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INFLUENCE OF CONCRETE-PICTORIAL-ABSTRACT (CPA) APPROACH TOWARDS THE ENHANCEMENT OF MATHEMATICAL CONNECTION ABILITY OF ELEMENTARY SCHOOL STUDENTS

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INFLUENCE OF CONCRETE-PICTORIAL-ABSTRACT (CPA) APPROACH TOWARDS THE ENHANCEMENT OF MATHEMATICAL CONNECTION ABILITY OF ELEMENTARY SCHOOL STUDENTS

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. The study aims to examine the influence of CPA learning approach towards the enhancement of mathematical connection ability of elementary school students. The present research is a quasi experiment using pretest and posttest design controls in Mathematics applied to 39 elementary school students in Purwakarta, West Java, Indonesia. The results of the study show that mathematical connection ability of elementary school students who were taught using CPA learning approach is enhanced than the elementary school students who were taught using conventional learning as a whole group of high and low achiever students according to their mathematical prior ability. Abstrak: Penelitian ini bertujuan untuk menguji pengaruh pendekatan pembelajaran CPA terhadap peningkatan kemampuan koneksi matematika siswa sekolah dasar. Penelitian ini adalah kuasi eksperimen dengan menggunakan kontrol desain pretest dan posttest dalam Matematika diterapkan kepada 39 siswa sekolah dasar di Purwakarta, Jawa Barat, Indonesia. Hasil penelitian menunjukkan bahwa kemampuan koneksi matematika siswa sekolah dasar yang diajarkan menggunakan pendekatan pembelajaran CPA lebih meningkat daripada siswa sekolah dasar yang diajar menggunakan pembelajaran konvensional secara keseluruhan dalam kelompok siswa berprestasi tinggi dan rendah sesuai dengan kemampuan matematika mereka sebelumnya.
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EduHumaniora: Jurnal Pendidikan Dasar | p-ISSN 2085-1243 | e-ISSN 2579-5457
Vol. 10 No.2 Juli 2018 | Hal 61-71
Putri, Misnarti, dan Saptini: Influence of Concrete-Pictorial-Abstract (CPA) Approach 61
INFLUENCE OF CONCRETE-PICTORIAL-ABSTRACT (CPA)
APPROACH TOWARDS THE ENHANCEMENT OF
MATHEMATICAL CONNECTION ABILITY OF ELEMENTARY
SCHOOL STUDENTS
Hafiziani Eka Putri
1
, Misnarti
2
, Ria Dewi Saptini
3
Universitas Pendidikan Indonesia, Kampus Purwakarta
Abstract:. The study aims to examine the influence of CPA learning approach towards the
enhancement of mathematical connection ability of elementary school students. The present research
is a quasi experiment using pretest and posttest design controls in Mathematics applied to 39
elementary school students in Purwakarta, West Java, Indonesia. The results of the study show that
mathematical connection ability of elementary school students who were taught using CPA learning
approach is enhanced than the elementary school students who were taught using conventional
learning as a whole group of high and low achiever students according to their mathematical prior
ability.
Keyword: Mathematical connection ability, CPA learning approach.
Abstrak: Penelitian ini bertujuan untuk menguji pengaruh pendekatan pembelajaran CPA terhadap
peningkatan kemampuan koneksi matematika siswa sekolah dasar. Penelitian ini adalah kuasi
eksperimen dengan menggunakan kontrol desain pretest dan posttest dalam Matematika diterapkan
kepada 39 siswa sekolah dasar di Purwakarta, Jawa Barat, Indonesia. Hasil penelitian menunjukkan
bahwa kemampuan koneksi matematika siswa sekolah dasar yang diajarkan menggunakan
pendekatan pembelajaran CPA lebih meningkat daripada siswa sekolah dasar yang diajar
menggunakan pembelajaran konvensional secara keseluruhan dalam kelompok siswa berprestasi
tinggi dan rendah sesuai dengan kemampuan matematika mereka sebelumnya.
Kata Kunci: Kemampuan koneksi matematis, pendekatan pembelajaran CPA.
INTRODUCTION
The background of the present
study is the existence of gap between the
expectation of curriculum and the reality of
mathematics learning in Indonesia
elementary schools. Stated in the
curriculum objective, Indonesia elementary
school students who are learning
mathematics are expected to master
mathematical ability, for example,
mathematical connection ability
(Depdiknas, 2006; Permendikbud Number
81A Year 2013). The importance of
mathematical connection ability is stated
clearly by NCTM (2000) that mathematical
connection helps students to develop their
perspective, seeing mathematics as an
1
UPI Kampus Purwakarta, Email: hafizianiekaputri@upi.edu
2
SDN 37 Tanjungpandan
3
UPI Kampus Purwakarta
integrated part of a group of topics and
admitting relevance and application both in
and outside the classroom. However, the
reality based on previous studies show that
mathematical connection ability of
Indonesia elementary school students is
still low (Suwandari, 2015; Saptini, 2016).
Based on this situation, improvement
should be made to enhance students’
mathematical connection ability.
The enhancement of elementary
school students’ mathematical connection
ability can be done through appropriate
learning approach. Elementary school
students who are, in average, between 7-12
years old are still in concrete operational
stage as stated by Piaget (in Desmita, 2008)
62 EduHumaniora: Vol. 10 No. 2, Juli 2018
that in this particular age, the students think
logically about the concrete events and
classify materials into different shapes.
Mathematics learning which is taught to the
students in elementary school must use
something real and concrete so that students
are expected to be able to understand the
learning better especially to be able to
understand the connection in the daily use.
One of the learning approaches thought to
be appropriate to enhance the mathematical
connection ability of elementary school
students is CPA approach.
Concrete-Pictorial-Abstract is an
approach in mathematics learning that is
conducted in stages. Every stage is
developed upon the previous stage and
because of that it should be conducted in
order.CPA learning consists of three
learning stages i.e. learning through
physical manipulation of concrete
materials, learning through pictorial
representation of concrete materials
manipulation, and problem solving using
abstract notation (Witzell, 2005).The
findings of study by Witzel (2005) upon
students of grade six and seven who are
identified to have learning algebra
difficulty conclude that students who learn
to solve algebra equation transformation
through CPA approach obtain higher test
result than the students of class control
(who received traditional learning).It is in
line with the study of Putri (2015) which
concludes that the implementation of CPA
approach in mathematics learning can
enhance the ability of mathematical
representation, spatial sense, and self-
efficacy of students who are going to be
elementary school teachers.
Some previous studies found
interaction between the learning approach
implemented and the mathematical prior
ability with the achievement or
enhancement of certain mathematical
ability. For example, study conducted by
Surya (2013) shows that there is a
significant interaction between contextual
learning approach and students group of
Mathematical Prior Ability (MPA) to the
improvement of visual thinking
representation ability. Furthermore, study
of Dwirahayu (2012) states that there is an
interaction between explorative learning
strategy and students’ MPA to the
improvement of visualization ability.
Therefore, it is important to investigate the
influence of the implementation of CPA
approach in mathematics learning in every
level of MPA group to enhance the ability
of students’ mathematical connection
ability. According to the elaboration above,
the writer is motivated to conduct a study
focused on the effort to enhance
mathematical connection ability of
elementary school students through CPA
approach learning. Hopefully, the findings
of the present study could give benefit both
for teachers in conducting mathematical
learning in class and for other researchers
who are interested in developing
mathematical learning.
LITERATURE REVIEW
Mathematical connection ability
Mathematical connection ability
can be defined as an ability to see the
interrelationship of mathematical concept
and an ability to see the interrelationship
between mathematics with other fields both
with other subjects and with daily life.
Bruner stated that in mathematics, every
concept relates to the other concept, so is
the others, for example between postulates,
between thories, between topics, and
between other mathematics branches for
example between algebra and geometry
(Ruseffendi, 1988). Therefore, in order for
students to be able to succeed more in
learning mathematics, students should be
given more opportunity to observe the
interrelationship.
There are two general mathematical
connections i.e. modeling connections and
mathematical connections. Modeling
connection is the relationship between the
situation of problems occured in the daily
life or in other disciplines with its
mathematical representation; meanwhile,
Putri, Misnarti, dan Saptini: Influence of Concrete-Pictorial-Abstract (CPA) Approach 63
mathematical connection is the relationship
between two equivalent representations and
between the solving process of each
representation. Lestari (2014) states that
mathematical connection ability is a high
level thinking ability which should be
developed because in mathematics
learning, every concept is interrelated to the
other concept. This mathematical
connection ability will give a more
meaningful learning for the students so that
it could help the students in understanding
the interrelationship between concepts.
Mathematical connection ability is
classified into three types, they are: 1)
Connection between mathematical
concepts i.e. mathematical material or topic
that has many connection on one to another.
This connection between mathematical
concept can help students to be able to
connect the topics; 2) Connection with
other disciplines outside mathematics i.e.
mathematics relates to the other subjects
that students have known or will know such
as physics, economy, social study, and
science; 3) Connection with the real world
or the daily life i.e. indicating that
mathematics can be connected to the
problem solving of daily life issues (NCTM
in Harahap, 2015).
Indicator of mathematical
connection ability according to NCTM
(2003) includes identifying and using the
interrelationship between mathematical
ideas, understanding how mathematical
ideas connected and developed one to
another so it is tied completely, identifying
and using mathematics in the concept
outside mathematics.In line with that,
Sumarmo (2005) suggests some indicators
to asses students’ mathematical connection
ability, they are: 1) identifying equivalent
representation of same concept; 2)
identifying the relationship of a
representation’s mathematical procedure to
the equivalent representation procedure; 3)
use and assess the interrelationship between
mathematical topics and interrelationship
outside mathematics; and 4) use
mathematics in the daily life.The indicators
of mathematical connection ability used in
the present study refer to the indicators
made by NCTM in which the connection
ability that will be assed is by connecting
mathematical topics, connecting
mathematics with other subject, and
connecting mathematics with the real world
or the students’ daily life. To evaluate
students’ mathematical connection ability,
scoring guide called as holistic scale from
North Carolina Department of Public
Instruction (in Ratnaningsih, 2003) is used
as shown in the table below:
Table 1. Scoring Guideline of Mathematical
Connection Questions
Students’ Respond to the
Questions
Score
No answer/answers not relevant
to the questions/no anwers
correct.
0
Only few of the questions
correctly answered.
1
Almost all aspect of questions
correctly answered.
2
All aspect of questions
fully/clearly and correctly
answered.
3
Concrete-Pictorial-Abstract (CPA)
Learning Approach
Concrete-Pictorial-Abstract (CPA)
learning approach is also usually called as
Concrete-Representational-Abstract (CRA)
approach or Concrete-Semiconcrete-
Abstract (CSA) approach. Neglecting the
names, those three learning approaches are
similar and initially are based on the
thinking of Bruner in 1960. The CPA
approach consists of three learning stages,
they are: (1) Physical manipulation of
concrete materials, (2) Pictorial
representation from concrete manipulation,
and (3) Problem solving using abstract
notation (Witzell, 2005). In line to that,
Cooper (2012) explains three orderly stages
from the learning with CPA approach, they
are: (1) Concrete stage is the first stage
which involves students physically
interacting by manipulating concrete
64 EduHumaniora: Vol. 10 No. 2, Juli 2018
materials (manipulative), (2) pictorial stage
is a transition stage which involves students
working by representing concrete model
which usually in the form of drawing
circles, dots, counting, or geometric
picture, and (3) Abstract stage is the final
stage where a mathematical concept
modeled simbolically using numbers,
variables, and other mathematical symbols.
The three aforementioned learning
stages using CPA approach is a unity which
implementation support one another.It is in
line with Riccomini et al. (in Yuliawaty,
2011) who states that the three learning
stages in CPA approach are integrated, may
not be seen separately to ensure students’
learning success. Miller and Mercer (in
Sousa, 2007) similarly state that each stage
in CPA approach constructs the previous
learning to enhance the conceptual
knowledge and retention in mathematics
learning.
Learning with CPA approach
provides a conceptual framework to create
a meaningful relationship between the
stages of concrete, pictorial, and abstract
understanding.The learning stages within
CPA approach according to Flores (2010)
are as follows: (1) Choose concrete
materials (manipulative) which will be used
to introduce conceptual understanding of a
lesson that will be learned by the students;
(2) Guide the students to independently
participate in using the concrete materials
(manipulative) by giving them direction
and sign; (3) Change the use of the
manipulative materials with pictures or
paintings; (4) Use strategy that could help
the students memorizing previous learning
previously conducted. It functions as a
transition process from the use of pictures
or paintings to the use of numbers or
symbols; and (5) Encourage the students to
only use numbers or symbols in completing
mathematics exercise given, and this
activity is focused on mastery.
In the learning process with CPA
approach, it is very possible that students
with higher prior mathematical ability
(students with high and middle
mathematical prior ability) feel no need to
use the concrete materials to create
manipulative pictures from them or the
students feel no need to create pictures to
make connection with the abstract concept.
However, the activities are needed to be
conducted because some experience shows
that the aforementioned learning activities
allow students to internalize the problem
solving process and help them to master the
skill to imitate process. It is in line with
Anstrom (in Yuliawaty, 2011) who states
that such activities conducted in the
learning with CPA approach will firm the
knowledge background and ensure the
students to do reasoning approach and
create connection to solve more complex
issues. Learning process with real activity
by using concrete materials and pictures
when introducing new concepts in
mathematics is very important to be
conducted. It is in line with Sousa (2007)
that the development of cognitive strategy
llike the stages of learning with CPA
approach needs to be conducted to inform
students regarding the technique to solve
mathematical issue not only to find the
answer.
In the learning process with CPA
approach, there is a manipulative aspect
which is called to contain advantage and
trick at the same time. As a fun factor,
learning by using manipulative materials
can enhance students’ diposition and
attitude in class (Goracke in Cooper, 2012).
However, the use of manipulative materials
has a tricky potential when the students
consider the learning as a playing activity
rather than as an opportunity to enhance
their understanding on mathematics (Mc
Neil and Jarvin in Cooper, 2012).
The use of manipulative materials
in mathematics learning is an important
process to help students understand the
mathematics concepts they are learning. It
is in line with Lidinillah (2009) that
mathematics learning should be able to
bridge chidren’s thinking ability which still
in the level of concrete operational with
Putri, Misnarti, dan Saptini: Influence of Concrete-Pictorial-Abstract (CPA) Approach 65
mathematics which conceptually abstract.
Therefore, manipulative materials can be
used as a bridge to connect concrete to
abstract way of thinking. Moreover
Hartshom et al. (in Yuliawaty, 2011) states
that implementing idea in mathematics is
difficult because some mathematics
concepts are very “abstract”. One of
practical ways to give a meaningful
learning experience is by using the
manipulative materials.
The importance of the use of
manipulative materials is also stated by
Skemp (in Turmudi, 2012) that
manipulative materials in mathematics
learning can be used as a foundation to
learn more abstract lesson. It is in line with
Bruner (in Suwangsih, 2012) who states
that interaction with manipulated concrete
materials can help students memorizing and
implementing mathematics ideas learned in
solving the issue in real world correctly.
Similarly, Brownell (in Suwangsih, 2012)
states that the use of concrete materials to
be manipulated can help students to
understand the meaning of concept and skill
learned. Therefore according to the
aforementioned studies, the use of concrete
materials to be manipulated can be very
meaningful in the effort to develop and
enhance students’ mathematical ability.
After conduct a learning with CPA
approach, it is possible to happen that
students cannot solve the abstract problem
successfully. To the case like this, teacher
should identify the reason why the students
do not understand the concept.Ricomini
(2010) advices, as follows: (1) teach again
the concept in concrete level; (2) teach
again the concept in representation level;
(3)give students opportunity to speak with
their own words in explaining the solution
and how they got it. Therefore, it suggests
that if the students do not show pictorial
level mastery, the learning is back to the
concrete level. Likewise, when the students
do not show abstract level mastery, the
learning is back to the pictorial level.
Learning with CPA approach is
advantageous for most students and has
been proven effective to help students who
have difficulty in learning mathematics
because this CPA approach is gradual from
learning through real materials to through
pictures then through symbols (Jordan,
Miller, and Mercer in Sousa, 2007).
Students often feel frustrated when teacher
gives a mathematical issue only in abstract
form. Teacher needs to develop concept to
control mathematical content and to
provide teaching which allow students to
process new learning more meaningfully
and effectively.
Some studies support the effectivity
of the approach, for example the study of
Witzel (2005) to the six and seven grade
students who are identified to have
difficulty in algebra learning. Students who
learn how to solve algebra equation
transformation through CPA approach
obtain higher test result than the class
control students (received traditional
teaching). In addition, students using CPA
approach make less prosedural mistake
when solving algebraic variables (Wizel,
Mercer, and Miller in Sousa, 2007).
Findings in Yuliawaty (2011)
suggest that the enhancement of
understanding ability and mathematical
problem solving of junior high school
students who receive learning through CPA
approach is better than students who
receive conventional learning.Moreover,
study of Yeo et al. (2005) which conducted
to students of Tamasek Junior College
Singapore found that the learning with CPA
approach could help the students who have
difficulty in visualizing and connecting to
correct a certain integral statement and
evaluating volume in higher level of
calculus. The findings in Putri (2015)
concluded that the ability of mathematical
representation, spatial sense, and self-
efficacy of students who are going to be
elementary school teachers in a public
university in West Java, Indonesia with
CPA approach enhance than the students
who receive conventional learning.
Therefore, it can be concluded that
learning with CPA approach is a learning
66 EduHumaniora: Vol. 10 No. 2, Juli 2018
approach which pays attention to the order
of the three learning stages, starting from
the use of concrete materials, then students
are given opportunity to create pictorial
representation from the concrete materials,
and finally students work in abstract
notation. It is expected that by doing the
three learning stages, students will
understand the mathematical concepts
clearly and correctly and feel direct
advantage when learning mathematics.
The interrelationship between
mathematical connection ability and the
Concrete-Pictorial-Abstract (CPA)
approach
The indicators of mathematical
connection ability that will be assed in the
present study are students’ ability to relate
mathematics topics, to relate mathematics
with other subjects, and to relate
mathematics with real world or the
students’ daily life. To enhance the
abilities, CPA approach is used in the
learning in the class. The first stage in
learning with CPA approach is learning
with concrete materials. The use of
concrete materials will certainly help
students to enhance their ability to relate
mathematics with the real world. The
second stage is pictorial, in this stage the
students are given opportunity to make
representation of real materials by using
pictures. Pictures that the students make
function as a connecting bridge between
real world and abstract mathematics. The
learning in this stage will help students to
enhance their ability to relate mathematics
with other subjects of other disciplines. The
last learning stage is abstract stage, in this
stage students will learn using abstract
mathematical symbols. The learning in this
stage will give opportunity to students to
enhance their ability to relate mathematical
topics. Therefore, every learning activity
designed within CPA approach is predicted
to be able to enhance students’
mathematical connection ability.
RESEARCH METHOD
The present study is a quasi
experiment study with pretest and posttest
control groups. Ruseffendi (1998: 45)
describes such study as follows:
O X O
O O
Note: O = Test (pretest and posttest)
X = Mathematics learning using
CPA approach
The study is conducted with two
learning groups that are learning group with
CPA approach as an experiment group and
conventional learning as a control group.
The population in the present study
is all elementary school students in
Purwakarta district, West Java province,
Indonesia.The sample is the fifth grade
students of SDN 10 Ciseureuh and SDN
Cibatu of Purwakarta district. Subjects
whom becoming sample of the present
study are 39 students which consist of 20
students in experiment group and 19
students in control group. Students in the
experiment group will receive learning with
CPA approach and students in the control
group will receive conventional learning.
The instrument used is mathematical
conection ability test. The test is given in
the beginning and in the end of learning
both for the experiment and control groups
of students.
FINDING AND DISCUSSION
The ehancement of students’
mathematical connection ability is
observed from the normalized gain (N-
gain). The enhancement criterion is
classified based on the criterion suggested
by Hake (in Meltzer, 2002) as follows:
Table 2. Criterion of Students’ Mathematical
Connection Ability
Enhancement
Interval
Enhancement
Criterion
N-gain> 0,7
High
0,3< N-gain 0,7
Middle
N-gain 0,3
Low
Putri, Misnarti, dan Saptini: Influence of Concrete-Pictorial-Abstract (CPA) Approach 67
The recapitulation of analysis result
of students’ N-gain based on the learning
both as a whole and according to
Mathematical Prior Ability (MPA) is
presented in the following table.
Table 3. Enhancement Average of Mathematical
Connection Ability Based on MPA and as a Whole
Average
Whole
MPA Group
High
Middle
Low
0,738
0,822
0,711
0,736
0,735
0,771
0,897
0,382
The data in the table describes that
the average enhancement of students’
mathematical connection ability seen as a
whole does not show any difference.
Students with high and low MPA who
receive CPA learning approach obtain
higher mathematical connection ability
than the students who receive conventional
learning. Students with middle MPA
experience opposite situation, the
enhancement of students’ mathematical
connection ability is higher in students with
conventional learning than students with
CPA learning approach. Based on the N-
gain criterion, it is known that the average
enhancement of mathematical connection
ability as a whole is high in both types of
learning for students with high and middle
MPA. Meanwhile for the low MPA
students, enhancement of mathematical
connection ability is high in the CPA
learning approach and middle in
conventional learning.
To test the significance of the
average difference of mathematical
connection ability enhancement, test of
average difference is conducted. The result
of test of average difference with level of
significance 95% shows that if observed as
a whole and from high MPA students
group, there is no significant difference of
mathematical connection ability between
students with CPA and conventional
learning. In middle MPA students group,
the result of test of average difference
shows a significant difference in which the
mathematical connection ability of students
who receive conventional learning is
significantly better than students who get
CPA learning approach. Test of average
difference of mathematical connection
ability of students with low MPA shows a
significant difference in which
mathematical connection ability of students
with CPA learning approachis significantly
better than students with conventional
learning. Therefore, it can be stated that the
implementation of CPA approach
significantly influences the mathematical
connection ability of students with low
MPA.
Discussion
Descriptive study shows that the
enhancement of mathematical connection
ability in students who receive
CPAlearning approachis better than
students who receive conventional
learning, except to the group of middle
MPA students where oppossite finding
derived. This finding is in line with Witzel
(2005) who concluded that the sixth and
seventh grade students in United States who
get CPA approach learning achieve better
learning result in algebra than students who
get conventional learning. Putri (2015) also
concluded that the mathematical
representation ability of students who are
going to be elementary school teacherswith
CPA learning approach is better than
students with conventional learning.
The enhance of mathematical
connection ability of students who get
CPAlearning approach isbetter than
students with conventional learning
because the learning stages in CPA
approach give opportunity to the students to
develop new knowledge by creating a
connection with their prior knowledge. It is
in line with Miller and Mercer (in Sousa,
2007) which suggest that every stage in
CPA develop previous learning to motivate
students’ learning, memorizing ability, and
to construct conceptual knowledge.
The first stage in CPA learning
approach is the use of concrete materials
68 EduHumaniora: Vol. 10 No. 2, Juli 2018
which can be manipulated that can help
students to prepare themselves to learn
more abstract concept.It is in line with
Skemp (in Turmudi, 2012) which suggests
that the use of manipulativematerials in
learning can serve as a foundation for
further learning in more abstract level.
In line with this, Bruner (in
Suwangsih, 2012) states that interaction
with manipulated concrete materials will
firm the concept understanding and help
students to memorize ideas learned easier
and to apply it in real situation
appropriately. Furthermore, the second
stage in CPA learning is pictorial where
students are trained to transfer their
mathematical ability from concrete to
symbolic representation (abstract). In the
learning process in this stage, students are
trained to represent their various
mathematical ideas by making pictures,
diagrams, graphics, tables, symbols, or
mathematic models, designing problem
solving in mathematics, making questions
or explanation verbally and written with
their own language related to the process
and result of mathematical problem
solution derived.It is in line with NCTM
(2000) which states that the representation
of physical materials, pictures, diagrams,
graphics, and symbols help students to
communicate their thoughts.Therefore, it is
learning circumstance through such stages
in CPA learning approach that makes
students able to enhance their mathematical
connection ability more than students with
conventional learning.The highest
enhancement of mathematical connection
ability is achieved by students from high
MPA group.This situation is very possible
to happen because according to Arends (in
Yumiati, 2015), the ability to learn new
ideasdepends on earlier previous ability and
existing cognitive structure. It is in line with
Sweller (1998) who states that students
with high MPA have low cognitive burden
because they have already had many
schemesabout mathematic concept which
ease them to master mathematic ability.
Therefore, it is unsurprising that students in
high MPA group get higher achievement
and enhancement of mathematical
connection ability than students in middle
or low MPA groups. It is certainly
influenced by theirprior ability which eases
them to learn something new.
The achievement and enhancement
of mathematical connection ability of
students with low MPA are better than
middle MPA group in CPA learning
approach. It is possible to happen because
stages in CPA learning approach give
opportunity for the low achiever students to
understand the concept that is taught more
easily. Learning stages which are begun by
manipulating concrete materials continued
by representing concrete models which are
usually pictorial such as circles, dots, or
geometric pictures, and continued to
abstract stage in which mathematic concept
is modelled symbolically to help students
retain memory of a concept then develop
description of that concept understanding
process to their way of thinking.It is in line
with Jordan, Miller, and Mercer (in Sousa,
2007) who state that CPA approach is
benefitting most student and proven very
effective in helping students who have
mathematics learning difficultiesbecause
this CPA approach moves gradually from
the real materials to pictures and then to
symbols.Therefore, it is obvious that the
three stages in CPA give students with low
ability opportunity to develop their
mathematical connection ability.
It is observed that the low
achievement and enhancement of
mathematical connection ability of middle
MPA students than low MPA students are
because during the learning process, many
middle achiever studentsseemed to be less
enthusiast in the learning process of
manipulating concrete materials and
making pictures out of them; meanwhile, it
is different with the low achiever students,
they learned very enthusiastically when
given opportunity to manipulate concrete
materials and then to make pictures of
them. Low achiever studentsare often seen
Putri, Misnarti, dan Saptini: Influence of Concrete-Pictorial-Abstract (CPA) Approach 69
to try to use the concrete materials to
understand and to solve issues presented in
the worksheet, sometimes they are seen and
heard asking question to the high
achieverstudents, and discuss their thoughts
to their friends in the group. Middle
achiever students are trapped in a condition
where they feel no need to follow the
learning stages by using manipulative
materials, while the use of manipulative
materials (concrete materials) will be very
helpful in firming mathematical concept
understanding. It is in line with Lidinillah
(2009) who state that manipulative
materials can be used as a bridge to connect
concrete to abstract way of thinking.Similar
to Kelly (2006), manipulative materials can
be used to help students understand the
process of problem solving related to
mathematical concept or topic.
Based on the interview result,
students from low MPA group feelmore
helped in understanding mathematical
concept by having a learning opportunity
manipulating the concrete materials
directly and turning them into
pictures.These activities help firming their
understanding about mathematical concept
they are learning. Meanwhile, middle
achiever students tend to ignore the
learning process by using concrete
materials, they feel that the learning with
concrete materials is playing activity. This
middle MPA students’ pressumption could
trap the students to notseriously study so
that the enhancement of mathematical
connection ability of middle MPA students
is lower than low MPA students.The
situation is in line with Mc Neil and Jarvin
(in Cooper, 2012) which state that the use
of manipulative materials contains a tricky
potential for the students when they
consider it to be a playing activity to fill the
time rather than to provide an opportunity
to enhance their mathematical
understanding.
From all the explanation above, it
can be concluded that the implementation
of CPA learning approach can be used as
one of the techniques to enhance students’
mathematical connection ability especially
for students with low mathematical prior
ability. The use of manipulative materials
(concrete) that are varied and challenging
the students’ way of thinking is also needed
so that the students do not trap in a monoton
and boring learning situation.
CONCLUSION
All in all, according to the findings
and discussion of the present study, it can
be concluded that CPA approach is very
good to be implemented to enhance
students’ mathematical connection ability
especially for students with low
mathematical prior ability (MPA).The
enhance of mathematical connection ability
of students with low MPA who receive
CPA learning approach is high; meanwhile,
the enhancemet of mathematical
connection ability of students who receive
conventional learning is found in students
with middle MPA.
REFERENCE
Cooper, E. T. (2012). Using Virtual
Manipulatives with Pre-service
Mathematics Teachers to Create
Representational Models.
International Journal for
Technology in Mathematics
Education, Vol 19, No 3. [Online].
Diakses dari https://ehis-ebscohost-
com.ezp.lib.unimelb.edu.au/eds/pdf
viewer
/pdfviewer?vid=6&sid=cd03d495-
1f99-4ec2-90d5-85ac8c67257b
%40se ssionmgr115&hid=5.
Desmita. (2008). Psikologi Perkembangan.
Bandung: ROSDA.
Depdiknas. (2006). Kurikulum Tingkat
Satuan Pelajaran. Jakarta:
Depdiknas.
Dwirahayu, G. (2012). Pengaruh Strategi
Pembelajaran Eksploratif terhadap
Peningkatan Kemampuan
Visualisasi, Pemahaman Konsep
Geometri, dan Karakter Siswa.
(Disertasi). Sekolah Pascasarjana,
70 EduHumaniora: Vol. 10 No. 2, Juli 2018
Universitas Pendidikan Indonesia,
Bandung.
Flores, M. M. (2010). Using the Concrete
RepresentationalAbstract
Sequence to Teach Subtraction
With Regrouping to Students at
Risk for Failure. Journal: Remedial
and Special Education,Volume 31
Number 3 May/June 2010 195-207.
[Online]. Diakses dari
http://resource binder
802a.wikispaces.com/file/view/Eff
ective+Math+ Strategies+CRA.pdf.
Harahap, T. H. (2015). Penerapan
Contextual Teaching and Learning
untuk Meningkatkan Kemampuan
Koneksi dan Representasi
Matematika Siswa Kelas VII-2
SMP Nurhasanah Medan Tahun
Pelajaran 2012/2013. Jurnal
EduTech, 1(1), hlm. 1-19. Dipetik
Maret 25, 2016, dari
http://jurnal.umsu.ac.id/index.php/e
dutech/article/download/273/pdf_2.
Kelly, C. A. (2006). Using Manipulative in
Mathematical Problem Solving: A
Performance Based Analysis.
[Online]. Diakses dari
http://www.math.
edu/tmme/tmmevol3no2_colorado_
pp184_193.pdf.
Lestari, K. E. (2014). Implementasi Brain-
Based Learning untuk
Meningkatkan Kemampuan
Koneksi dan Kemampuan Berpikir
Kritis serta Motivasi Belajar Siswa
SMP. Jurnal Pendidikan Unsika,
2(1), hlm. 36-46. Dipetik Maret 25,
2016, dari
http://journal.unsika.ac.id/index.ph
p/judika/article/view/120.
Lidinillah. (2009). Alat Peraga Manipulatif
dalam Pembelajaran Pemecahan
Masalah Matematika di Sekolah
Dasar. [Online]. Diakses dari
http://abdulmuizlidinillah.wordpres
s.com.
Meltzer, D.E. (2002). Addenum to:The
Relationship between Mathematics
Preparation and Conceptual
Learning Gain in Physics: A
Possible “Hidden Variable” in
Diagnostics Pretest Scores.
[Online]. Diakses dari
http://www.physics.iastate.edu/per/
docs/Addendum_on_normalized_g
ain.pdf#search=%22meltzer%2C%
202002%2C%20gain%2C%20a%2
0possible%20hidden%20variable%
22.
NCTM. (2003). Program for Initial
Preperation of Mathematics
Specialists. [Online]. Diakses
darihttp://www.ncate.org/LinkClic
k .aspx?fileticket =%2Frfx
5Ju56RY%3D&tabid=676.
NCTM. (2000). Using the NCTM 2000
Principles and Standards with The
Learning from
Assessmentmaterials. [Online].
Diakses dari http://www wested.org
/lfa /NCTM2000.PDF.
Permendikbud. Nomor 81A Tahun 2013
tentang Implementasi Kurikulum
2013.
Putri, H.E. (2015). Penagaruh Pendekatan
Concrete-Pictorial-Abstract (CPA)
terhadap Peningkatan Kemampuan
Representasi Mathematis, Spatial
Sense, dan Self-Efficacy Mahasiswa
Calon Guru Sekolah Dasar.
(Disertasi). Sekolah Pascasarjana,
Universitas Pendidikan Indonesia,
Bandung.
Putri, H. E. (2015). The Influence of
Concrete-Pictorial-Abstract (CPA)
Approach to The Mathematical
Representation Ability
Achievement of the Pre-Service
Teachers at Elementary School.
International Journal of Education
and Research, 3(6), P.113-126.
Ratnaningsih, N. (2003). Mengembangkan
Kemampuan Berpikir Matematik
Siswa SMU melalui Pembelajaran
Berbasis Masalah. Bandung:
(Tesis) PPs UPI. Tidak diterbitkan.
Ruseffendi, E. T. (1988). Pengantar
kepada Membantu Guru
Mengembangkan Kompetensinya
Putri, Misnarti, dan Saptini: Influence of Concrete-Pictorial-Abstract (CPA) Approach 71
dalam Pengajaran Matematika
untuk Menigkatkan CBSA.
Bandung: Tarsito.
Ruseffendi, E. T. (1998). Statistika Dasar
untuk Penelitian Pendidikan.
Bandung: IKIP Bandung Press.
Saptini, D. R. (2016). Penerapan
Pendekatan CPA untuk
Meningkatkan Kemampuan
Koneksi Matematis Siswa SD.
Purwakarta: (Skripsi) UPI Kampus
Purwakarta. Tidak diterbitkan.
Sousa, D. A. (2007). The Concrete-
Pictorial-Abstract Approach.
[Online]. Diakses dari
http://www.logan
schools.org/mathframework/CPA.p
df.
Sumarmo, U. (2005). “Pembelajaran
Matematika untuk Mendukung
Pelaksanaan Kurikulum Tahun
2002 Sekolah Menengah”. Makalah
pada Seminar Pendidikan
Matematika 7 Agustus 2005
Universitas Negeri Gorontalo,
Gorontalo.
Surya, E. (2013). Peningkatan Kemampuan
Representasi Visual Thinking pada
Pemecahan Masalah Matematis
dan Kemandirian Belajar Siswa
SMP melalui Pembelajaran
Kontekstual. (Disertasi). Sekolah
Pascasarjana, Universitas
Pendidikan Indonesia, Bandung.
Suwandari, D. W. (2015). Penerapan
Pendekatan Kontekstual untuk
Meningkatkan Kemampuan
Koneksi Matematis Siswa SD.
Purwakarta: Skripsi UPI Kampus
Purwakarta. Tidak diterbitkan.
Suwangsih, E. (2012). Teori-teori Belajar
dalam Pembelajaran Matematika.
Subang: Royyan Press.
Sweller, J. (1998). Cognitive Load during
Problem Solving: Effect on
Learning. Cognitive Science, 1(2),
hlm.257-285.
Turmudi. (2012). Matematika Landasan
Filosofis, Didaktis, dan Pedagogis
Pembelajaran Matematika untuk
Siswa Sekolah Dasar. Jakarta:
Direktorat Jenderal Pendidikan
Islam, Kementerian Agama RI.
Witzel, W. S. (2005). Using CRA to Teach
Algebra to Students with Math
Difficulties in Inclusive Settings. A
Contemporary Journal, 3(2), hlm.
4960, 2005 .[Online]. Diakses dari
https://ehis-ebscohost-com.ezp.
lib.unimelb.edu.au/eds/pdfviewer/p
dfviewer?vid=7&sid= cd03d495 -
1f99-4ec2-90d5-
85ac8c67257b%40
sessionmgr115&hid=116.
Yeo, C. J. Wong, C. M., dan Ho, F. H.
(2005). Concrete-Pictorial-
Abstract (CPA) Approach for
Teaching Volume of Revulotion in
Advanced Level Mathematics.
[Online]. Diakses dari
http://www.ijstr.org/final-
print/may2018/Concrete-pictorial-
abstract-Approach-On-Students-
Attitude-And-Performance-In-
Mathematics.pdf
Yuliawaty, L. (2011). Pembelajaran
Matematika dengan Pendekatan
CRA (Concrete-Representational-
Abstract) untuk Meningkatkan
Kemampuan Pemahaman dan
Pemecahan Masalah Matematik
Siswa SMP. (Tesis). Sekolah
Pascasarjana, Universitas
Pendidikan Indonesia, Bandung.
Yumiati. (2015). Meningkatkan
Kemampuan Berpikir Aljabar,
Berpikir Kritis Matematis, dan Self-
Regulated Learning Siswa Melalui
Pembelajaran CORE. (Disertasi).
Sekolah Pascasarjana, Universitas
Pendidikan Indonesia, Bandung.
... This high-level ability aims to develop students' analytical, evaluation, and creative power. Mathematical connection skills are part of higher-order thinking skills that need to be mastered by students (Hendriana, Rohayati, & Sumarmo, 2017; The Ministry of Education and Culture of Indonesia, 2003;Putri, Misnarti, & Saptini, 2018;Sumarmo, 2010). ...
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Diakses dari http://resource binder 802a.wikispaces.com/file/view/Eff ective+Math+ Strategies+CRA.pdf. Harahap, T. H. (2015). Penerapan Contextual Teaching and Learning untuk Meningkatkan Kemampuan Koneksi dan Representasi Matematika Siswa Kelas VII-2 SMP Nurhasanah Medan Tahun Pelajaran 2012/2013. Jurnal EduTech, 1(1), hlm. 1-19. Dipetik Maret 25, 2016, dari http://jurnal.umsu.ac.id/index.php/e dutech/article/download/273/pdf_2.
Addenum to:The Relationship between Mathematics Preparation and Conceptual
  • D E Meltzer
Meltzer, D.E. (2002). Addenum to:The Relationship between Mathematics Preparation and Conceptual