Teaching multiplication of numbers from 1 to 10 stkip surya students using matematika gasing slide iicma2013.josephine+sulis

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
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.

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?

Theoretical Framework:
Matematika GASING

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:
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

2 3
1

4

5

Critical Point in GASING

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.

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.

Research Method: An Overview of the Conjectured
Local Instructional Theory
1. Multiplication Concept
Activity

Main
Goals
Concept Students
of
understand
Multipli- the concept
cation.
of
multiplicati
on.

Description of Activity







Questions to be Asked Conjectures of Students’
Answers
The teacher introduces  How to explain
 Multiplication is when
the meaning of
multiplication;
you add repeatedly; 1
multiplication using
what is 1 x 5; how
x 5 means there are 5
real objects such as
about 2 x 5?
number 1’s or 1
containers and
 Explain 6 x 3 and 3
number 5; 2 x 5 means
markers (concrete).
x 6 using real
there are 2 number 5’s
The teacher explains
objects! What can
or 5 number 2’s.
the mathematical
be deduced from
 Same results so same
writings of
multiplication 6 x
meaning; same results
multiplication
3 and 3 x 6? Do
but not the same
(abstract).
they have the same
concretely; same
The teacher points out
meaning?
results but different
the commutative law
calculations.
for multiplication.

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
Gampang, Asyik, dan Menyenangkan

Multiplication Concept

□
3 x 6 3 □
6 x 3 6

3

= 3 + 3 + 3 + 3 + 3 + 3 = 18

6

= 6 + 6 + 6 = 18


Activity

Main Goals


MultipliStudents

cation of
understand, able
Number 1. to compute and
teach

multiplication of
number 1.



multiplication of
number 1 concretely.
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.

Questions to be Asked






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?

Conjectures of
Students’ Answers
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.

Multiplication of Number 1
Concrete

1 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

Abstract
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

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

Activity
Multiplication of
two same
numbers.

Main Goals
Students
understand,
able to
compute and
teach
multiplication
of two same
numbers.







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.

Questions to be Asked





Conjectures of
Students’ Answers
Based on what you
 Yes – one or two
have learnt about
students are encouraged
multiplication of
to come forward and
numbers 1, 10, 9, 2
explain using concrete
and 5 so far; anyone
objects; no.
knows how to explain  Just write down the
multiplication of two
results.
same numbers?
 Just memorize the
How about the abstract
results; by recognizing
stage?
the patterns.
How would you
memorize the results?

Multiplication of Two Same Numbers
Concrete
1x1
2x2
3x3
4x4
5x5
6x6
7x7
8x8
9x9
10 x 10

= 1 □1 = 1
= 2 □2 = 2 + 2 = 4
= 3 □3 = 3 + 3 + 3 = 9
= 4 □4 = 4 + 4 + 4 + 4 = 16
= 5 □5 = 5 + 5 + 5 + 5 + 5 = 25
= 6 □6 = 6 + 6 + 6 + 6 + 6 + 6 = 36
= 7 □7 = 7 + 7 + 7 + 7 + 7 + 7 + 7 = 49
= 8 □8 = 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 64
= 9 □9 = 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 = 81
= 10 □10 = 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 = 100


Abstract
1 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


Mental Calculation
1
2
3
4
5

x
x
x
x
x

1
2
3
4
5

= 1 (multiplication of number 1)
3+3=66+3=9
= 9
= 16
4 + 4 = 8  8 + 8 = 16

9 x 9 = 81 (multiplication of number 9)
10 x 10 = 100 (multiplication of number 10)

Mental Calculation
1 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


Research Method: An Overview of the Conjectured Local
Instructional Theory
Activity

Main Goals


Multipli- Students

cation of
understand,
number 3. able to
compute and
teach

multiplication
of number 3.



Questions to be Asked Conjectures of
Students’ Answers
The teacher explains  How do you
 For examples: 1 x 3
multiplication of
explain
means there is 1 box
number 3
multiplication of
containing 3 markers
concretely.
number 3
or 3 stones, 2 x 3
concretely?
means there are 2
the mathematical
 How about the
boxes containing 3
writings of
mathematical
markers or 3 stones
multiplication of
writings?
each, etc.; no answers.
number 3 (abstract).  How would you
 Just write down the
The teacher
memorize the
results.
encourages students
results?
to find a way to
results; by
memorize the
recognizing the
results easily.
patterns; by using
fingers.

Concrete
1x3
2x3
3x3
4x3
5x3
6x3
7x3
8x3
9x3
10 x 3

= 1 □3
= 2 □3
= 3 □3
= 4 □3
= 5 □3
= 6 □3
= 7 □3
= 8 □3
= 9 □3
= 10 □3

=3
=3+3=6
=3+3+3=9
= 3 + 3 + 3 + 3 = 12
= 3 + 3 + 3 + 3 + 3 = 15
= 3 + 3 + 3 + 3 + 3 + 3 = 18
= 3 + 3 + 3 + 3 + 3 + 3 + 3 = 21
= 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 24
= 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 27
= 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 30


Abstract
1x3=3
2x3=6
3x3=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


Mental Calculation
1x3=3
2x3=6
3x3=9
4x3=
5 x 3 = 15

3 + 3 = 6  6 + 6 = 12

9 x 3 = 27
10 x 3 = 30

Mental Calculation
1x3=3
2x3=6
3x3=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

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)


A Card Game

18

21

24

6

7

8


Research Method: An Overview of the Conjectured Local
Instructional Theory
Activity
Multiplication of
number
8.

Main Goals
Students
understand,
able to
compute and
teach
multiplication
of number 8.







The teacher explains 
multiplication of
number 8 concretely.
the mathematical

writings of
multiplication of
number 8 (abstract). 
The teacher
encourages students
to find a way to
memorize the
results easily.

Questions to be Asked Conjectures of
Students’ Answers
How do you explain  For examples: means
multiplication of
there is 1 box
number 8
containing 8 apples or
concretely?
8 bananas, means
How about the
there are 2 boxes
mathematical
containing 8 apples or
writings?
8 bananas each, etc.;
How would you
no answers.
memorize the
 Just write down the
results?
results.
results; by recognizing
the patterns; by using
fingers; by using card
games; by singing
songs.

Concrete
1 x 8 = 1 □8 = 8
2 x 8 = 2 □8 = 8 + 8 = 16
3 x 8 = 3 □8 = 8 + 8 + 8 = 24
4 x 8 = 4 □8 = 8+ 8 + 8 + 8 = 32
5 x 8 = 5 □8 = 8 + 8 + 8 + 8 + 8 = 40
6 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 = 64
9 x 8 = 9 □8 = 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 72
10 x 8 = 10 □8 = 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 80


Abstract
1x8=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


Mental Calculation
1x8=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


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.

Result and Analysis:
Teaching Experiment
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: Analysis Retrospective
I.

Tests to measure computational
competence
Tests 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

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

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.

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
Results and Analysis
 Teaching Experiment
 Restrospective Analysis
Conclusion
References

Conclusion
1.
2.

3.

4.

5.

Matematika GASING helped students to understand the concept of
multiplication better.
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.
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.
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.
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.

References
[1] Beishuizen, M. and Anghileri, J., 1998, Which Mental Strategies in the Early
Number Curriculum? A Comparison of British Ideas and Dutch Views,
British Educational Research Journal Vol. 24 No. 5, 519-538.
[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.

References
[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.

Teaching multiplication of numbers from 1 to 10 stkip surya students using matematika gasing slide iicma2013.josephine+sulis

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