EXPLICIT DIRECT INSTRUCTION VERSUS DISCOVERY LEARNING 1
CHAPTER4
STATISTICAL ANALYSIS
In this chapter, a detailed statistical analysis of the independentgroups will be done to establish whether there are any significantdifferences in their means. This will be done using variousstatistical tests with the aim of identifying whether the Discoveryteaching method and the Direct Instruction method affect the learningoutcomes in different ways. This chapter will be categorized into twomajor parts:
Firstly, I will first conduct several independent sampling tests tocompare the means of the two independent groups. For this part of theanalysis, independent samples will be conducted to compare the meansusing a ttest statistical analysis model to compare the means of anormally distributed interval dependent variable for the twoindependent groups 4^{th} and 6^{th} period. Forevery unit test, this paper will first discuss a period 4 testfollowed by a period 6 test. An interpretation of the test resultswill be provided. A comparison of the results derived from the twogroups will also be provided at the end of each unit test result.
Secondly, I will use an ANOVA test to compare the independentvariables and make conclusions based on the two tests. This test willidentify the variability between these independent groups andestablish whether the difference is large relative to the variabilitywithin the same groups. This will help us conclude that the means ofthese two groups from which the data were drawn are significantlydifferent or not. This ANOVA test will be done on the Fall Final Testfor both period 4 and 6. Moreover, the results of the statisticaltests done to validate the results will also be discussed in details.
Part 1: t – test
UNIT 0 TEST:
The period 4 and period 6 students both took unit 0 test. The testresults have been run on SPSS version 24 independent ttest todetermine the mean of the outcome. The results for each period classhave been discussed accordingly.

Period 4
The following are the results after computingindependent ttest for period 4 students to ascertain any genderdifferences in mean performance on Unit Test 0:
Group Statistics 

GENDER 
N 
Mean 
Std. Deviation 
Std. Error Mean 

UNIT 0 TEST 
Male 
15 
85.000 
17.7764 
4.5898 
Female 
14 
82.857 
16.4778 
4.4039 
Period 4 students taking unit 0 test had a mean of 85 for males and82.86 for female students. The male recorded higher mean score thanthe females. The standard deviation differences were minimal, though,at 1.29.
Independent Samples Test 

Levene`s Test for Equality of Variances 
ttest for Equality of Means 

F 
Sig. 
t 
df 
Sig. (2tailed) 
Mean Difference 
Std. Error Difference 
95% Confidence Interval of the Difference 

Lower 
Upper 

UNIT 0 TEST 
Equal variances assumed 
.255 
.618 
.336 
27 
.739 
2.143 
6.38 
10.94 
15.23 
Equal variances not assumed 
.337 
26.999 
.739 
2.143 
6.36 
10.91 
15.20 
Our computed test statistic t is t=0.34 with a df of df=27. Thepvalue (Sig (2tailed)) corresponding to the given degreesof freedom and test statistic is p =0.74. Since our p> .01is greater than the test preferred significance level α =0.05, the null hypothesis is accepted and we conclude that the thereis no significant difference between the mean score for males andfemale scores. Based on the results above, we can conclude that therewas no significant mean difference in the mean scores between maleand female students (t_{27.00} = 0.34, p>0.05).The average test score for the male students was (15.2315.19) =0.04more than the average score for the females.

Period 6
Group Statistics 

GENDER 
N 
Mean 
Std. Deviation 
Std. Error Mean 

UNIT 0 TEST 
Male 
16 
85.25 
12.1299 
3.0325 
Female 
18 
85.78 
13.3485 
3.1463 
Period 6 students taking unit 0 test had a mean of 85.25 for malesand a mean of 85.78 for the female students. The females, however,have a higher mean (85.79) score compared to men mean score (85.25),hence statistically insignificant. The standard deviation is alsominimal at 3.03 for girls and 3.15 for men.
Independent Samples Test 

Levene`s Test for Equality of Variances 
ttest for Equality of Means 

F 
Sig. 
t 
df 
Sig. (2tailed) 
Mean Difference 
Std. Error Difference 
95% Confidence Interval of the Difference 

Lower 
Upper 

UNIT 0 TEST 
Equal variances assumed 
.045 
.833 
.12 
32 
.905 
.528 
4.395 
9.48 
8.43 

Equal variances not assumed 
.12 
31.98 
.905 
.528 
4.3698 
9.429 
8.37 
Our computed test statistic t is t= 0.12 with a df of df=32. Thepvalue (Sig (2tailed)) corresponding to the given degreesof freedom and test statistic is p =0.91. Since our p> .01is greater than the test preferred significance level α =0.05, the null hypothesis can be accepted. Based on our resultsabove, we can conclude that there was no significant difference inthe mean scores between male and female students (t_{31.99}= 0.12, p>0.05). The average test score for the malestudents was (8.428.37) = 0.05 more than the average score for thefemales. We conclude that the there is no significant differencebetween the mean score for boys and girls scores.
Comparison Period 4 and 6 Results:
From Unit 0 test results for Period 4 and 6 classes above, we canconclude that the period 4 class showed that there was no significantmean difference in the mean scores between male and female students(t_{27.00} = 0.34, p>0.05). The average testscore for the male students was (15.2315.19) = 0.04 more than theaverage score for the females. The period 4 male students had a meanscore of 85.00 while female students had a meanscore of 82.86, there is no significant difference compared to period6 mean scores for male and female students, 85.25 and 85.78,respectively. The period 6 class results indicated that therewas no significant mean difference in the mean scores between maleand female students (t_{31.98} = 0.12, p>0.05).Theaverage test score for the male students was (8.428.37) = 0.05 morethan the average score for the females.
The results of the unit 0 test for period 4 and 6 indicate that thereis no statistically significant difference between the mean testscores for both classes. In other words, since period 4 classstudents have no statistically significantly higher mean scores onthe unit tests than the 6^{th} period, the hypothesis holds.The research hypothesis stated that the period 4 class will show ahigher level of mastery and understanding of the mathematical conceptpresented during the semester. Our results confirm that thehypothesis is true.
UNIT 1 TEST

Period 4
The period 4 and period 6 students both took unit 1 test. The test results have been run on SPSS version 24 independent ttest to determine the mean of the outcome. These are the results for the test: Group Statistics 

GENDER 
N 
Mean 
Std. Deviation 
Std. Error Mean 

UNIT 1 TEST 
Male 
15 
79.87 
19.09 
4.93 
Female 
14 
75.14 
26.17 
7.00 
Period 4 students taking unit 1 test had a mean of 79.87 for malesand 75.14 for female students. The standard deviation differences are7.08, relatively large difference. The standard error means arehowever not as large 4.93 and 7.00.
Independent Samples Test 

Levene`s Test for Equality of Variances 
ttest for Equality of Means 

F 
Sig. 
t 
df 
Sig. (2tailed) 
Mean Difference 
Std. Error Difference 
95% Confidence Interval of the Difference 

Lower 
Upper 

UNIT 1 TEST 
Equal variances assumed 
2.94 
.098 
.56 
27 
.581 
4.72 
8.47 
12.65 
22.092 

Equal variances not assumed 
.55 
23.69 
.586 
4.72 
8.56 
12.95 
22.398 
Our computed test statistic t is t= 0.56 with a df of df=27. Thepvalue (Sig (2tailed)) corresponding to the given degreesof freedom and test statistic is p =0.58. Since our p =.58 is greater than the test preferred significance level α =0.05, the null hypothesis is accepted.
Based on our results above, we can conclude that there was nosignificant mean difference in the mean scores between male andfemale students (t_{23.69} = 0.55, p>0.05).The average test score for the male student’s was 0.31 more thanthe average score for the female students.

Period 6
Group Statistics 

GENDER 
N 
Mean 
Std. Deviation 
Std. Error Mean 

UNIT 1 TEST 
Male 
16 
73.56 
19.78 
4.95 

Female 
18 
81.50 
18.45 
4.35 

Independent Samples Test 

Levene`s Test for Equality of Variances 
ttest for Equality of Means 

F 
Sig. 
t 
df 
Sig. (2tailed) 
Mean Difference 
Std. Error Difference 
95% Confidence Interval of the Difference 

Lower 
Upper 

UNIT 1 TEST 
Equal variances assumed 
.68 
.42 
1.2 
32 
.24 
7.94 
6.56 
21.29 
5.43 

Equal variances not assumed 
1.21 
30.88 
.24 
7.94 
6.59 
21.4 
5.5 
Our computed test statistic t is t= 1.21 with a df of df=32. Thepvalue (Sig (2tailed)) corresponding to the given degreesof freedom and test statistic is p =0.24. Since our p>.0001 is greater than the test preferred significance level α =0.05, the null hypothesis is accepted. Based on our results above, wecan conclude that there was no significant mean difference in themean scores between male and female students (t_{30.878}= 1.21, p>0.05). The average test score for the malestudents was (5.425.49) = 0.08 less than the average score for thefemales.
Comparison Period 4 and 6 Results:
From Unit 1 test results for Period 4 and 6 classes above, wecan make our conclusion From Unit 0 test results for Period 4 and 6classes above, we can make our conclusion The period 4 class showedthat there was no significant mean difference in the mean scoresbetween male and female students (t_{23.69} = 0.55,p>0.05). The average test score for the male students was0.31 more than the average score for the females.
The period 6 class results indicated that There was no significantmean difference in the mean scores between male and female students(t_{30.878} = 1.21, p>0.05).The averagetest score for the male students was (5.425.49) = 0.08 less thanthe average score for the female students.
The results of the unit 1 test for period 4 and 6 indicated thatthere was no statistically significant difference between the meantest scores for both classes. In other words, since period 4 classstudents have no statistically significantly higher mean scores onthe unit tests than the 6^{th} period, the hypothesis doesnot hold. The research hypothesis states that the period 4 class willshow a higher level of mastery and understanding of the mathematicalconcepts presented during the semester. The unit 1 test independentttest indicates that there was little difference between the twomethods of teaching.
UNIT 2 TEST
The period 4 and period 6 students both took unit 2 test. The testresults have been run on SPSS version 24 independent ttest todetermine the mean of the outcome. The results for each period classwould be discussed.

Period 4
Group Statistics 

GENDER 
N 
Mean 
Std. Deviation 
Std. Error Mean 

UNIT 2 TEST 
Male 
15 
89.80 
8.80 
2.27 
Female 
14 
86.93 
16.23 
4.34 
Period 4 students taking unit 0 test had a mean of 89 for males and86 for female students.
Independent Samples Test 

Levene`s Test for Equality of Variances 
ttest for Equality of Means 

F 
Sig. 
t 
df 
Sig. (2tailed) 
Mean Difference 
Std. Error Difference 
95% Confidence Interval of the Difference 

Lower 
Upper 

UNIT 2 TEST 
Equal variances assumed 
8.068 
.008 
.60 
27 
.56 
2.87 
4.80 
6.98 
12.73 

Equal variances not assumed 
.59 
19.73 
.56 
2.87 
4.9 
7.35 
13.1 
Our computed test statistic t is t= 0.60 with a df of df=27. Thepvalue (Sig (2tailed)) corresponding to the given degreesof freedom and test statistic is p =0.56. Since our p>.0001 is greater than the test preferred significance level α =0.05, the null hypothesis is to be accepted. Based on our resultsabove, we can conclude that there was no significant mean differencein the mean scores between male and female students (t_{19.73}= 0.59, p>0.05). Moreover, the average test score for themale students was (12.7313.10) = 0.37 less than the average scorefor the females.

Period 6
Group Statistics 

GENDER 
N 
Mean 
Std. Deviation 
Std. Error Mean 

UNIT 2 TEST 
Male 
16 
86.31 
15.25 
3.81 
Female 
18 
93.61 
15.01 
3.54 
Independent Samples Test 

Levene`s Test for Equality of Variances 
ttest for Equality of Means 

F 
Sig. 
t 
df 
Sig. (2tailed) 
Mean Difference 
Std. Error Difference 
95% Confidence Interval of the Difference 

Lower 
Upper 

UNIT 2 TEST 
Equal variances assumed 
.23 
.63 
1.4 
32 
.17 
7.299 
5.196 
17.88 
3.29 

Equal variances not assumed 
1.4 
31.40 
.17 
7.299 
5.20 
17.90 
3.30 
In comparison, period 6 students taking unit 0 tests had a mean of85.25 for males and a mean of 85.78 for the female students. Thestandard deviation differences were also minimal. The independentsamples test shows a pvalue of 0.83 which is also greater than 0.05,hence not statistically significant.
Our computed test statistic t is t=1.41 with a df of df=32. Thepvalue (Sig (2tailed)) corresponding to the given degreesof freedom and test statistic is p =0.17. Since our p = .17 isgreater than the test preferred significance level α = 0.05,the null hypothesis is accepted and we conclude that the mean scorefor boys and girls is not significantly different.
Based on our results above, we can conclude as follows:

There was a significant mean difference in the mean scores between male and female students (t_{31.404} = 1.40, p< .001).

The average test score for the male students was (3.293.30) = 0.01 less than the average score for the females.
Comparison Period 4 and 6 Results:
From Unit 2 test results for Period 4 and 6 classes above, wecan make our conclusion The period 4 class results showed that therewas no significant mean difference in the mean scores between maleand female students (t_{19.73} = 0.59, p>0.05).Theaverage test score for the male students was (12.7313.10) = 0.37less than the average score for the females. The period 4 malestudents had a mean score of 89.80 while femalestudents had a mean score of 86.93. There was no significantdifference compared to period 6 mean scores for male and femalestudents and the means were 86.31 and 93.61 respectively.
The period 6 class results showed that there was a significant meandifference in the mean scores between male and female students(t_{31.404} = 1.40, p< .001).The averagetest score for the male students was (3.293.30) = 0.01 less thanthe average score for the females.
The results of the unit 2 test for period 4 and 6 indicate that thereis no significant difference between the mean test scores for bothclasses. In other words, since period 4 class students had nosignificantly higher mean scores on the unit tests than the 6^{th}period, the hypothesis does not hold. The research hypothesis statedthat the period 4 class will show a higher level of mastery andunderstanding of the mathematical concepts presented during thesemester. But the results of the tests for unit 2 done to test thesuccess of both discovery and instructional classes’ showed theopposite.
UNIT 3 TEST
The period 4 and period 6 students both took unit 3 test. The testresults have been run on SPSS version 24 independent ttest todetermine the mean of the outcome. The results for each period classhave been discussed accordingly.

Period 4
Group Statistics 

GENDER 
N 
Mean 
Std. Deviation 
Std. Error Mean 

UNIT 3 TEST 
Male 
15 
73.87 
21.39 
5.53 
Female 
14 
75.29 
25.54 
6.83 
Period 4 students taking unit 3 test had a mean of 73.87 for malesand 75.29 for female students.
Independent Samples Test 

Levene`s Test for Equality of Variances 
ttest for Equality of Means 

F 
Sig. 
t 
df 
Sig. (2tailed) 
Mean Difference 
Std. Error Difference 
95% Confidence Interval of the Difference 

Lower 
Upper 

UNIT 3 TEST 
Equal variances assumed 
.695 
.412 
.16 
27 
.87 
1.42 
8.73 
19.32 
16.49 

Equal variances not assumed 
.16 
25.46 
.87 
1.42 
8.78 
19.49 
16.65 
Our computed test statistic t is t= 0.16 with a df of df=27. Thepvalue (Sig (2tailed)) corresponding to the given degreesof freedom and test statistic is p =0.87. Since our p>.0001 is greater than the test preferred significance level α =0.05, the null hypothesis is accepted. Based on our results above, wecan conclude that there was no significant mean difference in themean scores between male and female students (t_{25.46}= 0.16, p>0.05). Moreover, the average test score for themale students was (16.4916.65) = 0.16 less than the average scorefor the females.

Period 6
Group Statistics 

GENDER 
N 
Mean 
Std. Deviation 
Std. Error Mean 

UNIT 3 TEST 
Male 
16 
67.88 
18.92 
4.73 
Female 
18 
75.67 
16.45 
3.88 
In comparison, period 6 students taking unit 3 tests had a mean of67.88 for males and a mean of 75.67 for the female students.
Independent Samples Test 

Levene`s Test for Equality of Variances 
ttest for Equality of Means 

F 
Sig. 
t 
df 
Sig. (2tailed) 
Mean Difference 
Std. Error Difference 
95% Confidence Interval of the Difference 

Lower 
Upper 

UNIT 3 TEST 
Equal variances assumed 
1.44 
.24 
1.29 
32 
.208 
7.79 
6.06 
20.14 
4.56 
Equal variances not assumed 
1.27 
29.98 
.21 
7.79 
6.12 
20.28 
4.70 
Our computed test statistic t is t= 1.29 with a df of df=32. Thepvalue (Sig (2tailed)) corresponding to the given degreesof freedom and test statistic is p =0.21. Since our p>.0001 is greater than the test preferred significance level α =0.05, the null hypothesis is accepted and we conclude that the thereis no significant difference between the mean score for boys andgirls scores. Based on our results above, we can conclude that therewas no significant mean difference in the mean scores between maleand female students (t_{29.984} = 1.27, p>0.05).The average test score for the male students was (4.564.70) = 0.14less than the average score for the females.
Comparison Period 4 and 6 Results:
From Unit 3 test results for Period 4 and 6 classes above, wecan make our conclusion The period 4 class indicated that there wasno significant mean difference in the mean scores between male andfemale students (t_{25.46} = 0.16, p>0.05).The average test score for the male students was (16.4916.65) =0.16 less than the average score for the females. The period 4 malestudents had a mean score of 73.867 whilefemale students had a mean score of 75.286, there is no significantdifference compared to period 6 mean scores for male and femalestudents, 67.88and 75.67, respectively.
The period 6 class indicated that there was no significant meandifference in the mean scores between male and female students(t_{29.984} = 1.27, p>0.05).The average testscore for the male students was (4.564.70) = 0.14 less than theaverage score for the females.
The results of the unit 3 test for period 4 and 6 indicate that thereis no significant difference between the mean test scores for bothclasses. Moreover, since period 4 class students have nosignificantly higher mean scores on the unit tests than the 6^{th}period, the hypothesis does not hold. The research hypothesis statesthat the period 4 class showed a higher level of mastery andunderstanding of the mathematical concepts presented during thesemester. Our results showed that there was no much difference in thestandard deviations for the two classes. The hypothesis does nothold.
UNIT 4 TEST
The period 4 and period 6 students both took unit 4 test. The testresults have been run on SPSS version 24 independent ttest todetermine the mean of the outcome. The results for each period classhave been discussed.

Period 4
Group Statistics 

GENDER 
N 
Mean 
Std. Deviation 
Std. Error Mean 

UNIT 4 TEST 
Male 
15 
75.53 
21.94 
5.66 
Female 
14 
76.43 
26.34 
7.04 
Period 4 students taking unit 4 test had a mean of 75.53 for malesand 76.43 for female students.
Independent Samples Test 

Levene`s Test for Equality of Variances 
ttest for Equality of Means 

F 
Sig. 
t 
df 
Sig. (2tailed) 
Mean Difference 
Std. Error Difference 
95% Confidence Interval of the Difference 

Lower 
Upper 

UNIT 4 TEST 
Equal variances assumed 
2.15 
.15 
.10 
27 
.921 
.895 
8.98 
19.32 
17.53 

Equal variances not assumed 
.099 
25.4 
.92 
.895 
9.04 
19.49 
17.70 
Our computed test statistic t is t= 0.10 with a df of df=27. Thepvalue (Sig (2tailed)) corresponding to the given degreesof freedom and test statistic is p =0.92. Since our p>.0001 is greater than the test significance level α = 0.05,the null hypothesis is to be accepted. Based on our results above, wecan conclude that there was no significant mean difference in themean scores between male and female students (t_{25.40}= 0.10, p>0.05). Moreover, the average test score for themale students was (17.5317.70) = 0.17 less than the average scorefor the females. Based on these findings we can conclude that thethere is no significant difference between the mean score for boysand girls scores.

Period 6
Group Statistics 

GENDER 
N 
Mean 
Std. Deviation 
Std. Error Mean 

UNIT 4 TEST 
Male 
16 
74.31 
15.499 
3.8748 
Female 
18 
79.17 
20.957 
4.9397 
Independent Samples Test 

Levene`s Test for Equality of Variances 
ttest for Equality of Means 

F 
Sig. 
t 
df 
Sig. (2tailed) 
Mean Difference 
Std. Error Difference 
95% Confidence Interval of the Difference 

Lower 
Upper 

UNIT 4 TEST 
Equal variances assumed 
1.125 
.297 
.76 
32 
.453 
4.85 
6.39 
17.87 
8.163 
Equal variances not assumed 
.77 
31.0 
.445 
4.85 
6.28 
17.658 
7.95 
Our computed test statistic t is t= 0.76 with a df of df=32. Thepvalue (Sig (2tailed)) corresponding to the given degreesof freedom and test statistic is p =0.45. Since our p =.45 isgreater than the test significance level of α = 0.05, thenull hypothesis is to be accepted. Based on our results above, wecan conclude that there was no significant mean difference in themean scores between male and female students (t_{31.039}= 0.77, p>0.05). Moreover, the average test score for themale students was (8.16307.95) = 0.21 more than the average scorefor the females. Based on these findings we can conclude that thethere is no significant difference between the mean score for boysand girls scores.
Comparison Period 4 and 6 Results:
From Unit 4 test results for Period 4 and 6 classes above, wecan make our conclusion The period 4 class showed that there was nosignificant mean difference in the mean scores between male andfemale students (t_{31.039} = 0.77, p>0.05).Theaverage test score for the male students was (8.16307.95) = 0.21more than the average score for the females.
The period 6 class showed that There was no significant meandifference in the mean scores between male and female students(t_{31.039} = 0.77, p>0.05). The averagetest score for the male students was (8.16307.95) = 0.21 more thanthe average score for the females. On the other hand, the period 4male students had a mean score of 75.53 whilefemale students had a mean score of 76.43, there is no significantdifference compared to period 6 mean scores for male and femalestudents, 74.31 and 79.17, respectively. The results of theunit 4 test for period 4 and 6 indicate that there is nostatistically significant difference between the mean test scores forboth classes. In other words, since period 4 class students have nostatistically significantly higher mean scores on the unit tests thanthe 6^{th} period, the hypothesis does not hold. The researchhypothesis states that the period 4 class will show a higher level ofmastery and understanding of the mathematical concepts presentedduring the semester. Our results show otherwise.
FALL FINAL TEST
The period 4 and period 6 students both took the final fall test. Thetest results have been run on SPSS version 24 independent ttest todetermine the mean of the outcome. The results for each period classhave been discussed accordingly.

Period 4
Group Statistics 

GENDER 
N 
Mean 
Std. Deviation 
Std. Error Mean 

FALL FINAL TEST 
Male 
15 
170.67 
21.23 
5.48 
Female 
14 
160.93 
28.05 
7.50 
Period 4 students taking Fall Final Test had a mean of 170.67 formales and 160.93 for female students.
Independent Samples Test 

Levene`s Test for Equality of Variances 
ttest for Equality of Means 

F 
Sig. 
t 
df 
Sig. (2tailed) 
Mean Difference 
Std. Error Difference 
95% Confidence Interval of the Difference 

Lower 
Upper 

FALL FINAL TEST 
Equal variances assumed 
.826 
.37 
1.06 
27 
.299 
9.74 
9.2 
9.13 
28.61 

Equal variances not assumed 
1.05 
24.2 
.31 
9.74 
9.29 
9.42 
28.89 
Our computed test statistic t is t= 1.01 with a df of df=27. Thepvalue (Sig (2tailed)) corresponding to the given degreesof freedom and test statistic is p =0.30. Since our p>0.33is greater than the test preferred significance level α =0.05, the null hypothesis is accepted. Based on our results above, wecan conclude that there was no significant mean difference in themean scores between male and female students (t_{24.194}= 1.05, p>0.05). Moreover, the average test score for themale students was (28.6128.90) = 0.29 less than the average scorefor the females. Based on these findings we conclude that the thereis no significant difference between the mean score for boys andgirls scores.

Period 6
Group Statistics 

GENDER 
N 
Mean 
Std. Deviation 
Std. Error Mean 

FALL FINAL TEST 
Male 
16 
158.13 
46.91 
11.73 
Female 
18 
160.33 
26.18 
6.17 
Independent Samples Test 

Levene`s Test for Equality of Variances 
ttest for Equality of Means 

F 
Sig. 
t 
df 
Sig. (2tailed) 
Mean Difference 
Std. Error Difference 
95% Confidence Interval of the Difference 

Lower 
Upper 

FALL FINAL TEST 
Equal variances assumed 
.96 
.34 
.17 
32 
.86 
2.21 
12.84 
28.36 
23.94 

Equal variances not assumed 
.17 
22.9 
.87 
2.21 
13.25 
29.63 
25.21 
Our computed test statistic t is t= 0.17with a df of df=32. Thepvalue (Sig (2tailed)) corresponding to the given degreesof freedom and test statistic is p =0.86 Since our p = .86 isgreater than the test preferred significance level α = 0.05,the null hypothesis is accepted. Based on our results above, we canconclude that there was no significant mean difference in the meanscores between male and female students (t_{22.90} =0.17, p>0.05). Moreover, the average test score for themale students was (23.9425.21) = 1.27 less than the average scorefor the females. Based on those findings we can conclude that thethere is no significant difference between the mean score for boysand girls scores.
Interpretation & Analysis:
From Fall Final test results for Period 4 and 6 classes above, we canmake our conclusion the period 4 class had a pvalue of 0.33while period 6 classes had a pvalue of 0.86.The period 4 classes have a statistically significantly lower meanscore on the Fall Final tests than the period 6 class. The period 4male students had a mean score of 170.67 whilefemale students had a mean score of 160.93, there is no significantdifference compared to period 6 mean scores for male and femalestudents, 158.13 and 160.33, respectively. The results of theFall Final test for period 4 and 6 indicate that there is nosignificant difference between the mean test scores for both classes.In other words, since period 4 class students have no statisticallysignificantly higher mean scores on the unit tests than the 6^{th}period, the hypothesis does not hold. The research hypothesis statesthat the period 4 class will show a higher level of mastery andunderstanding of the mathematical concepts presented during thesemester.
Part 2: Analysis of Variance (ANOVA)
This section provides an alternative test, ANOVA with repeatedmeasures, to substantiate the finding from the Ttest. The use of theANOVA with repeated measures test will be done to determine whetherDiscovery teaching method and the Direct Instruction method affectthe learning outcomes in different ways. ANOVA has been done for allthe 6 tests done period 4 and 6 students. ANOVA test has been doneonly for confirmation purposes.
The following are the results run for Period 4 and 6 students finalfall tests outcome:
WithinSubjects Factors 

Measure: MEASURE_1 

Fall Final 
Dependent Variable 
1 
Period4FallFinal 
2 
Period6FallFinal 
Descriptive Statistics 

Mean 
Std. Deviation 
N 

Period4 Fall Final 
165.39 
25.06 
28 
Period6 Fall Final 
157.29 
38.77 
28 
Looking at the two means for period 4 and period 6 students (165.39and 157.29 respectively), there is no significant statisticaldifference between the two means. Again, the standard deviations forthe two classes, 25.06 and 38.77 are not significantly large and thevariance is 13.71.
Multivariate Tests^{a} 

Effect 
Value 
F 
Hypothesis df 
Error df 
Sig. 
Partial Eta Squared 

Fall Final 
Pillai`s Trace 
.039 
1.099^{b} 
1.00 
27.00 
.304 
.039 

Wilks` Lambda 
.961 
1.099^{b} 
1.00 
27.00 
.304 
.039 

Hotelling`s Trace 
.041 
1.099^{b} 
1.00 
27.00 
.304 
.039 

Roy`s Largest Root 
.041 
1.099^{b} 
1.00 
27.00 
.304 
.039 

a. Design: Intercept Within Subjects Design: Fall Final 

b. Exact statistic 
Mauchly`s Test of Sphericity^{a} 

Measure: MEASURE_1 

Within Subjects Effect 
Mauchly`s W 
Approx. ChiSquare 
df 
Sig. 
Epsilon^{b} 

GreenhouseGeisser 
HuynhFeldt 
Lowerbound 

Fall Final 
1.00 
.00 
0 
. 
1.00 
1.00 
1.00 
Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix. 

a. Design: Intercept Within Subjects Design: Fall Final 

b. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of WithinSubjects Effects table. 
Tests of WithinSubjects Effects 

Measure: MEASURE_1 

Source 
Type III Sum of Squares 
df 
Mean Square 
F 
Sig. 
Partial Eta Squared 

Fall Final 
Sphericity Assumed 
920.16 
1 
920.16 
1.099 
.304 
.039 
GreenhouseGeisser 
920.16 
1.00 
920.16 
1.099 
.304 
.039 

HuynhFeldt 
920.16 
1.00 
920.16 
1.099 
.304 
.039 

Lowerbound 
920.161 
1.000 
920.161 
1.099 
.304 
.039 

Error (Fall Final) 
Sphericity Assumed 
22608.339 
27 
837.346 

GreenhouseGeisser 
22608.339 
27.00 
837.346 

HuynhFeldt 
22608.339 
27.00 
837.346 

Lowerbound 
22608.339 
27.00 
837.346 
Tests of WithinSubjects Contrasts 

Measure: MEASURE_1 

Source 
Fall Final 
Type III Sum of Squares 
df 
Mean Square 
F 
Sig. 
Partial Eta Squared 
Fall Final 
Linear 
920.161 
1 
920.161 
1.099 
.304 
.039 
Error (Fall Final) 
Linear 
22608.339 
27 
837.346 
From the tests of within subjects contrasts table, p = 0.304. Thisimplies that p is greater than 0.304. There is no statisticalrelationship between the subjects of study.
Tests of BetweenSubjects Effects 

Measure: MEASURE_1 

Transformed Variable:Average 

Source 
Type III Sum of Squares 
df 
Mean Square 
F 
Sig. 
Partial Eta Squared 
Intercept 
1457700.446 
1 
1457700.45 
1126.828 
.00 
.977 
Error 
34928.054 
27 
1293.63 
Estimated Marginal Means
Fall Final 

Measure: MEASURE_1 

Fall Final 
Mean 
Std. Error 
95% Confidence Interval 

Lower Bound 
Upper Bound 

1 
165.39 
4.74 
155.68 
175.11 
2 
157.27 
7.33 
142.25 
172.32 
Profile Plots
Results Summary
Part 1 of the analysis conducted independent samples ttest tocompare the means of a normally distributed interval dependentvariable for the two independent group’s 4^{th} and 6^{th}period class students. All the ttest for the units (unit 0,unit 1, unit 2, unit 3, unit 4 and fall final tests) revealed thatthe period 4 class has a statistically significantly lower mean scoreon the Fall Final tests than the period 6 classes. The tests revealedthat the hypothesis “that the period 4 class will show a higherlevel of mastery and understanding of the mathematical conceptspresented during the semester is not true.
Part 2 of the analysis was also a comparison test, ANOVA. The testwas only done for Fall Final test results. These results firmlyconfirmed that there is no significant relationship betweeninstructional methods of teaching and discovery methods of teachingfor period 4 and period 6 classes.
CHAPTER5
OUTPUT INTERPRETATION
CONCLUSION
The results from the study do not support our hypothesis. Thehypothesis suggested that the 4^{th} period class will show ahigher level of mastery and understanding of the mathematicalconcepts presented during the semester. The results also failed tosupport the alternative hypothesis that the 6^{th} periodsubjects will not have as higher level of understanding of the reasonwhy and the deeper concepts and therefore have lower performance ratelike with the 4^{th} period class. The period 4 students didnot perform much better in the results, it was expected that theperiod 4 class would report greater means for all the units ascompared to the same units for period 6 students. However, the ttestresults indicate that the period 4 class has a statisticallyinsignificantly different mean score for all the tests undertaken bythe period 4 and 6 students.
Scholars need to realize that a single type of instructions cannotfit all the learners since every student have own differences,strengths, and weaknesses which impact on the instructions. Insteadof forcing the students into the instructional technique, it isimportant to change the teaching techniques to suit the individualneeds of every student. Kyllonen and Lajoie demonstrated that thehighly structured instructional presentations were more beneficial tothe special needs students while the stronger curriculum were moreimportant to the abled students (Kirschner, Sweller, and Clark,2006). Direct instructions have been found to minimize the motivationof the learners in engaging in the task, which could potentiallyimpact their performance negatively.
Limitations of Study
There are some factors that may be a limitation to this study. First,this study depended on a convenience sample. Since the data wasextracted from Buchanan High School during the 2016 fall semester,the demographics may not apply to other parts of the country. Theresults may also vary with the class makeup, the subject being taughtand the demographical composition of the classes. Moreover, somestudents may have undergone training regarding the mathematicalsubjects at home, church or any attachments they may be having. Somestudents are also more intelligent than others they have the abilityto grab lessons fasters, whether they are undergoing instructionalcurriculum or discovery method.
Even though the lessons were deemed appropriate for high schoolstudents, the mathematical models behind the tests may have beensophisticated for the learners to discover on their own. Again,students are likely to copy one another’s result, which leads tobiased results. The students who do not have a clear idea how tocomplete their tasks are likely to copy from the students who havesuccessfully cleared their tests.
Educational Implications
Despite the few limitations, the research has shown the comparativeadvantages of using one system over the other does not significantlyimpact on the learners. Although the results showed that directinstruction method of teaching did improve learners’ ability toperform as compared to the discovery learning. This canbe attributedto explicit nature of the lesson and the manner in which presentationto the students was done. When basic concepts are being taught, thensuch an approach is ideal. The direct instruction teaching styleshould not be fully teacherdriven. The teacher has to interact withthe learners accordingly, instead of giving lectures in a lecturingstyle.
Discovery learning method on the other hand advocates for the reducedguidance of the students. This is very important since the studentslearn from several approaches, with the lecturers using a blend ofmultiple pedagogical techniques to impart the message.
Future Research
Despite the fact that numerous studies have been conducted with theaim of establishing and understand the relationship between discoveryand instructional techniques in the classroom, there are still someareas that need to be looked at in the future studies. Newer andbetter research and data analysis tools are being developed and thereis room for further studies in this subject. This study does not giveroom for special needs learners like the disabled persons. This meansthat future studies should also include the disabled and otherlearners with special needs since they also form a critical part ofthe learning population. The learners who English are the secondlanguage, like the Chinese, also needs to be considered in thedemographics since it significantly impacts on the outcome. Futurestudies also need to consider students from diverse socioeconomic andsociocultural backgrounds.