciated. May God richly bless you. In the same vein, I express my appreciation and thanks to the Head of Department and lecturers of the Department of Mathematics Education, Winneba, for their tireless dedication to duty.
I cannot forget to make mention of my parents, Mr. and Mrs. Osei and my siblings
Mr. Richard Osei, Mrs. Kate Osei, Miss Haggah Osei, Madam Ama Serwaa and
Madam Ama Konadu who in diverse ways contributed immensely to my educational pursuit. May their efforts, prayers and contributions be richly rewarded.
My special thanks also go to the Management of the participating schools for granting me permission to carry out the study in their schools. My proud gratitude also goes to the mathematics instructors and the entire students in the participating schools for their support and participation during the fieldwork and for accepting to be part of the study.
Finally, to my dear wife, Mrs. Dorcas Osei who has been a pillar and tower behind me throughout this thesis and my entire education. When all hope was lost, she encouraged me to carry on and never give up. Thank you dear. Once again, I say thanks to all my loved ones, God richly bless you all.
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TABLE OF CONTENTS
Title Page
DECLARATION iii
DEDICATION iv
ACKNOWLEDGEMENT v
LIST OF TABLES xii
LIST OF FIGURES xiii
ABSTRACT xiv
CHAPTER ONE: INTRODUCTION 1
1.0 Overview 1
1.1 Background of the Study 1
1.2 Statement of the Problem 5
1.3 Purpose of the Study 8
1.4 Objectives of the Study 8
1.5 Research Questions 8
1.5.1 Research Hypothesis 9
1.6 Significance of the Study 9
1.7 Limitations of the Study 10
1.8 Organization of the Study 10
1.9 Operational Definition of Terms 11
CHAPTER TWO: REVIEW OF RELATED LITERATURE 13
2.0 Overview 13
2.1 Theoretical Framework: Constructivist Theory 14
2.2 Teaching Approaches 15
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2.2.1 Traditional Teaching 16
2.2.1.1 Advantages of the Traditional Teaching 19
2.2.1.2 Disadvantages of the Traditional Teaching 21
2.2.2 Constructivist Teaching or Pedagogy 23
2.2.2.1 The Epistemological Base of Constructivist Teaching 24
2.2.2.2 Implications of Constructivism for Learning and Teaching 25
2.2.2.3 Studies on Constructivist Teaching Approach 28
2.2.3 Constructivist and Traditional Ideas about Teaching and Learning 31
2.3 Teaching Approaches used by Teachers. 33
2.4 Review of Research works on Students’ Perceptions about Constructivist
Teaching 35
2.5 Conceptual Understanding of Algebra 36
2.5.1 The Concept of a Variable. 36
2.5.2 Algebraic Terms and Expressions 38
2.5.3 Solving Algebraic Expressions and Equations 40
2.5.4 The Concept of Equation / Equality 42
2.5.5 Formulating Equations from Context Problems 43
2.5.6 Modeling Equations from Verbal Representations 45
2.6 Students' Difficulties in Solving Algebraic Problems 46
2.7 Summary of the Literature Reviewed 49
CHAPTER THREE: RESEARCH METHODOLOGY 50
3.0 Overview 50
3.1 Research Design 51
3.2 Population 54
3.3 Sample and Sampling Technique 56
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3.4 Research Instruments 57
3.4.1.1 Test (Algebra Concept Achievement Test (ACAT) 57
3.4.1.2 Classroom and Lesson Observation Checklist 58
3.4.1.3 Questionnaires 59
3.4.1.4 Interview Guide 60
3.5 The Pilot Study 61
3.5.1 How the Pilot Study was conducted 61
3.6 Content Validation of Instruments 63
3.6.1 Achievement Test 63
3.6.2 Classroom and Lesson Observation Checklist 63
3.6.3 Questionnaires and Structured Interview Questions 64
3.7 Reliability of the Instrument 64
3.7.1 Achievement Test 64
3.7.2 The Lesson Observation Schedule 65
3.7.3 Questionnaire and Interview Guide 65
3.8 Treatment Procedure 65
3.8.1 Constructivist Teaching Approach as the Treatment Procedure 65
3.9 Data Analysis Procedures 69
3.9.1 Quantitative Data Analysis 69
3.9.2 Qualitative Data Analysis 70
3.9.3 Data Analysis Procedures for Research Questions 70
3.9.3.1 Research Question One 71
3.9.3.2 Research Question Two 71
3.9.3.3 Research Question Three 72
3.9.3.4 Research Question Four 72
3.10 Assumptions of the Study 73
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3.11 Logistical and Ethical Considerations 74
3.12 Chapter Summary 75
CHAPTER FOUR: DATA ANALYSIS AND DISCUSSIONS OF RESULTS 76
4.0 Overview 76
4.1 Demographic Characteristics of Participants 77
4.1.1 Background Information of Participating Teachers 77
4.1.1.1 Gender of Participating Teachers 77
4.1.1.2 Teaching Experience of Participating Teachers 78
4.1.1.3 Teaching Qualification of Participating Teachers 78
4.1.2 Background Information of Participating Students 79
4.2 Data Analysis by Research Questions 80
4.3 Summary of Statistical Findings 103
4.3.1 Participation of the Study 104
4.3.2 Descriptive Statistics 104
4.3.3 Inferential Statistics 104
4.4 Chapter Summary 105
CHAPTER FIVE: SUMMARY, CONCLUSIONS AND RECOMMENDATIONS 107
5.1 Overview 107
5.2 Revisiting the Purpose, the Objectives and Research Questions of the Study 107
5.3 Summary of Key Findings 109
5.4 Achieving the Purpose of the Study 110
5.5 Linking the Study Results to the Theoretical Framework of the Study 111
5.6 Conclusions 112
5.7 General Recommendations 114
5.8 Suggestions for Future Research 116
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REFERENCES 118
APPENDIX 1 Lesson Observation Schedule 133
APPENDIX 2 Teachers’ Questionnaire 134
APPENDIX 3 Interview Questions for Teachers and Students 135
APPENDIX 4 Test Instrument - Pre-test Questions 136
APPENDIX 5 Marking Scheme for Pretest Questions 138
APPENDIX 6 Test Instrument - Post-test Questions 140
APPENDIX 7 Marking Scheme for Post-test Questions 142
APPENDIX 8 Letter of Introduction from the Mathematics Department 144
APPENDIX 9 Letter of Consent to the District Director of Education 145
APPENDIX 10 Hypothesis Testing Summary by Mann-Whitney's Test 146
APPENDIX 11 Normal Q-Q Plots 147
APPENDIX 12 Students' Questionnaire 148
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LIST OF TABLES
2.1: A Comparison between Traditional and Constructivist-Based Classrooms 31
3.1: Benchmarks for Teaching and Learning 53 3.2: Population for Control and Experimental Schools. 55
3.3: Summary of Sampling Technique and Sample Size 56
3.4: Reliability Statistics for the Four Conceptual Areas 65
4.1: Gender of participants 79
4.2: A summary for behaviours, skills, and approaches exhibited by the participants during lesson observations (n = 6). 82
4.3: Mean Percentage Scores for each of the Conceptual Areas in Algebra 87
4.4: Descriptive Statistics of Control and Experimental Groups 90
4.5: Independent Samples Test of Pretest and Posttest Scores 91
4.6: Test of Normality for Experimental and Control Groups 92
4.7: Results of the paired samples t-test on the pre-test and post-test achievement of students in the experimental and control groups. 94
4.8: Response of Participants on Collaborative Learning 97
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LIST OF FIGURES
3.1: Basic Design of the Study 52
4.1 Gender of Participating Teachers 77
4.2 Participants’ Years of Teaching Experience 78
4.3: The Participants’ Teaching Qualification 79
4.4: Teaching Approaches Mostly Preferred by Participants 80
4.5: Agree and Disagree Responses by Categories in Percentage 100
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ABSTRACT
The purpose of this study was to investigate the effects of the Constructivist Teaching Approach (CTA) on senior high school students’ performance in algebra in the Gomoa East District. The study was conducted in two senior high schools in the Gomoa East District of Ghana with a six-week intervention programme. Participants consisted of one hundred and forty (140) form two (2) students from the two participating schools with students from one of the schools serving as the control group whilst students from the other school served as the experimental group. The study adopted a convergent parallel mixed method approach which combined elements of qualitative and quantitative data collection and analysis. The Algebra Concept Achievement Test (ACAT) and questionnaires served as the main instruments for the quantitative approach, whilst the qualitative approach employed interviews and lesson observations. The analysis of the pre-test and post-test scores were based on quantitative data collected using descriptive and inferential statistics. Results from the paired sample t-test showed that there was a statistically significant improvement in the scores between students who were taught using the constructivist approach than those who were taught using the traditional approach, (M = 78.51, SD = 19.25), t (69) = 20.2, p < 0.05). Also, analysis of data from the questionnaire and interviews administered to the students to solicit their views about the constructivist method of teaching revealed that, the method made lessons more interesting, practical and easy to understand although some students agreed otherwise. It was therefore recommended that the classroom practice which was more teacher centred needed to be critically examined and teaching should aim at encouraging collaborative learning and constructive discourse. Teachers should therefore incorporate the use of constructivist teaching approach in their classroom teaching.
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CHAPTER ONE INTRODUCTION 1.0 Overview
Chapter one is an introductory part of the study which presents the background to the study, the statement of the problem, purpose of the study, research questions, delimitations and limitations of the study, significance of the study, operational definition of terms, and organization of the study.
1.1 Background of the Study
One of the national objectives of education is to prepare the child for life after school.
Teaching and learning of mathematics plays a major role in preparing the child adequately to fit into the society. A good foundation in algebra is very essential for the success of the child in life. For example, Cai and Moyer (2012) found that algebra has been characterised as the most important gatekeeper in mathematics.
Algebra is one of the topics in the Core Mathematics Syllabus for Senior High School
(SHS) students to study. It involves algebraic expressions, linear equations in one or two variables and quadratic equations as reported by Curriculum Research
Development Division (CRDD, 2007). The study of Algebra is an important field of study, playing considerable roles in mathematical thinking, and is a mathematical language expressing itself in symbols, tables, words and graphs (Stacey &
MacGregor, 2000). Students need to comprehend the symbols and the manipulations done with symbols in order to interpret the letters used in different situations and the structural properties of the algebraic equations to solve them (Kieran, 2007).
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In Ghana, mathematics is one of the important subjects that is to be learnt by all students from the basic school to the higher level of education. However, performance in the subject is of major concern. Many other researchers (Rakes, Valentine,
McGatha & Ronau, 2010; Hiebert & Carpenter, 1992; Angelo & Cross, 1993; Smith,
2001) share the same views about the poor performance in mathematics. Most of the reasons are related to curriculum and methods of teaching rather than the students’ lack of capacity to learn (Carnine, 1991; Jones, Wilson & Bhaswani, 1997). For example, it has been a traditional phenomenon that, students only begin learning algebra when they enter senior high school (MacGregor, 1991). Linchevski and
Herscovics (1996) found in their study that learners at this level experience serious problems in understanding pre-algebraic concepts and that the teaching of algebra should not wait till learners get to high school. This makes most of the students exhibit weak understanding in mathematics especially in comprehending algebraic concepts. Some studies shows that students face challenges and make common errors and mistakes in comprehending the concepts of equation and variable (Gurbuz &
Akkan 2008) in mathematical interpretation of letters or a variety of notations (Dede
& Peker, 2007) and in solving equations appropriate for problems (Yenilmez & Avcu,
2009). Therefore, the researcher believes that in order to address learners’ performance in mathematics, it is necessary to explore learners’ understanding in algebra. Luneta and Makonye (2010) concluded that algebraic errors have negative impact in learning calculus, which constitutes another algebraic component of
Mathematics.
With regards to pedagogy, Airasian and Walsh (1997) argue that the existing mode of teaching of mathematics in schools has not fulfilled the needs of the vast majority of our students, and that not enough instructional stress is put on the students thinking
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abilities. Fleisch (2008) also posits that teachers’ instructional methods are believed to contribute to learners’ errors and poor performance in algebra. Studies have consistently highlighted teachers’ instruction as an important variable to influence the performance of learners in mathematics and could also be a contributing factor not only to the learners’ poor performance but also the reason for the learners’ errors in this subject (Anku, 2014; Osafo-Affum, 2001; Graven, 2004).
It is worth noting that the field of education has undergone a significant shift in thinking about the nature of human learning and the conditions that best promote the varied dimensions of human learning. In psychology, there has been a paradigm shift in designed instruction from behaviorism to cognitivism and now to constructivism
(Cooper, 1993). The idea was to change the focus of the classroom from teacher dominated to student-centred using a Constructivist Approach. Constructivist teaching practices in Science and Mathematics classrooms are intended to produce much more challenging instruction for students and thus, produce improved meaningful learning.
These changes have led to instruction in which students are expected to contribute actively to mathematics lessons by explaining their mathematical reasoning to each other and constructing their own understanding of mathematical concepts.
Studies on Constructivist Teaching Approach – both universally and locally have shown such a constructivist-based approach to be promising (Gravemeijer, 1993;
Ginsburg – Block & Fantuzzo, 1998) and its positive effects have been found for both students’ performance and motivation. Quite a number of research works in science related fields such as Chemistry attest to the effectiveness of the constructivist teaching approach. For example, research by Akku, Atasoy and
Geban (2003) studied under the title “Effectiveness of Instruction Based
Constructivist Approach on Understanding Chemical Equilibrium Concepts” on
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seventy-one (71) 10th grade students from two chemistry classes revealed that the students who used the constructivist principles-oriented instruction earned significantly higher scores than those taught by traditional instruction in terms of achievement related to chemical equilibrium concepts. Added to this, Kim (2005) found that using constructivist teaching methods for the 6th Graders resulted in better learner achievement than traditional teaching methods. The Kim’s (2005) study also found that learners preferred constructivist methods over traditional ones. Doğru and
Kalender (2007) compared science classrooms using traditional teacher-centred approaches to those using student-centred constructivist methods. In the initial test of learner performance, which was administered immediately after the intervention, the researchers found no significant difference between traditional and constructivist methods. However, in the follow-up assessment, which occurred 15 days later, learners who learned through constructivist methods showed better retention of knowledge than those who learned through the comparative traditional methods.
In the area of mathematics, Bhutto (2013) researched on the effect of teaching of algebra through social constructivist approach on 7th Graders’ learning outcomes in
Sindh in Pakistan and found that the experimental group that was taught through social constructivist approach excelled in achieving statistically significant learning outcomes than the control group that was taught through traditional one-way teaching.
In Ghana, research by Andam (2015), published in the Advances in Research Journal on using the constructivist approach of solving word problems involving algebraic linear equations at Mansoman Senior High School, in the Amansie West District of
Ghana concluded that, “until the intervention stage, students did not know that they
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could learn better from their colleagues. The intervention led to students developing more positive attitudes towards mathematics in general. In spite of all these from the findings of this study, regardless of the challenges associated, the constructivist approach of teaching promoted students’ active participation in the teaching and learning process and improved their performances in solving algebraic word problems” (p. 113).
The studies mentioned in the above discussion provided some motivation and justification for the current study to be conducted. Given this background, Dhlamini
(2012, p. 241) proposes that “schools should also see the need to train learners to become effective in collaborative learning settings” and further suggested that instruction that promotes collaborative skills of learners ought to be designed.
The current study therefore examined the effects of the constructivist teaching approach on the performance of form two senior high school mathematics learners in algebra. In the light of the national search for teaching approaches that can improve mathematics performance of learners in Ghana, the researcher believed that the outcome of the current study is timely, since the traditional teaching approach could not serve its intended purpose. The results of this study may be of importance for those interested in empowering teachers to meet the challenges of the new curriculum.
1.2 Statement of the Problem
Several studies show that students face challenges and make common errors and mistakes in comprehending the concepts of equation and variable (Gurbuz and Akkan
2008), in mathematical interpretations of letters or a variety of notations (Dede and
Peker, 2007) and in solving equations appropriate for problems (Yenilmez and Avcu,
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2009). The importance of algebra taught as an integral part of mathematics in most national curricula cannot be underestimated. For example, the National Council of
Teachers of Mathematics (NCTM, 2000) recommended introducing algebra and algebraic reasoning in elementary and middle grades throughout the study of mathematics. It is also felt that mathematics community is concerned about the knowledge required for effective teaching of algebra (Ball & Thames, 2010).
Warren (2008) reports that, learners in developing countries and some developed countries experience difficulties with proper understanding of algebraic concepts, especially algebraic variables. This is because of the mechanical way teachers teach it without explaining real meaning in social context. This obviously leads to poor learning and open opportunities for learners to make errors. Teachers’ instructional methods are believed to contribute to learners’ errors and lack of conceptual understanding in learning Algebra (Fleisch, 2008). Algebra is a powerful problem- solving tool (Nickson, 2000), hence understanding it is central to students’ ability to do mathematics. It follows from this that, in order to improve students’ performance in mathematics in general, the teacher should enhance a profound understanding and acquisition of algebraic concepts and thinking skills of their learners.
However, it has been observed that many students struggle to cope with learning
Algebra especially in algebraic word problems and equations (Kieran, 1992). There is a large bulk of research bearing on students’ algebra learning, particularly on students’ misconceiving of various concepts in school algebra (Gurbuz & Akkan,
2008; Dede & Peker, 2007; Yenilmez and Avcu, 2009). It has been typically indicated from most literature that students lacked the relevant understanding of operational symbols (Sfard’s 1991), simplifying expressions (Welder 2012) and equality and
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equation solving (Capraro et al 2008). For these reasons, there has been a great attention paid to addressing students’ difficulties in algebra in recent times.
Although some students have good arithmetic background and can solve problems using lengthy arithmetic procedures that they come up with themselves, they are hesitant to use algebraic methods (Mayer & Hegarty 1996). . Added to this, most of them do have some type of language deficiency in reading and writing. For these students, algebraic concepts such as word problems on linear equations in one variable and the concept of algebraic expressions are particularly difficult because the students may have trouble understanding and properly using the information given
(Kieran, 1992; Geary, 1994). Limited research and literature are available these days on teaching word problems and algebraic expressions for students with difficulties, however, using recognized methods such as the constructivist approach can help make the study of algebraic word problems and expressions more enjoyable for the students
(Maccini, McNaughton & Ruhl, 2000). The use of constructivist approach to teach the underlying principles of algebra, for example, translating word problems into linear equations in one variable and evaluating algebraic expressions will link concrete representations of mathematical ideas to abstract symbolic representation to enhance conceptual understanding (Capps & Pickreign, 1998).
It is therefore considered that examining the effect of the constructivist teaching approach on high school students’ achievement in algebra would fill an important gap in the literature. This research therefore examined the effects of the constructivist teaching approach on high school students’ performance in Algebra. The algebraic concepts used in the study pertained to variables, expressions, equations, and word problems.
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1.3 Purpose of the Study
The study sought to carry out a quasi-experimental study to determine the effect of the
Constructivist Teaching Approach (CTA) on senior high school students’ performance in algebra in the Gomoa East District of Ghana. The study also aimed at finding the students’ perceptions on the use of CTA in teaching algebra.
1.4 Objectives of the Study
The objectives of this study were to:
1. Identify the teaching approaches used by Senior High School mathematics teachers in their lessons. 2. Investigate the effects of the teaching approaches often employed by the teachers on students’ performance in algebra. 3. Investigate the effect of the constructivist teaching approach on students’ achievement in algebra. 4. Find out students’ perceptions of the constructivist teaching approach on their performances in Algebra?
1.5 Research Questions
Considering the objectives stated above, the following research questions guided the study:
1. What teaching approaches do Senior High School mathematics teachers often employ in their lessons? 2. What is the effect of the teaching approaches often used by teachers on students’ performance in Algebra? 3. What is the effect of the constructivist teaching approach on students’ achievement in Algebra? 4. What are students’ perceptions of the constructivist teaching approach on their performances in Algebra?
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1.5.1 Research Hypothesis
The following hypothesis was tested for research question three (3).
Null Hypothesis (푯푶): There is no significant difference in achievement between students taught by the constructivist teaching approach and those taught by the traditional approach?
Alternate Hypothesis (푯ퟏ): There is significant difference in achievement between students taught by the constructivist teaching approach and those taught by the traditional approach?
1.6 Significance of the Study
There are a considerable number of studies on students’ achievement in arithmetic
(Cai, 1995; Cobo & Batanero, 2004). Comparatively, few studies address the issue of students’ achievement in algebra. They too pay attention to some isolated topics such as variables, word problems, equations, or expressions. Little or no attempt has been made to study the interrelated nature of the students’ algebraic achievement in more than one conceptual area.
More detailed exploration of students’ algebraic achievement is a crucial prerequisite for any further attempt to improve the quality of mathematics education. It is essential to determine the perceptions held by students regarding their teaching and learning practices and what they actually do in their respective classrooms. The researcher addressed these issues by inquiring into students’ performance in the basic building blocks in algebra: variables, expressions and equations. Word problems introduced a context where the above three components could link to a solution model.
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It is hoped that the results of this study will bring to bear the various teaching approaches employed by teachers in teaching mathematics and some of the factors which either make teachers reject constructivist teaching or accept and employ it in their instruction. Knowledge of these factors would be used by teachers to improve on their teaching practices and approaches that would enhance teaching and learning by understanding.
School managers and teachers may also use this research to monitor and predict the performance of students in relation to the teaching approaches adopted and employed during instruction. Finally, educational policy makers, as well as curriculum planners and developers, could use the findings of this research during training workshops on teaching mathematics effectively in Ghana.
1.7 Limitations of the Study
The study was conducted in two senior high schools selected in the Gomoa East
District. As such, results of the study might only apply to the schools under study.
Based on the findings from these few selected schools under the study, the researcher could not draw conclusive and generalizable conclusions. Further research is required on this problem, probably, using more learners in order to obtain a more robust evidence for the findings. Added to this, the caliber of students admitted by the schools had students with weak background in Mathematics and English which in turn had influence on the study.
1.8 Organization of the Study
The study was structured in five (5) chapters: chapter one centered on the introduction and problem definition as well as its purpose and research questions. Relevant
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literature review was also presented in chapter two and based on the subheadings related to the study. The research design and methodology were then described in chapter three. Chapter four centred on the analysis and discussion of the results obtained from the various data sources used for the study. The summary of findings, discussions, recommendations, and suggestions for further studies into the problem, based on the findings of this study were also discussed in chapter five.
1.9 Operational Definition of Terms
The following operational definitions were used for the current study:
Constructivism: A philosophy or an educational approach that is based on the premise that those who are engaged in a learning process construct their own understanding of the world through their experiences.
Constructivist Teaching Approach: The approach that promotes learners active participation in the learning process of selected algebraic topics with the view to encourage them to construct their own mathematical knowledge to enhance their achievement in algebra.
Small Group Instruction: This is a conductive learning environment where a small group of students were given clear and explicit instructional objectives to increase understanding and course content. Using this, students are able to discuss and express questions and concerns on lesson taught.
Form Two: A second year class in the formal school system in senior high school education in Ghana.
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Disadvantaged Learners: A group of learners from populations with low social status and low educational achievement.
Algebraic Errors: The phrase ‘algebraic errors’ refers to the mistakes that learners tend to make when they solve certain tasks in algebra. For this study, an error was regarded as a mistake in the process of solving an algebraic problem algorithmically, procedurally or by any other method.
A Mathematical Variable: An entity that can take various values in any particular context. The domain of the variable may be limited to a particular set of numbers or algebraic quantities.
A Mathematical Equation: An equation could be regarded as a statement that asserts the equality of two expressions usually written as linear array of symbols that are separated into left and right sides and joined by an equal sign.
Group Work: An arrangement in which two or more participants (learners) worked together in a form of discussing algebraic tasks to achieve shared solutions.
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CHAPTER TWO REVIEW OF RELATED LITERATURE
2.0 Overview
In this chapter, the literature reviewed largely centred on the objectives set for the study; which were:
1. To identify the teaching approaches used by Senior High School mathematics
teachers in their lessons.
2. To investigate the effects of the teaching approaches often employed by the
teachers on students’ performance in algebra.
3. To investigate the effect of the constructivist teaching approach on students’
achievement in algebra.
4. To find out students’ perceptions of the constructivist teaching approach on
their performances in Algebra?
Furthermore, a background study of the theory underpinning this study was also examined. The background study of the theory reviewed past work by researchers who have studied the implementation of constructivist teaching approach to improve students’ achievement in mathematics. Research works on the various teaching approaches used by teachers was also examined. The last section of the chapter provided the needed review on the four conceptual areas of algebra - mathematical variables, algebraic expressions, algebraic equations and word-problems used in the research.
The research works reviewed in this study came from three major sources: 1) academic books from the university library 2) peer-reviewed journal articles from
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digital databases including ERIC, Google Scholar and 3) other online electronic resources.
2.1 Theoretical Framework: Constructivist Theory
The theoretical framework used for the study was based on the Constructivist Theory of teaching and learning of mathematics. Constructivist theory is considered to originate in the work of two early twentieth-century contemporary epistemological theorists, Jean Piaget in 1976 and Lev Vygotsky in 1986, whose cognitive theories of learning were developed as reactions to behaviorism which was the dominant science at the time (Cholewinski, 2009; Wyer Jr., 2014).
Hein (1991) refers to the constructivist theory as the idea that learners construct knowledge by constructing meaning individually and/or socially when they are in the learning process. This view has two fundamental aspects. First, we have the learner as the main person constructing knowledge in the learning process. Second, there is no knowledge independent of the meaning attributed to the experience by the learner or a community of learners. Simply, this knowledge depends on peoples’ experiences and their thinking. Therefore, constructivism as a theory has shown how useful it is, in allowing researchers to make sense of others’ experiences.
The constructivist theory was preferred in this study because it specified instructional methods that assisted students to actively explore complex topics/ environments.
The constructivist theory of learning underpins a number of important approaches, these include: situated learning, concept mapping, collaborative instructional approach, anchored instruction, problem-based learning, cognitive apprenticeship, discovery learning, and scaffolding (Cholewinski, 2009; Jackson, 2006; Jia, 2010;
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Lai-chong & Ka-ming, 1996; Rowe, 2006; Wu, Hwang, Su, & Huang, 2012). This study focused on the collaborative instructional approach. This instructional approach was relevant to the study since the researcher was particularly interested in developing a social classroom environment based on Vygotsky’s constructivist learning approach (Collaborative Learning) to enhance second year high school
Mathematics learners’ achievement in algebra, by exposing their difficulties and subsequently providing a treatment for the observed difficulties. This aim was explored in terms of four conceptual areas of algebra, namely, the mathematical variables, mathematical expressions, mathematical equations and solving word- problem.
2.2 Teaching Approaches
Teaching is not only combination of the teachers’ own behaviours/arrangement in the classroom, but also a connection of many intertwining relationships among teachers, students, and the mathematics content in classroom instruction (Boaler, 2002b; Frank,
Kazemi & Battey, 2007; Lampert, 2001, Kilpatrick, Swafford & Findell, 2001). It is also a process of engaging together in generating mathematical meaning (Boaler,
2002; Franke et al., 2007; Kilpatrick et al., 2001). Effective teaching expects to continually elevate students’ mathematical competencies and the level of student’s involvement in learning, also determines the quality of teaching (Kilpatrick et al.,
2001). Research shows that different concepts of teaching, learning and classroom cultures influence the ways in which teachers teach and how students learn (Dossey,
1992). According to Marx and Collopy (1995), teachers’ teaching styles directly influence students’ learning. Teachers who are sensitive to their students’ learning are more likely to change their practices and such changes are more likely to improve students’ learning (Irwin & Britt, 1994).
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Recent education reforms require teachers to depart from the traditional practice of knowledge transmission to constructivist teaching where students are encouraged to construct knowledge through inquiry (Beck et al., 2000; Levitt, 2002). Constructivist classrooms allow students to actively participate in the learning activities to construct their knowledge thus, keep them engaged during a longer period of time (Schraw &
Lehman, 2001). Since knowledge construction requires connection with prior knowledge, constructivist teaching draws on students’ prior knowledge and experiences (Driscoll, 2005). Rich and authentic contexts need to be provided for students for them to link school learning with the world outside school (Jonassen,
1999). Teacher education programs should include frequent opportunities of constructivist teaching experiences for pre-service teachers in order for them to gain content and pedagogical skills (Haney & McArthur, 2002).
2.2.1 Traditional Teaching
In looking at the traditional approach to teaching, a discussion on direct instruction, didactic instruction, and features of behaviourism are included since these in essence describe traditional teaching.
The behaviourist approach still remains mainstream in the educational field; from the concepts or adoption of behaviourism (Wenger, 1998). Behaviourist theories emphasise behaviour modification through stimulus – response connections and selective reinforcement (Fang & Chung, 2005). The intended behaviour is reinforced by repetitive practicing and praising of correct answers (Wei & Eisenhart, 2011). The behaviourist theory explains learning as passively receiving stimulus or information rather than mentally processing such information (Fang & Chung, 2005). These theories completely ignore issues of meaning; particularly social meaning. They
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address issues such as learning through rewards and practices and assess student learning based on observable behaviour (Fang & Chung, 2005).
The key assumption of behaviourists that students learn what was taught or transmitted. As long as the knowledge was clearly communicated and received, then this knowledge could be generalised in other circumstances (Boaler, 2002). Multiple opportunities for drill and practice should be offered to reinforce certain behaviours
(Boaler, 2002).
Traditional teaching is based on behaviourism where the focus is on drill and practice
(Fang & Chung, 2005), speed and accuracy of answers, with an outcome of automatic recall (Trotman, 1999). The teaching is limited to the classroom context and the teacher has limited freedom from schools to arrange their teaching activities (Chi,
1999). The teacher is assumed to know all mathematics for students’ learning (Begg,
1992).
Researchers agree that traditional teaching promotes teacher centred learning, where teachers control all the teaching discourse (McCarthey & Peterson, 1995). Thus, teacher-centred and quiet classrooms normally appear during the delivery of instructions in the classrooms. Teacher is seen as a provider of information and learners are passive recipients of the information (Even & Tirosh, 2008), and often need to give up their individual decision making in obedience to the demands of the classroom teacher (Boaler & Greeno, 2000).
The role of the teacher in the traditional approach is to adopt a clear and coherent presentation of instruction (Trotman, 1999), such as:
o Lecturing through the “write and talk” method (Threlfall, 1996);
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o giving very clear and detailed instructions for the procedures before the
beginning of an activity, (Fang & Chung, 2005);
o Correcting immediately students’ incorrect statements (Threlfall, 1996);
o Ensuring that students know what to do in each stage (Sosniak, Ethighton &
Varelas, 1994).
Teachers follow the syllabus to transmit knowledge (Livingstone & Izard, 1994), monitor students’ progress (Frederiksen, 1984), give regular tests (Werry, 1989) and ensure that students retain this knowledge until examinations are over (Livingstone &
Izard, 1994). Therefore, skill-based task would be given with an expectation of a uniformity of learning (Windschitl, 1999a), with an emphasis on rote memorization of mathematics rules (Wei & Eisenhart, 2011). Teachers use formulae and encourage students to use particular rules or formulae in most mathematics problem solving
(Bennett, 1976; Silver, Smith & Nelson, 1995). The teaching emphasis is on content
(Threlfall, 1996) with a speedy transmission of facts and knowledge (Even & Tirosh,
2008), such as:
o Using basal texts in mathematics and many worksheets (McCarthey &
Peterson, 1995);
o Separating mathematical subject matter into small objectives within a
sequence of task (Begg, 1996);
o Asking convergent or factual questions for which they have prepared answers
already and assessing students’ work within the narrow domain of each unit
(Carr & Ritchie, 1992);
o Focussing on the product of a student’s work rather than including the
processes (Trotman, 1999).
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2.2.1.1 Advantages of the Traditional Teaching
Advantages of the traditional approach to teaching include:
1. Teachers can cover more mathematical content within a limited time.
2. Firm teacher discipline leads to good self-discipline by students (Bennett, 1976).
3. Students may feel more secured in a structured teaching environment (Bennett,
1976).
4. Students may perform better under traditional teaching rather than from a
constructivist approach (Mousley, Clements & Ellerton’s, 1992; Weltman &
Whiteside, 2010).
Some benefits of the traditional approach of teaching are that students more easily learn and apply the rules/procedural knowledge in similar situations and receive senses of achievement. It is easier for teachers to adopt these approaches than relational methods because less knowledge and technique is required in instrumental teaching (more rule/procedures) and students are finds it easier to write right answers on paper (Skemp, 2006).
Teaching approaches with great emphasis on procedures and memory are still commonly adopted in many classrooms (Pesek & Kirshner, 2000; Wei & Eisenhart,
2011). Leung and Park (2002) argued that “procedural teaching does not necessarily imply rote learning or learning without understanding” (p. 127). They researched nine mathematics teachers in each place – Hong Kong and Korea. They found that most of the teachers adopted procedural teaching strategies but conveyed conceptual and procedural understanding to students and they also found support from Ma’s work
(Ma, 2010). Added to this, teaching through procedures of core concepts and
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repetitive practices might benefit the high mathematics achievements of Asian students (Leung & Park, 2002). Leung and Park (2002) perceived that conceptual and procedural understandings are connected (Heibert & Lefervre, 1986), especially, when students conduct repetitive practices (Dahline & Watkins, 2000; Leung & Park,
2002) that provide various challenges (Leung & Park, 2002).
Direct instruction is the instructional approach which is most prevalent in traditional classrooms. This approach entails reviewing, teaching and practicing that which was taught. The write and talk method is mainly used in direct instruction classrooms. The teaching strategies of direct instruction place an emphasis on the teachers’ explanation of the content; also called explicit teaching (Zhang, 2002). The learning theories associated with direct instruction strategies do not come from a single theory but may be viewed as a combination of behaviourism, and the information processing and transmission theories from the cognitive theory (Zhang, 2002). A direct instruction has five steps:
1. learning new ideas from old experiences,
2. clearly explaining the content of the teaching material,
3. helping students to do practice in time, or guided practice,
4. adjusting mistakes from feedback, and
5. allowing students to complete their assignment individually (Zhang, 2002).
Didactic or instrumental approaches also commonly appear in the traditional classrooms. Students’ participation in a didactic classroom normally is governed by textbooks, procedures and rules related to memorization and procedure duplication.
They rarely negotiate or develop ideas, procedures or creativeness (Boaler & Greeno,
2000). Students’ learning in the didactic and instrumental approaches, then applying them (Boaler & Greeno, 2000; Silver et al., 1995; Skemp, 1976, 2006). Didactic
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approaches place an emphasis on memorization and procedural practice, but rarely develop mathematical ideas (Boaler & Greeno, 2000). To some extent, instrumental approaches are similar to didactic approaches, but they place more emphasis on procedures and ignore the understanding behind the rules/procedures (Skemp, 1976,
2006).
Wenger (1998) also argued that if teachers regard knowledge as learning pieces of fact, then naturally they would present knowledge in a high structured manner. From that perspective, direct lecturing will be the teaching strategy (Wenger, 1998). Then, the most efficient way is probably impart knowledge through demonstration and practice. This can be seen in the traditional mathematics classrooms in Boaler’s
(1997, 2002) research.
To sum up, all students learned passively from teachers’ explanations in the traditional approaches, direct instruction, didactic approaches and instrumental approaches (Boaler & Greeno, 2000; Skemp, 1976, 2006; Zhang, 2002). This traditional approach often combines teacher centred views of learning (McCarthey &
Peterson, 1995), teaching strategies of a behaviourist approach and monitoring of class events (including decisions of classrooms learning task, tests given or students’ learning progress). However, there is more meaning processing in the direct instruction (Zhang, 2002), but not much understanding processing in the didactic and instrumental approaches (Boaler & Greeno, 2000; Skemp, 1976, 2006).
2.2.1.2 Disadvantages of the Traditional Teaching
The traditional teaching approach may inhibit students’ freedom to think. According to Fletcher (2008), the traditional method of instruction discourages learners from engaging in higher algebraic thinking in mathematics lessons. It fails to focus on
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mental processes (Romberg, 1993; Trotman, 1999). Other disadvantages include over- emphasis on rote learning, insufficient emphasis on creative expression (Bennett,
1976), concerns with academic standards and competition (Bennett, 1976) and use of external rewards such as grades. For example, external rewards were used when teachers reinforced right answers, corrected wrong ones and evaluated by right answers (Kamii, 1985). Hagg (1991) also argued that, the behaviourist teaching practice may result in students regarding learning with little enthusiasm or intellectual tension and it may fail to cater for average students. It has been suggested that the emphasis on “rule following” rather than “rule learning” is anti-mathematical
(Neyland, 1994).
These learning behaviours may lead to many students developing negative feelings toward passively receiving abstract knowledge (Boaler & Greeno, 2000).
Additionally, they can result in students developing over-dependency on the authority of the teachers (Boaler & Greeno, 2000). Some scholars have viewed African learners as being passive learners with a heavily reliance on teachers’ instructions (Agyei &
Voogt, 2011). The limitation of the behaviourist approach becomes more apparent, particularly in the teaching of higher-order skills (Neyland, 1994).
Research conducted by Baker, Czarnocha, Prabhu (2004) showed that when using the traditional curriculum, with its focus on computational modelling of procedural knowledge, the knowledge students acquired was not long term. In summary, in traditional teaching, students work on graded exercises, memorize content and formulas, and are continuously tested throughout a unit of work and at the end of the unit. No emphasis is placed on processes, so hence the need for an alternative approach to teaching.
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2.2.2 Constructivist Teaching or Pedagogy
Another approach to teaching is from a cognitive perspective, i.e., constructivism.
Spector (2010), defined the constructivist approach of teaching as a teaching process based on the understanding that knowledge is constructed by the knower based on his/her internal mental process. According to Fang and Chung (2005), an individual’s reasoning and cognitive growth is emphasized from perspectives of cognitive psychology. Brown (2005) suggested that constructivist approach is accepted as the most relevant method of teaching and that educational policies, educational models, and educational practices should focus on constructivist teaching.
Constructivism offers unique instructional strategies characterized by emphasis on active learning, social interaction and peer mentoring, and reflection. Such multi- dimensional engagement with the learning content deepens understanding, cements conceptual grasp, and fosters a plethora of higher-order cognitive and socio emotional skills (Jonassen, Howland, Moore & Marra, 2004). Constructivism is based on the premise that we all construct our own perspective of the world, based on individual experiences and internal knowledge. Learning is based on how the individual interprets and creates the meaning of his or her experiences. Knowledge is constructed by the learner and since everyone has a different set of experiences and perceptions, learning is unique and different for each person.
Constructivist theorists believe that learning is a process where individuals construct new ideas or concepts based on prior knowledge and/or experience. Each of us generates our own mental models, which we use to make sense of our experiences. We resolve conflicts between ideas and reflect on theoretical
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explanations. Learning, therefore, is simply the process of adjusting our mental models to accommodate our new experiences. Jonassen, Howland, Moore and Marra
(2004) argued that meaningful learning occurs when learners are active, constructive, intentional, cooperative and working on authentic tasks. They further asserted that human learning is a naturally active mental and social process. This theory is used to focus on preparing people to problem solve. Therefore, to be successful, the learner needs a significant base of knowledge upon which to interpret and create ideas. The constructivist theory of learning underpins a number of important approaches, these include: situated learning, concept mapping, collaborative instructional approach, anchored instruction, problem-based learning, cognitive apprenticeship, discovery learning, and scaffolding. (Cholewinski, 2009; Jackson, 2006; Jia, 2010; Lai-chong &
Ka-ming, 1996; Rowe, 2006; Wu, Hwang, Su, & Huang, 2012). This study focused on the collaborative instructional approach.
Based on the studies of Brooks & Brooks (1999) and Fosnot (1989), there are propositions that students construct their own understanding by using prior knowledge to interpret information. In this view, the effective use of authentic resources can aid knowledge construction, and that peer discussion and negotiation is critical to the constructive process. It is the researcher's belief that these are feasible guidelines to be implemented among students to assess whether constructivist activities will improve performance and enhance students’ problem solving skills.
2.2.2.1 The Epistemological Base of Constructivist Teaching
Constructivism is an epistemological view of knowledge acquisition that emphasizes knowledge construction rather than knowledge transmission and the recording of
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information conveyed by others. It is anchored on cognitive psychology but from a practical perspective has roots in the “progressive model of John Dewey (1933). It is aligned with active learning and promotes comparison of new ideas with prior knowledge (Goldin, 1990; Piaget, 1973; Steffe, 1991; Von Glasersfeld, 1997;
Vygotsky, 1978). Constructivism involves learners’ interpretation of knowledge and understanding from the experiences encountered as active learners (Slavin, 2000). In addition, Von Glasersfeld (1996: 19) states that “for whatever things we know, we know only insofar as we have constructed them as relatively viable permanent entities in our conceptual world”. Communication and justification of ideas are important in helping learners develop problem-solving skills (Piaget, 1973). For constructivism, the teacher is a guide, facilitator, and co-explorer who encourages learners to question, challenge and formulate their own ideas, opinions and conclusions (Ciot,
2009; Cannelle & Reif, 1994; Ismat, 1998).
There is much importance in facilitating correct mathematical language, justifying and sharing ideas with others (Ball & Bass, 2000). Learners can construct meaning in
Mathematics from others or from use of individual objects (Von Glasersfeld, 1997).
The act of solving one’s own problems (Wood, Cobb & Yackel, 2000) as well as the process of question-asking concerning various strategies applicable to the
Mathematics topics can increase learners’ mathematical abilities and thereby reduce the number of errors they commit (Carpenter, Fennema, Fuson, Hiebert, Human,
Murray & Wearne, 1994).
2.2.2.2 Implications of Constructivism for Learning and Teaching
The constructivist view of learning has had a most noticeable influence on curriculum thinking in science and mathematics education since 1980 (Wubbels & Brekemans,
1997). This view has important consequences for the development of new teaching
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and learning approaches that focus on students’ understanding of science and mathematics concepts rather than recall of facts and formulae.
The constructivist approach to learning is based on the idea that the learner constructs his or her own knowledge through negotiation of meaning (Hand, Treagust, & Vance,
1997). Tobin and Tippins (1993) suggested that constructivism has been used as a referent for building a classroom that maximizes student learning. In such a classroom, the teacher takes account of what students know, maximizes social interactions between learners so that they can negotiate meaning, and provides a variety of sensory experiences from which learning is built.
Duit and Confrey (1996) noted the following five assumptions shared by mathematics and science educators for reorganizing the curriculum and teaching to improve learning in school science and mathematics from a constructivist perspective: first, more emphasis is usually given to the applicability of science and mathematics knowledge in situations in which students are interested; second, introduction into the curriculum of issues of meta-knowledge about science and mathematics is needed; third, extinguishing students' everyday conceptions is impossible and inadvisable; fourth, constructivist approaches are student-centred; and, fifth, the norms and patterns of classroom interaction are a fundamental influence on the effectiveness of reform efforts. They also suggested that innovation processes could be implemented in terms of developing new media, including science textbooks, revising traditional content structures, and using a range of constructivist teaching strategies.
A very clear example of implications of constructivism for teaching and learning was provided in a study reported by Yager (1995). This study focused on the science,
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technology and society (STS) instructional approach and works through in-service and other training programs to introduce teachers to the constructivist approach in implementing reform. The research aim was to explore both teacher changes and student learning outcomes. In his study, Yager compared 133 teachers involved with the constructivist program with 48 teachers involved in another in-service training program, but not one using constructivist principles. Results indicated that teachers using constructivist principles had increases in teacher confidence, higher levels of using constructivist techniques, and more student-centered classrooms. In terms of student achievement, there were significant differences in 105 more traditional classrooms. The students in constructivist classrooms had a significant advantage over students in traditional classrooms in these domains: concept, process, application, creativity, attitude, and the world view domain. In other words, students had higher scores in all six of the domains that were tested.
Yager’s work seems to demonstrate significant benefits to both teachers and students for the use of constructivist teaching. Teachers exhibited more confidence, while there were positive effects on students’ learning outcomes. Yager’s study is very credible because it worked with a larger population than other studies on constructivist teaching and had the added benefit of a qualitative method. This is very significant and helpful in working with more traditional educators and school districts, demonstrating to them in a measurable fashion how constructivist teaching can improve both teaching and learning.
Another quality of a constructivist teaching is its interactive nature. Authentic student- student and student-teacher dialogue are very important in a constructivist classroom.
Belenky, Clinchy, Goldberger, and Tarule (1986) inform us that constructivists
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distinguish didactic talk when participants report experiences but no new understanding occurs, from a real talk where careful listening creates an environment within which emerging ideas can grow. Perhaps this defines the difference between teacher talk in a direct instruction classroom and purposeful talk by students in a student-centred constructivist classroom where meaningful discussion occurs and meanings emerge. Belenky et al (1986) explained that in "real talk", domination is absent, while reciprocity, cooperation, and collaborative involvement are prominent.
Consequently, constructivist activities in the classroom that focus on speaking and listening promote not only constructivist thought but also important connections between teacher and students.
2.2.2.3 Studies on Constructivist Teaching Approach
Research examining the use of constructivist teaching approach has been done in multiple contexts and under a variety of lenses. Past studies advocating for constructivist pedagogy (for example, Cobb, 1996; Dangel, 2011; Fox, 2001; Phillips,
1995) in science-related disciplines have been duly documented. Previous researches on constructivist teaching approach and students’ perspectives of the approach were outlined in this study. The scope of the review of the papers covered the general overview of the papers, methodology, results and discussions of the research findings of the various research works.
Research by Andam (2015), published in the Advances in Research Journal on using the constructivist approach of solving word problems involving algebraic linear equations at Mansoman Senior High School, in the Amansie West District of Ghana concluded that “until the intervention stage, students did not know that they could learn better from their colleagues. The intervention led to students developing a more positive attitude towards mathematics in general. In spite of all these from the
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findings of this study, regardless of the challenges associated, the constructivist approach of teaching promoted students’ active participation in the teaching and learning process and improved their performance in solving algebraic word problems”.
Tellez (2007) reviewed major reform efforts in curriculum and pedagogy to establish that “the importance of constructivism in educational theory and research cannot be underestimated”. Several studies support constructivist approaches in science-related disciplines (for examples, see, Cobb, 1996; Dangel, 2011; Fox, 2001; Phillips, 1995).
Also, Chin, Duncan and Hmelo-Silver (2007) cited several studies supporting the success of the constructivist problem-based and inquiry learning methods. For example, Chin et al. (2007) described a project called GenScope, which was an inquiry-based science software application. Students, who were in the experimental group using the GenScope software, showed significant gains over the control groups. The largest gain was shown by the students who were enrolled in the basic courses.
Chin et al. (2007) cited a study by Geier on the effectiveness of inquiry-based science for middle school students as demonstrated by their performance on high-stakes standardized tests. The improvement was 14% for the first cohort of students and
13% for the second cohort. Chin et al. (2007) also found that inquiry-based teaching methods greatly reduced the achievement gap for African-American students.
Guthrie, Taboada and Humenick (2004) compared three instructional methods for third-grade reading: a traditional approach, a strategies instruction only approach, and an approach with strategies instruction and constructivist motivation techniques
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including student choices, collaboration, and hands-on activities. The constructivist approach, called Concept-Oriented Reading Instruction (CORI), resulted in better student reading comprehension, cognitive strategies and motivation.
Kim (2005) found that using constructivist teaching methods for the 6th Graders resulted in better learner achievement than traditional teaching methods. The Kim’s
(2005) study also found that learners preferred constructivist methods over traditional ones. However, Kim (2005) did not find any difference in student self-concept or learning strategies between those taught by constructivist and those taught in traditional methods.
Doğru and Kalender (2007) compared science classrooms using traditional teacher- centred approaches to those using student-centred constructivist methods. In the initial test of learner performance, which was administered immediately after the intervention, Doğru and Kalender (2007) found no significant difference between traditional and constructivist methods. However, in the follow-up assessment, which occurred 15 days later, learners who learned through constructivist methods showed better retention of knowledge than those who learned through the comparative traditional methods.
Bhutto (2013) researched on the effect of teaching of algebra through social constructivist approach on 7th Graders’ learning outcomes in Sindh in Pakistan and found that the experimental group that was taught through social constructivist approach excelled in achieving statistically significant learning outcomes than the control group that was taught through traditional one-way teaching.
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2.2.3 Constructivist and Traditional Ideas about Teaching and Learning
In the constructivist classroom, the focus tends to shift from the teacher to the learners (Brooks & Brooks, 1999). One of the teacher's biggest responsibilities becomes that of ‘asking good questions’. Again, in the constructivist classroom both teacher and learners think of knowledge not as inert factoids to be memorized but as a dynamic and ever-changing view of the world we live in and the ability to successfully stretch and explore that view (Brooks & Brooks, 1999).
When comparing the traditional teaching methods to the constructivist-based teaching method, Applefield et al. (2001) stated that in the traditional approach a bottom-up strategy, which involves isolating the basic skills, teaching occurs by separating and building these incrementally before tackling higher order tasks. This is an essentially objectivist and behavioral approach to instruction (teaching method) although cognitive information processing views often lead to similar instructional practices. However, constructivist-based teaching method turns this highly sequential approach on its head. Instead of carefully structuring the elements of topics to be learned, learning proceeds from the natural need to develop understanding and skills required for completion of significant tasks. The distinctions between the traditional teaching methods and the constructivist-based teaching methods are reflected in
Table 2.1.
Table 2.1: A Comparison between Traditional and Constructivist-Based Classrooms
The Traditional Classroom The Constructivist Classroom • Strict Adherence to fixed Curriculum • Focus on the pursuit of learner questions and interests
• Textbook and Workbooks oriented • The use of primary sources and manipulative materials
• Teacher is a provider and learners are • Learning is interactive and builds on passive recipients what learners already know
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• Teacher assumes a directive and • Teacher interacts and negotiate with authoritative role learners
• Assessment id through testing and • Assessment is through learner works, emphasis on correct answers observations, points of view and tests
• Knowledge is inert • Knowledge is dynamic and changes with experiences.
• Learners work individually and • Learners work in groups to facilitate independently self-construction of knowledge.
Source: Brooks and Brooks, 1993, p.17
Table 2.1 shows that there are significant differences in basic assumptions about knowledge, learners, and learning between traditional and constructivist approaches.
It is important to stress that constructivists do acknowledge that learners in the traditional classroom are also constructing knowledge but it is just a matter of the emphasis being on the learner and not on the teacher. In terms of the current study, learners’ errors which affected their performance and observed during a constructivist teaching approach (CTA), were meaningfully exposed because learners were given opportunities to be the constructors of their knowledge. This is in line with the last point in Table 2.1 on the constructivist section of the table. As learners verbalize their knowledge during active participation in the group discussion their errors are manifested.
However, in the traditional teaching approach and learning environment, learners’ errors could be observed after instruction through post-lesson activities because the teacher is the main player during instruction.
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2.3 Teaching Approaches used by Teachers.
According to Fletcher (2003) and Osafo-Affum (2001), irrespective of the level at which mathematics is taught; the role of the Ghanaian mathematics teacher has almost always been that of a lecturer and explainer, communicating the structure of mathematics systematically. Fredua-Kwarteng and Ahia (2015) stated that teaching and learning mathematics in Ghanaian classrooms is still dominated by the
“transmission” and “command” models. According to them the learning culture of mathematics in Ghanaian schools are such that: students learn mathematics by listening to their teacher and copying from the chalkboard rather than asking questions for explanations. Consequently, mathematics is learnt by bringing up facts, theorems or formulas instead of probing for meaning and understanding of mathematical concepts. Students hardly ask the logic or philosophy questions underlying those mathematical principles, facts, or formulas.
Students go to mathematics classes with the object to calculate something. Therefore, if the classes do not involve calculations they do not think that they are learning mathematics. So students learn mathematics with the goal to attain computational fluency, not understanding (Fredua-Kwarteng & Ahia, 2015).
In a research conducted on students’ perceptions of their teachers’ teaching of mathematics in Ghana which sampled the views of 358 students from 12 Junior High
Schools (12-14 years) randomly selected to complete a semi-structured questionnaire, the study revealed that students’ perceptions of their teachers’ teaching vary as the results established that both teacher-centred and student-centred teaching approaches were used by mathematics teachers. Students ascribed higher percentages to teacher- led activities than student-led activities.
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The study also established that teachers’ actions and inactions impact positively or negatively on students learning experiences as majority of the respondents reported that their learning experiences are to a larger extent controlled by that teacher.
Majority of the respondents indicated that their teachers normally tell them which questions to solve and which methods to use. Although the study was limited only to
12 schools, the findings provided a conceptual framework for further research into how students’ views could be used by both teachers and educational authorities in improving the teaching and learning of mathematics as students’ are in a better position to provide useful information regarding their teachers teaching and how it impacts on their learning. Among others, it was recommended that students’ ratings or evaluation of their teachers’ teaching should be considered in evaluating teachers’ teaching and effectiveness.
As a result, teaching and learning mathematics in the traditional methods do not motivate students; neither does it targets the development of understanding or support student-centred learning. Students are not involved in tackling problems with a number of possible alternative solutions.
In spite of its limitations, it is cheap and does not require much intensive advance lesson preparation on the parts of both teacher and students and can be conducted anywhere.
However, constructivist-based teaching method turns this highly sequential approach on its head. Instead of carefully structuring the elements of topics to be learned, learning proceeds from the natural need to develop understanding and skills required for completion of significant tasks.
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The above observations provided the basis for testing the impact of the constructivist teaching approach on senior high school students in the district. The method was expected to provide an intervention to expose learners’ errors and improve learners’ achievement in algebra through exploratory talk, class discussions, argumentation and support provided by their peers.
2.4 Review of Research works on Students’ Perceptions about Constructivist
Teaching (Using the Collaborative Learning Approach)
Proponents of collaborative learning claim that active exchange of ideas within small groups increases interest among participants and promotes critical thinking (Gokhale,
1995). Furthermore, Johnson and Johnson (1986) state that there is evidence that co- operative teams achieve higher levels of thought and retain information longer than students who work individually. According to Totten, Sills, Digby, and Russ (1991) shared learning gives students an opportunity to engage in discussion, take responsibility for their own learning and so become critical thinkers. Thus, research suggests that CL brings positive results such as deeper understanding of context, critical thinking, increased overall achievement in grades, improved self-esteem, and higher motivation to remain on task, more opportunities for personal feedback, celebration of diversity, group conflicts resolution and improved teamwork and social skills (Concept to Classroom, 2004).
Research based on interview and questionnaire administered on first year ESL students at the University of Botswana, surveyed students’ perceptions of
Collaborative Learning (CL). The research was aimed at providing depth and detail on students’ perceptions of what they have gained from the process and possibly indicate what areas might need to be improved or changed. Analyses of data revealed
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that most students claim to have derived academic benefits such as better comprehension and improved performance, and acquired generic skills – enhanced communication and problem-solving skills. About half of the respondents believe they gained social skills: they found CL enjoyable and made new friends. Most students agreed that CL practices should be encouraged and continued. It was concluded that students’ perception of CL at the University of Botswana is similar to findings in the stated literature. It was recommended that, in addition to focusing on academic benefits of CL, teachers should also be concerned with the social aspects of
CL.
2.5 Conceptual Understanding of Algebra
2.5.1 The Concept of a Variable.
In introducing algebra, first students encounter the concept of a variable, then algebraic terms and expressions, after which equations would follow. This sequence is based on the fact that equations involve expressions, while expressions in turn involve variables. Understanding of the concept of a variable is fundamental to the study of algebra. According to Van de Walle (2004, pg. 474), “a variable is a symbol that can stand for any one of a set of numbers or other objects.” In some cases, the referent set may have only one value, while it may have an infinite number of values in others, and the variable represents each one of them. Students need to develop a clear concept of a variable, that is, an understanding of how the values of an unknown change. A variable provides an algebraic tool for expressing generalizations. Unlike constants which can be defined in numerous ways, variables cannot be defined in terms of cardinal or ordinal values but can only be defined by number system reference (e.g. a
푠 = b + c) or by expression reference (푒. 푔, 푣 = ). 푡
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Research studies indicate that students experience a lot of difficulties in dealing with letters, in algebra. From the study conducted by Kuchemann (1981), six stages through which students’ progress in acquiring a mental model of a variable are identified. These are as follows:
1. Letter evaluated. In this case students avoid operating on a specific unknown
and as such simply assign a numerical value to the unknown from the outset.
The pupil may recall any number or recall the number fact about the expressed
relationship.
2. Letter not used. Here a pupil may just ignore the existence of the letter, or at
best acknowledge it, but does not give it meaning. For example, If a + b = 5, a
+ b + 2 =? and the pupil gives 7 as an answer.
3. Letter used as an object. In this case the child regards the letter as shorthand
for an object in its own right. For example, ‘3b’ as ‘3 balls’. At this level
pupils are able to regard expressions like 5 + 2a, p + 1 as meaningful.
4. Letter used as a specific unknown. The learner here regards the letter as a
specific but unknown number and can operate on it directly.
5. Letter as a generalized number. The letter is here regarded as representing, or
at least as being able to take several values rather than just one value.
6. Letter used as a variable. This is the final stage where the pupil sees the letter
as representing a range of unspecified values and understands that a
relationship exists between two such sets of values.
It was found in that study that a greater number of the student treated the letters as specific unknowns than as generalized numbers; despite the classroom experiences they had in representing number patterns as generalized statements. The majority either treated the letters as objects or ignored them. It is important that instruction is geared towards helping student construct a clear concept of a variable. Students need
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to have experiences with the different meanings and uses of variables so that they can become comfortable in dealing with them in different contexts. Activities that offer student opportunity to explore and investigate patterns and require them to make verbal formulations of rules that describe the observed patterns and finally generalize situations can play a significant role in the development of the concept of a variable.
While English and Warren (1998: 168-169) support this idea, they have however realized that “students find it easier to verbalize a generalization than to express it symbolically”. It is clear therefore that a clear understanding of variable is essential as variables provide the algebraic tool for expressing generalizations.
2.5.2 Algebraic Terms and Expressions
Mathematical knowledge is communicated through the symbolic mathematical language. This language uses numbers, letters and other conventional symbols. Austin and Howson (1979) assert that mathematical symbolism in its now internationally accepted form, is shorthand, the bulk of which has been devised by speakers of a few related languages. The use of this symbolism can accordingly cause considerable difficulties to those whose mother tongue has different structures. The use of this formal mathematical language requires the students to have a clear understanding of the relevant mathematical concept in order that she/he can translate into the correct symbolic notation, manipulate the symbols and then be able to translate back into meaningful concepts (Austin & Howson 1979).
Use of brackets in mathematics is also one of the complications in interpreting mathematical expressions or statements, as this structure is not present in ordinary language. Earlier research provides evidence that simplification of algebraic expressions creates serious difficulties for many students (Linchevski & Herscovics
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1996). Students experience serious problems in grouping or combining like terms.
Whereas in arithmetic, operations yield other numbers, in algebra operations may yield algebraic terms and/or expressions. For example,
Arithmetic Algebra
2 × 4 = 8 (8 𝑖푠 푎 푛푒푤 푛푢푚푏푒푟) 2 × 푎 = 2푎 ( 푎, 푡푒푟푚)
3 + 4 = 9 푎 + 푏 (푒푥푝푟푒푠푠𝑖표푛)
12 푎 = 2 , 푏 ≠ 0 (푎 푡푒푟푚) 6 푏
1 × 2 + 4 , 푐푎푛 푏푒 푠𝑖푚푝푙𝑖푓𝑖푒푑 푡표 푛푢푚푒푟𝑖푐 푡푒푟푚 푤ℎ𝑖푙푒 3(푥 + 푦) 푐푎푛 표푛푙푦 푏푒 3푥 + 3푦
When given a problem whose final answer is, say, 2x + 3, some student would go further to give 5x as their answer. This results from the fact that to these students 2x
+3 is not acceptable as the solution; to them a solution should always be a single term.
It is clear from this that students need to be helped to appreciate the dual nature of expressions. They should be able to see expressions as a process and as a product.
Expressions encapsulate a process as instructions to calculate a numerical value, but they are also a product as objects which can be manipulated in their own right
(French, 2002, pg. 24). As French puts it “failure to appreciate this dual nature of expressions is a major barrier to success in algebra” (French, 2002, pg. 24). Tall and
Thomas cited in French (2002, pg. 15) and Nickson (2000, pg.142) identified four obstacles frequently met by pupils in making sense of algebraic expressions. These are:
• The parsing obstacle;
• The expected answer obstacle;
• The lack of closure obstacle; and
• The process-product obstacle.
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To some pupils, the Plus (+) sign signals that they have to do some calculation; they expect to produce an answer. This is what is referred to as the expected answer obstacle. The way we read from left to right is also noted to influence pupils to interpret for example 3 + 2x as saying ‘add 3 and 2 and then multiply by x.’ This obstacle is what is termed the parsing obstacle. This obstacle also leads pupils into reading “ab as a and b” and thereby end up thinking that it is the same as a + b. When pupils show discomfort when they have to accept, say, 2x + 3 as a final answer after some algebraic manipulations is said to be due to the ‘lack of closure’ obstacle. To the pupils this is an incomplete answer. The process-product obstacle refers to pupils’ failure to appreciate the dual nature of algebraic expressions i.e. expressions can indicate an instruction and at the same time they can represent the result of the operations (French, 2002:15-16).
From the study that Bishop and Stump (2000) conducted, many pre-service teachers lacked full understanding of what algebra was and could not explain the kind of activities which may characterize a classroom that promotes algebraic reasoning. It is worth noting that when teachers’ knowledge about the role of letters in mathematics is rich, it is very likely that the necessary understandings will be passed over to their pupils. As Bishop and Stump, (2000:108). put it ‘‘teacher with a rich understanding of connections between mathematical ideas is more likely to reveal and represent them, at the same time, a teacher who lacks them is unlikely to promote deep insight in his or her students”.
2.5.3 Solving Algebraic Expressions and Equations
The problems with algebraic terms and expressions lead to further problems that are usually seen when students solve equations. Students who have insufficient
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conceptual knowledge about terms and expressions experience serious problems when they have to read and interpret the symbolic form of equations. Students are usually not able to make sense of the algebraic equations, as they do not really understand the structure of the relations in the equation (Kieran, 1992:397).
Kieran identified some complexities in the use of the word structure in the context of algebraic equations. According to Kieran (1992), surface structure, refers to simply the arrangement of different terms and operations that go to make up an algebraic (or arithmetic) expression or equation. As Nickson (2000:112) puts it, ‘‘Systemic structure refers to the properties of operations within an algebraic expression and the relationships between the terms of the expression that come from within the mathematical system’’. For example, rewriting 2 + 5(x + 2) as 5(x + 2) + 2 using the commutative law or as 5x + 12 using the distributive law and addition. There is also the notion of the structure of an equation, which incorporates the systemic structure and the relationship of equality (Nickson, 2000:112).
Solution of linear equations involves both procedural and structural operations.
Procedural operations refer to the arithmetic operations carried out on numbers to yield numbers, while structural operations refer to a set of operations carried out on algebraic expressions. For example, substituting p and q in to the algebraic expression p + q with real numbers to obtain 13 is a procedural operation while simplifying an expression such as 5p + q – 2p to yield an equivalent expression 3p + q is a structural operation (Kieran, 1992:392).
Algebraic manipulations include processes such as simplifying, expanding brackets, collecting like terms, factorizing, etc. In solving linear equations these processes are performed when transforming the original equation to its simpler equivalent forms.
Earlier research studies such as that by Kieran (1992), and the experience of the
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researcher as a mathematics teacher, reveal that most pupils do actually encounter difficulties when they are confronted with solving equations involving negative numbers. This is usually evident where brackets are involved, particularly when the brackets follow the minus sign. Other cases include where the like terms are collected and there is a gap in between them, with a minus sign before the other terms. Nickson
(2000:120) indicates that these result from a static view of the use of brackets and jumping off with the posterior operation by pupils. According to Nickson, pupils at this level could not realize that “926 – 167 - 167 was the same as 926 - (167 +167) as only two of the 27 pupils thought this was the case. “The jumping off with the posterior operation” refers to cases where pupils, in an attempt to collect like terms where the distance between the terms is involved, they tend to focus on the operation sign that follows the term. For example, 푥 + 7 – 2푥 – 3 may be simplified to
3푥 – 4.
Mastery of skills and knowledge required for correct manipulation of algebraic expressions plays a very significant role in pupils’ ability to solve linear equations correctly. French (2002:24) affirms that understanding and fluency in performing operations with negative numbers is an essential pre-requisite for learning algebra successfully. He notes, “a proper understanding of algebraic processes is inevitably very dependent on a corresponding understanding and facility with arithmetical operations” (French, 2002 p. 47).
2.5.4 The Concept of Equation / Equality
Equations are mathematical statements that indicate equality between two expressions. Many pupils at the elementary level and junior secondary level fail to interpret the equal sign as a symbol denoting the relation between two equal
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quantities. To them the sign is interpreted as a command to carry out a calculation.
Experience in classroom teaching has it that in solving for the unknown in an equation of the form, 7 + 3 = 푥 + 9. Pupils would respond as follows: 7 + 3 = 10 +
9 = 19.
Usiskin et al (2003:137) further illustrate this situation by indicating that when pupils are asked to find out what number would make the statement 7 + __ = 10 + 5 true, many would give the answer as 3, seeing 10 as the result after addition, ignoring the 5 on the right. In their workings, particularly those that involve extended computations; learners would calculate, for example, 13 + 45 + 7 as 13 + 45 = 58 + 7 =
65. This clearly indicates that pupils interpret the equal sign as the command to carry out the calculation; it does not represent the relation between the left-hand side and the right-hand side of the equation. The above-indicated problem suggests that teachers should emphasize the meaning of the equality and the role of the equal sign in an equation to the pupils. It is very important that learners are helped to develop proper interpretation of the equal sign as this understanding is essential in algebraic manipulations.
2.5.5 Formulating Equations from Context Problems
Mathematics is taught in schools to develop in the students, knowledge and skills that they require in solving problems they may encounter in their daily life. It has however been realized that when confronted with realistic mathematical problems, students often find it very hard to formulate the given problem situation into the symbolic mathematical language. This is mainly due to lack of correct interpretation of the question, which involves identification of variables and relationships that exist between those variables.
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As mentioned earlier, mathematics is sometimes considered a language, due to its strong lingual base, often spoken in symbolic notations. Mathematical terms are well defined and symbols are used to express the mathematical relationships in shorthand.
From previous studies, it has been realized that, when solving algebraic word problems pupils experience serious difficulties when interpreting a problem and translating it into the symbolic mathematical language.
MacGregor (1991:25) indicates that several researchers have confirmed that the sequential left to right translation from ordinary language to mathematical symbolism is a common procedure taught to pupils. The common error associated with this approach is the ‘reversal error’. The student-professor problem is the well-known example that illustrates this. The problem requires students to write an equation for this statement: “At this university there are six times as many students as professors” using S for the number of students and P for the number of professors (MacGregor
1991:19). From research reports, some responses to the student-professor problem were 6S = P which is actually wrong. Although this question was asked to university students, their wrong response is a result of their inherent misconceptions from early years in algebra learning. Kieran (1992:403) reckons “some semantic knowledge is often required to formulate these equations; but solvers only typically use nothing more than syntactic rules.” As English and Halford (1995) put it “students rely on direct syntactic approach to solving these problems, that is, they use a phrase-by- phrase translation of the problem into variables and equations” (English & Halford
1995:241). Berger and Wilde (1987:23) also affirm “algebra word problems have been a source of consternation to generations of students.” Even for those pupils who are most able in solving linear equations, the moment these equations are cloaked in a verbal cover story (Berge &Wilde 1987:123), they also struggle to solve such problems. From experiences that one has gathered as a mathematics teacher, it has
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become very clear that in order for pupils to become successful problem solvers, they need to be taught problem solving as a skill.
According to Constructivism effective learning occurs when pupils are actively involved in the process, therefore exposing learners to solving many real-life mathematical problems is very advantageous in developing their problem-solving skills.
2.5.6 Modeling Equations from Verbal Representations
Mathematics is a language for communication and a tool for new discovery. Like any language, it has grammatical rules and syntax structure that can be difficult for students to master (Esty, 1992). Students must have skills in reading comprehension and reasoning before an algebraic expression or equation can be derived. The use of language in classrooms is critical in developing these skills with Senior High students.
Before students learn to represent algebraic situations symbolically, they should have opportunities to discuss them in easily understood, everyday language, thus developing their conceptual understanding (Kieran & Chalouh, 1993).
According to Rosnick (1981), Students’ greatest difficulties in algebra are modeling equations from problem situations. Translating from verbal relational statements to symbolic equations, or from English to mathematics, causes students of all ages a great deal of confusion. Like the solving of equations, modeling equations from word problems can be taught with a procedural or conceptual emphasis.
Lodholz (1990) observed that writing equation from word problem is often a skill taught in contrived situations or isolation. Mathematical word problems that require students to write an expression that represents “5 more than 3 times a number;” when
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taught apart from opportunities for application, can cause students difficulty when interpreting meaningful sentence later. This gives students a procedural method for doing what, by nature, should be conceptual. Students may translate English sentences to mathematical expressions, simply moving from left to right. “Three less than a number” is interpreted by many students as “3 − 푥” since the words “less than”
(which means to subtract, they have always been told) follow the 3. Teachers must be aware of these misconceptions and address them in instruction (Lodholz, 1990).
Writing equations from word problems is a difficult skill for students, whether caused by cognitive misconceptions or literal translation. Students’ inclination to translate directly from English sentences to algebraic expressions may be augmented by the procedural method many teachers use when addressing this topics in class. It is not uncommon for teachers to encourage students to look for “key words” in word problems that signify a particular operation (Kroesbergen et al., 2004). According to
Wagner and Parker (1993), though looking for key words can be a useful problem- solving heuristic, it may encourage over-reliance on a direct, rather than analytical, mode for translating word problems into equations.
2.6 Students' Difficulties in Solving Algebraic Problems
According to Newman (1983), difficulty in problem-solving may occur at one of the following phases, namely, comprehension, strategy know-how, transformation process skill and solution. Schoenfeld (1985) suggested four aspects that contributed to problem-solving performance. These are the problem solver's, mathematical knowledge, knowledge of heuristics, affective factors which affect the way the problem solver views problem solving, and managerial skills connected with selecting and carrying out appropriate strategies
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Kroll (1993) in their study of problem-solving identified three major cognitive and affective factors; namely knowledge, control and beliefs, and effects that contributed to students difficulties in problem-solving. Lester (1994) on the other hand emphasized that difficulties experienced during problem-solving could also be caused by the problem solver's characteristics such as traits, disposition, and experiential background.
In the early 1970s research tended to attribute difficulties in solving problems to the various task variables such as content and context variables, structure variables, syntax variables and heuristic behaviour variables (Goldin & Mc Clintock, 1979).
However, Lester (1994) contended that there was a general agreement that problem difficulty is not so much a function of various task variables but rather a function of characteristics of the problem solver. In other words, the knowledge one possesses, one's disposition and one's experiential background often influence problem-solving performance. These were also evident in a study conducted in Singapore by Kaur
(1995) and Lee (2001). Kaur indicated that Singapore's students experienced problem- solving difficulties such as:
1. Lack of comprehension of the problem posed
2. Lack of strategy knowledge
3. Inability to translate the problem into a mathematical statement
Lee who conducted a local study on first year undergraduate students solving routine calculus problems found that the difficulties faced by the students were:
1. Lack of experience in defining problems
2. A tendency of rush toward a solution before the problem has been clearly
defined
3. A tendency to think convergent
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4. Lack of domain-specific knowledge
McGinn and Boote (2003) identified four primary factors that affected perceptions of problem difficulty. These were:
Categorization- ability to recognize that a problem fits into an identifiable category of problems which run a continuum from easily categorizable to uncategorizable
Goal interpretation- figuring out how a solution would appear which run a continuum from well-defined to undefined
Resource relevance- referring how readily resources were recognized as relevant from highly relevant to peripherally relevant, and
Complexity- performing a number of operations in a solution.
Mc Ginn and Boote (2003) suggested that the level of difficulty of the problem depended on the problem solvers perceptions of whether they had suitably categorized the situation, interpreted the intended goal, identified the relevant resources and executed adequate operations to lead toward a solution.
Not all the errors that students do make when solving word problems result from difficulties in representing and translating problem statements. Once the problem has been translated, problem-solving errors can and do still occur and these errors are often due to a bug (Lewis, 1981)
Sometimes, students get confused when they try to formulate a solution for an algebraic word problem. Kieran (1992) says that to solve a problem such as; when 2 is added to 4 times a certain number, the sum is 24; students would subtract 4 and divide by 2 using arithmetic. But solving the problem using algebra would require setting up an equation like 2 + 4푥 = 24. There are therefore two different kinds of thinking
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involved in these two contexts which would sometimes confuse students. In arithmetic, students think of the operations, they use to solve the problem whereas, in algebra, they must represent the problem situation rather than the solving operations.
This means apart from the difficulties encountered by students when translating word problems into algebraic language, there are other barriers such as inter interference from other systems, like not understanding the equal sign as a relationship, and' other misconceptions in simplifying algebraic expressions.
2.7 Summary of the Literature Reviewed
In studies that adopted the use of the constructivist approach in teaching and learning
(Cobb, 1996; Dangel, 2011; Fox, 2001; Phillips, 1995; Chin, Duncan & Hmelo-
Silver 2007), the approach proved to be a useful methodology in addressing the difficulties faced by the students in mathematics and science related subjects. Most of these research works employed Constructivist Learning theory which was meant to provide an alternative approach to address students’ difficulties and improve on their overall achievement in the subjects.
Literature on the teaching approach used by teachers affirmed the national culture of mathematics teaching which to a considerable extent was teacher centred. This approach was seen to limit students’ collaboration in the classroom and had largely not addressed the challenges students face in the subject. Therefore, an alternative teaching approach was deemed to be necessary to address students’ challenges in algebra.
For the purpose of this research report, the researcher aligned himself with the
Constructivist teaching and learning approach and identified the learning process as a
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collaboration between the learner, the teacher and other learners. The researcher was
of the view that when learners collaborated in the classroom by sharing ideas and
concepts, it had a lasting impact on their achievement and the Constructivist theory
provided the needed basis for this assertion. In addition, reviewed literature on
collaborative learning according to Totten et al (1991) showed that shared learning
gives students an opportunity to engage in discussion, take responsibility for their
own learning and so become critical thinkers. Other research works have also
suggested that collaborative learning brings positive results such as deeper
understanding of context, critical thinking, and increased overall achievement in
grades, improved self-esteem, and higher motivation to remain on task, more
opportunities for personal feedback, celebration of diversity, group conflicts
resolution and improved teamwork and social skills (Concept to Classroom, 2004).
Hence the literature reviewed on the constructivist teaching approach which
characterized the use of collaborative learning pedagogy provided a strong basis as
the alternative approach employed in the current study to improve students’
achievement in algebra.
CHAPTER THREE
RESEARCH METHODOLOGY
3.0 Overview
In this chapter, the main methodological constructs that were employed in the various
stages of the study were explained and later united together to create an overall
summary of the methodology. This included a review of the methods that were used
at different stages of the study and their validity and reliability, sampling procedures,
the pilot study, the main study, data collection instruments, data analysis methods and
intervention, ethical issues, and a chapter summary.
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3.1 Research Design
This study sought to investigate the effects of the constructivist teaching approach on
Senior High School students’ performance in Algebra in the Gomoa East District. The study adopted a convergent parallel mixed method approach which combined elements of qualitative and quantitative data collection and analysis. According to
Cresswell, Klassen, Plano Clark and Smith (2011), a mixed method research approach is the use of both quantitative and qualitative methods to collect data that answer the research questions. That is, researchers collect or analyze not only numerical data, which is customary for quantitative research, but also narrative data, which is the norm for qualitative research in order to address the research questions formulated for a particular study. A convergent parallel mixed method design was used and it is a design where quantitative data collection and analysis, and qualitative data collection and analysis occur at the same time and are then compared after the completion of the study (Johnson, Onwuegbuzie & Turner, 2007).
The mixed method design was selected because the strengths of the quantitative data and the qualitative data can be combined to obtain a more insightful understanding of the research problem (Bryman, 2006; Cresswell & Plano Clark, 2007). Moreover, the use of mixed method design also helped to achieve data triangulation. Triangulation simply refers to the application and combination of several research methods in the study of the same phenomenon (Boghan & Biklen, 2006). According to O’Donoghue and Punch (2003), triangulation is a “method of cross-checking data from multiple sources to search for regularities in the research data” (p.78). Altricher, Feldman,
Posch and Somekh (2008) contend that triangulation “gives a more detailed and
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balanced picture of the situation” (p. 147). Data triangulation in this current study therefore increased the credibility and validity of the results.
For the current study, the quantitative methodology involved a quasi-experimental design with a non-equivalent control group design. It sought to examine the difference in achievements of two groups (experimental and control group) to determine whether the constructivist teaching approach had an effect on improving students’ achievement in studying algebra. The quasi experimental design was preferred in this context because the study was characterized by the investigation of a cause
(instructional approach) and effect (achievement), manipulation (choice of instructional approach), and control (same unit being taught), and involved a non- random assignment of participants to two groups, experimental (treatment) and control groups. Data from the questionnaire administered to the students were also analysed quantitatively.
For the qualitative analysis, interviews and the classroom and lesson observations served as the main instruments. This was done to get a better insight into students’ perceptions about the use of the constructivist teaching approach and also to gain better understanding on the type of teaching approach commonly used by the teachers in the district. Trustworthiness and creditability were obtained by the use of these multiple data sources. The basic design of the study showing how each group was involved is shown in Figure 3.1.
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Target Selection Population of Class
Pre test Pre test
Treatment Control Group (Constructivist approach) (Traditional approach)
Post test Post test
Figure 3.1: Basic Design of the Study
A cursory look at Figure 3.1 revealed that the study consisted of three phases. These phases were the pre-test stage, treatment stage and the post-test stage. The first phase was the pre-test which was carried out simultaneously on all the groups before administering the treatment. The second phase was the treatment stage of which the experimental group was taught using the Constructivist approach while the control group was taught using traditional teaching approach. Finally, the third phase was the post-test to both groups after the six weeks of treatment. After the respondents went through these three phases, the test results were evaluated to determine whether the
Constructivist approach affected students’ achievement in Algebra.
Consequently, the variables of this study were categorized into independent variables and dependent variables. In this study, there were two independent variables, which were the approaches used in teaching and learning Algebraic concepts, thus, the
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Constructivist approach and the Traditional approach. The dependent variable of this study was students’ scores on the Algebra Concept Achievement Test (ACAT). These scores were analysed to establish whether a significant difference existed between the control group and experimental group or not.
The researcher also developed the following bench marks for learning and teaching.
The benchmarks are given in Table 3.1
Table 3.1: Benchmarks for Teaching and Learning
Benchmarks for…. Moving from…. Moving towards….
Passive absorption of Active Engagement with information information
Both individual activity and Individual activity Learning Collective work
Individual differences among Individual differences among students seen as problems students seen as resources
Varied teacher roles, from Teachers in information- information deliverer to deliverer role architect of educative experiences
Teachers structure Teachers doing most of the classrooms for individual work and shared work Teaching Concepts mentioned and Concepts developed and lessons not coherently elaborated and lessons not organized coherently organized
Teachers as founts of Teachers inclined to improve knowledge their practice continually
These bench marks influenced the use of the Constructivist teaching and learning approach as the main model for the research. The study employed the collaborative learning (group work) approach to help students understand the concepts of Algebra.
3.2 Population
The population for the study consisted of students and mathematics teachers from two senior high schools in the district. For the participating teachers, six mathematics
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teachers were selected from a target population of nine mathematics teachers in both schools. On the other hand, for the students, the target population consisted of all form two (2) classes in both schools comprising of two hundred and thirty (230) students. Whilst the accessible population were the General Arts and Home
Economics second-year classes consisting of one hundred and sixty-two students
(162) in both schools. The rationale for choosing these classes was that the researcher identified these classes as having serious challenges with mathematics, hence the study could help determine whether the treatment would help improve their performance in mathematics especially in Algebra. Table 3.2 provides the details of the population for the study.
Table 3.2: Population for Control and Experimental Schools.
Accessible Study School Target Population Population
Control Group (School A) 125 88
Experimental Group (School B) 105 74
Total 230 162
Source: Field Data, 2018.
Table 3.2 indicates that for the control school (School A), eighty-eight (88) students were available for the study. The total population for the form two classes stood at one hundred and twenty-five (125). For the experimental school (School B), seventy- four (74) students were available for the study. The total population for the form two classes were one hundred and five (105).
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3.3 Sample and Sampling Technique
Sampling refers to the process and techniques used to select the study participants.
Sampling reduces the cost of collecting data by working with a manageable and accessible group that is representative of the population (Welman et al., 2005). The participants in this study consisted of six mathematics teachers and one hundred and forty Form two students. The purposive sampling and the convenience sampling techniques were employed to select the sample of teachers and students for the study.
The students were divided into the control and the experimental groups. For the control group, out of the eighty-eight (88) students available for the study, seventy
(70) students participated fully in the study, whilst for the experimental group, out of the seventy four (74) students available for the study, seventy (70) participated fully in the study. Therefore the total sample size used for the study was one hundred and forty (140). Six (6) mathematics teachers were also purposively sampled for the study.
A summary of the number of students and teachers who participated in the study, sampling technique and sample size is presented in Table 3.3.
Table 3.3 Summary of Sampling Technique and Sample Size
Sampling Groups Sample Percentage (%) Sampling Technique (Two Schools) Size
Mathematics Teachers 6 4.1% Purposive
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Students 140 95.9% Convenient Sampling (using Intact Groups)
Total 146 100%
From Table 3.3, a total of one hundred and forty (140) students participated fully in
the study. Students from two different senior high schools in the Gomoa East District
were conveniently selected due to easy accessibility and geographical proximity to the
researcher. Six mathematics teachers were purposively selected from the two schools
with the purpose of helping the researcher get insight into their preferred teaching
methods.
The students chosen for the control and experimental groups were sampled from Form
two classes of the two schools. These classes were chosen for the study because the
students in these classes exhibited the relevant characteristics necessary for the study.
3.4 Research Instruments
The following research instruments were used for the study:
1. Test (Algebra Concept Achievement Test (ACAT).
2. Classroom and Lesson Observation Checklist
3. Questionnaire for teachers and students
4. Interviews with teachers and students
3.4.1 Description of the Research Instruments
3.4.1.1 Test (Algebra Concept Achievement Test (ACAT)
Questions on the achievement test were selected using the senior high school
teaching syllabus for mathematics curriculum (Ministry of Education, 2010) as a
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guide. The tests consisted of nineteen (19) test items under the following conceptual areas: variables, expressions, equations, and word problems. The test was prepared to obtain data about students’ understanding of the concepts. The following were the two main aspects that were considered in preparing the test items:
1. The items that were directly related to the conceptual understanding of
algebra. Students had to explain some basic properties in algebra or they had
to identify patterns or relationships and represent or interpret them
algebraically. Some of them contained algebraic manipulations. Problems
without a specific context pertaining to simplification of algebraic expressions,
evaluating expressions, and solving equations were examples of this group.
2. The next type of questions were word problems that students needed to
represent algebraically in order to solve them. These items usually appear in
day-to-day life. Most of them were contextual problems. In some of the short-
answer problems in the test, students had to provide and justify their answers
by using mathematical language or other representations. In this way, the
lapses of their conceptual understanding were easily identified.
3.4.1.2 Classroom and Lesson Observation Checklist
Observation is a useful tool to collect information on classroom events (Cohen et al.,
2011). Observation makes it possible for the researcher to directly see what people do rather than rely on what they say they do (Finn & Jacobson, 2008). The researcher developed an observation checklist to evaluate whether mathematics teachers applied constructivist approaches for understanding and the extent to which they adopted
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these approaches in their instructions. This helped to corroborate the findings from the questionnaires administered to the teachers.
Qualitative data was collected using observation protocols on what transpired during instruction in both the control and the experimental schools. It must be emphasized that the scope of classroom observations covered observation of the teacher, the learner and the instruction. The researcher used a notebook to record the feedback from lesson observations. Areas of focus during classroom observation were established by the researcher in line with the study research questions. The checklist was made up of ten items which were categorized into three groups. The first two questions centred on teachers’ use of constructivist activities for instruction, the next three also centred on the extent to which teachers allowed learners to interact in class whilst the last five questions focussed on how teachers involved learners in planning learning and assessment activities. The researcher indicated the response to each item on the checklist by placing a tick [√] against “Yes” or “No”. The observation checklist used in the study are presented in Appendix 1.
3.4.1.3 Questionnaires
Two questionnaires were designed for both teachers and students. The questionnaire designed and administered to the teachers specifically explored their views about common teaching approaches they adopted for instruction and helped to answer the research question one. This questionnaire instrument had two sections consisting of ten questions. Section A of the questionnaire sought for the demographic information on gender, length of teaching experience, and the qualification of the participant.
Questions 1 to 6 covered this section and participants were to indicate their response with a tick [√]. Section B (question 7 to 10) also sought to find out the type of
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instructional approaches teachers often employed during mathematics lessons. The teachers’ questionnaire is documented in Appendix 2.
Another questionnaire was also designed for the students and sought to identify students’ perspectives of collaborative learning. Learners were given a twenty-item questionnaire on their perceived experiences during the constructivist teaching approach in algebra lessons. Questions in this section were open ended and required participants to choose their responses by placing a tick [√] against the answer of their choice from a four point Likert scale from “strongly agree”, “agree”, “disagree”,
“strongly disagree”.
3.4.1.4 Interview Guide
Interviewing is a very powerful way of eliciting the ideas of people (Gay, Mills &
Airasian, 2012; Rubin & Babbie, 2008), their knowledge, values and attitudes (Gay et al., 2012; Tuckman, 1988). This study adopted face-to-face interviews to collect data from the sample. Teachers’ interviews were used to also corroborate the findings of the first research question about the common teaching approaches teachers often employed in teaching. This helped to get a better understanding to some of the answers provided in the questionnaire. Pre-structured open-ended questions were used to solicit responses (Best & Kahn, 2006). There were five questions in all. The first three questions centred on approaches often employed by teachers and their impact on their learners. Questions 4 and 5 focused on exploring the extent to which teachers adopted constructivist approaches in their instructions. The teacher interview schedule is documented in Appendix 3.
Another semi-structured interview questions was done for the students to address the research question: ‘What are students’ perception about the use Constructivist
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approach in teaching and learning of algebra?’ The semi-structured interview guide contained 15 items (See Appendix 4) was used to elicit information on the students’ impressions about the use of the constructivist approach to teaching algebra, whether or not they enjoyed learning with the constructivist teaching approach and environment, new things they learnt, their challenges and recommendations.
3.5 The Pilot Study
The study was piloted in one public secondary school located in a different district.
The pilot sample consisted of forty (40) second year mathematics learners. The pilot school shared similar characteristics with schools in the two schools in the main study in terms of poor learner performance in mathematics, being a public school governed by the same educational policies and reflecting similar socio-economic factors.
3.5.1 How the Pilot Study was conducted
The questions of the achievement tests were piloted to gather information about the validity and reliability of the test. The necessary structural changes were made to the test instrument before it was used for the main study. The categorization of questions during the pilot study can be found in appendix 2.
The students’ tests were marked and the facility index for each item was calculated
C using the formula: Facility Index = , where C is the number of students who N answered an item correctly and N is the total number of students in the sample
(Nunnally, 1972; Alderson et al. 1995; McAlpine, 2002). It was observed that the items that were too easy to answer gave fewer student errors. However, the response rate was low for the items that were difficult to answer. Therefore, a reasonable
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facility index in between 0.3 and 0.8 was selected for the items that were to be included in the final questions to be accepted for the main study. After determining the facility index for each test items, it became necessary to revise or eliminate some of the questions and the reasons for their elimination or revision are given in appendix 5.
The same content was taught in both the control and experimental groups. Both groups had equal number of instructional periods. During instruction questions were asked to test learners’ understanding and learners asked questions for clarity. Each learner was assessed and scored on each item on the pre-test and post-test. The scores obtained by learners in the pre-test were recorded by the researcher. The post-test scores obtained by learners were recorded by the researcher again. The scores from both the pre-test and post-test were analyzed using descriptive and inferential statistics.
After the revision of the test items, the final version of the test was prepared with a total of 19 items (See appendix 6). For example, in question 4(c), the word ‘expand’ was added to the question. The purpose of question 18 was to invite students to extend their thinking beyond the given situation and generalize the situation. Students were given instructions in the test to use algebraic methods to solve all the problems.
The wording of problem 12 was changed from a ‘true-false’ item to a ‘written response’ item since this will give more information about student work and their thinking. Also, a slight adjustment was made to question 9, since its facility value was greater than 0.8. The amended version of the test instrument used for the main study contained 19 items and has been presented in Appendix 6.
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3.6 Content Validation of Instruments
Validity in qualitative and quantitative research means appropriateness of the tools, processes and data. It does not mean certainty in the results but pursuit of maximum validity (Cohen et al., 2007). For instance, qualitative data is subjective to participants’ expressions with their perceptions and attitudes. Such data produces bias to some extent (Cohen et al., 2007). It is not possible to achieve perfect validity from qualitative (Cohen & Manion, 1994) and quantitative data (Cohen et al., 2000, 2007).
The content validity of the various test instruments were determined as follows:
3.6.1 Achievement Test
Content validity addressed how well the nineteen (19) items of the test samples were able to address the study objectives. To address the content validity of the test items, the test was given to two experienced mathematics teachers for a review of the test items. They confirmed that the content of the test adhered to the requirements of the senior high school mathematics syllabus and the tests were consistent with the objectives of the study.
3.6.2 Classroom and Lesson Observation Checklist
A pilot study was conducted prior to the commencement of the main study. The purpose of the pilot study was to examine the level of bias in the research process, and also to test the observation process (MacMillan and Schumacher, 2006; Johnson and
Christenson, 2012). The final lesson observation schedule was influenced by the findings of observation schedule that was conducted during the pilot study.
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3.6.3 Questionnaires and Structured Interview Questions
To ensure validity of the questionnaires, the constructed questionnaire was given to my supervisor and other two mathematics experts. These people assisted productively in the validation process. The experts were provided with copies of the questionnaires and they suggested changes which were considered so as to strengthen the validity of the questionnaires. The recommended changes that were implemented include linking all the questions to the objectives and the research questions such that there was a relationship between the previous and the next questions.
Validity of the interviews was determined with the help of data that had been collected through the questionnaires. Since both the questionnaire and the semi- structured interviews were used to answer the first and third research questions, data from both instruments were compared to establish their validity. To achieve this, convergent validity was examined. Convergent validity may be determined when data collected on the same variable from more than one instrument is compared (Johnson
& Christensen, 2012). The researcher used the same technique to determine the validity of data collected through the semi-structured interviews.
3.7 Reliability of the Instrument
Reliability as described in this study measured how consistent the test results were on a retest and were determined as follows:
3.7.1 Achievement Test
The twenty-two (22) test items were trial tested during the pilot study to establish their reliability. To test for reliability of the test, the study used the internal consistency technique by employing Cronbach’s Coefficient Alpha (훼) (Cronbach,
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1951). With a sample of forty (40) students, the value of 훼 = 0.847 was computed
for reliability of the test as presented in Table 3.4. The results confirmed that the test
was reliable to measure students’ achievement in algebra.
Table 3.4: Reliability Statistics for the Four Conceptual Areas
Reliability Statistics Cronbach's Alpha Based on Cronbach's Alpha N of Items Standardized Items .847 .878 22
3.7.2 The Lesson Observation Schedule
Reliability of the lesson observations was determined through a process of repeated
usage of the observation schedule. It was used during the pilot study as well as in the
main study.
3.7.3 Questionnaire and Interview Guide
Reliability of the questionnaires and interview guide was assessed through a test-
retest procedure that involved administering the same questionnaire and interview
guide to the same individuals under the same conditions over a period of time.
3.8 Treatment Procedure
This section discusses the treatment procedure used to remedy students’ difficulties in
understanding the four conceptual areas in algebra.
3.8.1 Constructivist Teaching Approach as the Treatment Procedure
During the treatment carried out for the experimental group, students were grouped
and each group was made up of four (4) students. The treatment lasted for six (6)
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weeks. The control group were taught the same algebraic concepts using the traditional teaching approach within the six-week period. At the end of the experiment, the same Algebra Concept Achievement Test (ACAT) was administered to the students to compare their achievement in algebra and to measure the effect of treatment.
The participating schools followed the schools’ guidelines on the construction of their school timetables, allowing two hours and forty minutes of teaching time for mathematics per week for the participating classes. It must be noted that this was not a researcher developed instructional timetable but it was a school generated schedule.
Both the control group and the experimental group wrote the pre-test and the post-test around the same period. The post-test had quite different questions as the pre-test although it was ensured that both tests tested the same concepts. The researcher invigilated both tests in both schools. The constructivist-based classroom was characterised by group learning approach. Learners were put in groups of at least four members in each group. In addition, the researcher appointed group leaders for each group. The researcher only provided explanations when required to do so. Most importantly, the potential of more robust engagement was exploited with worked-out examples in algebraic variables, expressions, equations and word-problems that were given to groups as worksheets.
The experimental group and the control group were exposed to identical worksheet tasks. However, the mode of presenting the worksheet tasks varied between the two groups. At the instruction phase of the lesson, the researcher divided the lesson into three stages: introduction; the body of the lesson; and conclusion. At the introduction stage, the researcher introduced the topic to class, explained the key terms and
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concepts, asked questions to assess learners’ prior knowledge of the topic, and established the basic errors and algebraic skills of learners.
At the body stage of the lesson, learners in their groups were given the example sheets to discuss the solution steps whiles the researcher monitored group discussion.
During this stage self-explanation activity and probing took place. At this stage of instruction, the researcher carefully monitored the group work and whole-class discussion. This was necessary so as to intervene and redirect the learners to correct their errors and misunderstandings. To do this, the teacher occasionally asked the learners probing question such as: “Why did you do it that way? Will it work if you did it the way your friend suggested? “What makes the answer given by a peer to be wrong?” “What is the correct way to do it?”
Students were then made to talk about it further in their group as the researcher would get back to them shortly. There were no specific rules that informed the researcher when to intervene or how extensive the intervention should be. Most significantly, the researcher was at liberty to make these decisions and these were made on the basis of the researcher’s knowledge of the subject matter and learners’ past experiences. The role of the researcher as a teacher was limited to guiding and facilitating rather than telling the learner. The researcher created a purposeful, intentional and collaborative learning environment that enabled learners to actively strive to achieve the cognitive objective.
The concluding stage of the lesson was meant for reflection of the lesson where group discussion of activity took place, and success rate of the lesson was evaluated. The lesson concluded with more tasks given as homework. At learning and performance phases one could hear different voices and sounds from the various groups like “I got it” and sometimes learners exhibited signs of frustration when they encountered
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challenges in their knowledge construction process with utterances such as “I don’t understand” and “your answer is wrong”. Learners’ gestures like nodding the head in agreement with the explanation given by peers in the group or the teacher, and their utterances such as “okay”, “now I know” indicated that knowledge constructions was taking place.
Furthermore, one unique feature of the Constructivist Teaching Approach lesson was that, it was difficult to identify who the teacher (researcher) was, as the researcher was moving around from group to group in order to monitor, assist and direct learners’ discussion. The researcher sometimes sat down with the learners in the group and watched as learners discussed the task assigned to them. Most critically, the learners sometimes did not seem to notice the presence of the teacher in their group and kept on discussing and talking with each other. If someone with traditional preconceived notions that classrooms of learning should be ordered, systematic and quiet had entered the classroom he or she would miss the dynamic learning that was occurring in that classroom and many other classrooms structured for cooperative learning and from constructivist philosophical perspective.
The classroom arrangement was such that one would not even determine which part of the classroom was the back and which one was the front part. In this Constructivist
Teaching Approach generated teaching environment, a teacher’s desk was not even seen. All learning activities in this constructivist lesson were centred on the learners.
Using the principles of cooperative learning and constructivist learning theory, the researcher carefully built a learning community in which teacher-learner and learner- learner interaction, which was subtly arranged, promoted knowledge construction and deep enduring learning that enabled learners’ errors to be exposed and treated during the lesson. The teacher realised that in order to empower learners to verbalise their
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prior knowledge so as to expose the errors they inhabit and treat them thereby
improving their performance in algebra, they must interact with one another as a
community of learners frequently and easily.
The unique features of Constructivist Teaching Approach which distinguished it from
the Traditional Teaching Method were: the group learning approach; the nature of
learner involvement and participation; the guiding and facilitating role of the
researcher; the learner-centred lesson; the social interactions that existed in the
classroom; availability of scaffolds and problem-solving tools; the manner in which
learners’ errors were exposed and treated during instruction; the prevalence of
interactive learning environment and learners critical responses of other learners
contributions through verbalisation, argumentation, and exploratory talk.
3.9 Data Analysis Procedures
Data collected from the achievement tests and questionnaire were analyzed using
quantitative methods whilst data from interviews and lesson observations were
analyzed using qualitative methods.
3.9.1 Quantitative Data Analysis
Data from the achievement tests and questionnaire were analyzed quantitatively. The
two statistical methods used in this analysis of the achievement tests were descriptive
statistics and inferential statistics. In using the inferential statistics to analyze the
quantitative data, the independent t-test and the dependent or paired samples t-test
were used to analyze participants’ scores related to performance on the dependent
variable (Gay et al., 2011). The dependent variable was learners’ Algebra
achievement posttest scores whilst the covariate was learners’ pre-test scores. Before
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performing the t-tests, the researcher evaluated the assumptions underlying the study namely, the assumption of normality and linearity of data distribution. Various statistical tools were also used to analyze certain aspects of the quantitative data. An alpha level of 0.05 was used for all statistical data analyzed.
The questionnaire designed and administered to the teachers specifically explored their views about common teaching approaches they adopted for instruction and helped to answer the research question one. The participating teachers were first asked to indicate the teaching approaches they preferred most. The teaching approaches mostly preferred by the participating teachers were analysed quantitatively.
For the questionnaire delivered to the students to solicit their perceptions on the constructivist approach, the set of the four point Likert scale used to gather the data were also analysed quantitatively.
3.9.2 Qualitative Data Analysis
Qualitative data was collected through the interviews and lesson observations conducted during the study. Data in the form of field notes gathered from lesson observations and those gathered from the interviews were treated as existing evidence to triangulate the findings with other sources of data.
3.9.3 Data Analysis Procedures for Research Questions
The data analysis procedures for each of the research questions have been described as follows:
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3.9.3.1 Research Question One
Research question one sought to investigate the teaching approach used by teachers in
teaching mathematics. Data from questionnaire administered to the teachers, lessons
observations, and interviews were used to gather data for the research question.
The questionnaire designed and administered to the participating teachers specifically
explored their views about common teaching approaches they adopted for instruction.
Data gathered from the questionnaire was analyzed quantitatively.
Added to this, an observation checklist was developed to evaluate the behaviours,
skills and approaches exhibited during instruction. Qualitative analysis of these
observations were then carried out by the researcher through the frequency
distribution developed from the data gathered.
Teacher interviews were also done to get a better understanding to some of the
answers provided in the questionnaire. Data from the teacher interviews was analyzed
qualitatively.
3.9.3.2 Research Question Two
Research question two sought to establish the effect of the teaching approaches
identified in research question one on the students’ achievement in Algebra. A
descriptive analysis was carried out on the data obtained through the pre-test and post-
test scores from the two groups with reference to the teaching approaches adopted by
the teachers.
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3.9.3.3 Research Question Three
The hypothesis for the study was tested in research question three. It sought to
examine whether there were any significant differences in achievement between
students taught by the constructivist teaching approach and those taught by the
traditional approach. The results of the experimental group before and after the
treatment were analyzed both descriptively using frequency tables and inferentially
using the T-test analysis. These results were compared to that of the control group
who did not receive the intervention. The difference in means obtained from the
responses before and after the intervention between the control and experimental
groups were computed and its significance established using t-test statistical analysis.
The t-test results were used to test the null hypothesis. An alpha level of 0.05 was
used for all statistical data. Triangulation of the quantitative data was achieved
through a qualitative analysis of data collected through the classroom and lesson
observation guide. This gave further confirmation of the treatment method and its
effect on learners’ achievement in algebra.
3.9.3.4 Research Question Four
Quantitative data for this research question was collected through questionnaire
administered to students in the experimental group. It sought to identify students’
perspectives of the treatment method through a twenty-item questionnaire on their
perceived experiences about the treatment method used during the algebra class. All
the items in the questionnaire were designed for a Likert scale response using a four-
interval scale of “strongly agree”, “agree”, “disagree”, “strongly disagree”.
In order to elicit more information from participants, semi-structured interview
questions were used in informal situations to interview some students who received
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the constructivist methodology. This was done to corroborate the findings gathered
from the questionnaire. Data obtained from the questionnaire were analyzed using
percentages and data from interviews were analyzed qualitatively.
3.10 Assumptions of the Study
This study was conducted with two main assumptions in mind. This was to ensure
that the resultant change in post-test scores in the groups would be attributed to the
treatment method. The first assumption was that the scores in the pre-test and post-test
were expected to be normally distributed in all four algebraic conceptual areas in an
instance that the CTA and TTM learners were homogeneous and that assessments
were to be done as honest as it had been planned.
The second assumption was that if the groups were homogenous and assessments
were carried out as honest as it were planned, the variances of TTM and CTA groups
were expected to be equal or near equal in all the sub variables under investigation
(equality of variances). A formal normality test was performed by using
Kolmogorov-Smirnov and Shapiro-Wilk. The Normal Q-Q plot was used to
determine the normality of the pre-test and post-test scores. The 푝-value for the
KolmogorovSmirnov and Shapiro-Wilk was used to determine the normality of the
pre-test and post-test scores. The null hypothesis for normality was that there was no
significant difference between the learners’ achievement in the pre-test and post-test
scores in the experimental group, while the alternate hypothesis suggested that there
would be significant difference between their achievement in the pre-test and post-test
in the experimental group.
If the 푝-value for both Kolmogorov-Smirnov and Shapiro-Wilk was more than 5%,
we fail to reject the null hypothesis. This meant the post-test and pre-test scores were
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normally distributed. Paired sample t-test could be applied only when the variables for the study were normally distributed. The normal Q-Q plot of posttest-pretest scores indicated that the experiment and control groups were normally distributed. The researcher used SPSS to perform a formal normality test by using Kolmogorov
Smirnov and Shapiro-Wilk to determine the linear relationship of data distribution graphically, using a scatter plot. The slope of the regression lines was roughly parallel and it was assumed that there was a linear relationship between the covariate.
3.11 Logistical and Ethical Considerations
All protocols in conducting a research were observed by the researcher. Ethical considerations included confidentiality of information, names, and sources. The researcher encouraged voluntary participation, arising from informed consent
(Mugenda & Mugenda, 2003). To access the required information, permission from relevant bodies such as the University of Education, Winneba Graduate School was first sought. Approvals from the Gomoa East District was also obtained to conduct research in their schools. Informed consent of principals was obtained using the relevant documentation. These documents included informed invitation letters to the principals to conduct the research in their schools and informed invitation letters to students for their participation. Participation was voluntary and participants had the right to withdraw from the study at any time. Nevertheless, all research activities of this study did not interfere with teaching and learning programs in each of the two participating schools.
Special emphasis was laid on confidentiality or anonymity of interviews in case of sensitive or gazette data. Since this research involved minors (learners under 18 years) who were vulnerable to emotional, verbal abuse, and psychological traumas, in
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order to protect them from harm, the normal existing teaching and learning condition or environment was maintained. No corporal punishment, verbal abuse, stigmatization, intimidation, prejudice or bullying was allowed. The normal security measures during school time was observed; school rules and regulations, and disciplinary codes were also enforced. During the reporting and discussion of data, none of the participants, schools, or communities were identified (pseudonyms were used) and participants were not judged or evaluated on their participation or non- participation. All the data that were collected had no names of the participants since each participant was given a unique code when the instruments were administered.
3.12 Chapter Summary
This study was an experimental research which employed both qualitative and quantitative methods as the overall design. The main research instrument in the quantitative phase was a test instrument (Algebra Concept Achievement Test) and questionnaires administered to both teachers and students while classroom observation checklist and students’ interviews served as the main instruments for the qualitative phase. The study used the constructivist teaching approach in improving learners’ achievement in algebra. The main study was conducted after a pilot trial of the test instrument. Students’ answers to the test, their written answers on the questionnaire, teachers’ questionnaire, interview transcripts, researcher notes and classroom observations were simultaneously used as multiple data sources to arrive at valid conclusions about the impact of the constructivist teaching approach.
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CHAPTER FOUR DATA ANALYSIS AND DISCUSSIONS OF RESULTS
4.0 Overview
This chapter presents the report and the analysis of the results of the study. The chapter comprises the analysis and discussions of the results of each of the research questions including the descriptive and inferential statistics of the hypotheses of the study and the overall summary of the research findings. The data that were obtained through the use of the four research instruments namely; Algebra Concept
Achievement Test, questionnaire, interview and lesson observations are presented, analysed, and discussed in this chapter. The sequence of the presentation and the discussion of the results obtained were discussed in accordance with the research questions formulated for the study. Interpretations of the data were made based on literature findings and theories.
The purpose of this study was to investigate the effects of the Constructivist Teaching
Approach (CTA) on Senior High School students’ achievement in algebra in the
Gomoa East District. The study was guided by the following research questions:
1. What teaching approaches do Senior High School mathematics teachers often
employ in their lessons?
2. What is the effect of the teaching approaches often used by teachers on
students’ performance in Algebra?
3. What is the effect of the constructivist teaching approach on students’
achievement in Algebra?
4. What are students’ perceptions of the constructivist teaching approach on their
performances in Algebra?
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The study tested this null hypothesis for research question three (3).
Null Hypothesis (푯푶): There is no significant difference in achievement between
students taught by the constructivist teaching approach and those taught by the
traditional approach?
4.1 Demographic Characteristics of Participants
4.1.1 Background Information of Participating Teachers
All teachers who participated in the study taught in the two schools that were located
in the district. Biographical information such as gender, teaching experience and
teaching qualification of the participants were determined in the study.
4.1.1.1 Gender of Participating Teachers
Data on gender of participating teachers in the study was gathered and presented in a
pie chart as shown in Figure 4.1.
17%
83%
Male Female
Figure 4.1: Gender of Participating Teachers
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A total of six (6) mathematics teachers from two Senior High Schools in the Gomoa
East District participated in the study. Out of the six (6) participants, five (83.3%)
were males whilst the remaining one, representing 16.7% was a female.
4.1.1.2 Teaching Experience of Participating Teachers
Participants were asked to indicate their teaching experience in years. The results are
presented in Figure 4.2.
16.70%
50%
33.30%
up to 2 years 3 to 4 years more than 5 years
Figure 4.2: Participants’ Years of Teaching Experience
The results in Figure 4.2 showed that three (50%) of the participating teachers had
teaching experience above 5 years, two (33.3%) had teaching experience of 3 to 4
years whilst only one (16.7%) of the participating teachers had a teaching experience
up to 2 years.
4.1.1.3 Teaching Qualification of Participating Teachers
Participating teachers were asked to indicate their teaching qualifications. The
responses given by the participants were organized and presented in a bar graph as
shown in Figure 4.3
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80%
70% 67%
60%
50%
40% 33% 30%
20% Number of Teachers (%) of Teachers Number 10%
0% BEd BSc Type of Teaching Qualification
Figure 4.3: The Participants’ Teaching Qualification
A casual look at Figure 4.3 indicated that four (66.7%) of the teachers had a
Bachelor’s Degree in Education (BEd) whilst the remaining two teachers,
representing 33.3% had Bachelor of Science Degree (BSc).
4.1.2 Background Information of Participating Students
The number of students who participated in the control and experimental groups of
the study is presented in Table 4.1.
Table 4.1: Gender of Participating Students
Gender Control Group Experimental Group Total
N % N % N %
Females 29 41 26 37 55 39
Males 41 59 44 63 85 61
Total 70 100 70 100 140 100
Source: Field work, 2018
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From Table 4.1, one hundred and forty (140) students were engaged in the study with
70 students in the control group and 70 students in the experimental group. Of the 70
students in the control group, 29, which represents 41%, were females and 41, which
represents 59%, were males. Again, concerning the experimental group of 70
students, 26, which represents 37%, were females while 44 representing 63% were
males.
4.2 Data Analysis by Research Question
4.2.1 Research Question One: What teaching approaches do Senior High School
mathematics teachers often employed in their lessons?
To find answers to this question, data were gathered through the administration of
questionnaires, lesson observation and teacher interviews.
Teaching Approaches Mostly Preferred by Participants
The participating teachers were first asked to indicate the teaching approaches they
mostly used. Figure 4.4 summarises their response.
60.00%
50.00%
40.00%
30.00% 50.00%
PREFERENCE PREFERENCE (%) 20.00% 33.30% 10.00% 16.70%
0.00% 0.00% 0.00% Write and Talk Peer Tutoring Problem solving Discussion Discovery TEACHING APPROACH
Figure 4.4: Teaching Approaches Mostly Preferred by Participants
Figure 4.4 revealed that the participating teachers’ preference and usage for the
different teaching methods varied. Half (50.0%) of them used the problem solving
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approach, 33.3% also used the discussion method and 16.7% used the write and talk approach. It was also revealed that none of the participating teachers used the peer tutoring and the discovery methods.
The problem solving approach was their most used teaching approach according to the six (6) participating teachers whilst peer tutoring and discovery approaches were not used as teaching approaches by the teachers.
Below are some of the remarks made by the participating teachers during the interviews:
Interviewee 2 remarked that:
Problem-solving is my most preferred teaching approach because it easily supports teaching and learning even in the absence of teaching and learning resources.
Interviewee 5 also remarked that:
Diverse learning skills are acquired by students when they are taught through the problem-solving approach.
Interviewee 3 stated that:
My school does not have the available resources to support teaching and learning by discovery and investigation, therefore I prefer the write and talk method to the other approaches.
The finding that the problem-solving approach (a constructivist approach) was the most used teaching approach amongst the participating teachers is in line with the finding that recent education reforms require teachers to depart from the traditional practice of knowledge transmission to constructivist teaching where students are encouraged to construct knowledge through inquiry (Beck, Czerniak & Lumpe, 2000;
Levitt, 2002). Also, Brown (2005) suggested that the constructivist approach is
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accepted as the most relevant method of teaching and that educational policies, education models, and education practices should focus on constructivist teaching.
After the participants completed the questionnaire, observations were made to investigate the extent of selection and adoption of constructivist approaches in the various classrooms. The responses “Yes” or “No” were used to indicate the prevalence of the behaviours, skills and approaches that were observed and presented in Table 4.2.
Table 4.2: A Summary of Behaviours, Skills, and Approaches Exhibited by the Participating Teachers During Lesson Observations (풏 = ퟔ)
No. Behaviour(s) / Skills / Frequency (%) Approaches Observed Yes No 1. The teacher uses activities that make 2 (33.3%) 4 (66.7%) understanding a more central and reachable objective in his/her class.
2. The teacher uses verbs such as explain, find 2 (33.3%) 4 (66.7%) evidence, derive formulas, generalise, deduce, suggest, in the lesson objectives.
3. The teacher allows students to interact 4 (66.7%) 2 (33.3%) constructively with one another in building and integrating new knowledge from experiences.
4. The teacher teaches explicitly and allows 5 (83.3%) 1 (16.7%) students to make connections to meaningful contexts outside the classroom.
5. The teacher encourages students to think 3 (50%) 3 (50%) beyond what they know.
6. The teacher encourages students to learn by 1 (16.7%) 5 (83.3%) doing through enquiry, discovery, observations, reflection and demonstration.
7. The teacher encourages students to actively 2 (33.3%) 4 (66.7%) participate in their own learning process and involve learners in deciding the topics they learn and plan assessment program with them.
8. The teacher provides students with rich on- 2 (33.3%) 4 (66.7%) going assessments and feedback that foster
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understanding of mathematics.
9. Teacher allows the students to evaluate each 1 (16.7%) 5 (83.3%) other’s work.
10. The teacher puts the students’ needs into 2 (33.3%) 4 (66.7%) consideration. Results from Table 4.2 indicate that almost all the teachers (83.3%) teach explicitly and sometimes allow students to make connections to meaningful contexts outside the classroom. Four (66.7%) teachers were observed to have allowed their students to interact constructively with one another in building and integrating new knowledge from experiences. Half (50%) of the teachers encouraged their students to think beyond what they already knew. Two (33.3%) of the participating teachers were observed using activities to make understanding a reachable objective in their class as against four (66.7%) who did not use activities.
It was observed that only one (16.7%) teacher occasionally encouraged students to learn by doing, through enquiry and discovery. This observation buttressed the point that most of the teachers were used to the traditional approach which was predominantly teacher-centred. Added to this, only one of the teachers was also observed to have allowed the students to evaluate each other’s work.
Added to this, the results also indicated that participants were aware of constructivist principles but did not totally put most of these principles into practice. For example, while 66.7% of the teachers did not use activities which make understanding a more reachable objective in their lessons whilst 88.3% of them allowed their learners to make meaningful connection to the outside world.
Results from teacher interviews were also used to corroborate some of these facts and were not different from the data gathered from the questionnaires. The following were some of the interview responses:
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On the interview question: Do you always involve your learners in deciding the topics they learn and plan assessment program with them? Please explain your answer. This question was asked to solicit for further information from the respondents on item number seven (7) in Table 4.2.
Interviewee 4 explained that:
I don’t see any need to involve my learners in deciding the topics they learn because I need to select the topics from the syllabus upon which they (students) can be assessed in the West African Senior School Certificate Examination (WASSCE).
Interviewee 6 shared the same view and explained that:
My students lack knowledge on other forms of assessment. Therefore they look up to me (their teacher) to decide for them what they should learn.
However, interviewee 5 had a different view and explained that:
I involve my learners in planning what they learn based on the syllabus because it makes them develop interest in what they will be learning in the term. This even motivates them to read ahead and therefore facilitating a smooth lesson delivery.
The idea of involving learners in planning stages is very noble and necessary for knowledge construction by learners. The reason teachers may not involve their learners in the planning of learning and assessment activities could be that they themselves were not fully involved in the planning and development of the curriculum. Carl (2005) stated that when the teacher is given the opportunity to make his or her voice heard in the early stages of the curriculum planning instead of just coming in as an implementer, teaching and learning can be optimized. Learner involvement in planning lessons and assessment activities is very crucial in constructivist teaching because it gives students that sense of responsibility. For
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example, when they are involved in deciding the assessment type, the duration and dates for each assessment at the beginning of the term, they get involved and would want to understand the purpose for each assessment.
Since knowledge construction requires connection with prior knowledge, constructivist teaching draws on students’ prior knowledge and experiences (Driscoll,
2005). Moreover, constructivists believe that knowledge is constructed when an individual attaches meaning to an experience or activity (Merriam et al., 2007). Rich and authentic contexts need to be provided for students for them to link school learning with the world outside school (Jonassen, 1999). Therefore, the majority
(83.3%) of participants’ responses that they are able to link mathematics to the real world is consistent with these findings.
The interview question, asking whether teachers allowed their learners to use their own methods in solving problems yielded five out the six participating teachers indicating that they did so. For example,
Interviewee 1 indicated that:
To me, the right procedure in arriving at the answer is very key. I allow my students to use their own methods in solving problems because mathematics answers could be arrived at through different means.
A similar view was expressed by interviewee 3 who remarked that:
Though I allow my students to use their own methods in solving problems, most of the times, I insist they use the formula I give them since it helps in solving the mathematics questions very fast.
Interviewee 5 simply remarked that:
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Students always need formulas to help them solve mathematics questions. Therefore, I insist that they use the formula I give them.
From the above comments made by some of the interviewees, the indication that the participating teachers allowed their learners to use their own methods in solving problems is a good practice. However, participants’ insistence that the learners use formula given by them (teachers) to arrive at correct answers contradicts constructivist views. This is because, in constructivist classroom, the teacher is a guide, facilitator and co-explorer who encourages learners to question, challenge and formulate their own ideas, opinions and conclusions (Ciot, 2009; Cannelle & Reif,
1994; Ismat, 1998).
Fosnot (1996) suggests that the classroom should be a community of discourse engaged in activity, reflection and conversation. Learners are responsible for defending, proving, justifying, and communicating their ideas to the classroom community. Therefore the participating teachers’ decision to allow their learners to interact with each other, share ideas and help each other during mathematics instructions is in line with constructivist views.
Moreover, the finding that teachers allow their students to make connections to meaningful contexts outside the classroom is also consistent with earlier findings from the questionnaire. As stated earlier, Jonassen (1999) stated that rich and authentic contexts need to be provided for students for them to link school learning with the outside world.
Lastly, the finding that four (66.7%) of the participating teachers did not put their students’ needs into consideration does not deviate from responses from the
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questionnaire where it was found out that involving learners in the planning of learning and assessment programs was not a priority among the respondents. This is contrary to the views of Carl (2005) who stated that when the teacher is given the opportunity to make his or her voice heard in the early stages of the curriculum planning instead of just coming in as an implementer, teaching and learning can be optimized. It could be concluded from the above discussions that most of the senior high school teachers in the Gomoa East District embrace and have preference for the ideas for constructivism, they still employ the traditional teaching approach for instruction.
4.2.2 Research Question Two: What is the effect of the teaching approaches often used by teachers on students’ performance in Algebra?
Quantitative data was gathered through the achievement test administered to the students. Students’ performances as a result of the instructional approaches with respect to each of the conceptual area were determined. A descriptive analysis was carried out on the data obtained through the pre-test scores from the two groups with reference to the teaching approaches adopted by the teachers. The data were organized and presented in Table 4.3.
Table 4.3: Mean Percentage Scores for each of the Conceptual Areas in Algebra
Conceptual Area Traditional Approach Constructivist Approach Pre-test Post-test Pre-test Post-test Variable 39.5 57.1 41.0 86.9 Algebraic Expressions 40.6 57.0 42.2 79.6 Equations 47.4 53.3 49.0 78.7 Word Problem 23.7 26.2 24.3 64.2
Results from Table 4.3 clearly revealed that the students had a serious challenge with word problems before the intervention was carried out. Solving word problems was
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difficult for students as it involves many steps in the solution process, resulting in mean scores below 25% for all students during the pre-test. This finding is consistent with the findings from a study conducted by Kaur (1995) and Lee (2001). Kaur
(1995) indicated that students experienced problem-solving difficulties such as lack of comprehension of the problem posed, lack of strategy knowledge and inability to translate the word problem into mathematical statements. Most of the students exhibited these clear signs in their attempts to solve the algebraic word problems during the pre-test. Apart from the difficulties encountered by students when translating word problems into algebraic symbols, there were other barriers such as not understanding the equal sign as a relationship and lack of comprehension of the problem posed in translating the problem into a mathematical statement.
Mean pre-test scores of 40.6% and 42.2% obtained by students who were taught through traditional and constructivist approaches respectively under algebraic expressions also revealed that many students had problems interpreting mathematical expressions or statements and simplifying them. Earlier research provided evidence that simplification of algebraic expressions creates serious difficulties for many students (Linchevski & Herscovics 1996). Students experience serious problems in grouping or combining like terms. They did not demonstrate understanding of expressions as a process and as a product. Expressions encapsulate a process as instructions to calculate a numerical value, but they are also a product as objects which can be manipulated in their own right (French, 2002). The traditional approach adopted by the teachers could not offer a lasting solution to address such misconceptions of students.
Students’ performance in equations were comparatively better compared to the other conceptual areas. Questions on equations were relatively easier for the students
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compared to variables and expressions. The questions on equations were mainly on solving simple or linear systems of equations in which students had to use a particular algorithm. They had to use elimination, substitution, or working backward methods that have a prescribed procedure. One exception was question 15 where they had to decide the answers with or without carrying out the algorithm.
The questions under “variables” were easier and solving many of them did not need to follow algorithms. Some of these problems did not require deeper thinking strategies. The questions were mostly about students’ knowledge of basic definitions.
When students misconstrue or misuse the standard mathematical practices, then we say that an error has occurred. At this stage, we only can hypothesize whether this error had really occurred due to a robust misconception or it simply was a momentarily lapse of concentration on the part of the student. Sometimes, errors seem to point towards having all the features of a misconception. However, this is a hypothesis which can be tested by listening to students. For example, some errors disappeared when students were asked to work out the problems again in the interviews. Therefore, it could be assumed that these errors may have occurred due to a simple lapse of concentration, forgetfulness, or any other reason other than due to a deeply rooted misconceptions.
This outcome is an indication that the teaching approaches adopted by the participating teachers did not help improve the performance of the students in learning algebra.
4.2.3 Research Question Three: What is the effect of the constructivist teaching approach on students’ achievement in Algebra?
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This research question sought to examine whether there were any significant differences in achievement between students taught by the constructivist approach and those taught by the traditional approach. The null hypothesis formulated for this question was:
Null Hypothesis (푯푶): There is no significant difference in achievement between students taught by the constructivist approach and those taught by the traditional approach?
Data from pre-test and post-test scores of students from the Algebra Concept
Achievement Test were used to answer the research question. The descriptive statistics obtained from the achievement test of the control and experimental groups are presented in Table 4.4.
Table 4.4: Descriptive Statistics of Control and Experimental Groups
Experimental Group Control Group Statistic Pre-test Post-test Pre-test Post-test
Mean 39.37 78.51 38.09 47.65
Standard Deviation 9.78 9.25 18.86 18.79 n 70 70 70 70
Source: Analysis of Field Data (2018)
A cursory look at Table 4.4 shows that the mean pretest scores of the experimental group was 39.4 with standard deviation (SD) 9.8 while the post-test mean score was
78.5 with SD of 19.25. These results suggest that there was a gain mean score in the experimental group as a result of the instruction that was implemented in this group.
A comparable gain which was less than that of the experimental group was observed in the control group.
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For the control group, the mean pre-test scores was 38.1 with SD 18.9 and the mean
of the post-test score was 47.7 with SD 18.8. This observation also suggests that there
was an increase in learners’ algebraic achievement in the control group. However,
the observed increase in the control group was comparatively less than the gain score
that was observed in the experiment group.
An inferential statistic was then carried out to determine whether the observed mean
differences were statistically significant. Two assumptions were tested using the
independent sample t-test to determine whether the two groups were equivalently
positioned prior to the commencement of the experiment. This test was conducted as a
formal test for the pre-test of the two groups since they were unrelated. The results are
shown in Table 4.5.
Table 4.5: Independent Samples Test of Pretest Scores
Levene's Test for Equality of t-test for Equality of Means Variances Mean 95% confident Level Sig. (2- F Sig. t df Differe SED tailed) nce Lower Upper Equal var 10.82 .001 -.338 138 .736 -1.27 3.76 -8.71 6.16 assumed Equal var Pre not -.338 127.86 .736 -1.27 3.76 -8.71 6.17 test assumed Equal var - not -2.307 128.03 .023 3.74 -16.03 -1.23 8.63429 assumed
Source: Analysis of Field Data (2018)
From Table 4.5, the homogeneity of variance as assessed by Levene’s test for
equality of variances provided a p-value of 0.001, which was less than 5% and hence
was interpreted to be significant. This result suggested that the assumption of
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homogeneity of variance was not met. Hence a non-parametric test (Mann-Whitney
U Test) was then conducted to correct this anomaly. Results from the Mann-
Whitney’s U test revealed that the pre-test scores for the experimental group, which was observed to be 39.37 (see Appendix 10) was not significantly higher than the control group with a mean score of 38.09, 푝 = 0.907 with a mean difference of 1.28.
Since the p value was greater than 0.05 it was therefore reasonable to accept the null hypothesis and reject the alternate hypothesis. This meant that there was no significant difference between the means of the pre-test scores of both the experimental and control schools. Hence the two groups (experimental and control) were considered to be equivalently positioned prior to the commencement of the experiment. In this context, the two groups were considered to be equivalent in terms of the tendency of participants to perform poorly when they attempted to solve Form two algebraic tasks in the classroom.
The normal Q-Q plot of post-test-pre-test scores was used in testing whether there was a normal distribution between the test scores in pre-test and post-test of the two groups. The results of the normal Q-Q (see Appendix 11) indicated that the pre-test and post-test scores of the control and experimental groups were approximately normally distributed.
In addition, a formal normality test was performed by using Kolmogorov-Smirnov and Shapiro-Wilk and has been reported in Table 4.6.
Table 4.6: Tests of Normality for Experimental and Control Groups Kolmogorov-Smirnova Shapiro-Wilk Test Group Statistic df Sig. Statistic df Sig. Pretest Control .145 70 .001 .916 70 .000
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Experimental .150 70 .001 .925 70 .000
Control .111 70 .033 .941 70 .003 Posttest Experimental .123 70 .010 .917 70 .000
Source: Field Data (2018) a. Lilliefors Significance Correction
The results from the Shapiro-Wilk test (p< 0.05) (Shapiro & Walk, 1965; Razali &
Wahl, 2011) as reported in Table 12 and a visual inspection of their pictograms, normal Q-Q plots (See Appendix 11) showed that the test scores were approximately normally distributed for both pre-test and post-test scores for both control and experimental group with a skewness of 0.360 (푆퐸 = 0.287) and a kurtosis of
−1.284 (푆퐸 = 0.566) for the pretest score of the control group. The experimental group also recorded a skewness of 0.368 (푆퐸 = 0.287) and a kurtosis of
−1.153 (푆퐸 = 0.566). For the post-test scores, the control group recorded a skewness of 0.190 (푆퐸 = 0.287) and a kurtosis of −1.19 (푆퐸 = 0.566) whilst the experimental group also recorded a skewness of −0.043 (푆퐸 = 0.287) and a kurtosis of −1.499 (푆퐸 = 0.566). All the z-scores are within ±1.96 indicating that the scores were normally distributed. Hence the assumption of normality was satisfied.
The null hypothesis for normality test was that the difference in the post-test and pre- test scores would not be normally distributed while the alternate asserted that the difference in the post-test and pre-test scores would be normally distributed. Since the p-values for both Kolmogorov-Smirnov and Shapiro-Wilk were less than 0.05 the null hypothesis was therefore rejected, meaning the difference of the post-test and pre-test scores were normally distributed. These p-values were less than 0.05, meaning that the residuals were normally distributed.
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To ascertain whether or not the difference observed in the means were statistically significant, a paired samples t-test was conducted to test the null hypothesis that there was no significant difference between the pre-test and post-test scores of students in the experimental and control groups. Table 4.7 presents the results of the paired samples t-test on the pre-test and post-test achievement of the control and experimental group respectively.
Table 4.7: Results of the Paired Samples T –Test on the Pre-Test and Post-Test Scores of Students in the Experimental and Control Groups
Std. Error Group Test Mean Difference Std. Dev. Mean t df Sig.
Experimental Pre-test 39.10 17.01 0.84 20.19 69 0.000 Post-test
Control Pre-test 9.56 8.06 0.60 15.45 69 0.000 Post-test
Source: Analysis of Field Data, 2018
From Table 4.7, the paired sample t-test results showed the mean score difference, M
= 39.10, SD = 17.01 and p-value = 0.000 between the post test and pre-test as statistically significant for the experimental group. The results indicated a statistically significant increase in the students’ achievement scores from the pre-test mean score of 39.37 with SD of 9.78 to the post-test mean score of 78.51 with SD of 9.25, t (69)
= 20.2, p < 0.05. The effect size, measured by Cohen’s d was found to be 2.56 with a correlation coefficient of 0.79 indicating a large effect size (Cohen, 1988).
This showed a very large effect on students’ achievement in algebra using the constructivist approach. Also, the results implied that after the students had gone
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through the intervention, they improved in their understanding and achievement of the concepts in algebra. Thus, constructivist teaching approach had a positive impact on the students’ achievement in solving algebraic problems.
Similarly, a paired samples t‐test was employed to compare the pre-test and post test scores for the students taught with traditional teaching approach (control group). The paired sample t-test was examined to find out if the mean score difference of 9.56 with SD of 8.06 between the post-test and pre-test of the control group was statistically significant. This was done to assess the effect of traditional method on students’ achievement in learning algebra. The results from Table 4.7 indicated that there was also an increase in the students’ achievement from the pre-test mean score of 38.09 with SD of 18.86 to the post-test mean score of 47.65 with SD 18.79), t (69)
= 15.45, p = 0.000 < 0.05. To determine whether the increase was significant, the effect size, measured by Cohen’s d was found to be 0.51 with a correlation coefficient of 0.25 indicating a low effect size (Cohen, 1988).
This indicated that the effect size, 0.79 for the experimental group was found to be larger than that of the control group recording an effect size of 0.25. From this result, it could be deduced that students also gained slightly from traditional teaching approach of learning algebraic concepts. This outcome is an indication that a well- structured traditional approach of teaching can also improve students’ performance in learning algebra.
Based on the above findings, the claim that ‘There is no difference in achievement between students who were taught using the traditional approach and students who were taught using the constructivist approach” is rejected.
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We therefore conclude that, ‘There is a significant difference in achievement between students who were taught using the traditional approach and students who were taught using the constructivist approach”.
According to Hartshell, Herron, Fang, and Rathod (2009), traditional or teacher- centred instructional methods fail to prepare students to attain high achievement levels in mathematics. The researcher observed from the scripts that were marked that, after students have been taught using the traditional approach, most of them were able to state formulae for solving the questions correctly but did not have an understanding of how to apply the stated formulae correctly to find answers to the questions; thus, they engaged in rote memorization of the formulae and rules (Wei &
Eisenhart, 2011). Fletcher (2008) states that the traditional method of instruction discourages learners from engaging in higher algebraic thinking in mathematics. On the contrary, the researcher observed that after students have been taught through the constructivist approach, students quickly developed an in-depth understanding of the concept and therefore found it easier to use it to find answers to the questions.
D’costa (2010) opines that constructivist teaching fosters critical thinking and creates motivated and independent learners. The finding above does not deviate from the study findings of other researchers like Christopher and Merek (2009), Ryan (2006) and Yoder and Hochevar (2005) who have also presented evidence that students’ exam scores are higher when taught with constructivist approaches than when taught with traditional approaches.
4.2.4 Research Question Four: What are students’ views about the constructivists teaching approach used to improve their performance in
Algebra?
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This research question sought to find out students’ views about the constructivist teaching method. A twenty-item questionnaire and structured interview questions were used to solicit the views of the students.
4.2.4.1 Questionnaires Administered to the Students Seventy (70) copies of the questionnaire were distributed to the students taught with the constructivist approach. Sixty-eight (68) out of the seventy (70) copies were completed by the students and were used for the analysis and the results shown in
Table 4.8.
Table 4.8: Responses of Participating Students on the Constructivist Teaching Approach
1 4 2 3 Strongly Strongly Working in pairs and groups…. Agree Disagree Agree Disagree 1 helped 20 40 5 3 understanding/comprehension (29.4%) (58.8%) (7.4%) (4.4%) 2 fostered exchange of knowledge, 31 32 3 2 information and experiences (45.6%) (47.1%) (4.4%) (2.9%) 3 20 31 15 2 made problem-solving easier (29.4%) (45.6%) (22.1%) (2.9%) 4 23 28 12 5 stimulated critical thinking (33.8%) (41.2%) (17.6%) (7.4%) 5 promoted a more relaxed 21 29 14 4 atmosphere (30.9%) (42.6%) (20.6%) (5.9%) 6 18 32 14 4 received useful/helpful feedback (26.5%) (47.1%) (20.6%) (5.9%) 7 16 33 15 4 got fresh insight (23.5%) (48.5%) (22.1%) (5.9%) 8 focused on collective efforts rather 19 36 8 5 than individual effort (27.9%) (52.9%) (11.8%) (7.4%) 9 greater responsibility – for myself 24 34 7 3 and the group (39.7%) (50.0%) (10.3%) (4.4%) 10 enabled learners to help weaker 36 25 3 4 learners in the group (52.9%) (36.8%) (4.4%) (5.9%) 11 26 27 13 2 enhanced communication skills (38.2%) (39.7%) (19.1%) (2.9%) 12 16 36 10 6 improved performance (23.5%) (52.9%) (14.7%) (8.8%) 13 made learners actively participated 23 28 13 4 in the teaching/learning process (33.8%) (41.2%) (19.1%) (5.9%) 14 20 28 13 7 was fun (29.4%) (41.2%) (19.1%) (10.3%) 15 made new friends 16 20 21 11
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(23.5%) (29.4%) (30.9%) (16.2%) 16 20 29 8 11 fostered team spirit (29.4%) (42.6%) (11.8%) (16.2%) 17 wasted time explaining things to 14 15 23 16 others (20.6%) (22.1%) (33.8%) (23.5%) 18 difficult getting members to 19 24 12 13 actively participate in tasks (27.9%) (35.3%) (17.6%) (19.1%) 19 39 23 3 3 should be encouraged/continued (57.4%) (33.8%) (4.4%) (4.4%) 20 maximum group size should be 40 17 6 5 four (58.8%) (25.0%) (8.8%) (7.4%)
Source: Field Data (2018) Percentages are indicated in brackets
Considering the combined responses of both agree and disagree responses from the results of Table 4.8, the results revealed that, sixty-three (92.7%) of respondents agreed that the constructivist approach helped fostered exchange of knowledge, information and experiences. Out of the number, Sixty-two (91.2%) were of the view that working in pairs or groups should be encouraged and continued whilst sixty
(92.6%) believed that the approach brought greater responsibilities to themselves and enabled learners to help weaker learners in the group. Sixty (88.2%) of the respondents believed that working in pairs and groups helped with understanding and comprehension. Fifty-one (75.0%) of the respondents also noted that the approach made learners to actively participate in the teaching and learning process. This finding does not deviate from the study of researchers like Bonwell and Eison (1991) who stated that collaborative learning is a strategy that involves students in doing things and thinking about the things they do. They opined that the approach emphasizes the active participation of learners. Concerning the maximum number of group size for collaborative learning, fifty-seven (83.8%) of the respondents were of the view that the maximum number of students in a group should be four. Just as experts differ on the make-up of groups, they also debate about the most effective size for small groups. Slavin (1987) claims that having two or three members per
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group produces higher achievement than groups with 4 or more members. Antil et al
(1997) concluded that most teachers who use CL prefer pairs and small groups of three and four. Unquestionably, there seems to be a trend towards a maximum of four students in a group. Elbaum et al (1997) suggest that we have dialogues with students about their preferences for group composition and expected outcomes. This is very important if we actually are adopting a learner-centred approach.
It was also obvious from the responses of the participants that some agreed to the negative aspects of collaborative learning. Forty-three (63.2%) of the respondents were of the opinion that collaborative learning made it difficult getting members to actively participate in tasks whilst twenty-nine (42.7%) of the respondents agreed that the approach wasted a lot of time explaining concepts to others. These findings corroborated some of the ideas by some critics of collaborative learning like Randall
(1999) who cautioned against abuse and overuse of group work. She identified one of its weakness as that of placing too much burden on some students. She argued that in mixed-ability groups, the result is often that stronger students are left to teach weaker students and do most of the work. Other critics claim that CL is detrimental to students who benefit more from learning alone. Yet others recommend that we negotiate more with students to determine how they learn best and apply these ideas to the way we structure classes. These are valid points that every committed teacher who uses CL must consider and address accordingly.
For clarity of analysis, the items in the questionnaire were grouped into four themes:
1. academic benefits, 2. social benefits, 3. generic/ life-long learning skills, and negative aspects of collaborative learning. Items 1,2,6,7,10,12,13,19 and 20 represented academic benefits; items 5, 14, and 15 as social benefits; items 3,4,8,9,11,
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and 16 were generic skills; and items 17 and 18 represent negative aspects of collaborative learning. Students’ responses according to the four themes have been presented in Figure 4.5.
AGREE AND DISAGREE RESPONSES BY CATEGORIES IN PERCENTAGE
AGREE DISAGREE % % 82.50 77.70 % 65.70 % % 52.90 47.10 % 34.30 % % 22.30 17.50
ACADEMIC BENEFITS SOCIAL BENEFITS GENERIC SKILLS NEGATIVE ASPECTS
Figure 4.5: Agree and Disagree Responses by Categories in Percentage
From Figure 4.5, it could clearly be seen that 82.5% agreed that collaborative learning has academic benefit. That is, the approach helped students to understand the concepts; fostered exchange of knowledge, information and experiences; enabled learners to help weaker learners in the group; improved performance and made learners participate actively in the teaching and learning process. 77.7% agreed that collaborative learning enabled students acquire generic skills such as making problem solving easier, stimulating critical thinking, helped students to focus on collective efforts rather than individual efforts, enhanced communication skills and fostered team spirit. This is in line with research works conducted by proponents of collaborative learning who claimed that active exchange of ideas within small groups increases interest among participants and promotes critical thinking (Gokhale, 1995).
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Furthermore, Johnson & Johnson (1986) state that there is evidence that co-operative teams achieve higher levels of thought and retain information longer than students who work individually. According to Totten, Sills, Digby, and Russ (1991) shared learning gives students an opportunity to engage in discussion, take responsibility for their own learning and so become critical thinkers. Thus, research suggests that collaborative learning brings positive results such as deeper understanding of context, critical thinking, increased overall achievement in grades, improved self-esteem, and higher motivation to remain on task, more opportunities for personal feedback, celebration of diversity, group conflicts resolution and improved teamwork and social skills (Concept to Classroom, 2004).
Concerning social benefits, 65.7% of the respondents agreed that the approach promoted a more relaxed atmosphere, helped students to interact and make new friends and was fun to use collaborative learning. Social constructivists claim that for knowledge to be internalized and a framework established, a social communication must first take place. It is this discourse that leads to the conceptual framework in which to relate the new knowledge (Bruffee, 1992). As MacGregor states,
“Knowledge is shaped, over time, by successive conversations, and by ever-changing social and political environments” (MacGregor, 1990 p.96).
Quite a number of respondents agreed to the negative aspects of collaborative learning. It should be pointed out that, in terms of the negative aspects, 57.4% of the respondents disagreed that it is a waste of time explaining things to others whilst
63.2% agreed that it is difficult getting members to actively participate in tasks. Thus, the latter largely accounts for why almost half of the students (52.9%) agree regarding the negative aspects of CL.
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It could be concluded that many students (82.5%) believed that CL has academic benefits whilst 77.7% of the respondents believed the approach helped them to acquire generic skills. About two-thirds of the respondents (65.7%) agreed that CL has social benefits and only less than half of the students (47.1%) disagree regarding the negative aspects of CL.
In order to corroborate the findings from the questionnaire, some students were interviewed to solicit their views on collaborative learning. There were five (5) items on the interview guide which focused on students’ experiences and opinions of the use of constructivist approach and reflected their views about their participation in the lesson. Some of the responses made by the students during the interviews are reported below:
The approach gives me more understanding (Interviewee 3).
This comment corroborate with the findings of Blunck and Yager (1990), which found out that students in classes taught with a constructivist approach improved more in their understanding of the nature of science when compared to students in classes taught with a textbook oriented approach.
Working in small groups is a good way to learn and quite exciting. It is more interesting than lecture method (Interviewee 5).
This response suggested that, when teachers actively engage students in the learning process, they learn better and constructively.
The approach makes lessons more practical. It has helped me to think critically and has improved my oral presentation skills; even though sometimes there are conflicts within the group. It also helps in learning to work with others hence helps in good interpersonal skills (Interviewee 6).
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The view from the student suggested that, when mathematics is taught in a more practical way, it becomes easier to understand every concept. In this regard, Baviskar,
Hartle & Whitney (2009) opined that constructivist learning environment need enough resources which are needed for practical work to enhance students’ learning.
Other responses from some of the interviewees were:
I think it should be continued because it encourages team work and exchange of ideas. It gets us talking and we get to practice presentation before class presentation (Interviewee 2).
I feel it is okay but sometimes working in a group leads to people becoming lazy, as they feel the rest of the group will work because the mark is shared. (Interviewee 1).
To me it was not fun. Some students chose their friends and behaved like their group was better than others. (Interviewee 4).
The findings in this study indicated that just over half of the respondents found the constructivist approach enjoyable. This implied that almost half of the students found the class boring. The question now was, how then could we make constructivist learning activities more fun? This is a question every practitioner of constructivist learning should answer. In answering this question, we must ensure that we set objectives that take care of this factor and get input from our students. Course materials should incorporate some fun activities in order to make lessons livelier.
4.3 Summary of Statistical Findings
This section summarises the statistical findings from the statistical analyses.
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4.3.1 Participation of the Study
The data analysis that was presented in this report covered the one hundred and forty
(140) participating students. That is, Experimental Group (EG = 70); and Control
Group (CG = 70) who participated fully in the study. In total, 140 students representing 86.4% students and six (6) teachers participated fully in the study.
4.3.2 Descriptive Statistics
The means and standard deviations were used as the descriptive statistics to analyze the data for the study. On percentage improvement in the achievement test, it was discovered that the mean performance of learners in the experimental group improved from 39.4% in the pre-test to 78.5% in the post-test representing over 90.0% increase in scores whiles that of the control group improved from 38.1% to 47.7% representing 25.2% increase. This meant that Constructivist teaching was found to be better than the Traditional teaching in improving the performance of learners in the achievement tests.
4.3.3 Inferential Statistics
Inferential statistics used to test the two assumptions of the study were: the non- parametric Mann-Whitney U Test; and Kolmogorov-Smirnov and Shapiro-Wilk test for the assumption of normality of test scores. Results from the Mann-Whitney’s U test revealed that the pre-test scores for the experimental group, which was observed to be 39.37 ± 25.20, was not significantly higher than the control group (38.09 ±
18.86), 푝 = 0.907 with a mean difference of 1.28. Since the p value was greater than
0.05 it was therefore reasonable to accept the null hypothesis and reject the alternate hypothesis. This meant that there was no significant difference between the means of the pre-test scores of both the experimental and control schools. Hence the two
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schools (experimental and control) were considered to be equivalently positioned prior to the commencement of the experiment.
The formal normality test by Kolmogorov-Smirnov and Shapiro-Wilk performed yielded a p-value less than 0.05 respectively. Hence, the null hypothesis was therefore rejected, meaning the difference of the post-test and pre-test scores were normally distributed. The paired samples t-test performed corroborated Kolmogorov-Smirnov and Shapiro-Wilk’s result. The result showed that the assumption of normality of test scores was not violated and hence the post-test-pretest scores in the experimental and control group were normally distributed.
On hypothesis testing, the paired samples t-test was used to test the hypothesis of the study. It was found that there was statistically significant improvement in achievement scores of the experimental group from 39.37 to 78.51 (푝 = 0.000 <
0.05) with an effect size of 0.79 compared with that of the control group with and effect size of 0.25. Since the p-value was found to be less than 0.05, it was reasonable to reject H0 of no effect in favour of H1. It was therefore concluded that the constructivist teaching approach was effective in improving learner’s achievement in algebra.
4.4 Chapter Summary
This chapter presented, analysed and discussed the data collected from the achievement tests, lesson observations and the questionnaire administered to both teachers and students. The descriptive statistics (mean and standard deviation) and inferential statistics (independent samples t-test and paired samples t-test) were employed as statistical techniques with the help of the use of IBM SPSS software to
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analyse the data. From the analysis of the results, the study revealed that learners who received the constructivist teaching intervention significantly improved in the four algebraic concepts than learners who were taught through the traditional teaching approach.
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CHAPTER FIVE SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
5.1 Overview
This study was motivated by the desire to search for pedagogical solution to the
perennial trend of poor performance of learners in Mathematics in Ghana. Poor
performance in this study was linked to the algebraic errors that learners do in a
second year mathematics classroom. This study created an opportunity for the
researcher to investigate the effectiveness of a teaching method in improving learners’
performance in Mathematics by reducing the errors they commit in algebra. The study
sought to find out the comparative impact of teaching that is based on constructivist
learning theory, and the traditional teaching approach on secondary school learners’
achievement in algebra.
First and foremost, in this chapter the purpose, objectives and research questions set
for the study were revisited. Afterwards, a summary of the key findings of the study
were also outlined. Recommendations regarding future research in this study area
were also identified. Lastly, the conclusions to the study were discussed.
5.2 Revisiting the Purpose, the Objectives and Research Questions of the Study
The main purpose of this study was to investigate the comparative effects of a
constructivist teaching approach (CTA) and the traditional teaching methods (TTM)
on form two learners’ achievement in algebra. The participants of the study were
drawn from two senior high schools in the Gomoa East district, which were
considered to be poorly performing in form two Mathematics as a result of the
numerous errors the learner’s did in performing some algebraic tasks. The objectives
of the current study were to:
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1. Identify the teaching approaches used by Senior High School mathematics
teachers in their lessons.
2. Investigate the effects of the teaching approaches often employed by the
teachers on students’ performance in algebra.
3. Investigate the effect of the constructivist teaching approach on students’
achievement in algebra.
4. Find out students’ perceptions of the constructivist teaching approach on their
performances in Algebra?
It was possible to conclude that the aim of the current study and its associated objectives have all been achieved (see analysis and discussions in Section 4.2). It is the researcher’s view that this research has substantially provided evidence to support the view that: (1) the constructivist teaching approach has a greater potential to enhance the reduction of learners’ errors in algebra and also improve the performance of learners in Mathematics when it is compared with the traditional teaching approach; and, (2) the study can serve as a useful point of reference for those who are attempting to improve the teaching and learning of Mathematics in secondary schools, particularly in Ghana.
The current study explored the following research questions:
1. What teaching approaches do Senior High School mathematics teachers often
employ in their lessons?
2. What is the effect of the teaching approaches often used by teachers on
students’ performance in Algebra?
3. What is the effect of the constructivist teaching approach on students’
achievement in Algebra?
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4. What are students’ perceptions of the constructivist teaching approach on their
performances in Algebra?
The analysis and discussions done in Chapter four provided evidence to support the
notion that the research questions of the current study have been answered, and that
the objectives of the study have been achieved.
5.3 Summary of Key Findings
A summary of the key findings from the four main research questions employed in the
study is presented below:
1. Results from lesson observations and interviews revealed that the “write and talk”
method of teaching was the most often used teaching approach. Participants
explained that though they had preferences for constructivist approaches, factors
such as unavailable teaching and learning resources, inadequate classroom
infrastructure and time allocation worked negatively against them in their attempt to
employ constructivist approaches for classroom instruction. Participants failed to
adopt activities that made understanding a more reachable goal in their classrooms.
Moreover, only a few of the participants adopted constructivist principles that
encouraged their learners to learn by doing through enquiry, discovery,
observations, reflection and demonstration.
2. Students had serious challenges with word problems. Results indicated mean scores
below 25% for all students during the pre-test. However, students performed better
during the post-test with a mean percentage score of 64.20% in word problem when
the intervention was carried out. Students’ performance in equations were
comparatively better compared to the other conceptual areas.
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3. The test results revealed that there was a significant difference in the mean
achievement score between students who were taught using the traditional teaching
approach and those taught using the constructivist teaching approach. The post-test
mean achievement score for students in the experimental group was greater than
that of students in the control group. The effect size of 2.56 with a correlation
coefficient of 0.79 showed a significant effect in achievement scores for the
experimental group. The finding above did not deviate from the study findings of
other researchers like Christopher and Merek (2009), Ryan (2006) and Yoder and
Hochevar (2005) who have also presented evidence that students’ exam scores are
higher when taught with constructivist approaches than when taught with traditional
approaches.
4. The study revealed a slight gain in performance for students taught using the
traditional teaching approach. This outcome could be an indication that a well-
structured traditional approach of teaching could also improve students’
performance.
5. Results from the data gathered through students perception of collaborative learning
(CL) revealed that many students (82.5%) believe that CL has academic benefits
whilst 77.7% agreed that CL enables students acquire generic skills. Some of the
students agreed that CL helps in the acquisition of lifelong learning skills and has
academic benefits than they do social skills. Quite a number of respondents agreed
about the negative aspects of CL.
5.4 Achieving the Purpose of the Study
The purpose of the current study was re-stated in Section 5.2 (see also, Section 1.3).
The design used for the study made it possible to conduct the investigation in which
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two types of instructional approaches were investigated in terms of students’ achievement during algebra lessons. It also made it possible to create two comparative learning environments to conduct the investigation that would enable the achievement of the study aim. The results of the study, which have been discussed in the preceding sections, suggested that the aim of this study has been achieved.
5.5 Linking the Study Results to the Theoretical Framework of the Study
The theoretical framework of the current study were presented in Section 2.1.
Evidence from constructivist research studies indicate that instructions based on constructivist learning theory are preferred (Johri, 2005). Tellez (2007: p.553) found that “the importance of constructivism in educational theory and research cannot be underestimated”. Studies by Phillips (1995), Cobb (1996), Fox (2001) and Dangel
(2011) support constructivist approach in science-related disciplines. Traditional teaching methods are becoming less tenable to stimulate conceptual understanding as they have ignored the fact that the knowledge, which the learners discover by themselves, is more enduring than the knowledge transmitted to them by the teacher or someone else. Constructivism recognizes that learning is a cognitive process involving construction and reconstruction of ideas.
As a learning theory, constructivism recognizes the learner as a meaning maker rather than a passive recipient of factual knowledge and conceived learning as process where meaning is modified on the grounds of evidence. Fundamentally, the constructivist approach to teaching recognizes the social interaction in the teaching and learning process. Empirical studies conducted by Tellez (2007), Phillips (1995), Cobb (1996),
Fox (2001), Dangel (2011), Guthrie et al. (2004), Kim (2005), Doğru and Kalender
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(2007), and Bhutto (2013), which are reviewed in Section 2.5 indicated that constructivist teaching methods have more positive effect on learners’ performance in
Mathematics and Science than traditional teaching methods. Looking closely at the findings of previous empirical studies side by side with the findings of the current study, there is credible evidence that learners’ achievement in algebraic concepts can be modified by using constructivist teaching approach as an effective method of teaching.
Although each of the empirical studies reviewed in Section 2.5 implemented a different method of constructivist teaching in comparison with traditional method, their results indicated that the learners who received constructivist instruction showed significant gain on their academic achievements than those who received traditional instruction. It was also found that in situations where no significant difference was found between the achievement of the constructivist group and traditional group, it was discerned from qualitative evidence that the learners and teachers who applied the constructivist methods showed preference to the constructivist approach over the traditional approach. The results of the statistical tests for this research indicated a significant achievement in scores between the learners who received constructivist- based instruction than the learners who received traditional instruction in the four conceptual areas in algebra. This corroborated most of the research findings reviewed in literature under Section 2.5.
5.6 Conclusions
As indicated in the earlier chapters of the study, various scholars have stated that constructivist approaches to teaching are more effective in preparing students to attain
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high achievement than the traditional approaches because they allow learners to actively and creatively construct knowledge. It was discovered in this study that though most senior high school mathematics teachers in the Gomoa East District embrace and have preference for the ideas of constructivism, they still employ the traditional “write and talk” approach for instruction.
The study further revealed that after students have been taught through the constructivist approach, students quickly developed an in-depth understanding of the concept and therefore found it easier to use it to find answers to the questions. This corroborated the findings from D’costa (2010), who opines that constructivist teaching fosters critical thinking and creates motivated and independent learners.
Added to this, after a careful review of the data from the questionnaire soliciting students’ perception about collaborative learning, it was clear that collaborative learning definitely enhances learning in several ways. Students in this study acknowledged the many benefits they derived from the practice to include academic, social, among others. The findings indicated that students’ responses were similar to stated literature that collaborative learning facilitates the acquisition of academic, social, and generic skills (Gokhale 1995; Totten et al 1991; Ingleton 2000; Radencich and McKay 1995; Slavin 1987; Antil et al 1997).
However, teachers failure to adopt and select constructivist approaches for classroom instruction was based on several factors including unavailable teaching and learning resources, inadequate classroom infrastructure and time allocated for instruction. A critical look at these factors will put teachers, learners and other stakeholders in the field of education in a better position to make the construction of mathematical
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knowledge a reality. Finally, the vital role of algebra cannot be underestimated since it covers most of the topics in mathematics and other subjects.
5.7 General Recommendations
In the light of the findings of this study, the researcher recommended the following teaching and learning practices:
1. Mathematics teachers should provide to the learners’ ample opportunities to
discover and construct their own knowledge rather than the learners absorbing
the teachers’ own ideas. It is important for teachers to note that all knowledge
emanates as a hypothetical construction. No individual constructs knowledge
for another. The knowledge that the learner constructs by himself is more
meaningful than that the one that is transmitted to him by the teacher or
someone else;
2. The teacher’s role should be that of an instructor, guide and facilitator.
Teachers should always explain the purpose and usefulness of a task before
students carry out the task. This will arouse the learners’ interest;
3. Mathematics teaching should aim at encouraging a collaborative learning
approach and constructive mathematical discourse in their classroom
instruction;
4. It is important to pay close attention to group dynamics and maintain both
general and focused observations as the groups work. To get groups to work
productively, the teacher should appoint a group leader who will coordinate
group activities and a secretary who will record transactions. A group leader
could perform both tasks in very small groups. Roles could be rotated among
group members if the group is to work together for some time. Teachers could
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review each group’s progress by checking the group’s record of activities,
monitor participation and progress and intervene when the need arises;
5. Group and individual performances should be made components of the final
assessment. This will motivate students to actively participate in learning.
Although uniform marks are generally given in group work, students should
be alerted to the fact that lack of active participation in group project could
lead to the award of lower mark for the individual. An awareness of this will
increase the level of participation by members;
6. Furthermore, it is important that teachers regularly obtain feedback from
students on various aspects of the teaching-learning process. In fact, learner
feedback is vital to constructivist-based teaching and learning. Practitioners
should get formal and informal feedback from their students as often as is
feasible;
7. It is not enough to receive feedback from our students; we must act on the
feedback in order to sustain increased learning outcomes. Collaborative
Learning takes time to be accepted by both students and staff. It needs to be
carefully explained, structured, and the students well-prepared;
8. This research confirms previous findings that collaborative learning has many
benefits, such as improved learning skills, as well as some negative aspects,
such as difficulty in getting some students to participate. The negative aspects
can be successfully be dealt with. Although, there is no ‘the perfect
methodology’, collaborative learning is one instructional method that
significantly facilitates the acquisition of academic, social and generic skills;
9. Teacher educators for mathematics should organize and create awareness
among other teachers that the traditional instruction is becoming less and less
relevant to achieving the goals of mathematics education in this modern
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dispensation. Mathematics teachers should be encouraged to teach
mathematics linking theories of constructivism with classroom practice;
10. Teacher training institutions should also place emphasis on constructivist
theories in order to ensure that pre-service teachers have acquired the pre-
requisite skills to effectively teach their learners within the domains of
constructivism by the time they get into classrooms.
11. Lastly, the government should make every effort to provide schools with the
necessary resources to facilitate constructivist teaching and learning.
Availability of adequate teaching materials like textbooks, computers, graph
boards. Classroom space and other infrastructure will encourage or motivate
teachers to select and adopt constructivist approaches for classroom
instruction.
5.8 Suggestion for Future Research
First, it is recommended that further studies be undertaken on the same subject covering the privately owned schools and other public schools in the District which were not included in this study and the results compared with the findings of this research.
The study also investigated the effect of using the traditional write and talk approach and the collaborative learning approach. It did not delve deep into comparing the effect of using other constructivist approaches such as problem-solving, investigation, discovery, small-group work, reflective and peer tutoring, and other traditional methods like the lecture method of teaching. As such, the researcher suggests a
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comprehensive study of the relationship between the use of the other constructivist approaches named above and other traditional methods as well.
,,
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APPENDIX 1 Lesson Observation Schedule No. Behaviour(s)/skills/approaches Frequency (%) observed Yes No 1. The teacher uses activities that make understanding a more central and reachable objective in his/her class.
2. The teacher uses verbs such as explain, find evidence, derive formulas, generalise, deduce, suggest, in the lesson objectives.
3. The teacher allows students to interact constructively with one another in building and integrating new knowledge from experiences.
4. The teacher teaches explicitly and allows students to make connections to meaningful contexts outside the classroom.
5. The teacher encourages students to think beyond what they know.
6. The teacher encourages students to learn by doing through enquiry, discovery, observations, reflection and demonstration.
7. The teacher encourages students to actively participate in their own learning process and involve learners in deciding the topics they learn and plan assessment program with them
8. The teacher provides students with rich on-going assessments and feedback that
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foster understanding of mathematics.
9. Teacher allows the students to evaluate each other’s work.
10. The teacher puts the students’ needs into consideration.
APPENDIX 2 Teachers’ Questionnaire
Section A: Background Information Please supply your background information. Indicate your choice by placing a tick [√] in the bracket [ ]. 1. Name of school………………………………… 2. What is your gender? Male [ ] Female [ ] 3. What is/are your current teaching subject(s)? Mathematics only [ ] Mathematics and other subjects [ ] (Specify) ……………………………………. 4. Which classes do you teach? JHS 1 Only [ ] JHS 2 only [ ] JHS 3 only [ ] JHS 1-3 [ ] Other (specify)……………………………. 5. What is your teaching experience in years? Up to 2 years [ ] 3 to 4 years [ ] More than 5 years [ ] 6. What is your teaching qualification(s)? Bachelor’s Degree in Education (B.Ed.) [ ] Bachelor of Science Degree (BSc) [ ]
Other (Specify) ……………………………………..
Section B: Instructional Approaches Often Employed By Senior High School Mathematics Teachers
7. Teachers use different approaches in teaching. What would your preferred teaching approach emphasize?
Write and Talk [ ] Peer Tutoring [ ] Problem-solving [ ]
Discussion [ ] Discovery [ ] Other (specify) [ ]
8. Please give reason(s) for your choice
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………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… APPENDIX 3 Interview Questions to Teachers
1. Do you agree that mathematics teaching should emphasize understanding over result? Could you please briefly explain your answer?
2. What does your actual teaching style emphasize? (Conventional or Constructivist?)
3. Which of the following teaching approaches do you mostly use for your classroom instruction?
Write and Talk, Peer Tutoring, Problem-solving, Discussion, Discovery.
Please explain why you use this approach very often in your instruction.
4. Do you always involve your learners in deciding the topics they learn and plan the assessment program with them? Please explain your answer?
5. Do you allow your learners to use their own methods in solving problems? Do you insist that they use the formula given them to arrive at answers? Please explain your answer. Interview Questions to Students
1. How do you prefer to learn, alone or in a group?
2. Why do you prefer to learn alone or in group?
3. Does the use of constructivists makes learning more interesting? How?
4. Do you think you can perform much better in mathematics if your teachers use
constructivists approach? Please explain
5. Would you like most of your mathematics lessons to be taught using constructivist
approach?
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APPENDIX 4 Test Instrument - Pre-test Questions Learner Code:
This is a non-evaluative assessment. Your performance in this assessment will have no bearing on your CASS marks. The assessment is designed to help you with algebra, by helping your teacher understand the mistakes you make, as well as why you make them. Instructions: 1. Answer all questions. 2. Use algebraic methods to solve all the problems. 3. Time: one hour
1) Kennedy sells b Oranges. Portia sells six times as many Oranges as Kennedy. An Orange costs 40 Pesewas. a) Name a variable in this problem. b) Name another variable in the problem. c) Name something in the problem that is not a variable. 2) There are k female scouts in a parade. There are 9 females in each row. Write an algebraic expression to find out how many rows of female scouts are marching in the parade.
3) What does yz mean? Write your answer in words. 1 6 4 푟 (6−푡) 푥 4) Simplify a) 푎 × b) − c) (5푥 − 3)2 d) − e) 푔 ( ) f) 푎 푥−1 푥+1 4 2 푦 푦푎+푦푏
푦+푦푑 푥 푥 5) Simplify: + 6) Subtract 5x from 15. 7) Multiply 푚 + 6 푏푦 5. 푦 푧 1 1 8) The letter n represents a natural number. What is more, 표푟 ? How do you 푛 푛+1 know? 9) Which is larger y or t in 푦 + 2푡 + 3. Explain. 10) Bernard is 14 years old now and his father is 40 years old. How many years will it be until Bernard’s father is twice as old as Bernard? 11) Shirts cost 푠 cedis each and pants cost 푝 cedis a pair. If I buy 5 shirts and 3 pairs of pants, explain what 5s + 3p represents.
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12) The equation 5W = 4R describes the relationship between W, the number of white cars produced and R, the number of red cars produced by a car company. Next to each of the following statements place a T if the statement follows from the equation, an F if the statement contradicts the equation, and a U if there is no certain connection. a) There are 5 white cars produced for every 4 red cars b) The ratio of red to white cars is five to four. c) More white cars are produced than red cars. 13) Mr. Osei shared his stamp collection with his two sons and the daughter: John, Benedict and Tracy. Tracy received 5 times the number of stamps than John did, and 4 less stamps than those received by Benedict. The whole quantity received by John and Benedict is 22 stamps. How many stamps did Mr. Osei give to each child?
14) Solve for y. 4푦 + 25 = 73 15) Consider solving the linear system: 푎 + 푏 = 5 and 푎 − 푏 = 7 a) To eliminate a, do you add or subtract the two equations? b) To eliminate b, do you add or subtract the two equations? c) Will you obtain the same solution if you add or subtract the two equations? Explain. 16) Solve the following linear system of equations. 2푥 + 푦 = 2 and 3푥 − 2푦 = 3 17) Starting with a number, if you multiply it by 3 and then add 27, you get 45. What number did you start with? 18) Francis has a phone plan. He pays ¢10.00 each month plus ¢0.10 each minute of long distance calls. One month, Francis made 100 minutes of long distance calls and his bill was ¢20.00. In the next month, he made 300 minutes of long distance calls and his bill was ¢40.00. Francis said, “If I talk 3 times as long it only costs me 2 times as much!” Will Francis’s rule always work? Explain your reasoning?
19) Senanu decided to buy a football with his five friends. Each friend agreed to pay the same amount and Senanu paid the balance of ¢25. The total cost of the football was ¢80. How much did each friend pay?
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APPENDIX 5 Marking Scheme for Pretest Questions
Question Answer Details Marks Number 1 (a) b A1 for identifying b as a variable in 1 the question
(b) 6b A1 for identifying 6b as another 1 variable in the question
(c) 40 A1 for identifying 40 as not a 1 variable in the problem [3] 2 푘 A1 for Number of girl’s scouts in the 1
9 parade divided by the number of [1] 푘 girls in each row = 9
3 푦 × 푧 or 푦 multiplied by z A1 for writing 푦 × 푧 or 푦 multiplied 1 by 푧 [1] 1 4. (a) 1 A1 for cancelling a × = 1 1 푎 (b) 2(푥 + 5) M1 for finding L.C.M 2
푥2 − 1 A1 for simplifying
(c) 25푥2 − 30푥 + 9 A1 for Expanding (5푥 − 3)2 1
푟 − 12 + 2푡 M1 for finding L.C.M (d) 2 4 A1 for simplifying
(e) 푔푥 A1 for multiplying g 푏푦 푥 only and 1 푦 dividing by y
(f) 푎 + 푏 A1 for simplifying 1 1 + 푑 [8]
푥(푧 + 푦) 5. A1 for correct expression 1 푦푧 [1]
15 − 5푥 6. A1 for correct expression 1
[1]
5푚 + 30 A1 for correct expression 7. 1 [1]
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1 𝑖푠 푏𝑖푔푔푒푟 8. 푛 2 A1 for correct expression [2] A1 for correct explanation
y is larger, give reasons 9. A1 for correct answer and 1 [1] explanation
M1 for 40 + 푥 = 2(14 + 푥) In 12 years’ time 10. M1 for Simplifying 3
A1 for correct answer [3]
11. The total amount paid for 5 1 shirts and 3 pants A1 for correct explanation [1]
12. (a) T 1 A1 for correct answer (b) T 1 A1 for correct answer (c) T 1 A1 for correct answer [3] 13. John received 3 Stamps Benedict received 19 Stamps M1 for either 푥 + 푦 = 22 표푟 4 Tracy received 15 Stamps 5푥 = 푦 − 4 [4] A1 for each correct answer 14. 푦 = 12 1 M1 for simplifying and dividing by 4 [1]
15 (a) Subtract the two equations A1 for correct answer 1
(b) Add the two equations A1 for correct answer 1 (c) Yes, with reasons A1 for correct answer and explanation 1 M1 for either eliminating 푥 표푟 푦 [3] 16. 푥 = 1 , 푦 = 0 M1 for simplifying A2 for the values of 푥 푎푛푑 푦 ALT: Accept the Method of 4 Substitution [4]
Let the number be 푥 17. The number is 6 M1 for 3푥 + 27 = 45 3 M1 for simplifying [3] A1 for correct answer A1 for correct explanation No with reasons 18. 1
M1 for 5푥 + 25 = 80 [1]
M1 for simplifying 19. 3 ¢11 A1 for correct answer [3]
TOTAL MARKS [45]
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APPENDIX 6 Test Instrument - Post-test Questions Learner Code: This is a non-evaluative assessment. Your performance in this assessment will have no bearing on your CASS marks. The assessment is designed to help you with algebra, by helping your teacher understand the mistakes you make, as well as why you make them. Instructions: 1. Answer all questions. 2. Use algebraic methods to solve all the problems. 3. Time: one hour
1) Fill in the blank spaces
(i) In −3푥, _____ is constant and _____ is variable.
(ii) In 10푎푏, _____ is constant and _____ are variables.
15푚푛 (iii) In , _____ is constant and _____ are variables. 푘
2) A certain number is added to six and the result is doubled. Write an expression for the statement.
3) For the algebraic expression 푥. 푥2 = 푥3 Explain in English what is being said.
6 4 푦 (6−푚) 4) Simplify a) 푥2 + 푥 + 2푥2 b) − c) (2푥 − 4)2 d) − 푥+4 푥−4 4 2
2푥 푚푥+푚푦 e) 푦 2 ( ) f) 푦 푚+푚푥
푎−푏 2푎 5) Simplify: + 6) Subtract 10x from 45. 7) Multiply 푦 − 8 푏푦 6. 2 3
8) Is it true that 푥2 = 2푥 − 1? Explain your answer.
9) Which is larger k or n in 푘 = 2푛 + 3. Explain.
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10) Kofi and Kojo were given GH¢380.00 to share. Kojo had GH¢75.00 more than Kofi. If Kofi’s share is y, find Kojo’s share.
11) Shirts cost 푠 cedis each and pants cost 푝 cedis a pair. If I buy 5 shirts and 3 pairs of pants, explain what 5s + 3p represents.
12) The equation 6W = 4R describes the relationship between W, the number of white cars produced and R, the number of red cars produced by a car company. Next to each of the following statements place a T if the statement follows from the equation, an F if the statement contradicts the equation, and a U if there is no certain connection. a) There are 6 white cars produced for every 4 red cars b) The ratio of red to white cars is six to four. c) More white cars are produced than red cars.
13) Mr. Osei shared his stamp collection with his two sons and the daughter: John, Benedict and Tracy. Tracy received 5 times the number of stamps than John did, and 4 less stamps than those received by Benedict. The whole quantity received by John and Benedict is 22 stamps. How many stamps did Mr. Osei give to each child?
14) Solve the equation. 5푥 + 2 = 2푥 + 17
15) Consider solving the linear system: 푥 + 푦 = 6 and 푥 − 푦 = 8 a) To eliminate y, do you add or subtract the two equations? b) To eliminate x, do you add or subtract the two equations? c) Will you obtain the same solution if you add or subtract the two equations? Explain.
16) Solve the following linear system of equations. 2푥 − 푦 = 10 and 푥 + 3푦 = 5
17) When a certain number is subtracted from 10 and the result is multiplied by 2, the final result is 4. Find the number.
18) Francis has a phone plan. He pays ¢10.00 each month plus ¢0.10 each minute of long distance calls. One month, Francis made 100 minutes of long distance calls and his bill was ¢20.00. In the next month, he made 300 minutes of long distance calls and his bill was ¢40.00. Francis said, “If I talk 3 times as long it only costs me 2 times as much!” Will Francis’s rule always work? Explain your reasoning?
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19) The combined ages of Jackie, Frank, and Elsie’s is 25 years. Frank is 5 years older than Jackie and Elsie’s age is 3 times Jackie’s age. Determine the age of each person.
APPENDIX 7 Marking Scheme for Post-test Questions Question Answer Details Marks Number 1 (i) -3 and 푥 A1 for identifying -3 as a constant 1 and 푥 as a variable in the question
(ii) 10 and a,b A1 for identifying 10 as a constant 1 and 푎, 푏 as variables in the question
(iii) 15 and 푚, 푛, 푘 A1 for identifying 15 as a constant 1 and 푚, , 푘 as variables in the question [3]
2 2(6 + 푥) A1 for correct expression 1 [1] 3 푥 multiplied by the square of 푥 A1 for correct explanation 1 is equal to the cube of 푥 [1] 4. (a) 3푥2 + 푥 A1 for correct answer 1
M1 for simplifying (b) 2푥 − 40 2 A1 for correct answer 푥2 − 16
2 2 A1 for Expanding (2푥 − 4) (c) 4푥 − 16푥 + 16 1
M1 for finding L.C.M (d) 푦 − 12 + 2푚 2 A1 for simplifying 4
(e) A1 for multiplying 푦2 푏푦 2푥 only 2푥푦 1 and dividing by y
푥 + 푦 (f) 1 + 푥 A1 for correct simplification 1
[8] 5. 7푎 − 3푏 A1 for correct expression 1 6 [1] 6. 45 − 10푥 A1 for correct expression 1 [1] 7. 6푦 − 48 A1 for correct expression 1 [1] 8. No A1 for correct reason 2 A1 for correct explanation [2]
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9. k is larger, give reasons A1 for correct answer and 1 explanation [1]
10. Kojo’s share GHC 227.50 M1 for 푥 + 75 + 푥 = 380 M1 for Simplifying 3 A1 for correct answer [3] 11. The total amount paid for 5 shirts and 3 pants A1 for correct explanation 1 [1] 12. (a) T A1 for correct answer 1 (b) T A1 for correct answer 1 (c) T A1 for correct answer 1 13. John received 3 Stamps [3] Benedict received 19 Stamps M1 for either 푥 + 푦 = 22 표푟 Tracy received 15 Stamps 5푥 = 푦 − 4 4 A1 for each correct answer [4] 14. 푥 = 5 M1 for grouping like terms and for 1 correct simplification [1] 15 (a) Add the two equations A1 for correct answer 1
(b) Subtract the two equations A1 for correct answer 1
(c) Yes, with reasons A1 for correct answer and 1 explanation [3]
푥 표푟 푦 16. 푥 = 5 , 푦 = 0 M1 for either eliminating M1 for simplifying 푥 푎푑 푦 A2 for the values of ALT: Accept the Method of 4 Substitution [4]
푥 17. The number is 8 Let the number be 2(10 − 푥) = 4 M1 for 3 M1 for simplifying [3] A1 for correct answer
A1 for correct explanation 1 18. No with reasons [1]
M1 for 푥 + 푥 + 5 + 3푥 = 25 Jackie is 4 years 3 M1 for simplifying 19. Frank is 9 years [3] Elsie is 12 years A1 for correct answers
TOTAL MARKS [45]
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APPENDIX 8
LETTER OF INTRODUCTION FROM THE MATHEMATICS DEPARTMENT
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APPENDIX 9
UNIVERSITY OF EDUCATION, WINNEBA Faculty of Science Education Department of Mathematics Education
Letter of Consent to the District Director of Education at Gomoa East District
Hope College, P. O. Box GP 18169, Accra. 25th September, 2017
The District Director, Ghana Education Service, Gomoa East District, Gomoa Afransi.
Dear Director,
Subject: Seeking Approval to Conduct a Research in Two Senior High Schools in the District.
I am Samuel Addo Osei, a final year MPhil student from University of Education, Winneba. My thesis supervisor is Dr. Nyala Joseph Issah. For the final thesis in my MPhil program, I am hoping to conduct a research study which seeks to search for responsive instructional method to address learners’ errors in algebra – by exposing the errors and thereby providing a treatment for the observed errors. I have selected two Senior High Schools in the Gomoa East District to collect data for this study.
The purpose of this study is to search for a responsive instructional method to address senior high school students’ conceptual understanding of algebra (pertaining to variables, expressions, equations, and word problems). The study will address learners’ errors in algebra in terms of exposing the errors and thereby providing a treatment for the observed errors.
I wish to administer a test instrument to about two hundred SHS 2 students in the two schools I have selected for the study. I would like to request for an approval from the district to allow me to conduct the research in these two schools in the district.
This study has been reviewed by University of Education, Winneba’s Ethical Review Office and will be conducted in such a way that it will not conflict with the academic work of the schools. If you would like more information, please contact me through email, [email protected] or cell phone numbers, 024-214-1305/020-350-6600.
Hoping for a positive response from your outfit. Thank you for your consideration.
Yours sincerely, ______
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Samuel Addo Osei (MPhil. Mathematics Education Student)
APPENDIX 10 Hypothesis Testing Summary by Mann-Whitney's Test
Figure 1: Independent Samples Mann-Whitney U Test on Pre-test Scores
Figure 4.6: Independent Samples Mann-Whitney U Test on Post-test Scores
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APPENDIX 11 Normal Q-Q Plots
Figure 4.7: Normal Q-Q Plots Control Group
Figure 4.8: Normal Q-Q Plots Experimental Group
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APPENDIX 12 Questionnaire on Collaborative Learning Administered to Students
1 4 2 3 Working in pairs and Strongly Strongly Agree Disagree groups…. Agree Disagree 1 helped
understanding/comprehension 2 fostered exchange of knowledge, information and experiences 3 made problem-solving easier 4 stimulated critical thinking 5 promoted a more relaxed
atmosphere 6 received useful/helpful
feedback 7 got fresh insight 8 focused on collective efforts
rather than individual effort 9 greater responsibility – for
myself and the group 10 enabled learners to help weaker
learners in the group 11 enhanced communication skills 12 improved performance 13 made learners actively participated in the teaching/learning process 14 was fun 15 made new friends 16 fostered team spirit 17 wasted time explaining things
to others 18 difficult getting members to
actively participate in tasks 19 should be
encouraged/continued 20 maximum group size should be
four
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