978-602-17465-1-6

Transcription

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.
Surabaya State University
Himpunan Matematika Indonesia
(Indonesia Mathematics Society/ IndoMS)
Published by:
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,
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
April, 22nd-23rd, 2013
This paper has been presented at
The First South East Asia Design/ Development Research (SEA-DR)
International Conference
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 )
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:H.A.A.vanEerde@u
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
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
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
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,
u.nl
3Sriwijaya University
Email:
[email protected],
Email:
[email protected]
m
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
u.nl
3 Utrecht University
Email:
[email protected]
1 Universitas Negeri
Dwi Sulistya
Kusumaningrum 1 Jakarta,
, Meiliasari 2
Email:dwisulistya1989@g
mail.com
2 Universitas Negeri
Jakarta
Email:melisari79@gma
il.com
Elika Kurniadi1,
Ratu Ilma Indra
Email:
2
Putri , Yusuf
[email protected]
Hartono3, Meike J.
d,
Abels4
Email:ratu.ilma@yaho
o.com
Email:y.hartono@unsr
i.ac.id
4Uttrecht University
1State University of
Evangelista L.W.
1
2
Palupi , Zulkardi ,
Surabaya
Darmawijoyo3,
Email:
4
Meike J. Abels
evangelista.palupi@ya
hoo.com
Email:
[email protected]
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
1STKIP PGRI
Lubuklinggau
Email:
[email protected]
2Universitas Negeri
Jakarta
m
3Universitas Negeri
Jakarta
Email:
[email protected]
m3
Design of Social
Network Learning
Model
114
Hatta University
Email:
[email protected]
2Bung Hatta University
Email:
alirantelaumbanua@yaho
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
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
om
Development of
Context-Based
Mathematics Learning
Tool for Fifth-Grade
Elementary School
Students
153
1Bung
viii
P-18
Hendra Lesmana
Email:Hendra4thenext@y
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:dwi_juniati@yahoo.
com
1Universitas Tadulako
Email:
kertayasa06match@yaho
o.co.id
2Universitas Tadulako
Email:sukayasa08@yahoo
.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:karinka1803@gmai
l.com
2Sriwijaya
University
Email:ratu.ilma@yaho
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:Lisnanipcmtk@yah
oo.com
m
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:somakim_math@ya
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]
Email:
[email protected]
Email:
yayasanbudi_s@yahoo
.com
Email:
meldariani.pcmtk2011@g
mail.com
Email:
[email protected]
Email:
[email protected]
m
Email:
meilani.safitri@ymail.
com
Email:
[email protected]
Email:
[email protected]
m
PGRI University of
Palembang
Email:misda_lina68@ym
ail.com
STKIP Al Hikmah
Email:Lutfi.format@gmail
.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
Jakarta
Email:
[email protected]
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]
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:petra.suwasti@s
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:nisaa_cute02@yaho
o.co.id
Indra Email:
[email protected]
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
pramanika.pcmtk2011@y
ahoo.com
Email:
[email protected]
3 PGRI University of
Palembang
Email:
[email protected]
om
1 Surabaya State
University
Email:P.Astutipuji@g
mail.com
State University of Jakarta
Email:
[email protected]
Risnawati1,
Zubaidah Amir2,
Defi3
1State
University of
Malang
Email:
[email protected]
Surabaya
Email:
[email protected]
Surabaya
Email:
[email protected]
4Utrecht University
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]
Riau
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]
Riau
Email:
[email protected]
Riau
P-45
Rully Charitas Indra Surya College of
Prahmana
Education (STKIP
Surya)
Email:rully.charitas@s
tkipsurya.ac.id
Designing
Division
Operation Learning in
The Mathematics Of
Gasing
391
P-46
Sakinah
Fajri1,
PGRI Padang
Email:
[email protected]
Email:
[email protected]
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
Surabaya
Email:
xiii
P-49
Sylvana N.
Sumarto1, Zulkardi2,
Darmawijoyo3,
Frans van Galen4
[email protected]
Surabaya
Email:
[email protected]
4Utrecht University
1 State
University of
Malang, 1 St Albertus
Senior High School
Email:
sylvananovilia@yahoo
.com
Email:
[email protected]
Email:
darmawijoyo@yahoo.
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
Email:
[email protected]
1STMIK
Surabaya
Email:
[email protected]
2STMIK Surabaya
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]
Surabaya
Email:
[email protected]
4 Utrecth University
Email:
[email protected]
Riau
Email:
[email protected]
Riau
Email:
[email protected]
Riau
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
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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.
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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
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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.
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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
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
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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
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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
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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.
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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.
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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). Examining Teachers’ Role in
Relation to Their Beliefs and Expectations about Students’ Thinking in Design
Research. In Sembiring, R. K., Hoogland, K., & Dolk, M., (Eds), A Decade of PMRI in
Indonesia, Bandung, Utrecht: APS International
Gravemeijer (1994). Gravemeijer, K.P.E. (1994). Developing Realistic Mathematics
Education. Netherlands: Technipress
Gravemeijer, K.P.E and Cobb, P. (2006). Design Research from a Learning Design
Perspective. In Dekker, van den, Gravemeijer, K., Mc Kenny, S., & Nieven, N.
(Eds). Educational Design Research (pp. 17-51). London: Rontledge.
Johar, R. and Afrina, M. (2011). The Teachers’ Effort to Encourage the Students’
Strategies to Find the Solution of Fraction Problem in Banda Aceh. In
International Conference of School Effectiveness and Improvement (ICSEI) at
Limassol, Cyprus, Jan 4th till 7th, 2011.
van den Heuvel-Panhuizen, M. (1996). Assesment Realistic Mathematics Education.
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.
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Transcription

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