Effects of Problem Solving Approach On Mathematics Achievement of Diploma in Basic Education Distance Learners at University of Cape Coast, Ghana
Effects of Problem Solving Approach On Mathematics Achievement of Diploma in Basic Education Distance Learners at University of Cape Coast, Ghana
Effects of Problem Solving Approach On Mathematics Achievement of Diploma in Basic Education Distance Learners at University of Cape Coast, Ghana
BY
E83F/23814/2011
NOVEMBER 2014
DECLARATION
I confirm that this thesis is my original work and has not been presented for a
degree in any other university/institution for consideration. This research has
been complemented by works duly acknowledged. Where text, data, graphics,
pictures or tables have been borrowed from other works including the Internet,
the sources are specifically accredited and references cited in accordance and
in line with anti-plagiarism regulations.
Signature:. Date:
Arthur, Benjamin Eduafo
E83F/23814/2011
We confirm that the work reported in this thesis was carried out by the
candidate under our supervision as University supervisors.
Signature:.. Date:
Dr. Marguerite K. Miheso-OConnor
Department of Educational Communication & Technology
School of Education, Kenyatta University
Signature: . Date:
Prof. Joanna O. Masingila
Department of Mathematics and Teaching & Leadership Program
Syracuse University
Syracuse, New York USA
ii
DEDICATION
This study is dedicated to my wife, Araba Gyakyewa Arthur and my three
and Benjamin Eduafo Arthur Jnr. for their lovely care and support.
iii
ACKNOWLEDGEMENT
Ghana for the fatherly love, care, and support he gave to me during my
studies. I wish also to express my profound thanks to all staff of Centre for
Continuing Education, especially staff of the Examinations Unit, for the their
(USA) (my two supervisors), the Chair, Dr. M. Kiio and the entire staff of the
Dei-Mensah, Mr. Daniel Agyirfo Sakyi, Rosemary Twum, Mr. Paul Nyagome,
Paul Agyei Mensah, P., Daisy, Nina Afriyie, Isaac Kwabena Otoo and Tahir
Ahmed Andzie whose input and critical review of my work made this write up
both in Kenya (Kayole and Komarock) and in Ghana (Chapel Hill church of
Christ, Cape Coast) and all my close friends in Kenya and in Ghana.
v
1.11 Organization of the Thesis .......................................................................24
1.12 Operational Definition of Terms ..............................................................25
1.13 Chapter Summary ....................................................................................25
CHAPTER TWO:..27
REVIEW OF RELATED LITERATURE ..................................................27
2.1 Introduction. ...................................................................................27
2.2 Distance Education ....................................................................................27
2.2.1 The Challenges of Distance Learners ......................................28
2.3 Learning Theories and Students Achievement .........................................31
2.3.1 Adult Learning Theory ............................................................31
2.3.2 Radical Constructivist Theory .................................................35
2.3.3 Social Constructivist Theory....................................................38
2.4 Cooperative and Collaborative Learning and Learners Achievement......39
2.4.1 Cooperative Learning...............................................................40
2.4.2 Collaborative Learning ............................................................42
2.5 Pre-service Teachers Knowledge about the Nature of Mathematics,
Teaching and Learning, and Knowledge for Teaching
Mathematics .............................................................................47
2.5.1 Pre-service Teachers Knowledge about the Nature of
Mathematics, and Mathematics Teaching and Learning .........47
2.5.2 Pre-service Teachers Knowledge for Mathematics Teaching
..................................................................................................50
2.6 A Problem-solving Approach to Teaching and Learning Mathematics ....54
2.6.1 A Problem-Solving Approach and Mathematics Achievement
..................................................................................................55
2.6.2 Developing Mathematical Understanding Through a Problem
Solving Approach ....................................................................59
2.6.3 Impact of a Problem-Solving Approach on Achievement of
Learners in Mathematics .........................................................64
2.7 Chapter Summary ......................................................................................73
vi
CHAPTER THREE:..74
RESEARCH DESIGN AND METHODOLOGY .......................................74
3.1 Introduction.. ..................................................................................74
3.2 Research Design.........................................................................................74
3.2.1 Treatment and Control Procedures ..........................................76
3.2.2 Briefing and Debriefing for Facilitators ..................................77
3.2.3 Variables ..................................................................................79
3.3 Location of the Study .................................................................................80
3.4 Target Population .......................................................................................81
3.5 Sampling Technique and Sample Size .......................................................82
3.5.1 Sampling Techniques ...............................................................83
3.5.2 Sample Size..............................................................................84
3.6 Construction of Research Instruments .......................................................86
3.6.1 Pre-test and post-test questions ................................................86
3.6.2 Questionnaires..........................................................................87
3.6.3 Classroom and Lesson Observation Schedule .........................88
3.6.4 Written Interview Protocol ......................................................89
3.7 The Pilot Study ..........................................................................................90
3.7.1 Validity ....................................................................................91
3.7.2 Reliability.................................................................................92
3.8 Data Collection Techniques .......................................................................93
3.9 Data Analysis and Presentation .................................................................95
3.10 Logistical and Ethical Consideration .......................................................97
3.11 Chapter Summary ....................................................................................98
CHAPTER FOUR: ....99
PRESENTATION OF FINDINGS, INTERPRETATION AND ...............99
DISCUSSION.99
4.1 Introduction. ...............................................................................99
4.2 Biographic Data of Respondents ...............................................................99
4.2.1 Gender ..................................................................99
4.2.2 Learner Respondents Age .....................................................100
vii
4.2.3 Teaching Experience..101
viii
4.4.2.1 Mathematics learning is to understand why a method works
rather than to only learn rules ................................................133
4.4.2.2 Students should learn mathematics in groups .....................134
4.4.2.3 Students should ask questions during mathematics lessons135
4.4.2.4 A problem-solving method is the best way of teaching
mathematics........137
4.4.2.5 Students should often be confronted with novel problems .138
4.4.2.6 Mathematics learning is creation of conditions to stimulate
self- learning ..........................................................................140
4.4.2.7 Mathematics learning as use of worksheet to learn ............142
4.4.2.8 Cooperative work in groups is good for efficient learning of
mathematics ...........................................................................143
4.4.2.9 Learning mathematics means learners discovering for
themselves......144
4.5 Effects of a Problem-Solving Approach on Facilitators Perceptions about
Learning and Teaching of Mathematics ................................147
4.5.1 Effects of Problem-Solving Approach on Perceptions about
Learning Mathematics .........................................................147
4.5.1.1 Learning mathematics in groups .........................................148
4.5.1.2 Teachers encouraging students to ask questions .................149
4.5.1.3 Students solving problems on their own .............................150
4.5.1.4 Encouraging students to deduce general principles from
practical work.........................................................................151
4.5.1.5 .Mathematics teaching means encouraging students to discuss
their Solutions ........................................................................152
4.5.1.6 Cooperative group work and group presentation is good for
mathematics learning .............................................................154
4.5.1.7 Students learn more from problems that do not have
procedure for solution ............................................................155
4.5.1.8 It is important for students to argue out their answers ........156
ix
4.5.1.9 Solving mathematical problems often entails use of
hypotheses, tests and re-evaluation ........................................158
4.5.1.10 Students learn mathematics from seeing different ways of
solving the problem................................................................159
4.5.2 Effects of Problem-Solving Approach on Respondents
Perception about Teaching of Mathematics ..........................159
4.5.2.1 Conventional approach .......................................................160
4.5.2.2 Whole class teaching by a teacher is more effective than
facilitating ..............................................................................161
4.5.2.3 Teachers ought to create an environment to stimulate students
to construct their own conceptual knowledge .......................162
4.5.2.4 Mathematics Teacher to Consciously Facilitate Problem
Solving in Class..............162
4.5.2.5 Allowing Students to Discover for Themselves Leads to ...164
Incompletion of Syllabus .......................................................164
4.6 Challenges in the Adoption of a Problem-Solving Approach in the
Teaching and Learning of Mathematics. ...............................................165
4.7 Chapter Summary ....................................................................................173
CHAPTER FIVE:....174
SUMMARY OF FINDINGS, CONCLUSIONS AND
RECOMMENDATIONS174
xi
LISTS OF TABLES
Table 1.1 : Mathematics Achievement of First-year and Third-year
UCC- CCE Distance Learners over a Ten year
Period, 2001-2010.. 10
Table 2.1 : Contrast between a conventional (Traditional)
Approach and learning through a problem solving
Approach. 59
Table 3.1 : Distribution of UCC-CCE Study Centres in the Ten
Regions of Ghana.. 81
Table 3.2 : Distribution of UCC-CCE Study Centres by Zones... 83
Table 3.3 : Randomly Selected Centres from Randomly Selected
Regions... 84
Table 3.4 : Percentage Distribution of Level of Cognitive
Learning Domains in Pre- intervention and Post-
intervention Test Items... 85
Table 3.5 : Summary of Sampling Technique and Sample Size... 86
Table 3.6 Percentage Distribution of Level of Cognitive
Learning Domains in Pre- intervention and Post-
intervention Test Items... 87
Table 4.1 : Respondents Type of School and Teaching Class.. 105
Table 4.2 : Respondents Type of School and School Section.. 105
Table 4.3 : Results for Knowledge (Pre-intervention and Post-
intervention. 107
Table 4.4 : Results for Comprehension (Pre-intervention and Post-
intervention) 108
Table 4.5 : Pre-intervention Test and Post-intervention Test
Scores- on Application... 109
Table 4.6 : Results on Analysis (Pre-intervention and Post
intervention) ................... 110
Table 4.7 : Pre-intervention Test Mean Scores for the Four levels
of Cognitive Learning Domains... 111
Table 4.8 : Paired Sample t-Test for the Pre-intervention Test
Scores. 112
Table 4.9 : Post-intervention test Mean Scores for Four Levels of
Cognitive Learning Domains.. 112
Table 4.10: Paired Sample t-Test for the Post- intervention Test
Scores.. 113
Table 4.11: Perceptions on Learning Mathematics are to know the
Rules 116
Table 4.12: Perceptions on Learning Mathematics Means finding
Correct Answers. 117
Table 4.13: Perceptions on Mathematics Learning Means to See a
Correct Example for Solution and then Trying to Do
the same...... 119
Table 4.14: Perceptions on Learning Mathematics Means to Cram
xii
and Practice Enough... 120
Table 4.15: Perceptions on Learning Mathematics Means to get
Right Answers in Mathematics... 122
Table 4.16: Perception on Mathematics Learning is to Learn a Set
of Algorithms and Rules that Cover all Possibilities.. 124
Table 4.17: Perceptions on Mathematics Learning are for the
Gifted 125
Table 4.18: Perceptions on Mathematics Learning as Learning
Rules and Methods by Rote 127
Table 4.19: Perceptions on Mathematics Learning as Learning
Formal Aspect as Early as Possible 128
Table 4.20: Perceptions on Mathematics Learning as Learning
through Conventional Approach. 129
Table 4.21: Perceptions on Mathematics Learning as Giving
Students Notes to Copy 130
Table 4.22: Perception on Mathematics Learning is to Understand
why a Method Works than to Learn Rules. 134
Table 4.23: Perceptions on Mathematics Learning as Students
Learning Mathematics in Groups... 135
Table 4.24: Perceptions on Mathematics Learning as Encouraging
Students to Ask Questions 136
Table 4.25: Perceptions on a Problem-solving Method is the Best
Way of Teaching Mathematics... 138
Table 4.26: Perceptions on Mathematics Learning as Confronting
Learners with Novel Problems when Learning
Mathematics 139
Table 4.27: Perceptions on Mathematics Learning as Teacher
Creating Conditions to Stimulate self-learning.. 141
Table 4.28: Perceptions on Mathematics Learning as Using of
Worksheet to Learn. 142
Table 4.29: Perceptions on Mathematics Learning as Cooperative
Work in Groups.. 143
Table 4.30: Perceptions on Mathematics Learning as Learners
Discovering for Themselves... 144
xiii
LIST OF FIGURES
Figure 1.1 : Conceptual framework model of the study................... 23
Figure 3.1 : Nonequivalent (pre-test and post-test) Quasi
Experimental Design. 75
Figure 4.1 : Respondents by Gender 100
Figure 4.2 : Respondents by Ages 101
Figure 4.3 : Respondents on Teaching Experience 101
Figure 4.4 : Respondents on Highest Academic Qualification 102
Figure 4.5 : Respondents by Mode of Admission to UCC-CCE.. 103
Figure 4.6 : Respondents on Class Size.. 106
Figure 4.7 : Learning mathematics in groups... 148
Figure 4.8 : Using Questioning technique 149
Figure 4.9 : Students Solving Problems on their Own. 150
Figure 4.10 : Teacher to Encourage Students. 152
Figure 4.11 : Discussion of Solutions 153
Figure 4.12 : Cooperative Group Work.. 154
Figure 4.13 : No Procedure for Solution 156
Figure 4.14 : Arguing On Answers 157
Figure 4.15 : Solving Mathematical Problems... 158
Figure 4.16 : Different ways of Learning Mathematics. 159
Figure 4.17 : Conventional Approach 160
Figure 4.18 : Whole Class Teaching is Effective than Facilitating 161
Figure 4.19 : Creation of an Environment for Conceptual
understanding Construction.. 162
Figure 4.20 : Teacher to Facilitate Problem Solving in Class 163
Figure 4.21 : Students to Discover for themselves Leads to
Incompletion of syllabus... 164
xiv
ABBREVIATIONS AND ACRONYMNS
CM Conventional Method
DL Distance Learner
Technology
UG University of Ghana
Examination
xv
ABSTRACT
The mathematics achievement of students pursuing Diploma in Basic
Education (DBE) degree in Ghanas University of Cape-Coast (UCC) distance
education programme from 2001 to present has been consistently low. This
study sought to determine the effects of a problem-solving approach
intervention on the mathematics achievement of DBE UCC distance learners
(DLs) in Ghana. The study employed a mixed research design, using a sample
of 506 DBE UCC first year DLs and eight facilitators. Study instruments
included before intervention and after intervention test items, questionnaires
and interview schedules. The study was guided by four objectives: (1) to
determine the difference a problem-solving approach made on UCC DBE DLs
achievement scores in mathematics, (2) to establish the change in DBE UCC
DLs perception about mathematics teaching and learning before and after
learning mathematics through a problem-solving approach, (3) to establish the
effects of a problem-solving approach on DBE UCC DLs mathematics
facilitators perceptions about mathematics teaching and learning and (4) to
determine the challenges faced by facilitators in adoption of a problem-solving
approach in teaching mathematics. Results for the first objective were
analyzed using mean, standard deviation and t-test statistics. It was found in
general that the experimental group performed slightly significantly above the
control group. Specifically the experimental group performed better in
knowledge and application than the control group. However, there was no
significant difference in the performance of the two groups in comprehension
and analysis. ANOVA result was used to analyze objective two. The study
found that a problem-solving approach significantly changed majority of the
pre-service prospective elementary mathematics teachers instrumentalist
driven views of mathematics teaching and learning to a problem solving
driven views or perceptions. A descriptive analysis conducted on facilitators
perceptions after learning through a problem-solving approach depicted a
multidimensional views about mathematics teaching and learning before a
three days training workshop and a problem-driven view after the training
workshop. The study discovered through descriptive analysis that the
facilitators could not fully put their developed problem solving driven view of
mathematics teaching and learning into practice as a result of several
mitigating factors including non-availability of non-routine problem solving
activity textbooks and limited teaching time. The study therefore recommends
among others a complete overhauling of Ghanas UCC-CCE mathematics
curriculum for pre-service prospective elementary teachers to include the use
of a problem-solving approach to teach mathematics, intensive retraining of
mathematics teacher trainers in UCC-CCE on the use of a problem-solving
approach in teaching mathematics and the need for a robust change of
Ghanas first and second cycle schools mathematics syllabi and textbooks to
promote and sustain the use of a problem-solving approach in teaching
mathematics.
xvi
CHAPTER ONE
1.1 Introduction
purpose of the study, significance of the study, delimitation and limitation of the
study, and theoretical and conceptual frame-work of the study. The context of
the study is the University of Cape Coast Centre for Continuing Education
(UCC-CCE) in Ghana.
Mathematics is crucial not only for success in school, but in being an informed
Hanushek, & Kain, 2005). Current technology and scientific advancement being
beyond low level comprehension and mere memorization of facts and formulae
1
if they are to become problem solvers of the future. Trainee teachers therefore
mathematics.
school teacher (Elementary and Junior High school teacher), needs a strong
driven by teachers ability to understand and use SCK to carry the task of
familiar and unfamiliar situations will likely lead the student to perform below
mathematics that enables learners to apply their skills to both familiar and
unfamiliar situations, thereby giving them the ability to use tested theories and
they are referring to mathematical tasks that have the potential to provide
2
intellectual challenges that can enhance learners mathematical development
and hence improve their performance in mathematics. Such tasks also promote
explicit beliefs about teaching and learning, about learners, and about context to
prospective mathematics teacher (for the purpose of this study, the distance
classrooms during DBE distance learners face to face sessions is like what
group of learners who usually sit quietly trying to make sense of what the expert
3
as a cooperative venture among learners (distance learners) who will be
problem solving, teaching for problem solving, and teaching through problem
skill).
Lambdin, dos Santos & Raymond, 1994). Making problem solving an integral
4
exploring, testing, and verifying (p. 154). Specific characteristics of a problem-
al., 1994)
et al., 1991)
back and let the pupils (learners) make their own way (Lester et al.,
1994).
teacher passively playing the supportive role of teaching by assisting the learner
5
type of learning supports constructivism and social-constructivism knowing and
(Lester et al., 1994, p.154). Furthermore, Bay (2000) explains teaching via
that may lead to the development of higher-order cognitive skills that are rarely
defines the role of the teacher as a facilitator of learning rather than a transmitter
of knowledge and the learner, as a manager and director of their own learning.
6
The learners or problem solvers have to follow the framework suggested by
Polyas (1957) How to Solve It book, that are presented in four phases or
Ghana has for the past ten years embarked upon distance education programmes
in four out of its seven public universities: University of Cape Coast (UCC),
7
University of Ghana (UG), Kwame Nkrumah University of Science and
at all levels more accessible and relevant to meet the learning needs of
UCC-CCE.
of Education and mandated to produce graduate teachers for the 2nd Cycle
initial admission of 155 learners to the Faculty of Education, the University has
Social Science and Arts and four Schools Agriculture, Physical Science,
Biological Science and Business; and it now has 54 programmes with a student
The University has faced increasing pressure, over the years, from qualified
provide opportunities for these applicants to pursue higher education, the UCC
8
established the Centre for Continuing Education (a distance education unit) in
1997 and was in full operation in 2001. The Centre has been established,
primarily among other reasons, to train more professional teachers for all levels
has 30 study centres spread across the ten regions in Ghana (with a centre at
least in each region). The main focus of the Centre is directed at among others,
Degree and Masters Degree and also to increase access to the Diploma in Basic
teachers).
The DBE programme in the centre is organized in such a way that students
attend face-to-face (FTF) sessions fortnightly and the rest of the time used by
the students to learn in the comfort of their own homes. Programme delivery is
by print mode.
9
distance learners have to study mathematics every year in the three years they
spend on the DBE distance programme. The mathematics course outline for the
UCC-CCE requires that all DBE learners on the UCC-CCE distance education
believed that distance learners will be able to teach mathematics confidently and
present has been consistently low. That is, the majority of these DLs by this
study are scoring below a cut-off mark of 55% in Mathematics (Table 1.1)
10
Although a 50% mark (grade letter D) by UCC-CCE standards is a pass, in this
study the bench mark for a pass is 55%, a grade letter of D+. The grading
learners over the years is as a result of the same instructional approach used in
instructional approach).
behavourial learning perspective and it has been a popular technique used for
instructional process, the content is delivered to the entire class and the teacher
tends to emphasize factual knowledge. In other words, the teacher delivers the
lecture content and the students listen to the lecture. In this approach, the
learners are passive receivers of knowledge. It has been found in most studies
11
that the conventional lecture approach in classroom is of limited effectiveness in
both teaching and learning and also in such a lecture students assume a purely
passive role and their concentration fades off after 15-20 minutes.
against this background that this study was carried out with the aim of assessing
UCC-CCE distance learners in mathematics from the year 2001 to 2010. This
facilitators in adopting teaching and learning strategies that will help to improve
the situation.
was ranked among the lowest in Africa and the world (Ghana was ranked 44th
12
out 45 participatory countries calls in 8th grade mathematics), calls for some
overhauling of the mathematics curriculum of both the Basic and Second cycle
2020 (captioned The First Step) states in the guidelines for formulation and
the vision will substitute teaching methods that promote inquiry and problem-
solving for those based on rote learning. This is one of the medium-term (1996-
2000) policies under education that is yet to materialize and is long overdue.
In UCC-CCE, several approaches have been adopted over the years to improve
FTFs contact time. Yet the low performance of learners continues to persist.
This persistence in low performance of learners may mean that the real source
and solution to the problem has not been systematically established, at least in
the context of UCC-CCE. This study contends that a possible solution to the
approach, which is the essence of this study. The teaching approach used
13
predominantly by UCC-CCE mathematics facilitators over the years seems to
analysis, synthesis, evaluation) may have usually not been reached by this
They posit that students were able to answer questions (in TIMSS assessment,
2003 and 2007) that required recall of facts and procedures (lower levels of
cognitive learning domains) and not deep conceptual knowledge (higher levels
improving the quality of instruction (Ball, 1991; Ma, 1999; Sherin, 1996).
14
achievement and thereby improve their ultimate achievement (performance) in
mathematics.
learners in mathematics. The general objective of the study was to determine the
The study was guided by four specific objectives. The first objective was based
analysis). The reason for using Blooms taxonomy is that it provides a clear
guide to use in the evaluation of both a set examination and candidates scores
approach.
15
3. To establish the effects of a problem-solving approach on facilitators
Based on the objectives, the study was directed by two null hypotheses as stated
below:
problem-solving approach.
16
learning in UCC, where mathematics teaching has been predominantly through
mathematics.
Distance learners who have benefited from this study may apply this approach
Division (TED) of the Ghana Education Service (GES) insight into the benefits
and the challenges associated with the use of a problem-solving approach and
text books for use by basic, secondary schools and teacher training colleges in
Ghana.
This study was limited to only the first-year Diploma Basic Education distance
learners in the UCC-CCE. This group of learners will have an additional two
17
years ahead of them to learn mathematics using any one of these approaches or
both depending on the outcome of the study. As a result of limited time for the
study, only one unit out of the six (6) units (thirty-six sessions) of the
1.8.2 Limitation
As a result of financial and time constraints, the study was limited to the UCC-
CCE distance education programme in Ghana. An optimum sample size for the
The facilitators and distance learners in both the control and treatment
18
The facilitators in the treatment group, after training, will be committed to
their facilitators in the treatment group with distance learners and their
This study drew its theoretical framework from constructivism which has two
constructs or builds new ideas or concepts based upon current and past
19
the traditional focus on individual learning to addressing collaborative and
constructed when individuals engage socially in talk and activity about shared
problems or tasks (Jones, 1996), and that knowledge is interwoven with culture
What these theories are suggesting is that a learners mind is not like an empty
vessel that has to be filled with knowledge but that a learner is an active learner
new knowledge from known and related experiences and also through social
that teaching should be learner centered rather than teacher centered, and in a
20
placed on learning through social interaction, and the value placed on cultural
background.
find solutions and answers. They make conjectures, explain their reasoning,
validate their assertions, and discuss questions from their own thinking and that
of others.
problems which have the capacity to engage all students in class, and devise
and structured by the teacher. Students learn by: developing shared meanings
21
participation, and social interaction that constructs contexts, knowledge, and
meanings.
knowledge and meaning and probe learners to go over the limit of their
understanding.
play more active roles in and accept more responsibilities for their own learning,
hence justifying the suitability of the use of the two theories as theoretical
22
Independent Intervening Dependent
variable Variable variable
Levels of Cognitive
A problem-solving teaching learning domains
Mathematics achievement
Views about the nature of
teaching and learning of
mathematics
Learning environment
The central thesis of the study was that first, a teaching-approach (problem-
approach) will affect UCC-CCE facilitators and distance learners views about
views about the nature of teaching and learning of mathematics. The expected
solving approach), but are unknown to the study and therefore cannot be
controlled by the study, are facilitators and learners attitudes towards teaching
This thesis was organized into five chapters. Chapter One presents background
operational terms have been contextually defined in this chapter. Chapter Two
highlights the methods that will be appropriate for the study. These methods
include the research design, study variables, location of study, target population,
methods of data analysis used in the study. The results are explained and
of findings and suggestion for additional research are outlined in Chapter Five.
24
1.12 Operational Definition of Terms
performance score.
transmitter of knowledge.
Education Degree.
failures.
It has been discussed in this chapter that distance learners in UCC-CCE distance
for this study). It has also been indicated that the instructional approach most
25
mathematics UCC-CCE facilitators use in teaching UCC-CCE distance
learners. The statement of the studys problem has been stated and objectives
for guiding the study generated. The significance of the study has been
displayed and its purpose explained. This chapter has outlined why the study is
necessary and how it will be carried out. Theoretical models for the study have
26
CHAPTER TWO
2.1 Introduction
also be reviewed. This chapter also includes a review of literature on: what
growing (Daniel, 1996; Jung, 2005, UNESCO, 2002) and may be viewed as an
attend the institutions to study. However, due to various reasons, some people
27
may not be able to attend on-campus lessons despite the desire for further
education.
developed countries, show that teacher training at a distance may reach large
national education systems. However distance learners are faced with a lot of
challenges.
Distance learners find themselves lost at sea when faced with the demands of
distance education, where they have to find their own way through the subject
manner:
28
Imagine yourself on a space odyssey, about to descend on to a new
mapping it and finding your way across and within it. That would be a
daunting task. Yet in many respects, that is similar to that task that
expected to explore and gain mastery of it. They have to find out about
the substance and structure of the subject, the main issues it addresses.
(pp. 107-108)
the challenge for the distance teacher is how to assist the new
that they dont become lost or mired in conceptual swamps and abort the
mission, and what can be done to enable distance learners to find their
way about the new subject and fulfill their own expectations. (pp. 107-
108)
Apart from providing the student with physical access to education, distance
29
much as the responsibility to learn rests with the student, the responsibility to
enable the student to access the required subject matter squarely rests with the
higher education system, are a daunting task for some distance learners. The
experiences of some distance learners in their past school days might have
caused them to develop negative images about the learning of mathematics such
as: the subject being difficult; a subject on the mind of only the teacher;
competitive; subject for only the gifted; a subject only made up of correct
Teacher education distance learners are mainly adults. Research indicates that
most adult learners are attracted to distance education programme because they
receive total support from their employers in terms of receiving pay increases
workplace (O Lawrence, 2007). Adult learners are believed to know their own
standards and expectation and therefore, no longer need to be told, nor do they
require approval and reward from person in authority in order to perform. The
30
implication of this argument is that all adult learners are self-directed. Self-
individuals take the initiative, without the help of others in planning, carrying
out, and evaluating their learning experiences. This constitutes the concept of
andragogy. In practice, not all adult learners are self-directed. How adults learn
adults).
Several learning theories have been used to justify the use of a problem-solving
differences between the ways adults and children learn, Knowles (1990)
popularized the concept of andragogy (the art and science of teaching adults to
learn), contrasting it with pedagogy (the art and science of teaching children).
31
Knowles (1990) posits a set of assumptions about adult learners, namely, adult
learners:
are ready to learn when they assume new social or life roles,
Inherent in these assumptions are implications for practice that are suggested by
skill levels,
32
In a submission, Speck (1996) notes that the following important points of adult
Adults will commit to learning when the goals and objectives are
needs.
Adults want to be the origin of their own learning and will resist learning
Adult learners need to see that the professional development learning and
Adult learners need direct, concrete experiences in which they apply the
Adults need to receive feedback on how they are doing and the results of
activities that allow the learner to practice the learning and receive
33
evaluation. Small-group activities provide an opportunity to share, reflect,
planning.
Coaching and other kinds of follow-up support are needed to help adult
Having developed the adult learning style (cycle of experiential learning) model
over many years prior, Kolb (1984) published his learning styles model. Kolb's
learning theory sets out four distinct learning styles (or preferences) that are
assimilated and distilled into abstract concepts producing new implications for
One can deduce from the ongoing discussions on adult learning that adult
learners should be seen as active learners (not as passive learners) who are
34
experiences. This practice of adult learners learning by feeling (concrete
DBE distance learners style of learning mathematics. Hence, the need for this
experiences and their ideas. The basic idea in this theory is that learning is an
constructor. In the constructivist classroom the teacher becomes a guide for the
learners are intrinsically motivated to generate, discover, build and enlarge their
35
Researchers such as Fosnot (1989) and Brooks and Brooks (1999) suggest that a
the learner. From this perspective, the learner is viewed as an active, not passive
resources and social interaction (Eggen & Kauchak, 2003). According to Brown
(2004), central to the notion of constructivism is the view that experience and
knowledge are filtered through the learners perceptions and personal theories.
critical thinking skills which are attributes needed by the learners to improve
Learners control their learning. This simple truth lies at the heart of the
36
This radical constructivist learning that promotes deeper construction of
evaluation), This learning gap explains the recurring low performance by the
distance learners in mathematics and therefore this study examined how this
that are more complex, powerful and abstract than those learners possess before
This radical constructivist learning paradigm suggests that distance learners who
are adult learners learn best when learning is: active, self-directed, based on
37
knowledge and experiences gained. This manner of learning is best experienced
result of discussions and interactions in the group and among groups. Many
studies argue that discussion plays a vital role in increasing students ability to
test their ideas, synthesize the ideas of others, and build deeper understanding of
what they are learning (Corden, 2001; Nystrand, 1996; Reznitskaya, Anderson
& Kuo, 2007; Weber, Maher, Powell & Lee, 2008). Large and small group
opportunities to talk with one another and discuss their ideas increase their
ability to support their thinking, develop reasoning skills, and to argue their
increases through offering more chances for learners to talk together (Barab,
Dodge, Thomas, Jackson, & Tuzun, 2007; Hale & City, 2002; Weber, Maher,
38
(engagement here denotes active participation and mental inclusion). Jarwoskis
revealed some insights into how learners think and learn while interacting with
These authors suggest that research should focus on the potential for small-
solving approach.
39
2.4.1 Cooperative Learning
common task (Johnson, & Johnson, 1999, and Siegel, 2005). According to
learners, with the teacher, and with themselves, they have opportunities to
explore, organize, and connect their thinking (p. 95). Group diversity in terms
similar vein, Bruner (1985) contends that cooperative learning methods improve
cooperative learning shows that learners benefit academically and socially from
achievement (Cohen, 1994; Davidson, 1989; Devries & Slavin, 1978; Johnson,
Johnson & Stanne, 2000; Reid, 1992; Slavin, 1990). Academic benefits include
(Lonning, 1993; Watson, 1991). Social benefits include more on-task behaviors
40
and helping interactions with group members (Burron, James, & Ambrosio,
1993; Gillies & Ashman, 1998; McManus & Gettinger, 1996), higher self-
means of increasing learners achievements and that group goals and individual
than learners who were not exposed to cooperative learning (Reid, 1992).
classroom, Johnson and Johnson (1989) report of the cooperative learning group
group. This finding may mean that the intervention of a cooperative learning
improved the academic achievement of the students. In another study and using
classroom. The result revealed among others that the teachers prior experiences
41
and teaching context influenced the implementation of cooperative learning
mathematics classroom (of 17 students- 9 males and 8 females) about the effects
mathematical concepts. These few examples confirm the earlier assertion that
Collaborative learning has been described as the use of small groups through
which learners work together to accomplish shared goals and to maximize their
own and others potential (Johnson, Johnson & Holubec, 1994; Gokhale, 1995).
responsible for one another's learning as well as their own and that the success
Johnson (1999), there is persuasive evidence that cooperative teams who work
longer than learners who work separately as individuals. For his part, Vygosky
42
(1978) claims that learners are capable of performing at higher intellectual
levels when asked to work in collaborative situations than when asked to work
individually. The peer support system makes it possible for the learner to
internalize both external knowledge and critical thinking skills and to convert
and situated learning (Handal, 2002; Murphy, 1997). For example, in a study to
individually. It was also found that both groups did equally well on the drill-
and-practice test.
structured activity, and individuals are accountable for their own learning
collaborative learning, each group is accountable for its learning. There seems
43
from its potential of developing learners into critical thinkers thereby improving
feedback. Today's job market and teaching profession demand people with good
and collaborative learning activities can help learners develop and sharpen up
these skills.
are critics of small group learning. For example Randall (1999), cautions against
abuse and overuse of group work. According to Randall (1999), the many
making can easily be dominated by the loudest voice or by the student who talks
the longest.
44
Although, Randalls (1999) and Gokhales (1995) arguments seem challenging,
there are solutions to the issues they have raised. In cooperative and
In support of assigning rolls, Esmonde (2009) suggests that these roles may be
The issue here is not competition but sharing of knowledge. For instance,
effective. Forman (1989) suggests that learners must have mutual respect for
identities. Such roles, Esmonde (2009) continues, provide all students the
preventing the discussion from being dominated by a few. The teacher therefore
every individual in the group. The teacher should also assist the learners to set
45
up rules or learning norms that will guide their activities in group learning.
thinkers can be put into one group so that they can be assigned extension work,
and during whole group discussion given the opportunity to share their
knowledge.
manage the class well to ensure learning outcomes will produce the desired
1996) and providing opportunities for all students to talk, listen, read, write, and
46
the effects the use of a problem-solving approach, an approach that is
mathematics.
education institutions, hold sets of beliefs more traditional than progressive with
discipline based on: rules and procedures to be memorized, that there is usually
one best way to arrive at an answer and that it is made up of completely right
47
Southwell & Khamis, 1992), and that mathematics learning requires neatness
and speed, and that there is usually a best way to solve a problem (Civil, 1990);
and that ability in mathematics learning is innate (Foss & Kleinsasser, 1996).
Nisbert and Warren (2000) for instance surveyed 389 primary teachers with
regards to their views on mathematics teaching and learning. They found that
primary teachers hold limited views about what mathematics is. They found that
primary teachers hold the view of mathematics as static and mechanic, rather
service learners beliefs about mathematics. The study employed pre-test and
that instigate cognitive conflict was implemented. The study discovered that
many of the learners held traditional beliefs about mathematics. It was found
that majority of the learners failed to hold progressive beliefs. However, the
their beliefs, thoughts and understanding. The implication is that if these trends
48
Improving learners achievement in mathematics may require quality teaching
mathematics. They stress the need for teacher education to understand that
knowledge and beliefs constitute these domains and what form teacher
education programmes should take in order to educate teachers so that they can
how they work and when and where to use them. Simply put, according to Ma
(PUFM) know how, and also know why (p. 108). Teachers should have the
49
of teachers knowledge, Shulman (1986) classifies teachers knowledge into
(1989a) proposes a detailed analytical model of the six different types and
this study are relevant to be examined. The different types are: subject matter
implement high-quality instruction (Ball, 1990; Ball & Bass, 2000; Ball &
Cohen, 1999; Hill, Schilling & Ball, 2004; Ma, 1999). What then is
In their papers, Schoenfeld (1981), Shulman (1987) and Ball (1991) take
50
performance indicator for assessing teachers mathematics achievement
(performance). For example, several people have a strongly held belief that
that is characterized by a number of factors, including its extent and depth, its
knowledge about mathematics as a whole and its history (Ernest, 1989a). Ernest
for teaching mathematics and that the major goal of teaching mathematics is to
learner. For Ernest, whatever means of instruction are adopted, the teacher
needs a substantial knowledge base of the subject in order to plan for instruction
and to understand and guide learners responses. Furthermore, he argues that the
51
Many prospective elementary school teachers (like UCC-CCE distance
learners) often assume that their own schooling mathematics knowledge will
supply the subject matter needed to teach young children (Feiman-Nemser &
skills (Ball & Fieman-Nemser, 1988). Yet researchers have highlighted the
Providing a solution to the issue raised, Haggarty (1995) suggests that a possible
solution is one in which learners, as active learners, take full responsibility for
their own weaknesses and take appropriate action. How feasible this suggestion
suggests that another possible solution is that tutors identify the most likely
areas of learners uncertainty and build those into programme for discussion
52
It may, however, be helpful for mathematics teacher educators to distinguish
with what this study intends to investigate. The substantial knowledge base of
the teacher, who in this study is the UCC-CCE distance learner, is weak and
education, Thompson (1984) and Ernest (1989b) argue that any attempt in
improving the quality of mathematics teaching and learning must begin with an
53
(p.54). A realization of a pedagogical knowledge such as the use of a problem-
education can help teachers develop a conception of teaching and learning that
This issue on reforms in mathematics teaching and learning requires the use of
understand the problem, devise a plan for solving the problem, carry out your
plan, and look back. In this teaching approach, learners are expected to learn to
strategies include using diagrams, looking for patterns, listing all possibilities,
mathematics problems, but also how to logically work our way through
54
problems we may face. The memorizer can only solve problems that he or she
has encountered already, but the problem solver can solve problems that he or
she has never been encountered before. According to Rav (1999), the essence of
environment for learners on their own, to explore problems and to invent ways
mathematics through problem solving base their pedagogy on the notion that
solve those problems, and in the process of solving the problems, they construct
been put into practice with three examples: using elementary, middle, and
that learners in all the three cases had no formal instructions on how to solve the
55
(Schroeder & Lester, 1989). The understanding and skills demonstrated by
learners in each case of the study supports the claim that problem solving is a
processes.
and its alternative solutions usually takes a longer time than the demonstration
approach used fewer problems and spent more time on each of them, compared
explain underlying reasoning for getting an answer) and fewer recall questions
The study by Hiebert and Wearne (1993) suggests that a judicious use of time
classroom, Allevato and Onuchic (2007) suggest and explain the following
Form groups and hand out the activity. The teacher presents the problem
to the learners, who, divided into small groups, read and try to interpret
56
and understand the problem. It should be emphasized that the
problem has not yet been presented in class. The problem proposed to
the learners, which we call the generative problem, is what will lead to
Observe and encourage. The teacher no longer has the role of transmitter
learners.
reading and interpretation; as well as those that might arise during the
57
Record solutions on the blackboard. Representatives of the groups are
Plenary session. The teacher invites all learners to discuss solutions with
their colleagues, to defend their points of view and clarify doubts. The
the teacher according to the program for that grade level) is formally presented.
58
developed in the search for reasonable answers to the problem given. The steps
also define a more-challenging role for the teacher. This approach to teaching is
approach in mathematics.
one of the most enjoyable and satisfying intellectual experiences one can have.
For Heibert and Wearne (1993) understanding mathematics so well that one
knows how it works confers an unparalleled sense of esteem and control (p. 4-
5). How can learners develop understanding of the mathematics that they are
learning?
engage in it develop, extend, and enrich their understanding (Heibert & Wearne
better solution methods and providing information for learners just at the right
time is crucial (Hiebert, Carpenter, Fennema, Fuson, Human, Murray Oliver &
problem solving related tasks that will arouse and sustain the interest of the
In Selecting Quality Task for Problem Based Teaching, Marcus, and Fey
(2003), argue that designing activities that will keep learners busy throughout
the standard class period is relatively easy, but making sure such activities lead
to learning important mathematics is much more difficult (p. 55). They further
argue that finding and adapting problem tasks that engage learners and lead
task for teachers (p. 55). To ensure selection of quality tasks for a problem-
approach teaching, Marcus and Fey (2003) suggest four questions that need to
Will selected tasks be engaging and problematic yet accessible for many
thinking?
61
These questions suggested by Macus and Fey (2003) seem to be calling for a
mathematics curriculum that is problem-based (task driven) and that will enable
solving approach.
increase focus on problem solving and modeling in countries from the West as
well as the East (the extent that this approach has been embraced in most
solving experiences will make learners be able to use and apply mathematical
become more engaged and enthused in lessons, and finally, learners will
problems in the context that make sense to the learner: if a learner does not
have a good sense of what he or she knows, he or she may find it difficult to be
62
information must be effectively applied to new problem situations (Fosnot,
for adding common fractions may be needed when it comes to the addition of
mathematics relationally and not by rote. This view of why it is important for
ensures that everything one knows about the topic will be useful. When solving
The student should also be prepared to learning mathematics actively rather than
In UCC-CCE, distance learners like most pre-service teachers learners are most
63
fails to promote critical thinking skills (Brown-Lopez & Lopez, 2009).
and critical thinking skills may improve (Hines, 2008) and thereby may improve
problem-solving approach is a rather new approach, it has not been the subject
of much research. Although less is known about the actual mechanisms learners
(Schroeder & Lester, 1989). One key question that needs addressing when
whether learners can explore problem situations and invent ways or employ
64
Many researchers (e.g., Carpenter, Franke, Jacobs, Fennema, & Empson, 1998)
indicate that learners can explore problem situations and invent ways to solve
the problems. For example, Carpenter et al. (1998) found that many first-,
second-, and third-grade learners were able to use invented strategies to solve a
problem. They also found that 65% of the learners in their research sample used
an invented strategy before standard algorithms were taught. By the end of their
study, 88% of their sample had used invented strategies at some point during
their first three years of school. They also found that learners who used invented
their knowledge to new situations than were learners who initially learned
standard algorithms.
1998; Cai, 2000) have also found evidence that middle school learners are able
to use invented strategies to solve problems. For example, when U.S. and
Chinese sixth-grade learners were asked to determine if each girl or each boy
gets more pizza when seven girls share two pizzas and three boys share one
pizza equally, the learners used eight different, correct ways to justify that each
65
Collectively, the aforementioned studies not only demonstrate that learners are
capable of inventing their own strategies to solve problems, but they also show
mathematics.
One of the studies that have informed this study is the work of Kousar (2010).
the population of the study. The learners of a grade 10 class of the Government
Pakistan Girls High School Rawalpindi were selected as a sample for the study.
Using a sample size of 48 learners Kousar (2010) equally divided them into an
conducted. The experimental group was then taught over a period of six weeks
control group continued with the instructional approach that they had prior to
being identified as the control group. After the intervention, an assessment was
used to see the effects of the intervention. A two-tailed t-test was used to
analyze the data, which revealed that both the experimental and control groups
66
significantly on the assessment following the intervention. Kousar concluded
that the results obtained were in line with for example those by Farooq (1980),
and Chang, Kaur, and Lee (2001), all cited in Kousar (2010).
Though the study is not about learners who are pre-service or in-service
teachers and therefore mature adults, it is worth learning from the findings.
There are several things Kousar noted that have informed this study. Kousar
claims that for authentic results, the teachers of the problem-solving approach
group should be provide training for at least one months duration. In the study,
the issue of training of the facilitators in the treatment group will be taken care
of. What seems significantly different about this study compared to Kousars is
that the analysis of performance of the learners in this study was not only
summative but this study went further to analyze the effects of a problem-
approach had on both Blooms lower and higher levels of cognitive domains.
In another study carried out by Adeleke (2007), the study specifically examined
67
Procedural Learning Strategy (PLS). A sample of 124 science learners assigned
into CLS, PLS and Conventional Method (CM) groups were involved in the
study making use of pre-test post-test control group design. Findings of the
in the two learning strategies. But a significant difference was recorded in the
performance of boys when comparing the two groups, also in the performance
of girls in the two groups. Adeleke (2007) concluded that when the training of
Learning Strategies, boys and girls will perform equally well without significant
insights. Hallagan et al. (2009) reported that initially, although most learners
68
squares with additional single items as exemplified by x2+1, than with multiple
distance learners.
replication design (an experimental design with treatment group and control
with parallel pre-test and post-test. The roles are then switched and the
experiment reproduced with the control becoming treatment group and the
within subject analysis, there were significant differences among the pre-test
and post-test 1 and 2 results. That is, learners in the control groups, who were
69
instructed using a procedural approach from weeks 1 to 6, demonstrated higher
gains than the experimental groups who were immersed in social constructivist
groups who were exposed to social constructivist activities alone. This contrary
finding could be as a result of the assessment not reflecting the manner in which
the learners were taught. According to Gay and Airasian (2000), a test cannot
accurately reflect students achievement if it does not measure what the student
was taught and was supposed to learn (p.163). Assessment should reflect the
purpose and objectives of instruction (Thompson & Briars, 1989). This study
instruction.
The ongoing research findings have shown that learners are capable of inventing
problems. However, research has also shown that some invented strategies are
not necessarily efficient strategies (Cai, Moyer & Grochowski, 1999; Carpenter
et al., 1998; Resnick, 1989). For example, in a study by Cai et al. (1999), a
arithmetic average: The average of Eds ten test scores is 87. The teacher
throws out the top and bottom scores, which are 55 and 95. What is the average
of the remaining set of scores? (pp. 5-6). One student came up with an unusual
strategy to solve it. In this solution, the student viewed throwing away the top
70
and bottom scores as taking 15 away from each of the other scores. By
and procedures but, based on their level of understanding, learners also should
procedures (Henningsen & Stein, 1997; Perry, Vanderstoep & Yu, 1993; Stigler
problem and its alternative solutions usually takes longer than the demonstration
must also decide what aspects of a task to highlight, how to organize and
orchestrate the work of the learners, what questions to ask to challenge those
with varied levels of expertise, and how to support learners without taking over
the process of thinking for them and thus eliminating the challenge (NCTM,
71
support for learners mathematical exploration, but not so much support that
will take over the process of thinking for their learners (e.g., Ball, 1993;
There are no specific, research-based guidelines that teachers can use to achieve
and it is unlikely that research will ever be able to provide such guidelines.
facilitators in UCC-CCE in Ghana and this study will seek to address it.
72
solution methods, they are better able to apply mathematical knowledge in new
learning. Most of the studies have been carried out outside Ghana and Africa.
subject content knowledge (SCK) and how it can affect the desired learners
also been discussed in-depth. Various theories and studies aimed at providing
73
CHAPTER THREE
3.1 Introduction
The purpose of this chapter is to describe the methodology of the study.
solve the research problem. In this chapter, the research design, the research
variables, location of the study, the target population, sampling techniques and
data collection techniques, logistical and ethical considerations for the study are
discussed.
was used for this study. According to Creswell (2009), the problems addressed
by social and health science researchers are complex, and use of either
gained from the combination of both quantitative and qualitative research than
Cohen, Marion, & Morrison, 2004; Greene, Caracelli & Graham, 1989; Strauss
& Corbin, 1990) argue that use of both forms of data and data analysis allow
74
results of a sample and to gain a deeper understanding of the phenomena of
interest. These presentations and synthesis justify the use of both quantitative
teaching approach) whereas the control group did not. The control group was
used to establish a baseline for reading achievement in this study. This design
was used since the study was conducted in a classroom setting and was not
possible to assign subjects randomly to groups. In addition this design was used
control-group design was used to investigate objective (1) of the study (Figure
3.1).
RE O1 X O3
RC O2 X O4
KEY
RE = treatment group RC = control group
X = treatment (a problem-solving approach)
X = no treatment
O1 = O2 = pre-test O3 = O4 = post-test
75
In addition to the quantitative procedures, qualitative design was used to
studied (Gosling & Edwards, 1995; Strauss & Corbin, 1992). In so doing, the
researcher learned more about the participants and the research setting (Bogdan
& Biklen, 1998; Eisner, 1991; Patton, 1990). Questionnaires for learners, as
Steinmetz, 1991). The research designs for this study are discussed in detail in
two parts: the research design for learners (distance learners) and the research
changes before or after treatment. Control and treatment groups were used for
this study. Eight (8) out of the thirty (30) study centres were selected for this
design (details of the selection are discussed under sampling). Four of the
selected centres were assigned to the control group and the remaining four to the
treatment group. The control group was taught by their teachers using the
Each of the groups was tested before treatment (pre-test). After they were taught
groups were observed while they engaged in learning. Also, a randomly selected
principle indicates that the variation that may be caused by extraneous factors
can all be combined under the general heading of chance. By applying this
2004).
generated outside the context of the classroom. This study was conducted in
the research and development team. By this design the teachers, as well as the
researcher, were together made responsible for the quality of the learners
mathematics education.
77
The facilitators (tutors from UCC-CCE) who taught the treatment group
students were the facilitators who handled the four randomly selected study
attend an orientation workshop for three days (Appendix A). They were
informed about the purpose of the study and were provided with guidelines for
implementation of the study. They were also given the opportunity to discuss
These facilitators were provided with copies of a consent form that indicated
that their participation is voluntary and that they were free to withdraw from the
participate and to withdraw from the research once it has started, as well as the
Eight (8) mathematics course tutors who handle learners from four study centres
were invited for a three days of an intensive orientation workshop on the use of
78
problem-solving approach, teamed up with the researcher as a resource person
for the workshop. Topics treated during the workshop included: Introduction to
role of an instructor and learners in problem solving). These topics treated were
their own meaning of what it means to teach and learn mathematics through a
solving questions and materials, such as worksheets that they would use in
The researcher kept notes of the activities as well as comments made by the
compare and assess these participants views about teaching and learning
3.2.3 Variables
79
the researcher may compare an independent variable to see its impact on
a dependent variable,
Most quantitative research, according to Creswell (2009), falls into one or more
of these three categories. This study used all three of the categories. The
variable or outcome variable was the variable that depends on the independent
The study was carried out in Ghana, specifically the University of Cape Coast
study centres for the Diploma in Basic Education (DBE) programme. These
centres are spread throughout the ten regions in Ghana. UCC-CCE is selected
80
for the study because the centre is leading in the running of DBE distance
education programmes for teachers in Ghana. Table 3.1 shows the distribution
of the spread of the study centres throughout the ten regions of Ghana
(Appendix B).
study is the aggregate of cases about which the researcher would like to make
generalizations and it is the units from which the information is required and
81
subjects for a study. According to Amedahe (2002), researchers usually sample
The target population for this study was UCC-CCE-DBE learners. The
accessible population was all DBE first-year learners. These first-year DBE
learners have two more years ahead of them to complete the programme and
therefore will have more time to adjust to any change in approach to teaching
examination for them to write and therefore their facilitators would not rush
them through the learning process with the aim to complete the course outline.
Being first-year learners, they may see this approach as a normal approach in
teaching mathematics at the university level and may therefore not feel reluctant
population to make conclusions about the population. Apart from the pragmatic
designs suffice. The first is probability sampling design that is based on random
selection where each population element is given a known non zero chance of
82
(Keppel, 1991). The second is non-probability sampling which is arbitrary (non-
random) and subjective (Cooper & Schindler, 2001). This study employed a
Probability sampling ensures the law of Statistical Regularity which states that
if on average the sample chosen is a random one, the sample will have the same
To ensure even spread in the locations of the centres that the study was
conducted, the thirty study centres of UCC-CCE were grouped into three zones:
regions) were randomly drawn from the southern zone, from which the
experimental groups were selected. Cape Coast, Obiri Yeboah and Swesbu
study centres were selected from the Central region and Takoradi study centre
was selected from Western region. Four study centres: Techiman study from
Brong-Ahafo region (Northern zone), Obuasi study centre from Ashanti region
(Middle zone), Ada study centre from Eastern region (Middle zone), and
83
Jasikan study centre from Volta region (Middle zone) were selected and
In all, eight out of thirty study centres, representing 26.7% of the study centres,
were used for the study. Four of the eight selected study centres were assigned
enough to give a confidence interval of desired width; as such the size of the
sample must be chosen by some logical process before a sample is taken from
the population. The sample size for this study (using a finite population of 5,060
(controlling any sampling error that may arise during selection) 10% of the total
Mills and Airasian (2009) for a quota sampling technique. Thus, the study used
eight study centres randomly drawn from the thirty study centres (that is, three
approximated to four since equal numbers of centers are needed for the control
84
and treatment groups) and a sample size of 506. This sample size of 506 was
The treatment and control groups were each assigned a sample size of 253
learners. Sample size for each selected study centre was computed
Cn
nc 253
Tc
Where,
Tc total number of first year students in selected treatment /control study centres
Table 3.4 Target population and Sample Size for Selected Control and
Experimental groups
The teachers/facilitators (eight in all) who teach the DBE students in the
selected study centres for the study were considered automatic participants of
85
the study. Henceforth, Table 3.5 summarizes the sampling techniques and
Facilitators 77 8 10 Purposive
questionnaire,
interview guide.
intervention after the treatment (Appendix D). The pre-test and post-test were
each comprised of short questions that required short answers. The questions
were set with objectives based on four of the six of Blooms (1956) levels of
86
discussing critical thinking. A table of specification was used to this effect to
are shown in Table 3.6. The percentage distribution was to establish whether
learners have developed deeper concepts and procedural skills across both the
numerical figures in the questions were different for the two tests. The questions
for the pre-test and post-test were set, typed, and copies run by the researcher to
ensure security, validity and reliability of test items. Learners were instructed to
show all of their work and not to erase or black out anything that they had
3.6.2 Questionnaires
87
F) in the treatment group. The responded questionnaires by the distance learners
were separately serially numbered for both experimental and control groups.
The responses of learners odd serially numbered were selected and used for
analyses of the study. The opinion type questionnaire used a four point Likert
Strongly agree.
The items in the questionnaire were divided into the following three sections:
The same questionnaire was administered (using the same group) in the post-
test survey after the study to determine any changes the use of a problem-
mathematics.
In this study, the observation method of data collection was to be used to assess
88
mathematics classrooms. The classroom observation protocol used (Appendix
what the facilitators and the distance learners do at the start, during and
The researcher was to sit in the participants class during their regular
mathematics time and used the observation protocol to record what was seen,
heard, and experienced during a teaching session (Gay & Airasian, 2000). There
was to be video and audio recording during teaching and learning sessions for
interview protocol (Creswell, 2009), was used to obtain more information from
treatment study centres. The written interview was done to ensure uniformity
and reliable questions and results. Additionally, the researcher could not depend
also be at all the four places all the time. The focus of the written interview was
89
in mathematics by the facilitators and distance learners. The researcher assured
The term pilot study refers to feasibility studies that are small scale version(s),
or trial run[s], done in preparation for the major study (Polit, Beck, & Hungler,
2001). One of the advantages of conducting a pilot study is that it might give
advance warning about where the main research project could fail, where
2001). The instruments for the study, especially the questionnaires and the
pretest questions were first analysed for consistency with the help of selected
in four study centres (three centres in the northern region and one centre Greater
distance learners in each selected pilot study centres. The pilot study centres
were not involved in the final research study. From the feedback obtained after
piloting, the study instruments were refined. The pre-intervention and post-
the questionnaire that proved difficult for student to respond to then were
90
3.7.1 Validity
from establishing the real causal relationships of a study. Gay and Airasian
causes, threats to internal validity were also controlled: the content validity of
used to identify the achievement domains being measured and to ensure that a
fair and representative sample of questions), and the content validity was also
reflects the true theoretical meaning of a concept. The study used questionnaire,
the randomization of the subjects to the treatment and control groups to ensure
91
and researcher expectancies) were minimized. Patton (2002) advocates the use
Mugenda and Mugenda (2003). They argue that the easiest way of assessing
suggestion of more than one instrument) which must measure the same concept
(p. 101).
3.7.2 Reliability
Alpha () formula (a formula used for estimating reliability of test items), and
each items scores of short answer essay test items were measured as to whether
it is correlated significantly with total scores, at either the 0.01 or 0.05 levels.
k 1
2
k
k 1 2
To test for reliability, the study used the internal consistency technique by
employing Cronbachs Coefficient Alpha test for testing a research tool. Internal
scores obtained from other times in the research instruments. Cronbachs Alpha
92
is a coefficient of reliability. It is commonly used as a measure of the internal
examinees.
0 and +1. According to Mugenda and Mugenda (2003), the coefficient is high
when its absolute value is greater than or equal to 0.7: otherwise it is low. A
consistency among the variables. This study will correlate items in the
Alpha of 0.8.
was taken from the Director of UCC-CCE to carry out research in UCC-CCE.
After a thorough discussion of the study and its benefits to the academic
treatment group from the four randomly selected study centres for a four-days
mathematics. The UCC-CCE complex office was used as the venue for the
93
Since a mixed method data analysis approach was used, the study adopted data
the findings.
The pre-test and post-test questions were administered with the help of six
senior members and six senior staff members in UCC-CCE. The completed
scripts were marked by the researcher and two mathematics lecturers in UCC,
I). The marking scheme was made of: B (factual mark), M (method mark), and
A (accuracy mark).
With the assistance of the six senior members and six senior staff members in
who took part in this study. The facilitators, as well as the sampled distance
after the post-test examination and handed them in. This arrangement
minimized any delay of receiving back any of the questionnaires and thereby
94
A written interview was conducted to help the researcher to get more
in the four sampled treatment study centres. This interview was conducted by
the researcher and his team of assistant researchers. The focus of the interview
problem-solving approach and the challenges and advantages associated with it.
The written interview was conducted after the study. The researcher assured
The data obtained from distance learners and mathematics facilitators through
from the field were organised and summarised to obtain a general sense of
information and to reflect on its overall meaning. The organized data was coded
into knowledge (K), comprehension (C), application (A) and analysis (S). The
coded data were used to generate a description of the setting or people as well as
categories or themes for analysis (Creswell, 2009). The description and themes
(e.g., frequency tables showing means, standard deviations, test statistics, and
findings and theories. Microsoft Excel and the latest (17th) version of Statistical
95
Package for Social Sciences (SPSS) programme were used to assist and enhance
the analyses.
distribution. The pre-test and post-test scores were further analysed using a t-
test statistics to determine if the treatment had effects on the groups. This was
mathematics.
approach. The perceptions for both the experimental and control groups before
responses before and after the intervention between the control and
ANOVA. The ANOVA results were used to test the null hypothesis there
96
were no changes in UCC-CCE DBE distance learners perceptions before and
entailed frequency distribution tables, percentages and graphs. The purpose was
The study was carried out with the available funds and within the available
facilitators teach were sought first. Special emphasis was laid on confidentiality
data. Permission was also sought from both the facilitators and the subjects
(distance learners used in the study) for the use of their videos and still pictures
in the study.
97
3.11 Chapter Summary
This chapter has introduced the research design and described both the study
instruments and pilot study used have been described. How the data were
meaning have been dealt with. Methods that were used to ensure validity and
reliability of data have been described. The chapter ended with a presentation of
data collection procedure, how data was analysed and the ethical issues
considered.
98
CHAPTER FOUR
DISCUSSION
4.1 Introduction
mathematics.
Although the biographic data were not central to the study, they helped to
and also the interpretations of the findings. The biographic data in this case
admission, type of school, teaching section, teaching class and class size. These
4.2.1 Gender
From Figure 4.1, the ratio of male to female learners in the control group is
approximately 3:2 and the ratio of male to female learners in the experimental
99
group is approximately 1:1. The ratio of males to females who answered the
questionnaire therefore stood at almost 1:1. Hence gender balance and therefore
56.3
50.3 49.7
60 43.7
50
Percentage
40
30
20
10
0
Male Female
age range of 24 35 years. This means the majority of the learners may remain
in the teaching profession for 20-30 years before reaching their retirement age.
100
86.7
90 83.3
80
70
2429 yrs
60
Percentage
30-35 yrs
50
40 36- 41yrs
30 42 - 47yrs
20 10.7 10.7 48+
10 4 1.3 1.3 0.7 0.7
0.7
0
Control Group Experimental Group
Furthermore, this age range is youthful and full of energy to work, learn and
experiment with new ideas. Therefore preparing them while young using
60 55.3
49
50 44
39 < 1 year
40
Percentage
1-5 yrs
30 6-10yrs
20 11-15 yrs
5.7 4 >15
10
1 0.3 0.7 1
0
Control Group Experimental Group
101
A high percentage of the learners respondents were novice teachers. Figure 4.3
shows that majority of learners (more than 90%) in the control group and the
experimental group have taught for five years or less. It may be possible that
majority of them may have taught for only a year or less since they are first year
pre-service learners. The teaching method used in teaching these learners in the
Figure 4.4 shows that over 80% of the learners in the control and experimental
group have completed the West Africa Senior Secondary School Certificate
Examination (WASSSCE).
102
These results indicate that most of the learners have been taught mathematics in
their views of mathematics learning and teaching may not differ from those who
are untrained provided they were taught and learned mathematics the same way
The UCC-CCE DBE distance learners were admitted via two methods - direct
and special entrance examination. Figure 4.5 shows that 95.3% and 96% for the
admission).
96
Special Entrance Exams 95.3
4 experimental
Direct 4.7 control
0 50 100
Percentage
103
These learners could not gain direct admission because they did not meet the
(certificates) shows that majority of them did not make a credit pass in either
are likely to have negative views or image about the teaching and learning of
teaching these learners mathematics should have the potential to change their
mathematics. Such an approach should also develop and arouse their interest to
learn mathematics, since at the end of their training they are going to teach the
subject.
Cross tabulations were done for type of school and both teaching class and
school section respectively. Table 4.1 shows that 56% and 64% of the
private schools whereas 44% and 36% of learners in the control and
104
Table 4.1 Respondents Type of School and Teaching Class
Teaching Class
Nursery Junior
and High
Kindergarten Classes Classes Secondary Total
Type of school (%) 1-3(%) 3- 6(%) (%) (%)
Experimental School Public 27.8 33.9 45.7 41.0 36.0
Type Private 72.2 66.1 54.3 59.0 64.0
Total 100.0 100.0 100.0 100.0 100.0
Control School Public 33.3 48.6 53.3 34.9 44.0
Type Private 66.7 51.4 46.7 65.1 56.0
Total 100.0 100.0 100.0 100.0 100.0
In addition Table 4.2 shows that over 80% of the control and experimental
School Section
Nursery Kinderg Primary Junior Total
arten High
Second
ary
Experimental School Public 11.8 33.9 37.4 43.6 36.0
type Private 88.2 66.1 62.6 56.4 64.0
(%) Total 100.0 100.0 100.0 100.0 100.0
Control School Public 22.2 39.2 50.3 48.3 44.0
type Private 77.8 60.8 49.7 61.7 56.0
(%) Total 100.0 100.0 100.0 100.0 100.0
prospective teacher has not experienced such an approach while teaching or has
not been trained with such an approach to teaching. In Ghana, most private
schools have small class sizes (at most 40 pupils in a class) whereas the
105
majority of the public schools have large class sizes at least 40 pupils in a class
(Awuah, 2014; Kraft, 1994). Awuah (2014) for example, agreeably advocates
for reduction in class sizes in Ghana to manageable levels for efficient transfer
of knowledge to the youth. From Figure 4.6, 63% and 64% of the respondents
for control group and experimental group, respectively, teach a class of less than
40 pupils.
A considerable number of the respondents, 23.3% and 25% for the control and
appropriate teaching approaches taking into consideration the large class sizes
106
4.3 Effects of a Problem-Solving Approach on UCC-CCE DBE Distance
Learners Achievement Scores in Mathematics
Students achievements in mathematics in most studies are usually measured in
Data obtained from the learners for both the pre-intervention test and post-
distribution tables for each of the levels of the first four levels of cognitive
learning domains. First, the study tested learners on the knowledge level of
cognitive learning domain and the results (Table 4.3) show that
Pre-test Post-test
SCORE Control Experimental Control Experimental
0 21.7 19.0 70.1 60.1
1 43.9 38.3 22.1 25.3
2 19.8 27.3 4.7 9.1
3 9.5 9.1 3.2 5.1
4 3.5 5.1 0.0 0.6
5 1.6 0.8 0.0 0.0
Total 100.0 100 100.0 100.0
However, in the post-intervention test, 3.2% and 5.7% of the learners in the
could not recall and apply most of the mathematical facts they were taught in
Senior High School, but those from the experimental group managed a higher
the same questions were set for the post-test as in the pre-test, in the post-test,
matching the facts to given examples as was in the pre-test. Therefore, this
result implies that learners thrived on guesswork in the pre-test but the
intervention did not adequately help learners change their mindset to enable
Secondly, the study tested learners on the comprehension level and the results in
Table 4.4 indicate that 49.14% and 51.1% of learners for the control and
experimental groups respectively scored 0. On the flip side are those who
were above average and it is evident that only 20.6% and 17% of learners for
108
However, in the post-test, there were 43.1% and 47.8% of learners for the
of those who were above average, 6.7% and 8.7% of learners for the control and
the two groups in the post-test was not encouraging. This is a demonstration of
The third level was on application and the summary of the performance. Results
in Table 4.5 show that 62.8% and 55.7% of learners for the control and
experimental groups, before for the intervention, respectively, and 79.1% and
71.5% of learners for the control and experimental groups, respectively, after
In terms of those who scored above average and scored at least 5 out of 9, there
were only 1.2% and 2.4% before the intervention, for the control and
109
experimental groups, respectively, and 0% and 1.6% after the intervention for
These results again demonstrated that in both the pre-test and post-test, learners
in the experimental group did not affect the learners adequately enough to
enable them apply the knowledge and skills gained to solve real life or novel
The last category dealt with analysis level of the Blooms first four levels of
cognitive learning. Table 4.6 shows the scores for both the pre-test and post-test.
In both the pre-test and post-test, over 90% of the learners in control and
experimental groups got a score of 0. This means they scored 0 or did not
Pre-test Post-test
SCORE Control Experimental Control Experimental
0 97.6 98.0 98.0 96.4
1 0.8 1.2 1.2 1.6
2 1.2 0.4 0.4 0.4
3 0.0 0.4 0.0 0.4
4 0.4 0.0 0.0 1.2
5 0.0 0.0 0.4 0.0
Total 100.0 100.0 100.0 100.0
The post-test did not show any remarkable improvement in performance for
both the control and the experimental groups despite the fact that they were
110
taught mathematics through a conventional approach and a problem-solving
approach respectively. This indicates that the skills required at the analysis level
The study further developed the means for the scores for the control and
experimental groups in both the pre-test and post-test. The results in Table 4.7
show the mean scores for the pre-test. The values suggest there may be
Table 4.7 Pre-intervention Tests Mean Scores for the Four Levels of
Cognitive Learning Domains
Pair Cognitive Learning Domain Group N Mean
1 Knowledge Control 253 1.3399
Experimental 253 1.4545
2 Comprehension Control 253 1.1344
Experimental 253 1.0356
3 Application Control 253 0.8656
Experimental 253 1.0198
4 Analysis Control 253 0.0474
Experimental 253 0.0316
Based on the means, the study sought to establish whether there was a
significant difference between the control and experimental groups and a paired
t-test was done. This was informed by the assumption that for an experimental
research the entry behavior for control and experimental groups ought to be the
same. The t-test results for the four pairs are as shown in Table 4.8.
111
Table 4.8 Paired Sample t-test for the Pre-intervention Test Scores
Cognitive Learning sig.
Domain Between Groups t df (2 tailed)
Knowledge Control - Experimental -1.113 252 0.267
Comprehension Control - Experimental 0.763 252 0.446
Application Control - Experimental -1.227 252 0.221
Analysis Control -Experimental 0.589 252 0.556
The t values -1.113, 0.763, -1.227 and 0.589 for knowledge, comprehension,
application and analysis respectively, were all not significant at P = 0.05. This
implies that there was no significant difference between the pre-intervention test
scores for the control and experimental groups. Thus the learners entry
Similarly, the study examined the post-test means scores outlined in Table 4.9.
The mean scores for the four pairs showed some difference with the
experimental scores having higher scores than those for the control group. The
112
study was premised on the null hypothesis that there is no significant difference
between the mean scores for the control and experimental groups. The study
conducted a paired t-test to test this hypothesis and the results are shown in
Table 4.10.
Table 4.10 Paired Sample t-test for the Post-intervention Test Scores
Based on the result (shown in Table 4.10) the t values -2.925 and 0.375 for
-0.487 and 1.129 for comprehension and analysis levels, respectively, were not
comprehension, application and analysis. These four cognitive domains are the
113
inform the teacher on what to do and why to do it when preparing a lesson and
setting questions to evaluate his/her teaching and the extent to which learners
have learnt the set objectives. In general, the experimental group performed
group performed better in knowledge and application than the control group.
skills were not clearly understood and mastered by the learners. Learners were
pre-test and post-test were consistent with that obtained by Hallagan, Rule, and
Carlson (2009). The significant changes realized in the study attest to the
mathematics (Hiebert & Wearne, 1993; Marcus & Fey, 2003; NCTM, 1991;
elementary teachers do not come to teacher education perceiving that they know
mathematics, and about schools. Their view about mathematics, about teaching
for the control and experimental groups before intervention and after
were captured both before and after the intervention and there were eleven (11)
statements with responses based on a 4-point Likert scale. Each perception was
perceptions before and after the intervention. The results are summarized in
115
4.4.1.1 Learning Mathematics is to know the rules
The first instrumentalist driven view the study sought to compare was the view
commit them to memory. Result in Table 4.10 shows that 88.6% and 89.3% for
know the rules and follow them strictly. This perception of mathematics
learning held by learners is likely to affect their performance if they were unable
and direct their teaching of mathematics. Learners might have developed this
mathematics.
F 0.245 14.607
Sig. 0.621 0.000
group. These scores before intervention and after intervention were subjected to
116
ANOVA (Table 4.11) that revealed that before intervention (F = 0.245 > 0.05)
there was no significant difference while after intervention (F = 14.607 < 0.05)
the ANOVA results (Table 4.11) indicate a difference resulting from the
Therefore, on the basis of this perception the study rejected the null hypothesis
approach.
Table 4.12 results show that 77.7% and 78.7% of the learners in the control and
intervention there was slight changes in the results with 74.3% and 69.4% of the
F 0.107 0.214
Sig. 0.743 0.644
117
This implies that a large number of learners view the learning of mathematics as
always finding the correct answers. In the same light, the ANOVA result
indicates the control and experimental groups before intervention (F = 0.107 >
0.05) and after intervention (F = 0.214 > 0.05), respectively, had no significant
difference at P=0.05 significance level. This result implies that the intervention
this perception the study accepted the hypothesis that there were no differences
A majority of the learners, 85.3% and 82.6% of the learners in the control and
the best way to learn mathematics is to see an example of the correct method for
solution, either on the blackboard or in a text-book, and then try to do the same
yourself. This is attributable to the fact that most learners learn mathematics by
118
Table 4.13 Perception of Learning Mathematics Means to See a Correct
Example for Solution and then Trying to do the same.
Before Intervention After Intervention
Responses Control Experimental Control Experimental
Strongly Disagree 5.3 5.7 5.3 7.0
Disagree 9.3 11.7 7.0 14.3
Agree 37.3 36.3 40.7 31.3
Strongly Agree 48.0 46.3 47.0 47.3
F 0.446 0.268
Sig. 0.505 0.605
makes them loose track of alternative ways of solving the problem, using
familiar ideas or procedures. After the intervention, there was a reduction of the
respectively, agreeing to this fact. However, upon finding out if the difference
was statistically significant, Result in Table 4.13 presents the ANOVA for both
before intervention (F = 0.446 > 0.05) and after intervention (F = 0.268 > 0.05)
the blackboard or in a text book, and then try to do the same yourself. Again, it
this perception the study accepted the hypothesis that there were no differences
119
4.4.1.4 Learning Mathematics means to cram and practice enough
Table 4.14 results show that before intervention a majority, 90.6 % and 91.3%,
of learners in the control group and the experimental group, respectively, before
intervention, perceive that if one can cram and practice enough mathematics,
they are used to. A student teacher who holds such a view in learning
mathematics.
F 4.519 3.287
Sig. 0.034 0.070
reforms cannot take place unless teachers (which may include prospective
teachers) deeply ingrained perceptions about mathematics and its teaching and
learning change. The results in Table 4.14 also show that after intervention there
120
were 89.7% and 78.6% of learners in both the control group and the
the difference was statistically significant, Table 4.14 result indicates that before
This means the learners held different levels of perceptions before the
0.05) however, as the result Table 4.14 indicates the change in perception
registered was not statistically significant. This implies that the intervention of a
difference. Therefore, on the basis of this perception the study accepted the null
approach.
believe that all problems in mathematics have an answer and that there is only
one answer and one correct solution method. The question one may ask is if a
student copies and gets an answer correct, can one conclude that the student
121
Similarly, the results in Table 4.15 show that 79.6% and 78.6% of learners in
agreed that those who get right answers in mathematics have understood what
F 0.460 4.554
Sig. 0.498 0.033
experiences they have had in mathematics classes and from the attitudes and
Table 4.15 show that after intervention, 79.0% and 70.3% of learners in the
control group and experimental groups, respectively, agreed to the view that
those who get right answers in mathematics have understood what they have
difference between the control and experimental groups in both before and after
intervention (F= 0.460 > 0.05), learners in the control and experimental groups
were of the same view that learning mathematics means to get right answers in
122
after intervention (F = 4.554 < 0.05) there was statistical significance. This
result implies that the intervention of a problem-solving approach that was used
Therefore, on the basis of this perception, the study rejected the null hypothesis
approach.
best way to arrive at an answer. Table 4.16 shows that 56.7% and 58% of
123
Table 4.16 Perception on Mathematics Learning is to learn a Set of
Algorithms and Rules that cover all Possibilities
Before Intervention After Intervention
Control Experimental Control Experimental
Responses
Strongly Disagree 13.0 14.7 11.7 13.7
Disagree 30.3 27.3 36.0 29.3
Agree 43.0 47.3 38.7 41.0
Strongly Agree 13.7 10.7 13.7 16.0
F 0.217 0.470
Sig. 0.642 0.493
After the intervention, there were 52.4% and 57% in the control and
mathematics. The ANOVA results (F = 0.217 > 0.05) indicate that before
that learning mathematics means to learn a set of algorithms and rules that cover
all possibilities. This means learners in the control group and experimental
group entry views on this statement are not the same. After learning
0.05). Therefore, on the basis of this perception the study accepted the null
approach.
124
4.4.1.7 Mathematics learning is for the gifted
With regard to the perception that mathematics learning is for the gifted, Foss
teachers hold the view that ability to learn mathematics is innate. Contrary to
the above sentiments, Table 4.17 shows that only 18% and 17.4% of learners in
the fact that mathematics learning is for the gifted. After intervention there were
16.4% and 17.4% agreeing that mathematics learning is for the gifted. This
means for both groups, both before intervention and after intervention the
F 3.549 0.018
Sig. 0.060 0.894
This high held view by learners that mathematics learning is not only for the
learning shows the preparedness of learners to learn and compete equally with
ANOVA results (Table 4.17) show that there was no significant difference
before intervention (F = 3.549 > 0.05) and after intervention (F = 0.018 > 0.05)
125
in the perception that mathematics is for the gifted in both groups. The study
concluded that the control and experimental groups were of the same view.
Therefore, on the basis of this perception the study accepted the null hypothesis
approach.
mathematics rules and methods by rote is the key to knowing and understanding
mathematics. Several studies confirm this view as one of the predominant views
1993; Nisbert & Warren, 2000; Southwell & Khamis, 1992). Upon this
backdrop results in Table 4.18 shows that 74.3% and 74% of learners in the
fact that learning mathematics rules and methods by rote is the key to knowing
to this perception.
126
Table 4.18 Perceptions on Mathematics Learning as Learning Rules and
Methods by Rote
Before Intervention After Intervention
Responses Control Experimental Control Experimental
Strongly Disagree 8.0 10.3 10.0 14.0
Disagree 17.7 15.7 17.3 22.3
Agree 44.0 46.7 45.0 42.3
Strongly Agree 30.3 27.3 27.7 21.3
F 0.588 6.375
Sig. 0.444 0.012
Looking at the percentages for the control and experimental groups before
0.588 > 0.05) indicates that there was no significant difference in perception
between the control group and the experimental group. However, after
intervention, the ANOVA results show that the change realized in learners
perception is statistically significant (F= 6.375 < 0.05) to warrant that the
on the basis of this perception, the study rejected the null hypothesis that there
127
suggest that the majority of learners, especially those in the control group, still
F 1.028 6.485
Sig. 0.311 0.011
Conversely, after intervention 77.7% and 64.3% of the learners in the control
The ANOVA results for before intervention (F = 1.028 > 0.05) show that there
experimental groups. After the intervention the results (F = 6.485 < 0.05)
study rejected the null hypothesis that there were no differences in UCC-CCE
128
4.4.1.10 Conventional approach is the best way to teach students to solve
mathematics problems
The results in Table 4.20 show that before intervention, there were 63.3% and
64.7% of the learners in the control and experimental groups, respectively, who
agreed that conventional approach is the best way to teach students to solve
F 0.222 20.388
Sig. 0.637 0.000
Conversely, after the intervention there were 61.7% and 49% of the learners in
the control and experimental groups, respectively, who agreed to the perception.
further analysis.
The ANOVA results for pre-test (F = 0.222 > 0.05) show that there is no
groups. After the intervention, the results (F = 20.388 < 0.05) signify a
rejected the null hypothesis that there were no differences in UCC-CCE DBE
problem-solving approach.
views, the results in Table 4.21 show that before intervention there were 69%
and 71.7% of the learners in the control and experimental groups, respectively,
agreeing to the view that students should be given notes to copy when learning
mathematics.
F 0.389 18.202
Sig. 0.533 0.000
learning is not the same as copying notes. The doing and talking promotes
130
learning and retention. The recording of the thinking processes and procedures
is itself copying notes for keep. After learners had learned through conventional
71.7% and 54.7% of the learners in the control and experimental groups,
after the intervention necessitated further analysis. The ANOVA results for
After the intervention, the results (F = 18.202 < 0.05) signify a difference in
Therefore on the basis of this perception the study rejected the null hypothesis
approach.
and routinely achieve his or her goals due to some kind of obstacle or challenge.
The ability to solve problems is considered to be one of the most complex and
individuals must first become aware of a difference between the current state of
affairs and the state of affairs that corresponds to the satisfaction of their goals.
131
problem. This is also called problem finding. Individuals then need to engage
sub-goals and steps through which the problem may be solved (also called
those sub-goals until the situation reaches a satisfactory state. Throughout the
necessary, reconsider their goals and actions. For instance, individuals may face
may have to reconsider their understanding of the problem or the actions they
captured both before and after the intervention. There were 11 statements with
132
perceptions of the control and experimental groups before and after the
intervention.
process wherein students encounter a problem a question for which they have
apply to get an answer (Schoenfeld, 1992). They must then read the problem
carefully, analyze it for whatever information it has, and examine their own
mathematical knowledge to see if they can come up with a strategy that will
help them find a solution. Based on this bright backdrop the study found that for
why a methods works (conceptual learning) than to learn rules by heart, there
were 73.3% and 75.6% of learners for the control group and experimental
group, respectively, before the intervention who agreed to this perception. After
the intervention, there were 65.6% and 81.3% of learners for the control group
and experimental group, respectively, who agreed (Table 4.22). The before
133
Table 4.22 Perception on Mathematics Learning is to Understand why a
Method Works than to Learn Rules
Before Intervention After Intervention
Responses Control Experimental Control Experimental
Strongly Disagree 8.3 6.7 13.3 6.7
Disagree 18.3 17.7 21.0 12.0
Agree 38.3 39.3 32.3 43.0
Strongly Agree 35.0 36.3 33.3 38.3
F 0.509 12.347
Sig. 0.476 0.000
The ANOVA results show that unlike before intervention (F = 0.509 > 0.05),
there was after intervention a significant difference (F = 12.347 < 0.05) in this
intervention coupled with the significant difference in Table 4.22 suggest that a
Therefore, on the basis of this perception, the study rejected the null hypothesis
approach.
level because peers discuss on friendly and equal terms. The results in Table
4.23 show that 85% and 84.3% of learners in the control and experimental
groups, respectively, before the intervention, agreed that students should learn
134
mathematics in groups. However, after intervention 17.7% and 91.1% of
learners in both control and experimental groups, respectively, agreed. The big
percentage drop in learners in the control group response after the intervention
may be as a result that their expectation of learning in groups was not met
F 0.022 621.320
Sig. 0.882 0.000
The ANOVA results shown in Table 4.23 affirms that before intervention (F =
0.022 > 0.05) there was no significant difference between the control and the
experimental groups. However, after intervention the results (F = 621.32 < 0.05)
study rejected the null hypothesis that there were no differences in UCC-CCE
problems and problem solving often involve interaction with other individuals.
135
A person may be asked to solve a problem for another person and may need to
one of the actions necessary to solve the problem (OECD, 2012). The results in
Table 4.24 show that 90.6% and 92.7% of learners from the control and
the view that students should ask questions during mathematics lessons. After
the intervention, there were 64% and 92.3% of learners in the control and
those in agreement in the control group and therefore the study sought to
F 1.582 72.560
Sig. 0.209 0.000
The ANOVA results (Table 4.24) show that before intervention (F = 1.582 >
control and experimental groups. After the intervention, The ANOVA results (F
136
basis of this perception the study rejected the null hypothesis that there were no
Researchers the world over are looking for ways by which one may use problem
solving as a teaching tool. The Principles and Standards for School Mathematics
In this way, new ideas, techniques and mathematical relationships emerge and
and reinforce the need to understand and use various strategies, mathematical
properties, and relationships. In tandem with the above sentiments, the study
mathematics. The results in Table 4.25 show that before intervention, there were
77% and 84% of the learners in the control and experimental groups,
respectively, who agreed. Contrary, after intervention, there were 14.3% and
47% of the learners in the control and experimental groups, respectively, who
experimental and control both before and after intervention. This big drop in
percent in both groups may be as a result that the learners were exposed to more
137
Table 4.25 Perceptions on a Problem-solving Method is the Best Way of
Teaching Mathematics
Before Intervention After Intervention
Responses Control Experimental Control Experimental
Strongly Disagree 5.3 5.0 45.0 24.0
Disagree 17.7 11.0 40.0 29.0
Agree 47.3 51.3 9.0 26.0
Strongly Agree 29.7 32.7 5.3 21.0
F 2.438 78.305
Sig. 0.119 0.000
The ANOVA results Table 4.25 indicate that before intervention (F = 2.438 >
method of teaching. Learners in both the control and experimental groups were
almost of the same view before intervention that effective mathematics learning
was used in teaching. Therefore, on the basis of this perception, the study
rejected the null hypothesis that there were no differences in UCC-CCE DBE
problem-solving approach.
solving, one has to reflect on the situation in order to identify the proper
arrangement of decisions and actions that may lead to a solution (OECD, 2012).
Thus, the use of a novel problem in teaching and learning mathematics through
4.26 show that before intervention there were 69.3% and 66.3%% of the
learners in the control and experimental groups, respectively, who agreed that
students should often be confronted with novel problems. After the intervention
there were 19.3% and 33.7% of the learners in the control and experimental
cases after the intervention, confirm or support the reasons given for the drop in
F 0.021 42.579
Sig. 0.886 0.000
Indeed there is a noticeable change in both the experimental and control both
before and after intervention. The ANOVA results in Table 4.26 (F = 0.021 >
139
difference between the control group and the experimental group views about
mathematics learning being effective when students are often confronted with
novel problems. This means the two groups had the same perception that
approach that was used in teaching. Therefore, on the basis of this perception
the study rejects the null hypothesis that there were no differences in UCC-
CCE DBE distance learners perceptions before and after learning mathematics
class by the teacher (Allevato & Onuchic, 2007; Hiebert & Wearne, 1993) to
show that before intervention, 76.3% and 86% of learners in the control and
their own.
140
Table 4.27 Perceptions on Mathematics Learning as Teacher Creating
Conditions to Stimulate self-learning
Before Intervention After Intervention
Responses Control Experimental Control Experimental
Strongly Disagree 7.0 6.3 41.3 30.3
Disagree 16.7 7.7 43.0 36.7
Agree 41.6 46.0 11.0 15.3
Strongly Agree 34.7 40.0 4.3 17.7
F 4.948 29.221
Sig. 0.026 0.000
However, after intervention there were 15.3% and 23% (perhaps students in
both groups were not stimulated adequately to learn mathematics own their
own) of the learners in the control and experimental groups respectively who
before intervention and after intervention, they are both large. The ANOVA
experimental groups for both categories. The study thus concludes that the
perception the study accepts the null hypothesis that there were no differences
141
4.4.2.7 Mathematics learning as use of worksheet to learn
The results in Table 4.28 indicate that before the intervention 82.7% and 85.3%
worksheets.
F 0.470 6.017
Sig. 0.493 0.014
After intervention, only 18.4% and 31% of learners in the control and
worksheets not adequately used or not at all used during the intervention. It is
evident that the difference between the control and experimental groups is larger
ANOVA results. The ANOVA results before intervention (F = 0.470 > 0.05)
indicate that intervention learners in both groups had the same perception about
Table 4.28 results (F = 6.017 < 0.05) show a statistical significant difference in
Therefore, on the basis of this perception the study rejected the null hypothesis
approach.
With regard to learning mathematics in groups, results in Table 4.29 show that
91% and 93% respondents in the control and experimental groups, respectively,
agreed prior to intervention. After intervention, there were only 19.7% and 28%
who agreed. The drop in percentages in both groups may be that the learners
F 5.010 28.224
Sig. 0.026 0.000
intervention and after intervention, those after intervention are large but this
notwithstanding the ANOVA for both before intervention (F = 5.010 < 0.05)
between the control and experimental groups for both categories. The study thus
143
concludes that the intervention of a problem-solving approach that was used in
teaching did not have considerable effects on this perception. Therefore on the
basis of this perception the study accepted the null hypothesis that there were
learners in both groups. Before intervention, the learners perceived the view of
learning mathematics; hence 72% and 75.3% of the learners for the control and
experimental groups respectively agreed with the perception (see Table 4.30).
F 0.664 5.082
Sig. 0.416 0.025
After the intervention, there were 16% and 33.3% of the learners in the control
and experimental groups, respectively, who agreed that students should discover
144
for themselves the meanings of mathematical concepts. The drop in percentages
mathematics by discovering for themselves was not fully met during the
intervention (F = 0.664 > 0.05) indicate that learners in both groups had the
themselves. However, after the intervention ANOVA result (F = 5.082 < 0.05)
teachers should allow them to discover for themselves the desired conceptual
perception the study rejects the null hypothesis that there were no differences
learning when the new approach is able to positively affect any retrogressive
perceptions about mathematics teaching and learning that they might have
145
developed during their elementary and secondary school days. The study
realized that before intervention, the entry perceptions of both groups were, on
the majority, the same and were of more instrumentalist driven perceptions than
teaching approach for the control group, not much significant changes were
and learning. On the other hand, the experimental group who received a
(1991) who postulates that the task of modifying long-held, deeply rooted
The study also revealed that a problem-solving approach used could not
146
4.5 Effects of a Problem-Solving Approach on Facilitators Perceptions
about Learning and Teaching of Mathematics
The results are discussed based on these two categories: learning and teaching.
It is the position of this study that a teachers perceptions about the learning of
mathematics can be a barrier to and impede acceptance of any novel ideas and
For example, it has been proposed that one factor that has influenced the lack of
beliefs about mathematics teaching and learning (Stigler & Hiebert, 1999). The
ability of such a new approach to change the teachers perceptions can ensure its
147
4.5.1.1 Learning mathematics in groups
The NCTM (1989) asserts that small group; cooperative learning can be used to
and the making of mathematical connections. Thus, the use of group work
done in groups. Before training, 75% strongly agreed to the statement while
after training all 100% strongly agreed that learners should be taught
have influenced their perception, thus the change. Lester (2013) suggests that
when research concerns are with classroom instruction, we should give attention
148
to groups and whole classes, and that small groups can serve as an appropriate
Sound classroom discourses are orchestrated by the questions that a teacher asks
the teacher does more of the questioning than the learners. This study
questions and results shown in Figure 4.8 indicate that there were 25% and 75%
who agreed or strongly agreed for both before and after training, respectively,
mathematics.
149
4.5.1.3 Students solving problems on their own
them on their own. In this light, Stanic and Kilpatrick (1989) and NCTM (1989)
note that if mathematics respondents present a problem and develop the skills
needed to solve that problem; it will be more motivational than teaching the
skills without a context. Such motivation gives problem solving special value as
a vehicle for learning new concepts and skills or the reinforcement of skills
already acquired. The results presented in Figure 4.9 indicate that before
allowed to solve problems on their own. However, after the training workshop,
150
The study underscores that before training, the majority of the respondents
mathematics as good and interesting. However after the training and realizing
the respondents who were in agreement before intervention gave up and opted
for disagreement. This result is congruent with McIntosh and Jarretts (2000)
assertion that some teachers do not have ready attitudes to take up a problem-
adaptable. In addition, Resnick (1987) expressed the belief that school should
focus its efforts on preparing people to be good adaptive learners, so that they
can perform effectively when situations are unpredictable and task demands
change. With regards to this background, the study established the respondents
practical work. The results in Figure 4.10 show that before training, all the
respondents (50% agreed and 50% strongly agreed) agreed that teachers should
151
Figure 4.10: Teacher to Encourage Students
However after the training 62.5% agreed or strongly agreed while 37.5%
respondents were made to discover mathematical rules on their own during the
benefit from comparing, reflecting on, and discussing multiple solution methods
encouraging students to share and compare their own thinking and problem-
their solutions with their peers during mathematics lessons. The results shown
in Figure 4.11 indicate that before the training, 87.5% of the respondents agreed
that the teacher should encourage students to discuss their solutions with their
After training, 100% of the respondents agreed to this perception after they had
seen the importance of allowing learners to discuss their solutions with their
peers in groups. The study attributes these results to effects of the problem-
research supports the value of using comparison and contrast to promote general
153
learning - identifying similarities and differences in multiple examples has
The Education Alliance (2006) noted the use of cooperative learning strategies
mathematical concepts (Leigh, 2006). In the study, before training (Figure 4.12)
all the respondents (50% agreed and 50% strongly agreed) agreed that
154
However after training, 12.5% indicated disagreement to the statement. There
was also a notable increase from 50% to 75% of the respondents that strongly
good for efficient learning of mathematics. These finding are in line with Reid
(1992) who alleges that cooperative group work in mathematics learning goes a
4.5.1.7 Students learn more from problems that do not have procedure
for solution
There is a growing awareness that many students are successfully learning how
there is a concern that the system may not be providing students with the
With this backdrop, the study established that before the training 50% of the
respondents disagreed and 50% agreed that students learn more from problems
that do not have a given procedure for their solutions. Conversely, after the
training, all 100% of the respondents strongly agreed (see Figure 4.13).
155
Figure 4.13: No Procedure for Solution
students who have been taught mathematics through a procedural approach tend
to score more highly in the areas related to computation than those related to
must encourage students to justify what they say and do to reveal their thinking
and logic (Pirie & Kieren, 1992). In their submission, McCrone, Martin,
Dindyal, and Wallace (2002) argue that if teachers focus on the problem
structure and the justification of answers by students, their students will have a
a better sense of the need for proving. Therefore this study sought the
156
respondents perception on whether it is important for students to argue out their
answers. The results presented in Figure 4.14 show that before training, 12.5%
student to argue why his or her answer in mathematics is correct while 87.5%
After training, all of the respondents (100%) indicated that they strongly agreed
Featherstone, (1992) who assert that one prominent method for producing a
157
reflect on their individual learning processes, and have more learner autonomy
(Christensen, 2003).
After training, all respondents (100%) strongly agreed with the perception as
158
4.5.1.10 Students learn mathematics from seeing different ways of solving
the problem
There were 62.5% of respondents who agreed and 37.5% who strongly agreed
to the perception that students learn mathematics from seeing different ways of
problems (McIntosh & Jarrett, 2000). This study therefore sought to identify the
A conventional approach has been the approach that the majority of teachers use
However, before training, 75% of the respondents disagreed with the statement
After training, 62.5% disagreed that a conventional approach was the best.
There were 37.5% that agreed to the statement that a conventional approach is
the best way to teach students to solve mathematical problem. The result show
the best way to teach students to solve mathematical problem. This result
160
to a problem-solving approach; rather it strengthened their belief in
conventional approach.
this regard, the study established respondents perceptions that whole class
teaching is more effective than facilitating. The results shown in Figure 4.18
shows that all respondents, before and after training, had 25% and 75% agreeing
than facilitating.
The results show no change of perception before and after the problem solving
161
4.5.2.3 Teachers ought to create an environment to stimulate students to
construct their own conceptual knowledge
Figure 4.19, all the respondents before training agreed that teaching
However, after the training, there were 12.5% of the respondents who disagreed
knowing when it is appropriate to intervene, and when to step back and let
162
learners make their own way (Lester et al., 1994). With regard to a mathematics
the respondents, before training, that agreed and 37.5% that strongly agreed to
solving in class. However, after training all (100%) respondents strongly agreed
163
4.5.2.5 Allowing Students to Discover for Themselves Leads to
Incompletion of Syllabus
rules for themselves by mathematics teachers will lead to not completing the
syllabus. Results in Figure 4.21 indicate that before training, 75% of the
students to discover rules for themselves will lead to not completing the
syllabus, while 25% of them were of a contrary thought that it would lead not
The issue of not completing the syllabus or concern about content coverage has
been the anxiety of many mathematics teachers hence making them teach
(McIntosh & Jarrett, 2000). After training, the results (from Figure 4.21) show
164
that all the respondents agreed that allowing students to discover rules for
with its own challenges. According to McIntosh and Jarrett (2000), many
They further argue that even if they encountered problem solving in their
college methods courses, once in the classroom, they often conform to the
conventional methods that hold sway in most schools. McIntosh and Jarrett
noted that teachers today have failed to be agents of change because teachers are
often caught between daily pressure from colleagues, parents, and others to
perform highly on standardized tests that measure basic skills, not performance
of standards-based material).
In this section, the study will discuss questions that focused on the negative
approach in their academic ladder. The results in Figure 4.22 show that 87.5%
165
of the respondents first encounter problem solving at the college level while
87.5
100
80
Percentage
60
40 12.5
20
0
secondary college
the techniques used for teaching at the primary and secondary levels directly
impact how one perceives the teaching and learning process. If the
solving.
The study went further to established how frequently the respondents used
problem solving approach and the results in Figure 4.23 indicate that 87.5%
100 87.5
80
Percentage
60
40
12.5
20 0
0
Never at times always
This signifies that the use of a problem-solving approach has a long way to go
solving is not easy since many of were taught by remembering facts whether or
not they were related to each other, whether or not were interested in the
subject, and in some instances were taught by rote. In fact, many teachers may
say that problem solving in their particular subject area is not possible, not
helpful, or only possible in limited parts of the subject matter. For example in a
study conducted in Ethiopia, Bishaw (2011) established that teachers have low
level beliefs regarding the use of a problem-solving approach and that teachers
167
This study established from respondents on what they regarded as negative
applicable to small class size, pegs high demand on the teacher, require lots of
study established that 75.0% of the respondents said the approach wastes time,
only applicable to small class size and pegs high demand on the teacher
causes a delay in the completion of the syllabus. They felt that it takes a
approach which spills out of the government stipulated school terms and thus
168
they were concerned that parts of the mathematics curriculum will need to be
omitted.
Pertaining to class size, the respondents expressed the fear that learners have
never met open-ended problems before and with the growing class size the
facilitators will not effectively reach all the students to ensure they have
in more open problem-solving situations, some learners will feel insecure. They
algorithms to practice and copy because of their inability to reach all of them in
a class lesson.
the teacher, the respondents noted that this approach comes with lots of new
demands on the facilitator that bring about discomfort because it is new to most
is not possible to use this approach to teach without first experiencing the
approach as a student.
The other challenge mentioned was that this approach requires a lot of resources
to implement. The 62.5% of the respondents who agreed to this challenge noted
169
that a problem-solving approach requires the use of different teaching resources
to facilitate learning, which are not readily available in the study centres. Some
cited that they only have one set of reference materials (a non-activity oriented
text book or module) and varying the novel questions will be challenging
respondents who noted that when most of the work is done in groups it becomes
that it kills the spirit of competition because learners work harder when they
know that when they get correct answers alone they will also do well in
The study went further and investigated the challenges respondents encountered
mathematics. The challenges listed included: lack of time to prepare for the
lecture, the teaching module they used was not activity oriented too many
170
100
100
90
80 75
70 62.5 62.5
Percentage
60
50 50
50
37.5
40
30
20
10
0
lacked time teaching to many slow external negative rejected by
to prepare module not learners to method that examination student colleagues
for the activity manage in delays pressure reception
lecture oriented groups completion
of syllabus
In relation to time, 50% of the respondents said they did not have enough time
to prepare for the lecture, citing the complexity involved in finding good
They also mentioned the teaching modules approved by the institutions. All of
the respondents noted that the modules were not designed with activities that
promote the use of a problem solving approach. The study could not supply the
were guided during the workshop to write out activity oriented problems from
171
addition, they were supplied with set of hand-outs containing problems for use
Class size was also a challenge with 75% of the facilitators mentioning that the
registering this concern, while 50% of the respondents noted rejection of the
approach from fellow facilitators. That is, fellow mathematics facilitators in the
same study centre they teach with refused to use the approach to teach
slow method that delays completion of the syllabus and external examination
The findings in the results above are in agreement with those obtained by
Anderson (2005). For example, Anderson (2005) found that teachers agreed
they needed considerable support in the form of time and resources so that they
172
4.7 Chapter Summary
This chapter has considered results, interpretation and discussions. The study
improving their achievement in mathematics. It was also found that the views of
study also observed that before the training workshop, the facilitators who teach
what mathematics teaching and learning consists of. Their views were divided
solving approach during the workshop, their views became more oriented
stage could not fully put into practice their newly formed perceptions about
173
CHAPTER FIVE
SUMMARY OF FINDINGS, CONCLUSIONS AND
RECOMMENDATIONS
5.1 Introduction
Students achievement in mathematics is largely dependent on the instructional
approach. Therefore the purpose of this study was to investigate the effect a
DBE DLs perceptions about mathematics teaching and learning before and
and discussed based on the above objectives using Microsoft Excel and
Statistical Package for Social Sciences (SPSS) package. It also tested and
the study, draws conclusions, makes recommendations and suggests areas for
further research.
This section recapitulates the findings of the study thematically where each
hypothesis.
174
The difference a problem-solving approach made on UCC-CCE DBE DLs
mathematics. The study further developed the means for the scores for the
control and experimental groups in both the pre-test and post-test which
suggested a differences between the pairs. For the pre-test, a paired t-test was
done premised on an assumption that for any experimental research the entry
behavior for control and experimental groups ought to be the same. The results -
1.113, 0.763, -1.227 and 0.589 for knowledge, comprehension, application and
analysis respectively, were all not significant at = 0.05. This implies that there
was no significant difference between the pre-intervention test scores for the
control and experimental groups. Thus the learners entry behaviors were found
to be similar before the intervention. The study also examined the post-
intervention test means scores which suggested some difference. The study
175
again conducted a paired t-test and the results -2.925 and 0.375 for knowledge
and 1.129 for comprehension and analysis levels, respectively, were not
approach
perceptions before and after the intervention. Findings in the percentage scores
176
Effects of a problem-solving approach on UCC-CCE mathematics facilitators
perceptions
This objective was analysed using percentages which revealed that before the
after being trained on the use of the intervention, their views became more
wastes time, the teaching modules are not activity oriented, there is the use of an
examination driven syllabus, there was a short time for professional or in-
domains. The study foresees that with the full implementation of a problem-
solving approach, students are likely to operate on all the first four levels of
learning domains.
mathematics. However, after being taught using the intervention, their views
classrooms. Accordingly Barkatsas and Malone (2005) say that initial teacher
education has the power to influence the teaching practices and therefore
178
problem-solving approach has a great potential of transforming the
solving approach were more perceptual (time wasting, large class size, other
teachers not embracing the approach) than structural (module lacks problem
learners.
5.4 Conclusions
This study which was about the effects of a problem-solving approach on the
Cape Coast has resulted in four (4) main conclusions based on the findings.
Firstly, based on the findings that the first year DBE distance learner in Ghanas
a short time and without non-routine activity oriented text books), performed
implemented has the potential of making learners perform better across all the
179
In objective two the study established that the intervention of a problem-solving
Thirdly, from the findings of objective three that the facilitators, before being
learning, nonetheless, after being trained for three days, their perceptions
that with longer and intensive training, a problem-solving approach has a great
Based on the findings from the fourth objective, the challenges encountered
comprised time wasting, only applicable to small class size, puts high
180
with an examination driven curriculum, and lacking in individual evaluation,
students mathematics.
Therefore basing on the findings of this study the study concludes that the
5.5 Recommendations
Based on the findings of this study, recommendations are made in two areas -
181
5.5.1 Policy Recommendation
The study, based on its findings, makes a policy recommendation into two parts:
for UCC-CCE Mathematics Teacher Educators and Administrators, and for the
Division (TED).
i. It has been mentioned previously in this study that central to raising student
students. Students who receive sound and appropriate instruction acquire more
knowledge than their peers without such instruction. Therefore, the study
182
ii. It has also been revealed in this study, as well as confirmed with other studies,
teaching and learning that are predominantly instrumentalist driven. Since this
study found that a problem-solving approach was able to change some of their
iii. Based on the fact that the three-day training workshop for the facilitators in a
last longer so that facilitators can be totally immersed in the new teaching
formed problem solving driven views to the fullest in the classroom during FTF
problem-solving approach.
183
wasting, large class size, other teachers not embracing the approach) than
will train the mathematics facilitators in the real classroom context using
reflective practices or the use of case study. In addition, the study also
training workshops. This will offer opportunities for others to learn how they
Vision 2020 of Ghana outlines that the government should substitute teaching
methods that promote inquiry and problem-solving for those based on rote
learning. It is upon this background and based on the studys findings that this
184
study has the following recommendations to the Ghana Education Service
i. The curriculum division of the GES should improve and transform the
and encouraging them to reflect and describe what they believe may be the
v. It is clear from this study that a focus is needed on teachers practice and
185
TED and mathematics educators in UCC should collaborate to give regular
recommended.
This chapter has dealt with summary of the study findings, implications of the
has also opened a new field for the study of mathematics using a problem-
187
REFERENCES
boys and girls? Essays in Education Volume 21, pp. 1-7. Retrieved
from http://www.usca.edu/essays/vol212007/adeleke.pdf
11.org/document/get/453.
of Teachers of Mathematics.
net.au/documents/RP42005.pdf
188
Auger, W.F., & Rich, S.J. (2007). Curriculum theory and methods: Perspectives
Awuah, P. (2014, March 16). Class sizes must be reduced. Ghana News
http://www.ghanaweb.com/GhanaHomePage/NewsArchive/artikel.ph
p?ID=303510
Press.
189
Ball, D. L. & Bass, H., (2000). Interweaving content and pedagogy in teaching
Ball, D. L., & Feiman-Nemser, S. (1988). Using textbooks and teachers guides:
Barab, S., Dodge, T., Thomas, M. K., Jackson, C. & Tuzun, H. (2007). Our
designs and the social agendas they carry. Journal of the Learning
190
Bay, J. M. (2000). Linking problem solving to student achievement in
1(2), 8 - 13.
388638).
Ben-Chaim, D., Fey, J. T., Fitzgerald, W. M., Benedetto, C., & Miller. J.
273.
Bilgin, I., Senocak, E., & Sozbilir, M. (2008). The effects of problem-based
view/653826(1).
191
Bloom, B. S. (1956). Taxonomy of educational objectives, handbook 1:
29(1), 41-62.
Brooks, J., & Brooks, M. (1999). In search of understanding: The case for
Curriculum Development.
Brown, G. (2004). How students learn: Key guides for effective teaching in
192
Bruner, J. (1985). Vygotsky: An historical and conceptual perspective. In J. V.
Burron, B., James, L., & Ambrosio, A. (1993). The effects of cooperative
707.
Cai, J., Moyer, J. C., & Grochowski, N. J. (1999). Making the mean
Carpenter, T. P., Megan L. F., Jacobs, V. R., Fennema, E., & Empson, S. B.
Chang, S.C., Kaur, B., Koay, P.L. & Lee, N.H. (2001). An exploratory analysis
193
Christensen, T. K. (2003). Finding the balance: Constructivist pedagogy in
243.
Cobb, P., Wood T., Yackel E., Nicholls, J., Wheatley, G., Trigatti, B., &
22, 329.
Cohen, L., Manion, L., & Morrison, K. (2004). Research methods in education
Cooper, D. R., & Schindler, P. S. (2001). Business research methods (7th ed).
Boston: McGraw-Hill.
Crespo, S. (2000). Seeing more than right and wrong answers: Prospective
194
Creswell, J. W. (2009). Research design: Qualitative, quantitative, and mixed
Publications, Inc.
York: MacMillan.
195
Dyson, A. H. (2004). Writing and the sea of voices: Oral language in, around
and about writing. In R.B. Rudell & N.J. Unrau (Eds.), Theoretical
models and processes of reading (5th ed.). (pp. 146-162), Newark, DE:
Eggen, P. D., & Kauchak, D.P. (2003). Teaching and learning: research-based
Publishing Company.
Ely, M., Anzul, M., Friedman, T., Garner, D., & Steinmetz, A. M. (1991).
ex.ac.uk/
196
Esmonde, I. (2009). Ideas and identities: Supporting equity in cooperative
1043.
Virginia: NCTM.
197
Feiman-Nemser, S., McDiarmid, G., Melnick, S., & Parker, M. (1987).
Connection, 5, 29-36.
Gay, L., R., & Airasian, P. (2000). Educational research: competencies for
analysis and application (6th ed.). Upper Saddle River, New Jersey:
198
Gllies, R. (2002). The residual effects of cooperative learning experiences: A
mathematics teaching and learning. New York, NY: Simon & Schuster
Macmillan.
Ltd.
199
Hallagan, J. E., Rule, A, C., & Carlson, L. F. (2009), Elementary school pre-
from http://www.math.umt.edu/tmme/vol6no1and2/TMME_vol6nos1
and2_ article15_pp.201_206.pdf
Hale, M. S., & City, E. A. (2002). But how do you do that?: Decision making
for the seminar facilitator. In J. Holden & J.S. Schmit (Eds.), Inquiry
200
Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray, H., ...
Hiebert, J., & Wearne D. (1993). Instructional task, classroom discourse, and
Hines, M.T. (2008) African American children and math problem solving in
http://www.nationalforum.com/Electronic%20Journal%20Volumes/Hin
es,%20Mack%20African%20American%20and%20Mathematical%20Pr
oblem%20Solving.pdf
2007.
201
Johnson, D. W., & Johnson, R. T. (1999). Learning together and alone:
Curriculum Development.
methods.htm
Jones, G. (1996). The constructivist leader. In J. Rhoton, & P. Bower (Eds), Issues in
Knowles, M. (1990). The Adult Learner: A neglected species ( 4th ed.). Huston:
Gulf Publishing.
202
Knowles, M. (1984). The modern practice of adult education: Andragogy
Education.
Group.
professionals? What are the issues and how are they resolved? In D.
203
Lararowtiz, R., Hertz-Lazaraowitz, R., & Baird, J. (1994). Learning science in
Lazarowitz, R., Baird, J., & Bowlden, V. (1996). Teaching biology in a group
447-462.
from http://scimath.unl.edu/MIM/files/research/HillenK.pdf
Lester, F. K.Jr., Masingila, J. O., Mau, S. T., Lambdin, D. V., dos Santon, V.
M., & Raymond, A.M. (1994). Learning how to teach via problem
204
Liljedahl, P. (2005). Mathematical discovery and affect: The effect of aha!
36(2&3), 219-236.
30, 1087-1101.
Higher Education.
Marcus, R., & Fey, J. T. (2003). Selecting quality tasks for problem-based
205
Marland, P. (1997). Towards more effective open and distance teaching.
Mathematics, Inc.
http://problembasedlearninginmath.wikispaces.com/file/view/McIntosh+
and+Jarrett+problem+solving.pdf
206
Murphy, E. (1997). Characteristics of constructivist learning and teaching.
elmurphy/emurphy/cle3.html.
Myers, C. & Jones, T.B. (1993). Promoting active learning: Strategies for the
http://www.nctm.org/news/content.aspx?id=25713
Nisbert, S., & Warren, E. (2000). Primary school teachers beliefs relating to
13(2), 3447.
207
Nystrand, M. (1996). Opening dialogue: Understanding the dynamics of
College Press.
/104.htm
http://www.oecd.org/edu/school/programmeforinternationalstudentasses
smentpisa/33694881.pdf
Publishing. http://dx.doi.org/10.1787/9789264128859-en
Perry, M., Scott, W., VanderStoep, S. W. & Yu, S. L. (1993). Asking questions
208
Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How
Polit, D. F., Beck, C. T., & Hungler, B. P. (2001). The importance of pilot
541.
209
Resnick, L. B. (1987). Learning in school and out. Educational Researcher, 16,
13-20.
Reznitskaya, A., Anderson, R.C., & Kuo, L. (2007). Teaching and learning
99(3), 561-574.
Rivkin, S. G., Hanushek, E. A., & Kain, J. F. (2005). Teachers, schools, and
Rowan, B., Correnti, R., & Miller, R. J. (2002). What large-scale survey
Pennsylvania.
MathsTeachingAndLearning.pdf
210
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving,
Shephan, M., & Whitenack, J. (2003). Establishing classroom social and socio
of Teachers of Mathematics. .
211
Siegel, C. (2005). Implementing a research-based model of cooperative
Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Strawhun, B.
301.
Space, the first and final frontier: Proceedings of the 15th annual
212
Star, J. R. (2002b). Re-conceptualizing procedural knowledge in mathematics.
Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the
Publications, Inc.
http://math-articles.blogspot.com/2009/09/mathematics-through-
problem-solving.html
The Education Alliance. (2006). Closing the achievement gap: Best practices in
213
Thompson A. G. (1984). The relationship of teachers conceptions of
UNESCO (2002). Open and distance learning: Trend, policy and strategy
org/image/0012/001284/128463e.pdf
Van Teijlingen, E., & Hundley, V. (2001). The importance of pilot studies.
Surrey.
Van Zoest, L., Jones, G., & Thornton, C. (1994). Beliefs about mathematics
214
Vinner, S. (1994). Traditional mathematics classrooms: Some seemingly
from http:/www.univie.ac,at/constructivism/EvG/papers/127/pdf
Weber, K., Maher, C., Powell, A., & Lee, H. S. (2008). Learning opportunities
com/static/pdf/93/art%253A10.1007%252Fs10857-013-
Whicker, K., Nunnery, J., & Bol, L. (1997). Cooperative learning in the
91, 42-48.
215
Wu, M., & Zhang D. (2006). An overview of the mathematics curricula in the
216
APPENDICES
APPENDIX A
Day One
Day Two
9.00-9:30 am (Benjamin Arthur)
Registration, opening, Introductions, Programme Run
down
217
1:00-2:45pm (Joanna Masingila)
Handout on cooperative learning, role of instructor
and student in problem solving
Activity 1.4 The mathematics in the pages of a
Newspaper
Debrief about learner and facilitator activity
Watch video of facilitator leading a class discussion
Day Three
218
APPENDIX B
MAP OF GHANA
219
APPENDIX C
Study Centre: ..
1.1 Identify and match the property for integers that is illustrated in each example.
f. 7 + -7 =0 p. distributive property
1.2 Write in your own words the meaning of the statement: the set of
natural numbers are closed under the operation multiplication.
220
1.3 Explain the relationship between the set of integers and the set of
rational numbers?
1.4 Why is the number 1 not a prime number?
1.5 Evaluate and give a reason for your answer to the mathematical
sentence: 14 26 2
1.6 A certain stock registered the following gains and losses in a week:
First it rose by 7 points, then it dropped 13 points, then it gained 8 Points,
then it gained another 6 points, and finally lost 8 points. Write a
mathematical expression that uses addition as the only operation, and then
find the net change in what the stock was worth during the week?
1.7 It takes Mamuna and Kojovi one-third of an hour and half an hour
respectively to walk round the school field. When will be their first time of
meeting if they should all start at 6.30 am from a starting point?
5
1.8 Round off the decimal fraction of to the nearest ten-thousandth
13
without using any calculating instrument.
1.9 Formulate a rule for finding the sum of the page numbers of a news
paper with 30 pages.
221
APPENDIX D
1.3 What is the relationship between the set of integers and the set of
rational numbers?
Explain your answer.
1.4 What number can be described as an even prime? Give a reason to your
answer.
222
1.5 Evaluate with reasons the mathematical sentence: 14 36 2 3
1.6 A certain stock registered the following gains and losses in a week:
First it dropped by 7 points, then it rose 13 points, then it gained 8 points,
then it lost another 6 points, and finally gained 8 points. Write a
mathematical expression that uses addition as the only operation, and then
find the net change in what the stock was worth during the week?
1.7 It takes Mamuna and Kojovi one-third of an hour and half an hour
respectively to walk round the school field. When will be their first time
of meeting if they should all start at 6.30 am from a starting point?
1.8 Thirteen women shared 5 litres of oil equally. What litres of oil, to the
nearest ten-thousandth litre(s) did each get? Do not use any calculating
instrument.
1.9 Formulate a rule for finding the sum of the page numbers of a news
paper with 30 pages.
223
APPENDIX E
3. Age: (A) 24-29 years (B) 30-35 years (C) 36-41 years
4. Experience in teaching:
(A) Less than one year (B) 1-5 years (C) 6-10 years (D) 11-15 years
(A) Nursery (B) Kindergarten (C) Primary (D) Junior High School
9. Class:
(A) Less than 40 (B) 41-50 (C) 51-60 (D) 61-70 (E) Above 71
224
SECTION B: DISTANCE LEARNERS KNOWLEDGE ABOUT
TEACHING AND LEARNING OF MATHEMATICS
225
24 Learners should ask questions during mathematics 1 2 3 4
lessons.
A problem-solving method (student-centered) is effective
25 to actively involve learners in the mathematics learning 1 2 3 4
process
26 Learners should often be confronted with novel problems
to solve. 1 2 3 4
27 Learners should be given notes to copy when learning
mathematics. 1 2 3 4
The teacher should create conditions to stimulate learners
28 to learn mathematics on their own. 1 2 3 4
29 Learners should learn mathematics by working with other
learners using worksheet. 1 2 3 4
30 Cooperative work in groups is good for efficient learning
of mathematics. 1 2 3 4
Learners should discover for themselves, the desired
31 conceptual knowledge in the learning process during the
learning of mathematics. 1 2 3 4
226
APPENDIX F
3. Age: (A) 30-35 years (B) 36-41 years (C) 42-47 years
4. Experience in teaching:
(A) Less than one year (B) 1-5 years 2 (C) 6-10 years (D) 11-15 years
(A) 1-2 years (B) 3-4 years (C) 4-5 years (D) 6-7 years
227
SECTION B: MATHEMATICS FACILITATORS KNOWLEDGE
ABOUT TEACHING OF MATHEMATICS
228
SECTION C: MATHEMATICS FACILITATORS KNOWLEDGE
ABOUT LEARNERS REASONING WHEN LEARNING
MATHEMATICS
229
APPENDIX G
Study centre:
Topic: ...
Class:.
Class size:
Date:
Time: ..
(iii).
(iv).
230
6. How are the learners learning in groups?
(i) Individually
(ii) Collaboratively
(iii)Cooperatively
(iv) Interactively
7. How are the learners coping with the questions on the worksheets?
(i) Interesting
(ii) Easy
(iii)Manageable
(iv) Challenging
10. What role did the teacher played during the lesson? Teacher.
(i) Provided answers to problems. [ ]
(ii) Encouraged learners to solve the problems with little help.[ ]
(iii)Assisted some groups to solve the problems. [ ]
(iv) Challenged the thinking of learners in his/her rounds from group
to group by asking them questions. [ ]
12.
13.
14.
231
APPENDIX H
FACILITATORS
2. How often have you been using the problem solving approach?
i) Never ii) at times iii) always
232
5. Did you encounter the following problems while teaching through a
problem-solving approach (choose all appropriate answers)
a. Lacked time to prepare for the lecture
b. The teaching module we use is not activity oriented
c. To many learners to manage them in groups
d. Slow method that caused a delay in completion of the syllabus
e. External examination pressure
f. Negative student reception
g. Rejection of a problem-solving approach by colleagues
7. What would you recommend that will enable other UCC-CCE mathematics
facilitators to implement a problem-solving approach in their mathematics
teaching?
233
APPENDIX I
TEST
1.1 Identify and match the property for integers that is illustrated in each example.
SOLUTION
f. 7 + -7 =0 l. additive inverse
1.2 Write in your own words the meaning of the statement: the set of natural
numbers are closed under the operation multiplication.
SOLUTION
The product of any two or more natural numbers is also a natural number
1.3 Explain the relationship between the set of integers and the set of rational
numbers
234
SOLUTION
SOLUTION
The number one (1) is not a prime because it has only one factor.
A prime number has two factors, one (1) and the number itself.
1.5 Evaluate and give a reason for your answer to the mathematical
sentence: 14 26 2
14 26 2 14 13 M1
27 A1
1.6 A certain stock registered the following gains and losses in a week:
First it rose by 7 points, then it dropped 13 points, then it gained 8 points,
then it gained another 6 points, and finally lost 8 points. Write a
mathematical expression that uses addition as the only operation, and then
find the net change in what the stock was worth during the week?
235
SOLUTION SCORING RUBRICS
7 ( 13) 8 6 ( 8) B1
7 13 8 6 8) M1
0 A1
1.7 It takes Mamuna and Kojovi one-third of an hour and half an hour
respectively to walk round the school field. When will be their first time of
meeting if they should all start at 6.30 am from a starting point?
1
of an hour 20 mins B0.5
3
1
of an hour 30 mins B0.5
2
1.9 Formulate a rule for finding the sum of the page numbers of a newspaper
with 30 pages.
236
SOLUTION SCORING RUBRICS
2S 31 31 31 ..... 31 M1
2S 31 30
31 30
S M1
2
S 465 A1
237
APPEN DIX J
Post-test of student in
Implementation of a PSCs and equivalent
problem-solving approach non-selected PSCs
by facilitators in PSCs implementation of a
problem-solving
TIME LINE
Provisional Registration
1. Fee payment 1/09/11 31/10/11 2 Months
Working on the Concept
proposal
Assignment of Supervisors
2. Preparation of proposal 1/11/11 31/03/12 5 Months
Defense of the proposal at the
department
Proposal approval by the
3. Graduate School 1/04/12 30/05/12 2 Months
Substantive Registration by
Graduate School
Preparation and testing of
4. instruments 1/06/12 31/08/12 3 Months
Obtaining relevant research
permits
Piloting of instruments
Data Collection/Field work
5. 1/09/12 30/04/13 8 Months
6. Data analysis & Thesis writing 1/05/13 1/3/14 9 Months
Notice of submission
7. Submission 1/03/13 31/03/14 1 Month
8. Oral presentation / thesis 1/06/14 30/06/14 1 Months
defense
9. Thesis editing Revision and 1/07/14 31/08/14 2 Month
submission of corrected thesis
10. Submission of final thesis 1/09/14 31/09/14 1Month
Total 34 Months
239
APPENDIX L
Expenditure GH GH
Transport and Travelling 1,000.00
Training programme 6,550.00
Internet Services 500.00
Stationery 500.00
Drafting Preparation 1,000.00
Communication Telephone 500.00
Printing 2,000.00
Personal Emolument for Assistance 4,000.00
4 @ GH1,000
6 Final Copies @ GH500 3,000.00 19,000.00
10% Contingencies 1,900.00
GRAND TOTAL 20,900.00
240