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**CHAPTER THREE METHODOLOGY AND RESEARCH DESIGN**

**Introduction**

The purpose of this study was to investigate the influence of refreshment module of pre-calculus mathematics on Applied Calculus 1 achievement. Refreshment module of pre-calculus mathematics was delivered in intervention method of Meta-cognitive with Co-operative learning (MCL), Co-operative Learning (CL), and Traditional Lecture (L) methods. The study was intended to investigate the effects of Meta-cognitive with Co-operative learning (MCL) and Co-operative learning (CL) methods on pre-calculus mathematics achievement among male and female first year pre-engineering university students in Ethiopia. This chapter describes the methodology that was used in this study. The chapter presents population and sample, the research design, the experimental conditions, procedures and methods of analyses. It also includes discussions that addressed reliability of test instrument with its validity and ethical issue of participant of the study. The study adopted the quasi-experimental quantitative research approach.

**Research Method**

**Population and sample**

This study was conducted in four higher learning institutions of Ethiopia. In Ethiopia students who join a governmental university, are randomly assigned by Minister of Education for each 29 governmental universities except two science and technology universities. Among 29 government universities in 2016/2017, four universities which have equal status of facilities like, library, dormitory, learning classrooms, were randomly selected to participate in the study. Hence, the population of this study included all pre-engineering first year students in those universities in 2016/2017. There were more than ten sections in the selected universities. Simple random sampling, specifically lottery method was used to select one section from each university. As a result, 200 pre-engineering university students who studied pre-engineering courses were taken as the participants of the study. In Ethiopian context, one class accommodates 55 to 65 students. For this study, in each group 50 students of 25 males and 25 females were selected from sampled class by simple random sampling technique.

According to Thyer (2011), quasi-experimental research attempts to determine causal relationships by applying a treatment to one group and comparing experimental group with a control group. Quasi–experimental research is used extensively in education where the subjects are not randomly and it allows the research to occur in its natural setting (Thyer, 2011). Such quasi experimental research in education has to occur in school based research at the beginning of class year (Ross & Morrison, 2003) . Ross and Morrison (2003) state that it is a common application to use Pre-testing or analysis of prior achievement to establish group equivalence and be exposed to two or more similar sections of students to alternative intervention strategies and compare them on designed dependent measures during the year.

As it was a quasi-experimental study, for its sample one section of pre-engineering first year students was randomly selected from each university. Those classes were the classes in which the selected instructors had been assigned.

To implement this study, for selected section that were going to take refreshment module, tutorial class was arranged by the assigned instructors after they had been trained and oriented about intervention. In order to make the tutorial class more effective and attractive, the regular class rooms of Applied Calculus 1 instructors were participated. All of the three participant instructors who taught intervention module of pre-calculus mathematics for pre-engineering students in this study were males, who had the same levels of preparation of education in teaching mathematics (MSc) with more than 5 years of teaching experience in their university.

At the beginning two instructors who would teach experimental groups were exposed to MCL and CL instructional methods and informed and practiced for one-week with researcher. The participating students were informed that the purpose of this study was to achieve good mark (i.e. letter A or B) on Applied Calculus 1 and examine different learning strategies that help in the improvement of prior knowledge of their pre-calculus achievement.

**Experimental conditions**

One class in each of the four universities was assigned randomly to one of the following four subjects. There were subjects in one of the universities who did not take or participate in the review of pre-calculus mathematics (control group) (n= 50). On the other hand, there were subjects in other university who took part in the intervention of review of pre-calculus mathematics through traditional /lecture/ methods (n=50), and also there were the third group in other university who participated in the intervention of review of pre-calculus mathematics through co-operative learning method (n=50), while the fourth group of subjects in one university who participated in the intervention of review of pre-calculus mathematics through meta-cognitive with co-operative learning.

The three groups that took the refreshment module of pre-calculus mathematics were different from one another in terms of the intervention method and materials used. The MCL group was asked meta-cognitive questions by the instructor and students in this group used meta-cognitive question sheet in co-operative learning setting. The CL group students studied co-operatively without using meta-cognitive question sheet, whereas the T group studied in the traditional lecture method.

**Design of the Study**

The method of study used in this research is the quasi–experimental design that identifies whether there is similarity between a comparison group and the treated group with respect to baseline (pre-intervention) characteristics. As stated by White and Sabarwal (2014), quasi– experimental design was conducted to revise an intervention of pre-calculus mathematics in view of understanding teaching-learning process in relation to meta-cognitive with co-operative learning and co-operative learning approach alone on pre-calculus mathematics achievement of pre-engineering first year students.

This research method has been used to assess the influence of revising pre-calculus mathematics, mainly focus on basic algebra (real number, interval, absolute value, polynomials, radicals, and fractional expressions), equations and inequalities (solving linear equations and inequalities in one variable, solving equations and inequalities involving absolute value, simultaneous equations), functions (definition of a function, domain, range). Trigonometry (radian and degree measures, trigonometric functions, identities and equations), exponential and logarithmic functions in the view of understanding teaching-learning process in relation to co-operative learning approach and meta-cognitive with co-operative learning on pre-calculus mathematics achievement of pre-engineering first year students. It has been planned to find out the influence of the independent variable on the dependent one at each of the levels in both cases of moderate variables (male and female).

The independent variable of this study was the intervention method with three categories:

1 Reviewing Pre-Calculus Mathematics through Meta-cognitive with Co-operative Learning intervention method (MCL).

2 Reviewing Pre-Calculus Mathematics through Co-operative Learning intervention method (CL).

3 Reviewing Pre-Calculus Mathematics through Traditional intervention method (T).

The moderator variable was the gender with two categories: 1. Male. 2. Female. The

dependent variable was Pre-Calculus Mathematics and Applied Calculus 1 achievement. The study was designed to investigate the influence of pre-calculus mathematics refreshment module on Applied Calculus 1 and to compare three intervention methods: (1) Meta-cognitive with co-operative learning intervention: (2) Co-operative learning intervention, and: (3) Traditional intervention lecture method.

**Materials and Instruments**

A major instrument used in this research was the students’ pre-calculus mathematics achievement test which had been prepared by researcher and materials that had been used such as manual/module, teacher’s action plan as well as a meta-cognitive question sheet.

**Intervention material**

The instrumental materials used in this study to identify the influence of students’ pre-calculus mathematics achievement on Applied Calculus 1, which was pre-calculus mathematics module, adapted from First Year Survival Guide of Mc Master University (Lovric, 2009) Manual/module. Pre-calculus mathematics’ topics chosen for this study were basic algebra, equations and inequalities, functions, exponential and logarithmic functions, and trigonometry; function because these topics of pre-calculus mathematics are the background for the mathematical concepts, problems, issues and techniques that appear in the calculus course. Pre-calculus mathematics has been common language for understanding and describing many aspects of the physical world of science and engineering (Flashman, 2000). Concept of function is, without doubt, one key background tool for the calculus and applied calculus. According to Flashman (2000), being familiar with those pre-calculus concepts and functions which were specified are the crucial base and terms for the calculus, that help applied calculus students to have background knowledge of numbers and variables, equations and functions and applications which are used to relate the quantities included. To investigate the refreshment module of those pre-calculus topics on Applied Calculus 1 achievement, each instructor carried out the intervention for 32 sessions of 50 minutes each, which was about 26.6 hours in the respective universities they were assigned. Explaining the topic was the first procedure of the instructor and next he delivered the allotted exercise for a session (50 minutes) insuring that all of the students in each small group would arrive at the same level of understanding with respect to his objective. A set of meta-cognitive question sheet (See Appendix 4) was set by the researcher based on the meta-cognitive components (planning, monitoring, and evaluation).

**Pre-Calculus Mathematics achievement test**

**Pre-test and posttest**

Dimitrov and Rumrill (2003) state that most of the time Pre-test is used to identify level of understanding of research participant students before instruction. Pre-tests can be used to identify if there are a knowledge gaps that may not be expected in students’ learning and it helps to generate ideas for a future lesson including further instruction of refreshment (Kelly, 2017). Forming an effective pre-test helps to identify areas of students’ strengths and weaknesses that can be improved through different intervention method (Kelly, 2017). Posttest measures students’ learning. pre-test-posttest is a tool that is used most often to measure changes resulted from experimental approach to compare groups in educational research (Dimitrov & Rumrill, 2003). For this study, one of the reasons of providing the Pre-tests was to compare its results with the outcome of the posttest. For this reason, researcher developed a test (pre and post) which were administered to both experimental and control groups. And the researcher used similar test items for both pre- and post-tests. The Pre-test and posttest were similar and the same in content and procedures to all groups (See Appendix 5).

The Pre-test was administered at the beginning of the program to evaluate prior knowledge extent of recalling those selected concepts of pre-calculus mathematics and identify the students’ prior knowledge. Pre-test was scored and analyzed by descriptive statistics such as percentage, mean, St.D, and inferential statistics like ANOVA and independent T-test. It has been thought that identifying the pre-engineering students background knowledge regarding pre-calculus mathematics achievement mainly on topics of pre-calculus mathematics such as basic algebra, equations and inequalities, functions, exponential and logarithmic functions, and trigonometric functions. The above descriptive and inferential statistics helps to see the role of revising pre-calculus mathematics on Applied Calculus 1 achievement result and correlation between the intervention methods and genders with posttest and Applied Calculus 1 achievement result. The posttest was scored and analyzed by descriptive statistics like percentage and mean and inferential statistics such as ANOVA, independent sample T-test. Moreover, the effect size of intervention was computed. The posttest also helps to determine if there were mean scores difference between the MCL, CL, and T groups after treatment and moderator variables (the female with female students of each group, male with male students of each group, male with female students of the whole groups and significance difference). The pre-calculus mathematics achievement test contained sixteen types of test format that demanded 100 short answers was used to minimize probability of cheating. These questions were prepared based on blooms taxonomy proportional to each of selected content.

**Implementation of the three intervention methods**

To explore effective intervention method that assist to improve achievement of pre-engineering students on basic pre-calculus mathematics the three intervention methods were implemented as follows:

The control groups did not attend pre-calculus mathematics refreshment and T group students learned the course in traditional lecture Method. On the other hand, MCL and CL students were assigned into heterogeneous small groups and each small group was randomly selected to form the group which contained three male and three female (total six) students. The other students in MCL and CL classes formed in small group by applying the same procedure. In this treatment, students were informed about the procedure that they would go through after a week from that moment. Students were informed about the use of intervention; this means that they would be exposed to an intervention method that would help them become more effective managers of their own recalling of pre-calculus mathematics in focused areas and learning activities. That took place for 8 weeks i.e. it was implemented for 32 periods through each intervention method.

Meta-cognitive with Co-operative Learning Method: According to Literacy (2012), in co-operative setting, meta-cognitive strategy is: the strategy that help students to become more strategic thinkers on the existing mathematical problems and process information by asking self-questions and working with other peer students. Dealing with this method, the instructor may motivate students to develop and examine how to focus on existing mathematical problems.

During the first two or three classes instructor can be exemplary to show how to use meta-cognitive question sheet, and expected to encourage and direct students to use self-addressed meta-cognitive questions. And instructor considers four types of meta-cognitive questions into his lesson plan that gives opportunity for students to practice those meta-cognitive questions during their learning tasks. Examples of those four types of meta-cognitive questions as follows:

a) Comprehensive questions (e.g. What are the issues raised as a problem?); b) Connective questions (e.g. In what ways the existing problems are similar or different from problems those solved previously?); c) Strategic questions (e.g. What is the simplest and appropriate strategy that helps to solve existing problem?); and lastly, d) Reflective questions (e.g. Does the solution of existing problem is satisfactory? If not, is there any other way to solve it?

Based on the idea mentioned above, the instructor introduced the processes of MCL approach to the students. Then, the discussion was made regarding the importance and the role of this procedure to enhance their achievement in mathematics. The instructor was expected to spend some time in introducing the concepts explicitly that how students become meta-cognitive thinkers within this learning environment. And made them be informed why they would learn meta-cognitive strategies, and how they could use these approaches in solving real-life problems. At the end of discussion on MCL method, students were assigned to groups. Each group was formed by selecting three female and three male students randomly as mentioned in the sections above. After the group arrangement, the students were provided with activities in their groups to solve the problem in a way they were oriented. Each group member was provided with specific activity that he or she could play the role as expected in the group like asking questions, summarizing, recording and presenting. The role was cyclical among the group members and each member was expected to be aware of his or her own role. The role of the group member who was assigned to ask questions would be asking meta-cognitive questions which were listed in the question sheet. The role of the summarizer was to deal with oral questions with respect to the main ideas and key points of the lesson. The role of the recorder was to write-down the steps of solution, the explanations, and the justifications of that solution. The solution was finally presented, explained, and justified to the whole class by the presenter. Each role was displayed several times in the classroom and the procedure was cyclical.

Structures’ of MCL in each small group by Think pair share method as follows

In co-operative learning, an instructor introduced the stages of co-operative learning method and made discussion with the students regarding the significance of applying this method in mathematics lesson. To form the small heterogeneous group, the instructor of the CL method followed the same procedure that was applied in the small group formation in MCL method. In this intervention, students got the chance to make discussions with their partners concerning activities provided to them. They also answered the questions, and exchanged their attempts to the neighbor students and made discussions with them to have common understanding with respect to responses that they had provided. This procedure occurred in entire groups in the class. The students able to evaluate their attempts by gathering information through experience sharing among each other at the time of discussion. The instructor also got a chance to assess the students’ understanding towards the content of the lesson.

At the end of this procedure, the instructor evaluated students’ achievement to ensure whether the students carefully attend the effectiveness of their group members and commemorate their group work achievement together. For the next class, the instructor and students followed the same process and the roles of students would be cyclical as mentioned in the MCL method.

The structure of CL in each small group in ‘Think pair share method’ is presented as follows:

Traditional (T) method: Groups under traditional method attended their tutorial class as they did at the formal class. In other words, the instructor taught the students as usual as he practiced in normal class and the students were not provided with group work activities and meta-cognitive questions.

N.B: There were two control groups from which one was used to compare groups without intervention while the other was used to compare instructional intervention method with traditional lecture method.

After two months delivering intervention of pre-calculus mathematics, at the last session, the students in three groups (MCL, CL, and T) were given posttest of pre-calculus mathematics and at the end of semester, the researcher collected the results of Applied Calculus 1 out of 100% of each group at the four universities.

**Analysis of the Experimental Study Findings**

The pre-calculus mathematics achievement test was recorded by the researcher and analysis was made to visualize extents of students’ background knowledge of basic pre-calculus mathematics and to decide whether there were any statistically significant differences among the four groups regarding the dependent variables. The statistical tools used under this procedure were descriptive statistics, T-test, and one-way analysis of variance (one-way ANOVA). The tools were used to compare the four mean scores on posttest of pre-calculus mathematics’ achievement and on the results of Applied Calculus 1 that they had scored at their normal classroom.

As stated in the manual of Pallant (2010), the one-way ANOVA works for analyzing variance in quantitative data by a single dependent variable. ANOVA is an extension of T-test that is used to identify the significant difference of means. ANOVA itself has two types of tests such as priori contrasts test that take place before the experiment and post hoc test that is applied after the experiment. The researcher applied T-test and ANOVA post hoc tests to analyze the data.

As remarked by Pallant (2010), ANOVA post hoc adjustment with Tukey or Scheffe is used most commonly. ANOVA Hochberg’s GT2 can be used if there is difference among group sizes and Games-Howell can be used if a group variance is less than 0.05 (Levenu test gives p value < 0.05). Tukey is used for homogeneous data. When data violet homogeneity, use « Tamhane’s T2 » because it is the most used test statistics by statisticians (Gupta, 1999).

Based on the above theory, ANOVA was used in this research first to compare posttest pre-calculus mathematics achievement of the three groups and the relationship between pre-calculus mathematics refreshment in respect of Applied Calculus 1 achievement was analyzed using spearman correlation coefficient. Then, ANOVA Post Hoc Test was used with multiple comparison technique to compare male students against male students’ Pre-Calculus Mathematics achievement and Applied Calculus 1 across the four groups. With ANOVA, if the significance level is less than 0.05, then there must be significance difference between two groups. But the difference between these groups is specifically unknown in ANOVA. In order to identify the differences, T-Test was applied. In SPSS20, independent sample T Test method was used to compare the mean of pre and posttest one independent variable. For each treatment of independent variables, the differences between values were computed.

The same method was applied to make comparison among students with respect to gender and Pre-Calculus Mathematics achievement and Applied Calculus 1 across the four groups. The statistical analyses in all cases were computed at 0.05 levels of significances.

One technique to judge the efficiency of a given intervention is the effect size that enables us to measure both the enhancement in students’ accomplishment for a group of students and the variation of students’ achievement expressed on a standardized scale (Coe, 2002). The effect size provides information about which intervention is worth having, specifically valuable to measure the effectiveness of a particular intervention, in relation to some comparison (Coe, 2002).

**Measures to Ensure Validity and Reliability**

Before Pre-test was administered, a pilot test was carried out to check the validity of research tools. For pilot test, six instructors who have been teaching Applied Calculus 1 were selected randomly from nearest universities to researcher’s university for convenience, those that were 75 km away did not participate to take refreshment module of pre-calculus mathematics to control Pre-test extraneous variable that come from information exchange.

As shown in Table 1 all participant instructors in conducting pilot test have a similar level of education (MSc degree) and more than six years teaching experience of applied calculus mathematics. There were three purposes to be achieved through the pilot test. First, pilot test was applied to test module material and instrument based on blooms taxonomies. Second, pilot test was used to test content validity and third, pilot test was used to test reliability of the test instruments.

For this purpose, the test instrument was evaluated based on the assertion of Bloom’s Taxonomy by the instructors who selected for the pilot study. This assertion was applied to assess students’ understanding of topics and their application of higher order thinking skills (DiDonato-Barnes & Fives, 2013). As stated by Abduljabbar and Omar (2015), one should consider the six stages of Bloom’s Taxonomy that start with the simplest of knowledge, then comprehension, application, analysis, synthesis and, finally, evaluation. And also an assessment of a given topic of study should be related directly to the amount of class time that allots to cover the objectives and the proportion of summative evaluation (DiDonato-Barnes & Fives, 2013).

In addition, a pilot study was applied to check content validity. Content validity is the mechanism that helps to evaluate the degree to which elements of an assessment instrument is appropriate to and representative for a particular purpose. The researcher applied a pilot test to evaluate the effectiveness and the coverage of content validity which can be used as self-evident measurement that shows the breadth of literature and test instruments (Rahmantya, 2009).

According to Rahmantya (2009), content validity can be evaluated by testing with an eye to decide the establishment of the sampled domain. As mentioned in Rahmantya (2009), content validity ratio (CVR) is a quantitative index for assessing content validity. Therefore, prepared test instrument was given for the instructors who have MSc in mathematics education. It was given to evaluate content validity of the test instrument items by CVR based on quantitative approach to content validity (Rahmantya, 2009). Those instructors were six instructors who have been teaching applied calculus 1 for first year pre-engineering. A purposive sampling technique was used to select the instructors. The instructors were asked independently to judge if the test items reflect the content domain of the study. With N judges, of which ′ ′ have judged the knowledge required for the item to be essential, = ( − /2)/( /2). where CVR indicates that Content Validity, ′ ′ indicates number of subject matter evaluator expertise that rates test items need modification or item is essential, N indicates that total number of subject matter evaluator expertise. After instructors’ suggestion six (CVR < 0.4) questions were corrected and modified in Pre-test and posttest (see Appendix 1).

The researcher applied test-retest mechanisms to evaluate the reliability of the test instrument. It is believed that test-retest reliability is administered by providing the same test to the same variable on two different occasions within short time interval. Test instrument is thought that there is no considerable difference in the achievement being evaluated between the time intervals of test-retest. The time interval that has been given plays a critical role to measure the validity of pre-and post-test instrument (Muijs, 2004). Whenever we evaluate the same thing twice, the correlation between observations partially depend on the time interval between the two occasions. If the time gap is short, the correlation is high; if the time gap is long, the correlation is low. This is because correlation and given time interval of observations are inversely proportional to each other (Muijs, 2004). This means when the time is closer, there will be mere similar factors that contribute to error (Muijs, 2004). A correlation of test-retest reliability is statistically quantified in the interval of zero and one where 1 being highly correlated in the test and the retest (Muijs, 2004). Perfection is ideal and most researchers accept a lower level, i.e. highly related when all items tend to measure 0.7, 0.8 or 0.9, depending upon the particular field of research.

Based on the principle above, to see the reliability of test instruments of this study, 20 pre-engineering students in 2015/2016 were randomly selected from one of non-participated universities and the test–retest was conducted in 30 minutes’ interval and its correlation was 0.998 (see Appendix 2) which indicates that test instrument is reliable.

**Table of Contents**

**ABSTRACT **

**ACKNOWLEDGEMENT **

**LIST OF FIGURES **

**LIST OF TABLES **

**LIST OF APPENDICES **

**ACRONYMS **

**CHAPTER ONE **

**1. INTRODUCTION**

**1.1. Background **

1.2. Statement of the Problem and Research Questions

1.3. General Objective of the Study

1.4. Specific Objectives of the Study

1.5. Hypotheses

1.6. Significance of the Study

1.7. Structure within Conceptual Frame Work of this Study

1.8. Operational Definitions

**CHAPTER TWO ****LITERATURE REVIEW**

**2.1. Introduction **

2.2. Mathematics in Engineering

2.3. Intervention Method

2.4. Gender Difference in Mathematics

2.5. Pre-Calculus Concepts

2.6. Some Studies about Meta-cognitive with Co-operative Learning Strategies

**CHAPTER THREE ****METHODOLOGY AND RESEARCH DESIGN**

**3.1. Introduction **

3.2. Research Method

3.3. Design of the Study

3.4. Materials and Instruments

3.5. Analysis of the Experimental Study Findings

3.6. Measures to Ensure Validity and Reliability

3.7. Permission to Conduct Research at an Institution (Ethical Issue)

**CHAPTER FOUR ****RESULTS AND DISCUSSION**

**4.1. Introduction **

4.2. Result fromPre-test

4.3. Research Question Number One

4.4. The Experimental Study Results

**CHAPTER FIVE ****DISCUSSION AND CONCLUSION**

**5.1. Introduction **

5.2. Interpretation ofPre-test

5.3. Discussion

5.4. Summary and Conclusions

**6. References**

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