**CHAPTER ONE**

**INTRODUCTION**

**Background of the Study**

The knowledge of mathematics is paramount in the success of every man in his numerous day to day activities in life. Mathematics education holds the potency of making individuals to relate mathematics knowledge to everyday problem being encountered and hence develop the individuals to a level that they are intellectually and economically stable. Right from the pre-historic days of the early human societies to the present “hitec” age, mathematics has played a fundamental role in the economic development of many countries of the world (Popoola, 2002).

In every country, regardless of the level of economic, scientific and technological development, mathematics has to be taught to a number of scientists, technical specialists, scientific researchers etc. The service of these professionals will be continuously required for the well-being of the people and for the development of the society. Nigeria as a country needs to strive for scientific and technological breakthrough in order to cater for her domestic and international needs to enable her assert her greatness among the United Nations member states.

The pertinent virtue of mathematics as well as its contributions to the development of mankind has earned the subject the prominence it enjoys among other school subjects. The importance accorded to mathematics in the school curriculum from primary to secondary levels reflects accurately the vital role played by the subject in contemporary society. It is a core subject in the primary and secondary school certificate curriculum. Also a credit pass in mathematics at the senior secondary school certificate examination is needed as a pre-requisite for admission into the tertiary institutions in Nigeria. It is in realization of this that many countries resort to making specially comprehensive and well-programmed efforts towards the effective teaching and learning of mathematics and sciences at all levels of their educational system through the development and implementation of innovative programmes and projects (Azuka, 2000). Unfortunately, pupils’ and students’ performance in this all important subject have not been impressive. To support this assertion, Agwagah (2004) lamented that in spite of all the important roles mathematics play in the development of mankind; its achievement has been very poor. In the same vein Ugwuanyi (2009) stated that even though the indispensability of mathematics in the development of our society has been universally acknowledged, the out put of its teaching and learning is still not encouraging. Also (WAEC 2005 & 2009) revealed dishearting and alarming poor performance of students in mathematics. However, this situation is not peculiar to Nigeria, it depict the general trend world wide as International Studies on Education (ISE) revealed that secondary school students in developing countries lag behind in mathematics (Fennema & Sherman in Popoola, 2002).

The dishearting high failure rate in mathematics at pre-tertiary school level has bothered the minds of many researchers, authors and mathematics educators and even the government and attempts are being made to proffer some solutions. Hence, many researchers, mathematics educators and concerned individuals have been considering ways and means of ensuring effective teaching and learning of mathematics in schools that can enhance students’ achievement, interest and retention. This leads to the various teaching methods adopted by mathematics teachers in teaching mathematics in our schools. For the purpose of this work, studies were grouped according to the treatment under investigation and category names were derived from these groupings. These teaching method types (i.e innovatives / experimental) that was tested against control were grouped into five categories as adapted from Haas (2002). These are:

- Cooperative learning

- Technology aided instruction

- Problem-based learning

- Manipulatives, Models and Multiple representations

- Direct instruction.

- Cooperative learning is a method of instruction characterized by students working together in small groups to reach a common goal. It is generally understood to be learning which takes place in environment where students work collaboratively in small groups by sharing ideas while working on a given task (Okebukola in Eniayeju, 2010). It is a discovery method in which small groups are used. Cooperative learning category is characterized by; Students’ positve interdependence, Individual and group accountability, Appropriate collaborative skills, Group processing, Heterogeneous groups, Teachers’ supervision and evaluation (Duplass, 2006).
- Technology Aided Instruction: This is a method of instruction that is characterized by using computer soft ware, Video tapes, power point projectors, hand-held calculator etc to enhance the teaching of mathematics. Kersina (2003) noted that, “technology” is essential in teaching and learning of mathematics, it influences the mathematics that is taught and enhance students’ achievement
- Problem-Based Learning: This is teaching through problem solving where students apply a general rule (deduction) or draw new conclusion or rules (induction) based on information presented in the problem. Problem based learning is a focused, experiential learning organzied around the investigation and resolution of messy, real world problems. Students learn through facilitated problem solving that centers on a complex problem. The teaching methods under this category should involve student ability to think critically, analyze and solve complex real world problems, to find, evaluate and use appropriate learning resources; to demonstrate effective communication skills and to use content knowledge and intellectual skills to become continual learners.

- Manipulatives, Models and Multiple Representations: Multiple representations are ways to symobolize, to describe and to refer to the same mathematical entity. It includes graphs and diagrams, tables and grids fomulas, concrete models etc. This is a method of teaching characterized by teaching the students techniques or strategies for generating or manipulating representations of mathematics content and processes, whether concrete, symbolic or abstract.
- Direct Instruction: This is a method of instruction characterized by teaching through establishing a direction and rationale for learning by relating new concepts to previous learning, leading students through a specified sequence of instruction based on predetermined steps that introduce and reinforce a practice and feedback relative to how well they are doing (Haas, 2002). This is recognized as the most common way that most mathematics teachers operate. According to Rosenshine(1979), direct instruction has the following characteristics: an academic focus; a teacher centered foucus; little student choice of activity, use of large groups rather than small groups for instruction; and use of factual questions and controlled practice in instruction.

In all the categories grouped above, the primary research studies that were put under each category in this study reflected the outlined characteristic feature of that particular category. This categorization helped in generalizing the research findings of this study. In line with this grouping, Pillmer & Light (1980) argued that grouping of studies according to their characteristics is an essential step in assessing the range of generalizability of research findings. Also Hedges (1982) stated that the basic assumptions are that the investigator has a priori grouping of studies and a scheme for classifying studies that are likely to produce similar results. Often this will take the form of set of categories into which studies can be cross classified by two or more sets of categories. These method type categories outlined here should not be considered mutually exclusive because one method may contain another. Several researches have been carried out on the effect of these teaching methods on students’ achievement, interest and retention either separately or collectively.

Achievement according to new Webster’s dictionary (1995), means to reach a required standard of performance, to carryout successfully. In the context of this study, achievement refers to cognitive achievement of students which is measured in terms of passes in teacher-made test/standardized test in mathematics. Hence, the researcher upholds the view of Ajua (2006) that student’s academic achievement entails successful academic progress attained through effort and skill. It involves the determination of the degree of attainment of the individuals in tasks, courses or programmes to which the individuals were sufficiently exposed. The academic of primary school pupils / secondary school students in mathematics has not been encouuraging.

Obodo (2004) stated that throughout Nigeria, at all levels of education – primary, secondary and tertiary, the performance of pupils/students in mathematics is at a very poor state. This is evident as the result of students in mathematics at the graduating class of each level of education kept on deteriorating year in year out. Statistics shows that mass failure in mathematics examination is real and the trend of students’ performance have been on a fluctuating decline (Ali 2000; Betikiu 2002; Agwagh, 2004; WAEC 2005, 2007, 2009 & 2010; NECO 2009). This low achievement in mathematics may be attributed to students’ lack of interest in the subject.

Interest according to Imoko & Agwagah (2006) is a subjective feeling of concentration or persisting tendency to pay attention and enjoy some activities or content. It is the feeling of intentness, concern or curiosity about an object (Obodo, 2004). It can also be regarded as the condition of being eager to know or learn about something. Interest is an important variable in the teaching and learning of mathematics. This is because when a pupil/student become interested in an activity he/she is likely to be more deeply involved in that activity. Okigbo & Okeke (2011) has this to say, “Though some children may be intellectually and physically capable of learning, they may never learn until their interest is stimulated” (p.I01). s Once the interests of the students are stimulated, they will continue to learn as long as the teacher is capable of sustaining their interest in the subject matter. They also said that interest is a mother of attention, once there is direct interest, attention is guaranteed and learning is assured.

Generally, there is a low interest in the study of mathematics and mathematics related courses at all levels of education in Nigeria (Obodo, 2004). Kurumeh (2007) stated that student’s fear and hate/dislike mathematics. The low interest in mathematics emanates from anxiety and fear. The phobia pupils/students have for mathematics causes them to dislike this important subject. Phobia has been observed by Aprebo in Okigbo (2010) to be an academic sickness whose virus has not yet been fully diagnosed for an effective treatment in the class and the symptoms of this phobia is usually expressed on the faces of mathematics students in their classes. Researchers like Glimer in Kurumeh (2007) attributed this situation to the fact that mathematics is foreign, difficult and abstract and the method of teaching it too, is foreign. The chief examiner’s report of WAEC (2009 & 2010) suggested that teachers should help students develop interest in mathematics and improve their achievement by reducing the abstractness of mathematics, and remove their apathy and fear of the subject. This gave birth to varieties of teaching methods that researchers and mathematics educators have been using in teaching mathematics in order to find their effects on students’ interest in mathematics. However, low interest and poor achievement in mathematics by the students may be as a result of their inability to retain what is learnt.

Retention has been described as the process of maintaining the availability of a replica of the acquired new means or repeat performance by a learner with an acquired piece of knowledge (Ausbel, Novak & Henesiana in Nneji, 2010). Retention also refers to the ability to remember or utilize already acquired knowledge or skills. It refers to skill or knowledge or competences a learner acquired and retained from a learning situation after forgetting has taken place (Ezeoano, 2008). It is the capacity to remember something, skills, knowledge, habits, attitudes or other responses initially acquired. Retention plays an important role for what is learned to be effectively applied. The teacher is usually faced with the task of how to help pupils/students improve on their ability to assimilate and retain what they have learnt.

Students’ poor retention in mathematics may not be unconnected with rote learning that is prevalent in schools. However, mathematics concepts cannot be learnt properly by mere memorization through rote learning as human beings have limited capacity for memorization. Rote learning can even be traced back to the colonial era. The teaching of mathematics (arithmetic) during the colonial era was by rote learning of some basic processes and computation, and so, the learners produced could only solve problems mechanically through the use of memorized formula without knowing the “how” and “why” of their solution. This kind of learning is still observable now especially in lower primary school pupil where a pupil can learn multiplication table from 2×1 to 12×12 through rote leaning. He/she can stand up and recite correctly all of them from beginning to the end (i.e. from 2×1 to 12×12). But when an independent question is asked say 8×7, the pupil may not get the answer easily rather he/she may start from 8×1 to meet up with 8×7 before a correct answer can be given. This is due to the fact that he/she does not know the how and why 8×7 will give 56.

Enough explanation with practical examples should be given to these pupils to bring out the how and why of certain solutions of mathematics concepts so as to boast their retentive ability. This is also applicable to secondary school students when they learn about formulae and their applications in solving problems. The derivation of the formular should be exposed to them so that they can understand why the formular is, as it is.

Dulton in Ezeamenyi (2004) asserts that failure to provide enough application to real life activities, social usage cum poor teaching techniques are strong limiting factors to students’ retention in mathematics. In support of this, Nneji (2009) stated that retention depends mainly on teaching strategy adopted by the teacher. In the same vein, Azuka (2009) made case for the adoption of instructional methods that promote students’ involvement and activity in the teaching of secondary school mathematics so as to enhance students’ retentiveness. Several teaching methods or strategies in teaching mathematics at both primary and secondary schools have been explored by researchers and their effect on pupils/students’ retention tested. Therefore, it is essential for pupils/students to master mathematics concepts, be interested in it, retain learnt concepts, and hence strive to achieve higher in it. The research studies on the effect of teaching methods on students’ academic achievement, interest and retention have been necessitated so as to achieve the above. These researchers reported conflicting result in the magnitude of the effect size and hence no consensus was reached on the most effective taeaching method category. The only way to arrive at a consensus on the most effective method category is therefore, to integrate the results of the previous studies so as to bring out a composite view of the mean effect of teaching methods on achievement, interest and retention. But this may not be an easy job since the statistical tools used by different researchers may not be the same. Buttressing this point, Adeleke (1988) explained that it is not an easy task to integrate all studies especially when varieties of statistics have been used. Hence the way out is to review all the related studies using meta-analytic procedure.

Meta-analysis is defined as the study of a large body of studies using statistical procedures for the purpose of integrating, synthesizing and making sense of them (Glass, 1976). It is the analysis of analyses. It is a statistical technique for combining the findings from independent studies. The objective of meta-analysis is to allow forquantitative analysis of reviewed research literature. The conduct of meta-analysis follows certain procedures which are: Identification of the problem; Literature search; Reading and coding studies; Quantifying study finding – effect size calculation; Statistical analysis of effect size; and Interpretation of results.

Identification of the problems: As with the traditional research one must first identify an area of investigation. However with meta-analysis it is important that the area in question has been researched to some extent. There is no set number of studies that are needed but one must remember in identifying an area of investigation that the purpose of the meta-analysis is to provide a consensus of past research.

Literature search: An extensive literature search is necessary when conducting a meta-analysis prior to this search, the researcher need to establish criteria for inclusion or exclusion of a particular study. This is a very important step in the meta-analysis procedure as it provides considerable subsequent quantitative values used for analysis. Examples of literature search are computer search, journal search or theses and dissertations.

Reading and Coding Studies: It is important to establish a coding sheet for research utilizing a meta-analysis design. The coding sheet is analogous to the instruments that ae used in the collection of data in traditinal research. This coding sheet provides guidance within the research and also establishes validity and power to the meta-analysis design.

Quantifying the Findings- Effect size calcations: One of the problems with meta-analysis design relates to comparing variable findings of differing units of measure. This has been often referred to as comparing applies and oranges. If these two variables were to be compared, one might look for a common unit of measure such as categorizing them as fruit or detemining their kilocalorie values. But for research that may compare an experiment group with control group, the use of effect size calculation are employed to reduce the values to standard deviation units (Thomas & Nelson, 1996). With the standard units of measure established, the argument of “apples and oranges” is addressed.

Statistical Analysis: The meta-analytic research utilizes a variety of statistical methods. But with the established standard units of measure as mentioned above, comparative statistical procedures can be run in the same way as traditional studies.

Interpretation of Results: The interpretation of results is strengthered with the statistical procedure mentioned above. Mann (1990) notes that clear cut findings often emerge from studies whose previous findings were literally scattered. Do to the quantitative nature of a meta-analytic review, its application of a scientific method allows for a greater objective analysis and less bias than the process of traditional review (Hoffert, 1997).

Therefore meta-analysis procedure focuses on combining results from different studies that address the same issues in the hope of identifying patterns among study results, sources of disagreement among those results, or other interesting relationship that may come to light in the context of multiple studies. As mentioned earlier, this is normally done by identification of a common measure of effect size, of which a weighted average might be the output of a meta-analysis. The weighting might be related to sample size within the individual studies. Meta-analysis is aimed at more powerfully estimating the true effect size as opposed to a less precise effect size derived in a single study under a given set of assumptions and conditions. In other words it combines several studies and will therefore be less influenced by local findings than single studies will be. Meta-analytic studies of this nature have been carried out by Ovute (1999) who reviewed 37 research studies on the effects of discovery learning on students’ achievement in science using meta-analysis procedure and successfully arrived at a generalized conclusion. Also, Ukwungwu (2001) did meta-analysis of empirical studies on gender related differences in achievement and interest in sciences where he integrated 82 independent studies and was able to arrive at a reasonable generalized conclusion. In the same vein, Ajuar (2006) reviewed 44 research findings on environment and students’ achievement in science and made significant and generalized conclusion. Ugbaja (2012) reviewed 36 studies on the effect of instructional method/strategies on students’ achievement scores in science subjects and made generalizable conclusion. Although attempts have been made at integrating research studies in sciences as stated above, only few have been done in mathematics and they are not even done in Nigeria. Thus, the present study is geared towards filling this gap so that a study of this kind can be done in mathematics in Nigeria.

**Statement of the Problem**

Evidences abound of lots of effort by researchers in Nigeria to determine the effects of different teaching methods on students’ achievement, interest and retention in mathematics. Many of these researchers report conflicting results in the magnitude of the mean effect on students’ achievement, interest and retention of both treatment and control groups when they are exposed to different teaching methods. Due to these inconsistencies in research reports, one cannot categorically state the effect of each teaching method in enhancing achievement, interest and retention that is the best or highest. Therefore, attempt should be made to integrate these studies in one researchwork. This kind of integration of studies have been done by some researchers. However, all of these researchers did their meta-analysis on sciences generally but not in a specific subject area like mathematics. Meta-analysis techniques are needed so as to arive at a consensus opinion on the effect of teaching methods on students’ academic achievement, interest and retention in mathematics in Nigeria. The problem of this study therefore, is to determine the magnitude of the mean effect of teaching methods on students’ achievement, interest and retention in mathematics in Nigeria using meta-analysis procedure.

**Purpose of the Study**

The main purpose of this study is to use meta-analysis procedure to determine the overall effect of teaching method on students’ achievement, interest and retention in mathematics in Nigeria. Specifically, the study tends to:

- Identify the results of the previous studies on the effect of teaching methods on students’ academic achievement in mathematics.
- Determine the effect size for each of the studies examined on the effect of teaching methods on students’ academic achievement in mathematics.
- Determine the mean effect size for all the studies examined on the effect of teaching methods on students’ academic achievement in mathematics.
- Determine the mean effect size for each teaching method categories of studies on students’ academic achievement in mathematics.
- Determine the variation of effect size with primary and secondary school levels for studies on achievement.
- Identify the results of the previous studies on the effect of teaching methods on students’ interest in mathematics.

- Determine the effect size for each of the studies examined on the effect of teaching methods on students’ interest in mathematics.
- Determine the mean effect size for all the studies examined on the effect of teaching methods on students’ interest in mathematics.
- Determine the mean effect size for each teaching method category of studies on students’ interest in mathematics.
- Determine the variation of effect size with primary and secondary school levels for studies on interest.
- Identify the results of the previous studies on the effect of teaching methods on students’ retention in mathematics.
- Determine the effect size for each of the studies examined on the effect of teaching methods on students’ retention in mathematics.
- Determine the mean effect size for all the studies examined on the effect of teaching methods on students’ retention in mathematics
- Determine the mean effect size for each teaching method category of studies on students’ retention in mathematics.
- Determine the variation of effect size with primary and secondary school levels for studies on retention.

**Significance of the Study**

The integration of results from past studies can help to get accurate representation of the population than the estimate provided by individual studies. This work will therefore be significant as it will help to provide a composite picture of the relationship between teaching method and students’ achievement, interest and retention in mathematics in Nigeria. More valid basis for theory and practice of teaching mathematics could be formulated from this kind of integration.

The theoretical significance of this study was based on the theoretical model of random effect model. This is a common model used to synthesize heterogeneous research. Random effects model assumes that there is variation in the true correlation being estimated in each study. It is significant because when studies are drawn (at random) from a population for trial, random effects estimates the variability.

Practically, it is hoped that the results of this study could be of immense importance to mathematics teachers, students, researchers, curriculum planners, policy makers, and mathematics text book authors. Successful integration of this study will help the teachers to identify and use the best appropriate method that had greather effect size for teaching a mathematical concept which can increase students’ academic achievement, generate and sustain their interest and improve their retentive ability in mathematics.

The result of this study could help in capturing the students’ interest in mathematics and hence improve their academic performances when the teacher employs the teaching method with large effect sizes.

To researchers, this study will serve as a data bank and will be useful while searching for literature for their empirical studies. It could also reduce the duplication of researches in mathematics education since the result of this study was subjected to higher statistical technique to determine the mean effect of the methods. The policy makers could utilize the information in this study to state goals why some teaching methods should be adopted as regards the teaching and learning of a topic in mathematics.