teaching multiplication of numbers from 1 to 10 stkip surya students using matematika gasing slide...
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Table of Content
• Background
• Research Questions
• Theoretical Framework
Matematika GASING
Multiplication of Numbers from 1 to 100 using Matematika GASING
• Research Method
Design Research and Participants
An Overview of the Conjectured Local Instructional Theory
1. Concept of Multiplication
2. Multiplication of Number 1
3. Multiplication of Two Same Numbers
4. Multiplication of Number 3
5. Multiplication of Number 8
• Results and Analysis
Teaching Experiment
Restrospective Analysis
• Conclusion
• Reference
Background
• Multiplication is one of the most important mathematical topics to be learnt due to its many applications in our daily life.
• Students are still having difficulties in learning multiplication and division (Raharjo et al., 2009). They do not remember basic multiplication (multiplication of two numbers where each number is of one digit) which means multiplication of numbers 1 to 10.
Table of Content
• Background
• Research Questions
• Theoretical Framework
Matematika GASING
Multiplication of Numbers from 1 to 100 using Matematika GASING
• Research Method
Design Research and Participants
An Overview of the Conjectured Local Instructional Theory
1. Concept of Multiplication
2. Multiplication of Number 1
3. Multiplication of Two Same Numbers
4. Multiplication of Number 3
5. Multiplication of Number 8
• Results and Analysis
Teaching Experiment
Restrospective Analysis
• Conclusion
• Reference
Research Questions
1. What is the learning trajectory of multiplication of numbers from 1 to 10 like using Matematika GASING?
2. How competent are STKIP Surya students in multiplication of numbers from 1 to 10?
3. How capable are they of teaching multiplication of numbers 1 to 10 using Matematika GASING?
Table of Content
• Background
• Research Questions
• Theoretical Framework
Matematika GASING
Multiplication of Numbers from 1 to 100 using Matematika GASING
• Research Method
Design Research and Participants
An Overview of the Conjectured Local Instructional Theory
1. Concept of Multiplication
2. Multiplication of Number 1
3. Multiplication of Two Same Numbers
4. Multiplication of Number 3
5. Multiplication of Number 8
• Results and Analysis
Teaching Experiment
Restrospective Analysis
• Conclusion
• Reference
Matematika GASING
• GASING stands for Gampang, ASyIk dan menyenaNGkan, which is translated as easy, fun and enjoyable (Surya, 2012).
• There are three stages in learning mathematics using GASING: concrete, abstract, mental calculation.
Theoretical Framework:Matematika GASING
Table of Content
• Background
• Research Questions
• Theoretical Framework
Matematika GASING
Multiplication of Numbers from 1 to 100 using Matematika GASING
• Research Method
Design Research and Participants
An Overview of the Conjectured Local Instructional Theory
1. Concept of Multiplication
2. Multiplication of Number 1
3. Multiplication of Two Same Numbers
4. Multiplication of Number 3
5. Multiplication of Number 8
• Results and Analysis
Teaching Experiment
Restrospective Analysis
• Conclusion
• Reference
Theoretical Framework:Multiplication of Numbers from 1 to 10 in Matematika GASING
• Multiplication is a mathematical operation which involves adding a number to itself a certain number of times.
• The result of adding a to itself b number of times is called the product of a by b. It is written as a × b or a.b or ab. It is also often called as “a times b”.
• The GASING Critical Point for Multiplication
Critical Point in GASING
1 2 3 45
1. Multiplication concept.2. Multiplication of numbers 1, 10, 9, 2 and 5.3. Multiplication of two same numbers.4. Multiplication of numbers 3 and 4.5. Multiplication of numbers 8, 7 and 6.
Table of Content
• Background
• Research Questions
• Theoretical Framework
Matematika GASING
Multiplication of Numbers from 1 to 100 using Matematika GASING
• Research Method
Design Research and Participants
An Overview of the Conjectured Local Instructional Theory
1. Concept of Multiplication
2. Multiplication of Number 1
3. Multiplication of Two Same Numbers
4. Multiplication of Number 3
5. Multiplication of Number 8
• Results and Analysis
Teaching Experiment
Restrospective Analysis
• Conclusion
• Reference
Research Method:Design Research and Participants
Design Research
• There are three phases in conducting design research: (1) preliminary design, (2) teaching experiment, (3) retrospective analysis.
• This method allows researchers to analyze the actual process of students’ learning and mental activities performed when participating in the instructional activities in a classroom (Bustang, et al., 2013).
Participants
• 14 first year undergraduate students at the matriculation mathematics class of the academic year 2013 - 2014 at STKIP Surya of the study program of the Computer and Information Technology.
Table of Content
• Background
• Research Questions
• Theoretical Framework
Matematika GASING
Multiplication of Numbers from 1 to 100 using Matematika GASING
• Research Method
Design Research and Participants
An Overview of the Conjectured Local Instructional Theory
1. Concept of Multiplication
2. Multiplication of Number 1
3. Multiplication of Two Same Numbers
4. Multiplication of Number 3
5. Multiplication of Number 8
• Results and Analysis
Teaching Experiment
Restrospective Analysis
• Conclusion
• Reference
Activity Main
Goals
Description of Activity Questions to be Asked Conjectures of Students’
Answers
Concept
of
Multipli-
cation.
Students
understand
the concept
of
multiplicati
on.
The teacher introduces
the meaning of
multiplication using
real objects such as
containers and
markers (concrete).
The teacher explains
the mathematical
writings of
multiplication
(abstract).
The teacher points out
the commutative law
for multiplication.
How to explain
multiplication;
what is 1 x 5; how
about 2 x 5?
Explain 6 x 3 and 3
x 6 using real
objects! What can
be deduced from
multiplication 6 x
3 and 3 x 6? Do
they have the same
meaning?
Multiplication is when
you add repeatedly; 1
x 5 means there are 5
number 1’s or 1
number 5; 2 x 5 means
there are 2 number 5’s
or 5 number 2’s.
Same results so same
meaning; same results
but not the same
concretely; same
results but different
calculations.
Research Method: An Overview of the Conjectured Local Instructional Theory
1. Multiplication Concept
Gampang, Asyik, dan Menyenangkan
Multiplication Concept
2 boxes containing 3 bananas written as 2 □3 2 x 3
3 boxes containing 5 pineapples written as 3 □5 3 x 5
Multiplication Concept
6 x 3 6 □3 = 3 + 3 + 3 + 3 + 3 + 3 = 18
3 x 6 3 □6 = 6 + 6 + 6 = 18
Gampang, Asyik, dan Menyenangkan
Table of Content
• Background
• Research Questions
• Theoretical Framework
Matematika GASING
Multiplication of Numbers from 1 to 100 using Matematika GASING
• Research Method
Design Research and Participants
An Overview of the Conjectured Local Instructional Theory
1. Concept of Multiplication
2. Multiplication of Number 1
3. Multiplication of Two Same Numbers
4. Multiplication of Number 3
5. Multiplication of Number 8
• Results and Analysis
Teaching Experiment
Restrospective Analysis
• Conclusion
• Reference
Multipli-
cation of
Number 1.
Students
understand, able
to compute and
teach
multiplication of
number 1.
The teacher explains
multiplication of
number 1 concretely.
The teacher explains
the mathematical
writings of
multiplication of
number 1 (abstract).
The teacher
encourages students to
pick up on the pattern
of the results of
multiplication of
number 1; this helps
students to memorize
the multiplication
easily.
Anyone knows how to
explain multiplication
of number 1
concretely?
What is the next stage
after explaining
concretely? How to
teach multiplication of
number 1 at this stage?
What can be deduced
from the results of this
multiplication so that
theycan be memorized
easily?
Yes – one or two
students are
encouraged to come
forward and explain
using concrete objects;
no.
Abstract stage – just
write the numbers and
the multiplication
results; no answers.
The results are the
numbers themselves;
there is a difference by
1; just add 1 to each
result in ascending
order; no answers.
Activity Main Goals Description of Activity Questions to be Asked Conjectures of
Students’ Answers
Research Method: An Overview of the Conjectured Local Instructional Theory
2. Multiplication of Number 1
Concrete1 x 10 = 1 □10 = 10
2 x 10 = 2 □10 = 10 + 10 = 20
3 x 10 = 3 □10 = 10 + 10 + 10 = 30
4 x 10 = 4 □10 = 10 + 10 + 10 + 10 = 40
5 x 10 = 5 □10 = 10 + 10 + 10 + 10 + 10 = 50
6 x 10 = 6 □10 = 10 + 10 + 10 + 10 + 10 + 10 = 60
7 x 10 = 7 □10 = 10 + 10 + 10 + 10 + 10 + 10 + 10 = 70
8 x 10 = 8 □10 = 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 = 80
9 x 10 = 9 □10 = 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 = 90
10 x 10 = 10 □10 = 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 = 100
Gampang, Asyik, dan Menyenangkan
Multiplication of Number 1
Abstract1 x 10 = 10
2 x 10 = 20
3 x 10 = 30
4 x 10 = 40
5 x 10 = 50
6 x 10 = 60
7 x 10 = 70
8 x 10 = 80
9 x 10 = 90
10 x 10 = 100
Gampang, Asyik, dan Menyenangkan
Multiplication of Number 1
Mental Calculation (Seeing the pattern)
1 x 10 = 10
2 x 10 = 20
3 x 10 = 30
4 x 10 = 40
5 x 10 = 50
6 x 10 = 60
7 x 10 = 70
8 x 10 = 80
9 x 10 = 90
10 x 10 = 100
Gampang, Asyik, dan Menyenangkan
Multiplication of Number 1
Table of Content
• Background
• Research Questions
• Theoretical Framework
Matematika GASING
Multiplication of Numbers from 1 to 100 using Matematika GASING
• Research Method
Design Research and Participants
An Overview of the Conjectured Local Instructional Theory
1. Concept of Multiplication
2. Multiplication of Number 1
3. Multiplication of Two Same Numbers
4. Multiplication of Number 3
5. Multiplication of Number 8
• Results and Analysis
Teaching Experiment
Restrospective Analysis
• Conclusion
• Reference
Activity Main Goals Description of Activity Questions to be Asked Conjectures of
Students’ Answers
Multipli-
cation of
two same
numbers.
Students
understand,
able to
compute and
teach
multiplication
of two same
numbers.
The teacher explains
multiplication of two same
numbers concretely.
The teacher explains the
mathematical writings of
multiplication of two same
numbers (abstract).
The teacher encourages
students to find a way to
memorize the results
easily.
Based on what you
have learnt about
multiplication of
numbers 1, 10, 9, 2
and 5 so far; anyone
knows how to explain
multiplication of two
same numbers?
How about the abstract
stage?
How would you
memorize the results?
Yes – one or two
students are encouraged
to come forward and
explain using concrete
objects; no.
Just write down the
results.
Just memorize the
results; by recognizing
the patterns.
Research Method: An Overview of the Conjectured Local Instructional Theory
3. Multiplication of Two Same Numbers
Gampang, Asyik, dan Menyenangkan
Concrete
1 x 1 = 1 □1 = 1
2 x 2 = 2 □2 = 2 + 2 = 4
3 x 3 = 3 □3 = 3 + 3 + 3 = 9
4 x 4 = 4 □4 = 4 + 4 + 4 + 4 = 16
5 x 5 = 5 □5 = 5 + 5 + 5 + 5 + 5 = 25
6 x 6 = 6 □6 = 6 + 6 + 6 + 6 + 6 + 6 = 36
7 x 7 = 7 □7 = 7 + 7 + 7 + 7 + 7 + 7 + 7 = 49
8 x 8 = 8 □8 = 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 64
9 x 9 = 9 □9 = 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 = 81
10 x 10 = 10 □10 = 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 = 100
Multiplication of Two Same Numbers
Gampang, Asyik, dan Menyenangkan
Abstract1 x 1 = 1
2 x 2 = 4
3 x 3 = 9
4 x 4 = 16
5 x 5 = 25
6 x 6 = 36
7 x 7 = 49
8 x 8 = 64
9 x 9 = 81
10 x 10 = 100
Multiplication of Two Same Numbers
Gampang, Asyik, dan Menyenangkan
Mental Calculation
1 x 1 = 1 (multiplication of number 1)
2 x 2 = 4 (multiplication of number 2)
3 x 3 =
4 x 4 =
5 x 5 = 25 (multiplication of number 5)
9 x 9 = 81 (multiplication of number 9)
10 x 10 = 100 (multiplication of number 10)
3 + 3 = 6 6 + 3 = 9
4 + 4 = 8 8 + 8 = 16
9
16
Multiplication of Two Same Numbers
Gampang, Asyik, dan Menyenangkan
Mental Calculation1 x 1 = 1
2 x 2 = 4
3 x 3 = 9
4 x 4 = 16
5 x 5 = 25
6 x 6 = 36
7 x 7 = 49
8 x 8 = 64
9 x 9 = 81
10 x 10 = 100
6 x 6 = 36
7 x 7 = 49
8 x 8 = 64
Multiplication of Two Same Numbers
Table of Content
• Background
• Research Questions
• Theoretical Framework
Matematika GASING
Multiplication of Numbers from 1 to 100 using Matematika GASING
• Research Method
Design Research and Participants
An Overview of the Conjectured Local Instructional Theory
1. Concept of Multiplication
2. Multiplication of Number 1
3. Multiplication of Two Same Numbers
4. Multiplication of Number 3
5. Multiplication of Number 8
• Results and Analysis
Teaching Experiment
Restrospective Analysis
• Conclusion
• Reference
Activity Main Goals Description of Activity Questions to be Asked Conjectures of
Students’ Answers
Multipli-
cation of
number 3.
Students
understand,
able to
compute and
teach
multiplication
of number 3.
The teacher explains
multiplication of
number 3
concretely.
The teacher explains
the mathematical
writings of
multiplication of
number 3 (abstract).
The teacher
encourages students
to find a way to
memorize the
results easily.
How do you
explain
multiplication of
number 3
concretely?
How about the
mathematical
writings?
How would you
memorize the
results?
For examples: 1 x 3
means there is 1 box
containing 3 markers
or 3 stones, 2 x 3
means there are 2
boxes containing 3
markers or 3 stones
each, etc.; no answers.
Just write down the
results.
Just memorize the
results; by
recognizing the
patterns; by using
fingers.
Research Method: An Overview of the Conjectured Local Instructional Theory
4. Multiplication of Number 3
Gampang, Asyik, dan Menyenangkan
Concrete1 x 3 = 1 □3 = 32 x 3 = 2 □3 = 3 + 3 = 63 x 3 = 3 □3 = 3 + 3 + 3 = 94 x 3 = 4 □3 = 3 + 3 + 3 + 3 = 125 x 3 = 5 □3 = 3 + 3 + 3 + 3 + 3 = 156 x 3 = 6 □3 = 3 + 3 + 3 + 3 + 3 + 3 = 18 7 x 3 = 7 □3 = 3 + 3 + 3 + 3 + 3 + 3 + 3 = 21 8 x 3 = 8 □3 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 249 x 3 = 9 □3 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 27
10 x 3 = 10 □3 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 30
Multiplication of Number 3
Gampang, Asyik, dan Menyenangkan
Abstract1 x 3 = 3
2 x 3 = 6
3 x 3 = 9
4 x 3 = 12
5 x 3 = 15
6 x 3 = 18
7 x 3 = 21
8 x 3 = 24
9 x 3 = 27
10 x 3 = 30
Multiplication of Number 3
Gampang, Asyik, dan Menyenangkan
Mental Calculation1 x 3 = 3
2 x 3 = 6
3 x 3 = 9
4 x 3 =
5 x 3 = 15
9 x 3 = 27
10 x 3 = 30
Multiplication of Number 3
3 + 3 = 6 6 + 6 = 12
Gampang, Asyik, dan Menyenangkan
Mental Calculation1 x 3 = 3
2 x 3 = 6
3 x 3 = 9
4 x 3 = 12
5 x 3 = 15
6 x 3 = 18
7 x 3 = 21
8 x 3 = 24
9 x 3 = 27
10 x 3 = 30
Multiplication of Number 3
Gampang, Asyik, dan Menyenangkan
The Song of The Multiplication of Number 3
• Tiga enam sembilan dua belas
• Lima belas delaapan belas
• Dua satu dua puluh empat
• Dua tujuh itu perkalian tiga
(music: Bintang Kecil - the Little Star)
Multiplication of Number 3
Table of Content
• Background
• Research Questions
• Theoretical Framework
Matematika GASING
Multiplication of Numbers from 1 to 100 using Matematika GASING
• Research Method
Design Research and Participants
An Overview of the Conjectured Local Instructional Theory
1. Concept of Multiplication
2. Multiplication of Number 1
3. Multiplication of Two Same Numbers
4. Multiplication of Number 3
5. Multiplication of Number 8
• Results and Analysis
Teaching Experiment
Restrospective Analysis
• Conclusion
• Reference
Activity Main Goals Description of Activity Questions to be Asked Conjectures of
Students’ Answers
Multipli-
cation of
number
8.
Students
understand,
able to
compute and
teach
multiplication
of number 8.
The teacher explains
multiplication of
number 8 concretely.
The teacher explains
the mathematical
writings of
multiplication of
number 8 (abstract).
The teacher
encourages students
to find a way to
memorize the
results easily.
How do you explain
multiplication of
number 8
concretely?
How about the
mathematical
writings?
How would you
memorize the
results?
For examples: means
there is 1 box
containing 8 apples or
8 bananas, means
there are 2 boxes
containing 8 apples or
8 bananas each, etc.;
no answers.
Just write down the
results.
Just memorize the
results; by recognizing
the patterns; by using
fingers; by using card
games; by singing
songs.
Research Method: An Overview of the Conjectured Local Instructional Theory
5. Multiplication of Number 8
Gampang, Asyik, dan Menyenangkan
Concrete1 x 8 = 1 □8 = 82 x 8 = 2 □8 = 8 + 8 = 163 x 8 = 3 □8 = 8 + 8 + 8 = 244 x 8 = 4 □8 = 8+ 8 + 8 + 8 = 325 x 8 = 5 □8 = 8 + 8 + 8 + 8 + 8 = 406 x 8 = 6 □8 = 8 + 8 + 8 + 8 + 8 + 8 = 48 7 x 8 = 7 □8 = 8 + 8 + 8 + 8 + 8 + 8 + 8 = 56 8 x 8 = 8 □8 = 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 649 x 8 = 9 □8 = 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 7210 x 8 = 10 □8 = 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 80
Multiplication of Number 8
Gampang, Asyik, dan Menyenangkan
Abstract1 x 8 = 8
2 x 8 = 16
3 x 8 = 24
4 x 8 = 32
5 x 8 = 40
6 x 8 = 48
7 x 8 = 56
8 x 8 = 64
9 x 8 = 72
10 x 8 = 80
Multiplication of Number 8
Gampang, Asyik, dan Menyenangkan
Mental Calculation1 x 8 = 8
2 x 8 = 16
3 x 8 = 24
4 x 8 = 32
5 x 8 = 40
6 x 8 = 48
7 x 8 = 56
8 x 8 = 64
9 x 8 = 72
10 x 8 = 80
6 x 8 = 48
Multiplication of Number 8
Table of Content
• Background
• Research Questions
• Theoretical Framework
Matematika GASING
Multiplication of Numbers from 1 to 100 using Matematika GASING
• Research Method
Design Research and Participants
An Overview of the Conjectured Local Instructional Theory
1. Concept of Multiplication
2. Multiplication of Number 1
3. Multiplication of Two Same Numbers
4. Multiplication of Number 3
5. Multiplication of Number 8
• Results and Analysis
Teaching Experiment
Restrospective Analysis
• Conclusion
• Reference
Result and Analysis: Teaching Experiment
• Conjectured local instructional theory was used as a guidance for the teacher and the researchers to refer to whilst conducting the actual experiment.
• An episode of teaching the concept of multiplication:Teacher : What is multiplication?Raja : Repetitive addition.Every students agreed that multiplication is repetitive addition. The teacher asked the next question.Teacher : What does it mean by 3 x 5 ? Raja & Nyong : Keep adding number 3 for 5 times.Ferry : Add number 5 for 3 times.There are obviously two different understandings.
Continued episode:•The teacher then explained 3 x 5 by showing three cards with
pictures of 5 bananas on each card to illustrate multiplication 3 x 5.
•The teacher also explained using picture cards that 2 x 5 and 5 x 2 had same results but different meanings.
Result and Analysis: Teaching Experiment
Table of Content
• Background
• Research Questions
• Theoretical Framework
Matematika GASING
Multiplication of Numbers from 1 to 100 using Matematika GASING
• Research Method
Design Research and Participants
An Overview of the Conjectured Local Instructional Theory
1. Concept of Multiplication
2. Multiplication of Number 1
3. Multiplication of Two Same Numbers
4. Multiplication of Number 3
5. Multiplication of Number 8
• Results and Analysis
Teaching Experiment
Restrospective Analysis
• Conclusion
• Reference
Result and Analysis: Analysis RetrospectiveI. Tests to measure computational
competenceTests Results:
• Written Test (100 questions): Average score 94 out of 100 Average time 5 minutes 6
seconds• Oral Test (50 questions)
Average score 42.50 out of 50 Average time 3 minutes 8
seconds
A Few Points of Observations:
•The results in I indicates that students could do multiplication of numbers
from 1 to 10 well both orally and in writings
•The result of written test in II contrasted the result of microteaching test. This
showed that students tended to find it more difficult to write the learning
process materials as opposed to delivering them orally
•There was a student who performed poorly during microteaching test, it was
discovered that she did not even understand addition - a material taught and
supposed to be mastered prior to multiplication. In this case she should go
back and restart from addition.
II. Tests to measure capability of teaching
Tests Results:• Written Test (5 questions, 60
minutes):Average score 43.57 out of 100
• Microteaching Test (20 minutes each)Average score 83.04 out of 100
Table of Content
• Background
• Research Questions
• Theoretical Framework
Matematika GASING
Multiplication of Numbers from 1 to 100 using Matematika GASING
• Research Method
Design Research and Participants
An Overview of the Conjectured Local Instructional Theory
1. Concept of Multiplication
2. Multiplication of Number 1
3. Multiplication of Two Same Numbers
4. Multiplication of Number 3
5. Multiplication of Number 8
• Results and Analysis
Teaching Experiment
Restrospective Analysis
• Conclusion
• References
Conclusion
1. Matematika GASING helped students to understand the concept of multiplication better.
2. Matematika GASING was able to create a fun and exciting environment when learning multiplication of numbers from 1 to 10; students were enthusiastic in participating in the activities.
3. Students were able to calculate and perform mental calculation of multiplication of numbers from 1 to 10 as well as teaching the materials relatively well.
4. The revised local instructional theory should contain added activities such as drilling at the end of every session as well as activities that focus on the emphasis of the ways of memorizing multiplication of numbers from 1 to 10.
5. Further research can be done to implement and test the revised local instructional theory. Other research can be done to investigate the problems students have with their mathematical writing skills.
Table of Content
• Background
• Research Questions
• Theoretical Framework
Matematika GASING
Multiplication of Numbers from 1 to 100 using Matematika GASING
• Research Method
Design Research and Participants
An Overview of the Conjectured Local Instructional Theory
1. Concept of Multiplication
2. Multiplication of Number 1
3. Multiplication of Two Same Numbers
4. Multiplication of Number 3
5. Multiplication of Number 8
• Results and Analysis
Teaching Experiment
Restrospective Analysis
• Conclusion
• References
References
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[2] Bustang, Zulkardi, Darmawijoyo, Dolk, M. and van Eerde, D., 2013, Developing a Local Instruction Theory for Learning the Concept of Angle through Visual Field Activities and Spatial Representations, International Education Studies Vol. 6 No. 8, 58-70.
[3] Ibrahim and Suparmi, 2012, Pembelajaran Matematika Teori dan Aplikasinya, SUKA-Press, Yogyakarta.
[4] Gravemeijer, K., 2009, Local Instruction Theories as Means of Supports for Teachers in Reform Mathematics Education, Mathematical Thinking and Learning Journal Vol. 6 No. 2, page 105 – 128.
[5] Gravemeijer, K. and van Eerde, D., 2009, Design Research as a Means for Building a Knowledge Base for Teachers and Teaching in Mathematics Education, The Elementary School Journal Vol. 109 No. 5, 510-524.
[6] Raharjo, M., Waluyati, A., Sutanti, T., 2009, Pembelajaran Operasi Hitung Perkalian dan Pembagian Bilangan Cacah di SD, Depdiknas: Pusat Pengembangan dan Pemberdayaan Pendidikan dan Tenaga Kependidikan (PPPPTK) Matematika.
[7] Reys, B. J., 1985, Mental Computation, The Arithmetic TeacherVol. 32 No. 6 (1985), 43-46.
[8] Surya, Y. and Moss, M., 2012, Mathematics Education in Rural Indonesia, Proceeding in the 12th International Congress on Mathematics Education: Topic Study Group 30, 6223-6229.
[9] Surya, Y., 2013, Modul Pelatihan Matematika GASING SD Bagian 1, PT. Kandel, Tangerang.
[10] Van den Akker, J., Gravemeijer, K., McKenney, S., and Nieveen, N., 2006, Educational Design Research, Routledge, Taylor and Francis Group, Abingdon.
References