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
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
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.
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 2014 Writer,
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.
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
2.3.6. 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
3.7.5. Hypotheses Test 45
CHAPTER IV
4.1. Research Result Description 46
4.1.1. The Description of Students’ Mathematical Representation
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
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
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
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 111
Appendix 8 The Blue Print of Student’s Mathematical Representation
Ability Test 114
Appendix 9 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
CHAPTER 1 INTRODUCTION
1.1The 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.
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, orpsychomotor?
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.
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.
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.
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
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.
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.
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
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.
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