TABLE OF CONTENTS
Title page – – – i
Certification – – – ii
Approval page – – – iii
Dedication – – – iv
Acknowledgements – – – v
Table of contents – – – vii
List of tables – – x
List of appendices – – – xi
Abstract – – – xii
CHAPTER ONE: INTRODUCTION
Background of the study – – 1
Statement of problem – – 11
Purpose of the study – – 13
Significance of the study – – 13
Research questions – – 14
Hypotheses – – 15
Scope of the study – – 15
CHAPTER TWO: LITERATURE REVIEW
Conceptual Framework – – 16
Performance Status of Students in Mathematics and Geometry- 17
Philosophical foundation of the Pre-NCE programme 19
Models of teaching and learning – – 21
The advance organizer model – – 24
The concept attainment model – 27
Retention of knowledge – – 31
Schematic diagram showing the relationship between key variables of
the study – – – 32
Theoretical Framework
Bruner’s learning theory – – 32
Ausubel’s learning theory – – 33
Constructivism – – – 36
Review of Related Empirical Studies
Studies on models of teaching – – 39
Studies on AOM and CAM – – 39
Studies on performance of students in mathematics and geometry 43
Studies on retention – – 44
Summary of Literature Review – – 47
CHAPTER THREE: RESEARCH METHOD
Research design – – 51
Area of the study – – 52
Population of the study – – 52
Sample and sampling techniques – – 53
Instrument for data collection – 53
Validation of the instrument – – 53
Reliability of the instrument – – 55
Experimental procedure – – 55
Method of data analysis – – 58
CHAPTER FOUR: RESULTS 59
Summary of findings – – 72
CHAPTER FIVE: DISCUSSION OF RESULTS, CONCLUSION,
IMPLICATIONS AND RECOMMENDATIONS
Discussion of results – – 74
Conclusion – – 78
Implications – – 79
Recommendations – – 80
Limitations of the study – – 81
Suggestions for further studies – – 82
References – – 83
Appendixes – – 91
LIST OF TABLES
Table 1: Mean achievement scores of students taught with CAM, AOM
and Conventional method – – 59
Table 2: Mean achievement scores of male and female students taught
with CAM and AOM – – 60
Table 3: Mean retention scores of students taught with CAM,
AOM and Conventional method. – – 61
Table 4: Mean retention scores of male and female students taught with
CAM and AOM – – 62
Table 5: Analysis of covariance (ANCOVA) for achievement difference
between the groups – – 63
Table 6: Scheffe Post Hoc analysis for the groups – 64
Table 7: Analysis of covariance (ANCOVA) for achievement difference
between male and female in their groups – 65
Table 8: Scheffe Post Hoc analysis of mean difference between male
and female in the groups – 66
Table 9: Analysis of covariance (ANCOVA) for the interaction effect
between gender and instructional models – 67
Table 10: Analysis of covariance (ANCOVA) for the difference in the
retention test means scores of the groups – 68
Table 11: Scheffe Post Hoc analysis of difference in the mean retention
scores for the groups – – 69
Table 12: Analysis of covariance (ANCOVA) for the difference in the
mean retention scores for male and female students in the
groups – – 70
Table 13: Scheffe Post Hoc analysis of difference in the mean retention
scores between male and female in the groups. – 71
Table 14: Analysis of covariance for the interaction effect between
gender and instructional models in the retention
of students 72
APPENDICES
Appendix A: AOM Lesson plan – – – 91
Appendix B: CAM Lesson plan – – – 121
Appendix C: Pre-PNGAT – – – 156
Appendix D: Post-PNGAT – – – 162
Appendix E: Table of specification for the instrument – 168
Appendix F: Calculation of reliability of the instrument – 169
Appendix G: Calculation of the test re-test stability measure
of the instrument – – – 171
Appendix H: Public Colleges of Education in Benue
and Kogi States – – 173
Appendix I: Recent performance statistics of students in SSCE – – 174
Appendix J: Performance profile of the Pre-NCE students in geometry in Kogi State College of Education, (KSCOE), Ankpa between 2005 and 2010 – 175
Appendix K: Calculation of item analysis indices of the instrument 176
Appendix L: Training guide on the advance organizer and concept attainment models of instruction – 178
appendix M: Evidence of validation of instrument – 185
Abstract
This
study investigated the effectiveness of two instructional models – the Concept
Attainment Model (CAM) and Advance Organiser Model (AOM) on the achievement and
retention of Pre-NCE students in geometry. Four research questions and six
hypotheses guided the study. The design of the study was pre-test pos-test
equivalent control group design or quasi experimental design. The study was
carried out in Kogi and Benue
States in the present
North Central Zone of Nigeria. The population of the study was 1100 Pre-NCE
students in both the State and Federal-owned colleges of education in the two
states. Three out of the four public colleges of education in the two states
were randomly selected for the study. The total number of students in their
intact classes who offered Pre-NCE geometry in these colleges was 830. This
formed the sample for the study. 402 (48.4%) of the students were male and 428
(51.6%) were female. The instrument used for data collection was Pre-NCE
Geometry Achievement Test (PNGAT). PNGAT had two versions Pre-PNGAT and Post-
PNGAT which were the same except for the swapping of some of the questions.
PNGAT was subjected to both face and content validation and item analysis.
Using Kuder Richardson (K-R) 20, the internal consistency was found to be 0.74.
Using Pearson r, the test retest stability measure was found to be 0.96. Pre-
PNGAT was administered on the groups before treatment started while Post- PNGAT
was administered at the end of the 5 week treatment period. After 2 weeks of
administration of Post-PNGAT, it was again administered on the groups as a
retention test. Scores from Pre-PNGAT, Post- PNGAT and the retention test were
analysed using means and analysis of covariance (ANCOVA). Some of the major
findings from the analysis were (i) AOM and CAM were more effective than the
conventional method for achievement (ii) CAM was more effective than AOM for
achievement; (iii) CAM and AOM were more effective than conventional method for
retention (iv) CAM was more effective than AOM for retention. Based on the findings,
the implications were highlighted and recommendations were made towards better
achievement and retention of Pre-NCE students in geometry.
CHAPTER ONE
INTRODUCTION
Background to the Study
Mathematics is often defined or described from the point of view of its utility, especially by those who only make use of the subject. However, a simple definition which seems to capture the overall essence of the subject of mathematics is that it is the study of quantity, structure, space and change (Agwagah, 2008). Mathematics is one of the core subjects in primary and secondary school levels of education in Nigeria. Performance in mathematics at the Senior Secondary School Certificate Examination (SSCE) is used to determine those who are qualified to enter the tertiary levels of education especially those taking courses in the sciences and science related disciplines like engineering, survey, medicine and so on.
Many institutions of higher learning have introduced post-Joint Admission and Matriculation Board (Post-JAMB) screening examination into their admission policy. In many of such screening exercises, mathematics features among the subject areas where candidates are tested. In some Colleges of Education running Preliminary Nigerian Certificate in Education (Pre- NCE) programme, admission into the programme is through screening examination in which mathematics forms part of the examination. The NCE certificate is awarded by Colleges of Education on successful completion of a 3-year programme. It is the minimum teaching qualification in Nigeria for primary and junior secondary school levels of education (Federal Republic of Nigeria, 2004).
The Pre-NCE programme is a remedial outfit mounted by National Commission for Colleges of Education, (NCCE) and run by some Colleges of Education. It is designed to redress the deficiencies of candidates who either cannot make the required number of credit passes (including Mathematics and English) at SSCE for direct entry into the NCE programme or for candidates who have the required number of passes at SSCE but cannot pass the screening exercise for entry into the NCE programme. The Pre-NCE programme is for duration of one academic session. The progamme is aimed at addressing the challenges of weak students in colleges of education (Junaid, 2008).
The Pre-NCE programme is widely spread across colleges of education in the country. During the one year programme, students are exposed to prescribed number of courses that can remedy their deficiencies for entry into the NCE programme. Furthermore, following the recently introduced policy by JAMB that all Pre- NCE students must write the Unified Entrance Examination, the Pre- NCE programme is now saddled with additional responsibility of preparing the students in the various subject areas.
It is therefore clear that mathematics is given huge consideration and plays prominent role at all levels of education system. One may ask: why are we so concerned about mathematics? Of all the disciplines in school curricula, why are we so concerned about the failure and/or success of our children in mathematics? The answer is premised on the fact that it is one of the subjects that pervades all endeavours of humanity – sciences, arts, engineering, medicine, astronomy, etc. It is in consideration of the versatility of mathematics that James (2005) asserted that it is not only an academic, a scientist, an engineer that needs mathematics but that even a shop keeper, a housewife, a sportsman, etc. needs it. Thomaskutty and George (2007) agreed with James when the authors asserted that mathematics should not be seen as a classroom discipline only. From the point of view of science and technology, mathematics takes a unique position. In fact, the tremendous achievement of science and technology towards industrialized economy and human survival seems to be buoyed up by many far reaching mathematical contributions.
From the foregoing, it is clear that mathematics plays enormous role in the affairs of humanity. But despite this role of mathematics, researches have showed that performance of students in the subject is generally poor (Okereke, 2006; Uloko and Usman, 2007; Galadima and Yusha’u, 2007). Recent performance statistics of students in mathematics emanating from WAEC (see Appendix I) is indicative that performance of students in the subject remains all-time low.
The scenario, certainly, is assuming alarming and scandalous proportion, resulting into a situation where more candidates opt for Pre-NCE programme than those going straight into NCE and degree programmes. This situation is, however, not peculiar to Nigeria; every country seems to be infected by the poor performance syndrome. Close to two decades ago in Britain, Bow (1993) feared that the worry about poor enrollment in physics and chemistry seemed to be overshadowed by the concern for a bleaker future of mathematics. In America, Raizen and Michelsohn (1994) reported that in comparism with other countries, American children lag behind in achievement tests especially in mathematics and science areas.
As if the general poor performance in mathematics is not worrisome enough, another dimension to the problem is students’ performance in geometry which is one of the major components of secondary school and Pre-NCE mathematics. Over the past two decades or so, WAEC has been disturbed about students’ performance in geometry. WAEC (2000,2003,2005,) reported that students did not only perform poorly in geometry questions but avoided questions in geometry in certificate examinations. The worry heightened and assumed attitudinal dimension when WAEC (2005) reported that geometry is one of the topics in secondary school mathematics towards which students have shown negative attitude.
The situation at the Pre-NCE level is very similar if not worse. This is obvious because, as has been noted earlier, the Pre-NCE is a representation of mainly weak students The four courses in mathematics offered at this level are Number and Numeration, Algebra, Geometry and Statistics. The researcher, at one time or the other has taught all these courses. Over the years, the internal examination records have always indicated that students perform relatively poorly in geometry (see appendix J). From Appendix J, it is obvious that geometry poses a problem for the Pre- NCE students. Since a pass in mathematics is one of the conditions for placement into NCE programme, there is the fear that the poor performance in geometry may be partly responsible for non-placement of many Pre-NCE candidates. The need therefore arises to probe into a better technique in the teaching of geometry at this level to ameliorate the problem.
Geometry, as a branch of mathematics, is not only crucial in understanding properties and relationships of points, lines and figures in space but its study can enhance the understanding of many phenomena in several other areas of knowledge, especially in the sciences. This, perhaps, informs the inclusion of geometry in the Pre-NCE mathematics curriculum by National Commission for Colleges of Education (NCCE). Educational researchers have attempted to probe into the causative factors of poor performance of students in mathematics. While some researchers have reported that hard curriculum content is a causative factor, others reported deficiencies in the teacher factor. For example, Habor-Peters (2003) reported lack of mastery of mathematics contents by mathematics teachers while Adeniyi (1998) and Ojo (2002) reported that poor quality of instructional techniques was responsible for the poor performance of students. Agashi (2005) reported that hard content of the curriculum and teacher ineffectiveness were partly responsible for the poor performance of students. A good number of these findings are teacher related and this casts some doubts on the effectiveness of the mathematics teacher especially in the area of instructional techniques.
There are ample research evidence supporting various instructional techniques as enhancing teaching and learning and better achievement of students in mathematics (Harbor-Peters, 2003; Okigbo, Osuafor, 2008; Eze, 2008; Bawa and Abubakar,2008; Tyarbee and Imoko,2009). There may be no doubt that some of such techniques are being used in our schools by some mathematics teachers. But despite the reported efficacy of such techniques and the possibility of their use in instruction, the poor performance syndrome persists.
This situation may not be unconnected with the fact that over the years educational psychologists have been worried over all manner of instructional techniques many of which are teacher – centred and address only the cognitive behaviour of the learner at the expense of the affective and psychomotor domains. The traditional method of teaching is characterized by this major flaw. The concept of traditional method or conventional method of instruction is derived from the concept of traditional education. The chief business of traditional education is to transmit to a next generation those skills, facts and standards of moral and social conduct that adult deem to be necessary for the next generation-s materials and social success (Dewey, 1938). As beneficiaries of this scheme, which educational progressivist John Dewey described as being imposed from above and from outside, the students are expected to docilely and obediently receive and believe these fixed answers. Teachers are the instruments by which this knowledge is communicated and these standards of behaviour are enforced.
Traditional education is communicated through various modes or techniques. Historically, the primary educational techniques of traditional education were simple oral recitation and rote memorization (memorization with no effort at understanding the meaning). From this brief description of traditional education and the technique of communication, one can observe that it is teacher-centered and leaves no room for students’ active participation in the learning process. It is widely criticized for its focus on teaching instead of learning. From the earlier primary traditional techniques of simple recitation and rote memorization other techniques which are equally characterized by students’ passivity in the learning process have been in use over the years. All these techniques which main focus is teacher as chief information giver and students as passive recipients may be said to be traditional or conventional method of instruction. In the context of this work, the conventional method is any other method used in teaching mathematics at the Pre-NCE level except the CAM and AOM.
The conventional method of teaching has therefore been widely criticized by the progressive psychologists. These psychologists advocated that effective learning can hardly take place under such a setting. For instance, Jerome Bruner emphasized the importance of understanding the structure of a subject being studied, the need for active learning as the basis for true understanding and the value of inductive reasoning in learning. Bruner believed students must actively identify key principles for themselves rather than relying on teachers’ explanations. Teachers must provide problem situations, stimulating students to question, explore and experiment, a process called discovery learning. Thus Bruner (1966) believed that classroom learning should take place through inductive reasoning, that is, by using specific examples to formulate a general principle. It is at this point of inductivism that David Paul Ausubel (Brunner’s contemporary) disagreed.
Ausubel (1960) believed that people acquire knowledge primarily through reception rather than discovery; thus learning should progress not inductively from examples to rules but deductively from the general to the specific or from the rules to examples. Ausubel’s strategy always began with an advance organizer – a technique which this work addresses – which is a kind of conceptual bridge between new material and students’ current knowledge. In a concerted effort to ‘decode’ the work of Bruner, Ausubel and other leading educational psychologists, Joyce and Weil (1980) developed more than 20 models of teaching in an attempt to chart a new course to instruction. Educational researchers in mathematics do not seem to have adequately explored these models in terms of their efficacy in instruction. This has created the need for this study with a focus on the relative efficacy of these models in teaching geometry at the Pre-NCE level.
The Advance Organizer Model (AOM) is a teaching model credited to the theory of David Paul Ausubel. Ausubel (1960) maintained that the primary process in learning is subsumption in which new material is related to relevant ideas in the existing cognitive structure on a substantive, non-verbatim basis. Mayer (2003) defined advance organizer as information that is presented to organize and interpret new incoming information. Advance organizers may be described as statements or devices used in the introduction of a topic which enable learners to orient themselves to the topic, so that they can locate where any particular bit of input fits in and how it links with what they already know. Advance organizers are introduced in advance of learning itself and are also presented at a higher level of abstraction, generality and inclusiveness. These core components of advance organizer are indicators of the emphasis this strategy places on the development of the cognitive structure of the learner and that it serves as an umbrella which houses all the components of the lesson. The substantive content of a given organizer or series of organizers is selected on the basis of its suitability for explaining, interpreting and interrelating the materials they precede. This strategy therefore satisfies the substantive as well as the programming criteria for enhancing the organization and strength of cognitive structure (Ausubel, 1963).
Ausubel and Fitzgerald (1962) believed that slow learners benefit most from the use of advance organizer. Such students require additional assistance in learning how to structure their thinking. There may be no doubt that a good number of Pre-NCE students fall into this category of learners given the general deficiency in mathematics. Secondly, Ausubel (1963) discovered that organizers are more useful when working with factual materials than they are when dealing with abstraction. Thirdly, Ausubel (1963) believed that advance organizer enhances retention. These reasons may have informed the need for this study involving advance organizer as a teaching model.
The Concept Attainment Model (CAM) is credited to the theory of Jerome Brunner and its design is to help students learn concepts through systematic categorization and classification of learning materials or objects in an attempt to determine the rationale behind the categories. By using this method, students will learn the material much better as the learners figure it out for themselves. As well as learning the material better and remembering it longer, the students will learn how to learn by using this model. It encourages creative thinking, communication and independent learning (Bruner, 1966).
From the descriptions of Advance Organiser Model (AOM) and Concept Attainment Model (CAM) above, it is clear that they are constructivist-oriented. Constructivism is built on the philosophy and cognitive psychology that human beings are not passive recipients of information. Learners actively take knowledge, connect it to previously assimilated knowledge and make theirs by constructing their own interpretation (Cheek, 1992). It is also clear from the descriptions that their use in instruction may enhance retention, which is one of the variables of interest in this study. The Encarta Dictionaries defines retention as “the ability to remember things” In education, these “things’ are the values, attitudes, skills acquired by the learner in the process of learning. Educational psychology is so much concerned about retention of what has been learned as it serves as a basis for sustaining the desired change in the behaviour of the learner. This underscores the reasons why retention of knowledge has become an important area of research in education, as it has the potential to inform instructional practices and school learning goals (Bahrick, 2000). Implicit in Bahrick’s submission is the fact that instructional techniques are determinants of the extent of retention of knowledge. Semb and Elli (1994) were of the same view that one factor that determines retention of school knowledge is the nature of the instructional approach. It is clear therefore that retention and achievement of students are related variables and this informs the rationale of this study to investigate which of the two models is better in terms of these variables.
One other variable that is of interest to this study is gender. Gender is an ascribed attribute that differentiates feminine from masculine socially (Lee, 2001). Gender difference in mathematics in particular and science in general has over the years become such a controversial issue that educational researchers have become so weary about and have no common ground to land. For example, Ezeugo and Agwagah (2000), Okereke (2001), Steen (2003); Onwiodukit and Akinbobola (2005), Eriba (2005), Abuh (2005), Okereke, (2006), Ogunkunle (2007), Uloko and Imoko (2007), Nzewi (2009) reported that males are superior to females in mathematics in terms of achievement. Others (e.g. Ozofor, 2001; Kurumeh, 2004) sharply disagreed contending that females are superior. There is, however, research evidence (Okigbo and Osuafor, 2008) that improved instructional techniques can close the gender gap in achievement in mathematics. Can the use of AOM and CAM close this gap in the study of geometry at the Pre-NCE level? This question forms part of the focus of this study.
Statement of the Problem
