IMPROVEMENT OF STUDENT MATHEMATICAL REASONING ABILITY IN VII GRADE SMP NEGERI 1 MEDAN BY APPLYING REALISTIC
MATHEMATICS EDUCATION (RME) ON THE SUBJECT OF FRACTION ACADEMIC YEAR 2012/2013
By:
Misna Fitriyani I.D Number 408111012
Bilingual Mathematics Education Study Program
A THESIS
Submitted to Fulfill the Requirement for Getting the Degree of Sarjana Pendidikan
MATHEMATICS DEPARTMENT
FACULTY OF MATHEMATICS AND NATURAL SCIENCES STATE UNIVERSITY OF MEDAN
By:
Misna Fitriyani I.D Number 408111012
Bilingual Mathematics Education Study Program
A THESIS
Submitted to Fulfill the Requirement for Getting the Degree of Sarjana Pendidikan
MATHEMATICS DEPARTMENT
FACULTY OF MATHEMATICS AND NATURAL SCIENCES STATE UNIVERSITY OF MEDAN
PREFACE
Give thankfulness to Allah SWT that gives the God’s mercy and gift so that writer can finish this thesis. The title of this thesis is “Improvement of Mathematical Reasoning Ability in VII Grade SMP Negeri 1 Medan by Applying Realistic Mathematics Education (RME) on The Subject of Fraction Academic Year 2012/2013”. This thesis was arranged to satisfy the requirement to obtain the Degree of Sarjana Pendidikan from Faculty Mathematics and Natural Science in Universitas Negeri Medan.
In arranging this thesis, the writer gets many assistances and supports from many persons. Therefore, in this chance the writer wants to say thank you very much to the Rector of Universitas Negeri Medan, Prof. Ibnu Hajar, MS and his staffs, to the Dean of FMIPA Universistas Negeri Medan, Prof. Drs. Motlan, M.Sc, Ph.D and also the Dean assistant I, II, and III. And don’t forget to say thank you to Prof. Dr. Mukhtar, M.Pd as a leader of Mathematics Department, Drs. Syafari, M.Pd as a leader of Mathematics Education Study Program and as my academic supervisor, Drs. Yasifati Hia, M.Si as a secretary of Mathematics Department and to Prof. Dr. Herbert Sipahutar, M.Sc as a Coordinator of Bilingual Class.
Special thanks to my thesis supervisor, Dr. Hasratuddin, M.Pd. An intellectual person that give me many sciences, knowledge, guidance, experiences, assistances in preparing, doing, and finishing this thesis. And then thanks a lot for Prof. Dr. Asmin, M.Pd, Dr. E. Elvis Napitupulu, M.S, and Dr. Edi Syahputra, M.Pd, who’s the persons responsible for my thesis. And thanks a lot also for all of lectures and staffs in FMIPA Univeristas Negeri Medan.
vi
lovely sisters, Zakiyah Khairani Siregar and Putri Akhiriyani Siregar that give me motivation and pray also. Writer doesn’t forget to say thank you so much to special person in my heart that give me motivation, support, pray and love, and help, namely Bambang Imansyah. And thank you so much to my friends in Bilingual Mathematics Education 2008, special to my lovely friends Siti Rafiah Rangkuti, Emil Hani, Hot Tiarma Sianipar, Farah Diba, Eva Puspita Sari, Yanti Rambe, Efrida Fitri, Togu M.B, and Siti Rahmadani.
Writer also say thank you to Drs. H. Ahmad Siregar, MM as the Headmaster of SMP Negeri 1 Medan, and Mrs. Elliati, S.Pd as a mathematics teacher who help the writer as long as research activities.
Considering that, this thesis of course has many weaknesses. Therefore, the writer needs some suggestions to make this thesis be better. Finally, writer wishes and expects that this thesis can be useful and can help to improve our knowledge especially in education.
Medan, January 2013 Writer,
BIOGRAPHY
iii
Improvement of Student Mathe matical Reasoning Ability in VII Grade SMP Negeri 1 Medan by Applying Realistic Mathe matics Education (RME)
On The Subject of Fraction Acade mic Year 2012/2013
Misna Fitriyani (408111012) ABSTRACT
The research objectives are: 1) To know is there an improvement in ability of student mathematical reasoning who studied by realistic mathematics education; 2) To find out whether by following realistic mathematics learning, student become more active interacting in learning process; 3) To find out the description of student response to mathematic for students that follow realistic mathematics learning. This research is Classroom Action Research that taken in four cycles. Each of cycle consists of four phases, namely: (1) planning phase, (2) action phase, (3) observation phase, (4) reflection and evaluation phase. This research has done in SMP Negeri 1 Medan. The research subject is students in VII Archimedes grade SMP Negeri 1 Medan, with the number of students are 29 students that consist of 13 students are boys and 16 students are girls. While, the research object is all of activities that happened in learning process to improve student mathematical reasoning ability by applying realistic mathematics education. The research results are: 1) based on observation result, student more interactive in learning process because student builds knowledge by involving
students’ experience. This thing is shown by the average of students’ activity
score is 2,81 means good level; 2) The interview result is shown that student faced constraints in reasoning activities. But, by realistic approach that supported by the use of visual aids and worksheet helps student in understanding the matter, finding concept, and improving reasoning ability; 3) The effectiveness of realistic approach implementation in improving reasoning ability can be seen from test result, that is shown in reasoning test I classical achievement only 10.71% with the average is 11.43. In reasoning test II increase to be 85.71% with the a verage is 37.64. Because there exists students are not achievement yet, therefore the action is continued. So that, in reasoning test III classical achievement until 100% with the average is 40.28 and in the reasoning test IV the average is 45.21with class ical achievement is 100% also. Thereby, it can conclude that the implementation of realistic approach can improve student mathematical reasoning ability.
Improvement of Student Mathe matical Reasoning Ability in VII Grade SMP Negeri 1 Medan by Applying Realistic Mathe matics Education (RME)
On The Subject of Fraction Acade mic Year 2012/2013
Misna Fitriyani (408111012) ABSTRAK
Tujuan dari penelitian ini adalah: 1) Mengetahui apaka h ada peningkatan kemampuan penalaran matematika bagi siswa yang belajar dengan pendekatan realistic; 2) Menemukan apakah dengan mengikuti pelajaran berbasis matematika realistik, siswa menjadi lebih aktif berinteraksi dalam proses pembelajaran; 3) Menemukan gambaran respon siswa terhadap matematika untuk siswa yang mengikuti pembelajaran matematika realistik. Penelitian ini merupakan Penelitian Tindakan Kelas yang dilaksanakan sebanyak empat siklus. Masing- masing siklus terdiri dari empat tahap, yaitu: (1) tahap perencanaan, (2) tahap pelaksanaan, (3) tahap observasi, (4) tahap refleksi dan evaluasi. Penelitian ini dilaksanakan di SMP Negeri 1 Medan. Subjek penelitian adalah siswa kelas VII Archimedes SMP Negeri 1 Medan dengan jumlah siswa 29 orang, yang te rdiri dari 13 orang siswa laki- laki dan 16 orang siswa perempuan. Sedangkan objek penelitian adalah aktifitas yang terjadi dalam proses pembelajaran untuk meningkatkan kemampuan penalaran matematika siswa dengan menerapkan pendidikan matematika realistik. Hasil yang diperoleh dari penelitian ini meliputi: 1) Berdasarkan hasil observasi, siswa lebih interaktif dalam proses pembelajaran karena membangun sendiri pengetahuan yang ingin dicapai dengan melibatkan pengalaman siswa. Hal ini berdasarkan rata-rata nilai kegiatan siswa yaitu 2,81 yang dikategorikan baik ; 2) Hasil interview menunjukkan bahwa siswa banyak mengalami kesulitan dalam aktifitas bernalar. Namun, dengan pendekatan realistik didukung penggunaan alat peraga dan worksheet membantu siswa dalam memahami materi, menemukan konsep, serta meningkatkan kemampuan bernalar; 3) Keefektifan penerapan pendekatan realistic dalam meningkatkan kemampuan penalaran dapat dilihat dari hasil tes, yang menunjukkan pada tes penalaran I ketuntasan klasikal hanya 10,71% dengan rata-rata 11,43. Pada tes penalaran II meningkat menjadi 85,71% dengan rata-rata 37,64. Karena masih ada siswa yang belum mencapai ketuntasan maka tindakan dilanjutkan, sehingga pada tes penalaran III ketuntasan klasikal mencapai 100% dengan rata-rata 40,28 dan pada tes IV rata-rata menjadi 45,21 dengan ketuntasan klasikal juga 100%. Dengan demikian dapat disimpulkan bahwa penerapan pendekatan realistik dapat meningkatkan kemampuan penalaran matematika siswa.
vii
CONTENTS
Page
Authentication sheet i
Biography ii
Abstract iii
Preface v
Table of Contents vii
Table List x
Figure List xi
Appendix List xii
CHAPTER I. INTRODUCTION
1.1 Problem Background 1
1.2 Problem Identification 7
1.3 Problem Restriction 7
1.4 Problem Formulation 7
1.5 Research Objectives 8
1.6 The Benefits of Research 8
CHAPTER II. LITERATURE REVIEW
2.1 Theoretical Framework 9
2.1.1 The Essence of Learning 9
2.1.2 Mathematics Learning 10
2.1.3 Mathematical Reasoning Ability 11
a. The Types of Reasoning 13
b. The Indicator of Mathematical Reasoning 15 2.1.4 Reasoning Ability in Mathematics Learning 16
2.1.5 Realistic Mathematics Education (RME) 17
2.1.5.2 The Principle of Realistic Mathematics Education 21 2.1.5.3 The Step of Realistic Mathematics Education 22 2.1.5.4 The Benefits and The Weakness of Realistic
Education Implementation 24
2.1.5.5 Design of Realistic Mathematics Education Lesson 26 2.1.5.6 The Relationship between Realistic Mathematics
Educations with the Improvement of Reasoning Ability 30
2.1.6 The Lesson of Fraction Matter 31
2.1.6.1 The Explanation and Symbol of Fraction 31
2.1.6.2 The Kinds of Fraction 31
2.1.6.3 The Operation of Fraction 33
2.2 Conceptual Framework 37
2.4 Action Hypothesis 40
CHAPTER III. RESEARCH METHODOLOGY
3.1 Location and Research Time 41
3.2 Subject and Research Object 41
3.3 Operational Definition 41
3.4 Research Design 42
3.4.1 The Type of Research 42
3.4.2 Research Procedures 42
3.4.3 Planning of Action 43
3.4.4 The Schedule of Research Planning 45
3.5 The Instrument of Data Collection 46
3.5.1 The Instrument of Student Mathematical Reasoning Ability 46
3.5.2 Observation Sheet 47
3.5.3 Interview Test 47
ix
CHAPTER IV. RESULT AND DISCUSSION
4.1 Research Result 51
4.1.1 Planning Action 51
4.1.2 Implementation/Acting phase result 51
4.1.3 Observation’s Result 57
4.1.4 Mathematical Reasoning Ability Test Result 63
4.1.5 Reflection Phase 66
4.1.6 Interview Result 69
4.2 Result Discussion 70
4.3 The Weakness of Research 73
CHAPTER V. CONCLUSION AND SUGGESTION
5.1 Conclusion 74
5.2 Suggestion 75
REFFERENCES 76
FIGURE LIST
Page
Figure 2.1.1 The Draft of Inductive Reasoning Process 14 Figure 2.1.2 The Draft of Deductive Reasoning Process 15 Figure 2.1.3 Concept and Applied Mathematization 19
Figure 2.1.4 Levels of Models in RME 19
Figure 2.1.5 A Model for Designing RME Lesson Materials 29
Figure 2.1.6.1 Fraction 31
Figure 2.1.6.2 Equivalent Fraction 31
Figure 2.1.6.3 Addition of Fraction 34
Figure 2.1.6.4 Example of Addition of Fraction 34
Figure 2.1.6.5 Substraction of Fraction 35
Figure 2.1.6.6 Multiplication of Fraction 35
Figure 2.1.6.7 Division of Fraction 36
Figure 2.3 Scheme of Conceptual Framework 39 Figure 3.4 The Main Procedure of Classroom Action Research 42 Figure 4.1.1 The Result of Teacher’s Activities Observation 61 Figure 4.1.2 The Observation Result of Students’ Activities 63 Figure 4.1.3 Chart of Mathematical Reasoning Ability Test I Result 64 Figure 4.1.4 Chart of Mathematical Reasoning Ability Test II Result 64 Figure 4.1.5 Chart of Mathematical Reasoning Ability Test III Result 65 Figure 4.1.6 Chart of Mathematical Reasoning Ability Test IV Result 66 Figure 4.2.1 The Improvement Graph of Students’ Mathematical
xii
APPENDIX LIST
Page
Appendix 1 Lesson Plan I (Cycle I) 79
Appendix 2 Lesson Plan II (Cycle II) 83
Appendix 3 Lesson Plan III (Cycle III) 88
Appendix 4 Lesson Plan IV (Cycle IV) 93
Appendix 5 The Blueprint of Reasoning Ability Test I 99 Appendix 6 The Blueprint of Reasoning Ability Test II 100 Appendix 7 The Blueprint of Reasoning Ability Test III 102 Appendix 8 The Blueprint of Reasoning Ability Test IV 103
Appendix 9 Reasoning ability test I 105
Appendix 10 Reasoning ability test II 107
Appendix 11 Reasoning ability test III 109
Appendix 12 Reasoning ability test IV 111
Appendix 13 The Answer Key of Reasoning Ability Test I 114 Appendix 14 The Answer Key of Reasoning Ability Test II 116 Appendix 15 The Answer Key of Reasoning Ability Test III 118 Appendix 16 The Answer Key of Reasoning Ability Test IV 120 Appendix 17 Result of Mathematical Reasoning Ability Test 123
Appendix 18 Guided Student Worksheet (GSW) I 129
Appendix 19 Guided Student Worksheet (GSW) II 132
Appendix 20 Guided Student Worksheet (GSW) III 134
Appendix 21 Guided Student Worksheet (GSW) IV 137
Appendix 28 Assessment Rubric of Reasoning test II 174 Appendix 29 Assessment Rubric of Reasoning test III 176 Appendix 30 Assessment Rubric of Reasoning test IV 178 Appendix 31 Observation Sheet of Learning Process I 180 Appendix 32 Observation Sheet of Learning Process II 183 Appendix 33 Observation Sheet of Learning Process III 186 Appendix 34 Observation Sheet of Learning Process IV 189
Appendix 35 Result of Students’ Interview 192
Appendix 36 Field Notes Result 199
Appendix 37 Examples of Student’s strategy move from “model of”
to “model for” 205
Appendix 38 Photos Documentation 207
1
CHAPTER I INTRODUCTION
1.1 Proble m Background
Education world is still bumped into many problems, especially education world in Indonesia. And unconscious, the problems appear are effected quality of Indonesian education also from bad to worse. This thing is proved by Indonesian is got in 33rd position of 45 countries in Third International Mathematics Sciences Study (TIMSS) in 2003. In 2006, Program for International Students Assessment (PISA), that it gives score to how well immediacy of students 15 years old in facing life, Indonesian get level 50 of 57 countries in sciences, reading, and mathematics (The world Bank, 2011).This fact proof that there is not an appropriate evaluation yet, about whether problem that is really happened.
Unconsciously, one of the basis problem of Indonesian education world is education paradigm mistake that base all of education implementation systems. Still many used traditional method in learning system cause the quality of education become low, and finally student’s achievement has not improved. Why it can be called like that? It is because the use of traditional method such as conventional learning method not make student as learning subject. Student is more accepted lesson than acting to find the knowledge. This thing also happens in mathematics learning in Indonesian. Whereas, based on the appendix of National Minister of Education rules (Permendikanas) Number 20 year 2006 about standard of content (Wijaya, 2012 : 16), said that mathematics learning has goals, namely:
2) Using reasoning in pattern and characteristic, doing mathematics manipulation in generalizing, arranging proof, or explaining idea and mathematics statement.
3) Problem solving that include understanding problem ability, designing mathematical model, solving model and conjecturing solution that is gotten.
4) Communicate idea with symbol, table, diagram, or other media to make clear situation or problem.
5) Have attitude appreciate mathematics use in life, namely have curiosity, attention, and interest in learning mathematics, and tough attitude and self-confidence in problem solving.
Based on the goal of mathematics learning, can be said that learning mathematic not only enough be able to do calculation in mathematic, but must be mathematics learning become meaningful learning where students can use his ability and curiosity independently, and not look mathematics as an abstract thing. Mathematics should be able to imagined by student, so that student can understand mathematics concept well. In the meantime, Keith Delvin is presented the four faces of mathematics, namely; (1) mathematics as computation, formal reasoning, and problem solving; (2) mathematics as a way knowing; (3) mathematics as a creative medium; (4) applications of mathematics (NCTM, 2000:16).
From the four faces of mathematics above, one of ability that so hoped to appear by learning mathematics this time is reasoning ability. Reasoning ability is one of very important competence that students must have it in achieving optimal mathematics learning outcome. Reasoning activity, many involve student’s critical thinking ability in facing a problem. Improving reasoning ability means
that improving student’s ability in submits conjecture, take conclusion based on the fact and relevant source, doing mathematics manipulation, and generalizing.
Mathematical reasoning is one of method that is given to develop
3
analytic inclined record pattern, structure, and regularity in real world and symbolic things. He will give question whether the pattern is accidentally, whether it has reason so that easy to conjectured and proved. Mathematical
reasoning process must be invested to students early. From students’ experience
early with mathematics, important to help them to understand mathematics that statement always has a reason. Such as question “Why do you say that it is true?”
and “Is there one of you have another answer?” will help students look up that the
question need proof. In addition, while the evidence can be received logically, thus it will become an enough argument in mathematics class. This thing will become the first step to realize that mathematical reasoning is based on assumptions and specific rules latter. Thereby, mathematical reasoning is a mind habitual like other habit. Therefore, it must be developed consistently using various context, know both of reasoning and authentication are fundamental aspects in mathematics.
From the interview with Mrs. Eliyati, one of teacher mathematics in SMP NEGERI 1 MEDAN, explain that students mathematical reasoning ability in SMP NEGERI 1 MEDAN are not smooth yet. Still many students have low mathematical reasoning ability. Its impact is seen in national examination that still not enough high yet. This thing is caused by learning implementation not many engaging students yet. Therefore, very important a learning activity that invite students many involve, thinking much of and more motivated to learn mathematics.
concept of the fractional number and equivalent fractions and on developing the operations of addition and multiplication along with their respective inverses
(D’augustine, 1973:182).
Delivered in PMRI National training for Junior high school teacher in Yogyakarta (2010), the aims of fraction learning in Junior High School can be said as follows:
1. Solving contextual problem and finding fraction concept from contextual problem that solved.
2. Understanding fraction concept, explaining the intertwining between concept and implementing fraction concept, flexible, accurately, efficiently, and correctly in problem solving.
3. Using reasoning in the pattern and property, doing manipulation and making generalization about fraction.
4. Communicating the concept and the use of fraction
5. Having appreciated attitude of the use of fraction in daily life.
However, fraction concept is not a simple concept. The uniqueness of fraction, it is different with natural number and integer, some times make it difficult to understood by student (Pitkethley & Hunting, 1996), and make it difficult to introduced to student (Clarke, et al., 2007) in Wahyu (2010). For example:
Based on procedure above, appearing questio ns that need to contemplate such as (1) is it possible student that not yet study about division fraction procedure be able to solve the problem?, (2) does student understand the division of fraction procedure?, (3) what is the problem and the solution mea ningful for student?. The questions need to think well, because delivery the procedure above can make student find difficulties in understanding fraction concept.
5
As an example, Pearn and Stephens (2004) in Wahyu (2010) do a research and find that student still often use the thinking way of integer while solving problem that related with arranging fraction. Student often look the difference between numerator and denominator to determine which fraction is bigger or smaller. Such as: when student is asked to compare fraction .
One of learning approach that engages student’s contribution, the use of life context and interactivity is RME (Realistic Mathematics Education). RME comes from Netherland that developed by Freudenthal (1973, 1991). He has a certain view that mathematic in human life activity. While Verschaffel and Corte
(1996) in Turmudi (2009 : 9) give a term to it as “ mathematics as human sense-making and problem solving activity”. So, The RME theory focuses on guided
reinvention through mathematizing and takes into account students’ informal
solution strategies and interpretations through experientially real context problems. The heart of this reinvention process involves mathematizing activities in problem situations that are experientially real to students. It is important to note that reinvention is a collective, as well as individual activity, in which whole-class discussions centering on conjecture, explanation, and justification play a crucial role (Treffers, 1991; Gravemeijer, 1994, 1999 in Department of Mathematic
Educations’ Ewha Womans University).
RME (Realistic Mathematics Education) gives meaning that mathematics education process is bundle process about what is mathematic, how does student
learn mathematic, and how mathematics must be taught. So, hoped by RME
In a research about professional development, teachers that given assignment to compare realistic approach with nonrealis tic that generally used now, had a nation that in realistic approach students appear more dominant and more effective than daily learning that have done by teacher. Students in this new paradigm actively build understanding and their mathematics knowledge by interacting between students (Turmudi, 2009 : 118). The thing give evidence that learning with Realistic Mathematics Education make students become more interesting and more active in mathematic learning. Thereby, more often students active in learning process then more often too students construct his own knowledge to solve problem. Its mean that students will try to understand and use his logic to solve problem by knowledge that students have until produce mathematic theory. As simple example, from the previous problem that has given
, if teacher gives mathematics knowledge early like this, student will be confused why division operation can be multiplication operation, and why the result the greater becomes.
But, by realistic mathematics approach student is given contextual problem that guided student to reinvention the concept why division operation of fraction can be multiplication operation. Such as give the problem as follows:
“Mother buys cooking oil 1½ liters. Because she wants to give some her cooking oil to neighborhood, then mother infuses the oil to small bottle
size ½ liters. How many bottles that can receive the oil?”
Form the contextual problem above, can described that student can imagine the problem and try to use student’s thinking logic or reasoning in solving the problem. Because, from the infuses oil process to bottles, student is
easier to understand why the result is 3. Student can understand that and have different unit with 3. That is why the use of context need in learning process.
From the example above, it can conclude that Realistic Mathematics Education (RME) invites students to develop step-by-step tool and mathematic comprehension to more formal level.
7
Based on description of the background, then writer interest to do a research with title “Improvement of Student Mathematical Reasoning Ability in VII Grade SMP Negeri 1 Medan by Applying Realistic Mathematics Education (RME) on The Subject of Fraction Academic Year 2012/2013”.
1.2 Proble m Identification
From the background description is obtained problem identification in this study are as follow:
1. Student’s involvement in learning process of mathematics is still less. 2. Mathematics knowledge is not built from meaningful life context and
relevant to students so that students cannot construct his informal skill to be formal skill.
3. Students reasoning ability in mathematics learning is low because it does not make to be habitual early.
1.3 Proble m Restrictions
According to problem statement and research question above then researcher will instruct to the case intended. Researcher makes limitation of the study on improvement ability of students reasoning in mathematics learning by realistic approach.
1.4 Proble m Formulation
Based on the background that have described above, the problem in this research is formulated as follow:
1. How to implement realistic mathematics approach so that be able to improve of students’ mathematical reasoning ability?
2. How is the improvement of students’ mathematical reasoning ability who studied by realistic mathematics approach?
1.5 Research Objectives
According to research question that proposed in this study, therefore researcher is formulated the goal of this research, namely:
1. To know is there an improvement in ability of students mathematical reasoning who studied by realistic mathematics education.
2. To find out whether by following realistic mathematics learning, students become more active interacting in learning process.
3. To find out the description of student response to mathematic for students that follow realistic mathematics learning.
1.6 The Benefits of Research
The result of this research later expected to provide benefits for students and teacher.
1. For students
a. By existence of this research expected can help students to prefer mathematic.
b. It can grow up motivation and interest of students in learning mathematic.
c. Train and improve mathematical communication and reasoning ability of students.
d. Train students’ proficiency to cooperation and interaction with other person.
e. Train students to braver to give argument in learning process.
f. Invite students to more dominant involve and more active in learning.
2. For teacher
As a consideration in choosing appropriate mathematics learning model to improve mathematical reasoning ability of students.
74
CHAPTER V
CONCLUSION AND SUGGESTION
5.1 Conclusion
Based on all of this classroom action research implementation, include learning process, the analysis result, and observation result concluded some cases as follow:
1. Based on the teaching and learning process that have done in this research and the result of observation activity, to get the best result of realistic mathematics approach implementation, the using of context is also supported by visualization like using visual aids and figures. After teacher gives contextual problem, teacher must give student time to understanding the problem. After that, teacher guides student to make a description of the problem based on their experiences in daily life and then, let student to find the solution by using their own model. If students learn in group, teacher also give time to students to compare and discuss the answer with friends. Moreover, the last teacher guides students to discuss together and decides solution the most appropriate one. Then, make the conclusion so that constructed a mathematics concept. Finally, students find out the knowledge that expected.
2. The implementation of realistic mathematics approach can improve mathematical reasoning ability. It can be seen from the result of mathematical reasoning ability test. From cycle I, still under of fifty persen, means the mathematical reasoning ability of students still very low. Then, it is improving become middle level in cycle II. In cycle III improve on high level and in cycle IV achieve more than ninety persen or in very high level.
good criteria also. From the mathematical reasoning ability test, in cycle I only three students that get minimum standard. In cycle II improve become more than eighty persen classically. Moreover, in cycle III and IV all of students have achieved the minimum standard.
5.2 Suggestion
Based on the conclusions that have concluded from all activities in learning process, there are some recommended suggestions, namely:
1. It is suggested to the teacher to more often train the student mathematical reasoning ability in learning process.
2. It is suggested to the teacher to use contextual problem in teaching mathematics and combined with appropriate visual aids, guided student worksheet (GSW) or the figures to help the students construct their own knowledge from informal to formal knowledge. So that, students can reinvention the mathematics concept especially in fraction subject.
3. It is suggested to the students to making the best use of student’s experiences in daily life to build their own knowledge in mathematics. Because by building student’s own knowledge from informal to formal knowledge, student has done thinking process that can help to develop student’s reasoning ability.
4. It is suggested to teacher also to improve stimulation and motivation in the use of interactivity and the use of students’ contribution.
