UNDERSTANDING MAXIMA-MINIMA PROBLEMS IN DIFFERENTIAL
Calculus THROUGH
MATLAB ASSIMILATION EMPLOYING COOPERATIVE LEARNING
(by Engr.
Rolando Jasmin Quitalig)
SUBMITTED BY:
MURAYAMA, Cheycel (Aguirre)
BS Civil Engineering – 2
2010108011
INTRODUCTION
|
A.
STATEMENT
OF THE PROBLEM
The
researcher wanted to investigate the effectiveness of the technology in
enhancing learning and upholding more appropriate tools for understanding
maxima-minima problems in differential calculus through MATLAB assimilation
employing cooperative learning
Specifically,
it seeks answer to the following:
1. What
are the mean performance scores in the pre-assessment and post-assessment of
the three classes?
1.1
Conventional Class (CC);
1.2
MATLAB-Assisted Class (MAC);
1.3
MATLAB-Assisted Class Employing Cooperative Learning (MCL).
2. Is
there a significant difference in the pre-assessment between the means of the:
2.1 Conventional
Class (CC) and MATLAB-Assisted Class (MAC);
2.2 Conventional
Class (CC) and MATLAB-Assisted Class Employing Cooperative Learning (MCL);
2.3 MATLAB-Assisted
Class (MAC) and MATLAB-Assisted Class Employing Cooperative Learning (MCL).
3. Is
there a significant improvement in the students’ performance under:
3.1
Conventional Class (CC);
3.2
MATLAB-Assisted Class (MAC);
3.3
MATLAB-Assisted Class Employing Cooperative Learning (MCL).
4. Is
there a significant difference in the post-assessment between the means of the:
4.1 Conventional
Class (CC) and MATLAB-Assisted Class (MAC);
4.2 Conventional
Class (CC) and MATLAB-Assisted Class Employing Cooperative Learning (MCL);
4.3 MATLAB-Assisted
Class (MAC) and MATLAB-Assisted Class Employing Cooperative Learning (MCL).
5. What
are the implications of the result of this study?
B.
HYPOTHESES
The
following are the null hypotheses incorporated in this study and they were all
tested at a 0.05 level of significance:
1. There
is no significant difference in the pre-assessment between the means of the:
1.1 Conventional
Class (CC) and MATLAB-Assisted Class (MAC);
1.2 Conventional
Class (CC) and MATLAB-Assisted Class Employing Cooperative Learning (MCL);
1.3 MATLAB-Assisted
Class (MAC) and MATLAB-Assisted Class Employing Cooperative Learning (MCL).
2. There
is no significant improvement in the students’ performance under:
2.1
Conventional Class (CC);
2.2
MATLAB-Assisted Class (MAC);
2.3
MATLAB-Assisted Class Employing Cooperative Learning (MCL).
3. There
is no significant difference in the post-assessment between the means of the:
3.1
Conventional Class (CC) and MATLAB-Assisted Class (MAC);
3.2
Conventional Class (CC) and MATLAB-Assisted Class Employing Cooperative
Learning (MCL);
3.3
MATLAB-Assisted Class (MAC) and MATLAB-Assisted Class Employing Cooperative
Learning (MCL).
C.
SIGNIFICANCE
OF THE STUDY
The
study intended to give assistance to Calculus courses and the following were
some of its beneficiaries:
1. Students;
2. Teachers;
3. School
Administrators;
4. Researchers
and Readers.
D.
SCOPE
AND LIMITATIONS
The
study was only limited to the freshmen engineering students enrolled in MATH134
(Calculus 2, a 4-unit course, six (6) hrs a week) and MATH011L (MATLAB, a
1-unit course, four and a half (4½) hrs a week) at Mapúa
Institute of Technology (MIT), Intramuros, Manila. At that point, three (3)
groups were selected through purposive random sampling.
RELATED LITERATURE
|
A.
REFORMS
AND TRENDS
The
union of theories, ideas, and interpretation lead to a certain perspective that
considers issues and trends for more sensible reforms. Vrasidas and McIsaac
(2001) argue that for successful technology integration, there is a need to
shifting pedagogical approaches and reform of teacher education programs.
The
researcher considers the bearing of the new paradigms for unleashing the
enhancement of learning from tied and confined environment. The comparison of
traditional set-up in teaching against the contemporary method is also
presented in this study. Taking into account the obvious implications of the
new trends, paradigm shift espoused the idea of setting the learning
environment centered to the students. Moreover, the high regard on paradigm
shift is encouraged so as technology and contemporary philosophies on learning
will be linked to the vantage of views of the students.
B.
COOPERATIVE
LEARNING
The
fused of contemporary philosophies to underlying principles would establish
foundation for continuous academic developments. It will be energized by the
educators who subscribed to the significance of the endeavour made by
constructivists’ advocates. Teachers who are engaged in the learning process of
the students often consider strategies that will be fitted to the learners’
need. Based on these readings, efforts have been made to create learning
intensive in a social context.
Jaworski
and Potari (1998), quoted by Nickson (2000), used the concept of Teaching Triad
in their research wherein the interactions within a mathematics classroom and
the teacher’s thinking motivates the lesson. They define Teaching Triad as a
framework developed in order to help identify many factors that characterize
the teaching mathematics classroom. The three (3) elements of the triad are:
1. Management
of Learning (ML) which provides a description of the teacher’s role in the
classroom as it is constituted by the teacher and the learner.
2. Sensitivity
to Students (SS), which concern’s the teacher’s knowledge of learners and the
attention to their needs.
3. Mathematical
Challenge (MC) describes the challenges that are presented to learners to bring
about mathematical thinking and activity.
Studies have shown that the inevitable
learning gains of the students on this type of strategy can be further achieved
when they are exposed I problem solving. Confronting, understanding, and
solving problems are the composition of the heart of mathematics.
C.
TECHNOLOGY
AND COOPERATIVE LEARNING
Computer-based
instruction with cooperative learning provides students at different levels an
opportunity to work together. It also helps teachers to meet various needs of
students. The increasing availability of computer-related technologies in
classrooms has prompted the investigation of their influence on processes of
conceptual development and conceptual change (Becker, 1991 on Windschitl &
Andre, 1998). The ability of simulations to portray phenomena and allow users
to interact with the dynamics of a model system for representing processes,
such as photosynthesis or the functioning of human cardiovascular system,
creates an arguably unique way of helping learners to conceptualize (Windschitl
& Andre, 1998). Their study supported the idea that computer-based
simulations offer a suitable cognitive environment in a constructive
instruction perspective.
D.
CALCULUS
REFORMS
The
principal aim of calculus reform is to use active, constructivist learning to
shift calculus education away from just providing skills in symbolic
manipulation to providing deeper conceptual understanding. However, other aims
also operate. For example, many strands of calculus reform seek to motivate
students by contextualizing the problems in real-world examples and using
computer technologies.
Calculus
is a rich subject with a varied cultural history. It serves not only as a basis
of mathematical modelling and problem-solving in applications, but also as a
natural pinnacle of the beauty and power of mathematics for the vast majority of calculus students
who take it as their final mathematics course (Tall et al, 2001). The
researcher made emphasis on calculus since the course is considered as strenuous
for the learners.
STATISTICAL TREATMENT
OF DATA
|
The study employs
the concepts raised in the field of Statistics and the following were some
employed by the researcher:
A.
MEAN
This formula was used to find the mean performance
score of the three (3) classes in the pre-assessments and post-assessments in
the Problem 1 under the Statement of the Problem.
B.
ANALYSIS
OF VARIANCE (ANOVA)
Analysis
of Variance (ANOVA) was used to rift the total variation of the pre-assessments
and post-assessments for the three (3) groups.
C.
t –
TEST FOR INDEPENDENT SAMPLES
It was
used to compute and identify whether there is a significant difference in the
pre-assessments and post-assessments between the means in Problem 2 and Problem
4 under the Statement of the Problem.
D.
t –
TEST FOR DEPENDENT SAMPLES
To determine whether there was a significant
improvement in the student’s performance which was also based upon on Problem 3
under the Statement of the Problem, this formula was utilized.
E.
PERCENTAGE
FORMULA
This formula was utilized to
determine the rate of each studied variables.
F.
WEIGHTED
MEAN
To affirm the statistical
claims of this study which was raised in Problem 5 under the Statement of the
Problem, this qualitative measure was done.
ANALYSIS AND
INTERPRETATION OF DATA
|
In this part, the
researcher interprets whatever data he gathered together with a depth analysis
on the facts given. Basically, it was also based on the problems raised in
Statement of the Problem.
A.
What
are the mean performance scores in the pre-assessment and post-assessment of
the three classes?
1.1
Conventional Class (CC);
1.2
MATLAB-Assisted Class (MAC);
1.3
MATLAB-Assisted Class Employing Cooperative Learning (MCL).
TABLE 1
Mean Performance Scores in the Pre-Assessment
and Post-Assessment
of the Students in the Experimental and
Control Groups
GROUP
|
Pre-Assessment
|
Post-Assessment
|
||
Mean
|
Standard Deviation
|
Mean
|
Standard Deviation
|
|
CC
|
7.57
|
1.79
|
18.20
|
2.93
|
MAC
|
8.03
|
1.61
|
19.80
|
3.38
|
MCL
|
7.93
|
1.89
|
23.47
|
2.87
|
7.84
|
20.49
|
|||
Based on the table above, it showed that
the mean score of the experimental groups were somewhat higher than the control
group. It only implies that the experimental groups performed slightly better
than the control group with of course respect to the pre-assessment. It can be
deduced from the mean score of the three groups to the piece of fact that
students have low understanding regarding the course since the assessment was
administered prior to the instruction.
B.
Is
there a significant difference in the pre-assessment between the means of the:
2.1Conventional Class (CC) and MATLAB-Assisted
Class (MAC);
2.2Conventional Class (CC) and MATLAB-Assisted
Class Employing Cooperative Learning (MCL);
2.3MATLAB-Assisted Class (MAC) and
MATLAB-Assisted Class Employing Cooperative Learning (MCL).
TABLE 2
Analysis of Variance of the Data for Pre-Assessments’
Mean Scores
Source of Variation
|
Sum of Squares
|
Degress of Freedom
|
Mean Square
|
Computed t
|
Critical t
|
Column Means
|
3.62
|
2
|
1.81
|
0.58
|
3.11
|
Error
|
272.20
|
87
|
3.13
|
||
TOTAL
|
275.82
|
89
|
Based on the table
above, it showed that the critical value was somewhat higher than the computed
t-value. It was merely based on a fact that there was an enormous increase of
the means on the post-assessments, approximately twice that of group’s mean
scores on the pre-assessments. Consequently, it is very clear that the strategy
utilized on the corresponding classes is proven to be effective.
C.
Is
there a significant improvement in the students’ performance under:
3.1 Conventional Class (CC);
3.2 MATLAB-Assisted Class (MAC);
3.3 MATLAB-Assisted Class
Employing Cooperative Learning (MCL).
·
CONVENTIONAL
CLASS (CC)
TABLE 3
Difference of Student’s Mean Scores Between
Pre-Assessment and Post-Assessment
Under the Conventional Class (CC)
Assessments
|
Mean Scores
|
Mean Difference
|
Standard Deviation
|
Computed t value
|
Critical t value
|
Decision
|
Interpretation
|
PRE
|
7.57
|
10.63
|
1.79
|
20.82
|
1.70
|
Reject
HO
|
Significant
|
POST
|
18.20
|
2.93
|
Since the computed
t-value exceeded the critical t-value at 5% level of significance, then there
was a significant difference in the pre-assessment scores under the
Conventional Class (CC). It only means that the conventional teaching method
was an effective approach
·
MATLAB-ASSISTED
CLASS (MAC)
TABLE 4
Difference of Student’s Mean Scores Between
Pre-Assessment and Post-Assessment
Under the MATLAB-Assisted Class (MAC)
Assessments
|
Mean Scores
|
Mean Difference
|
Standard Deviation
|
Computed t value
|
Critical t value
|
Decision
|
Interpretation
|
PRE
|
8.03
|
11.77
|
1.61
|
21.47
|
1.70
|
Reject
HO
|
Significant
|
POST
|
19.80
|
3.38
|
The computed
t-value is relatively higher than the critical t-value which the researcher
considered the improvement of the performance of the students in the
experimental group as a very significant. It only proves that MATLAB-Assisted
approach is an effective tool in increasing the potentials of learners in
deeply understanding the concepts in Differential Calculus.
·
MATLAB-ASSISTED
CLASS EMPLOYING COOPERATIVE LEARNING (MCL)
TABLE 5
Difference of Student’s Mean Scores Between
Pre-Assessment and Post-Assessment
Under the MATLAB-Assisted Class Employing
Cooperative Learning (MCL)
Assessments
|
Mean Scores
|
Mean Difference
|
Standard Deviation
|
Computed t value
|
Critical t value
|
Decision
|
Interpretation
|
PRE
|
7.93
|
15.53
|
1.893
|
34.012
|
1.70
|
Reject
HO
|
Significant
|
POST
|
23.47
|
2.874
|
Obviously, the
computed t-value is greater than the critical t-value at 5% level of
significance. The difference was found to be significant by the researcher.
Therefore, contemporary philosophy and technology instruction were very
effective in increasing the learning capabilities of the students in Differential
Calculus.
D. Is
there a significant difference in the post-assessment between the means of the:
4.1 Conventional Class (CC) and MATLAB-Assisted
Class (MAC);
4.2 Conventional Class (CC) and MATLAB-Assisted
Class Employing Cooperative Learning (MCL);
4.3 MATLAB-Assisted Class (MAC) and
MATLAB-Assisted Class Employing Cooperative Learning (MCL).
TABLE 6
Analysis of Variance of the Data for Post-Assessments’
Mean Scores
Source of Variation
|
Sum of Squares
|
Degrees of Freedom
|
Mean Square
|
Computed t
|
Critical t
|
Column Means
|
437.42
|
2
|
218.71
|
23.23
|
3.11
|
Error
|
819.07
|
87
|
9.42
|
||
TOTAL
|
1256.49
|
89
|
The researcher
deduced that the mean scores of the three (3) groups for post-assessment differ
significantly. It was merely based on a fact that there was an enormous
increase of the means on the post-assessments, approximately twice that of
group’s mean scores on the pre-assessments. Consequently, it is very clear that
the strategy utilized on the corresponding classes is proven to be effective.
·
CONVENTIONAL
CLASS (CC) AND MATLAB-ASSISTED CLASS (MAC)
TABLE 7
Difference of Student’s Mean Scores Between
CC and MAC
Class
|
Mean Scores
|
Mean Difference
|
Standard Deviation
|
Computed t value
|
Critical t value
|
Decision
|
Interpretation
|
CC
|
18.20
|
1.60
|
2.929
|
1.96
|
1.699
|
Reject
HO
|
Significant
|
MAC
|
19.80
|
3.377
|
Based on the data
above, the researcher found that MATLAB-Assisted strategy was better than the
conventional method. That idea was deduced by statistical analysis, through a
computed t-value of 1.96 against its critical value of 1.699, which actually
reveals that the means are considerably different at a 5% level of
significance.
·
CONVENTIONAL
CLASS (CC) AND MATLAB-ASSISTED CLASS EMPLOYING COOPERATIVE LEARNING (MCL)
TABLE 8
Difference of Student’s Mean Scores Between
CC and MCL
Class
|
Mean Scores
|
Mean Difference
|
Standard Deviation
|
Computed t value
|
Critical t value
|
Decision
|
Interpretation
|
CC
|
18.20
|
5.27
|
2.929
|
7.03
|
1.699
|
Reject
HO
|
Significant
|
MCL
|
23.47
|
2.874
|
Given the
preceding table, a mean difference of about 5.27 was noted and proven
significant by the relative difference of the computed t-value against its
critical value at 5% level of significance. The researcher has concluded that
using these results cooperative learning in MATLAB-Assisted class provides a
very satisfactory output compared to the conventional class.
·
MATLAB-ASSISTED
CLASS (MAC) AND MATLAB-ASSISTED CLASS EMPLOYING COOPERATIVE LEARNING (MCL)
TABLE 9
Difference of Student’s Mean Scores Between
MAC and MCL
Class
|
Mean Scores
|
Mean Difference
|
Standard Deviation
|
Computed t value
|
Critical t value
|
Decision
|
Interpretation
|
MAC
|
19.80
|
3.67
|
3.377
|
4.53
|
1.70
|
Reject
HO
|
Significant
|
MCL
|
23.47
|
2.874
|
The computed
t-value of 4.53 which is greater than the critical t-value of 1.70 gives an
idea that the null hypothesis has to be rejected. This leads to the conclusion
that there is a significant difference in the post-assessments between the
means of the MATLAB-Assisted Class (MAC) and the MATLAB-Assisted Class
Employing Cooperative Learning (MCL) groups.
E.
What
are the implications of the result of this study?
Throughout the experimentation process, it was clearly
noticed by the researcher that both MATLAB-Assisted Class (MAC) and MATLAB-Assisted
Class Employing Cooperative Learning (MCL) were more active and intrusive than
the Conventional Class (CC). The researcher believes that there is an important
part or component of the MATLAB assimilation that is not necessarily on the
progress itself rather to more dynamic contribution showed by the students in
each of the classes
CONCLUSIONS
|
Based on the findings of the
study, the following conclusions were drawn:
1.
Explicit figures in favor of the post-assessments’
performance were then exhibited by the mean performance scores in the
pre-assessments and post-assessments of the students under the Conventional
Class (CC), MATLAB-Assisted Class (MAC), and the MATLAB-Assisted Class
Employing Cooperative Learning (MCL).
2.
The classes under the Conventional Class (CC),
MATLAB-Assisted Class (MAC), and the MATLAB-Assisted Class Employing
Cooperative Learning (MCL) were initially comparable on the onset of the study.
3.
The strategy used in this study led to the improvement
of learning the maxima-minima in any of the Calculus courses.
4.
The treatment employed among the experimental
groups resulted to a better performance of the students.
5.
The MATLAB-Assisted Class Employing Cooperative
Learning (MCL) group produced a better output than the Conventional Class (CC)
as well as the MATLAB-Assisted Class (MAC).
6.
Student’s reasoning through discussion,
clarification of the ideas and the evaluation of the other ideas were developed
by exposing them to MATLAB in a social context.
RECOMMENDATIONS
|
Given the preceding data, findings, and analysis, the
researchers propose the following recommendations:
1.
The students should engage themselves in
emerging their prospective through analytical discussions among their peers.
2.
The teachers should keep themselves on improving
their teaching styles through adequate trainings.
3.
Faculty should engage to the idea of
implementing technology with contemporary philosophy on their respective
education.
4.
For those teaching maxima-minima in Calculus
courses, they should be encouraged to use MATLAB-Assisted Class Employing
Cooperative Learning (MCL) or any available software that can accommodate the
said program.
5.
The gaining of having mathematical
programs/software should be made available.
6.
Proper authorities or administrators should
funded qualified faculty to international trainings related to mathematics
establishing technology and different contemporary philosophies.
7.
Future research studies may acquire to
investigate the other variables of the study. For instance the greater number
of samples to be considered, longer span of time, interventions of faculty,
different teachers and group selection as well as the size.
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