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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

Learning

Average

Whole

MPA Group

High

Middle

Low

CPA

0,738

0,822

0,711

0,736

Conventional

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–

Representational–Abstract

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.

49–60, 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.