# 978-602-17465-1-6

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978-602-17465-1-6

PROCEEDINGS The First South East Asia Design/ Development Research (SEA-DR) International Conference “Design Research for Change and Innovation” April, 22nd-23rd, 2013 Organized By: Master Program on Mathematics Education Sriwijaya University Supported By: Freudenthal Institute of Science and Mathematics Education (FISME) Utrecht University, The Netherland. Master Program on Mathematics Education Surabaya State University Himpunan Matematika Indonesia (Indonesia Mathematics Society/ IndoMS) Published by: Master Program on Mathematics Education Sriwijaya Uiversity Palembang, South Sumatra 1st Issue April 2013 Issue Proceeding of the First South East Asia Design/ Development Research (SEA-DR) International Conference 2013 Reviewers: Prof. Dr. Zulkardi, M.I.Komp., M.Sc. [et.al]Palembang, Master Program on Mathematics Education, Sriwijaya University Editors: Prof. Dr. Zulkardi, M.I.Komp., M.Sc. - Palembang, Master Program on Mathematics Education Sriwijaya University, 2013 The proceeding can be accessed at: http://eprints.unsri.ac.id/ ISBN : 978-602-17465-1-6 978-602-17465-1-6 Process editing of all the articles in proceedings is conducted by the Team Reviewer The First South East Asia Design/ Development Research (SEA-DR) International Conference 2013 from Master Program on Mathematics Education, Sriwijaya University ii PROCEEDINGS The First South East Asia Design/ Development Research (SEA-DR) International Conference “Design Research for Change and Innovation” April, 22nd-23rd, 2013 This paper has been presented at The First South East Asia Design/ Development Research (SEA-DR) International Conference “Design Research for Change and Innovation” Master Program on Mathematics Education, Sriwijaya University, Palembang-South Sumatra, April, 22nd-23rd, 2013 Keynote Speakers: 1. 2. 3. 4. Prof (Ass.). Dr. M.L.A.M (Maarten) Dolk (Utrecht University, The Netherlands) Prof (Ass.). Dr.H.A.A. (Dolly) van Eerde (Utrecht University, The Netherlands) Prof (Ass.). Dr. Wang Qiyun (National Institute of Education, Singapore) Prof. Dr. Zulkadi, M.I.Komp., M.Sc. (Sriwijaya University) Plenary Discussion: 1. 2. 3. 4. 5. Mr.F.H.J. (Frans) van Galen (Utrecht University, The Netherlands) Dr. Agung Lukito, MS (Surabaya State University) Prof. Dr. Ahmad Fauzan, M.Pd, M.Sc (Padang State University) Prof. Dr. Didi Suryadi, M.Ed (Indonesia University of Education) Dr. Ratu Ilma Indra Putri, M.Si (Sriwijaya University) Paper Reviewers: 1. 2. 3. 4. 5. 6. 7. Prof. Dr. Zulkadi, M.I.Komp., M.Sc. (Sriwijaya University) Dr. Ratu Ilma Indra Putri, M.Si (Sriwijaya University) Dr. Somakim, M.Pd (Sriwajaya University) Evangelista L.W. Palupi, S.Pd., M.Sc (Surabaya State University) Sylvana Novilia Sumarto, S.Si., M.Sc (Malang State University) Bustang, S.Pd., M.Sc (Makassar State University) Moch. Lutfianto, S.Pd., M.Pd (STKIP Al-Hikmah ) Master Program on Mathematics Education Sriwijaya University, South Sumatera 2013 iii Foreword Design and Developmental Research has been the momentum in educational research. Various innovative development and design of research have evolved so rapidly, especially in South East Asia Region. The development of teaching materials, learning trajectory, assessment material, media, and teacher training development/educational workshop, nowadays, are the result of the design/developmental researches. Several researchers have good methods in developing but others still need more knowledge and information to improve their research quality. Hence, it needs scientific forum to share the theory and experience from the experts about design/developmental research. Regarding the mentioned fact, Faculty of Teacher Training and Education, Sriwijaya University Master Program on Mathematics Education in collaborated with Utrecht University, State University of Surabaya, and the Indonesian Mathematical Society (INDO-MS) organized a regional conference in South East Asia, namely “The First South East Asia-Design/Development Research (SEA-DR) Conference 2013” on the last 22nd23rd April 2013. The goal of the first SEA-DR conference is to give contribution for the development of education in general and mathematics education in particular. This first SEA-DR Conference has theme: "Design Research for Change and Innovation”. I would like to take this opportunity to address my sincere gratitude to the esteemed keynote lecture and plenary panel, Utrecht University, National Institute Education of Singapore, Surabaya State University, Indonesia University of Education, Padang State University, and other parties for their endless supports, so that the conference could be successfully held. My special thanks are also for all participants and the paper presenters from universities, schools and others institutions in South East Asia. And the last, we present this book for all researchers and educators in the world. We welcome any inputs and criticisms to improve this program for the next time. See you on the SEA-DR Conference 2014. Head Committee Prof. Dr. Zulkadi, M.I.Komp., M.Sc. iv CONTENTS Cover ....................................................................................................................................................... i Publishing Statement ........................................................................................................................ ii Paper Reviewers .................................................................................................................................. iii Foreword................................................................................................................................................. iv Contents ................................................................................................................................................... v Code Name Institution Title Page K-1 Dolly van Eerde Freudenthal Institute SME, Utrecht University, the Netherlands Email:[email protected] u.nl Design Research: Looking Into The Heart of Mathematics Education 1 K-2 Frans van Galen Contexts and Models in Mathematics Education 12 K-3 Ratu Ilma Indra Putri Freudenthal Institute for Science and Mathematics Education (FISME), Utrecht University Email:[email protected] l Sriwijaya University Email: [email protected] K-4 Zulkadi K-5 Maarten Dolk K-6 Wang Qiyun P-1 Achmad Dhany F. 1 , Ummy Salmah 2 The Role of Model in Design Research at Sriwijaya University Sriwijaya University Design research: Email: Why, How and [email protected] What? Some examples from personal experiences Utrecht University, The Creating A Culture of Netherlands Mathematizing: Email: [email protected] Allowing Students to be Young Mathematicians National Institute of Supporting Group Education, Singapore Collaboration in An Email: Online Learning [email protected] Environment: An Educational Design Research Case 1 State University of Surabaya Email: [email protected] 2 State University of Makassar The Development of Students Worksheet Using PMRI Approach on Materials of Rectangle and 22 29 30 31 1 v Email: [email protected] Square for the VII Grade Students of Junior High School P-2 Adelia Pramarista STKIP Surya Email: [email protected] .id Learning Design on Whole Number Addition Using Gasing Method 11 P-3 Ahmad Wachidul Kohar 1 , Abdul Haris Rosyidi 2 1 State University of Surabaya Email: [email protected] m 2 State University of Surabaya Email: [email protected] m The Development of Mathematics Learning Instruments Integrating Multiple Intelligences on Topics of Cuboid And Cube for The Eighth Grade Students of Junior High School 20 P-4 Ardoni Padang State University Email: [email protected] The Learning Medium Engineering Based on Expert System for Information Services Sources Course in the Study Program of Library and Information Science 32 P-5 Azizah Husin Sriwijaya University Email: [email protected] id Implementation of Experiential Learning Methods on Environmental Lesson for Elementary School 42 P-6 Bustang 1 , Dolly van Eerde 2 , Zulkardi 3 , Darmawijoyo 4 1State Reinventing the Concept of Angle by Spatial Representations and Physical Activities 50 P-7 Cecil Hiltrimartin Quality of Students Problem Solving 60 University of Makassar Email: [email protected], , 2Utrecht University, Email:[email protected] u.nl 3Sriwijaya University Email: [email protected], 3Sriwijaya University Email: [email protected] m Sriwijaya University Email: vi [email protected] om Worksheet Designed by Junior High School Mathematics Teachers in Gunung Megang P-8 Devi Nofriyanti 1 , Ratu Ilma Indra Putri 2 1 Sriwijaya University Email: [email protected] 2 Sriwijaya University Email: [email protected] Developing Instructional Materials of Tangents to Two Circles Using Multimedia With PMRI Approach in 8th Grades 65 P-9 Dwi Afrini Risma 1 , Dolly van Eerde 2 , Mieke Abel 3 , Ratu Ilma Indra Putri 4 1STIKIP Developing Students’ Spatial Ability through Spatial Visualization and Spatial Orientation Tasks 74 Developing students’ Spatial Visualization on Volume Measurement 84 Map as A Tool to Support the Development of Spatial Ability 95 How Can the Students Of 5th Grade Make Up A System to Locate An Object: Level Of Child’s Developmental Thinking Of Locate A Point 104 P-10 P-11 P-12 Meranti Email: [email protected] m 2Utrecht University Email:[email protected] u.nl 3 Utrecht University Email: [email protected] 4 Sriwijaya University Email: [email protected] 1 Universitas Negeri Dwi Sulistya Kusumaningrum 1 Jakarta, , Meiliasari 2 Email:[email protected] mail.com 2 Universitas Negeri Jakarta Email:[email protected] il.com 1Sriwijaya University Elika Kurniadi1, Ratu Ilma Indra Email: 2 Putri , Yusuf [email protected] Hartono3, Meike J. d, 2Sriwijaya University Abels4 Email:[email protected] o.com 3Sriwijaya University Email:[email protected] i.ac.id 4Uttrecht University Email: [email protected] 1State University of Evangelista L.W. 1 2 Palupi , Zulkardi , Surabaya Darmawijoyo3, Email: 4 Meike J. Abels [email protected] hoo.com 2 Sriwijaya University Email: [email protected] 3 Sriwijaya University vii P-13 Fadli1, Basuki Wibawa2, Zulfiati Syahrial3 P-14 Fazri Zuzano1, AliranTelaumbanua 2 P-15 Febrian1, Monica Wijers2, Dwi Juniati3, Agung Lukito4 P-16 Fitriana Rahmawati P-17 Friska Riyanti Email: [email protected] 4Utrecht University Email:[email protected] 1STKIP PGRI Lubuklinggau Email: [email protected] 2Universitas Negeri Jakarta Email:[email protected] m 3Universitas Negeri Jakarta Email: [email protected] m3 Design of Social Network Learning Model 114 Hatta University Email: [email protected] 2Bung Hatta University Email: [email protected] o.co.id 1Surabaya State University Email: [email protected] 2Utrecht University Email: [email protected], 3Surabaya State University Email: [email protected], 4Surabaya State University Email: [email protected] STKIP PGRI Bandar Lampung Email:[email protected] o.id Development of Education Game Based Interactive Maths Teaching Media on Probability for Year Eleven Students 124 Developing Students’ Initial Understanding of Area Measurement Through The Unit: Teaching Experiment on A Lesson with Cashewnut Cookie Context Conducted in Third Grade Students of Elementary School 132 Logical Reasoning Elementary School Students in Debit Measurement Learning Based On Pendidikan Matematika Realistik Indonesia (PMRI) 144 State University of Padang Email:[email protected] om Development of Context-Based Mathematics Learning Tool for Fifth-Grade Elementary School Students 153 1Bung viii P-18 Hendra Lesmana Sriwijaya University Email:[email protected] ahoo.com Developing Computer Based Non Routine Problem to Train The Student Employing Fundamental Mathematical Capabilities at 9th Grade SMP N 2 Belitang III 162 P-19 Hongki Julie1, 1,2Sanata Dharma University, Email: [email protected] 3Surabaya State University Email:[email protected] com 1Universitas Tadulako Email: [email protected] o.co.id 2Universitas Tadulako Email:[email protected] .co.id State University of Surabaya [email protected] First Cycle Developing Teaching Materials for Integers in Grade Four with Realistic Mathematics Education 172 Design Research in PMRI: Flash Media to Support the Fourth Grade Student Learn Great Common Divisor (GCD) Materials 183 Using Array Representations to Show the Commutative Property of Multiplication 188 1Sriwijaya Learning Design Using Pendidikan Matematika Realistik Indonesia (PMRI) Approach For the Topic Surface Area And Volume Of Cuboid For 2nd Grade Junior High School 197 St. Suwarsono2, Dwi Juniati3 P-20 I Ketut Kertayasa1, I Wayan Sukayasa2 P-21 Ismi Ridha AsySyifaa P-22 Karin Amelia Safitri1, Ratu Ilma Indra Putri2 University Email:[email protected] l.com 2Sriwijaya University Email:[email protected] o.com P-23 Koichi N. Tomita Kota Kinabalu, Japan Email: [email protected] m To Design Lesson Well We'll Reconfirm the Sequence of Teaching Fractions: How to Design Classroom Lesson 204 P-24 Lisnani1, Ratu Ilma Indra Putri2, Somakim3 1Sriwijaya Quadrilateral Materials Design with Fable "Catches Dog Cat" Tangram Puzzle for Students and Class II 209 University Email:[email protected] oo.com 2Sriwijaya University Email:[email protected] m 3Sriwijaya University ix P-25 Mardiah Harun1, Yullys Helsa2 P-26 Media harja1, Zulkardi2, Budi santoso3 P-27 Meldariani Roy1, Ratu Ilma Indra Putri2, Somakim3 P-28 Meilani Safitri1, Yusuf Hartono2, Somakim3 P-29 Misdalina P-30 Moch. Lutfianto Email:[email protected] hoo.com 1Elementary School Teachers Educational Programs Email: [email protected] m 2Padang State University Email: [email protected] 1SMA Plus Negeri 2 Banyuasin III Email: [email protected] 3 Sriwijaya University Email: [email protected] 3 Sriwijaya University Email: [email protected] .com 1 Sriwijaya University Email: [email protected] mail.com 2 Sriwijaya University Email: [email protected] 3 Sriwijaya University Email: [email protected] m 1 Sriwijaya University Email: [email protected] com 2 Sriwijaya University Email: [email protected] 3 Sriwijaya University Email: [email protected] m PGRI University of Palembang Email:[email protected] ail.com STKIP Al Hikmah Email:[email protected] .com The Development Of Mathematics Assessment Model in Elementary School 219 Teaching Materials Development Volume Objects to Play the Valid Blog 226 Student’s Ability to Construct Mathematical Ideas in The Topic of Coordinate Cartesian System In Primary School 236 Development of Learning Media Based Macromedia Flash about Triangle for Student Grade 7 Junior High School 241 Instructional Design of Integration Factor Based on Reciprocal Teaching Approach 251 Unfinished Student Answer in PISA Mathematics Contextual Problem 262 x P-31 Muchlishah Rosyadah1, Pinta Deniyanti S2, Meiliasari3 P-32 Nila Kesumawati P-33 Niniwati1, University of Jakarta Email: [email protected] m 2State University of Jakarta Email: [email protected] 3State University of Jakarta Email: [email protected] PGRI University of Palembang Email: [email protected] om Develop Inductive Reasoning on Pattern Numbers with A Realistic Mathematics Education Approach in The Ninth Grade Students In Mts AlKenaniyah, Jakarta 269 Development Mathematical Creative Thinking Ability Problems on The Topics of Fractions For 7 Grade Students 279 1Bung Developing ComputerBased Interactive Instructional Medium of Mathematics for Senior High School Grade at Three Dimensions Material 285 Novi SMAN 1 Penukal Utara1 1 Komariyatiningsih , Email: Ratu Ilma Indra [email protected] 2 Sriwijaya University Putri2, Email: Nila Kesumawati3 [email protected] 3 PGRI University of Palembang Email: [email protected] om Nyiayu Fahriza PGRI University of Fuadiah Palembang Email: [email protected] Communication Mathematics In Probability Of Students Grade XI IPA Using PMRI Approach 294 Design of Teaching Materials Based on Realistic Mathematic Education (RME) to Improve Students Creative Thinking Ability 299 P-36 Petra Suwasti STKIP Surya Email:[email protected] tkipsurya.ac.id The Use of Gasing Method for Teaching Two-Digit Subtraction for 2nd Grade Students of SDN Cihuni II Tangerang 306 P-37 Pramanika Arieyantini1, 1 Sriwijaya The Context of Animal Vegetative Propagation 314 Andika Putra. P-34 P-35 R2 1State Hatta University Email: [email protected] 2Bung Hatta University Email: [email protected] Ratu Email: University xi Ilma Indra Putri2, Nila Kesumawati3 P-38 Puji Astuti P-39 Puspita Sari P-40 Rahmah Johar1, Cut Khairunnisak2 P-41 1Syiah Kuala University Email: [email protected] h.ac.id 2STKIP Bina Bangsa Getsempena, Aceh Email:[email protected] o.co.id 1Sriwijaya University Indra Email: [email protected] 2 Sriwijaya University Email: [email protected] Rindu Alriavindrafunny1, Siti.M.Amin2, Agung Lukito3, Monica Wijers4 P-43 in Exponential Learning at Junior High School Rala Novita Sari1, Ratu Ilma Putri2 P-42 [email protected] ahoo.com 2 Sriwijaya University Email: [email protected] 3 PGRI University of Palembang Email: [email protected] om 1 Surabaya State University Email:[email protected] mail.com State University of Jakarta Email: [email protected] Risnawati1, Zubaidah Amir2, Defi3 1State University of Malang Email: [email protected] 2State University of Surabaya Email: [email protected] 3State University of Surabaya Email: [email protected] 4Utrecht University Email: [email protected] 1UIN Sultan Syarif Kasim Riau Email: [email protected] Learning One-Digit Decimal Numbers by Measurement Activity 323 An Empty Number Line to Develop Mental Arithmetic Strategies 334 Supporting Students In Learning Multiplication through Splitting Strategy 344 The Development of Learning Set Material Comparison Using Indonesian Realistic Mathematics Education (PMRI) to Determine The Ability of Students Representation 355 Understanding the Concept of Conservation of Area: Recomposing A Shape Will Preserve Its Area 362 Development of Learning Instrument of Opportunities Theory Using Problem Based 372 xii P-44 Risnawati1, Zubaidah Amir2, Defi3 2UIN Sultan Syarif Kasim Riau Email: [email protected] 3UIN Sultan Syarif Kasim Riau Email: [email protected] Instruction (PBI) Model for Students of Mathematics Education Program Education and Teachers’ Training Faculty, UIN Sultan Syarif Kasim Riau 1UIN Trigonometry Module Development Approach Using Aptitude Treatment and Interaction (ATI) for Education and Teacher Training Faculty Students At Mathematics Department of State Islamic University of Suska UIN Riau 381 Sultan Syarif Kasim Riau Email: [email protected] 2UIN Sultan Syarif Kasim Riau Email: [email protected] 3UIN Sultan Syarif Kasim Riau Email: [email protected] P-45 Rully Charitas Indra Surya College of Prahmana Education (STKIP Surya) Email:[email protected] tkipsurya.ac.id Designing Division Operation Learning in The Mathematics Of Gasing 391 P-46 Sakinah Fajri1, PGRI Padang Email: [email protected] 2 Sriwijaya University Email: [email protected] 3 Sriwijaya University Email: [email protected] 4 Utrecht University Email: [email protected] Surabaya State University Email: [email protected] m Introducing the Position of Numbers with Number Line Activities 399 First Cycle on Designing the Tangram Game Activities as an Introduction to The Concept of Area Conservation 406 1STKIP Educational Design Research: Developing Students’ Understanding of Area as The Number of Measurement Units Covering A Surface 416 Nurul Ratu Ilma Indra Putri2, Yusuf Hartono3, Frans van Galen4 P-47 Shofan Fiangga P-48 Susilahudin Putrawangsa1, Agung Lukito2, Siti M Amin3, Monica Wijers4 1 STKIP Hamzanwadi Selong Lombok Email: [email protected] om 3State University of Surabaya Email: xiii P-49 Sylvana N. Sumarto1, Zulkardi2, Darmawijoyo3, Frans van Galen4 [email protected] 2State University of Surabaya Email: [email protected] 4Utrecht University Email: [email protected] 1 State University of Malang, 1 St Albertus Senior High School Email: [email protected] .com 2 Sriwijaya University Email: [email protected] 3 Sriwijaya University Email: [email protected] com 4 Utrecth University Email: [email protected] Bung Hatta University Email: [email protected] Design Research: Ratio Table and Money Context as Means to Support the Development of Students’ proportional Reasoning 427 The Development of Interactive Mathematics Instructional Media for Application of Integral Materials 436 P-50 Syukma Netti P-51 Tanzimah PGRI University of Palembang Email:tanzimah.imah @ yahoo.com Learning Design on the Integers Addition and Reduction Using "Kakisambe" 444 P-52 Thyas Agie1, 1 Sriwijaya Developing Instructional Materials Complement of Set Based on Indonesian Version of Realistic Mathematics Education in 7th Grade of Junior High School 453 Design Research for Exploring Cognitive Process and Soft Skill Attributes Based on Personality Type Classification 461 Supporting Students’ First Conception about Addition of Integers Through Number Line Activities for Third 468 Ratu Ilma2 P-53 Tri Sagirani1, M.J. Dewiyani S.2 P-54 Weni Dwi Pratiwi1, Siti M. Amin2, Agung Lukito3, University Email: [email protected] m 2 Sriwijaya University Email: [email protected] 1STMIK Surabaya Email: [email protected] 2STMIK Surabaya Email:[email protected] du 1PGRI University of Palembang Email: [email protected] m xiv Frans Van Galen4 P-55 Zubaidah Amir1, Risnawati2, Defi3 2State University of Surabaya Email: [email protected] 3State University of Surabaya Email: [email protected] 4 Utrecth University Email: [email protected] 1UIN Sultan Syarif Kasim Riau Email: [email protected] 2UIN Sultan Syarif Kasim Riau Email: [email protected] 3UIN Sultan Syarif Kasim Riau Email:[email protected] Grade Primary School Constructivism-Based Development Workbook Lectures on Linear Program in The Department of Mathematics Education, Faculty and Teaching Tarbiyah UIN Riau Suska 477 xv 1st SEA-DR PROCEEDING ISBN : 978-602-17465-1-6 Code: P-40 SUPPORTING STUDENTS IN LEARNING MULTIPLICATION THROUGH SPLITTING STRATEGY Rahmah Johar1, Cut Khairunnisak2 Syiah Kuala University1, STKIP Bina Bangsa Getsempena, Aceh2 [email protected], [email protected] Abstract Most Indonesian textbooks do not support students to develop and discuss a variety of multiplication strategies. The books mostly introduces multiplication tables, let students memorize them, and then continues by ask students use multiplication algorithm. As a result, the students can apply that algorithm but they did not know the reason behind that algorithm. Realistic Mathematics Education for Indonesia (PMRI) has designed PMRI textbooks. Some of topics are developed by using Design Research. There are two cycles in this research; each of the cycles consists of preparation, teaching experiment, and a retrospective analysis phase. The contexts and activities used in the first cycle were revised and changed to be more meaningful for Indonesian students. The changes then were employed in the second cycle. Our findings suggest that to support students in developing a splitting strategy, we need some meaningful activities such as developing multiplication network, multiplication by tens, and various contexts consisting of onedigit and two-digit multiplications Keywords: Splitting strategies, design research, multiplication INTRODUCTION In a 'traditional' learning process, the teacher takes control of each activity. The teacher extensively directs, explains, and questions the students, followed by working on paper-and-pencil assignments. Many teachers, included Indonesian’ teachers transmit knowledge in the form of rules and tricks and learners have to practice those rules with straightforward and simple problems. In practicing time, the teacher asks student to discuss how to implement their explanation to the similar problem. If there is any mathematical thinking happening in the classroom, it is often the teacher who is doing that thinking. Students are not really invited to think (Dolk et.al, 2010; Widjaya, Dolk, and Fauzan, 2010; Johar and Afrina, 2011). This condition is occurred due to the good ideas of Indonesian curriculum did not supported by neither teaching strategies of teachers nor text book. Most Indonesian textbooks do not support students’ strategies to obtain the result of multiplication in many ways. They mostly introduce multiplication as a repeated addition through multiplication tables, and then continued using the multiplication algorithm. For example, when someone asks students to answer 7x6 = …., the students have memorized multiplication table and will start by reciting 1x6=6, 2x6=12, 3x6=18, 4x6=24, 5x6=30, 6x6=36, 7x6=42, and sometimes they are miscalculating. They cannot make some multiplication networking, which is stringing strategy. The example stringing strategy is 7x6 = 5x6 + 2x6 = 30 + 12 = 42, or 7x6 = 3x6 + 3x6 + 1x6 = 18 + 18 + 6 = 42. While the students answer multiplication 344 1st SEA-DR PROCEEDING ISBN : 978-602-17465-1-6 problems with the bigger number, for instance 26x4, the students use standard algorithm of multiplication. At this condition, the student can apply that algorithm but they did not know the reason behind that algorithm. According to Realistic Mathematics Education (RME) was introduced by Freudenthal in the Netherlands in 1968 as cited in Gravemeijer (1994), students should experience mathematics as a human activity. It is an activity of solving problems, looking for problems, and organizing a subject matter. In addition, van den HeuvelPanhuizen (1996) suggests mathematics must be taught in the order in which the students themselves might be inventing it. In this line, van den Heuvel-Panhuizen (2005) explained that mental calculation is the backbone of the primary school number strand. Mental calculation is considered as insightful calculation in which the children make use of memorized number facts and the properties of numbers and operations. Thus mental calculation is not simply doing calculations in the head. It is more a matter of using the head for making the calculations. Writing down intermediate steps or using the empty number line may be part of it. The basic strategies for mental calculation are: stringing, splitting and varying. For example 1) a stringing strategy; 6x48 = … It is 3×48=144; 3×48=144; 144+144=288, b) a splitting strategy; 6x48 = … It is 6×40=240; 6×8=48; 240+48=288, and 3) a varying strategy; 19×25= …. It is (20×25)–(1×25)=475. The splitting strategy is simpler than the stringing strategy because the splitting strategy use tens. Students need experiences using both splitting strategy and stringing strategy. To develop RME in Indonesia, Indonesian mathematicians and mathematics educator have designed “Realistic Mathematics Education in Indonesia” or Pendidikan Matematika Realistik Indonesia (PMRI), known as the Indonesian location of RME. Dutch consultants from the APS Netherlands and Freudenthal Institute (FI) of Utrecht University have been involved in the coaching the PMRI team. The team that consists of teachers educator and teachers started a pilot program of PMRI from 2001 until now. The PMRI team wants to change mathematics education in such a way that most children will be able to do and enjoy mathematics to develop their mathematics knowledge, skills, and strategies. Dolk et.al (2010) suggested that such changes start in classroom. Creating learning community in the classroom asks for another role of teachers and students. One aspect of these new norms is related to the teacher’s expectations and beliefs about students’ mathematical thinking. In summer 2007, the PMRI team decided to start working on a complete mathematics textbook series for primary education. The team developed a trajectory in which the production of learning materials goes a long side with workshops for prospective authors, with the implementation of design research, and with a further development of the didactical framework for PMRI (Amin et.al, 2010). Multiplication is one of the topic in PMRI textbook that introduced since second grade. At the third grade of textbook, students start to develop mental calculation using splitting strategies for multiplication in one digit by two digits. Mental arithmetic is back in favour (Pepper, in Beishuizen, Meindert, 1998) and many authors offer suggestions for everyday mental activities. A common viewpoint is to stimulate students own mental methods and to discuss several approaches and procedures. The expectation is that--when practiced on a regular basis--they will develop or adopt more efficient and flexible strategies in due course. 345 1st SEA-DR PROCEEDING ISBN : 978-602-17465-1-6 We are as one of the authors of PMRI textbooks have designed the some learning trajectories for primary school students. Students are guided to solve many realistic problems about multiplication with various strategies, not introduce a formal algorithm immediately for the third grade 3 student. This research aimed to develop a local instruction theory in PMRI textbooks to support students in learning multiplication through splitting strategy. RESEARCH METHODOLOGY The main goal of this research is to investigate how to support students in learning multiplication through splitting strategy. This research used a type of research method namely design research for achieving the research goal. Design research is a type of research methods aimed to develop theories about both the process of learning and the means that are designed to support that learning. There are three phases of conducting a design research. Those three phases are preparation and design phase, teaching experiment, and retrospective analysis (Gravemeijer & Cobb, 2006). There are two cycles in this design research. The preparation of the first cycle is held on November to December, 2010. At that time one of the authors came to Freudenthal Institute, Universiteit Utrecht. Then the topic of multiplication is tried it out for several schools, which are on January 31st in SDN 48 Banda Aceh and on February 9th 2011 in SDN 7 Banda Aceh. Last, at 8th to 15th of March 2011 it is held in Hof ter Weide school and St. Dominiccus school in Utrecht, the Netherlands. At the second cycle, the authors of PMRI text book continue to design text book since June to November 2011 as a Final Draft. Further more on September to November 2012 the learning trajectory for multiplication in PMRI textbook for third grade is implemented at SDIT Nurul Ishlah Banda Aceh on 29th of Augustus 2012 to 19th of September 2012 and 20th to 22th of March 2013. This teaching experiment is held for seven meetings. During teaching experiment, the research team consisting of teacher educators and teachers designed the investigation, observed the class at work, discussed their observations, analyzed the data, and planned the following day’s investigation. The resident teacher was present and worked closely together with the research team during all these steps. Data collection was gained through interviewing teacher and students, observing activities in classroom, and collecting students’ works. In this research the video data provides the primary data. The video recorded activities and discussion in the whole class and in some groups of students, and also recorded interviews with teacher and some students. The written data was provided as an addition to the video data. In this research, the written data includes student’s work, observation sheet, assessments result, and some notes gathered during the experiment. . RESULTS AND DISCUSSION Result of the First Cycle Preparation of this cycle is held on November to December, 2010. At that time one of the authors came to Freudenthal Institute, Universiteit Utrecht. She looks for many resources about teaching mathematics for third and fourth grade and discusses them with the experts of RME at Freudenthal Institute. In this preparation step, the 346 1st SEA-DR PROCEEDING ISBN : 978-602-17465-1-6 Hypothetical Learning Trajectory (HLT) designed for the teaching experiment multiplication for the third grade of primary school as follow. 1) Figure of group of tomatoes and two-direction-network of multiplication 2) The number of car’s wheel and the number of tennis balls in boxes 3) Students are able to use many ways to split multiplication of 2-digits and 1-digit numbers 4) Rearranging seats in a party and calculate how many seat are there 5) The number of candy in the inside of a tube with 10 candies and the outside of a tube, for the number of student who get them 6) Multiplication of two-digit and one-digit numbers using the rectangle visualization 7) Multiplication of 2-digits and 1-digit using mental calculation without the rectangle Due to all of Indonesian students at the third grade have learned about algorithm of multiplication of two-digit and one-digit numbers on September to October every year, the researcher choose to implement the stringing strategy of multiplication of two-digit and one-digit numbers. Before teaching experiment, the researchers and teachers revise context from ‘becak’ to car, design the lesson plans, and prepare some media. Then we made some conjectures about students’ strategy to solve the problem. On January 31st, the teacher of SDN 48 Banda Aceh taught about stringing strategy of multiplication to count the number of car’s wheel (see figure 1 a) and the total of tennis balls in 17 boxes, each box has 6 balls. Figure 1. a. The car’s wheel Problem and b. students strategy The strategy of many groups of students is repeated addition one by one car, or two by two cars, and so on. Only one group of student has the pattern mentally (see figure 1 b). For the second problem, about ‘the total of tennis balls in 17 boxes, each box has 6 balls, the students’ strategy almost the same. There is one group implements multiplication algorithm to solve that problem, however they didn’t implement that algorithm correctly (see figure on the right side). In this class, there is no student made connection between repeated addition and multiplication symbol. 347 1st SEA-DR PROCEEDING ISBN : 978-602-17465-1-6 The researchers want to know whether the other students in the different schools have the same strategy to solve the problem above. On February 8th 2011 the teacher at SDN 7 Banda Aceh taught about stringing strategy of multiplication to count the number of car’s wheel and the number of tennis balls. The SDN 7 students’ strategy are similar to the SDN 48 students’ strategy, which are the repeated addition, without multiplication symbol as a stringing strategy or splitting strategy. On February 9th 2011 the researcher taught at SDN 7 Banda Aceh about stringing strategy of multiplication. The context is arranging the seats to count easier the number of seats. The teacher hung the media on the board (see figure on the right side). Firstly, the students add seats row by row. To develop their strategy, teacher gave some small stones as a representative of seats, then teacher rows of seats,Each each asked student to arrange that seats for some paths so thatThere easyareto5 calculate. row has 16 seats. Is it enough group of student arranges them and split them into some parts then students draw 90 peoples?strategy How are you some stones as a circle, they count the number of seats asforstringing (see sure? figure 2 below). 5x5 =25 5x5 =25 5x6 =30 5x10 5x6 = =50 30 Figure 2. The strategies of students SDN 7 Banda Aceh There are only four stringing strategies of the Acehness students found. Next, researcher and some Aceh teachers join Exchange Program to Utrecht, Netherlands for three weeks on February 23rd to March 15th, 2011. One of our activities was give lesson at third grade of Primary School about multiplication. We changed the manipulative as a representative of seats; they are some pieces of paper, instead of small stones. We also changed the picture about 6 balls in a box. Due to the Dutch students at Hof ter Weide and St. Dominiccus school have learned about splitting strategies on February 2011, they got many stinging and splitting trategies to solve multiplication problems about the number of 14 cars’ wheels and number of 6 balls in 17 boxes, as the figure 3 and 4 follow. Figure 3. Hof ter Weide School Students’ Strategies 4x4 = 16 10x4 = 40 16+40 = 56 348 1st SEA-DR PROCEEDING 10x4 + 4x4 = 56 4x4 = 16 = 56 4x10 = 40 10x4 = 40 11x4 = 44 12x4 = 48 13x4 = 52 14x4 = 56 ISBN : 978-602-17465-1-6 10 x 4 = 40+4+4+4+4 = 56 4+4+4+4+4+4+4+4+4+4+4+4+4 +4 Figure 4. St. Dominiccus School Students’ Strategies ccording to the Utrecht students’ strategy, they found some stringing strategies and splitting strategy. Some of them represent repeated addition with multiplication symbol. Most of them use mental calculation to count the result of multiplication as whether stringing or splitting strategy. One interesting mental calculation is 17 x 6 = 10 x 6 + 7 = 102. He means 17 x 6 = 10 x 6 + 7 x 6 = 102, he counted quickly. In addition, there is one student at Hof ter Weide School use ratio table to solve that problem. Next day, the students solve the problem about arranging some seats; ‘There are 5 rows of seats, each row has 16 seats. Is it enough for 90 peoples? How are you sure? How many seats are there?’ They got many stinging and splitting strategies to solve multiplication problems for lesson 2 as the figure 5 follow. Figure 5. Hof ter Weide School Students’ Strategies The St. Dominiccus School students are more fluent than Hot ter Weide School students about implement splitting strategy. They have many various stringing strategies to find the result of 5 x 16. Then, that student decided 5 x 10 and 5 x 6 is the easiest one. The result of first cycle are; 1) the author of PMRI textbook need to put some activities about multiplication by 10 before splitting strategy and 2) the researcher have some conjectures of students’ splitting strategies. This result will be inserted in Hypothetical Learning Trajectory (HLT) for the teaching experiment of the second cycle. Result of the Second Cycle The second cycle of learning splitting strategy of multiplication had purposes to applying, evaluating, and revising the previous HLT. The activities used in the learning trajectory were adapted from PMRI draft book for the third grade students. Therefore, the focus of this second cycle is to know how the route of activities works. It was predicted that the researchers need to reform the design of learning route, according to how the students accept and understand it after the teaching experiment. According to what happened in the teaching experiment at the first cycle, the previous learning route that has been designed in the preparation phase was elaborated and revised. The teaching experiment was conducted in 8 days, each day has a different contexts/ problems. The retrospective analysis through teaching 349 1st SEA-DR PROCEEDING ISBN : 978-602-17465-1-6 experiment was elaborated in each stage of students’ learning route, instead of each day/activity. The designed HLT was used as guideline of the retrospective analysis. The aim of the retrospective analysis was that to explain how students acquire the basic concepts of splitting strategy of multiplication, thus could be generalized to be an instructional design. The activities for the second cycle are as follow: 1. One-Time-Less and One-Time-More Group of Tomatoes as Introduction of TwoDirection-Network of Multiplication 2. Counting Group of Objects: Recalling Repeated Addition as Multiplication 3. Developing Idea of Four-Direction of Multiplication Network 4. Seeds in Pouches and Matches in Glasses: Multiplying by Tens 5. Passengers of Tricycles: Two-Digit and One-Digit Multiplication 6. Rearranging Seats for Party: Stringing Strategy for One-Digit and Two-Digit Multiplication 7. Grouping the Stamps: Eliciting the Splitting Strategy of Multiplication 8. Multiplication of two-digits and one-digit numbers The discussion of this paper only focus on activity 5 to 8. We will discuss the students answer and their problem to get that answer Activity 5: Passengers of Tricycles: Two-Digit and One-Digit Multiplication The context used was about the total number of passengers accommodated by 13 tricycles, which picture presented in front of the classroom. As the starting point, with the aim at engaging the students in the activities, the teachers asked some students to conduct any question related to the tricycle. One of the student’s answers was “What is the total number of tricycles wheels?” The teacher then asked “What can you do to count it easily?” Through the guidance of the teacher, the students told that it can be easier to count it by grouping the tricycles, which consisted different number of tricycles. One group consisted of 2 tricycles, another group had 2 tricycles, and the remainder 3 groups consisted of 3, 5, and 1 tricycle. Figure 6. Students’ Strategy to Group Tricycles Figure 6 shows the strategy of grouping written by the teacher according to students’ answer. As shown in the figure, the tricycles were grouped in five groups with different number of tricycle. The number of wheels for group 1 (consisting 2 tricycles) is 2×3=6. At this rate, the numbers of wheels for the other groups are 2×3=6, 3×3=9, 5×3=15, 1×3=3. Thus, the total number of those wheels is 39. After class discussion that resulted in many ways of grouping (stringing strategy of multiplication) the tricycles, the students then were given a task to find the total number of passenger could be accommodated by those tricycles. At the first time, it seemed that most of the students had misunderstanding about the stringing strategy. Rather than grouping the tricycle and counting the number of passengers, it seemed that they split 26 (the total number of passengers) to become the other style of 350 1st SEA-DR PROCEEDING ISBN : 978-602-17465-1-6 multiplication, even repeated addition (see Figure 7, pointed by the arrow). From the figure, the researcher concluded that in order to make various strategies, the students reversed the order of number. Figure 7. Students’ Strategies in Calculating Passengers Accommodated by Tricycles On the next day, the teacher provoked students to recall the stringing strategy of multiplication. The teacher asked two students to answer the homework (given before) in front of the class. The task is about determining the total number of tennis balls in 17 containers if each container accommodated 6 balls. At the first, the students could not remember the idea. However, once they remembered it, those two students came up with many correct stringing strategies of multiplication on the whiteboard. Appropriate with one of the characteristics of RME, the teacher used the students’ strategies to assist the other students to grasp with the idea of stringing strategy of multiplication. Thus, the other students could pose some other strategy of stringing the multiplication. Activity 6: Rearranging Seats for Party: Stringing Strategy for One-Digit and TwoDigit Multiplication The context offered to extend students’ understanding of stringing strategy of multiplication was rearranging seats for party. Initially, the seats were arranged (close-set) in 5 rows of 16 seats. As the representation of the seats, the students had pebbles to be rearranged. The students’ task was to put ‘path’ that people can easily sit on any seats (including the middle part of the row). Figure 8. Students’ Correct Strategies to Rearrange the Seats 351 1st SEA-DR PROCEEDING ISBN : 978-602-17465-1-6 The students used objects (such as pen or ruler) as the ‘path’ to separate the seats. Then, they wrote the multiplication represented by those arrangements on a paper. At the Figure 8, the group of students proposed three different stringing strategies: 1) putting the ‘path’ exactly in the middle of the seats thus the multiplication become 5×8=40 and 5×8=40 2) moving the ‘path’ to the previous column thus they got 5×9=45 and 5×7=35, and 3) moving the ‘path’ to the next column thus the multiplication become 5×6=30 and 5×10=50 Meanwhile, another group used three ‘paths’, thus the multiplication was splitting into four. However, according to the first strategy on the right side of Figure 8, it seemed that at the beginning the group had confusion how to split the multiplication (since the students made some multiplication and then scratched it). Then, after some guidance from the teacher, how to relate the ‘path’ rearrangement and the multiplication, the students in that group could come up with many stringing strategies. Activity 7: Grouping the Stamps: Eliciting the Splitting Strategy of Multiplication Firstly, show previous students strategy about finding the number of tennis balls in 17 containers, and about rearrangement seats activity. Teacher ask question, which strategy is faster than the other?”. However, the students said that all multiplications were easy, because multiplication of five rows. After that, the students were given another new context, which is group of stamps. The stamps were arranged in a rectangular form, each ten columns of stamps were separated from the ones. The expectation from this activity was that the students could grasp the splitting strategy of multiplication, where the two-digit number was separated by tens and ones. The students’ solution for stamps problem as Figure 9 follow. 10 x 6 = 60 5 x 6 = 30 + 15 x 6 = 80 10 x 8 = 80 7 x 8 = 42 + 17 x 8 = 122 Figure 9. Examples of Students Splitting Strategy using Stamps’ Grouping The next problems are some rectangle is spitted by tens and ones. The students’ solution is as Figure 10 below on the left side. The last problems are the student draw by themselves to find the results of multiplication one-digit and two digits. Their solutions are as Figure 10 on the right side. 352 1st SEA-DR PROCEEDING ISBN : 978-602-17465-1-6 Figure 10. Student answer about groups of stamps According to the students’ worksheets, the researcher concluded that the students already understood about the splitting strategy guided by rectangle model. Activity 8: Multiplication of two-digits and one-digit numbers They can also elaborate it to the more formal strategy, where they have to make their own rectangle picture to represent the number split to tens and ones. Finally, the students used splitting strategy mentally, even though for some time it was difficult for them (the students asked to make drawing as the representation). Figure on the right side shows students’ answers when they were asked to write any multiplication of two-digits and one-digit numbers. They used mental splitting strategy to calculate the result. According to the informal interview with some students, all of them who were interviewed said that the splitting strategy is a lot easier and faster than their other strategies, such as standard algorithm of multiplication or stringing strategy. However, the researcher found that the students got difficulties in mentally calculating the multiplication of two one-digit numbers, for instance 7×6 or 8×7. Sometimes, the students even used their finger to add two numbers. CONCLUSIONS The data of this research show that the prerequisite of students is very important to continue lesson. The Utrecht’ student have learned step by step about addition and subtraction in many ways and many mental calculation strategies (van den HeuvelPanhuizen, 2005). When they come to multiplication, they implement addition to combine some stringing strategy. In the other hand, Aceh’ students got some difficulties to do it. They use their fingers and add numbers for many times. Aceh’s students need more time to make connection among multiplication. Some students fail to complete the multiplication network. They are familiar to memorize multiplication separately. Even, there is one Utrecht student understand that 5 x 17 = 10 x 8 + 5. It means, 5 x 17 = 5 x 16 + 5 = 10 x 8 + 5. They understand that if one number is multiplied by 2 then the other number divided by 2. 353 1st SEA-DR PROCEEDING ISBN : 978-602-17465-1-6 Until now, the authors of PMRI text books have published some books for first, second, and third grade student at Primary School. However, the teacher did not implement all of topics in PMRI text books for give lessons; due to the Indonesian government don not give instruction to use PMRI text books yet. This research at the third grade is very depending on the students’ experiences in the previous grade. The impact is the students got some difficulties to get the result of multiplication one digit and two digits, when they have ‘a not nice number’ of multiplications, such as 7 x 8. Sometimes, the students even used their finger to add two numbers. Recommendation of this research are; 1) Indonesian government give instruction to Indonesian teachers to implement PMRI textbook step by step, 2) teachers need coach by the expert teacher or by teacher educator to implement PMRI in the classroom, and 3) this HLT can implement to support students use the splitting strategies, this HLT will get the better impact if students have used PMRI textbook in the previous grade. References Amin, S. M., Julie, H., Munk, F., and Hoogland, K. (2010) The Development of Learning Materials for PMRI. In Sembiring, R. K., Hoogland, K., & Dolk, M., (Eds), A Decade of PMRI in Indonesia, Bandung, Utrecht: APS International. Beishuizen, M. (1998) Which Mental Strategies in the Early Number Curriculum? A Comparison of British Ideas and Dutch Views. British Educational Research Journal. Vol. 24 Issue 5, p519 Dolk, M., Widjaya, W., Zonneveld, E. & Fauzan, A. (2010). 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Utrecht, the Netherlands: Freudenthal Institute van den Heuvel-Panhuizen, M. and Wijers, M. (2005) Mathematics Standards and Curricula in the Netherlands. ZDM Journal Vol. 37 (4). Widjaja, W., Dolk, M., Fauzan, A. (2010). The Role of Contexts and Teacher’s Questioning to Enhance Students’ Thinking. Education Journal in Southeast Asia. Vol.33 No.2, 168-186. This paper can be cited as Zulkardi(2013). Proceeding The First South East Asia Design/Development Research (SEA-DR) International Conference, Sriwijaya University, Palembang, April 22nd-23rd. 354