Download - Universiti Pendidikan Sultan Idris

DEVELOPMENT OF MODULES FEATURING
ACCOMPANIED VIDEO BY USING
SCIENTIFIC CALCULATOR
LIM KIAN BOON
KOD PENYELIDIKAN: 04-15-0006-09
DAPENA
UNIVERSITI PENDIDIKAN SULTAN IDRIS
2009
iv
ABSTRACT
The study was to highlight the usage of scientific calculator Casio fx-570 ES for
solving Numerical Method. Firstly, the study aimed at investigating the students'
achievement after the scientific calculator Casio fx-570 ES was recommended.
Secondly, the study also aimed to diagnose the differences in the duration taken by the
students in calculation. The population involved was students of Bachelor of Science
(Mathematics) with education who registered for the Numerical Methods course at
Sultan Idris Education University. The samples were 38 participants who attended the
workshop of using scientific calculator in the Numerical Methods subject. Tests on
Numerical Method were given to participants before and after the application of
calculator. This study has proven that it has positive effect in enhancing the students'
achievement in Numerical Method. The findings also show that students need shorter
duration to solve the Numerical Method problems through the scientific calculator
Casio fx-570 ES.
V
ABSTRAK
Kajian ini dijalankan untuk menguji keberkesanan penggunaan kalkulator saintifik
Casio fx-570 ES dalam pengajaran kaedah berangka. Terdapat dua objektif dalam
kajian ini. Pertama, mengkaji perbezaan dalam pencapaian pelajar dalam kaedah
berangka setelah kalkulator saintifik Casio fx-570 ES diperkenalkan. Kedua, kajian
ini juga mengkaji perbezaan peruntukan masa dalam penyelesaian masalah kaedah
berangka. Populasi adalah terdiri mahasiswa-mahasiswi Ijazah Sarjana Sains
(Matematik) dengan Pendidikan tahun dua yang mengambil kursus kaedah berangka
di Universiti Pendidikan Sultan Idris. Sampel terdiri dariapda 38 orang peserta yang
hadir ke bengkel penggunaan kalkulator saintifik dalam subjek kaedah berangka.
Ujian tentang topik-topik dalam kaedah berangka akan diberi kepada peserta dalam
kajian ini sebelum dan selepas penggunaan kalkulator saintifik Casio fx-570 ES
diperkenalkan. Kajian ini telah membuktikan keberkesanan kalkulator saintifik Casio
fx-570 ES dalam kaedah berangka dari segi pencapaian mahasiswa-mahasiswi dan
peruntukan masa menyelesaikan masalah.
VI
TABLE OF CONTENT
Page
DECLARATION
ii
ACKNOWLEDGEMENT
iii
ABSTRACT
iv
ABSTRAK
v
TABLE OF CONTENT
vi
LIST OF TABLES
x
LIST OF ABBREVIATIONS
xiv
CHAPTER 1 INTRODUCTION
1.1
Theoretical Background
1
1.2
Problem Statement
3
1.3
Research
1.4
Research Question
4
1.5
Research Hypothesis
5
1.6
S
1.7
Definition
1.8
Obj
ecti
ignificance
of
ve
the
of
Study
Terms
4
5
6
1.7.1
Scientific calculator
6
1.7.2
Numerical integration
6
1.7.3
Roots
Limitations
of
nonlinear
of
the
equation
7
study
7
VII
CHAPTER 2
LITERATURE REVIEW
2.1
Introduction
8
2.2
Implementing calculator in mathematics education
10
2.3
Attitudes and perceptions on calculator use
12
2.4
Impact on curriculum
15
2.5
Roots of Nonlinear Equation
16
2.5.1 Bisection Method
16
2.5.2
False Position Method
18
2.5.3
Secant Method
20
2.6
2.7
2.5.4 Newton Raphson' s Method
21
2.5.5
22
Fixed Point Iteration Method
Numerical Integration
23
2.6.1
Trapezoidal Rules
23
2.6.2
Simpson's Rule 1/3
23
2.6.3
Simpson's Rule 3/8
24
Conclusion
CHAPTER 3
24
METHODOLOGY
3.1
Design of the study
25
3.2
Target population and sample research
26
3.3
Instruments of the study
27
3.4
Algorithm of scientific calculator for numerical method
28
3.4.1
Bisection Method
28
3.4.2
False Position Method
32
3.4.3
Secant Method
37
viii
3.4.4
Newton Raphson's Method
42
3.4.5
Fixed Point Iteration Method
46
3.4.6
Trapezoidal Rules
48
3.4.7
Simpson's Rule 1/3
50
3.4.8
Simpson's Rule 3/8
51
3.5
Procedures
3.6
Data analysis
CHAPTER 4
of
the
study
52
53
RESEARCH FINDINGS
4.1
Introduction
54
4.2
Analysis on the achievements of respondents in solving 55
Numerical Method problems by using scientific calculator
4.3
4.2.1
Bisection Method
55
4.2.2
False Position Method
56
4.2.3
Secant Method
57
4.2.4
Newton Raphson's Method
58
4.2.5
Fixed Point Iteration Method
59
4.2.6
Trapezoidal Rules
60
4.2.7
Simpson's Rule 1/3
61
4.2.8
Simpson's Rule 3/8
62
4.2.9
Overall
63
Analysis on the time taken in solving Numerical Method
64
problems by using scientific calculator
4.3.1
Bisection Method
64
4.3.2
False Position Method
65
4.3.3
Secant Method 66
4.3.4
Newton Raphson's Method
67
4.3.5
Fixed Point Iteration Method
68
4.3.6
Trapezoidal Rules
69
4.3.7
Simpson's Rule 1/3
70
4.3.8
Simpson's Rule 3/8
71
4.3.9
Overall
72
CHAPTER 5
CONCLUSION AND SUGGESTION
5.1
Introduction
74
5.2
Conclusion
74
5.3
Suggestion
76
REFERENCE
78
X
LIST OF TABLES
Table
No.
4.1
Title
Mean and standard deviation of students' marks for both pre-test
Page
55
and post-test (Bisection Method)
4.2
Paired sample t-test of students' marks for both pre-test and post-
55
test (Bisection Method)
43
Mean and standard deviation of students' marks for both pre-test
56
and post-test (False Position Method)
4.4
Paired sample t-test of students' marks for both pre-test and post-
56
test (False Position Method)
4.5
Mean and standard deviation of students' marks for both pre-test
57
and post-test (Secant Method)
4.6
Paired sample t-test of students' marks for both pre-test and post-
57
test (Secant Method)
4.7
Mean and standard deviation of students' marks for both pre-test
58
and post-test (Newton Raphson's Method)
4.8
Paired sample t-test of students' marks for both pre-test and posttest (Newton Raphson's Method)
58
xi
4.9
Mean and standard deviation of students' marks for both pre-test
59
and post-test (Fixed Point Iteration Method)
4.10
Paired sample t-test of students' marks for both pre-test and post-
59
test (Fixed Point Iteration Method)
4.11
Mean and standard deviation of students' marks for both pre-test 60
and post-test (Trapezoidal Rule)
4.12
Paired sample t-test of students' marks for both pre-test and post-
60
test (Trapezoidal Rule)
4.13
Mean and standard deviation of students' marks for both pre-test 61
and post-test (Simpson's 1/3 Rule)
4.14
Paired sample t-test of students' marks for both pre-test and post- 61
test (Simpson's 1/3 Rule)
4.15
Mean and standard deviation of students' marks for both pre-test 62
and post-test (Simpson's 3/8 Rule)
4.16
Paired sample t-test of students' marks for both pre-test and post- 62
test (Simpson's 3/8 Rule)
4.17
Mean and standard deviation of students' marks for both pre-test 63
and post-test (Overall)
4.18
Paired sample t-test of students' marks for both pre-test and post- 63
test (Overall)
4.19
Mean and standard deviation of duration taken for both pre-test and 64
post-test (Bisection Method)
4.20
Paired sample t-test of duration taken for both pre-test and post-test 64
(Bisection Method)
Xii
4.21
Mean and standard deviation of duration taken for both pre-test and 65
post-test (False Position Method)
4.22
Paired sample t-test of duration taken for both pre-test and post-test 65
(False Position Method)
4.23
Mean and standard deviation of duration taken for both pre-test and 66
post-test (Secant Method)
4.24
Paired sample t-test of duration taken for both pre-test and post-test 67
(Secant Method)
4.25
Mean and standard deviation of duration taken for both pre-test and 67
post-test (Newton Raphson's Method)
4.26
Paired sample t-test of duration taken for both pre-test and post-test 68
(Newton Raphson's Method)
4.27
Mean and standard deviation of duration taken for both pre-test and 68
post-test (Fixed Point Iteration Method)
4.28
Paired sample t-test of duration taken for both pre-test and post-test 69
(Fixed Point Iteration Method)
4.29
Mean and standard deviation of duration taken for both pre-test and 69
post-test (Trapezoidal Rule)
4.30
Paired sample t-test of duration taken for both pre-test and post-test 70
(Trapezoidal Rule)
4.31
Mean and standard deviation of duration taken for both pre-test and 70
post-test (Simpson's 1/3 Rule)
4.32
Paired sample t-test of duration taken for both pre-test and post-test 71
(Simpson's 1/3 Rule)
xiii
4.33
Mean and standard deviation of duration taken for both pre-test and 71
post-test (Simpson's 3/8 Rule)
4.34
Paired sample t-test of duration taken for both pre-test and post-test 72
(Simpson's 3/8 Rule)
4.35
Mean and standard deviation of duration taken for both pre-test and 72
post-test (Overall)
4.36
Paired sample t-test of duration taken for both pre-test and post-test 73
(Overall)
xiv
LIST OF ABBREVIATION
MOE
-
The Malaysian Ministry of Education
UPSI
-
Sultan Idris Education University
PMR
-
Penilaian Menengah Rendah
SPM
-
Sijil Pelajaran Malaysia
STPM
-
Sijil Tinggi Pelajaran Malaysia
ICT
-
Information and communications technology
NCTM
-
National Council of Teachers of Mathematics
SPSS
-
Statistical Package for the Social Sciences
N
-
Number of respondents
/iMpre
-
Pre-test score mean
juMpost
-
Post-test score mean
juTpre
-
Pre-test duration mean
juTpost
-
Post-test duration mean
CHAPTER 1
INTRODUCTION
1.1
Theoretical Background
For the past two decades in the twentieth century, we can see the advancement of
technological tool in mathematics education. Ranging from scientific calculator to
computer algebra systems and more recently engaging dynamic software changing the
way of mathematics taught. Previously, the teaching and learning of mathematics
focused mainly in accomplishing the objective knowledge of mathematics that is
commonly found in mathematics textbooks. Since the discovery of calculators, the
calculator has gradually evolved from a machine that could only perform simple fourfunction operations into one that can now also execute highly technical algebraic
symbolic manipulations instantly and accurately (Norton, P. & Sprague, D., 2001).
The latest versions of calculators provide students access to mathematical concepts
and experiences that are not possible with the usual paper-and-pencil methods.
Nevertheless, there are still many who are sceptical on its use since it may impair
students' mathematical ability and result in increased mathematical illiteracy.
2
According to Khoju, M., Jaciw, A. & Miller, G. I. (2005), results of researches on the
use of calculators in the classroom however, show that calculators are valuable
educational tools that empower students to reach higher mathematical understanding.
The Malaysian Ministry of Education (MOE) has given the consent to all
secondary schools nationwide to employ scientific calculators to serve as a tool in
teaching and learning for the subject mathematics and additional mathematics.
Scientific calculators are not just acceptable in the classrooms environments but also
extended to be use in the Malaysian national examinations such as Penilaian
Menengah Rendah (PMR), Sijil Pelajaran Malaysia (SPM) as well as Sijil Tinggi
Pelajaran Malaysia (STPM). Calculator use in national examinations does not mean
to evaluate the candidates' proficiency in using calculator but more to facilitate the
candidates in calculations (MOE, 2002).
Despite all of their benefits and capabilities, calculators will never be able to
replace the human mind when it comes to knowing how to read and understand a
problem situation, writing an appropriate equation for the problem, choosing which
operation to use in order to solve the problem, correctly interpreting the solution
displayed on the calculator, and determining the appropriateness of the answer.
Calculators, in combination with mental, paper-and-pencil, and estimate skills when
appropriate, comprise the tools to help students work through the computations and
manipulations necessary for solving problems.
3
1.2
Problem Statement
Many educators are apprehensive at the mere thought of implementing the use of
Casio fx-570ES calculator in teaching Numerical Method. Educators worry that the
use of scientific calculator may prevent students from learning the basic concept of
Numerical Method. They worry that the only mathematical skill that the students will
acquire upon completion of their Numerical Methods course is button-pressing.
Another concern is related to students' motivation to do mathematics.
Educators think that since Casio fx-570ES calculator does all the work, students will
be less stimulated and challenged. In addition, by using calculator, students may fail to
learn or may actually forget how to do basic mathematical procedures. Many
educators believe that learning Numerical Methods is a hard work, which normally are
associated with manual computations and manipulations. Scientific calculator
eliminates much of that work, making them appear nothing but a crutch for students
who are too lazy to perform the assigned mathematical tasks.
On the other hand, there are two studies Kastberg, S. & Leatham, K. (2005)
and Chaves, J. A., White, A. L. & Cheah, U. H. (2006) revealed that calculator is just
a tool to help students solve problems; they do not do the work for students. It is still
up to the students to read the problem, understand what is asked, determine the
solution and decide whether the answer makes sense. The use of calculators simply
allows teachers and students to spend more time on the non-computational parts of the
problem solving process.
4
As a result, it is necessary to conduct a research on students in order to gain
the information about the impact of using Casio fx-570ES calculator in learning
Numerical
Method.
Furthermore,
this
study
may determine
whether the
implementation of Casio fx-570ES calculator in the instruction will help the students
to understand better the concept of Numerical Method.
1.3
Research Objective
The purpose of this study was to ascertain the effectiveness of using scientific
calculator in solving the Numerical Method problem, which includes Roots of
Nonlinear Equation and Numerical Integration. There are two objectives to achieve in
this study which focusing on the aspects of students' marks and duration taken in
solving the problems by means of Numerical Method. Explicitly, the objectives of this
research are:i)
To investigate whether the students' marks increase after they apply the
scientific calculator in solving, the Numerical Method problems.
ii)
To diagnose whether the duration taken to solve the Numerical
Method problems reduces after applying the scientific calculator.
1.4
Research Question
The rationale of this study was to determine the effectiveness of using scientific
calculator in solving the Numerical Method problems. Specifically, the questions for
this research are:
5
i)
Is there a significant effect in students' marks in solving the Numerical
Method problems using the scientific calculator?
ii)
How the application of scientific calculator influences the duration
taken for students in solving the Numerical Method problems?
1.5
Research Hypothesis
The hypothesis of research is stated as follows:
i)
Ho : There is no significant difference in marks between pre-test and
post-test in solving the Numerical Method problems.
Hi : The mean marks for pre-test is less than the mean marks for posttest in solving the Numerical Method problems.
ii)
Ho: There is no significant difference in duration between pre-test and
post-test in solving the Numerical Method problems.
Hi: The mean duration for pre-test is greater than the mean duration
for post-test in solving the Numerical Method problems.
1.6
Significance of the study
This research focuses on the effectiveness of scientific calculator in helping the
students to solve the Numerical Method problems. When the Calculator is used
appropriately, it works as a learning tool. The implementation of scientific calculator
and graphing calculator can enhance the students' comprehension about mathematical
concepts. By using the scientific calculator, students can reduce the time they spent on
tedious calculations. Implementation of using calculator as a tool in mathematics
6
education has been strongly supported by Kissane (2006). According to Kissane
(2006), the usage of calculators helps students to understand the association between
the mathematical concepts as well as computing calculations.
Part of the uneasiness felt about the use of calculators in classrooms is a result
of the belief that mathematics is and should be hard work, where work that is normally
associated with manual computations and manipulations. Calculators can eliminate
loads of work, making them seem dissident. This leads to perceptions where public
regard calculator use makes students to be lazy and depending on calculator in solving
mathematical problems. However, the fact that learning mathematics doesn't evolve in
performing tedious manipulations but to understand the mathematical concepts as well
as apply them in real life situations.
1.7
Definition of Terms
1.7.1
Scientific Calculator
Scientific calculator in this study refers to Casio fx-570ES. This calculator consists of
403 functions including CALC function and SOLVE function that enable equation
calculations, integration or differential calculations, matrix calculations, vector
calculations, complex number calculations, and base-n calculation.
1.7.2
Numerical Integration
Numerical integration constitutes a broad family of algorithms for calculating the
numerical value of a definite integral, and by extension, the term is also sometimes
used to describe the numerical solution of differential equations. If a formula for the
7
integrand is known, but it may be difficult or impossible to find an antiderivative
which is an elementary function, then numerical integration shall be applied.
1.73
Roots of Nonlinear Equation
A nonlinear system is any problem where the variable or variables to be solved for
cannot be written as a linear combination of independent components. Roots of a
nonlinear equation refer to roots of a system which does not satisfy the superposition
principle, or whose output is not proportional to its input.
1.8 Limitations of the study
In this research, only two major topics in Numerical Method course are focus; which
are Roots of Nonlinear Equation and Numerical Integration. The implementation of
this calculator is used to ascertain the efficiency of using scientific calculator in
helping the students in the stated topics. Practically, this research is done in Sultan
Idris Education University (UPSI) in Perak, Malaysia and the sample for this study is
the mathematics students who enrolled for Numerical Method course in the second
semester session 2008/2009. This sample consist of 38 participants and they have
attended the workshop regarding the implementation of scientific calculator in the
subjects.
CHAPTER 2
LITERATURE REVIEW
2.1
Introduction
In the era of globalization, the explosion of technologies is impacting the world in
more ways than can be imagined. The use of technology in education is seen as a way
to produce a more educated knowledge-based work force. The integration of
technology into the teaching and learning of mathematics has also not escaped the
attention of educators. As a discipline, mathematics too is very much influenced by
the rapid development of information and communications technology (ICT) and
mathematics educators have been looking at ways to integrate ICT into the curriculum
over the last decade. Subsequently the principle of integrating ICT in mathematics
teaching and learning is no longer controversial but on the contrary it has come to be
embedded in the mathematics curricula of most countries in the world. Increasingly
the use of technology is now seen as essential in the teaching and learning of
mathematics in schools. Lately mathematics educators have been looking at
mathematics processes as a focus to improve the learning of mathematics. Previously,
9
the teaching and learning of mathematics focused mainly on the objective knowledge
of mathematics that is commonly found in textbooks. Deductive reasoning was thus
the main emphasis in classrooms to achieve the learning objectives. Formulas were
taught and students learnt how to apply the formulas to solve problems.
The term "information and communication technologies" (ICT) refers to forms
of technology that are used to transmit, process, store, create, display, share or
exchange information by electronic means. Technology has frequently been viewed as
a widely useful asset to education. This asset is said to help the teaching process to be
more effective and boost the students' comprehension about the basic mathematical
concepts. Chong, C.K, Sharaf Horani, & Daniel J (2005) had identified several
barriers that arise in implementing ICT in mathematics education. Some major
barriers hindering the implementation of ICT in mathematics teaching were:
•
Lack of time in school schedule for projects involving ICT
•
Lack of adequate technical support for ICT projects
•
Not enough teacher training opportunities for ICT Projects
However, it is reminded that technology should not replace the mathematics
teachers and it must not be the focus in the instructional development. The
mathematics teachers must acknowledge that the tools are used to help them to teach
and transmits the mathematical concept clearly as Samer Habre and Todd A.
Grundmeier (2007) suggested. Introduction of ICT in education in fact were feared to
reduce the role of the teacher in transmitting the knowledge. Teachers' role in a
classroom will never be decrease since the successful learning outcomes are actually
depending on the learner itself where the learner take control of the learning process
10
themselves. Noss & Pachler (1999) suggested that to exploit the efficiency of ICT to
the learning process teachers must develop higher order skills when it comes to
selecting and evaluating the appropriate resources. Therefore, it is always the role of
the teacher that made the transmission of knowledge better. Dalton (2007) stated that
using five standards of effective teaching and their indicators provide guidance for
teacher planning, relating, managing, designing, implementing, motivating, assisting
and assessing. The five standards of teaching are:
2.2
•
Teacher and Students Producing Together
•
Developing Language and Literacy
•
Connecting Learning to Student's Lives
•
Teaching Complex Thinking
•
Teaching Through Conversation
Implementing calculator in mathematics education
Calculators are tools for doing mathematical computations. However, when used
appropriately can also be a tool for learning mathematics. Calculators now come in a
number of sizes and styles, and they cover a tremendous range of capabilities,
functions and prices. Despite the myths of harmful consequences resulting from their
use, calculators are a pedagogical tool of great value. Tay Kim Gaik (2005) had
carried out a study about using calculator Casio fx~570MS in Numerical Methods, yet
without the video featured. Studies about the role of calculator had been done and
Kissane (2006) stated that students using a calculator have direct access to
representations of mathematical ideas which are conceptually powerful which is vital
11
for learning. However in previous era, these representations of mathematical ideas
were shown with a great deal of manual effort.
Students can show their comprehension about the mathematical concept better
with the help of calculators. And for that reason, students can in fact focus in
understanding the mathematical concept and not distressing about computing the
wrong answer with the aid of calculators. Indeed calculators are crucial to help student
to understand the basic mathematical concept and not to be afraid of mathematics due
to its rote computations. In order for the implementation of calculator in the
mathematics classroom to succeed, collaboration from all stages be it the students
itself or the authorities. In the study that had been done by Yahya Abu Hassan (2007),
he has stated that the implementation of ICT in the schools has been the government's
priorities outline. Yahya Abu Hassan (2007) suggested that in Malaysia, it is a must
for a school to have at least a computer laboratory. And therefore, Malaysia is
considered to have implemented the technologies in teaching and learning
Mathematics.
Implementation of using calculator as a tool in mathematics education has
been strongly supported by Kissane (2006). According to Kissane (2006), the usage of
calculators helps students to understand the association between the mathematical
concepts as well as computing calculations. Despite the myths about calculators in
problem solving in mathematics education, calculators are considered to be a
pedagogical tool in helping the students as well as influenced the students' attitudes in
mathematics (Mohd Lazim Abdullah, Wan Salihin Wong Abdullah & Abu Osman Md
12
Tap, 2005). However some students disagree with the usage of calculators in
mathematics education (Carmen M. Latterell, 2007).
Kissane (2006) also stressed out that in order for the implementation of the
calculators as a learning tool in mathematics classrooms; there are three requirements
regarding the teachers. Firstly, the teachers should understand and master the usage of
calculator before teaching and learning process takes place. Before implementing the
calculator as a learning tool, it is advisable for the mathematics teachers to understand
and master the functions of the calculator. Even though it is inevitably will take time
to master the calculators but this will reduce the mistakes done using the calculator in
performing calculations. Secondly, the teacher should accept the varying pattern for
learning and teaching mathematics. This is due to the changes in technologies in
global where the syllabus might be change and this results the changing in the way of
teaching and learning. Apart from that, teachers should be ready to accept the impact
of implementing the calculator in mathematics classrooms as the students will be
constructing the mathematical concepts by themselves instead of directly by their own
teachers. The third requirement is teacher should be ready to accept the changes of
teaching materials when implementing calculator use in mathematics classrooms. The
changes in teaching materials will occur since the focus will be not on textbooks. The
construction of mathematical concepts will be help by the usage of calculator.
2.3
Attitudes and perceptions on calculator use
Several studies regarding the effects of using calculators in mathematics classrooms
has been done (Mohd Lazim Abdullah et al., 2005; Carmen M. Latterell- 2007; Samer