statistics
TITLE:
To determine whether standard 3 students of San Fernando Boys’ Government School perform better in Mathematics than Mental Mathematics
STATEMENT OF TASK
Mathematics is a very important subject taught in all schools. Mental mathematics compliments the regular mathematics and therefore both are tested in primary schools. Mathematics is the written application of operation. It teaches students to think clearly, reason well and strategize effectively. Mental Mathematics is the ability to utilise mathematical skills to solve problems mentally. The marks scored by pupils generate statistics which are used by teachers to analyse a student’s performance and development of theories to explain the differences in performance. The Standard 3 class is where the transition from junior to senior level occurs where teachers expect the transference of concrete to abstract thinking would have occurred.
A common theory by many primary school teachers is ‘Students perform better in Mathematics than Mental math. Mental math is something that has to be developed and involves critical thinking. Mental math requires quick thinking and the student must solve the problem in their minds whereas in regular mathematics, the problem can be solved visually. Therefore, teachers should take these factors into consideration while testing and marking students in these areas.’
In this study, the statistics of 30 students of a standard 3 class of San Fernando Boys’ Government School will be analysed to determine the truth of this theory.
DATA COLLECTION METHODS
Mathematics and mental mathematics marks of term 1 of the class of 2013 were obtained from a Standard 3 teacher of San Fernando Boys’ Government School. The marks were divided into the form:
Mathematics Marks, x
Mental Mathematics marks, y
DATA PRESENTATION AND ANALYSIS
Sample number
Marks earned (x) x² 1
70
4900
2
86
7396
3
49
2401
4
66
4356
5
94
8836
6
68
4624
7
92
8464
8
96
9216
9
76
5776
10
71
5041
11
83
6889
12
59
3481
13
65
4225
14
98
9604
15
80
6400
16
68
4624
17
72
5184
18
85
7225
19
100
10000
20
69
4761
21
61
3721
22
80
6400
23
87
7569
24
95
9025
25
63
3969
=1933
=154087
Table 1: Sample data of marks earned by students in Mathematics examinations
Sample Number
Marks Earned (y) y² 1
40
1600
2
93
8649
3
26
676
4
53
2809
5
100
10000
6
46
2116
7
93
8649
8
86
7396
9
39
1521
10
46
2116
11
74
5476
12
60
3600
13
40
1600
14
86
7396
15
66
4356
16
45
2025
17
61
3721
18
87
7569
19
48
2304
20
55
3025
21
75
5625
22
62
3844
23
24
576
24
60
3600
25
33
1089
=1498
=101338
Table 2 of marks earned by students in Mental Mathematics
Calculation of Measures of Spread (mean, variance and standard deviation) of mathematics and mental mathematics marks.
Mean = = =77.32
Variance S² = = =185.098
Standard Deviation S =
=
=13.61
Mean =
=
= 59.92
Variance S² = = =463.11
Standard S =
Deviation =
=21.52
STEM AND LEAF DIAGRAM
Mathematics
Mental Mathematics
9 9 9 8 8 6 5 3 1 6 2 1 0 7 6 5 3 0 0 8 6 5 4 2 0
0
1
2
3
4
5
6
7
8
9
10
4 6
3 9
0 0 5 6 6 8
3 5
0 0 1 2 6
4 5
6 6 7
3 3
0
Key: 9 4 represents 49 marks
Key: 5 3 represents 53 marks
Figure 1: Stem and leaf diagram of marks earned in Mathematics and Mental Mathematics
Calculation of the median Q1, Q3, inter-quartile range, maximum and minimum values of mathematics and mental mathematics
Mathematics
Mental Mathematics
Lower Quartile Q1 =th term
=
=6.5th term
=
=67 marks
Median Q2 =th term
=
=13th term
=76 marks
Upper Quartile Q3 =th term
=
=19.5th term
=
=89.5 marks
Inter-quartile =Q3-Q1
Range =89.5-67
=22.5
Maximum value = 100marks
Minimum value =49marks
Lower Quartile Q1 =th term
=
=6.5th term
=
=42.5 marks
Median Q2 =th term
=
=13th term
=60 marks
Upper Quartile Q3 =th term
=
=19.5th term
=
=80.5 marks
Inter-quartile =Q3-Q1
=80.5-42.5
=38
Maximum value =100
Minimum value =24
BOX AND WHISKER DIAGRAM
Figure 2 of a stem and leaf diagram of mathematics and mental mathematics marks
Figure 3 of a bar graph representing marks earned in mathematics
Modal Class: 61-70 marks
Figure 4 of a bar graph representing marks earned in mental mathematics
Modal Class: 31-40, 41-50, 51-60 marks
DISCUSSION OF FINDINGS
The data collected of marks earned by a standard 3 class in mathematics and mental mathematics was statistically analysed and conclusion about trends were made.
Measures of central tendency were calculated for both mathematics marks and mental mathematics marks. It was found that students tend to perform better in mathematics than mental mathematics as the mean of marks earned in mathematics (77.32) was higher than that of mental mathematics (59.92).
The modal classes were also determined from the bar graphs. The modal class of marks earned in mathematics (61-70) were also higher than the modal classes of mental mathematics which had three different modal classes (31-40, 41-50, 51-60). This also helps to prove the theory that students perform better in mathematics than mental mathematics is true.
The measures of spread of both subjects were also calculated. Both the variance and standard deviation of mathematics: 185.098 and 13.61 respectively were lower than that of those of mental mathematics: 493.11 and 21.52 respectively. The variance and standard deviation marks calculated for mental mathematics were much higher than mathematics which shows that the marks for mental mathematics were spread over a larger range of values and also shows more consistency in mathematics scores than mental mathematics.
The lower quartile, median, upper quartile, inter-quartile range, maximum and minimum marks were determined from the stem and leaf diagram which helped for easy analysis of data. The median for marks earned in mathematics (76) were higher than the marks earned in mental mathematics (60). This indicates a better performance in mathematics than mental mathematics. From the box and whisker diagram (figure 2), the mathematics marks are more positively skewed that the mental mathematics marks which is almost symmetrical. Also, from the comparative bar charts, the mental mathematics marks are more negatively skewed than the mental mathematics marks. This enforces the theory that students perform better in mathematics than mental mathematics marks.
From these statistics, it is proven that students of San Fernando Boys’ Government perform better in Mathematics than Mental Mathematics. Therefore, statistics can be generated at other primary schools throughout Trinidad to determine whether students of other schools follow the same trend that students score higher marks in Mathematics than Mental Mathematics and results can be compared.
To determine whether standard 3 students of San Fernando Boys’ Government School perform better in Mathematics than Mental Mathematics
STATEMENT OF TASK
Mathematics is a very important subject taught in all schools. Mental mathematics compliments the regular mathematics and therefore both are tested in primary schools. Mathematics is the written application of operation. It teaches students to think clearly, reason well and strategize effectively. Mental Mathematics is the ability to utilise mathematical skills to solve problems mentally. The marks scored by pupils generate statistics which are used by teachers to analyse a student’s performance and development of theories to explain the differences in performance. The Standard 3 class is where the transition from junior to senior level occurs where teachers expect the transference of concrete to abstract thinking would have occurred.
A common theory by many primary school teachers is ‘Students perform better in Mathematics than Mental math. Mental math is something that has to be developed and involves critical thinking. Mental math requires quick thinking and the student must solve the problem in their minds whereas in regular mathematics, the problem can be solved visually. Therefore, teachers should take these factors into consideration while testing and marking students in these areas.’
In this study, the statistics of 30 students of a standard 3 class of San Fernando Boys’ Government School will be analysed to determine the truth of this theory.
DATA COLLECTION METHODS
Mathematics and mental mathematics marks of term 1 of the class of 2013 were obtained from a Standard 3 teacher of San Fernando Boys’ Government School. The marks were divided into the form:
Mathematics Marks, x
Mental Mathematics marks, y
DATA PRESENTATION AND ANALYSIS
Sample number
Marks earned (x) x² 1
70
4900
2
86
7396
3
49
2401
4
66
4356
5
94
8836
6
68
4624
7
92
8464
8
96
9216
9
76
5776
10
71
5041
11
83
6889
12
59
3481
13
65
4225
14
98
9604
15
80
6400
16
68
4624
17
72
5184
18
85
7225
19
100
10000
20
69
4761
21
61
3721
22
80
6400
23
87
7569
24
95
9025
25
63
3969
=1933
=154087
Table 1: Sample data of marks earned by students in Mathematics examinations
Sample Number
Marks Earned (y) y² 1
40
1600
2
93
8649
3
26
676
4
53
2809
5
100
10000
6
46
2116
7
93
8649
8
86
7396
9
39
1521
10
46
2116
11
74
5476
12
60
3600
13
40
1600
14
86
7396
15
66
4356
16
45
2025
17
61
3721
18
87
7569
19
48
2304
20
55
3025
21
75
5625
22
62
3844
23
24
576
24
60
3600
25
33
1089
=1498
=101338
Table 2 of marks earned by students in Mental Mathematics
Calculation of Measures of Spread (mean, variance and standard deviation) of mathematics and mental mathematics marks.
Mean = = =77.32
Variance S² = = =185.098
Standard Deviation S =
=
=13.61
Mean =
=
= 59.92
Variance S² = = =463.11
Standard S =
Deviation =
=21.52
STEM AND LEAF DIAGRAM
Mathematics
Mental Mathematics
9 9 9 8 8 6 5 3 1 6 2 1 0 7 6 5 3 0 0 8 6 5 4 2 0
0
1
2
3
4
5
6
7
8
9
10
4 6
3 9
0 0 5 6 6 8
3 5
0 0 1 2 6
4 5
6 6 7
3 3
0
Key: 9 4 represents 49 marks
Key: 5 3 represents 53 marks
Figure 1: Stem and leaf diagram of marks earned in Mathematics and Mental Mathematics
Calculation of the median Q1, Q3, inter-quartile range, maximum and minimum values of mathematics and mental mathematics
Mathematics
Mental Mathematics
Lower Quartile Q1 =th term
=
=6.5th term
=
=67 marks
Median Q2 =th term
=
=13th term
=76 marks
Upper Quartile Q3 =th term
=
=19.5th term
=
=89.5 marks
Inter-quartile =Q3-Q1
Range =89.5-67
=22.5
Maximum value = 100marks
Minimum value =49marks
Lower Quartile Q1 =th term
=
=6.5th term
=
=42.5 marks
Median Q2 =th term
=
=13th term
=60 marks
Upper Quartile Q3 =th term
=
=19.5th term
=
=80.5 marks
Inter-quartile =Q3-Q1
=80.5-42.5
=38
Maximum value =100
Minimum value =24
BOX AND WHISKER DIAGRAM
Figure 2 of a stem and leaf diagram of mathematics and mental mathematics marks
Figure 3 of a bar graph representing marks earned in mathematics
Modal Class: 61-70 marks
Figure 4 of a bar graph representing marks earned in mental mathematics
Modal Class: 31-40, 41-50, 51-60 marks
DISCUSSION OF FINDINGS
The data collected of marks earned by a standard 3 class in mathematics and mental mathematics was statistically analysed and conclusion about trends were made.
Measures of central tendency were calculated for both mathematics marks and mental mathematics marks. It was found that students tend to perform better in mathematics than mental mathematics as the mean of marks earned in mathematics (77.32) was higher than that of mental mathematics (59.92).
The modal classes were also determined from the bar graphs. The modal class of marks earned in mathematics (61-70) were also higher than the modal classes of mental mathematics which had three different modal classes (31-40, 41-50, 51-60). This also helps to prove the theory that students perform better in mathematics than mental mathematics is true.
The measures of spread of both subjects were also calculated. Both the variance and standard deviation of mathematics: 185.098 and 13.61 respectively were lower than that of those of mental mathematics: 493.11 and 21.52 respectively. The variance and standard deviation marks calculated for mental mathematics were much higher than mathematics which shows that the marks for mental mathematics were spread over a larger range of values and also shows more consistency in mathematics scores than mental mathematics.
The lower quartile, median, upper quartile, inter-quartile range, maximum and minimum marks were determined from the stem and leaf diagram which helped for easy analysis of data. The median for marks earned in mathematics (76) were higher than the marks earned in mental mathematics (60). This indicates a better performance in mathematics than mental mathematics. From the box and whisker diagram (figure 2), the mathematics marks are more positively skewed that the mental mathematics marks which is almost symmetrical. Also, from the comparative bar charts, the mental mathematics marks are more negatively skewed than the mental mathematics marks. This enforces the theory that students perform better in mathematics than mental mathematics marks.
From these statistics, it is proven that students of San Fernando Boys’ Government perform better in Mathematics than Mental Mathematics. Therefore, statistics can be generated at other primary schools throughout Trinidad to determine whether students of other schools follow the same trend that students score higher marks in Mathematics than Mental Mathematics and results can be compared.