## CHAPTER 4

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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 t-test statistical analysis model to compare the means of anormally distributed interval dependent variable for the twoindependent groups 4th and 6th 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 t-test todetermine the mean of the outcome. The results for each period classhave been discussed accordingly.

1. Period 4

The following are the results after computingindependent t-test 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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) 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. Thep-value (Sig (2-tailed)) corresponding to the given degreesof freedom and test statistic is p =0.74. Since our p&gt .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 (t27.00 = 0.34, p&gt0.05).The average test score for the male students was (15.23-15.19) =0.04more than the average score for the females.

1. 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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) 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. Thep-value (Sig (2-tailed)) corresponding to the given degreesof freedom and test statistic is p =0.91. Since our p&gt .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 (t31.99= -0.12, p&gt0.05). The average test score for the malestudents was (8.42-8.37) = 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(t27.00 = 0.34, p&gt0.05). The average testscore for the male students was (15.23-15.19) = 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 (t31.98 = -0.12, p&gt0.05).Theaverage test score for the male students was (8.42-8.37) = 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 6th 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

1. 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 t-test 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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) 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. Thep-value (Sig (2-tailed)) 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 (t23.69 = 0.55, p&gt0.05).The average test score for the male student’s was 0.31 more thanthe average score for the female students.

1. 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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) 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. Thep-value (Sig (2-tailed)) corresponding to the given degreesof freedom and test statistic is p =0.24. Since our p&gt.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 (t30.878= -1.21, p&gt0.05). The average test score for the malestudents was (5.42-5.49) = -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 (t23.69 = 0.55,p&gt0.05). 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(t30.878 = -1.21, p&gt0.05).The averagetest score for the male students was (5.42-5.49) = -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 6th 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 independentt-test 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 t-test todetermine the mean of the outcome. The results for each period classwould be discussed.

1. 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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) 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. Thep-value (Sig (2-tailed)) corresponding to the given degreesof freedom and test statistic is p =0.56. Since our p&gt.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 (t19.73= 0.59, p&gt0.05). Moreover, the average test score for themale students was (12.73-13.10) = -0.37 less than the average scorefor the females.

1. 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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) 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 p-value 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. Thep-value (Sig (2-tailed)) 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 (t31.404 = -1.40, p&lt .001).

• The average test score for the male students was (3.29-3.30) = -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 (t19.73 = 0.59, p&gt0.05).Theaverage test score for the male students was (12.73-13.10) = -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(t31.404 = -1.40, p&lt .001).The averagetest score for the male students was (3.29-3.30) = -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 6thperiod, 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 t-test todetermine the mean of the outcome. The results for each period classhave been discussed accordingly.

1. 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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) 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. Thep-value (Sig (2-tailed)) corresponding to the given degreesof freedom and test statistic is p =0.87. Since our p&gt.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 (t25.46= -0.16, p&gt0.05). Moreover, the average test score for themale students was (16.49-16.65) = -0.16 less than the average scorefor the females.

1. 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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) 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. Thep-value (Sig (2-tailed)) corresponding to the given degreesof freedom and test statistic is p =0.21. Since our p&gt.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 (t29.984 = 1.27, p&gt0.05).The average test score for the male students was (4.56-4.70) = -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 (t25.46 = -0.16, p&gt0.05).The average test score for the male students was (16.49-16.65) =-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(t29.984 = 1.27, p&gt0.05).The average testscore for the male students was (4.56-4.70) = -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 6thperiod, 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 t-test todetermine the mean of the outcome. The results for each period classhave been discussed.

1. 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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) 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. Thep-value (Sig (2-tailed)) corresponding to the given degreesof freedom and test statistic is p =0.92. Since our p&gt.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 (t25.40= 0.10, p&gt0.05). Moreover, the average test score for themale students was (17.53-17.70) = -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.

1. 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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) 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. Thep-value (Sig (2-tailed)) 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 (t31.039= -0.77, p&gt0.05). Moreover, the average test score for themale students was (8.1630-7.95) = 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 (t31.039 = -0.77, p&gt0.05).Theaverage test score for the male students was (8.1630-7.95) = 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(t31.039 = -0.77, p&gt0.05). The averagetest score for the male students was (8.1630-7.95) = 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 6th 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 t-test todetermine the mean of the outcome. The results for each period classhave been discussed accordingly.

1. 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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) 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. Thep-value (Sig (2-tailed)) corresponding to the given degreesof freedom and test statistic is p =0.30. Since our p&gt0.33is 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 (t24.194= 1.05, p&gt0.05). Moreover, the average test score for themale students was (28.61-28.90) = -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.

1. 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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) 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. Thep-value (Sig (2-tailed)) 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 (t22.90 =-0.17, p&gt0.05). Moreover, the average test score for themale students was (23.94-25.21) = -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 &amp Analysis:

From Fall Final test results for Period 4 and 6 classes above, we canmake our conclusion the period 4 class had a p-value of 0.33while period 6 classes had a p-value 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 6thperiod, 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 T-test. 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:

 Within-Subjects 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 Testsa Effect Value F Hypothesis df Error df Sig. Partial Eta Squared Fall Final Pillai`s Trace .039 1.099b 1.00 27.00 .304 .039 Wilks` Lambda .961 1.099b 1.00 27.00 .304 .039 Hotelling`s Trace .041 1.099b 1.00 27.00 .304 .039 Roy`s Largest Root .041 1.099b 1.00 27.00 .304 .039 a. Design: Intercept Within Subjects Design: Fall Final b. Exact statistic

 Mauchly`s Test of Sphericitya Measure: MEASURE_1 Within Subjects Effect Mauchly`s W Approx. Chi-Square df Sig. Epsilonb Greenhouse-Geisser Huynh-Feldt Lower-bound 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 Within-Subjects Effects table.

 Tests of Within-Subjects 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 Greenhouse-Geisser 920.16 1.00 920.16 1.099 .304 .039 Huynh-Feldt 920.16 1.00 920.16 1.099 .304 .039 Lower-bound 920.161 1.000 920.161 1.099 .304 .039 Error (Fall Final) Sphericity Assumed 22608.339 27 837.346 Greenhouse-Geisser 22608.339 27.00 837.346 Huynh-Feldt 22608.339 27.00 837.346 Lower-bound 22608.339 27.00 837.346
 Tests of Within-Subjects 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 Between-Subjects 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 t-test tocompare the means of a normally distributed interval dependentvariable for the two independent group’s 4th and 6thperiod class students. All the t-test 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 4th 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 6th periodsubjects will not have as higher level of understanding of the reasonwhy and the deeper concepts and therefore have lower performance ratelike with the 4th 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 t-testresults 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 teacher-driven. 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.

## CHAPTER 4

• Uncategorized

EXPLICIT DIRECT INSTRUCTION VERSUS DISCOVERY LEARNING 1

CHAPTER4

STATISTICAL ANALYSIS

This chapter has been categorized into three parts: For the firstpart of the analysis, independent samples t-test has been conductedto compare the means of a normally distributed interval dependentvariable for the two independent groups 4th and 6thperiod. For every unit test, the paper fist discusses period 4 testthen the period 6 test with the interpretation for each providedafter the results. A comparison of the results is provided at the endof every unit test results.

The second part of the analysis provides an alternative test,chi-test, to substantiate the findings from the T-test results. Theuse of the chi-square test has been done to determine whetherDiscovery teaching method and the Direct Instruction method affectthe learning outcomes in different ways. For the second part,chi-test has only been done on the final fall test for both period 4and 6. Note that the results have been discussed at the end of everystatistical output.

The third part of the analysis also gives an alternative test, ANOVA,to further prove beyond doubt the finding of the T-test and Chi-testabove. The ANOVA test has been done to determine whether Discoveryteaching method and the Direct Instruction method affect the learningoutcomes in different ways. Like in the second part, ANOVA has onlybeen done on the Fall Final Test for both period 4 and 6. Note thatthe results have been discussed at the end of every statisticaloutput.

Part 1: t – test

UNIT0 TEST:

The period 4 and period 6 students both took unit 0 test. The testresults have been run on SPSS version 24 independent t-test todetermine the mean of the outcome. The results for each period classhas been discussed accordingly.

1. Period 4

The following are the results after computingindependent t-test 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 for female students. The male recorded higher mean score than thefemales. The standard deviation differences were minimal, though, at1.29.

 Independent Samples Test Levene`s Test for Equality of Variances t-test for Equality of Means F Sig. t df Sig. (2-tailed) 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. Thep-value (Sig (2-tailed)) corresponding to the given degreesof freedom and test statistic is p =0.74. Since our p &gt .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 boys andgirls scores.

Based on our results above, we can conclude as follows:

• There was no significant mean difference in the mean scores between male and female students (t27.00 = 0.34, p &gt0.05).

Theaverage test score for the male students was (15.23-15.19) = 0.04more than the average score for the females.

1. Period 6

 Group Statistics GENDER N Mean Std. Deviation Std. Error Mean UNIT 0 TEST Male 16 85.250 12.1299 3.0325 Female 18 85.778 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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper UNIT 0 TEST Equal variances assumed .045 .833 -.120 32 .905 -.5278 4.3951 -9.4804 8.4248 Equal variances not assumed -.121 31.979 .905 -.5278 4.3698 -9.4289 8.3734

Our computed test statistic t is t= -0.12 with a df of df=32. Thep-value (Sig (2-tailed)) corresponding to the given degreesof freedom and test statistic is p =0.91. Since our p &gt .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 boys andgirls scores.

Based on our results above, we can conclude as follows:

• There was no significant mean difference in the mean scores between male and female students (t31.99 = -0.12, p &gt0.05).

• The average test score for the male students was (8.42-8.37) = 0.05 more than the average score for the females.

Comparison Period 4 and 6 Results :

From Unit 0 test results for Period 4 and 6 classes above, we canmake our conclusion The period 4 class showed that there was nosignificant mean difference in the mean scores between male andfemale students (t27.00 = 0.34, p &gt0.05).The average test score for the male students was (15.23-15.19) = 0.04more than the average score for the females. The period 4 malestudents had a mean score of 85.00 while femalestudents had a mean score of 82.86, there is no significantdifference compared to period 6 mean scores for male and femalestudents, 85.25 and 85.78, respectively.

The period 6 class results indicated that there was no significantmean difference in the mean scores between male and female students(t31.98 = -0.12, p &gt0.05). The averagetest score for the male students was (8.42-8.37) = 0.05 more than theaverage 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 6th period, the hypothesis holds.The research hypothesis states that the period 4 class will show ahigher level of mastery and understanding of the mathematicalconcepts presented during the semester. Our results confirm that thehypothesis is true.

UNIT1 TEST

1. Period 4

The period 4 and period 6 students both took unit 1 test. The testresults have been run on SPSS version 24 independent t-test todetermine the mean of the outcome. These are the results for thetest:

 Group Statistics GENDER N Mean Std. Deviation Std. Error Mean UNIT 1 TEST Male 15 79.867 19.0933 4.9299 Female 14 75.143 26.1736 6.9952

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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper UNIT 1 TEST Equal variances assumed 2.936 .098 .558 27 .581 4.7238 8.4648 -12.6446 22.0922 Equal variances not assumed .552 23.693 .586 4.7238 8.5578 -12.9507 22.3984

Our computed test statistic t is t= 0.56 with a df of df=27. Thep-value (Sig (2-tailed)) 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 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 as follows:

• There was no significant mean difference in the mean scores between male and female students (t23.69 = 0.55, p &gt0.05).

• The average test score for the male students was 0.31 more than the average score for the females.

1. Period 6

 Group Statistics GENDER N Mean Std. Deviation Std. Error Mean UNIT 1 TEST Male 16 73.563 19.7787 4.9447 Female 18 81.500 18.4495 4.3486 Independent Samples Test Levene`s Test for Equality of Variances t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper UNIT 1 TEST Equal variances assumed .678 .416 -1.211 32 .235 -7.9375 6.5571 -21.2939 5.4189 Equal variances not assumed -1.205 30.878 .237 -7.9375 6.5848 -21.3695 5.4945

Our computed test statistic t is t= -1.21 with a df of df=32. Thep-value (Sig (2-tailed)) corresponding to the given degreesof freedom and test statistic is p =0.24. Since our p &gt.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 as follows:

• There was no significant mean difference in the mean scores between male and female students (t30.878 = -1.21, p &gt0.05).

• The average test score for the male students was (5.42-5.49) = -0.08 less than the average score for the females.

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 (t23.69 = 0.55, p&gt0.05). The average test score for the male students was 0.31 morethan 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(t30.878 = -1.21, p &gt0.05).The average test score for the male students was (5.42-5.49) =-0.08 less than the average score for the females.

. The results of the unit 1 test forperiod 4 and 6 indicate that there is no statistically significantdifference between the mean test scores for both classes. In otherwords, since period 4 class students have no statisticallysignificantly higher mean scores on the unit tests than the 6thperiod, 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. The unit 1 test independent t-test indicates that there isnot much difference between the two methods of teaching.

UNIT2 TEST

The period 4 and period 6 students both took unit 2 test. The testresults have been run on SPSS version 24 independent t-test todetermine the mean of the outcome. The results for each period classhas been discussed accordingly.

1. Period 4

 Group Statistics GENDER N Mean Std. Deviation Std. Error Mean UNIT 2 TEST Male 15 89.800 8.8010 2.2724 Female 14 86.929 16.2313 4.3380

Period 4 students taking unit 0 test had a mean of 85 for males and82 for female students.

 Independent Samples Test Levene`s Test for Equality of Variances t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper UNIT 2 TEST Equal variances assumed 8.068 .008 .598 27 .555 2.8714 4.8025 -6.9824 12.7253 Equal variances not assumed .586 19.734 .564 2.8714 4.8972 -7.3527 13.0956

Our computed test statistic t is t= 0.60 with a df of df=27. Thep-value (Sig (2-tailed)) corresponding to the given degreesof freedom and test statistic is p =0.56. Since our p &gt.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 as follows:

• There was no significant mean difference in the mean scores between male and female students (t19.73 = 0.59, p &gt0.05).

• The average test score for the male students was (12.73-13.10) = -0.37 less than the average score for the females.

1. Period 6

 Group Statistics GENDER N Mean Std. Deviation Std. Error Mean UNIT 2 TEST Male 16 86.313 15.2522 3.8130 Female 18 93.611 15.0064 3.5370
 Independent Samples Test Levene`s Test for Equality of Variances t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper UNIT 2 TEST Equal variances assumed .233 .632 -1.405 32 .170 -7.2986 5.1958 -17.8822 3.2850 Equal variances not assumed -1.403 31.404 .170 -7.2986 5.2010 -17.9005 3.3033

In comparison, period 6 students taking unit 0 test 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 p-value 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. Thep-value (Sig (2-tailed)) 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 (t31.404 = -1.40, p &lt .001).

• The average test score for the male students was (3.29-3.30) = -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 (t19.73 = 0.59, p&gt0.05). The average test score for the male students was(12.73-13.10) = -0.37 less than the average score for the females.The period 4 male students had a mean score of 89.80while female students had a meanscore of 86.929,there is no significant difference compared to period 6 mean scoresfor male and female students, 86.31and 93.61,respectively.

The period 6 class results showed that there was a significant meandifference in the mean scores between male and female students(t31.404 = -1.40, p &lt .001).The average test score for the male students was (3.29-3.30) =-0.01 less than the average score for the females.

The results of the unit 2 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 6th 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 but the tests for unit 2 doneby discovery and instructional classes shows the opposite.

UNIT3 TEST

The period 4 and period 6 students both took unit 3 test. The testresults have been run on SPSS version 24 independent t-test todetermine the mean of the outcome. The results for each period classhas been discussed accordingly.

1. Period 4

 Group Statistics GENDER N Mean Std. Deviation Std. Error Mean UNIT 3 TEST Male 15 73.867 21.3938 5.5239 Female 14 75.286 25.5416 6.8263

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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper UNIT 3 TEST Equal variances assumed .695 .412 -.163 27 .872 -1.4190 8.7264 -19.3241 16.4860 Equal variances not assumed -.162 25.462 .873 -1.4190 8.7813 -19.4878 16.6498

Our computed test statistic t is t= -0.16 with a df of df=27. Thep-value (Sig (2-tailed)) corresponding to the given degreesof freedom and test statistic is p =0.87. Since our p &gt.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 as follows:

• There was no significant mean difference in the mean scores between male and female students (t25.46 = -0.16, p &gt0.05).

• The average test score for the male students was (16.49-16.65) = -0.16 less than the average score for the females.

1. Period 6

 Group Statistics GENDER N Mean Std. Deviation Std. Error Mean UNIT 3 TEST Male 16 67.875 18.9169 4.7292 Female 18 75.667 16.4460 3.8764

In comparison, period 6 students taking unit 3 test 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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper UNIT 3 TEST Equal variances assumed 1.441 .239 -1.285 32 .208 -7.7917 6.0635 -20.1426 4.5593 Equal variances not assumed -1.274 29.984 .212 -7.7917 6.1149 -20.2802 4.6969

Our computed test statistic t is t= -1.29 with a df of df=32. Thep-value (Sig (2-tailed)) corresponding to the given degreesof freedom and test statistic is p =0.21 Since our p &gt.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 as follows:

• There was no significant mean difference in the mean scores between male and female students (t29.984 = 1.27, p &gt0.05).

• The average test score for the male students was (4.56-4.70) = -0.14 less 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 (t25.46 = -0.16, p &gt0.05).The average test score for the male students was (16.49-16.65) =-0.16 less than the average score for the females. The period 4 malestudents had a mean score of 73.867while female students had a meanscore of 75.286,there is no significant difference compared to period 6 mean scoresfor male and female students, 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(t29.984 = 1.27, p &gt0.05).The average test score for the male students was (4.56-4.70) =-0.14 less than the average score for the females.

The results of the unit 3 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 6th 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. Our results show that thereis no much difference in the standard deviations for the two classes.The hypothesis does not hold.

UNIT4 TEST

The period 4 and period 6 students both took unit 4 test. The testresults have been run on SPSS version 24 independent t-test todetermine the mean of the outcome. The results for each period classhas been discussed accordingly.

1. Period 4

 Group Statistics GENDER N Mean Std. Deviation Std. Error Mean UNIT 4 TEST Male 15 75.533 21.9378 5.6643 Female 14 76.429 26.3430 7.0405

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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper UNIT 4 TEST Equal variances assumed 2.153 .154 -.100 27 .921 -.8952 8.9779 -19.3163 17.5259 Equal variances not assumed -.099 25.396 .922 -.8952 9.0362 -19.4909 17.7004

Our computed test statistic t is t= -0.10 with a df of df=27. Thep-value (Sig (2-tailed)) corresponding to the given degreesof freedom and test statistic is p =0.92. Since our p &gt.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 as follows:

• There was no significant mean difference in the mean scores between male and female students (t25.40 = 0.10, p &gt0.05).

• The average test score for the male students was (17.53-17.70) = -0.17 less than the average score for the females.

1. Period 6

 Group Statistics GENDER N Mean Std. Deviation Std. Error Mean UNIT 4 TEST Male 16 74.313 15.4993 3.8748 Female 18 79.167 20.9572 4.9397
 Independent Samples Test Levene`s Test for Equality of Variances t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper UNIT 4 TEST Equal variances assumed 1.125 .297 -.760 32 .453 -4.8542 6.3906 -17.8714 8.1630 Equal variances not assumed -.773 31.0 .445 -4.8542 6.2781 -17.6578 7.9495

Our computed test statistic t is t= -0.76 with a df of df=32. Thep-value (Sig (2-tailed)) corresponding to the given degreesof freedom and test statistic is p =0.45. Since our p =.45 isgreater than the test preferred significance level α = 0.05,the null hypothesis is accepted and we conclude that the there is nosignificant difference between the mean score for boys and girlsscores.

Based on our results above, we can conclude as follows:

• There was no significant mean difference in the mean scores between male and female students (t31.039 = -0.77, p &gt0.05).

• The average test score for the male students was (8.1630-7.95) = 0.21 more than the average score for the females.

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 (t31.039 = -0.77, p &gt0.05).The average test score for the male students was (8.1630-7.95) = 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(t31.039 = -0.77, p &gt0.05). The averagetest score for the male students was (8.1630-7.95) = 0.21 more thanthe average score for the females.

The period 4 male students had a mean score of 75.53while female students had a mean score of 76.43, there is nosignificant difference compared to period 6 mean scores for male andfemale students, 74.31 and 79.17, respectively. The results ofthe unit 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 6th 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.

FALLFINAL TEST

The period 4 and period 6 students both took the final fall test. Thetest results have been run on SPSS version 24 independent t-test todetermine the mean of the outcome. The results for each period classhas been discussed accordingly.

1. Period 4

 Group Statistics GENDER N Mean Std. Deviation Std. Error Mean FALL FINAL TEST Male 15 170.667 21.2289 5.4813 Female 14 160.929 28.0507 7.4969

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 t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper FALL FINAL TEST Equal variances assumed .826 .371 1.059 27 .299 9.7381 9.1971 -9.1329 28.6091 Equal variances not assumed 1.049 24.194 .305 9.7381 9.2869 -9.4211 28.8973

Our computed test statistic t is t= 1.01 with a df of df=27. Thep-value (Sig (2-tailed)) corresponding to the given degreesof freedom and test statistic is p =0.30. Since our p &gt0.33 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 as follows:

• There was no significant mean difference in the mean scores between male and female students (t24.194 = 1.05, p &gt0.05).

• The average test score for the male students was (28.61-28.90) = -0.29 less than the average score for the females.

1. Period 6

 Group Statistics GENDER N Mean Std. Deviation Std. Error Mean FALL FINAL TEST Male 16 158.125 46.9125 11.7281 Female 18 160.333 26.1781 6.1702
 Independent Samples Test Levene`s Test for Equality of Variances t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper FALL FINAL TEST Equal variances assumed .957 .335 -.172 32 .864 -2.2083 12.8362 -28.3548 23.9381 Equal variances not assumed -.167 22.90 .869 -2.2083 13.2522 -29.6289 25.2123

Our computed test statistic t is t= -0.17 with a df of df=32. Thep-value (Sig (2-tailed)) 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 and we conclude that the there is nosignificant difference between the mean score for boys and girlsscores.

Based on our results above, we can conclude as follows:

• There was no significant mean difference in the mean scores between male and female students (t22.90 = -0.17, p &gt0.05).

• The average test score for the male students was (23.94-25.21)= -1.27 less than the average score for the females.

Interpretation &amp Analysis:

From Fall Final test results for Period 4 and 6 classes above, we canmake our conclusion The period 4 class had a p-value of 0.33while period 6 class had a p-value of 0.86.The period 4 class 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 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 6th 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.

Part 3: ANOVA

This section provides an alternative test, ANOVA with repeatedmeasures, to substantiate the finding from the T-test and chi-squaretest above. The use of the ANOVA with repeated measures test has beendone to determine whether Discovery teaching method and the DirectInstruction method affect the learning outcomes in different ways.ANOVA has been done for all the 6 tests done period 4 and 6 students.ANOVA test has been done only for confirmation purposes.

The following are the results run for Period 4 and 6 students finalfall tests outcome:

 Within-Subjects Factors Measure: MEASURE_1 FallFinal Dependent Variable 1 Period4FallFinal 2 Period6FallFinal
 Descriptive Statistics Mean Std. Deviation N Period4 FallFinal 165.393 25.0619 28 Period6 FallFinal 157.286 38.7670 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 does not fall much apart.

 Multivariate Testsa Effect Value F Hypothesis df Error df Sig. Partial Eta Squared FallFinal Pillai`s Trace .039 1.099b 1.000 27.000 .304 .039 Wilks` Lambda .961 1.099b 1.000 27.000 .304 .039 Hotelling`s Trace .041 1.099b 1.000 27.000 .304 .039 Roy`s Largest Root .041 1.099b 1.000 27.000 .304 .039 a. Design: Intercept Within Subjects Design: FallFinal b. Exact statistic

 Mauchly`s Test of Sphericitya Measure: MEASURE_1 Within Subjects Effect Mauchly`s W Approx. Chi-Square df Sig. Epsilonb Greenhouse-Geisser Huynh-Feldt Lower-bound FallFinal 1.000 .000 0 . 1.000 1.000 1.000 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: FallFinal b. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table.

 Tests of Within-Subjects Effects Measure: MEASURE_1 Source Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared FallFinal Sphericity Assumed 920.161 1 920.161 1.099 .304 .039 Greenhouse-Geisser 920.161 1.000 920.161 1.099 .304 .039 Huynh-Feldt 920.161 1.000 920.161 1.099 .304 .039 Lower-bound 920.161 1.000 920.161 1.099 .304 .039 Error(FallFinal) Sphericity Assumed 22608.339 27 837.346 Greenhouse-Geisser 22608.339 27.000 837.346 Huynh-Feldt 22608.339 27.000 837.346 Lower-bound 22608.339 27.000 837.346
 Tests of Within-Subjects Contrasts Measure: MEASURE_1 Source FallFinal Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared FallFinal Linear 920.161 1 920.161 1.099 .304 .039 Error(FallFinal) 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 Between-Subjects 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 1457700.446 1126.828 .000 .977 Error 34928.054 27 1293.632

EstimatedMarginal Means

 FallFinal Measure: MEASURE_1 FallFinal Mean Std. Error 95% Confidence Interval Lower Bound Upper Bound 1 165.393 4.736 155.675 175.111 2 157.286 7.326 142.253 172.318

ProfilePlots

ResultsSummary

Part 1 of the analysis conducted independent samples t-test tocompare the means of a normally distributed interval dependentvariable for the two independent groups 4th and 6th-periodclass students. All the t-test for the units (unit 0, unit 1, unit 2,unit 3, unit 4 and fall final tests) revealed that the period 4 classhave a statistically significantly lower mean score on the Fall Finaltests than the period 6 class. The tests revealed that the hypothesis“that the period 4 class will show a higher level of mastery andunderstanding of the mathematical concepts presented during thesemester” is not true.

Part 2 of the analysis did a confirmation experiment, chi-square, onthe Fall Final test results for both period 4 and period 6 classstudents. The results further confirmed no significant relationshipbetween discovery and instructional methods of training.

Part 3 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

OUTPUTINTERPRETATION

CONCLUSION

The results from the study do not support our hypothesis. Thehypothesis suggested that the 4th-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 6th-periodsubjects will not have as higher level of understanding of the reasonwhy and the deeper concepts and therefore have lower performance ratelike with the 4th-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, theindependent t-test results indicate that the period 4 class have astatistically insignificantly different mean score for all the testsundertaken by the 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 low-abled 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

There are some factors that may be a limitation to this study. First,this study depended on a convenience sample. While 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. Again, some students mayhave undergone training regarding the mathematical subjects at home,church or any attachments they may be having. Some students are alsomore intelligent than others, they have the ability to grab lessonsfasters, whether they are undergoing instructional curriculum ordiscovery method.

Even though the lessons were presented to be appropriate for highschool students, the mathematical models behind the tests may havebeen sophisticated for the learners to discover by themselves. Again,students are likely to copy one another’s results, 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 great potential. As hasbeen shown by the results, direct instruction method of teaching didimprove learners’ ability to perform. This can be attributed toexplicit nature of the lesson and the manner in which presentation tothe students was done. When basic concepts are being taught, thensuch an approach is ideal. The direct instruction teaching styleshould not be fully teacher-driven. The teacher has to interact withthe learners accordingly, instead of giving lectures in a lecturingstyle.

The discovery learning method on the hand allows for reduced guidanceof the students. This is very important since the students learn fromseveral approaches, with the lecturers using a blend of multiplepedagogical techniques to impart the message. With the present stateand federal government, mandates are related to performance in theschools there are high chances that direct instruction will befavored.

Future Research

While the study has gone deep beyond the early studies to understandthe relationship between discovery and instructional techniques inthe classroom, there are still some areas that need to be looked atin the future studies. This study ignored the disabled personsfuture studies should also involve the disabled since they also forma critical part of the population. The learners who English are thesecond language, 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.