ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM.
Maya L Hutapea, 2014
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE
SUBJECT OF PYTHAGOREAN THEOREM
THESIS
Submitted to Fulfill The Requirement for Getting The Master of Postgraduate Mathematics Education
by
MAYA L HUTAPEA 1202122
Mathematics Education Department SCHOOL OF POSTGRADUATE STUDIES
INDONESIA UNIVERSITY OF EDUCATION
BANDUNG
2014
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Maya L Hutapea, 2014
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu
==================================================================
Analysis of Students’ Learning O sta les
on The Subject of Pythagorean Theorem
Oleh Maya L Hutapea
S.Pd Universitas Negeri Medan, 2012
Sebuah Tesis yang diajukan untuk memenuhi salah satu syarat memperoleh gelar Magister Pendidikan (M.Pd.) pada Jurusan Pendidikan Matematika
© Maya L Hutapea
Universitas Pendidikan Indonesia Juni 2014
Hak Cipta dilindungi undang-undang.
Tesis ini tidak boleh diperbanyak seluruhya atau sebagian,
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Maya L Hutapea, 2014
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu
Approved and authorized by the supervisors:
Supervisor IProf. Dr. H. Didi Suryadi, M.Ed NIP. 1958020111984031001
Supervisor II
Dr. Elah Nurlaelah, M.Si NIP. 196411231991032002
Acknowledge
Chairman of the Department of Mathematics Education
Drs. Turmudi, M.Ed., M.Sc., Ph.D NIP. 196101121987031003
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Maya L Hutapea, 2014
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
A Descriptive Exploratory Research on Grade Nine of Junior High School Students in Bandung
Maya L Hutapea 1202122 ABSTRACT
This research departed from many errors made by junior high school students in solving Pythagorean theorem problems. These errors can be seen from mistakes they have done in solving the given problem. Many methods and models of learning used by teachers from various sources, but the models are not necessarily an instruction that can be applied successfully in all types of students. It requires a depth analysis of learning obstacles in students to find the actual problems experienced by students. From the result of the analysis it will be designed an effective and efficient instructional material to eliminate the learning obstacles found. The learning obstacles will be classified into three learning obstacles, which are: ontogenic obstacles, epistemological obstacles, and didactical obstacles. This research using a descriptive exploratory to describe the symptom and the phenomena occur within students in solving the problem. The research data obtained from a test of Pythagorean theorem problems and the result of interviews that then extracted to the instructional material design in the mathematics textbooks. The analysis results will be gathered and concluded in the way to design an instructional material that based on the didactical design research. This instructional material was designed by creating a situation where students are able to think and to do discovery of their own knowledge. In every situation given, it has been prepared the anticipation of didactical and pedagogical that consists of students’ response and the feedbacks from the teacher. This instructional material is not a standard teaching material yet, means it will continue to be evaluated and improved in order to obtain the best result.
Keywords: learning obstacles, didactical design research, Pythagorean theorem, instructional material design
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Maya L Hutapea, 2014
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu
TABLE OF CONTENT
Page
DECLARATION i
PREFACE ii
ACKNOWLEDGEMENTS iii
ABSTRACT v
TABLE OF CONTENT vi
LIST OF PICTURES xii
LIST OF TABLES xiv
LIST OF ATTACHMENTS xv
CHAPTER I INTRODUCTION
1.1 The Background 1
1.2 The Problem Identification and The Problem Formulation 7 1.3 The Research Objectives 8 1.4 The Research Benefits 8 CHAPTER II STUDY OF LITERATURE
2.1 Learning Obstacles 9
2.1.1 Ontogenic Obstacles 10
2.1.2 Didactical Obstacles 11
2.1.3 Epistemological Obstacles 11
2.2 Didactical Design Research 11
2.2.1 Didactical 11
2.2.2 Design Research 15
2.2.3 Didactical Design Research 16
2.3 Relevant Learning Theories 17
2.3.1 Piaget Learning Theory 17
2.3.2 Gagne Learning Theory 19
2.3.3 Ausubel Learning Theory 20
2.3.4 The Pythagorean Theorem 20
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ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu CHAPTER III RESEARCH METHOD
3.1 Research Method 25
3.2 Purposive Sampling 25
3.3. Research Procedures 25
3.3.1 The Preparation Phase 25
3.3.2 The Implementation Phase 26
3.3.3 The Planning Phase 26
3.4 Research Instrument 28
3.4.1 The Data Collection Techniques 28
3.4.1.1 Test 28
3.4.1.1.1 The Characteristic of Problem One 28 3.4.1.1.2 The Characteristic of Problem Two 28 3.4.1.1.3 The Characteristic of Problem Three 29 3.4.1.1.4 The Characteristic of Problem Four 29 3.4.1.1.5 The Characteristic of Problem Five 29 3.4.1.1.6 The Characteristic of Problem Six 30 3.5 Data Analysis Procedure 31
3.5.1 Data Analysis Technique 31
3.5.1.1 Data Reduction 31
3.5.1.2 Presentation of Data 31
3.5.1.3 Conclusion Withdrawal 32
3.5.2 Data Analysis Plan 32
3.5.3 Validation Method 32
3.5.3.1 Credibility Test 32
3.5.3.2 Transferability Test 32
3.5.3.3 Dependability Test 33
3.5.3.4 Confirmability Test 33
CHAPTER IV RESEARCH RESULT AND DISCUSSION
4.1 The General Descriptions of The Data 35 4.2 The Category of Data Based on Ontogenic Obstacles 36 4.2.1 The Influence of Algebraic Thinking on Ontogenic
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Maya L Hutapea, 2014
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu
Obstacles 37
4.2.1.1 The First Obstacle in Problem Three 38 4.2.1.2 The Second Obstacle in Problem Three 40 4.2.1.3 The Third Obstacle in Problem Three 41 4.2.1.4 The Third Obstacle in Problem Three 42 4.2.2 The Influence of Geometry Ability on Ontogenic
Obstacles 43
4.2.2.1 The First Obstacle in Problem Five 45 4.2.2.2 The Second Obstacle in Problem Five 45 4.3 The Category of Data Based on Epistemological Obstacles 46 4.3.1 The Influence of Misconception on Epistemological
Obstacles 46
4.3.1.1 The First Obstacle in Problem One 47 4.3.1.2 The Second Obstacle in Problem One 49 4.3.1.3 The Third Obstacle in Problem One 50 4.3.1.4 The Third Obstacle in Problem One 50 4.3.1.5 The Third Obstacle in Problem One 51 4.3.2. The Influence of Implicit Information on
Epistemological Obstacles 53
4.3.2.1 The First Obstacle in Problem Three 54 4.3.2.2 The Second Obstacle in Problem Three 54 4.3.2.3 The Third Obstacle in Problem Three 55 4.3.3 The Influence of Visual Representation on
Epistemological Obstacles 56
4.3.3.1 The First Obstacle in Problem Three 57 4.3.3.2 The Second Obstacle in Problem Three 59 4.3.3.3 The Third Obstacle in Problem Three 59 4.4 Category of Data Based on Didactical Obstacles 63 4.4.1 The Influence of Verbalistis on Didactical Obstacles 63 4.4.1.1 The First Obstacle in Problem Two 66 4.4.1.2 The Second Obstacle in Problem Three 67
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Maya L Hutapea, 2014
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu
4.4.1.3 The Third Obstacle in Problem Three 68 4.4.1.4 The Fourth Obstacle in Problem Three 69 4.4.1.5 The Fifth Obstacle in Problem Three 69 4.5 The Students’ Outcome Space of The Problems 71 4.5.1 The Outcome Space of Ontogenic Obstacles 71 4.5.2 The Outcome Space of Epistemological Obstacles 73 4.5.3 The Outcome Space of Didactical Obstacles 75 4.6 The Extraction of Research Findings and Mathematics Textbooks 76 4.6.1 The Analysis of Instructional Material Design of
Pythagorean Theorem in BPM1 77
4.6.2 The Analysis of Instructional Material Design of
Pythagorean Theorem in BPM2 78
4.6.3 The Analysis of Instructional Material Design of
Pythagorean Theorem in BPM3 80
4.6.4 The Analysis of Instructional Material Design of
Pythagorean Theorem in BPM4 81
4.6.5 The Analysis of Instructional Material Design of
Pythagorean Theorem in BPM5 82
4.6.6 The Extraction of Research Findings and The Analysis
of Mathematics Textbooks 83
4.7 Theoretical Arguments of The Instructional Material Design 84 4.7.1 Theoretical Arguments of The Instructional Material Design
For The First Meeting 84
4.7.1.1 The Conceptual Arguments For Situation One 84 4.7.1.2 The Didactical Arguments For Situation One 85 4.7.1.3 The Conceptual Arguments For Situation Two 85 4.7.1.4 The Didactical Arguments For Situation Two 86 4.7.1.5 The Conceptual Arguments For Situation Three 86 4.7.1.6 The Didactical Arguments For Situation Three 86 4.7.1.7 The Conceptual Arguments For Situation Four 86 4.7.1.8 The Didactical Arguments For Situation Four 87
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ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu 4.7.2 Theoretical Arguments of The Instructional Material Design
For The Second Meeting 88
4.7.2.1 The Conceptual Arguments For Situation Five 88 4.7.2.2 The Didactical Arguments For Situation Five 88 4.7.2.3 The Conceptual Arguments For Situation six 89 4.7.2.4 The Didactical Arguments For Situation Six 89 4.7.2.5 The Conceptual Arguments For Situation
Seven and Eight 89
4.7.2.6 The Didactical Arguments For Situation
Seven and Eight 90
4.7.2.7 The Conceptual Arguments For Situation Nine 90 4.7.2.8 The Didactical Arguments For Situation Nine 91 4.7.3 Theoretical Arguments of The Instructional Material Design
For The Third Meeting 91
4.7.3.1 The Conceptual Arguments For Situation Ten 92 4.7.3.2 The Didactical Arguments For Situation Ten 92 4.7.3.3 The Conceptual Arguments For Situation eleven 92 4.7.3.4 The Didactical Arguments For Situation Eleven 92 4.7.3.5 The Conceptual Arguments For Situation Twelve 93 4.7.3.6 The Didactical Arguments For Situation Twelve 93 4.7.3.7 The Conceptual Arguments For Situation Thirteen 94 4.7.3.8 The Didactical Arguments For Situation Thirteen 94 4.7.4 Theoretical Arguments of The Instructional Material Design
For The Fourth Meeting 94
4.7.4.1 The Conceptual Arguments For Situation Fourteen 94 4.7.4.2 The Didactical Arguments For Situation fourteen 95 4.7.4.3 The Conceptual Arguments For Situation Fifteen 95 4.7.4.4 The Didactical Arguments For Situation Fifteen 95 4.7.4.5 The Conceptual Arguments For Situation Sixteen 96 4.7.4.6 The Didactical Arguments For Situation Sixteen 96 4.7.4.7 The Conceptual Arguments For Situation Seventeen 97
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Maya L Hutapea, 2014
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu
4.7.4.8 The Didactical Arguments For Situation Seventeen 97 4.7.4.9 The Conceptual Arguments For Situation Eighteen 98 4.7.4.10 The Didactical Arguments For Situation Eighteen 98 4.7.5 Theoretical Arguments of The Instructional Material Design
For The Fifth Meeting 98
4.7.5.1 The Conceptual Arguments For Situation Nineteen 98 4.7.5.2 The Didactical Arguments For Situation Nineteen 98 4.7.5.3 The Conceptual Arguments For Situation Twenty 99 4.7.5.4 The Didactical Arguments For Situation Twenty 99 4.7.5.5 The Conceptual Arguments For Situation Twenty One 99 4.7.5.6 The Didactical Arguments For Situation Twenty One 100 4.7.5.7 The Conceptual Arguments For Situation Twenty Two 100 4.7.5.8 The Didactical Arguments For Situation Twenty Two 100 CHAPTER V CONCLUSION AND RECOMMENDATIONS
5.1 Conclusion 101
5.2 Recommendation 103
5.2.1 Recommendations For Instructional Design 103 5.2.2 Recommendation For Advanced Research 103
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Maya L Hutapea, 2014
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
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LIST OF PICTURES
Page
Picture 1.1 Metapedadidaktik Prism 4
Picture 2.1 Metapedadidaktik Prism 12
Picture 2.2 The Elaboration of Metapedadidaktik Prism 12
Picture 2.3 The Pythagorean Theorem 21
Picture 2.4 GED Sample Question 22
Picture 2.5 Students Common Errors 23
Picture 3.1 The Chart of Research Plan 27
Picture 4.1 The Chart of The Research Data Linkage 34
Picture 4.2 Problem Three 37
Picture 4.3 The Answer of S87 40
Picture 4.4 The Answer of S68 40
Picture 4.5 The Answer of S70 41
Picture 4.6 The Answer of S99 41
Picture 4.7 The Answer of S11 42
Picture 4.8 The Answer of S34 42
Picture 4.9 The Answer of S23 43
Picture 4.10 Problem Five 44
Picture 4.11 The Answer of S26 45
Picture 4.12 Problem One 47
Picture 4.13 The Answer of S22 51
Picture 4.14 The Answer of S67 52
Picture 4.15 Problem Four 53
Picture 4.16 The Answer of S99 55
Picture 4.17 The Answer of S62 55
Picture 4.18 The Answer of S79 55
Picture 4.19 The Answer of S93 56
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Maya L Hutapea, 2014
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu
Picture 4.21 Problem Six 57
Picture 4.22 The Comparison of Angle-based
Right Triangle 58
Picture 4.23 The Answer of S92 58
Picture 4.24 The Answer of S62 59
Picture 4.25 The Answer of S66 59
Picture 4.26 The Answer of S79 59
Picture 4.27 The Answer of S77 60
Picture 4.28 The Answer of S86 61
Picture 4.29 The Answer of S74 61
Picture 4.30 The Answer of S52 62
Picture 4.31 The Answer of S51 62
Picture 4.32 The Answer of S61 63
Picture 4.33 Problem Two 64
Picture 4.34 The Right Triangle That Showed in
The Interview Session 65
Picture 4.35 The Answer of S47 68
Picture 4.36 The Answer of S88 71
Picture 4.36b The Venn Diagram of Research Results 75 Picture 4.37 The Analysis of Instructional Material
Design of Pythagorean Theorem in BPM1 78 Picture 4.38 The Analysis of Instructional Material
Design of Pythagorean Theorem in BPM2 79 Picture 4.39 The Analysis of Instructional Material
Design of Pythagorean Theorem in BPM3 80 Picture 4.40 The Analysis of Instructional Material
Design of Pythagorean Theorem in BPM4 81 Picture 4.41 The Analysis of Instructional Material
Design of Pythagorean Theorem in BPM5 82 Picture 4.42 Right Triangle in Exercise (1) 91 Picture 4.43 Right Triangle in Exercise (2) 91
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Maya L Hutapea, 2014
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
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LIST OF TABLES
Page Table 4.1 The Outcome Space of Ontogenic Obstacles 71 Table 4.2 The Outcome Space of Epistemological Obstacles 73 Table 4.3 The Outcome Space of Didactical Obstacles 75
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ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
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LIST OF ATTACHMENTS
Page Attachment 1 Preliminary Observation Results 1 107 Attachment 2 Preliminary Observation Results 2 109
Attachment 3 Soal 113
Attachment 4 Lembar Jawaban 115
Attachment 5 Penyelesaian 116
Attachment 6 Jawaban S11 120
Attachment 7 Jawaban S37 121
Attachment 8 Jawaban S10 122
Attachment 9 Jawaban S27 123
Attachment 10 Jawaban S22 124
Attachment 11 Jawaban S53 125
Attachment 12 Jawaban S67 126
Attachment 13 Jawaban S62 127
Attachment 14 Jawaban S51 128
Attachment 15 Jawaban S74 129
Attachment 16 Jawaban S81 130
Attachment 17 Jawaban S47 131
Attachment 18 Jawaban S69 132
Attachment 19 Jawaban S87 133
Attachment 20 Jawaban S88 134
Attachment 21 Jawaban S91 135
Attachment 22 Jawaban S65 136
Attachment 23 Jawaban S63 137
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Maya L Hutapea, 2014
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu
Attachment 25 Jawaban S87 139
Attachment 26 Jawaban S68 140
Attachment 27 Jawaban S70 141
Attachment 28 Jawaban S99 142
Attachment 29 Jawaban S11 143
Attachment 30 Jawaban S34 144
Attachment 31 Jawaban S23 145
Attachment 32 Jawaban S99 146
Attachment 33 Jawaban S62 147
Attachment 34 Jawaban S79 148
Attachment 35 Jawaban S93 149
Attachment 36 Jawaban S94 150
Attachment 37 Jawaban S99 151
Attachment 38 Jawaban S26 152
Attachment 39 Jawaban S92 153
Attachment 40 Jawaban S62 154
Attachment 41 Transkrip Wawancara Soal Nomor 1 155 Attachment 42 Transkrip Wawancara Soal Nomor 2 166 Attachment 43 Transkrip Wawancara Soal Nomor 3 170 Attachment 44 Transkrip Wawancara Soal Nomor 4 174 Attachment 45 Transkrip Wawancara Soal Nomor 5 177 Attachment 46 Transkrip Wawancara Soal Nomor 6 178
Attachment 47 Design Bahan Ajar (Pertemuan I) 179
Attachment 48 Design Bahan Ajar (Pertemuan II) 187 Attachment 49 Design Bahan Ajar (Pertemuan III) 200 Attachment 50 Design Bahan Ajar (Pertemuan IV) 211 Attachment 51 Design Bahan Ajar (Pertemuan V) 244
Attachment 52 Dokumentasi Foto 256
Attachment 24 The Scanlation of Students’ Answers 267
Attachment 25 Validasi Soal Oleh Ahli I 278
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ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
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Attachment 27 Validasi Soal Oleh Ahli III 300
Attachment 28 Validasi Soal Oleh Ahli IV 311
Attachment 29 Validasi Soal Oleh Ahli V 322
Attachment 30 Surat Izin Penelitian 333
Attachment 31 Surat Bukti Penelitian Dari Sekolah I 334 Attachment 32 Surat Bukti Penelitian Dari Sekolah II 335 Attachment 33 Surat Bukti Penelitian Dari Sekolah III 336
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Maya L Hutapea, 2014
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu CHAPTER I
INTRODUCTION 1.1 The Background
Science is a knowledge which consists of three main points, namely as process, product or method and attitude (Carin and Sund, 1989). Science is a mental and physical process in comprehending nature, within thinking process, mental activity, attitude and confidence in solving nature problems. The process is about skills in understanding nature, observing, manipulating, reasoning, using hypothesis, measuring, predicting, defining, and applying experiment. Science product is knowledge from creative process such as facts, concepts, theory, model and others that continuously developed along with the development of the science process alone.
Mathematics as one of the science taught in school attached to the points above. Mathematics in achieving knowledge through a long learning process accompanied with mathematical thinking process, reasoning, creative thinking, proving, communication, and solving problem. Mathematics learning sued to generate students who are active, creative, critics, flexible, and be able to communicate well. In junior high school level, students are expected doing well in recalling, interpreting information, abstracting, solving problem, generalizing, and reasoning. All these demands cannot be achieved when there are still obstacles in learning mathematics.
A preliminary observation held at grade nine of junior high school students on the subject of Pythagorean theorem. This school is located in a downtown which has 30-40 students in each class. The preliminary observation was conducted in an excellent class consists of middle and high achievement students. The Pythagorean theorem had been taught by the former teacher on the second semester in the previous year, with a note that students had been informed about what subject will be tested in the exam. From the preliminary observation results showed there are some visible difficulties from students‟ answers, namely:
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Maya L Hutapea, 2014
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu
This is shown in the students‟ answers that use the Pythagorean theorem incorrectly. (can be seen in the attachment 1).
2. Students is less flexible in manipulating Pythagorean theorem;
The students could answer the problem regarding the Pythagorean concept but when the problem presented in another context, students could not answer the problem. (can be seen in the attachment 2)
The points above show there are problems and obstacles in solving problems. However the problems were not only because the students did not prepare well in advance, but also there were some of them that already memorized the formula, yet they could not do well in the test when it comes to contextual problem, they seemed confused in doing so. Whereas the problems were really identical with the others, but depth understanding is needed to know what is given and how to manipulate what is given to answer the question.
The obstacles in solving the problem given is a product of a learning process, means students could not solve the problem because there are difficulties experienced by students in class. To solve this problem, it is not necessarily solved by giving students more and more problems to be solved, but needs a way to analyzing, why those obstacles happened in the first place. From the result, teacher need to analyze what is required to solve problem.
Learning obstacles in students are different; depend on the factor that cause it; such as physiological, social, emotional, intellectual and pedagogical. Prior knowledge is the instance of intellectual factor, which is greatly affect the success of student learning, because learning occurred in the process of coding and encoding of information in students‟ mind. For students who have a sufficient prior knowledge will easily accept new information and communicate them in learning experiences and have more confidence in doing so. As for the students who have less prior knowledge, they will have difficulties to accept new information due to the lack of experience of the prior knowledge, so that making students difficult in understanding the concept.
Learning obstacles occur because of mistakes that happened in class, whether that came from teacher or students. One of the mistakes that occur in
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ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
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class is the same treatment teacher gave to the different characteristic of students. Additionally, another error in class is from students that difficult in connecting their existing knowledge to the new incoming knowledge, and they memorized the formula alone without understanding the concept, so it is difficult for them to applying in different context. They seem uncomfortable with the learning styles that do not suit their characters.
To meet students‟ needs and to provide sufficient portion for each learning styles, it is required an instructional material that capable to fulfill the needs. Instructional material in question is not mere information transferred from what is in the text book without going through the process of contextualization and re-personalization. Re-contextualization is an attempt to put things into another context, so it will be familiar and intertwined and fused by the whole. Re-personalization is making something identifiable as belonging to a particular person. As stated by Suryadi and Turmudi (2011) that preparation of instructional material in general is based only on presentation models that are provided in the text books without going through the process of contextualization and re-personalization. In fact, the presentation in the mathematics text books, in the form of concepts description, proving, problem examples, they are the synthesis of a long process that ended in the process of contextualization and re-personalization.
This case refers to the need for a renewal of the instructional material prepared by teacher in presenting the material in class. Teacher cannot be a copycat to impose instructional material in the book without looking at the characteristics and obstacles of students who will be taught. As mentioned by the National Council of Educational Research and Training (2006):
Mathematics education relies very heavily on the preparation that the teacher has, in her own understanding of mathematics, of the nature of mathematics, and in her bag of pedagogical techniques. Textbook-centered pedagogy dulls the teacher‟s own mathematics activity.
In school, especially in terms of learning, students often make the teacher as a role model for them. Akinsola (2008) stated “Attitudes differ according to how learners perceive what they are taught and whoever is teaching them.” So
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ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
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consciously or not, teachers who daily interact with students in the learning process pass on their attitude they showed indirectly when teaching to students.
In the learning process, Brousseau (1997) defined three types of learning obstacles, namely: ontogenic obstacles, didactical obstacles, and epistemological obstacles. Ontogenic obstacles are obstacles due to differences in the level of knowledge of student‟s ability with teacher‟s level of knowledge in teaching. Didactical obstacles are caused by the lack of precise methods or approach that teacher use in teaching. The epistemological obstacles are caused by the mathematical concept.
Teachers nowadays are expected to act not only as teachers, but also as researchers. This term is not a mere thing, but the presence of a research is very important in creating a better learning. Teachers do research to obtain information and solutions to solve obstacles experienced in class. After obtaining enough information, teachers can begin to try an innovation in terms of models, media, and approaches, and results were evaluated to see whether the innovation is done to solve the problem. Existing problem may not necessarily be directly solved, or may not even solved at all, for that teachers should continue to conduct researches and development so that what is expected to be achieved.
In learning, there are three important elements, they are teacher, students, and material. The three elements will form a relation, which are students-teacher, teacher-material, material-students. These three relations form a study called Didactical Design Research (DDR).
Picture 1.1. Metapedadidaktik Prism (Suryadi, 2010)
As illustrated, Metapedadidaktik explains the relation between students, teacher, and material. This metapedadidaktik prism is modified from the
Student
Teacher
Material Teacher
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ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
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Kansanen didactical triangle by Suryadi that added a relation between students-teacher-material, which is called anticipation of didactical and pedagogical.
Didactical is emphasized in learning since the planning stage (Suryadi, 2010). The didactical analysis is conducted before learning, which focuses in relation between teacher, students, and material. The didactical analysis will be the referral to form a learning design.
Learning design is an instructional material design that will be used as a guide in the course of learning. It consists of activity stages to be carried out in learning. This activity is not only involved how teacher teach the material to students, but also how students respond to what is presented by teacher, and the material.
In the design of instructional material, teacher must be able to predict what response will arise, and anticipate what response given after that. This stage mentioned by Suryadi as anticipation of didactical and pedagogical. In this stage, the competence and ability of teacher is needed to build the structure of good instructional material design.
These three relations are not formed by the class itself, but must be designed by the teacher in advance. A design of learning scenario is regulating how the learning will be done. However, there are things that also occur spontaneously or beyond expectations, and teacher must improvise swiftly so that learning objectives still be achieved. In this case the teacher‟s competence and ability to design a good learning are indispensable. Suryadi named the capability as Metapedadidaktik.
Many methods and models of learning used by teachers from various sources, but the models are not necessarily an instruction that can be applied successfully in all types of students. Some things to note are the school facilities and infrastructure, the social background of students, the school environment, and students‟ ability to receive learning. NCTM (2000) stated: “Teachers should design the learning of mathematics which aims in a goal-oriented with a focus on the process. Directed learning orientation on competence development „mathematical thinking‟ and „mathematical disposition‟”. In line with it, Wardani
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et al. (2011) stated “Among teaching, students, and school level variables, the biggest role for improving mathematical creativity and disposition was the teaching approach”
In practice, it is difficult to find a suitable model to be applied to all students, especially in class with 40 or more students, too many characters to be met while the time owned is limited. Therefore, it requires an instructional material designed by teacher, so that teacher knows what is needed. Teacher is not stick with a particular model, but flexibly adapted, according to the needs of the diverse of students‟ characters. Teacher‟s instructional material design is expected to reduce the obstacles that occur and give comfort to the students in learning. Comfort in learning can be done in various ways.
According to interviews with several teachers in general, teachers prefer to teach high achievement students, because it is easier, that is why teacher tend to communicate with high achievement students more than others. Teachers involve smart students because they capable in doing everything that teacher told. It is not making low achievement students to be motivated, but the opposite, they feel neglected and ignored, so they feel lazy to learn, and only expect all the answers from high achievement students. Then, teachers need to be more considerate in dividing their attention to the students, so every student gets the same opportunity to learn.
Division of the group members are often done at random, which it is actually not a good idea, because it is possible if in one group, all the members are high achievement students, and certainly they can do their task better, and other group, perhaps all the members are low achievement students, and it will be very difficult for them to do the task, besides they do not believe in themselves, they also do not believe their friend, and have no one to ask, in the end, they will not do the task. Division should be a heterogeneous group, comprised the three of students level, the high achievement, the middle achievement, and the low achievement, so the high achievement student can guide other members to do the task. In the end, this way expected each member of the group get the same knowledge, as well as all the members should be able to explain the work they did
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by their own words. Also, the number of members in group should not more than four students, it will be better if it consist only three students, so there is no chance for students to let others working alone.
However, in the group study, not only students‟ roles are expected, but also the role of teacher to facilitate and motivate students in order to form a favorable atmosphere of group study. Teachers are expected not only monitor from the front of the class, but also go around directly to each group to guide students in completing the task. This method will help students in learning and by doing that teacher also be able to recognize better the character of their students.
Based on the above explanation, it requires a depth analysis of learning obstacles in students using didactical design research. From the result of analysis, an effective and efficient instructional material to achieve mathematic learning demands will be designed. This instructional material is not a standard teaching material yet, means it will continue to be evaluated and improved in order to obtain the best result.
1.2 The Problem Identification and The Problem Formulation
This research departed from many errors made by junior high school students in solving Pythagorean theorem problems. These errors can be seen from mistakes they have done in solving the given problem. The students‟ answers are not as expected, and students‟ learning outcomes are also far from satisfactory. Because this research is only up to the analysis of learning obstacles and instructional material design, then the didactical design research formulation used only to the extent of the analysis of didactical situation. In the analysis of the didactical situation, there are three important components which are didactical relations, pedagogical relations, and anticipation of didactical and pedagogical.
The problem formulations of this research are:
1. What are learning obstacles experienced by students in the learning of Pythagorean theorem?
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2. How is instructional material design which was made based on the didactical design research?
1.3 The Research Objections
1. Analyze the students‟ learning obstacles in the learning of Pythagorean theorem;
2. Create an instructional material design that is based on the didactical design research;
1.4 The Research Benefits
1. Provide analysis of the students‟ learning obstacles in the learning of Pythagorean theorem;
2. Provide an instructional material design that are based on the didactical design research;
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Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu CHAPTER III
RESEARCH METHOD 3.1 Research Method
The method used in this research is descriptive exploratory method, where the approach taken towards the object under study is a qualitative approach. This research uses descriptive method, because the data to be collected are not numbers, but rather the results of interviews, field notes and documents. This is done because the phenomena will be discussed in a complex and holistic. Therefore, this research uses test and interviews to match the empirical reality of the prevailing theories using descriptive methods. "Descriptive research is research that seeks to describe a symptom, event, incident that occur in the present." (Sujana and Ibrahim, 1989:65)
This research was conducted to obtain in-depth information about the learning obstacles experienced by students, the results of the analysis of learning obstacles will be used as a guideline in designing instructional materials that according to the three components of didactical design research (DDR).
3.2 Purposive Sampling
The purposive samples of this research are students of three state junior high schools of different clusters in Bandung.
3.3 Research Procedures
The research procedure is divided into three major phases, namely: the preparation phase, the implementation phase, and the final phase, described as follows:
3.3.1 The Preparation Phase
There are five steps in the preparation phase, they are: 1. Conducting library research to formulate the problem; 2. Preparing the proposal to be submitted on proposal seminar;
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4. Conducting the revision and validation of the research instruments; 5. Conducting the permits to perform research related to the parties;
3.3.2 The Implementation Phase
There are six steps in the implementation phase, they are:
1. Giving tests to the research samples (students that will be given the test have obtained learning about Pythagorean concept and have been told in advance what subject matters will be tested);
2. Conducting interviews with students and teachers to obtain in-depth information about possible learning obstacles in students;
3. Analyzing students’ answers to the questions given;
4. Analyzing learning obstacles based on the test results and the interviews; 5. Planning an anticipation of didactical, anticipation of pedagogical, and
anticipation of didactical and pedagogical.
6. Designing an appropriate instructional material with the three components of didactical design research.
3.3.3 The Final Phase
There are three steps in the final phase, they are:
1. Processing the data that has been obtained from the implementation of the research;
2. Drawing conclusions based on the discussion of the research results; 3. Preparing of research results in the form of a thesis.
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Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu Picture 3.1 The Chart of Research Plan
Revision and validation
Conclusion Processing and data
analysis
Discussion
Conducting test
Instructional material design Analysis of Learning
Obstacles Preparation of instrument:
1. Test
2. Interviews outlines
Analysis of students’ learning obstacles on the subject of Phytagoras Problem Formulation
Literature Study
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Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu 3.4 Research Instrument
Research instrument is an instrument used to obtain the information needed in the research. Arikunto (2002:136) stated that: "The instrument is a tool or a research facility used by researchers to collect data in order to work more easily and better results, in terms of more accurate, are comprehensive and systematic so that it will be more easily processed."
3.4.1 The Data Collection Technique
The data was collected using tests instruments and interviews. 3.4.1.1Test
The test is an instrument used to collect primary data in the form of a six-question essay tests (can be seen in attachment 3). The tests are described on the following explanation:
3.4.1.1.1 The Characteristic of Problem One
Communication skills can be seen in how students express or explain problem solving; of course this should be supported by the adequate understanding of the concept of the right triangles and the Pythagorean theorem. In this case students were asked to investigate whether the three sides of triangles are the Pythagorean triple or not along with the explanations. The important things to answer this problem are the knowledge of what is the definition of Pythagorean triple and investigate how the sides of a triangle can be called as a Pythagorean triple. This will bring the students to the knowledge of the characteristics of a right triangle, such as the characteristics of the hypotenuse or the longest side of triangle.
3.4.1.1.2 The Characteristic of Problem Two
Basically this problem is just a matter of repeating of Pythagorean theorem, but there were many students that could not answer this problem correctly. Students are required to write down the Pythagorean formula of a right triangle is known in the problem. The obstacles experienced by students is derived from the
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image of a right triangle is slightly different from a right triangle they usually see. Not a few students who assume that the Pythagorean formula involving only a, b and c (as in the textbook), the less of variation in right triangle name makes students glued to the formula of so that when the name of the given is different, student were confused in determining the Pythagorean formula. Likewise with a right triangle shape which are often given in books or often done by students is less varied make students tend to change by turning, tilting, matching the right triangle picture in the problem to compare to the one they learned.
3.4.1.1.3 The Characteristic of Problem Three
In the third problem, it is given enough information for students to do the next step to solve the problem, but it is required the knowledge of algebra, such as the term of variables and coefficients, as well as algebraic operations. This problem is made to show the students' understanding Pythagorean theorem that is connected with algebra.
3.4.1.1.4 The Characteristic of Problem Four
The fourth problem is slightly more complicated than the previous problem, since it involved algebra and geometry knowledge. The knowledge of geometry in question is knowledge of plane as: characteristics of square and the area of square. In this problem was not given any instructions that this problem can be solved by the Pythagorean theorem, so a lot of students who completed using the formula of the area of square.
3.4.1.1.5 The Characteristic of Problem Five
This problem involves geometry knowledge, which is the trapezoidal plane. But not only involve the plane, this problem would also like to see the extent to which students were able to see the possibility of solving problems by constructing another plane in a trapezoidal plane so that in the end the students
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can use the Pythagorean theorem. It is needed the knowledge of trapezoid circumference formula and the comparison of special angel-based right triangle. 3.4.1.1.6 The Characteristic of Problem Six
This problem requires a mastery of the comparison of special angel-based right triangles with angle based. But unfortunately when the problem was tested, many students failed to do a visual representation, which led students to the wrong problem solving and answers. Students also less of thorough and did not read the questions carefully. In addition, students also tend not to work on the question in the form of word problems.
3.4.1.2Interview
This research uses unstructured interviews as the primary data source, which the researcher do not use a structured interview guide to collect data systematically, but the researcher prepared the interview outlines that will be asked. This interview will be documented in a sound recording and photos. Because this is not a structured interview, the questions from the interviewer will flow according to the condition of the interview, the biased question can be seen after the interview. The questions that biased or ambiguous are discarded so as not to affect the validity of the data.
In the early stages, researcher will spend more time listening to what the respondent and not directly ask the problems studied in the research, this is done so that respondents feel comfortable and answered the questions honestly. Once researcher knows the situation of the respondent, then prepare more questions related to respondents' answers before and will be directed to the issues addressed in the research.
The steps of the interview to be conducted are as follows: i. Researcher prepares the interview outlines to be asked.
ii. Researcher determines the respondent that will be interviewed.
iii.Researcher determines the interview schedule according to the condition of the respondent.
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Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu iv.Researcher put interview results into an interim report.
v. Researcher analyzes the results obtained from the initial interview.
vi.Researcher plan the follow-up that will be done based on the results of the initial interviews that will be directed to the interview about students’ learning obstacles.
3.5 Data Analysis Procedure 3.5.1 Data Analysis Technique
Bogdan (in Sugiyono, 2010) stated “Data analysis is the process of systematically searching and arranging the interview transcripts, field note, and other materials that you accumulate to increase your own understanding of them and to enable you to present what you have discovered to others” So all of the data obtained through interviews and documentation, arranged, organized, sorted systematically to increase the understanding of the studied problems then passed on to others. Data Analysis of this research is the analysis of the model used by Miles and Huberman, namely:
3.5.1.1Data Reduction
In qualitative research, data and information received quite a lot, from time to time of data collection, the data obtained was getting more and more complicated. Therefore it is necessary to discard unimportant data, then select and focus on the data that are important in connection with the problem under study. In the data reduction phase, the sample will be categorized in certain aspects, making a visible pattern to the focus of attention of researcher.
3.5.1.2Presentation of Data
Presentation of data in qualitative research can be form of reviews, charts, graphs or other. This is done to simplify the researcher to conduct an analysis and comprehension of the data obtained. With the presentation of data can be seen the relation of each category, see the common thread that is made into a theory that
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are still hypothetical. This hypothesis should be tested first before it becomes a standard theory.
3.5.1.3Conclusion Withdrawal
Inferences will be made when the initial conclusion is proven valid and reliable. This conclusion must have strong evidence and expect a new finding that has not been studied previously.
3.5.2 Data Analysis Plan
Data analysis was performed during and after data collection continues over the triangulation technique is expected to produce a specific pattern of relationships which will generate new theory.
3.5.3 Validation Method 3.5.3.1Credibility Test
Credibility test is conducted to increase confidence/credibility of the data. Credibility test is done in the following ways:
i. Increase the persistence
Researcher examines and observes carefully the data, in order to obtain accurate data.
ii. Negative case analysis
Researcher look for the contrary data to the findings obtained. If no data is negative /contradictory, the findings can be said to be true.
iii.Using reference material
These references include documents, records or evidence that could explain the findings of the study, so the data obtained more convincing.
iv.Conduct member check
Checking the data back to the subject under study, so there is no misinterpretation of the answer or data provided by the subject.
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Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu 3.5.3.2Transferability Test
Transferability test is conducted by readers of research results. When readers understand the purpose of the research and were able to give an overview of the results of these studies with detail and clear, it can be said that research meet the standards of transferability. Sanafiah Faisal (in Sugiyono: 1990)
3.5.3.3Dependability test
This test is done by the supervisor, to see the whole process of the study conducted by researchers is true, not merely data manipulation. This process is not only seen from the results of the study, but conducted since researchers decide what problems to be studied until the conclusion of the study.
3.5.3.4Confirmability Test
Tests conducted after the results of the study concluded, the test is conducted by people who are experts, such as examiners.
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Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu CHAPTER V
CONCLUSION AND RECOMMENDATION 5.1 Conclusion
From the analysis result and the discussion on the research findings, it can be concluded as following:
1. The learning obstacles experienced by students in the learning of Pythagorean theorem are as follow:
a. The students tend to use the quick way to solve the problems without adequate understanding of the concept. This way directs students to use no problem solving at all and difficult to communicate their idea to the context of the problems. The students often use terms to memorize the concept that lead students to misconception and verbalism, like: ‘the sloping slide’ ‘the sum of’ ‘positive sign’ and others. Students tend to remember the term alone without further explanation regarding the concept. The students often memorize the concept but without lack of understanding, so that students did not do well in the problems with different context.
b. The students tend to fail in doing the problems with implicit information and the problem that requires visual representation in the process of solving the problem, this occurs because students usually work with the problem with explicit information and presented picture. It is also because of the lack of students’ understanding and geometry ability; students often make mistakes in doing visual representation, this leads students to incorrect answer. Based on the interview results, the students do not like the word problems or a problem with long questions. The students thought that this kind of problem will lead to complicated and long answers, so students tend not to answer the problems even before read the problem first.
c. Ontogenic obstacles, didactical obstacles and epistemological obstacles are found in the students’ answer of the given problems.
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This happened mostly because students forget easily the concept taught before and also the lack of prerequisites knowledge that make students difficult in doing problems with mathematical connection. Ontogenic obstacles occurred in the problem solving that needs algebraic thinking, didactical obstacles occurred because the using of term and quick solution that lead students to memorize alone. The epistemological obstacles occurred in the problems that need mathematical connection and implicit information. This research indicates that the most common learning obstacles found within students is epistemological obstacles, which supposedly caused by the less of flexibility in doing problems and the meaningless learning, that it also can be caused by the didactical obstacles.
d. Three of four students that had none of the three obstacles are from the high achievement students from each school that completed their answer correctly, eventhough there was mistake, it was only an error in calculating, while another student was the one who only wrote down the question in the answer.
e. There are 31 students that have all of those obstacles in their answers, seven students from the low cluster school, 9 students from the moderate cluster school and 15 students from the high cluster school. The students are from different levels of knowledge, but mostly they are from the low and the high achievement. The reason to this might be because high achievement students tend to solve the problem with their own way such as comparison or analogy but did not pay attention to the rule, while the low achievement students tend to fail in memorizing the formulas.
2. From the analysis result and extraction of mathematics textbooks, it is needed a didactical design that involves students actively in the learning process, such as discovery, reconstruction and work in varied problems. The instructional material design that is made based on the didactical
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design research are shown in the attachment 47 up to attachment 51 and described on page 84.
5.2 Recommendation
From the research that has been carried out and the analysis obtained, it can be presented recommendation as follow:
5.2.1 Recommendation For Instructional Design
Based on the learning obstacles occurred within the students, instructional that actively involve students in discovering the concept is needed. The instructional design need anticipation of obtsacles that might appear in the learning process, so that it can be prepared the feedback and the properties that can minimize or eliminate the obstacles. It will be better if in each situation, the approach used is regarding to eliminate those obstacles; also each meeting, there should be a situation where students are given interesting activities in order to make students grow attached and always look forward to the next meeting of learning mathematics. The design should bridge the learning process and overcome the gaps between the existing knowledge and the new knowledge so students have better learning experiences
5.2.2 Recommendation For Advanced Research
There are aspects that still need further research in this research, however because of the limited time and other reasons those aspects can be brought up in this research, such as implementation of the instructional material design. Therefore, there is wide opportunity for advanced researchers to examine the topics of mathematical matters and examine the design of the instructional materials in this research to the class, that obtained feedbacks in hoping would gain a new innovation of better mathematics learning.
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Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu CHAPTER V
CONCLUSION AND RECOMMENDATION 5.1 Conclusion
From the analysis result and the discussion on the research findings, it can be concluded as following:
1. The learning obstacles experienced by students in the learning of Pythagorean theorem are as follow:
a. The students tend to use the quick way to solve the problems without adequate understanding of the concept. This way directs students to use no problem solving at all and difficult to communicate their idea to the context of the problems. The students often use terms to memorize the concept that lead students to misconception and verbalism, like: ‘the sloping slide’ ‘the sum of’ ‘positive sign’ and others. Students tend to remember the term alone without further explanation regarding the concept. The students often memorize the concept but without lack of understanding, so that students did not do well in the problems with different context.
b. The students tend to fail in doing the problems with implicit information and the problem that requires visual representation in the process of solving the problem, this occurs because students usually work with the problem with explicit information and presented picture. It is also because of the lack of students’ understanding and geometry ability; students often make mistakes in doing visual representation, this leads students to incorrect answer. Based on the interview results, the students do not like the word problems or a problem with long questions. The students thought that this kind of problem will lead to complicated and long answers, so students tend not to answer the problems even before read the problem first.
c. Ontogenic obstacles, didactical obstacles and epistemological obstacles are found in the students’ answer of the given problems.
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Maya L Hutapea, 2014
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu
This happened mostly because students forget easily the concept taught before and also the lack of prerequisites knowledge that make students difficult in doing problems with mathematical connection. Ontogenic obstacles occurred in the problem solving that needs algebraic thinking, didactical obstacles occurred because the using of term and quick solution that lead students to memorize alone. The epistemological obstacles occurred in the problems that need mathematical connection and implicit information. This research indicates that the most common learning obstacles found within students is epistemological obstacles, which supposedly caused by the less of flexibility in doing problems and the meaningless learning, that it also can be caused by the didactical obstacles.
d. Three of four students that had none of the three obstacles are from the high achievement students from each school that completed their answer correctly, eventhough there was mistake, it was only an error in calculating, while another student was the one who only wrote down the question in the answer.
e. There are 31 students that have all of those obstacles in their answers, seven students from the low cluster school, 9 students from the moderate cluster school and 15 students from the high cluster school. The students are from different levels of knowledge, but mostly they are from the low and the high achievement. The reason to this might be because high achievement students tend to solve the problem with their own way such as comparison or analogy but did not pay attention to the rule, while the low achievement students tend to fail in memorizing the formulas.
2. From the analysis result and extraction of mathematics textbooks, it is needed a didactical design that involves students actively in the learning process, such as discovery, reconstruction and work in varied problems. The instructional material design that is made based on the didactical
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Maya L Hutapea, 2014
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu
design research are shown in the attachment 47 up to attachment 51 and described on page 84.
5.2 Recommendation
From the research that has been carried out and the analysis obtained, it can be presented recommendation as follow:
5.2.1 Recommendation For Instructional Design
Based on the learning obstacles occurred within the students, instructional that actively involve students in discovering the concept is needed. The instructional design need anticipation of obtsacles that might appear in the learning process, so that it can be prepared the feedback and the properties that can minimize or eliminate the obstacles. It will be better if in each situation, the approach used is regarding to eliminate those obstacles; also each meeting, there should be a situation where students are given interesting activities in order to make students grow attached and always look forward to the next meeting of learning mathematics. The design should bridge the learning process and overcome the gaps between the existing knowledge and the new knowledge so students have better learning experiences
5.2.2 Recommendation For Advanced Research
There are aspects that still need further research in this research, however because of the limited time and other reasons those aspects can be brought up in this research, such as implementation of the instructional material design. Therefore, there is wide opportunity for advanced researchers to examine the topics of mathematical matters and examine the design of the instructional materials in this research to the class, that obtained feedbacks in hoping would gain a new innovation of better mathematics learning.
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Maya L Hutapea, 2014
ANALYSIS OF STUDENTS’ LEARNING OBSTACLES ON THE SUBJECT OF PYTHAGOREAN THEOREM
Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu
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Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu
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Universitas Pendidikan Indonesia | repository.upi.edu | perpustakaan.upi.edu
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