CHAPTER
I
INTRODUCTION
1.1 Journal
THE POWER OF GAMES TO
LEARN MATHEMATICS
Ariyadi Wijaya
Mathematics Education Department, Yogyakarta State University
(ariyadiwijaya@hotmail.com; a.wijaya@uny.ac.id)
Abstract
The central
principle of Realistic Mathematics Education is that the learning of mathematics
needs to be laid on meaningful situation; either meaningful problems or activities.
Many research found that activities of game playing are meaningful to students
and can be effective tools for enhancing learning and understanding of complex subject
matter. Consequently, this article focuses on the benefit of games to support
the learning of mathematics and the role of teacher in conducting game-based
learning.
Keywords :
Realistic Mathematics Education, game-based learning, the role of teacher
INTRODUCTION
Many mathematics
concepts are directly taught as an isolated concept at the formal level of young
children (Castle & Needham, 2007; Kamii & Clark, 1997 and van de Walle
& Folk, 2005). Teaching and learning of mathematics mostly focuses on the algorithm
or procedure to solve problems at formal level of mathematics. However, in fact
young children have difficulty in understanding the concepts of mathematics in
the formal level (Van de Walle & Folk, 2005). Consequently, it is important
not to start the teaching and learning process of mathematics from formal
level. Freudenthal viewed mathematics as human activity, instead as subject
matter that must be transferred from teachers to students (Freudenthal, 1991).
This view underlies the development of Realistic Mathematics Education (RME).
The central principle
of RME is that mathematics should be meaningful to students. The meaningful
mathematics can be achieved by giving contextual problem as the starting point
of the teaching and learning. The term “realistic” in RME does not mean that
the problems have to be encountered in daily life, but it means that the problems
should be experientially real for student. In the process of teaching and learning,
students should be allowed and encouraged to invent their own strategies and
ideas. However, the different strategies and ideas of students have to be led
to the formation of mathematics concepts as the final goal of the teaching and
learning process.
The foundation of
mathematics education in kindergarten and elementary school needs to be laid on
doing meaningful activities or experiences, through which a connection is made
between informal knowledge and the formal concepts of mathematics (Buys &
de Moor, 2005 and Castle & Needham, 2007). Consequently, it is important to
give young children experience-based activities that embody some basic concepts
of mathematics. Experience-based activities are relevant with Freudenthal’s
idea that stresses mathematics as a human activity, instead of subject matter
that has to be transmitted (Freudenthal, 1991). Freudenthal (ibid)
proposed the need to connect mathematics to reality through problem situation
because experience-based activities could contribute to the emerging of
mathematical practices. For young children, game playing could be a problem
situation, which is experientially real for them and, therefore, can be used as
a starting point for their learning process. Game playing can form a natural part
of the experience-based and development-focused activities for the teaching and
learning of mathematics. In Indonesia, there are some (traditional) games that,
without any consideration, embody mathematics concepts. Consequently, the
central issue of this article is the use of games (especially Indonesian
traditional games) as experience-based activities for teaching and learning of
mathematics.
REALISTIC MATHEMATICS EDUCATION
Realistic Mathematics
Education was underlined by the idea of Hans Freudenthal that viewed
mathematics as human activity, instead as subject matter that must be transferred
from teachers to students (Freudenthal 1991). Based on Freudenthal’s idea, the
teaching and learning process of mathematics should be connected to contextual problems
that experientially real for students. In spite of the use of contextual
problem, the teaching and learning process should lead to the formation of
mathematical concepts through the process of mathematization. There are five
tenets of realistic mathematics education defined by Treffers (1987), namely:
1.
Phenomenological
explorationContextual problems are used
as the base and starting point for the teaching and learning process. The
teaching and learning process is not started from formal level but from a situation
that is experientially real for student.
2.
Using models and symbols for progressive mathematization
The aim of this tenet is bridging from
concrete level to more formal level using models and symbols.
3.
Using students’ own construction
The freedom for students to use their
own strategies could direct to the emergence of various solutions that can be
used to develop the next learning process. The students’ strategies in the
activities are discussed in the following class discussion to support students’
acquisition of the formal level of mathematics concepts
4.
Interactivity
The learning process of students is not merely an individual process, but it is also a social process. The learning process of students can be shortened when students communicate their works and thoughts in the social interaction emerged in classroom.
The learning process of students is not merely an individual process, but it is also a social process. The learning process of students can be shortened when students communicate their works and thoughts in the social interaction emerged in classroom.
5.
Intertwinement
The activities used in the teaching and learning process do not merely support learning for a single mathematics topic, but they also should support the learning process of other mathematics topics or concepts.
The activities used in the teaching and learning process do not merely support learning for a single mathematics topic, but they also should support the learning process of other mathematics topics or concepts.
GAME-BASED LEARNING
Learning occurs when
students process new information or knowledge in such a way that it makes sense
to them. In a supporting environment, students can discover meaningful
relationships between contextual problem situation and abstract ideas that will
lead to the process of internalization of concepts through the process of (guided)
reinventing, reinforcing, and connecting. However, student need attractive
contextual situation as the starting point of their learning process. Games can
be powerful and attractive contextual situation for students because one of the
natural characteristics of a
game playing is that it uses more action instead of word explanation. This characteristic will stimulate students’ personal motivation and satisfaction in the learning process. Hence, games can encourage attractive and active learning in which students are more active in their learning process (Garris, Ahlers, & Driskell, 2002).
game playing is that it uses more action instead of word explanation. This characteristic will stimulate students’ personal motivation and satisfaction in the learning process. Hence, games can encourage attractive and active learning in which students are more active in their learning process (Garris, Ahlers, & Driskell, 2002).
The other
characteristic of game playing that can benefit and support the teaching and
learning process is that many games are played in teams or groups. The teamwork
created in a game playing can foster collaboration and interactivity among
students (Kaptelin & Cole, 2002 and Sheffield, 2005). Furthermore, this
characteristic can accommodate multiple learning styles and skills of students.
Ricci, Salas & CannonBowers (1996) and Sheffield (2005) found that game
playing can reinforce mastery skills of students because game playing is
effective tool for enhancing learning and understanding of complex subject
matter. As an example is how Indonesian traditional games (i.e. benthik and
gundu) can support second graders’ learning of linear measurement (Ariyadi
Wijaya, 2008).
Despite the
aforementioned benefits of games in education, the use of games in the teaching
can learning process cannot stand alone or independent. The games need to be
followed by discussion in which the values and/or contents of the game are
directed to the formation of mathematics concept. It is in line with the
principles of experiential learning that was developed by Kolb. Kolb mentioned
four stages of experiential learning, namely: (1) concrete experience, (2)
reflective observation, (3) abstract conceptualization, and (4) active
experimentation (Kolb, 1984). Game playing serves as a concrete experience in
which students can develop strategies to solve problem-based game. The game playing is used as the base
for the observations and reflections that transform experiences into abstract
concepts. The strategies developed by students are directed to the formation of
mathematics concepts in the process of reflective observation and abstract
conceptualization. Active experimentation in the game-based learning can be the
application of mathematics concept formed in the reflective and abstract conceptualization
stage.
GAMES FOR MATHEMATICS LEARNING
The following are
examples of game that can be used in the learning of
mathematics concepts:
mathematics concepts:
1.
Gundu
The mathematics concepts embodied in the first part of Gundu game, namely when determining the order of the players. In this part all players have to throw their marble to a given hole or pole. The first player is a player whose marble is the nearest to the hole or pole.
Students can use two strategies when determine the nearest marble, namely by comparison (when the different distances of the marbles is obvious to observe) and measurement (for the close distances).
The mathematics concepts embodied in the first part of Gundu game, namely when determining the order of the players. In this part all players have to throw their marble to a given hole or pole. The first player is a player whose marble is the nearest to the hole or pole.
Students can use two strategies when determine the nearest marble, namely by comparison (when the different distances of the marbles is obvious to observe) and measurement (for the close distances).
2.
Dakon or congklak
Dakon
is a game that is played by two players and each player has a big hole as “a deposit
hole” and some small holes (the number of small holes depends on the type of
dakon). There are three versions of Dakon or Congklak, namely dakon with 10
holes, 12 holes and 16 holes. For the 10 holes dakon, there are 32 seeds or beads
that should be put in the small holes at the beginning of the game (4 seeds or
beads for each small hole). For the 12 holes dakon, there are 50 seeds or beads
that should be put in the small holes at the beginning of the game (5 seeds or beads
for each small hole). For the 16 holes dakon, there are 98 seeds or beads that
should be put in the small holes at the beginning of the game (7 seeds or beads
for each small hole).
The mathematics concepts embodied in Dakon game are counting and division. The counting emerges when students distribute the seeds or beads and count the final result in the “deposit hole”. At the beginning of the game, each player has to distribute his/her seeds or beads in the small holes equally. The concept of division can be developed when the teacher change the rule of the game and giving various numbers of seeds or beads to students.
The mathematics concepts embodied in Dakon game are counting and division. The counting emerges when students distribute the seeds or beads and count the final result in the “deposit hole”. At the beginning of the game, each player has to distribute his/her seeds or beads in the small holes equally. The concept of division can be developed when the teacher change the rule of the game and giving various numbers of seeds or beads to students.
3.
Ular tangga
In
Ular tangga game, the players have to throw a die and then move their pin in the
numbers of steps as shown by the top of the die. When a player’s pin stops at a
stair, the player can directly “jump up” the pin to the top edge of the stair.
On the other hand, a player has to “jump down” his/her pin to the head of snake
when his/her pin stops at the tail of a snake. The winner of the game is the
first player who arrives at number 100 as the final destination. Mathematics
concepts that are embodied in Ular tangga are counting, addition and subtraction.
The counting (i.e. counting one by one) emerges when a player/student moves
his/her pin step by step. Addition concept emerges when a player does not move
his/her pin step by step, but he/she adds the number shown by the die to the
current position of the pin. Addition concept can be focused or emphasized by
asking students about how many steps they get as an advantage when they get a
stair. On the other hand, the disadvantage when students meet a snake can be used
to emphasize the concept of subtraction.
THE ROLE OF TEACHER IN
THE GAME-BASED LEARNING
Game playing provides a natural situation for social interaction,
such as students’ agreement in deciding a strategy for the fairness of their
games (Ariyadi Wijaya, 2008). Interactivity as the fourth tenet of RME
emphasizes on students’ social interaction to support individual’s learning
process. The learning process of students is not merely an individual process,
but it is also a social process that both perform simultaneously (Cooke &
Buchholz, 2005; Lave & Wenger, 1991 in Lopez & Allal, 2007; Michelle
& Cobb, 2003 and Zack & Graves, 2002). The learning process of students
can be shortened when students communicate their works and thoughts in the social
interaction both in game playing and class discussion. As mentioned earlier,
game playing needs to be supported by a class discussion to develop students’
concrete experiences into mathematical concepts. Consequently, in the class
discussion the teacher plays an important role in orchestrating social
interaction to reach the objectives both for individual and social learning
(Cooke & Buchholz, 2005 and Doorman & Gravemeijer, in press). The roles
of teacher in the class discussion can be elaborated in the following manners:
1.
Providing students opportunities to present ideas
According
to the third tenet of RME, it is important to start the class discussion by using
students’ own construction, such as students’ strategies. The teacher,
as the orchestrator of class discussion, should stimulate students to present
their ideas as the starting point of class discussion.
2.
Stimulating social interaction
According
to Vigotsky in Zack & Graves (2001), social interaction is the core of learning
process because learners first construct knowledge in their interaction with people
and activity or context. Therefore, a teacher should be a good orchestrator in
provoking students’ social interaction. The teacher could provoke social
interaction by either making groups of students or asking questions. Generating
micro discussions in a macro discussion in the class discussion can be the
first step to stimulate the students to share and discuss their strategies. The
second strategy for stimulating social interaction is by posing appropriate
questions.
3.
Connecting activities
In
supporting students’ learning, it is important for the teacher to help children
communicate and develop their ideas by elaborating upon what they already know.
An example of this manner is when the teacher encourages students to perceive
the concept of measuring as covering.
4.
Eliciting the mathematical concepts
The
most important objective of a class discussion is transforming students’ concrete
experiences into mathematical concepts as mentioned by Cooke & Buchholz
(2005) and Kolb in de Freites & Oliver (2006). Transforming a concrete
experience into a mathematical concept can be facilitated by posing stimulating
problem or conflict.
5.
Asking for clarification
Asking
for clarification is important for the learning process because it can investigate
students’ reasoning about their idea or strategies that could reveal both difficulty
and achievement in students’ learning process. The following vignette is an
example of a critical part in a student’s learning process that was revealed
through asking clarification (Ariyadi, 2008).
Dea : No.
We should start from number “1”
Teacher :
Why do we start from number “1”?
Dea :
Because zero is nothing
The
teacher’s question is a kind of question for asking clarification and
reasoning. From Dea’s response, it seemed that Dea was still confused between
measuring and counting object (cardinality). Therefore, from this invention the
teacher could know which part of the learning process that should be developed.
Another advantage of asking clarification is when students’ reasoning gives information
about the strength of particular methods or strategies that could support
students’ learning process.
As a
summary, by asking clarification we can know how a weakness of some students’ progress
could be diminished by providing a proper guidance. How the strength of
particular methods or strategies offer an opportunity to develop students’
learning process also can be found by asking clarification.
CONCLUSIONS
In
general, the use of game playing in learning mathematics can give three
benefits, namely: (1) motivational benefit, (2) social benefit and (3)
conceptual benefit. Attractive is the characteristic of game playing that lead
to the first benefit, motivational benefit. The attractivity of game playing
can stimulate students’ personal motivation and satisfaction in the learning
process. Many games are played in groups or at least in pairs, therefore games
can give social benefit. Teamwork or at least communication created in a game
playing can foster collaboration and interactivity among students. The last
benefit, conceptual benefit, means that game playing can be an effective tool
for enhancing learning and understanding of mathematics concepts.
REFERENCES
Ariyadi
Wijaya. 2008. Indonesian Traditional Games as Means to Support Second Graders’
Learning of Linear Measurement. Master Thesis. Utrecht: UtrechtUniversity.
Castle,
K. & Needham, J. 2007. First Graders’ Understanding of Measurement. Early
Childhood Education Journal. Vol. 35: 215 – 221.
Doorman,
L.M. & Gravemeijer, K.P.E. in press. Emergent modeling: discrete graphs to support
the understanding of change and velocity. ZDM Mathematics Education.
Freudenthal, H. 1991. Revisiting Mathematics Education: China Lectures.
Dordrecht, The Netherlands: Kluwer Academics Publisher.
Garris,
R., Ahlers, R., & Driskell, J. E. 2002. Games, motivation, and learning: A research
and practice model. Simulation & Gaming, 33(4), 441–467.
CHAPTER II
ANALYSIS JOURNAL
2.1 Background
Thecentral
principle of Realistic Mathematics Education is that the learning of
mathematics needs to be laid on meaningful situation, either meaningful problems
or activities. Many research found that activities of game playing are
meaningful to students and can be effective tools for enhancing learning and
understanding of complex subject matter. Consequently, this article focuses on
the benefit of games to support the learning of mathematics and the role of teacher
in conducting game-based learning.
2.2
Research Question
How the benefit of games to support
the learning of mathematics and the role of teacher in conducting game-based
learning ?
2.3 Purpose
The purpose of journal is :
1.
To know the benefit of games to
support the learning of mathematics
2.
To know the role of teacher in
conducting game-based learning
2.4 Research Method
1.
Experiments with direct approach
to student
2.
Understanding of mathematics
learning method of experts
2.5
Research Result
A.
Tenets of realistic mathematics education
There are five tenets of realistic mathematics
education defined by Treffers (1987), namely:
1.
Phenomenological
explorationContextual problems are used
as the base and starting point for the teaching and learning process. The
teaching and learning process is not started from formal level but from a situation
that is experientially real for student.
2.
Using models and symbols for progressive mathematization
The aim of this tenet is bridging from
concrete level to more formal level using models and symbols.
3.
Using students’ own construction
The freedom for students to use their
own strategies could direct to the emergence of various solutions that can be
used to develop the next learning process. The students’ strategies in the activities
are discussed in the following class discussion to support students’ acquisition
of the formal level of mathematics concepts
4.
Interactivity
The learning process of students is not merely an individual process, but it is also a social process. The learning process of students can be shortened when students communicate their works and thoughts in the social interaction emerged in classroom.
The learning process of students is not merely an individual process, but it is also a social process. The learning process of students can be shortened when students communicate their works and thoughts in the social interaction emerged in classroom.
5.
Intertwinement
The activities used in the teaching and learning process do not merely support learning for a single mathematics topic, but they also should support the learning process of other mathematics topics or concepts.
The activities used in the teaching and learning process do not merely support learning for a single mathematics topic, but they also should support the learning process of other mathematics topics or concepts.
B.
Game-based learning
Games can be powerful and attractive
contextual situation for students because one of the natural characteristics of
a game playing is that it uses more action instead of word explanation. This
characteristic will stimulate students’ personal motivation and satisfaction in
the learning process.
C.
Games for mathematics learning
The following are
examples of game that can be used in the learning of
mathematics concepts:
mathematics concepts:
1.
Gundu
The mathematics concepts embodied in the first part of Gundu game, namely when determining the order of the players. In this part all players have to throw their marble to a given hole or pole. The first player is a player whose marble is the nearest to the hole or pole.
Students can use two strategies when determine the nearest marble, namely by comparison (when the different distances of the marbles is obvious to observe) and measurement (for the close distances).
The mathematics concepts embodied in the first part of Gundu game, namely when determining the order of the players. In this part all players have to throw their marble to a given hole or pole. The first player is a player whose marble is the nearest to the hole or pole.
Students can use two strategies when determine the nearest marble, namely by comparison (when the different distances of the marbles is obvious to observe) and measurement (for the close distances).
2.
Dakon or congklak
Dakon
is a game that is played by two players and each player has a big hole as “a deposit
hole” and some small holes (the number of small holes depends on the type of
dakon). The mathematics concepts embodied in Dakon game are counting and
division. The counting emerges when students distribute the seeds or beads and
count the final result in the “deposit hole”. At the beginning of the game,
each player has to distribute his/her seeds or beads in the small holes
equally. The concept of division can be developed when the teacher change the
rule of the game and giving various numbers of seeds or beads to students.
3.
Ular tangga
Mathematics
concepts that are embodied in Ular tangga are counting, addition and subtraction.
The counting (i.e. counting one by one) emerges when a player/student moves
his/her pin step by step. Addition concept emerges when a player does not move
his/her pin step by step, but he/she adds the number shown by the die to the
current position of the pin. Addition concept can be focused or emphasized by
asking students about how many steps they get as an advantage when they get a
stair. On the other hand, the disadvantage when students meet a snake can be used
to emphasize the concept of subtraction.
D.
The role of teacher in the
game-based learning
Game playing provides a natural situation for social interaction,
such as students’ agreement in deciding a strategy for the fairness of their
games (Ariyadi Wijaya, 2008). Interactivity as the fourth tenet of RME
emphasizes on students’ social interaction to support individual’s learning
process.
The roles of teacher in the
class discussion can be elaborated in the following manners:
1.
Providing students
opportunities to present ideas
2.
Stimulating social
interaction
3.
Connecting activities.
4.
Eliciting the mathematical
concepts
5.
Asking for clarification
BAB III
CONCLUSION
3.1 Suggestion
In general, the use of game playing
in learning mathematics can give three benefits, namely :
1.
Motivational benefit
2.
Social benefit
3. Conceptual benefit
Attractive
is the characteristic of game playing that lead to the first benefit,
motivational benefit. The attractivity of game playing can stimulate students’
personal motivation and satisfaction in the learning process. Many games are
played in groups or at least in pairs, therefore games can give social benefit.
Teamwork or at least communication created in a game playing can foster
collaboration and interactivity among students. The last benefit, conceptual
benefit, means that game playing can be an effective tool for enhancing
learning and understanding of mathematics concepts.
REFERENCES
Ariyadi
Wijaya. 2008. Indonesian Traditional Games as Means to Support Second Graders’
Learning of Linear Measurement. Master Thesis. Utrecht: UtrechtUniversity.
Castle,
K. & Needham, J. 2007. First Graders’ Understanding of Measurement. Early
Childhood Education Journal. Vol. 35: 215 – 221.
Doorman,
L.M. & Gravemeijer, K.P.E. in press. Emergent modeling: discrete graphs to support
the understanding of change and velocity. ZDM Mathematics Education.
Freudenthal, H. 1991. Revisiting Mathematics Education: China Lectures.
Dordrecht, The Netherlands: Kluwer Academics Publisher.
Garris,
R., Ahlers, R., & Driskell, J. E. 2002. Games, motivation, and learning: A research
and practice model. Simulation & Gaming, 33(4), 441–467.
