THE DIFFERENCE OF STUDENTS’ MATHEMATICAL REPRESENTATION ABILITY BY USING INSTRUCTION OF PROBLEM BASED LEARNING AND DIRECT INSTRUCTION IN GRADE X.
THE DIFFERENCE OF STUDENTS’ MATHEMATICAL
REPRESENTATION ABILITY BY USING INSTRUCTION
OF PROBLEM BASED LEARNING AND DIRECT
INSTRUCTION IN GRADE X
by:
Elfan Syahputra
IDN. 4103312002
Bilingual Mathematics Education
Thesis
Submitted in Fulfill of the Requirements for the degree of
Sarjana Pendidikan
DEPARTMENT OF MATHEMATICS
FACULTY OF MATHEMATICS AND SCIENCES
UNIVERSITAS NEGERI MEDAN
MEDAN
2015
THE DIFFERENCE OF STUDENTS’ MATHEMATICAL REPRESENTATION
ABILITY BY USING INSTRUCTION OF PROBLEM BASED LEARNING
AND DIRECT INSTRUCTIOIN IN GRADE X
Elfan Syahputra (IDN 4103312002)
ABSTRACT
This research is quasi-experiment. The purpose of this research is to
know whether students’ mathematical representation by using instruction of
problem based learning classroom is higher than direct instruction classroom at
SMA Swasta Panca Budi Medan.The population of this research is students of
SMA Swasta Panca Budi Medan which consists of 14 classes, whereas the sample
consists of 2 classes, namely, X-MS1 as experimental class consists of 32 students
and X-MSA as control class consists of 30 students. Experimental class used
Problem Based Learning and control class used Direct Instruction. Collecting data
technique of this research is mathematical representation test given in the end of
learning either in experimental class or control class. The type of this test is essay
test. Before doing hypothesis test, the normality and the homogeneity test should
be done. The result of those tests, sample was taken from normal distributed and
homogeneous population. The data analysis of experimental class by using t-test
with significance level α = 0.05, it was obtained that tcalculation > ttable then H0 is
rejected and Ha is accepted.It can be concluded that students’ mathematical
representation ability by using instruction of problem based learning is higher
than direct instruction in grade X. The research that has been done, researcher
suggested that Problem based learning can be as consideration to teachers in
enhancing senior high school students’ mathematical representation ability.
Teacher intends to use problem based learning, needed preparation and used time
effectively in its implementation. The result and instrument of this research can be
used as consideration to implement problem based learning in a different class
grades and subjects for the future researchers.
PREFACE
Alhamdulillah, praise and gratitude to Almighty God, Allah SWT for His
blessings and mercy that provide health and wisdom to finish this thesis with the
title “ The Difference of Mathematical Representation Ability by Using
Instruction of Problem Based Learning and Direct Instruction in Grade X.
I want to express special thanks to my beloved family, my father, Aspan
Effendi, my mother, Siti Elfina, and my blood sister, Siska Amelia and Fatia
Aulia, for their everlasting support, patience, love, and their prayer throughout
this process. I dedicate this thesis for them. My warm thanks go to my Big
Family, for their support prayer love and care.
Special thanks are extended to my supervisor, Mr. Prof. Dr. Mukhtar,
M.Pd. for his time, patience, a lot of advice, constructive criticism, questioning
and probing, and constant encouragement that were deeply appreciated and
valued. I also extend special thanks to Mr. Prof. Dr. Edi Syahputra, Mrs. Dra.
Katrina Samosir, M.Pd., and Mr. Denny Haris, S.Si., M.Pd. as my examiners who
gave me insightful suggestions from compiling proposal till this thesis
compilation was done. And also for my academic counselor, Mr. H. Banjarnahor,
M.Pd., I want to express my special thanks for his guidance and suggestion during
the lecture period.
I would like to thank Mr. Prof. Dr. Ibnu Hajar, M.Si. as the Rector of
Medan State University as well as the staff employees in rector’s office, Mr. Prof.
Dr. Motlan, M.Sc., Ph.D. as the dean of Mathematics and Sciences Faculty, Mr.
Prof. Dr. rer.nat. Binari M., M.Si. as the coordinator of bilingual class, Mr. Dr.
Edi Surya., as the chief of Mathematics Department, Mr. Zul Amry, Ph.D as the
chief of Study Program of Mathematics Department, Mr. Drs. Yasifati Hia, M.Si.,
and all of staff employees in Mathematics Department of Mathematics and
Sciences Faculty that were involved in this thesis compilation.
I also express my deep gratitude to the principal of SMA Swasta Panca
Budi Medan, mathematics teacher, Mrs. Ilma S.Pd, and all of teachers in SMA
Swasta Panca Budi Medan, that gave me permit to conduct the research, helped
me from the beginning till the research done, and gave me support throughout this
process. I also want to thank all of students in class MS-1 and MS-A for their
collaboration, support, and passion when I was teaching them.
I would like to thank my best friends, Abdul, Anggi, Dian, Dwi, Erlyn,
Elfan, Kiki, Falni, Mila, Lia, Meiva, Martyanne, Melin, Surya, Nelly, Petra, Riny,
Rully, Sartika, Sheila, Siti, Uli, and Mimi, and also especially for my hometown
best friends Eri Syahputra, Taufik Ismail, M. Yogie A.Md, friends of best keles
Risva, Mora ,friends of PPLT in Kisaran, and all my friends i did not mention for
their support and love throughout this process. It would not have been possible
without each of you.
I have done the best for finishing this thesis, but I realize that this thesis
has its shortcomings either in contents, grammar, or technical writing. Hence, I
hope the constructive criticism and suggestion from the reader for the perfection
of this thesis. I hope this thesis could be useful for further researchers and useful
in enrichment the knowledge.
Medan, December
Writer,
Elfan Syahputra
Idn. 4103312002
2014
BIBLIOGRAPHY
Elfan Syahputra was born in Medan, the 9th July 1992. Her father named
Aspan Effendi and her mother named Siti Elfina. He is the first of 3 children.
In 1997, he accepted in TK Swasta Free Methodist Medan, and graduated
in 1998. In 1998, he continued study to SD Free Methodist Medan and graduated
in 2004. In 2004, he continued study to SMP Swasta Al-Azhar Medan, and
graduated in 2007. In 2007, he continued study to SMA Swasta Panca Budi
Medan, and graduated in 2010. In 2010, she accepted in Bilingual Class
Mathematics Education, Department of Mathematics, Faculty of Mathematics and
Sciences, Universitas Negeri Medan, and graduated on Wednesday, December
10th 2014.
CONTENT
Agreement Sheet
ii
Bibliography
iii
Abstract
iv
Preface
v
Content
vii
List of Figure
x
List of Table
xi
List of Appendix
xii
CHAPTER 1 INTRODUCTION
1.1.
The Background of the Research
1
1.2.
Problem Identification
8
1.3.
Problem Limitation
8
1.4.
Problem Formulation
9
1.5.
Research Objective
9
1.6.
Research Benefit
9
1.7.
Operational Definitions
9
CHAPTER II LITERATURE STUDY
2.1.
Theoritical Framework
11
2.2.
Representation
11
2.2.1.
Representation in Mathematics
12
2.2.2.
Mathematical Representation Ability
15
2.3.
Problem Based Learning Model
19
2.3.1.
Learning Model
19
2.3.2.
Problem Based Learning Model
20
2.3.3.
Characteristics of Problem Based Learning Model
22
2.3.4.
Advantages and Weakness of Problem Based Learning
24
2.3.5.
Stages of Problem Based Learning
25
2.3.6.
Direct Instruction
26
2.4.
Statistics
27
2.4.1.
Collecting and Organizing Data
27
2.4.2.
Grapichal representation
27
2.4.3.
Averages-Mean, Median, and Mode
29
2.4.4.
Dispersion
30
2.4.5.
Grouped Frequency Distribution
30
2.4.6.
Mean, Median, and Mode of grouped data
31
2.5.
Conceptual Framework
32
2.6.
Hypothesis
33
CHAPTER III
3.1.
Research Location and Research Objects
34
3.2.
Population and Sample of Research
34
3.2.1.
Population
34
3.2.2.
Sample
34
3.3.
Research Variables
34
3.3.1.
Independent Variable
35
3.3.2.
Dependent Variable
35
3.4.
Type and Design of Research
35
3.4.1.
Type of Reseach
35
3.4.2.
Design of Research
35
3.5.
Procedure of Research
36
3.6.
Instrument of Research
37
3.6.1.
Initial Test
38
3.6.2.
Test of Student’s Mathematical Representation Ability
38
3.7.
Data Analysis Tecnique
43
3.7.1.
Calculating Mean Score
43
3.7.2.
Calculating of Deviation Standard
44
3.7.3.
Normality Test
44
3.7.4.
Homogenety Test
44
3.7.5.
Hypotheses Test
45
CHAPTER IV
4.1.
Research Result Description
4.1.1.
The Description of Students’ Mathematical Representation
46
Ability
46
4.1.2.
The Description of Mathematical Representation Test
48
4.2.
Analysis of Research Data
49
4.2.1.
Normality Test
49
4.2.2.
Homogeneity Test
50
4.2.3.
Compare Means Test (One-tailed)
51
4.2.4.
Analysis of Observation Sheet
53
4.3.
Research Discussion
56
CHAPTER V
5.1.
Conclusion
60
5.2.
Suggestion
60
REFERENCE
61
FIGURE LIST
Figure 1.1. The Question of Inital Test no.2
6
Figure 1.2. The Student Answer of Intial test no.1
6
Figure 1.2. The Student Answer of Intial test no.2
7
Figure 2.1. The Relationship Between Internal and External
Representation In Developing Child’s Understanding of
the Concept of Numeracy
14
Figure 2.2. Bar Graphs
28
Figure 2.3. Pie Charts
28
Figure 2.4. Line Charts
29
Figure 3.1. Procedure of Research
37
Figure 4.1. Histogram Post-test of Students’ Mathematical
Representation Ability
47
TABLE LIST
Table 2.1. Operational Form of Mathematical Representation Ability
17
Table 2.2. Indicator of Mathematical Representation Ability
18
Table 2.3. Stages of Problem Based Learning
25
Table 2.4. Syntax Of The Direct Instruction Learning Model
26
Table 3.1. The Blueprint of Mathematical Representation Ability Problem 39
Table 3.2. The Rubric of Mathematical Representation Ability Problem
40
Table 4.1. The Result Post-test of Mathematical Representation Ability
47
Table 4.2.Mean Percentage of Experimental and Control Class of
Each Indicator
49
Table 4.3 One-sample Kolmogorov Smirnov Test
50
Table 4.4 Homogeneity Variance Test
51
Table 4.5 Independent Sample t-test
52
Table4.6 Observation Sheet
53
APPENDIX LIST
Appendix 1
The Blueprint Of Mathematical Representation Ability
Initial Test
64
Appendix 2
Initial Test Of Mathematical Representation Ability
65
Appendix 3
Alternative Solution Of Mathematical Representation
Ability Initial Test
66
Appendix 4
Lesson Plan Direct Instruction Method
67
Appendix 5
Lesson Plan Instructional of Problem Based Learning
Model
81
Appendix 6
Worksheet
104
Appendix 7
Guidelines For Scoring of Mathematical Representation
Ablity Test
Appendix 8
Appendix 9
111
The Blue Print of Student’s Mathematical Representation
Ability Test
114
Test Of Mathematical Representation Ability (Post Test)
115
Appendix 10 Alternative Solution Of Mathematical Representation
Ability (Post Test)
117
Appendix 11 Posttest Score Of Experimental And Control Class
120
Appendix 12 Normality Test
122
Appendix 13 Homogeneity Test
123
Appendix 14 Independent Sample t-test
124
Appendix 15 Documentation
126
Appendix 16 T-table Value of t-distribution
129
CHAPTER 1
INTRODUCTION
1.1 The Background of the Research
The most important thing to increase the progress of a nation is human
resources. Indonesia is categorized as a developing country and the quality of
National Education has very wide impacts in all aspects of human’s life. By
education, human will be able to solve various problems and difficulties of life.
Therefore, the purposes of education are as a self-builder, shaper, and developer.
Globalization requires people to have adequate education in order to compete.
Unfortunately, from the following facts below, Indonesia still has some serious
educational problems, such as (1)Every minute, four children out of school;
(2)54% of teachers do not have sufficient qualifications to teach; (3)34% shortage
of
school
teachers;
(4)Uneven
distributions
of
teachers;
(5)Education
Development Index (EDI) is at 69th position out of 127 countries. (UNESCO,
2011)
The quality of education is the first indicator of Country Development
Rate. Therefore, educational development is the main factor for the national
development. As a developing country, the quality of education in Indonesia is
still low. It is caused by the learning quality that is not optimal yet. It is shown by
the low number of students’s learning outcomes in senior high school, especially
in mathematics. Generally in SMA Panca Budi Medan, 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 75. It is caused by the
teachers’s methods and the students’s activities. Based on that average grade’s
percentage, shown that teaching of mathematics has not been maximal yet to get a
good result. Which means still needs to be improved in order to minimize the
number of students whose grade is lower than KKM required by the school.
Nowadays, the government has been actively encouraging the education in
Indonesia. One of the government's efforts is curriculum development becomes
curriculum of 2013. This curriculum requires active students. The curriculum is
not knowledge-oriented only, but also on the attitude and psychomotor, it means
the education process is success. But when students show bad learning
achievements, attitude and psychomotor, it means that the educations process has
failed. It means that this curriculum requires teachers to measure the students’
characters, things that were never done, especially in SMA Panca Budi Medan.
Teaching activities in school is a part of the general educational activities,
which automatically will increase the students’ achievements. The higher number
of students who can reach the level of understanding and mastering the material,
the higher success of teaching is. Learning model recommended in the 2013
curriculum is problem-based learning, project-based learning and discovery
learning. With these models is expected the increasing grades of student learning
outcomes, is it cognitive, affective, or psychomotor?
In the learning process, teachers are required to encourage students to
learn actively, so learning becomes meaningful to students. In line with (Slameto,
2003) argues that in the process of teaching and learning, teachers should create a
lot of students’s activities in thinking and doing. Learning activities that are done
by the students themselves, the impressions will not go away, but thoughts,
processed and then released again in a different form. Or students will ask, ask
opinions, raises discussions with teachers. So students can run the command,
carry out a task, make charts, diagrams, and the essences of the lesson presented
by the teachers. When the students become active, then they have a
knowledge/science well.
Learning methematics will be meaningful to students if it is done in
accordance with the students’s initial knowledges. From the beginning of
knowledge, teachers provide materials/learning resources that correspond to the
basic competencies required. Then conditioned with the guidance of the teachers
to make students active in constructing their own knowledge. Learning will be
meaningful if teachers relate the new knowledge with some experiences which has
contained one of the important factors in learning mathematics.
Description above reinforce the researchers to conclude the learning
strategy in this model is one less variable trigger low student’s learning outcomes.
In an effort to improve students’s learning outcomes, required innovations in
learning mathematics. One step that can be done by the teachers as the learners
mentor is to choose the right learning model. The use of a less precise model of
learning can lead to be bored, lack of understanding the material, and finally may
decrease the motivations of participants in the study.
The main problem in formal education (school) is the low absorptive
capacity of the learners. This is proven by the result of the students’ learning
which is very low. Achievement is certainly the result of learning conditions that
are conventional and will not make the students aware of participants, how to
actually learn it. In other words, the learning process is still dominated by the
teachers and not provide access for the students to develop independently through
discovery in the process of thinking (Trianto, 2010).
Mathematics is one of those subjects that has a very close concepts related
with daily life. This means that the learning is not enough if just to teach
mathematics conceptually, but students also need to understand how to use the
concept significantly.
In learning mathematics, students must have comprehension, skills, and
knowledge which these aspects are known and can be done by teachers and
students. NCTM (in Effendi: 2012) states that the expected goals in learning
mathematics are to set of five standard process that must be owned by students
namely problem solving ability, communication ability, connection ability,
reasoning ability, and representation ability. As stated by Fadillah (2011) that
beside solving problem ability, reasoning, communication, and connection,
entering representation as component of standard process in Principles and
Standards for School Mathematics is very exact since for mathematical thinking
and communicating of mathematical ideas, the students need to represent on
various form of mathematical representation. Moreover, it can’t be denied that
mathematics objects are abstract so that to learn and understand abstract ideas
requires representation.
Meanwhile, Jones (in Fadillah: 2011) also explain three reason why
representation as a standard process, namely (1) Basic ability must be owned by
students to build a concept and mathematical thinking is doing translation on
various representations type smoothly; (2) Teacher should provide mathematical
ideas through various representations since the situation can provide enormous
influence to students in learning mathematics; and (3) Teachers should provide
various exercises to students since the students really need these exercises to build
their representations so that having ability and good in understanding the concept
and flexible can be used on problem solving.
This is consistent with the opinions expressed by Yuniawatika (2011)
which said that students can be encouraged to find and create various
representations that could be used as a thinking way in expressing their
knowledge from abstract to concrete and the situation can be concluded that
mathematical representations ability as a way to increase and express
mathematical thinking ability of students.
Rahmi (in Hutagaol: 2013) said that diagram, picture, table, chart,
mathematical statement, written text, also combination of all as representations
variety can be used by students in expressing mathematical ideas. Variety of
representations such as table, picture, graph, and another symbol are part of
mathematics that can’t be separated since mathematical representations is a part of
mathematics.
But based on last situation, students mathematical representations ability
in school is still less considered since many students who don’t understand the
mathematical representations ability.
As stated by Hudiono, (in Fadillah: 2011) in his research on learning
mathematics at Junior High School, concluded that the lack of teacher’s
knowledge and student’s learning habits using direct instruction has not been
possible to maximally develop representation power of students. Accordingly,
NCTM (in Fadillah: 2011) also said that mathematical representation ability of
students is very limited. It is shown by the student’s ability in solving
mathematical problems dominated in symbolic representations, so they pay less
attention to the other representations. As’ari (in Fatayati: 2012) also said the
students should not only understand the abstract ideas contained in mathematics
but also must express the abstract ideas in different forms of representation which
is easy to understand for the students such as in the form of images, symbols, and
words since the representation is one of the alternative forms can be used to solve
problems in mathematics.
From problem result which is addressed to some students in grade XI at
SMA Panca Budi Medan, it was found that the students’s mathematical
representation ability is still less. Shown by students’ answers, many of them have
not been able yet to draw a tabular (table frequency).
The following are questions and students’ answers which were given in
order to know the mathematical representation ability of students;
1. The scores of 45 students on a 20-point Science quiz are as follows:
17 20 15 18 19 16 11 10 15 16 12 12 13 14 11 10 14 13 12 11 13 15 14
10 15 16 17 17 18 20 20 18 19 19 18 17 16 15 12 12 13 14 15 19 20
a. Draw a frequency table for the grouped data !
b. How many students will pass the science quiz at the standard score > 15?
c. Determine the value of mean !
2. Graph for the number of students based on their ethnic background in
School Q
Figure 1.1 The Question of Initila Test no.2
Describe the brief of the overall pattern of the number of students based on
their ethnicities background ?
Figure 1.2 The Student Answer of Intial Test no.1
Figure 1.3 The Student Answer of Intial Test no.2
From 30 students who answer the question, can be seen that 60,5% of
them have not been able yet to build their visual representations in making table
exactly, while 75,35% 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.
Disinterest of student in mathematics subject caused by the student’s
ignorance of the usefulness of materials studied in mathematics into their daily
lives. In addition, teachers only focus on books when teaching.
A wide range of innovative learning strategies that are considered the
development of students’s cognitive abilities and independence. One model is
Problem Based Learning (PBL). PBL is a learning model that engages students to
solve a problem through the stages of the scientific method so that students may
learn the knowledge related to the problem and having skills to solve the problem.
Objectivities to be achieved in Problem Based Learning is a student’s ability to
think critically, analysis, systematic and logical to look for an alternative solution
through the exploration of the empirical data in order to develop a scientific
attitude (Sanjaya, 2008). So, the learning goals expected to be achieved is to
improve student’s learning outcomes and to develop a scientific attitude of the
student.
Problem Based Learning model begins with problem, then students deepen
their knowledge about what they have already known and what they need to know
to solve the problem. (Duch in Riyanto, 2010) states that: Problem Based
Learning is learning model that exposes learners to the challenge of “learning to
learn”. Students actively work in groups to seek the solutions of problems. The
problem is as a reference for students to formulate, analyze, and solve. Problem
Based Learning model is intended to develop students’s critical thingking,
analytical, and to find and to use appropriate resources for learning.
In this study, the role of teachers is asking problems, providing
encouragement, motivation, and teaching matherials, as well as providing the
necessary facilities for learners in the process of reasoning. In addition, teachers
also provide support the finding and intellectual development of students.
In the learning of problem based learning students are required to
undertake the process of solving problems presented by digging out as much. This
experience is indispensable in everyday of life where the growth of mindset and
work patterns of a person depends on how he positioned himself in the study.
Problem Based Learning is learning using a real problem (the fact) that is
presented at the beginning of learning. First step is understanding of the problem
so that the necessary reasoning abilities, and then probed for known solutions to
these problems.
Based on problems above, the writer is interested to do research with the
title “The difference of Students’s Mathematical Representation Ability by
using Instruction of Probelm Based Learning and Direct Instruction
in
Grade X ’’
1.2
Problem Identification
a. Mathematical representation ability of student is still low.
b. Student still have difficulties in solving mathematical represetation
tests.
c. Teachers learning model used is still less variation and the learning
process is still conventional.
1.3
Problem limitation
Based on the problems identification above, there is a wide scope of
issues, so this research is limited to know the following :
1. The subject material that will be taught is statistics
2. Teaching model that will be applied in this research is problem based
learning model
3. The ability will be measured is representation ability
1.4.
Problem Formulation
Problem formulation in this research is: “Is students’s mathematical
representation ability using problem based learning higher than direct instruction
in grade X SMA Panca Budi Medan Academic Year 2014/2015?”
1.5.
Research Objective
Research objective in this research is: To know is students’s mathematical
representation ability using problem based learning higher than direct instruction
in grade X SMA Panca Budi Medan Academic Year 2014/2015.
1.6.
Research Benefits
The expected benefits of this research are:
1. Mathematics teachers may used the material consideration and input, in
order to choose one of mathematics’s alternative learning model in school.
2. Prospective teachers may use the benefits of this research to solve the
problems and difficulties which is often raised in the school. So, they can
be the professional teachers.
3. Students may use the benefits of this research as references and
encouragements to improve the students’s mathematical representation of
students in order to learn mathematics.
4. Researchers may use the benefits of this research to increase the
knowledge about the problems and difficulties which is often arised in the
school.
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’s 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. Problem Based Learning is a learning method based on the principle which
is the issue (problem) may be used as a starting point to obtain or to
integrate the new science (knowledge). Thus, there is a problem which is
used as a meaning for students to learn something which can contribute
their knowledge.
CHAPTER V
CONCLUSION AND SUGGESTION
5.1 Conclusion
Based on the result of research and discussion can be conclude that
students’ mathematical representation ability by using instruction of probelm
based learning higher than direct instruction in grade X SMA Panca Budi Medan
5.2 Suggestion
Based on the result of research and the above conclusion, then researcher
submits some suggestions, as follow:
1.
Probelm based learning Model can be as consederation to teachers in senior
high school to develop students’ mathematical representation ability.
2.
For further researcher, result and instrument of this research can be used as
consideration to implement instruction of problem based learning in different
class level topic.
3.
Learning process of mathematics by using instruction of problem based
learning needs longer time since in its learning, students receive information
from teacher indirectly, so that it is needed preparation and used time
effectively in its implementation.
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REPRESENTATION ABILITY BY USING INSTRUCTION
OF PROBLEM BASED LEARNING AND DIRECT
INSTRUCTION IN GRADE X
by:
Elfan Syahputra
IDN. 4103312002
Bilingual Mathematics Education
Thesis
Submitted in Fulfill of the Requirements for the degree of
Sarjana Pendidikan
DEPARTMENT OF MATHEMATICS
FACULTY OF MATHEMATICS AND SCIENCES
UNIVERSITAS NEGERI MEDAN
MEDAN
2015
THE DIFFERENCE OF STUDENTS’ MATHEMATICAL REPRESENTATION
ABILITY BY USING INSTRUCTION OF PROBLEM BASED LEARNING
AND DIRECT INSTRUCTIOIN IN GRADE X
Elfan Syahputra (IDN 4103312002)
ABSTRACT
This research is quasi-experiment. The purpose of this research is to
know whether students’ mathematical representation by using instruction of
problem based learning classroom is higher than direct instruction classroom at
SMA Swasta Panca Budi Medan.The population of this research is students of
SMA Swasta Panca Budi Medan which consists of 14 classes, whereas the sample
consists of 2 classes, namely, X-MS1 as experimental class consists of 32 students
and X-MSA as control class consists of 30 students. Experimental class used
Problem Based Learning and control class used Direct Instruction. Collecting data
technique of this research is mathematical representation test given in the end of
learning either in experimental class or control class. The type of this test is essay
test. Before doing hypothesis test, the normality and the homogeneity test should
be done. The result of those tests, sample was taken from normal distributed and
homogeneous population. The data analysis of experimental class by using t-test
with significance level α = 0.05, it was obtained that tcalculation > ttable then H0 is
rejected and Ha is accepted.It can be concluded that students’ mathematical
representation ability by using instruction of problem based learning is higher
than direct instruction in grade X. The research that has been done, researcher
suggested that Problem based learning can be as consideration to teachers in
enhancing senior high school students’ mathematical representation ability.
Teacher intends to use problem based learning, needed preparation and used time
effectively in its implementation. The result and instrument of this research can be
used as consideration to implement problem based learning in a different class
grades and subjects for the future researchers.
PREFACE
Alhamdulillah, praise and gratitude to Almighty God, Allah SWT for His
blessings and mercy that provide health and wisdom to finish this thesis with the
title “ The Difference of Mathematical Representation Ability by Using
Instruction of Problem Based Learning and Direct Instruction in Grade X.
I want to express special thanks to my beloved family, my father, Aspan
Effendi, my mother, Siti Elfina, and my blood sister, Siska Amelia and Fatia
Aulia, for their everlasting support, patience, love, and their prayer throughout
this process. I dedicate this thesis for them. My warm thanks go to my Big
Family, for their support prayer love and care.
Special thanks are extended to my supervisor, Mr. Prof. Dr. Mukhtar,
M.Pd. for his time, patience, a lot of advice, constructive criticism, questioning
and probing, and constant encouragement that were deeply appreciated and
valued. I also extend special thanks to Mr. Prof. Dr. Edi Syahputra, Mrs. Dra.
Katrina Samosir, M.Pd., and Mr. Denny Haris, S.Si., M.Pd. as my examiners who
gave me insightful suggestions from compiling proposal till this thesis
compilation was done. And also for my academic counselor, Mr. H. Banjarnahor,
M.Pd., I want to express my special thanks for his guidance and suggestion during
the lecture period.
I would like to thank Mr. Prof. Dr. Ibnu Hajar, M.Si. as the Rector of
Medan State University as well as the staff employees in rector’s office, Mr. Prof.
Dr. Motlan, M.Sc., Ph.D. as the dean of Mathematics and Sciences Faculty, Mr.
Prof. Dr. rer.nat. Binari M., M.Si. as the coordinator of bilingual class, Mr. Dr.
Edi Surya., as the chief of Mathematics Department, Mr. Zul Amry, Ph.D as the
chief of Study Program of Mathematics Department, Mr. Drs. Yasifati Hia, M.Si.,
and all of staff employees in Mathematics Department of Mathematics and
Sciences Faculty that were involved in this thesis compilation.
I also express my deep gratitude to the principal of SMA Swasta Panca
Budi Medan, mathematics teacher, Mrs. Ilma S.Pd, and all of teachers in SMA
Swasta Panca Budi Medan, that gave me permit to conduct the research, helped
me from the beginning till the research done, and gave me support throughout this
process. I also want to thank all of students in class MS-1 and MS-A for their
collaboration, support, and passion when I was teaching them.
I would like to thank my best friends, Abdul, Anggi, Dian, Dwi, Erlyn,
Elfan, Kiki, Falni, Mila, Lia, Meiva, Martyanne, Melin, Surya, Nelly, Petra, Riny,
Rully, Sartika, Sheila, Siti, Uli, and Mimi, and also especially for my hometown
best friends Eri Syahputra, Taufik Ismail, M. Yogie A.Md, friends of best keles
Risva, Mora ,friends of PPLT in Kisaran, and all my friends i did not mention for
their support and love throughout this process. It would not have been possible
without each of you.
I have done the best for finishing this thesis, but I realize that this thesis
has its shortcomings either in contents, grammar, or technical writing. Hence, I
hope the constructive criticism and suggestion from the reader for the perfection
of this thesis. I hope this thesis could be useful for further researchers and useful
in enrichment the knowledge.
Medan, December
Writer,
Elfan Syahputra
Idn. 4103312002
2014
BIBLIOGRAPHY
Elfan Syahputra was born in Medan, the 9th July 1992. Her father named
Aspan Effendi and her mother named Siti Elfina. He is the first of 3 children.
In 1997, he accepted in TK Swasta Free Methodist Medan, and graduated
in 1998. In 1998, he continued study to SD Free Methodist Medan and graduated
in 2004. In 2004, he continued study to SMP Swasta Al-Azhar Medan, and
graduated in 2007. In 2007, he continued study to SMA Swasta Panca Budi
Medan, and graduated in 2010. In 2010, she accepted in Bilingual Class
Mathematics Education, Department of Mathematics, Faculty of Mathematics and
Sciences, Universitas Negeri Medan, and graduated on Wednesday, December
10th 2014.
CONTENT
Agreement Sheet
ii
Bibliography
iii
Abstract
iv
Preface
v
Content
vii
List of Figure
x
List of Table
xi
List of Appendix
xii
CHAPTER 1 INTRODUCTION
1.1.
The Background of the Research
1
1.2.
Problem Identification
8
1.3.
Problem Limitation
8
1.4.
Problem Formulation
9
1.5.
Research Objective
9
1.6.
Research Benefit
9
1.7.
Operational Definitions
9
CHAPTER II LITERATURE STUDY
2.1.
Theoritical Framework
11
2.2.
Representation
11
2.2.1.
Representation in Mathematics
12
2.2.2.
Mathematical Representation Ability
15
2.3.
Problem Based Learning Model
19
2.3.1.
Learning Model
19
2.3.2.
Problem Based Learning Model
20
2.3.3.
Characteristics of Problem Based Learning Model
22
2.3.4.
Advantages and Weakness of Problem Based Learning
24
2.3.5.
Stages of Problem Based Learning
25
2.3.6.
Direct Instruction
26
2.4.
Statistics
27
2.4.1.
Collecting and Organizing Data
27
2.4.2.
Grapichal representation
27
2.4.3.
Averages-Mean, Median, and Mode
29
2.4.4.
Dispersion
30
2.4.5.
Grouped Frequency Distribution
30
2.4.6.
Mean, Median, and Mode of grouped data
31
2.5.
Conceptual Framework
32
2.6.
Hypothesis
33
CHAPTER III
3.1.
Research Location and Research Objects
34
3.2.
Population and Sample of Research
34
3.2.1.
Population
34
3.2.2.
Sample
34
3.3.
Research Variables
34
3.3.1.
Independent Variable
35
3.3.2.
Dependent Variable
35
3.4.
Type and Design of Research
35
3.4.1.
Type of Reseach
35
3.4.2.
Design of Research
35
3.5.
Procedure of Research
36
3.6.
Instrument of Research
37
3.6.1.
Initial Test
38
3.6.2.
Test of Student’s Mathematical Representation Ability
38
3.7.
Data Analysis Tecnique
43
3.7.1.
Calculating Mean Score
43
3.7.2.
Calculating of Deviation Standard
44
3.7.3.
Normality Test
44
3.7.4.
Homogenety Test
44
3.7.5.
Hypotheses Test
45
CHAPTER IV
4.1.
Research Result Description
4.1.1.
The Description of Students’ Mathematical Representation
46
Ability
46
4.1.2.
The Description of Mathematical Representation Test
48
4.2.
Analysis of Research Data
49
4.2.1.
Normality Test
49
4.2.2.
Homogeneity Test
50
4.2.3.
Compare Means Test (One-tailed)
51
4.2.4.
Analysis of Observation Sheet
53
4.3.
Research Discussion
56
CHAPTER V
5.1.
Conclusion
60
5.2.
Suggestion
60
REFERENCE
61
FIGURE LIST
Figure 1.1. The Question of Inital Test no.2
6
Figure 1.2. The Student Answer of Intial test no.1
6
Figure 1.2. The Student Answer of Intial test no.2
7
Figure 2.1. The Relationship Between Internal and External
Representation In Developing Child’s Understanding of
the Concept of Numeracy
14
Figure 2.2. Bar Graphs
28
Figure 2.3. Pie Charts
28
Figure 2.4. Line Charts
29
Figure 3.1. Procedure of Research
37
Figure 4.1. Histogram Post-test of Students’ Mathematical
Representation Ability
47
TABLE LIST
Table 2.1. Operational Form of Mathematical Representation Ability
17
Table 2.2. Indicator of Mathematical Representation Ability
18
Table 2.3. Stages of Problem Based Learning
25
Table 2.4. Syntax Of The Direct Instruction Learning Model
26
Table 3.1. The Blueprint of Mathematical Representation Ability Problem 39
Table 3.2. The Rubric of Mathematical Representation Ability Problem
40
Table 4.1. The Result Post-test of Mathematical Representation Ability
47
Table 4.2.Mean Percentage of Experimental and Control Class of
Each Indicator
49
Table 4.3 One-sample Kolmogorov Smirnov Test
50
Table 4.4 Homogeneity Variance Test
51
Table 4.5 Independent Sample t-test
52
Table4.6 Observation Sheet
53
APPENDIX LIST
Appendix 1
The Blueprint Of Mathematical Representation Ability
Initial Test
64
Appendix 2
Initial Test Of Mathematical Representation Ability
65
Appendix 3
Alternative Solution Of Mathematical Representation
Ability Initial Test
66
Appendix 4
Lesson Plan Direct Instruction Method
67
Appendix 5
Lesson Plan Instructional of Problem Based Learning
Model
81
Appendix 6
Worksheet
104
Appendix 7
Guidelines For Scoring of Mathematical Representation
Ablity Test
Appendix 8
Appendix 9
111
The Blue Print of Student’s Mathematical Representation
Ability Test
114
Test Of Mathematical Representation Ability (Post Test)
115
Appendix 10 Alternative Solution Of Mathematical Representation
Ability (Post Test)
117
Appendix 11 Posttest Score Of Experimental And Control Class
120
Appendix 12 Normality Test
122
Appendix 13 Homogeneity Test
123
Appendix 14 Independent Sample t-test
124
Appendix 15 Documentation
126
Appendix 16 T-table Value of t-distribution
129
CHAPTER 1
INTRODUCTION
1.1 The Background of the Research
The most important thing to increase the progress of a nation is human
resources. Indonesia is categorized as a developing country and the quality of
National Education has very wide impacts in all aspects of human’s life. By
education, human will be able to solve various problems and difficulties of life.
Therefore, the purposes of education are as a self-builder, shaper, and developer.
Globalization requires people to have adequate education in order to compete.
Unfortunately, from the following facts below, Indonesia still has some serious
educational problems, such as (1)Every minute, four children out of school;
(2)54% of teachers do not have sufficient qualifications to teach; (3)34% shortage
of
school
teachers;
(4)Uneven
distributions
of
teachers;
(5)Education
Development Index (EDI) is at 69th position out of 127 countries. (UNESCO,
2011)
The quality of education is the first indicator of Country Development
Rate. Therefore, educational development is the main factor for the national
development. As a developing country, the quality of education in Indonesia is
still low. It is caused by the learning quality that is not optimal yet. It is shown by
the low number of students’s learning outcomes in senior high school, especially
in mathematics. Generally in SMA Panca Budi Medan, 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 75. It is caused by the
teachers’s methods and the students’s activities. Based on that average grade’s
percentage, shown that teaching of mathematics has not been maximal yet to get a
good result. Which means still needs to be improved in order to minimize the
number of students whose grade is lower than KKM required by the school.
Nowadays, the government has been actively encouraging the education in
Indonesia. One of the government's efforts is curriculum development becomes
curriculum of 2013. This curriculum requires active students. The curriculum is
not knowledge-oriented only, but also on the attitude and psychomotor, it means
the education process is success. But when students show bad learning
achievements, attitude and psychomotor, it means that the educations process has
failed. It means that this curriculum requires teachers to measure the students’
characters, things that were never done, especially in SMA Panca Budi Medan.
Teaching activities in school is a part of the general educational activities,
which automatically will increase the students’ achievements. The higher number
of students who can reach the level of understanding and mastering the material,
the higher success of teaching is. Learning model recommended in the 2013
curriculum is problem-based learning, project-based learning and discovery
learning. With these models is expected the increasing grades of student learning
outcomes, is it cognitive, affective, or psychomotor?
In the learning process, teachers are required to encourage students to
learn actively, so learning becomes meaningful to students. In line with (Slameto,
2003) argues that in the process of teaching and learning, teachers should create a
lot of students’s activities in thinking and doing. Learning activities that are done
by the students themselves, the impressions will not go away, but thoughts,
processed and then released again in a different form. Or students will ask, ask
opinions, raises discussions with teachers. So students can run the command,
carry out a task, make charts, diagrams, and the essences of the lesson presented
by the teachers. When the students become active, then they have a
knowledge/science well.
Learning methematics will be meaningful to students if it is done in
accordance with the students’s initial knowledges. From the beginning of
knowledge, teachers provide materials/learning resources that correspond to the
basic competencies required. Then conditioned with the guidance of the teachers
to make students active in constructing their own knowledge. Learning will be
meaningful if teachers relate the new knowledge with some experiences which has
contained one of the important factors in learning mathematics.
Description above reinforce the researchers to conclude the learning
strategy in this model is one less variable trigger low student’s learning outcomes.
In an effort to improve students’s learning outcomes, required innovations in
learning mathematics. One step that can be done by the teachers as the learners
mentor is to choose the right learning model. The use of a less precise model of
learning can lead to be bored, lack of understanding the material, and finally may
decrease the motivations of participants in the study.
The main problem in formal education (school) is the low absorptive
capacity of the learners. This is proven by the result of the students’ learning
which is very low. Achievement is certainly the result of learning conditions that
are conventional and will not make the students aware of participants, how to
actually learn it. In other words, the learning process is still dominated by the
teachers and not provide access for the students to develop independently through
discovery in the process of thinking (Trianto, 2010).
Mathematics is one of those subjects that has a very close concepts related
with daily life. This means that the learning is not enough if just to teach
mathematics conceptually, but students also need to understand how to use the
concept significantly.
In learning mathematics, students must have comprehension, skills, and
knowledge which these aspects are known and can be done by teachers and
students. NCTM (in Effendi: 2012) states that the expected goals in learning
mathematics are to set of five standard process that must be owned by students
namely problem solving ability, communication ability, connection ability,
reasoning ability, and representation ability. As stated by Fadillah (2011) that
beside solving problem ability, reasoning, communication, and connection,
entering representation as component of standard process in Principles and
Standards for School Mathematics is very exact since for mathematical thinking
and communicating of mathematical ideas, the students need to represent on
various form of mathematical representation. Moreover, it can’t be denied that
mathematics objects are abstract so that to learn and understand abstract ideas
requires representation.
Meanwhile, Jones (in Fadillah: 2011) also explain three reason why
representation as a standard process, namely (1) Basic ability must be owned by
students to build a concept and mathematical thinking is doing translation on
various representations type smoothly; (2) Teacher should provide mathematical
ideas through various representations since the situation can provide enormous
influence to students in learning mathematics; and (3) Teachers should provide
various exercises to students since the students really need these exercises to build
their representations so that having ability and good in understanding the concept
and flexible can be used on problem solving.
This is consistent with the opinions expressed by Yuniawatika (2011)
which said that students can be encouraged to find and create various
representations that could be used as a thinking way in expressing their
knowledge from abstract to concrete and the situation can be concluded that
mathematical representations ability as a way to increase and express
mathematical thinking ability of students.
Rahmi (in Hutagaol: 2013) said that diagram, picture, table, chart,
mathematical statement, written text, also combination of all as representations
variety can be used by students in expressing mathematical ideas. Variety of
representations such as table, picture, graph, and another symbol are part of
mathematics that can’t be separated since mathematical representations is a part of
mathematics.
But based on last situation, students mathematical representations ability
in school is still less considered since many students who don’t understand the
mathematical representations ability.
As stated by Hudiono, (in Fadillah: 2011) in his research on learning
mathematics at Junior High School, concluded that the lack of teacher’s
knowledge and student’s learning habits using direct instruction has not been
possible to maximally develop representation power of students. Accordingly,
NCTM (in Fadillah: 2011) also said that mathematical representation ability of
students is very limited. It is shown by the student’s ability in solving
mathematical problems dominated in symbolic representations, so they pay less
attention to the other representations. As’ari (in Fatayati: 2012) also said the
students should not only understand the abstract ideas contained in mathematics
but also must express the abstract ideas in different forms of representation which
is easy to understand for the students such as in the form of images, symbols, and
words since the representation is one of the alternative forms can be used to solve
problems in mathematics.
From problem result which is addressed to some students in grade XI at
SMA Panca Budi Medan, it was found that the students’s mathematical
representation ability is still less. Shown by students’ answers, many of them have
not been able yet to draw a tabular (table frequency).
The following are questions and students’ answers which were given in
order to know the mathematical representation ability of students;
1. The scores of 45 students on a 20-point Science quiz are as follows:
17 20 15 18 19 16 11 10 15 16 12 12 13 14 11 10 14 13 12 11 13 15 14
10 15 16 17 17 18 20 20 18 19 19 18 17 16 15 12 12 13 14 15 19 20
a. Draw a frequency table for the grouped data !
b. How many students will pass the science quiz at the standard score > 15?
c. Determine the value of mean !
2. Graph for the number of students based on their ethnic background in
School Q
Figure 1.1 The Question of Initila Test no.2
Describe the brief of the overall pattern of the number of students based on
their ethnicities background ?
Figure 1.2 The Student Answer of Intial Test no.1
Figure 1.3 The Student Answer of Intial Test no.2
From 30 students who answer the question, can be seen that 60,5% of
them have not been able yet to build their visual representations in making table
exactly, while 75,35% 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.
Disinterest of student in mathematics subject caused by the student’s
ignorance of the usefulness of materials studied in mathematics into their daily
lives. In addition, teachers only focus on books when teaching.
A wide range of innovative learning strategies that are considered the
development of students’s cognitive abilities and independence. One model is
Problem Based Learning (PBL). PBL is a learning model that engages students to
solve a problem through the stages of the scientific method so that students may
learn the knowledge related to the problem and having skills to solve the problem.
Objectivities to be achieved in Problem Based Learning is a student’s ability to
think critically, analysis, systematic and logical to look for an alternative solution
through the exploration of the empirical data in order to develop a scientific
attitude (Sanjaya, 2008). So, the learning goals expected to be achieved is to
improve student’s learning outcomes and to develop a scientific attitude of the
student.
Problem Based Learning model begins with problem, then students deepen
their knowledge about what they have already known and what they need to know
to solve the problem. (Duch in Riyanto, 2010) states that: Problem Based
Learning is learning model that exposes learners to the challenge of “learning to
learn”. Students actively work in groups to seek the solutions of problems. The
problem is as a reference for students to formulate, analyze, and solve. Problem
Based Learning model is intended to develop students’s critical thingking,
analytical, and to find and to use appropriate resources for learning.
In this study, the role of teachers is asking problems, providing
encouragement, motivation, and teaching matherials, as well as providing the
necessary facilities for learners in the process of reasoning. In addition, teachers
also provide support the finding and intellectual development of students.
In the learning of problem based learning students are required to
undertake the process of solving problems presented by digging out as much. This
experience is indispensable in everyday of life where the growth of mindset and
work patterns of a person depends on how he positioned himself in the study.
Problem Based Learning is learning using a real problem (the fact) that is
presented at the beginning of learning. First step is understanding of the problem
so that the necessary reasoning abilities, and then probed for known solutions to
these problems.
Based on problems above, the writer is interested to do research with the
title “The difference of Students’s Mathematical Representation Ability by
using Instruction of Probelm Based Learning and Direct Instruction
in
Grade X ’’
1.2
Problem Identification
a. Mathematical representation ability of student is still low.
b. Student still have difficulties in solving mathematical represetation
tests.
c. Teachers learning model used is still less variation and the learning
process is still conventional.
1.3
Problem limitation
Based on the problems identification above, there is a wide scope of
issues, so this research is limited to know the following :
1. The subject material that will be taught is statistics
2. Teaching model that will be applied in this research is problem based
learning model
3. The ability will be measured is representation ability
1.4.
Problem Formulation
Problem formulation in this research is: “Is students’s mathematical
representation ability using problem based learning higher than direct instruction
in grade X SMA Panca Budi Medan Academic Year 2014/2015?”
1.5.
Research Objective
Research objective in this research is: To know is students’s mathematical
representation ability using problem based learning higher than direct instruction
in grade X SMA Panca Budi Medan Academic Year 2014/2015.
1.6.
Research Benefits
The expected benefits of this research are:
1. Mathematics teachers may used the material consideration and input, in
order to choose one of mathematics’s alternative learning model in school.
2. Prospective teachers may use the benefits of this research to solve the
problems and difficulties which is often raised in the school. So, they can
be the professional teachers.
3. Students may use the benefits of this research as references and
encouragements to improve the students’s mathematical representation of
students in order to learn mathematics.
4. Researchers may use the benefits of this research to increase the
knowledge about the problems and difficulties which is often arised in the
school.
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’s 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. Problem Based Learning is a learning method based on the principle which
is the issue (problem) may be used as a starting point to obtain or to
integrate the new science (knowledge). Thus, there is a problem which is
used as a meaning for students to learn something which can contribute
their knowledge.
CHAPTER V
CONCLUSION AND SUGGESTION
5.1 Conclusion
Based on the result of research and discussion can be conclude that
students’ mathematical representation ability by using instruction of probelm
based learning higher than direct instruction in grade X SMA Panca Budi Medan
5.2 Suggestion
Based on the result of research and the above conclusion, then researcher
submits some suggestions, as follow:
1.
Probelm based learning Model can be as consederation to teachers in senior
high school to develop students’ mathematical representation ability.
2.
For further researcher, result and instrument of this research can be used as
consideration to implement instruction of problem based learning in different
class level topic.
3.
Learning process of mathematics by using instruction of problem based
learning needs longer time since in its learning, students receive information
from teacher indirectly, so that it is needed preparation and used time
effectively in its implementation.
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