SUMMARY, FINDINGS, CONCLUSION 
AND SUGGESTIONS

This chapter provides an overview of the significant aspects of the various stages of the study. This chapter includes study in retrospect, major findings of the study, educational implications and suggestions for the further research in the area. 
Study in Retrospect
This section tries to make a retrospective study of the title, variables, objectives, hypotheses, methodology, tools and statistical technique used for the study.
Restatement of the Problem
The study is entitled as INTERACTION EFFECT OF MATHEMATICAL BELIEFS AND ACHIEVEMENT MOTIVATION ON ACHIEVEMENT IN MATHEMATICS OF SECONDARY SCHOOL STUDENTS
Variables of the Study
	The independent and dependent variables selected for the present study are: 
	Independent variable
The independent variables used in the study are :
•	Mathematical Beliefs
•	Achievement Motivation 
	Dependent variable
The dependent variable used in the study is ‘Achievement in Mathematics’.
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
	The study was intended to investigate the main effect and interaction effect of Mathematical Beliefs and Achievement Motivation on Achievement in Mathematics among secondary school students. The investigator used survey method for the study. A precise description of sample, tool and statistical technique used for the study are described.
Sample of the study
The population for the study comprised of secondary school students of the Kerala state. The present study was conducted on a sample of 600 secondary school students selected from various schools of Kozhikode and Malappuram district of Kerala state, by using stratified sampling technique giving due to the representation of gender and socioeconomic status of secondary school students.
Tools used for the study
To measure the variable Mathematical Beliefs, investigator developed Mathematical Beliefs Inventory (Ganisha & Niranjana, 2019) with the help of supervising teacher. For measuring, the scale on Achievement Motivation in Mathematics (Vijayakumari & Sumangala, 2000) was adopted. The tool intended to measure the striving of secondary school pupils to attain expected goals by learning mathematics.
Statistical techniques used for the study
For the purpose of analyzing the data, investigator carried that statistical technique such as
•	Descriptive statistics
•	Test of significance difference between mean score of large 
     independent sample.(t – test )
•	One – way ANOVA
•	Two way ANOVA (2x3 factorial design)
Major Findings of the Study
	Important findings of the study are presented below:
1.	The value of mean, median and mode for Mathematical Beliefs for the total sample of secondary school students are 26.62, 27 and 29 which is approximately equal. The coefficient of skewness is -0.77 show that the distribution of the scores of Mathematical Beliefs of the secondary school students is slightly negatively skewed for the total sample. The measure of kurtosis for Mathematical Beliefs reveals that the distribution of scores of Mathematical Beliefs (K=0.63) is slightly leptokurtic in nature for the total sample of the secondary students. It can be concluded that the distribution of the variable Mathematical Beliefs of secondary school students follows normal distribution.
2.	 The value of mean, median and mode for Achievement Motivation for the total sample of secondary school students are 80.87, 80 and 82 which is approximately equal. The coefficient of skewness is 0.06 shows that the distribution of the scores of Achievement Motivation of the secondary school students is slightly positively skewed for the total sample. The measure of kurtosis for Achievement Motivation reveals that the distribution of scores of Achievement Motivation (K=0.50) is slightly leptokurtic in nature for the total sample of the secondary students. It can be concluded that the distribution of the variable Achievement Motivation for the total sample is approximately normal.
3.	The value of mean, median and mode for Achievement in Mathematics for the total sample of secondary school students are 34.04, 31 and 30 which is approximately equal. The coefficient of skewness is 0.82 shows that the distribution of the scores of Achievement in Mathematics of the secondary school students is slightly positively skewed for the total sample. The measure of kurtosis for achievement in Mathematics reveals that the distribution of scores of Achievement in Mathematics (K=0.55) is slightly leptokurtic in nature for the total sample of the secondary students. It can be concluded that the distribution of the variable achievement in Mathematics for the total sample is approximately normal.
4.	The t-value obtained for Mathematical Beliefs is 1.63, which is less than the tabled value at 0.05 level (1.96). Since the t-value obtained is less than the tabled value, it can be concluded there exists no significant difference in the mean scores of Mathematical Beliefs of boys and girls of secondary schools.
5.	The t-value obtained for Achievement Motivation is 1.82, which is less than the tabled value at 0.05 level (1.96). Since the t-value obtained is less than the tabled value, it can be concluded there exists no significant difference in the mean scores of Achievement Motivation of boys and girls of secondary schools.
6.	The t-value obtained for Achievement in Mathematics is 1.66, which is less than the tabled value at 0.05 level (1.96). Since the t-value obtained is less than the tabled value, it can be concluded there exists no significant difference in the mean scores of Achievement in Mathematics of boys and girls of secondary schools.
7.	The F-value obtained for Mathematical Beliefs (0.11) of secondary school students for subgroup based on Socio – Economic Status is less than the tabled value of F (3.01). Hence the difference in the mean scores of Mathematical Beliefs among High Socio–Economic Status group, Average Socio –Economic Status group and Low Socio–Economic Status group of secondary school students is not significant.
8.	  The F value obtained for Achievement Motivation (0.92) of secondary school students for subgroup based on Socio–Economic Status is less than the tabled value of F (3.01) for df (2,599) at 0.05 level of significance. Thus the mean scores of Achievement Motivation do not differ significantly among High Socio-Economic Status group, Average Socio-Economic Status group and Low Socio-Economic Status group of secondary school students. 
9.	The F value obtained for Achievement in Mathematics (10.11) of secondary school students for subgroup based on Socio –Economic Status is greater than the tabled value of F (4.64) for df (2,599) at 0.01 level of significance. Thus mean scores of Achievement in Mathematics differ significantly on the basis of Socio-Economic Status of secondary school students. Those secondary school students who belongs to High Socio-Economic status group scores high on Achievement in Mathematics than those students who belongs to Average and Low Socio-Economic status group. Those students who belongs to Average Socio-Economic status group scores high on Achievement in Mathematics than those students who belongs to Low Socio-Economic status.
10.	The main effect of Mathematical Beliefs on Achievement in Mathematics is significant as the obtained F value for Mathematical Beliefs on Achievement in Mathematics for total sample is 4.01 which is greater than the tabled value 3.86 for degrees of freedom (1, 594) required for significance at 0.05 level. It indicates that there exists significant difference in the mean scores of Achievement in Mathematics for High Mathematical Beliefs group and Low Mathematical Beliefs group.
11.	 The mean scores of two groups of Mathematical Beliefs reveals that the mean score of Achievement in Mathematics of High Epistemological Beliefs group (M=36.08, SD=14.22) compared to that of Low Mathematical Beliefs group (M=32.14, SD=14.05) is significantly greater. This indicates that those secondary school students who are having high Mathematical Beliefs, scores high on Achievement in Mathematics than those students who are having low Mathematical Beliefs.
12.	The main effect of Achievement Motivation on Achievement in Mathematics of secondary schools students for total sample is significant as the F value (8.18) obtained is greater than the tabled value at 0.01 level of significance. Therefore, there exists significant difference in the mean scores of Achievement in Mathematics of secondary school students for the High Achievement Motivation, Average Achievement Motivation and Low Achievement Motivation groups for the total sample. 
13.		The mean scores of Achievement in Mathematics for the average Achievement Motivation group (M=35.20) is higher than that of Low Achievement Motivation group (M= 29.76). Thus the two groups differ significantly. The mean scores of Achievement in Mathematics for Average Achievement Motivation group (M= 35.20) and High Achievement Motivation group (M = 36.86) do not differ significantly. It means that there is no difference in Achievement in Mathematics between Average Achievement Motivation group and High Achievement Motivation group. The mean score of Achievement in Mathematics for the High Achievement Motivation group (M= 36.86) and Low Achievement Motivation group (M = 29.76) differ significantly.
14.	The F value obtained for interaction effect of Mathematical Beliefs and Achievement Motivation on Achievement Mathematics for total sample is 0.77 which is less than the tabled value 4.64 for degrees of freedom (2, 594) required for significance at 0.05 levels. It indicates that the interaction effect of Mathematical Beliefs and Achievement Motivation on Achievement in Mathematics is not significant even at 0.05 level. Therefore, the mean scores of Achievement in Mathematics at two levels of Mathematical Beliefs do not vary significantly for High- Average- Low Achievement Motivation groups of secondary school students for total sample.
Conclusion
      	The distribution of variables Mathematical Beliefs and Achievement Motivation and Achievement in Mathematics among secondary school students follows normal distribution approximately. The results indicate that Mathematical Beliefs of secondary school students do not differ significantly with respect to gender and Socio–Economic Status. Achievement Motivation of secondary school students also do not differ significantly with respect to gender and Socio–Economic Status. But Achievement in Mathematics of secondary school students differ significantly on the basis of Socio-Economic Status but no significant difference exist with respect to gender.
	The results of two-way ANOVA with 2 x 3 factorial design indicates that the main effect of Mathematical Beliefs and Achievement Motivation on Achievement in Mathematics is significant and their interaction effect is not significant. 
Tenability of Hypotheses
The first hypothesis states that there exists significant gender difference in the mean scores of Mathematical Beliefs of secondary school students. The result showed that there is no significant difference in the mean scores of Mathematical Beliefs among secondary school students with respect to gender. Thus, the first hypothesis is rejected. 
The second hypothesis states that there exist significant difference in the mean scores of Mathematical Beliefs of secondary school students with respect to their Socio-Economic Status. In this study it was found that there exists no significant difference in the mean scores of Mathematical Beliefs of secondary school students with respect to their Socio-Economic Status. Hence, the second hypothesis is rejected.
The third hypothesis states that there exist significant gender difference in the mean scores of Achievement Motivation of secondary school students. In this study it was found there exist no significant gender difference in the mean scores of Achievement Motivation of secondary school students. Hence the third hypothesis is rejected. 
The fourth hypothesis states that there exist significant difference in the mean scores of Achievement Motivation of secondary school students with respect to their Socio-Economic Status. In this study it was found that there exist no significant difference in the mean scores of Achievement Motivation of secondary school students with respect to their Socio-Economic Status. Hence the fourth hypothesis is rejected
	The fifth hypothesis states that there exist significant gender difference in the mean scores of Achievement in Mathematics of secondary school students. The study reveals that there is no significant gender difference in the mean scores of Achievement in Mathematics of secondary school students. Hence fifth hypothesis is rejected.
The sixth hypothesis states that there exist significant difference in the mean scores of Achievement in Mathematics of secondary school students with respect to their socio-economic status. The study reveals that there is significant difference in the mean scores of Achievement in Mathematics of secondary school students with respect to their socio-economic status. Hence the sixth hypothesis is accepted.
	The seventh hypothesis states that 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. The results indicated that Achievement in Mathematics for High Mathematical Beliefs group and Low Mathematical Beliefs groups differ significantly. The results also showed that there exist significant difference in the mean scores of Achievement in Mathematics for High- Average –Low Achievement Motivation groups of secondary school students. Then, there exist significant main effect of Mathematical Beliefs and Achievement Motivation on Achievement in Mathematics of secondary school students. Thus the seventh hypothesis is accepted.
The eighth hypothesis states that 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. The results indicated that the mean scores of Achievement in Mathematics do not vary significantly for two levels of Mathematical Beliefs group and three levels of Achievement Motivation group. Thus the eighth hypothesis is rejected.
Educational Implications of the Study
The results of the study have various implications in educational field which may help curriculum developers and teachers to their attempt to enhance quality of education in Kerala. As a result of the findings in this research study, mathematics teachers need to be more aware of their students’ mathematical beliefs in order to help their students.  Teachers should attempt to foster positive beliefs about mathematics education among their students. There are  variety of ways that teachers can use the findings of this study to encourage the development of positive mathematical beliefs.  Teachers should not only teach their students mathematics, but specifically focus on teaching their students what mathematics is about. As students develop a more positive set of mathematical beliefs and develop a deeper understanding of what doing mathematics means, students may be better able to attain knowledge. The results indicated that the mathematical beliefs influence achievement significantly.  An increase in achievement of students is possible by increasing level of mathematical beliefs. Teachers should understand the mathematical beliefs held by the students related to Mathematics subject.
	Curriculum developers should understand the low mathematical beliefs affects learning. Curriculum developer can take up the challenge of helping students associate their productive mathematical beliefs with the activity of the course and the discipline. Sometimes student’s conceptual difficulties make difficulties in doing experiment and problem solving in mathematics. Teacher’s proper planning and execution of class by considering student’s mathematical beliefs can resolve this problem. Mathematics teacher can improve student’s mathematical beliefs in mathematics subject, through giving different tasks, activities, experiment or problem by considering student’s Mathematical Beliefs. Through this student’s dislikes about Mathematics subject can be reduced.
Mathematics is among the most misunderstood subjects. It is difficult due to its abstract nature,  but surely it can overcome through taking systematic effort. Teacher can strengthen student’s   mathematical beliefs and convince students that no one excels without effort, and that struggle is part of all kind of achievement. Teacher’s differential treatment can corrode student’s beliefs and help students to learn more effectively by understanding how they learn and how to manage their learning. It is the task of teachers to make the students aware of the value of mathematics task, its nature and learning. Greater emphasis should be given to student’s beliefs regarding mathematics and its learning equally with mathematics content.
Achievement Motivation is generally regarded as a drive to achieve targets and the process to maintain the drive. Motivation   provides an important foundation to complete behavior, such as planning, organization, decision making, learning and assessment. So teacher should know about factors influencing achievement motivation. It is important for both parents and educators, to understand why promoting and encouraging achievement motivation from an early age of imperative. If a student has high achievement motivation, knowing whether that the student is intrinsically or extrinsically motivated may be important in making predictions about the student’s academic career. Achievement  motives  influence  people’s  lives  in a  variety  of  ways,  although  more  attention  has focused  on  the  impact  of  fear  of  failure  than  on need for achievement. In academic contexts, fear of failure has been linked with decreased moral functioning and increased attention seeking behavior. 
Teacher should understand how to increase achievement motivation among students. Everyone likes getting rewards and offering our students the chances to earn them is an excellent source of motivation. Teacher should consider the personalities and needs of our students to determine appropriate rewards for our class. Teacher can conduct fieldtrips, bring in speakers, or even just head to the library for some research. Teachers should create a safe, supportive environment for students, affirming their belief in a student’s abilities rather than laying out the consequences of not doing things, students are much more likely to get and stay motivated to do their work. 
Suggestions for Further Research
The experience of conducting this study made the investigator to suggest the following further research in this area
•	Secondary school students are the samples of this study. The study can extended to other levels also.
•	The same study can be replicated on the basis of other components like creativity, problem solving ability, attitude of mathematics etc.
•	A study on pre-service teacher’s mathematical beliefs in relation to their achievement can be carried out.
•	Experimental study to understand the effectiveness of strategies on enhancing mathematical beliefs can be carried out.