THE DIFFERENCE OF STUDENTS MATHEMATICAL REPRESENTATION ABILITY BY USING PROJECT BASED LEARNING AND PROBLEM BASED LEARNING ON THE TOPIC OF STATISTICS IN GRADE X SMA NEGERI 1 PERCUT SEI TUAN.
i
THE DIFFERENCE OF STUDENTS' MATHEMATICAL REPRESENTATION
ABILITY BY USING PROJECT BASED LEARNING AND PROBLEM
BASED LEARNING ON THE TOPIC OF STATISTICS IN
GRADE X SMA NEGERI 1 PERCUT SEI TUAN
By:
Widi AuliaWidakdo
ID 4113111080
Mathematics Education Study Program
THESIS
Submitted to Qualify for Academic Title of
Sarjana Pendidikan
MATHEMATICS DEPARTMENT
FACULTY OF MATHEMATICS AND NATURAL SCIENCES
STATE UNIVERSITY OF MEDAN
MEDAN
2011
iv
PREFACE
Praise and great thanks to Allah SWT that gives the amazing grace, love,
strength and health so that writer can finish this thesis. The title of this thesis is
“The Difference Of Students' Mathematical Representation Ability By Using
Project Based Learning And Problem Based Learning On The Topic Of Statistics
In Grade X Sma Negeri 1 Percut Sei Tuan”. This thesis was arranged to satisfy the
requirement to obtain the Degree of Sarjana Pendidikan from Faculty
Mathematics and Natural Science in State University of Medan
In the completion of this thesis, the writer received support from various
parties, therefore it was appropriate writer big thanks to Mr. Prof. Dr. Asmin,
M.Pd as my thesis supervisor who has provided guidance, direction, and advice to
the perfection of this thesis. Thanks are also due to Prof. Dr. Mukhtar, M.Pd, Dr.
Asrin Lubis, M.Pd and Faiz Ahyaningsih, M.Si as author’s examiners who have
provided input and suggestion from the planning to the completion of the
preparation of the research of this thesis. Thanks are also extended to Dr. W.
Rajagukguk, M.Pd as academic supervisor and then thank you so much for all
author’s lecturer in FMIPA Unimed.
My thanks are extended to Prof. Dr. Syawal Gultom, M. Si. as rector of
Unimed, Prof. Drs. Motlan, M.Sc, Ph.D as Dean of Mathematics and Natural
Sciences Faculty and to coordinator of bilingual Prof. Dr. rer.nat. Binari
Manurung, M.Si, Dr. Edy Surya, M.Si. as Chief of Mathematics Department, Drs.
Zul Amry, M.Si, Ph.D as Chief of Mathematics Education Study Program, Drs.
Yasifati Hia, M. Si as Secretary of Mathematics Education, and all of employee
staff who have helped the author.
Thanks to Mr. Muliadi, S.Pd. M.Si as principle of SMA N 1 Percut Sei
Tuan, Mrs. Dra. Libes Doloksaribu, S.Pd as mathematics teacher and all teacher,
staffs and also the students in grade X IIS 1 and X IIS 3 SMA N 1 Percut Sei
Tuan who have helped writer conducting the research.
Especially I would like to express my gratitude to my dear father Mr. Puji
Widakdo and my dear mother Mrs. Dra. Erni Fianti, S.Ag continues to provide
v
motivation and prayers for the success of me completed this thesis. Special big
thanks to my beloved sister Firda Saufika, Nabila Nadra and Also my brother M.
Khair Arrayyan for giving support even moril or material and all my family for all
pray, motivation, and support until the end of my study.
I also thanks to my lovely second family of Himpunan Mahasiswa Islam
(HMI) which always help me and support in every condition without any
exception. Especially for Vivi, Taufik, Sapwan, Juli, Ilmi, Maryam, Widya, for all
of your crazy thing, Bang Elfan, Bang Dedi, Bang Imam, Bang Badzlan, Kak Imel
and Kak Mora for all of the suggestion and incredible advice. Thank you very
much for Rizki Ramadhana, Satoto, Hakim, Fatkhu, Zaki, also Anggun, Nadia,
and Putri Rizki for every helping you’ve given as my sisters and brothers in this
university. I love you and thanks for every spirit my freak friends Debby, Nelly
and Yohannes also my classmate in Bilingual Mathematics Education 2011.
At last, the author has finished this thesis in maximum level but author
realized there are some imperfections. For that, the author asks for building
comments and suggestions in order to reach the perfection of this thesis. The
author wishes that this thesis would be useful to improve the knowledge should
give a big effort to prepare this thesis, and the writer know that this thesis have so
many weakness. So that, the author needs some suggestions to make it this be
better. And big wishes, it can be improve our knowledge, understanding and
enrich the science education.
Medan, July
2015
Author,
Widi Aulia Widakdo
ID. 4113111080
vi
CONTENTS
Pages
Ratification Sheet
i
Biography
ii
Abstract
iii
Preface
iv
Contents
vi
List of Figure
ix
List of Table
x
List of Appendix
xi
CHAPTER I
INTRODUCTION
1.1.
Background
1
1.2.
Problem Identification
8
1.3.
Problem Limitation
8
1.4.
Problem Formulation
8
1.5.
Research Objectives
9
1.6.
Research Benefit
9
1.7.
Operational Definition
9
CHAPTER II
2.1.
LITERATURE REVIEW
Theoretical Framework
11
2.1.1. Mathematical Representation Ability
11
2.1.2. Project Based Learning
17
2.1.3. Problem Based Learning
21
2.1.4. Supporting Theory of Project Based Learning
25
2.1.5. Supporting Theory of Problem Based Learning
25
2.1.6. Summary of Subject Matter (Statistics)
29
2.1.6.1.
Ungrouped Data
29
2.1.6.2.
Grouped Data
30
vii
2.1.6.3.
Graphical Representation
30
2.2. Relevant Research
31
2.3. Conceptual Framework
32
2.4. Hypothesis
33
CHAPTER III
RESEARCH METHODOLOGY
3.1. Type of Research
34
3.2. Place and Time of Research
34
3.3. Population and Sample
34
3.4. Variable of Research
35
3.4.1. Independent Variable
35
3.4.2. Dependent Variable
36
3.5. Instrument of Research
36
3.5.1. Initial Test
36
3.5.2. Test of Students’ Mathematical Representation Ability
36
3.5.3. Test Validity
41
3.5.4. Test Reliability
42
3.5.5. Difficulty Level Index
43
3.5.6. Discrimination Power of the Test
43
3.6. Design of Research
45
3.7. Procedure of Research
45
3.8. Technique of Data Analyzing
48
3.8.1. Normality Test
48
3.8.2. Homogeneity Test
49
3.8.3. Hypothesis Test
50
CHAPTER IV RESULTS AND DISCUSSIONS
4.1 Research Results Description
4.1.1 The Description of Students’ Mathematical
52
52
Representation Ability
4.1.2. The Description of Matemathical Represention Test
53
viii
4.2. Analysis of Research Data
54
4.2.1 Normality Test
54
4.2.2 Homogeneity Test
55
4.2.3. Compare Means Test (One-tailed)
56
4.2.4 Analysis of Observation Sheet
58
4.3. Research Discussion
58
CHAPTER V CONCLUSION AND SUGGESTION
5.1 Conclusion
61
5.2 Suggestion
61
REFFERENCE
62
ix
LIST OF FIGURE
Figure 1.1.
The Question of Observation Question Number 2
Figure 1.2.
The Student’s Answer of Observation Question
Figure 1.3.
The Student’s Answer of Questionnaire
Figure 2.1.
The Relationship Between Internal and External
4
5
Representation Developing Child’s Understanding
Of The Concept Of Numeracy
13
Figure 2.2.
The Problem-Based Learning Cycle
22
Figure 2.3.
Learner knowledge and Zone of Proximal Development
27
Figure 2.4.
Bar Graphs
30
Figure 2.5.
Pie Charts
31
Figure 2.6.
Line Chart
31
Figure 3.1.
Procedure of Research
47
xi
LIST OF APPENDIX
Appendix 1.
The Blueprint of Mathematical Representation Ability
Initial Test
Appendix 2. Initial Test Of Mathematical Representation Ability
Appendix 3.
68
69
Alternative Solution of Mathematical Representation
Ability Initial Test
70
Appendix 4. Questionnaire of Student’s Opinion
71
Appendix 5. Lesson Plan of problem based learnimg model
72
Appendix 6. Lesson Plan of Project Based Learning
95
Appendix 7. Worksheet
104
Appendix 8. The Blueprint of Students’ Mathematical Representation
Ability Test
Appendix 9. Test of Mathematical Representation Ability (Post Test)
113
115
Appendix10. Alternative Solution of Mathematical Representation
Ability
118
Appendix11. Validation Sheet of Mathematics Problem
Solving Ability Test II
122
Appendix12. Observation Sheet Of Teacher’s Activities in
Experimental Class A: Project Based Learning
132
Appendix 13. Sheet Of Teacher’s Activities in Experimental
Class B: Prblem Based Learning
141
Appendix 14. Validity Of Post Test - Test In Trial Class
152
Appendix 15. Reliability Of Post Test - Test In Trial Class
155
Appendix 16. Discrimination Power Analysis And
Difficulty Level Index Of Pre – Test
Appendix 17. Attendance Of Students In Experimental
157
160
Appendix 18. Attendance Of Students In Experimental
Class I (Prblem Based Learning)
161
Appendix 19. Group Division Both Experiment Class
163
Appendix 20. posttest score of experimental and control class
164
xii
Appendix 21. Normality Test
167
Appendix 22. Homogeneity Test
168
Appendix 23. Independent Sample t-test
169
Appendix 24. Documentation
170
Appendix 25. T-table Value of t-distribution
176
Appendix 26. T-table Value of t-distribution
177
1
CHAPTER 1
INTRODUCTION
1.1.
Background
The National Education which based on Pancasila and the 1945
Constitution of the Republic of Indonesia was explained in Law Number 20 year
2003 about National Education System. The National Education functions is to
develop the capability, character, and civilization of the nation for enhancing its
intellectual capacity, and is aimed at developing learners' potentials so that they
become persons imbued with human values who are faithful and pious to one and
only God; who process morals and noble character; who are healthy,
knowledgeable, competent, creative, independent; and as citizens, are democratic
and responsible (Seameo, 2015).
Education gives knowledge, good thinking patterns, and a more systematic
framework. Education need logical thinking to connect the abstract part in the
mind to applied in solving problem of reality life. To construct this logical
thinking, it needs mathematics.
Mathematics subject is one of the principal subjects taught begin during
elementary school until to the university. Mathematics subject is also one of the
subjects tested in the national examination both at the elementary school, junior
high schools, as well as in senior high school.
Mathematics is a foundation and framework of the development of science
and technology. In everyday life we use and need mathematical concepts and
principles, as a tool in applications other disciplines as well as in the development
of mathematics itself. Seeing the importance of the role of mathematics in
everyday life, mastery of the subject areas of mathematics is a must.
Mathematics is one of the most important subjects that provide several
vital skills to the learners. The characteristics of math abilities also as principle
and process standards in mathematics that will be developed in the National
Council
of
Teachers
of
Mathematics
(NCTM,
2000)
are
problem
solving, reasoning, communication, connection, and representation. The five of
characteristics are the goal to be achieved in mathematics learning. So
2
mathematics is a learning that has final result more than a score in the final report,
but Cockroft (1982) said that ―
Mathematics can improve the ability of logical
thinking, accuracy, and spatial awareness, also gives effort the ability to solve
challenging problems‖.
Hudojo (2005: 64) said in his book that ―
hakekat matematika berkenaan
dengan ide-ide, struktur-struktur dan hubungan-hubungannya yang diatur
menurut urutan yang logis. Jadi matematika berkenaan dengan konsep-konsep
abstrak”. So it can be conclude that mathematics is a lesson that can improve the
way to think in life.
―
A representation is a configuration that can represent something else in
some manner‖ (Goldin, 2002: 208).
People develop representations in order to interpret and remember their
experiences in an effort to understand the world. Bruner (1966) found three
distinct ways in which people represent the world: (a) through action, (b)
through visual images, and (c) through words and language. He called these
kinds of representations enactive, iconic, and symbolic, respectively. Most
researchers agree that these three types of representations are important in
human understanding. (Salkind, 2007: 3)
Based on the explanation above, can be concluded that representation is a
term to make connection between abstract idea with logical thinking to
understanding mathematics, it needs representation.
Goldin and Shteingold (2001) wrote of two systems of representation.
―
External systems of representation include conventional representations that are
usually symbolic in nature. Internal systems of representation are created within a
person’s mind and used to assign mathematical meaning‖. Our numeration
system, mathematical equations, algebraic expressions, graphs, geometric figures,
and number lines are examples of external representations. These representations
have been developed over time and are widely used. External representations also
include written and spoken language. Examples of internal representations include
personal notation systems, natural language, visual imagery, and problem solving
strategies. Low ability of representation showing a lack of skilled students in
generating ideas, ask questions and respond to questions or opinions of others.
3
Based on explanation above, can be concluded that representation is one of
the important thing in understanding mathematics. Mathematics can be
understood if the students have good representation. So they able to describe,
interpret, express, symbolize or even modeling ideas, mathematical concepts and
the coherence among them and contained in a configuration, construction or
certain situations that appear in various forms in order to obtain clarity of
meaning, show understanding or looking for a solution of the problems.
In fact, our students in Indonesia has low quality in understanding
mathematics. It shows from the result of the survey of Program for International
Study Assessment (PISA) in 2012 showed that from 65 survey countries for
mathematics, reading and science skills, Indonesia was in 64th level with the mean
score of mathematics skill was 375 while the average of OECD (Organization for
Economic
Co-Operation
and
Development)
was
494
(http://www.theguardian.com/news/datablog/2013/dec/03/pisa-results-countrybest-reading-maths-science accessed on 7th of April, 2015).
Based on the observation of researcher did on January, 25th 2015 by doing
interview to the vice principle and giving questions and questionnaire to the
students, this problem also happened in SMA Negeri 1 Percut Sei Tuan. there are
many students who failed the examinations. Their grade is lower than KKM that
required by the school, it is about 65%, while the KKM in this school is 2.88 in
scale of 4,00 or 72 in scale of 100. By giving the questions about statistics to the
37 students of grade X at SMA Negeri 1 Percut Sei Tuan as follows:
1. The scores of 35 students on a mathematical quiz are as follows:
75 60 41 77 89 90 65 70 100 55 60 76 80 60 75 90 55 90 100 95 91 50 60 75
80 100 90 55 85 89 70 75 70 100 60
a. Prepare a frequency table for the grouped data.
b. How many students passed the science quiz if the standard score > 70?
c. Determine the value of mean
2. Bar chart for the number of students based on their ethnic background in
School A
4
Figure 1.1 The Question of Observation Question no.2
Describe in brief, the overall pattern the number of students based on their
ethnic background ?
Figure 1.2 The Student’s Answer of Observation Question
From 37 students who answer the question, can be seen that 66.67% of
them have not been able yet to build their visual representations in making table
exactly, while 70.27% of students also have not been able yet to build their
mathematical representations ability in equation or mathematical expression
aspects especially in making the equation. Mathematical model from initial
representation is also given and 65,49% of students have not been able yet to
represent their ideas or knowledge in writing the text form.
The mathematical representation ability of students has not satisfied yet
according to the observed results. This situation is caused by the lack of their
understanding in statistics and the lack of representing something from abstract to
concrete.
5
Based on the observation that did on January 22th, 2015 through
questionnaires were distributed randomly to 100 foreign students of SMA Negeri
1 Percut Sei Tuan, only 39% of students learn mathematics more than equal to 3
hours every week outside the school activity, For the example at house or course
place. 38% of students learn mathematics less than 3 hours every week, and 33%
never learn mathematics outside the school activity.
Figure 1.3 The Student’s Answer of questionnaire
Actually, 86% of students know mathematics is important and they need
mathematics in their daily life. But it doesn’t enough to motivate them in learning
mathematics. Based on observation, 55% students said that mathematics is
difficult, 45% said it is not so difficult if they learn it seriously and none said
mathematics is easy. 10% of students said it can be happen because they hate
mathematics, while 38% of students give explanation that they have difficulty in
understanding mathematics. 19% of students the difficulty of mathematics is
depend on the matter and 33% explain it because of teacher doesn’t explain
clearly.
6
It can be concluded that mathematical representation in SMA Negeri 1
Percut Sei Tuan is still low. There are many factors can lead to low mathematics
student learning achievements. Prasad (2008) said:
There are three dimensions – school environment, teacher-student relations
and value orientation among teachers’ influence the whole educational
process in the classroom situation. School environment is an external factor
and teacher-student relation is an internal factor. We know that values
among teacher decide and control both the factors.
It means, not only the students it self that can influence the student’s ability
in understanding mathematics, but also teacher and its environment. Teacher can
reduce this problem by giving innovative learning strategies that are considered
the development of students’ cognitive abilities and independence. One of them is
by giving ―
learning by doing‖ in teaching and learning process. Stalheim and
Smith (1998) said:
People have known for hundreds of years that they remember what they see
and do. A 2000 year old proverb states: ―
I hear and I forget. I see and I
remember. I do and I understand.‖ Experience has taught me the wisdom of
this proverb. Data given by Stice (1987) also supports this proverb. He
indicates that learners remember 10% of what they read, 26% of what they
hear, 30% of what they see, 50% of what they see and hear, 70% of what
they say and 90% of what they say as they do something.
One model that provides learning by doing is Project based Learning.
Project Based Learning is a teaching that is designed for complex problems in
which students conduct an investigation to understand, emphasizing long learning
activities, assignments given to students multidisciplinary, oriented products
(artifacts). According Mahanal (2009) PBL learning in general have guidelines
steps: planning, creating and processing.
In fact, a growing body of research suggests that students learn more deeply
and perform better on complex tasks if they have the opportunity to engage
in more ―
authentic‖ learning—projects and activities that require them to
employ subject knowledge to solve real-world problems. Studies have
shown a positive impact on learning when students participate in lessons
that require them to construct and organize knowledge, consider
alternatives, engage in detailed research, inquiry, writing, and analysis, and
to communicate effectively to audiences. (Newmann, 1996)
7
A project-based learning lesson provides students with the opportunity to
learn in an authentic, challenging, multidisciplinary environment, to learn
how to design, carry out, and evaluate a project that requires sustained
effort over a significant period of time, to learn to work with minimal
external guidance, both individually and in groups, to gain in self-reliance
and personal accountability. (Özdemir in Bas 2011: 10)
In this study, student will be guided to make project as data processing and
will be and these data will be presented in the form of a report in the form of
board and packaged as attractive as possible which will then be exhibited in a
mini exhibition to be presented to the visitors. This activity is expected to improve
the mathematical representation of students through activities that stimulate the
representation ability by presentation to the visitors. Giving their opinion about
their project orally, in writing in the form of words, symbols, or expressions of
mathematical notation, making graph, diagrams, tables or physical objects such as
report board.
In this observation, the observer will compare the learning model of
project-based learning with problem based learning model. PBL is one of model
that make active learning is occurred. Arends (2012: 396) said ―
the essence of
problem-based learning consists of presenting students with authentic and
meaningful problem situations that can serve as springboards for investigations
and inquiry‖.
PBL makes students work with classmates to solve complex and authentic
problems that help develop content knowledge as well as problem –
solving, reasoning, communication, and self-assessment skills. These
problems also help to maintain student interest in course material because
students realize that they are learning the skills needed to be successful in
the field (White, 2011: 1).
Montague (in Sajadi, Amiripour and Malkhalifeh, 2013: 2) defined
mathematical word problem solving as a process involving two stages: problem
"representation" and "problem execution". Both of them are necessary for
problem solving successfully. Successful problem solving is not possible without
first representing the problem appropriately. Appropriate problem representation
8
indicates that the problem solver has perceived the problem and serves to guide
the student toward the solution plan.
Based on the general description above, then the researcher has interested
to do research entitled ―
The Difference of Students' Mathematical
Representation Ability By Using Project Based Learning
And Direct
Instructional Learning Model on the Topic of Statistics in Grade X SMA
Negeri 1 Percut Sei Tuan‖.
1.2.
Problem Identification
Based on the explanation in the background, the problem identification:
a.
Student’s mathematical learning in SMA Negeri 1 Percut Sei Tuan outcomes is
still low.
b. Mathematical representation ability of students in SMA Negeri 1 Percut Sei
Tuan is still low.
c. Students of in SMA Negeri 1 Percut Sei Tuan still have difficulties in solving
mathematical represetation tests.
d.
Teacher learning model used is still less variation and the learning process is
still conventional.
1.3.
Problem limitation
For more directing this research so focused and specific to the problem in
this study in limited to the students’ mathematical representation ability on the
topic of statistics grade X in SMA Negeri 1 Percut Sei Tuan A. Y. 2014/2015 as
well as the learning model is applied in the model limit by Project Based Learning
and Problem Based Learning.
1.4.
Problem Formulation
Based on the above problem limitation, then the problem formulation in
this research: is there any difference students’ mathematical representation ability
taught by Project Based Learning with Problem Based Learning on the topic of
statistics in grade X SMA Negeri 1 Percut Sei Tuan A. Y. 2014/2015?
9
1.5.
Research Objective
The purpose of this research: to know any difference students’
mathematical representation ability taught by Project Based Learning
with
Problem Based Learning on the topic of statistics in grade X SMA Negeri 1
Percut Sei Tuan A. Y. 2014/2015?
1.6.
Research Benefits
The benefits of this research are:
1.
Being incoming material for researchers as mathematics teacher candidates to
apply Project Based Learning as mathematics’ alternative learning model in
school.
2.
For teachers and prospective teachers, this study could be a reference in
planning learning of statistics subject.
3.
For students, is expected to use Project Based Learning and can be used to
improve the students' mathematical representation ability.
4.
For school, is expected to be a source of information or contribute ideas for
improvement of mathematics teaching, especially in school where the
researcher conducted and the school in general.
1.7.
Operational Definitions
In order to avoid the differences of clarity meaning about important terms
contained in this research, the operational definitions will be noted as following :
1.
Mathematical
representation ability is
students’ ability to
express
mathematical ideas (problem, statement, definition, and so on) into form: (1)
Picture,
diagram,
graph,
or
table;
(2)
Mathematical
notation,
numerical/algebra symbol; (3) Written texts/words the interpretation of their
mind.
2.
Project Based Learning is Project based learning is a model that organizes
learning around projects.
10
3.
Projects are complex tasks, based on challenging questions or problems, that
involve
students
in
design,
problem-solving,
decision
making,
or
investigative activities; give students the opportunity to work relatively
autonomously over extended periods of time; and culminate in realistic
products or presentations.
4.
PBL is one of model that make active learning is occurred. Arends (2010)
said that PBL is a student centered approach that organizes curriculum and
instruction around carefully crafted ―
ill-structured‖ and real-world problems
situations. Learning is active rather than passive, integrated rather than
fragmented, and connected rather than disjointed.
61
CHAPTER V
CONCLUSION AND SUGGESTION
5.1 Conclusion
Based on the result of research and discussion can be conclude that: The
students’ representation ability that taught by using Problem Based Learning
Model is different with Project Based Learning Model on topic Statistics at grade
X SMA Negeri 1 Percut Sei Tuan A.Y. 2014/2015.
5.2 Suggestion
Based on the result of research and the above conclusion, then researcher
submits some suggestions, as follow:
1.
For teacher, both of project based and problem based learning can be used to
improve representation mathematical ability.
2.
For next researcher should use another model of learning to improve
mathematical representation ability or choose another topic of mathematics.
3.
For students, give more attention and follow the instruction that teacher has
given during the learning of mathematics. So that it will be useful effectively.
62
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THE DIFFERENCE OF STUDENTS' MATHEMATICAL REPRESENTATION
ABILITY BY USING PROJECT BASED LEARNING AND PROBLEM
BASED LEARNING ON THE TOPIC OF STATISTICS IN
GRADE X SMA NEGERI 1 PERCUT SEI TUAN
By:
Widi AuliaWidakdo
ID 4113111080
Mathematics Education Study Program
THESIS
Submitted to Qualify for Academic Title of
Sarjana Pendidikan
MATHEMATICS DEPARTMENT
FACULTY OF MATHEMATICS AND NATURAL SCIENCES
STATE UNIVERSITY OF MEDAN
MEDAN
2011
iv
PREFACE
Praise and great thanks to Allah SWT that gives the amazing grace, love,
strength and health so that writer can finish this thesis. The title of this thesis is
“The Difference Of Students' Mathematical Representation Ability By Using
Project Based Learning And Problem Based Learning On The Topic Of Statistics
In Grade X Sma Negeri 1 Percut Sei Tuan”. This thesis was arranged to satisfy the
requirement to obtain the Degree of Sarjana Pendidikan from Faculty
Mathematics and Natural Science in State University of Medan
In the completion of this thesis, the writer received support from various
parties, therefore it was appropriate writer big thanks to Mr. Prof. Dr. Asmin,
M.Pd as my thesis supervisor who has provided guidance, direction, and advice to
the perfection of this thesis. Thanks are also due to Prof. Dr. Mukhtar, M.Pd, Dr.
Asrin Lubis, M.Pd and Faiz Ahyaningsih, M.Si as author’s examiners who have
provided input and suggestion from the planning to the completion of the
preparation of the research of this thesis. Thanks are also extended to Dr. W.
Rajagukguk, M.Pd as academic supervisor and then thank you so much for all
author’s lecturer in FMIPA Unimed.
My thanks are extended to Prof. Dr. Syawal Gultom, M. Si. as rector of
Unimed, Prof. Drs. Motlan, M.Sc, Ph.D as Dean of Mathematics and Natural
Sciences Faculty and to coordinator of bilingual Prof. Dr. rer.nat. Binari
Manurung, M.Si, Dr. Edy Surya, M.Si. as Chief of Mathematics Department, Drs.
Zul Amry, M.Si, Ph.D as Chief of Mathematics Education Study Program, Drs.
Yasifati Hia, M. Si as Secretary of Mathematics Education, and all of employee
staff who have helped the author.
Thanks to Mr. Muliadi, S.Pd. M.Si as principle of SMA N 1 Percut Sei
Tuan, Mrs. Dra. Libes Doloksaribu, S.Pd as mathematics teacher and all teacher,
staffs and also the students in grade X IIS 1 and X IIS 3 SMA N 1 Percut Sei
Tuan who have helped writer conducting the research.
Especially I would like to express my gratitude to my dear father Mr. Puji
Widakdo and my dear mother Mrs. Dra. Erni Fianti, S.Ag continues to provide
v
motivation and prayers for the success of me completed this thesis. Special big
thanks to my beloved sister Firda Saufika, Nabila Nadra and Also my brother M.
Khair Arrayyan for giving support even moril or material and all my family for all
pray, motivation, and support until the end of my study.
I also thanks to my lovely second family of Himpunan Mahasiswa Islam
(HMI) which always help me and support in every condition without any
exception. Especially for Vivi, Taufik, Sapwan, Juli, Ilmi, Maryam, Widya, for all
of your crazy thing, Bang Elfan, Bang Dedi, Bang Imam, Bang Badzlan, Kak Imel
and Kak Mora for all of the suggestion and incredible advice. Thank you very
much for Rizki Ramadhana, Satoto, Hakim, Fatkhu, Zaki, also Anggun, Nadia,
and Putri Rizki for every helping you’ve given as my sisters and brothers in this
university. I love you and thanks for every spirit my freak friends Debby, Nelly
and Yohannes also my classmate in Bilingual Mathematics Education 2011.
At last, the author has finished this thesis in maximum level but author
realized there are some imperfections. For that, the author asks for building
comments and suggestions in order to reach the perfection of this thesis. The
author wishes that this thesis would be useful to improve the knowledge should
give a big effort to prepare this thesis, and the writer know that this thesis have so
many weakness. So that, the author needs some suggestions to make it this be
better. And big wishes, it can be improve our knowledge, understanding and
enrich the science education.
Medan, July
2015
Author,
Widi Aulia Widakdo
ID. 4113111080
vi
CONTENTS
Pages
Ratification Sheet
i
Biography
ii
Abstract
iii
Preface
iv
Contents
vi
List of Figure
ix
List of Table
x
List of Appendix
xi
CHAPTER I
INTRODUCTION
1.1.
Background
1
1.2.
Problem Identification
8
1.3.
Problem Limitation
8
1.4.
Problem Formulation
8
1.5.
Research Objectives
9
1.6.
Research Benefit
9
1.7.
Operational Definition
9
CHAPTER II
2.1.
LITERATURE REVIEW
Theoretical Framework
11
2.1.1. Mathematical Representation Ability
11
2.1.2. Project Based Learning
17
2.1.3. Problem Based Learning
21
2.1.4. Supporting Theory of Project Based Learning
25
2.1.5. Supporting Theory of Problem Based Learning
25
2.1.6. Summary of Subject Matter (Statistics)
29
2.1.6.1.
Ungrouped Data
29
2.1.6.2.
Grouped Data
30
vii
2.1.6.3.
Graphical Representation
30
2.2. Relevant Research
31
2.3. Conceptual Framework
32
2.4. Hypothesis
33
CHAPTER III
RESEARCH METHODOLOGY
3.1. Type of Research
34
3.2. Place and Time of Research
34
3.3. Population and Sample
34
3.4. Variable of Research
35
3.4.1. Independent Variable
35
3.4.2. Dependent Variable
36
3.5. Instrument of Research
36
3.5.1. Initial Test
36
3.5.2. Test of Students’ Mathematical Representation Ability
36
3.5.3. Test Validity
41
3.5.4. Test Reliability
42
3.5.5. Difficulty Level Index
43
3.5.6. Discrimination Power of the Test
43
3.6. Design of Research
45
3.7. Procedure of Research
45
3.8. Technique of Data Analyzing
48
3.8.1. Normality Test
48
3.8.2. Homogeneity Test
49
3.8.3. Hypothesis Test
50
CHAPTER IV RESULTS AND DISCUSSIONS
4.1 Research Results Description
4.1.1 The Description of Students’ Mathematical
52
52
Representation Ability
4.1.2. The Description of Matemathical Represention Test
53
viii
4.2. Analysis of Research Data
54
4.2.1 Normality Test
54
4.2.2 Homogeneity Test
55
4.2.3. Compare Means Test (One-tailed)
56
4.2.4 Analysis of Observation Sheet
58
4.3. Research Discussion
58
CHAPTER V CONCLUSION AND SUGGESTION
5.1 Conclusion
61
5.2 Suggestion
61
REFFERENCE
62
ix
LIST OF FIGURE
Figure 1.1.
The Question of Observation Question Number 2
Figure 1.2.
The Student’s Answer of Observation Question
Figure 1.3.
The Student’s Answer of Questionnaire
Figure 2.1.
The Relationship Between Internal and External
4
5
Representation Developing Child’s Understanding
Of The Concept Of Numeracy
13
Figure 2.2.
The Problem-Based Learning Cycle
22
Figure 2.3.
Learner knowledge and Zone of Proximal Development
27
Figure 2.4.
Bar Graphs
30
Figure 2.5.
Pie Charts
31
Figure 2.6.
Line Chart
31
Figure 3.1.
Procedure of Research
47
xi
LIST OF APPENDIX
Appendix 1.
The Blueprint of Mathematical Representation Ability
Initial Test
Appendix 2. Initial Test Of Mathematical Representation Ability
Appendix 3.
68
69
Alternative Solution of Mathematical Representation
Ability Initial Test
70
Appendix 4. Questionnaire of Student’s Opinion
71
Appendix 5. Lesson Plan of problem based learnimg model
72
Appendix 6. Lesson Plan of Project Based Learning
95
Appendix 7. Worksheet
104
Appendix 8. The Blueprint of Students’ Mathematical Representation
Ability Test
Appendix 9. Test of Mathematical Representation Ability (Post Test)
113
115
Appendix10. Alternative Solution of Mathematical Representation
Ability
118
Appendix11. Validation Sheet of Mathematics Problem
Solving Ability Test II
122
Appendix12. Observation Sheet Of Teacher’s Activities in
Experimental Class A: Project Based Learning
132
Appendix 13. Sheet Of Teacher’s Activities in Experimental
Class B: Prblem Based Learning
141
Appendix 14. Validity Of Post Test - Test In Trial Class
152
Appendix 15. Reliability Of Post Test - Test In Trial Class
155
Appendix 16. Discrimination Power Analysis And
Difficulty Level Index Of Pre – Test
Appendix 17. Attendance Of Students In Experimental
157
160
Appendix 18. Attendance Of Students In Experimental
Class I (Prblem Based Learning)
161
Appendix 19. Group Division Both Experiment Class
163
Appendix 20. posttest score of experimental and control class
164
xii
Appendix 21. Normality Test
167
Appendix 22. Homogeneity Test
168
Appendix 23. Independent Sample t-test
169
Appendix 24. Documentation
170
Appendix 25. T-table Value of t-distribution
176
Appendix 26. T-table Value of t-distribution
177
1
CHAPTER 1
INTRODUCTION
1.1.
Background
The National Education which based on Pancasila and the 1945
Constitution of the Republic of Indonesia was explained in Law Number 20 year
2003 about National Education System. The National Education functions is to
develop the capability, character, and civilization of the nation for enhancing its
intellectual capacity, and is aimed at developing learners' potentials so that they
become persons imbued with human values who are faithful and pious to one and
only God; who process morals and noble character; who are healthy,
knowledgeable, competent, creative, independent; and as citizens, are democratic
and responsible (Seameo, 2015).
Education gives knowledge, good thinking patterns, and a more systematic
framework. Education need logical thinking to connect the abstract part in the
mind to applied in solving problem of reality life. To construct this logical
thinking, it needs mathematics.
Mathematics subject is one of the principal subjects taught begin during
elementary school until to the university. Mathematics subject is also one of the
subjects tested in the national examination both at the elementary school, junior
high schools, as well as in senior high school.
Mathematics is a foundation and framework of the development of science
and technology. In everyday life we use and need mathematical concepts and
principles, as a tool in applications other disciplines as well as in the development
of mathematics itself. Seeing the importance of the role of mathematics in
everyday life, mastery of the subject areas of mathematics is a must.
Mathematics is one of the most important subjects that provide several
vital skills to the learners. The characteristics of math abilities also as principle
and process standards in mathematics that will be developed in the National
Council
of
Teachers
of
Mathematics
(NCTM,
2000)
are
problem
solving, reasoning, communication, connection, and representation. The five of
characteristics are the goal to be achieved in mathematics learning. So
2
mathematics is a learning that has final result more than a score in the final report,
but Cockroft (1982) said that ―
Mathematics can improve the ability of logical
thinking, accuracy, and spatial awareness, also gives effort the ability to solve
challenging problems‖.
Hudojo (2005: 64) said in his book that ―
hakekat matematika berkenaan
dengan ide-ide, struktur-struktur dan hubungan-hubungannya yang diatur
menurut urutan yang logis. Jadi matematika berkenaan dengan konsep-konsep
abstrak”. So it can be conclude that mathematics is a lesson that can improve the
way to think in life.
―
A representation is a configuration that can represent something else in
some manner‖ (Goldin, 2002: 208).
People develop representations in order to interpret and remember their
experiences in an effort to understand the world. Bruner (1966) found three
distinct ways in which people represent the world: (a) through action, (b)
through visual images, and (c) through words and language. He called these
kinds of representations enactive, iconic, and symbolic, respectively. Most
researchers agree that these three types of representations are important in
human understanding. (Salkind, 2007: 3)
Based on the explanation above, can be concluded that representation is a
term to make connection between abstract idea with logical thinking to
understanding mathematics, it needs representation.
Goldin and Shteingold (2001) wrote of two systems of representation.
―
External systems of representation include conventional representations that are
usually symbolic in nature. Internal systems of representation are created within a
person’s mind and used to assign mathematical meaning‖. Our numeration
system, mathematical equations, algebraic expressions, graphs, geometric figures,
and number lines are examples of external representations. These representations
have been developed over time and are widely used. External representations also
include written and spoken language. Examples of internal representations include
personal notation systems, natural language, visual imagery, and problem solving
strategies. Low ability of representation showing a lack of skilled students in
generating ideas, ask questions and respond to questions or opinions of others.
3
Based on explanation above, can be concluded that representation is one of
the important thing in understanding mathematics. Mathematics can be
understood if the students have good representation. So they able to describe,
interpret, express, symbolize or even modeling ideas, mathematical concepts and
the coherence among them and contained in a configuration, construction or
certain situations that appear in various forms in order to obtain clarity of
meaning, show understanding or looking for a solution of the problems.
In fact, our students in Indonesia has low quality in understanding
mathematics. It shows from the result of the survey of Program for International
Study Assessment (PISA) in 2012 showed that from 65 survey countries for
mathematics, reading and science skills, Indonesia was in 64th level with the mean
score of mathematics skill was 375 while the average of OECD (Organization for
Economic
Co-Operation
and
Development)
was
494
(http://www.theguardian.com/news/datablog/2013/dec/03/pisa-results-countrybest-reading-maths-science accessed on 7th of April, 2015).
Based on the observation of researcher did on January, 25th 2015 by doing
interview to the vice principle and giving questions and questionnaire to the
students, this problem also happened in SMA Negeri 1 Percut Sei Tuan. there are
many students who failed the examinations. Their grade is lower than KKM that
required by the school, it is about 65%, while the KKM in this school is 2.88 in
scale of 4,00 or 72 in scale of 100. By giving the questions about statistics to the
37 students of grade X at SMA Negeri 1 Percut Sei Tuan as follows:
1. The scores of 35 students on a mathematical quiz are as follows:
75 60 41 77 89 90 65 70 100 55 60 76 80 60 75 90 55 90 100 95 91 50 60 75
80 100 90 55 85 89 70 75 70 100 60
a. Prepare a frequency table for the grouped data.
b. How many students passed the science quiz if the standard score > 70?
c. Determine the value of mean
2. Bar chart for the number of students based on their ethnic background in
School A
4
Figure 1.1 The Question of Observation Question no.2
Describe in brief, the overall pattern the number of students based on their
ethnic background ?
Figure 1.2 The Student’s Answer of Observation Question
From 37 students who answer the question, can be seen that 66.67% of
them have not been able yet to build their visual representations in making table
exactly, while 70.27% of students also have not been able yet to build their
mathematical representations ability in equation or mathematical expression
aspects especially in making the equation. Mathematical model from initial
representation is also given and 65,49% of students have not been able yet to
represent their ideas or knowledge in writing the text form.
The mathematical representation ability of students has not satisfied yet
according to the observed results. This situation is caused by the lack of their
understanding in statistics and the lack of representing something from abstract to
concrete.
5
Based on the observation that did on January 22th, 2015 through
questionnaires were distributed randomly to 100 foreign students of SMA Negeri
1 Percut Sei Tuan, only 39% of students learn mathematics more than equal to 3
hours every week outside the school activity, For the example at house or course
place. 38% of students learn mathematics less than 3 hours every week, and 33%
never learn mathematics outside the school activity.
Figure 1.3 The Student’s Answer of questionnaire
Actually, 86% of students know mathematics is important and they need
mathematics in their daily life. But it doesn’t enough to motivate them in learning
mathematics. Based on observation, 55% students said that mathematics is
difficult, 45% said it is not so difficult if they learn it seriously and none said
mathematics is easy. 10% of students said it can be happen because they hate
mathematics, while 38% of students give explanation that they have difficulty in
understanding mathematics. 19% of students the difficulty of mathematics is
depend on the matter and 33% explain it because of teacher doesn’t explain
clearly.
6
It can be concluded that mathematical representation in SMA Negeri 1
Percut Sei Tuan is still low. There are many factors can lead to low mathematics
student learning achievements. Prasad (2008) said:
There are three dimensions – school environment, teacher-student relations
and value orientation among teachers’ influence the whole educational
process in the classroom situation. School environment is an external factor
and teacher-student relation is an internal factor. We know that values
among teacher decide and control both the factors.
It means, not only the students it self that can influence the student’s ability
in understanding mathematics, but also teacher and its environment. Teacher can
reduce this problem by giving innovative learning strategies that are considered
the development of students’ cognitive abilities and independence. One of them is
by giving ―
learning by doing‖ in teaching and learning process. Stalheim and
Smith (1998) said:
People have known for hundreds of years that they remember what they see
and do. A 2000 year old proverb states: ―
I hear and I forget. I see and I
remember. I do and I understand.‖ Experience has taught me the wisdom of
this proverb. Data given by Stice (1987) also supports this proverb. He
indicates that learners remember 10% of what they read, 26% of what they
hear, 30% of what they see, 50% of what they see and hear, 70% of what
they say and 90% of what they say as they do something.
One model that provides learning by doing is Project based Learning.
Project Based Learning is a teaching that is designed for complex problems in
which students conduct an investigation to understand, emphasizing long learning
activities, assignments given to students multidisciplinary, oriented products
(artifacts). According Mahanal (2009) PBL learning in general have guidelines
steps: planning, creating and processing.
In fact, a growing body of research suggests that students learn more deeply
and perform better on complex tasks if they have the opportunity to engage
in more ―
authentic‖ learning—projects and activities that require them to
employ subject knowledge to solve real-world problems. Studies have
shown a positive impact on learning when students participate in lessons
that require them to construct and organize knowledge, consider
alternatives, engage in detailed research, inquiry, writing, and analysis, and
to communicate effectively to audiences. (Newmann, 1996)
7
A project-based learning lesson provides students with the opportunity to
learn in an authentic, challenging, multidisciplinary environment, to learn
how to design, carry out, and evaluate a project that requires sustained
effort over a significant period of time, to learn to work with minimal
external guidance, both individually and in groups, to gain in self-reliance
and personal accountability. (Özdemir in Bas 2011: 10)
In this study, student will be guided to make project as data processing and
will be and these data will be presented in the form of a report in the form of
board and packaged as attractive as possible which will then be exhibited in a
mini exhibition to be presented to the visitors. This activity is expected to improve
the mathematical representation of students through activities that stimulate the
representation ability by presentation to the visitors. Giving their opinion about
their project orally, in writing in the form of words, symbols, or expressions of
mathematical notation, making graph, diagrams, tables or physical objects such as
report board.
In this observation, the observer will compare the learning model of
project-based learning with problem based learning model. PBL is one of model
that make active learning is occurred. Arends (2012: 396) said ―
the essence of
problem-based learning consists of presenting students with authentic and
meaningful problem situations that can serve as springboards for investigations
and inquiry‖.
PBL makes students work with classmates to solve complex and authentic
problems that help develop content knowledge as well as problem –
solving, reasoning, communication, and self-assessment skills. These
problems also help to maintain student interest in course material because
students realize that they are learning the skills needed to be successful in
the field (White, 2011: 1).
Montague (in Sajadi, Amiripour and Malkhalifeh, 2013: 2) defined
mathematical word problem solving as a process involving two stages: problem
"representation" and "problem execution". Both of them are necessary for
problem solving successfully. Successful problem solving is not possible without
first representing the problem appropriately. Appropriate problem representation
8
indicates that the problem solver has perceived the problem and serves to guide
the student toward the solution plan.
Based on the general description above, then the researcher has interested
to do research entitled ―
The Difference of Students' Mathematical
Representation Ability By Using Project Based Learning
And Direct
Instructional Learning Model on the Topic of Statistics in Grade X SMA
Negeri 1 Percut Sei Tuan‖.
1.2.
Problem Identification
Based on the explanation in the background, the problem identification:
a.
Student’s mathematical learning in SMA Negeri 1 Percut Sei Tuan outcomes is
still low.
b. Mathematical representation ability of students in SMA Negeri 1 Percut Sei
Tuan is still low.
c. Students of in SMA Negeri 1 Percut Sei Tuan still have difficulties in solving
mathematical represetation tests.
d.
Teacher learning model used is still less variation and the learning process is
still conventional.
1.3.
Problem limitation
For more directing this research so focused and specific to the problem in
this study in limited to the students’ mathematical representation ability on the
topic of statistics grade X in SMA Negeri 1 Percut Sei Tuan A. Y. 2014/2015 as
well as the learning model is applied in the model limit by Project Based Learning
and Problem Based Learning.
1.4.
Problem Formulation
Based on the above problem limitation, then the problem formulation in
this research: is there any difference students’ mathematical representation ability
taught by Project Based Learning with Problem Based Learning on the topic of
statistics in grade X SMA Negeri 1 Percut Sei Tuan A. Y. 2014/2015?
9
1.5.
Research Objective
The purpose of this research: to know any difference students’
mathematical representation ability taught by Project Based Learning
with
Problem Based Learning on the topic of statistics in grade X SMA Negeri 1
Percut Sei Tuan A. Y. 2014/2015?
1.6.
Research Benefits
The benefits of this research are:
1.
Being incoming material for researchers as mathematics teacher candidates to
apply Project Based Learning as mathematics’ alternative learning model in
school.
2.
For teachers and prospective teachers, this study could be a reference in
planning learning of statistics subject.
3.
For students, is expected to use Project Based Learning and can be used to
improve the students' mathematical representation ability.
4.
For school, is expected to be a source of information or contribute ideas for
improvement of mathematics teaching, especially in school where the
researcher conducted and the school in general.
1.7.
Operational Definitions
In order to avoid the differences of clarity meaning about important terms
contained in this research, the operational definitions will be noted as following :
1.
Mathematical
representation ability is
students’ ability to
express
mathematical ideas (problem, statement, definition, and so on) into form: (1)
Picture,
diagram,
graph,
or
table;
(2)
Mathematical
notation,
numerical/algebra symbol; (3) Written texts/words the interpretation of their
mind.
2.
Project Based Learning is Project based learning is a model that organizes
learning around projects.
10
3.
Projects are complex tasks, based on challenging questions or problems, that
involve
students
in
design,
problem-solving,
decision
making,
or
investigative activities; give students the opportunity to work relatively
autonomously over extended periods of time; and culminate in realistic
products or presentations.
4.
PBL is one of model that make active learning is occurred. Arends (2010)
said that PBL is a student centered approach that organizes curriculum and
instruction around carefully crafted ―
ill-structured‖ and real-world problems
situations. Learning is active rather than passive, integrated rather than
fragmented, and connected rather than disjointed.
61
CHAPTER V
CONCLUSION AND SUGGESTION
5.1 Conclusion
Based on the result of research and discussion can be conclude that: The
students’ representation ability that taught by using Problem Based Learning
Model is different with Project Based Learning Model on topic Statistics at grade
X SMA Negeri 1 Percut Sei Tuan A.Y. 2014/2015.
5.2 Suggestion
Based on the result of research and the above conclusion, then researcher
submits some suggestions, as follow:
1.
For teacher, both of project based and problem based learning can be used to
improve representation mathematical ability.
2.
For next researcher should use another model of learning to improve
mathematical representation ability or choose another topic of mathematics.
3.
For students, give more attention and follow the instruction that teacher has
given during the learning of mathematics. So that it will be useful effectively.
62
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