METHODOLOGY
Research methodology is a science of studying how research is done scientifically. It is a systematic way to solve research problem. It is also defined as the study of methods by which knowledge is gained. Research methodology aims to give the work plan of research. The study is an attempt to understand the interaction effect of Mathematical Beliefs and Achievement Motivation on Achievement in Mathematics of secondary school students. 'Methodology of a study is the totality of the procedures followed by the investigator to make it scientific and valid to the extent possible. As such it is very crucial that the success of any research depends on the method adopted and the tools and techniques employed for data collection and analyses.
The methodology adopted for the present study is described under the following major headings:
•	Variables of the study
•	Objectives of the study
•	Hypotheses of the study
•	Method used
•	Sample selected for the study
•	Tools used for data collection
•	Data collection procedure
•	Scoring and consolidation of data
•	Statistical  techniques used
Variables of the Study
The independent variables and dependent variable variables selected for the study are the following.
Independent Variables
The independent variables selected for the study are;
•	Mathematical Beliefs
•	Achievement Motivation 
Dependent Variable
	The dependent variable selected is Achievement in Mathematics. 
	The variables selected for the study are summarized in the Figure 2.

Figure 2 .Variables selected for the study
Objectives of the Study
The objectives of the study are:
•	To find out whether there exist any significant difference in the Mathematical Beliefs of secondary school students for the subgroups based on 
o	Gender
o	Socio-Economic Status
•	To find out whether there exist any significant difference in Achievement Motivation of secondary school students for the subgroups based on 
o	Gender
o	Socio-Economic Status
•	To find out whether there exist any significant difference in Achievement in Mathematics of secondary school students for the subgroups based on 
o	Gender
o	Socio-Economic Status
•	To study the main effect and interaction effect of  Mathematical Beliefs and Achievement Motivation on Achievement in Mathematics of secondary school students for the  total sample 
Hypotheses of the Study
The study was carried out to test the following hypotheses:
•	There exist significant gender difference in the mean scores of Mathematical Beliefs of secondary school students.
•	There exist significant difference in the mean scores of Mathematical Beliefs of secondary school students with respect to their Socio-Economic Status.
•	There exist significant gender difference in the mean scores of Achievement Motivation of secondary school students.
•	There exist significant difference in the mean scores of Achievement Motivation of secondary school students with respect to their Socio-Economic Status.
•	There exist significant gender difference in the mean scores of Achievement in Mathematics of secondary school students.
•	There exist significant difference in the mean scores of Achievement in Mathematics of secondary school students with respect to their socio-economic status.
•	There exist significant main effect of the independent variables namely: Mathematical Beliefs and Achievement Motivation on the dependent variable, Achievement in Mathematics of secondary school students for the total sample.
•	There exist significant interaction effect of the independent variables namely: Mathematical Beliefs and Achievement Motivation on the dependent variable,  Achievement in Mathematics of secondary school students for the total sample
Methodology 
Method Used
The purpose of the present study is to investigate the influence of Mathematical Beliefs and Achievement Motivation on Achievement in Mathematics of secondary school students.  Survey method was used by the investigator in order to collect necessary information.
Sample Selected for the Study
Selection of a suitable sample is an important part of any research work. A sample is a small portion of the population that is selected for observation and analysis, one can make certain inferences about the characteristics of the population from which it was drawn (Best & Kahn, 2014). 
A good sample must be as nearly representative of the entire population as possible and ideally. It must provide the whole of the information about the population as possible and ideally it must provide the whole of the information about the population from which the sample has been dawn (Koul, 2009).
The population of the study comprised of secondary school students in Kerala state who follows Kerala state syllabus. The sample of the study constituted 600 secondary school students studying in standard IX who were selected from Kozhikode and Malappuram districts of Kerala state.
Stratified sampling technique was conceived to be best suited for selection of the sample of the present study. Total numbers of 600 secondary school students were collected by giving due representation to gender and socio economic status. The breakup of the final sample is given in Table 2.

Table 2
The Breakup of the Final Sample 
Sample	Categories	Number of students	
Gender	Boys
Girls	300
300	
Socio-Economic Status	Low Socio-Economic status	333	
	Average Socio-Economic status	199	
	High Socio-Economic status	68	

	The list of the schools from which data was collected is given as Appendix I.
The factors or strata taken into consideration while selecting the sample are the following.
	Gender 
Gender plays a major role in the study. Most of the studies reviewed revealed that sex difference makes a change in their Mathematical Beliefs, Achievement Motivation, and Achievement in Mathematics.
	Socio-economic status
Socio-Economic Status plays a major role in the study. Most of the studies reviewed revealed that Socio-Economic Status makes a change in their Mathematical Beliefs, Achievement Motivation, and Achievement in Mathematics.
Tools used for Data Collection
The selection of a suitable tool is an aspect of any successful research work. An accurate tool can only measure accurate data. Accurate data is the back bone of any successful research.
	For the purpose of collecting data, the investigator used the following tools.  
•	Mathematical Beliefs Inventory (Ganisha & Niranjana, 2019)
•	Scale of Achievement Motivation in Mathematics (Vijayakumari & Sumangala, 2000 )
	The tools used for collecting data are described in detail under this section.
Mathematical Beliefs Inventory
Mathematical Beliefs Inventory was used to measure the Mathematical Beliefs of secondary school students. Mathematical Beliefs Inventory constructed and standardized by the investigator with the help of the supervising teacher was used to measure the Mathematical Beliefs of secondary school students. The Inventory was constructed by considering the four dimensions of Mathematical Beliefs such as nature of mathematics, mathematics teaching and learning, mathematics in daily life and competence in mathematics. The dimensions of Mathematical Beliefs were selected by considering the dimensions proposed by Grouws, Howald and Colengelo (1996), Zakaria and Musiran (2010) and Op’t Eynde and De Corte’s (2004). The investigator prepared Mathematical Beliefs Inventory which consisted of four dimensions such as the nature of mathematics, mathematics teaching and learning mathematics in daily life and competence in mathematics.
	Planning of the inventory
The first step in the construction and standardization of an inventory was planning of the inventory. After selecting the topic, the investigator had gone through the theoretical background and analyzed related studies. It can be found that Mathematical Beliefs have a major role in the teaching learning process. The purpose of the present study is to find out the main and interaction effect of Mathematical Beliefs and Achievement Motivation on Achievement in Mathematics of secondary school students. As Mathematical Beliefs is a variable which is strongly related to the inner feelings of an individual, the investigator decided to prepare an inventory to measure the Mathematical Beliefs of secondary school students.
Preparation of the inventory
The inventory was constructed on the basis of dimensions of Mathematical Beliefs selected from the studies of Grouws, Howald and Colengelo (1996), Zakaria and Musiran (2010) and Op’t Eynde and De Corte’s (2004). The various dimensions of Mathematical Beliefs identified for constructing Mathematical Beliefs Inventory are
•	Nature of Mathematics
•	Mathematics in daily life
•	Mathematics teaching and learning
•	Competence in Mathematics
The details of the dimensions are given below:
	Nature of mathematics
	According to this dimension, the students believe that mathematics as subject is the key to scientific knowledge and meaningful connections exist between mathematical concepts. Mathematics as a subject helps to develop scientific and logical thinking among the students. The students believe that mathematics is a dynamic field which is growing and ever changing in nature. Mathematics is viewed as a systematized, organized and exact branch of science which deals with abstract concepts (Grouws, Howald & Colengelo ,1996; Zakaria & Musiran ; 2010)
	Eg.  Mathematics is the high level language of science.
	Mathematics in daily life
	The students believe that mathematics is  a endeavor and mathematics as a school subject has high value in everyday life or future work. Students also believe that mathematics helps to understand the world around them in a better manner as it is the formal way of representing the real world. (Grouws, Howald & Colengelo, 1996)
	Eg.  Mathematics represents the real world.
	Mathematics teaching and learning
	In this dimension, Zakaria and Musiran (2010) describes mathematical beliefs. According to Zakaria and Musiran (2010) students believe that learning of mathematics is an active process and develops critical thinking. It enables the students to find solution to the problems in mathematics in their own way by understanding. The rules stated to mathematics subjects. They also believe that by adopting systematic methods every problems in mathematics can be solved.
Eg.  Mathematics problems can be solved by using different methods.
	Competence in mathematics
	Op’t  Eynde and De Corte (2004) explains that in order for students to become competent problem solvers they must develop a mathematical disposition where affect plays a major role. Students believe that the facts and concepts in mathematics can be learned easily if we understand the rules and procedures properly. They also have the notation that learning mathematics is not an inborn talent rather through hardworking anyone can master the concepts in mathematics.
	Eg. Concepts in mathematics can be comprehend through hardworking.
	Based on the above mentioned components, the investigator developed Mathematical Beliefs Inventory. The draft scale consists of 43 items related to Mathematical Beliefs.
	The component wise distribution of items in Mathematical Beliefs Inventory  is given in Table 3.
Table 3 
Component wise Distribution of Items in Mathematical Beliefs Inventory 
Sl. 
No.	Component	Item No. in inventory	
1	Nature of Mathematics	1,2,3,4,5,6,7,8,9,38,42,43	
2	Mathematics in daily life	10,11,24,28,29,30,41	
3	Mathematics teaching and learning	12,14,15,16,18,19,21,22,23,25,26,27	
4	Competence in Mathematics	13,17,20,31,32,33,34,35,36,37,39,40	
	
	Scoring procedure
The responses of Mathematical Beliefs Inventory has to be made in two options. i.e., ‘agree’ and ‘disagree’. The items are scored by giving a score of ‘1’ for ‘agree’ response and a score of ‘0’ for ‘disagree’ response. The total score obtained for each sample on Mathematical Beliefs Inventory is calculated to identify the score of Mathematical Beliefs of secondary school students.
	Pilot study
          For the pilot testing of the developed inventory, a sample of 200 secondary school students of IX standard was selected. While selecting the sample for pilot testing, due representation was given to gender of the students. The prepared Mathematical Beliefs Inventory was administered to the selected sample. Before administering the tool necessary instructions were given to the students regarding how to mark the response. The scores obtained in the pilot testing were subjected to item analysis. A copy of draft Mathematical Beliefs inventory (Malayalam) is given as Appendix II
	Item analysis
For the purpose of selection of items for the final Mathematical Beliefs Inventory, item analysis was caused out to ensure the quality of items. The preliminary inventory was administrated to a sample of 200 secondary school students selected through stratified sampling technique by giving representation to gender. The response sheets of 200 students were arranged in the rank order of total score obtained by them. The upper 27 percentage and lower 27 percentage was taken as the upper group and lower group respectively. The mean and standard deviation of the score obtained for the upper and lower groups were calculated. The t-value (critical ratio) for each item was calculated by using the following formula.
t = 
= The mean score on a given statement for the upper group
= The mean score on the statement for the lower group
n	= Number of cases 

The critical ratio (t-value) obtained for each item in Mathematical Beliefs   Inventory is given in Table 4.
Table 4   
Data showing Critical Ratio (t-value) of the Items in Mathematical Beliefs Inventory
Item No.	t-value	Item No.	t-value	
1	3.729	23	2.224	
2	0.811*	24	2.937	
3	2.751	25	3.567	
4	3.499	26	2.397	
5	2.284	27	3.376	
6	2.284	28	0.990*	
7	2.832	29	2.414	
8	2.169	30	3.273	
9	4.342	31	0.432*	
10	2.734	32	2.559	
11	2.747	33	2.043	
12	2.414	34	2.043	
13	3.171	35	0.320*	
14	0.546*	36	0.471*	
15	0.282*	37	2.465	
16	2.145	38	2.421	
17	2.148	39	3.702	
18	0.811*	40	2.606	
19	0.362*	41	2.142	
20	1.978	42	2.414	
21	2.043	43	3.624	
22	3.083			
*Indicates the rejected items 
 	Finalization of inventory
Items with a critical ratio greater than 1.96 were selected for the final Inventory. Thus out of the 43 items, 34 items were selected for the final Mathematical Beliefs   Inventory.
A copy of the final version of the Mathematical Belief Inventory (Malayalam and English version) is appended as Appendix III and Appendix IV.
 	Reliability of the tool
 	Reliability is the degree of consistency that the instrument or procedure demonstrates whatever it is measuring, it does so consistently (Best & Kahn, 2014). The investigator ensured the reliability of the tool by using Test-retest method. The inventory was administrated to a sample of 50 secondary school students. After three weeks the same inventory was administrated to the same sample of 50 secondary school students. The two sets of scores are correlated and the reliability coefficient obtained is 0.67. The index suggested that the tool was reliable. The reliability of the tool is also established by using split half method. The reliability coefficient obtained is 0.74 which ensured the reliability of Mathematical Beliefs Inventory.
Validity of the tool
Validity is that quality of a data gathering instrument or procedure that enables it to measure what it is supposed to measure (Best & Kahn, 2014). The validity of the tool is ensured through face validity by consulting with experts in the field of education. The criterion related validity of the tool was established by correlating by the scores of Mathematical Beliefs Inventory obtained from 50 secondary school students with that of the scores obtained by administering the Mathematical Beliefs Inventory (Zakaria & Musiran, 2010). The validity coefficient obtained for the tool is 0.69. The index suggested that the tool is valid.
Scale of Achievement Motivation in Mathematics 
	For measuring Achievement Motivation in Mathematics of secondary school students,the investigator adopted Scale of Achievement Motivation in Mathematics (Vijayakumari & Sumangala, 2000). The tool intended to measure the striving of secondary school pupils to attain expected goals by learning mathematics. The scale is a three point scale. The draft scale consists of 51 statements on seven constructs viz., Work ethic, Acquisitiveness, Dominance, Pursuit of excellence, Competitiveness, Status aspiration and Mastery. The final scale on Achievement Motivation in Mathematics consisted of 42 statements among which 36 statements are positive and six are negative.
	The details of the constructs incorporated in the scale on  Achievement Motivation in Mathematics are described below 
 	Work ethic
	Work ethic is considered as the desire to work hard and excel in Mathematics and is based on the reinforcement in the performance itself. 
	Eg. I like to do mathematical problems which others cannot. 
	Pursuit of excellence
	The pursuit of excellence is the competition with a standard of excellence and is the reward obtained by making the best performance in Mathematics. 
           Eg. I am satisfied with my excellence in Mathematics. 
	 Status aspiration.
	The desire to be a dominant member or leader among other students is incorporated under status aspiration. It is the reinforcement in climbing the social status hierarchy. 
	Eg. Others should consider me as a model in Mathematics learning. 
	Competitiveness
	It is the satisfaction obtained while competing with others. 
	Eg. I like to compete with students who are capable in Mathematics.
 	Acquisitiveness
	Motivation based on the reinforcing properties of position or possession attained by the individual is considered as acquisitiveness. 
	Eg. I desire to be accepted in my Mathematics class. 
	Mastery 
Mastery is the satisfaction obtained when succeeding in the study of difficult matters. 
Eg. I feel satisfaction when I am successful in completing difficult mathematical tasks. 
	Dominance 
It includes the desire to lead or to take initiative or to become a dominant member of the group. 
	Eg. I am a leader in all mathematical activities. 
	Scoring Procedure 
Students were asked to respond to the statements of the scale on Achievement Motivation in Mathematics in any of the three ways viz., Always (A), Sometimes (B), and Never (C). For a positive statements, the scores assigned are three, two, and one and for a negative statement, the scores  assigned are one, two, and three respectively for responses Always (A), Sometimes (B), and Never (C). The total score obtained for each sample in scale on Achievement Motivation in Mathematics were considered as the scores of Achievement Motivation in Mathematics of secondary school students.
Reliability 
	Reliability of the scale was established by developers of the scale by using the test-retest method and the value obtained is 0.796 (N = 40) and the interval between administration is three weeks. Cronbach's alpha coefficient was also was found to be 0.83 (N = 52). These indices indicate that the scale is reliable. 
	Validity 
          The developers of Scale of Achievement Motivation in Mathematics ensured the validity of scale by using face validity. The authors of the scale also ensured the construct validity and criterion related validity  of the tool. The coefficient obtained for construct validity is 0.721 and for criterion validity is 0.657. All these indices showed that the scale is reasonably valid to measure.
	A copy of scale of Achievement Motivation in Mathematics (Malayalam and English version) is given as Appendix V and VI.
Data Collection Procedure
Mathematical Beliefs Inventory was administered to the sample selected for the study. In order to collect data, the investigator asked permission from the Headmasters of secondary schools in advance to administer the tool. After getting permission, the investigator administered the two tools  among the secondary school students selected from Kozhikode and Malappuram districts of Kerala state. Before administering the tool the investigator explained the purpose of the study, directions for making responses and ensured secrecy of their response. The filled response sheets were collected back by the investigator. The marks of Achievement Test conducted by the school for first terminal examination were collected from respective class teachers of the selected sample.
The general data sheet was used to collect general information such as name, occupation, educational level, income and locale of residence. To calculate Socio-Economic Status scores of parents proper weightage was given to educational level, occupation and income as shown in the Table 5.
Table 5
Weightage Given to Educational Level, Occupation and Income for Calculation of the Socio-Economic Status Score
Education	Weightage	Occupation	Weightage	Income	Weightage	
SSLC/Below SSLC	5	Coolie	5	Below 50,000	5	
Plus-two/
Pre- degree	10	Agriculture	10	50,000-2 lakhs	10	
Degree	15	Business	15	2 - 3.5 lakhs	15	
Post graduation	20	Employed	20	3.5-5 Lakhs	20	
Professional	25	Professional	25	Above  5 lakhs	25	
Statistical Techniques Used
   	The statistical techniques used for the analysis of data are explained below.
Descriptive Statistics
	To know the basic properties of the variables mean, median, mode, standard deviation, skewness and kurtosis was calculated to the dependent and independent variables such as Mathematical Beliefs, Achievement Motivation and Achievement in Mathematics of secondary school students.
	Mean
	The mean is the most used method of describing the central tendency. The mean is found out using the formula

Where, 
X      = Mean
A     =  Assumed mean 
fi       =  frequency of the class interval
di      =  xi-A
c      =  length of the class interval
∑fi    =  Total number of scores
	Median
Median is a point in an array, above and below which one half of the scores fall. It is the measure position rather than the magnitude.
The median is calculated using the formula.
	Median = 
Where, 
	l = lower class boundary of the median class
	h = size of the median class interval 
	f = frequency corresponding to the median class
	N = Total number of observations i.e., sum of the frequencies 
	c =  Cumulative frequency preceding median class. 
	Mode
The mode is defined as the most frequently occurring score in a distribution.
	The mode was calculated using the formula.	
		Mode = 3 median – 2 mean
	Standard deviation
	Standard deviation is the most important measure of variability. The standard deviation is the square root of the average of the squares of deviations of the scores taken from mean, standard deviation is calculated using the formula.

Where,
  	f = frequency of the class interval
∑f = total number of scores
  	x = raw score
	Skewness
	A distribution is said to be skewed if the value of the mean, median &mode are different and there is symmetry between the right and left half of the curve such type of curve is inclined more towards the left or right of the centre of the curve.
Skewness was calculated by using the formula.

	Kurtosis
	The term kurtosis refers to the flatness or peakness of a frequency distribution as compared with the normal. The following formula for measuring kurtosis is 
Kurtosis = P75 –P25
                 2(P90 –P10)
Where,
 P75 = 75th percentile
P25= 25th percentile
P90 = 90th percentile
P10 = 10th percentile
Test of Significance of the Difference between Means of Large Independent   Sample (t test)
	Test of significance of the difference between mean of large independent sample were used to know whether there exist significant gender difference in the mean score of Mathematical Beliefs, Achievement Motivation and Achievement in Mathematics of secondary school students.
	The t value can be calculated using the formulae;

Where,
X1 = Mean of the upper group
X2 = Mean of the upper group
s1 = standard deviation of the upper group
s2 = standard deviation of the lower group
n1 = Sample size of the upper group
n2 = Sample size of the lower group
One way Analysis of Variance
	One way analysis of variance was carried to know whether there exist significant difference in the mean scores of Mathematical Beliefs, Achievement Motivation and Achievement in Mathematics of secondary school students for the subgroups based on Socio-Economic Status.
Two-way Analysis of Variance with (2x3) Factorial Design
	Analysis of Variance (ANOVA) has been defined as the separation of the variance ascribable to other groups (Fisher, 1950). In other words, the Analysis of Variance is used to test the significance of the difference between the means of a number of different populations. It is an effective way to determine whether the means of more than two samples are too different to attribute to sampling error. ANOVA is an inferential statistical procedure by which a researcher can test the null hypothesis that two or more population means are equal. If two in dependent variables are included simultaneously in an ANOVA, the analysis is called a two-way ANOVA.
	Two way analysis of variance with 2X3 factorial design was carried to study the main and interaction effect of the independent variables namely, Mathematical Beliefs and Achievement Motivation on the dependent variable Achievement in Mathematics of secondary school students. The 2X3 factorial design consist of two levels of Mathematical Beliefs and three levels of Achievement Motivation. Mathematical Beliefs were classified into High Mathematical Beliefs group and Low Mathematical Beliefs group. The Achievement Motivation was classified into High Achievement Motivation group, Average Achievement Motivation group and Low  Achievement Motivation group.
Classification Techniques
The classification techniques followed for the independent variables; Mathematical Beliefs and Achievement Motivation are described as follows:
Classification of Mathematical Beliefs
	The scores of Mathematical Beliefs of secondary school students was classified into two groups. i.e., High Mathematical Beliefs group and Low Mathematical Beliefs groups. The classification was done on the basis of the median value, i.e., 27 obtained for the scores of Mathematical Beliefs of secondary school students. Students whose score fall above and equal to median were considered as High Mathematical Beliefs group and whose score below the median was considered as Low Mathematical Beliefs group.
	Classification of Achievement Motivation
	The scores of Achievement Motivation of secondary school students was classified into three groups. i.e., High Achievement Motivation group, Average Achievement Motivation group  and Low Achievement Motivation group were considered for ANOVA. The mean and standard deviation of the variable was calculated for the total sample and those who score greater than or equal to  were taken as High Achievement Motivation group, those in between  and as Average Achievement Motivation group and below or equal to  as a Low Achievement Motivation group.