International Education Studies; Vol. 9, No. 7; 2016 ISSN 1913-9020 E-ISSN 1913-9039
Published by Canadian Center of Science and Education
145
Limited Intervention at Sub Concept of Fractions in the Object Conversion into Fractions
Henry Kurniawan1,2, Toto Nusantara2, Subanji2, Susiswo2, Iwan Setiawan3, Akbar Sutawidjaja2, Abdur Rahman As’ari2 & Makbul Muksar2
1 Politeknik Negeri Lampung, Indonesia 2 Universitas Negeri Malang, Indonesia 3 SMA Negeri 2 Martapura OKU Timur Sumatera Selatan, Indonesia
Correspondence: Henry Kurniawan, Program Studi Manajemen Informatika Jurusan Ekonomi dan Bisnis, Politeknik Negeri Lampung, Indonesia. E-mail: [email protected]
Received: December 18, 2015 Accepted: January 23, 2016 Online Published: June 28, 2016
doi:10.5539/ies.v9n7p145 URL: http://dx.doi.org/10.5539/ies.v9n7p145
Abstract This research is an exploratory study with a qualitative approach, which is based on interviews with a task-based the purpose of this study is to describe the understanding of elementary school students in interpreting sub concept fractions in changing of the object is given to fractions with limit intervention. While intervention on problems solving mathematical in a fraction of this is an attempt to change behavior, thoughts and feelings of a person to develop the students’ knowledge in achieving the objectives of the fractions learning experience.
Keywords: interpretation, fractions, object, limit intervention
1. Introduction This article discusses the learning of mathematics in elementary school (elementary school) about the basic concepts of fractions by using media/props are believed to provide pleasure and understanding to students. Learning by using media/props greatly assist the creation of learning in accordance with the demands of the curriculum in 2013, namely: fun, contextual and meaningful through the steps of learning to observe, ask, experimental/discovery, process the information and summarizing the results which were consistent with the objectives.
Learning activities to recognize the concept of ordinary fractions will be more meaningful when it is preceded by a story about the use of real objects e.g. eggs, apples, tomatoes, tofu, pancakes, followed by fractional block or shaded paper. To avoid misunderstanding in assessing the concept of fractions and arithmetic operations on these numbers we will need to pay close attention to sub concept of fractions. Sub concept is useful to guide the understanding and give command of the facts, operations and principles of the fractions as an integral part of mathematical objects.
According to Psychology Bruner (1966) learning will be more meaningful and more quickly achieve the goal if it starts from the stage of concrete (enactive) that uses the real object, then semi-concrete (econic) the object that replaced the image, and the last is abstract (symbolic) the grain which is only in the form of emblem/symbol that only in the form of letters, or numbers only. According to Bruner if students are experienced math learning for each topic with treatment by the third stage, the students will be able to develop knowledge far beyond what they received from the teacher.
In the process of teaching the fifth grade students of SDN 2 Merjosari, teachers beginned by fractions ½ using paper one sheet is cut into two equal parts, and they told his students “a piece of paper divided into two parts, then the result is ½”, herein if No students ask “why ½?, why not the paper into two pieces?” On learning activity it is seened that some of students do not understand the concept of fractions ½.
Then the learning continue after students acquired basic knowledge of the fractions concept, the teacher wanted to measure students’ reasoning and the ability to apply the basic concepts of fractions. Those students were asked to determine the fractional value as congruent parts of a whole based on the colors red, yellow and white as in the Figure 1 below:
www.ccsen
From Figuimage. Altknow that and the va
This examis not reco
Students abetween thbetween thmay incluinterpretatstudents ar
Based on tin interpreinterventiointerpretininterventio
Once the r
a) Contrfracti
b) Provi
2. Literatu2.1 Concep
Clarke, Rothe fractiounderstanddifficult to
Learning aword prob
net.org/ies
ure 1 above stthough some sthe fractional
alue 2/4 of the w
mple raises the pognized by the
are said to havhe various inthe representati
ude oral or wrtion and reprere the parts tha
this backgrounet the sub conon?” While theng sub fractioon.
research is don
ribute to the dions.
ide inputs to th
ure Review pt of Fractions
oche and Mitchons, which is d students’ leao be introduced
activities to recblems using con
Figure 1
tudents answerstudents know
value as part white color.
possibility of mteacher, then t
ve a thoroughterpretations oion of the differitten symbol, esentation of fat states do not
nd, the researcncept of fractie purpose of th
ons in changin
ne, the results o
development o
he readers who
s
hell (2007) stadifferent from
arning (Behr, Ld to the studen
cognize the concrete objects
Internation
1. Fraction valu
red fractionalpart of the ovgroup congrue
major problemthe conflict wi
understandingof fractions. Inerent fractionsobject manipu
fractions, othet have to be co
ch question is:ions in changhis study is tong the concep
of this study ar
of the interpre
o are interested
ates that the com the naturalLesh, Post, &
nts.
oncept of fractie.g. apple and
nal Education Stu
146
ue as part grou
value, is ½, 1verall concept,ent part, is the
ms/conflicts inll lead to misc
g of the fractin addition, hes (Cathcart, Poulation, image
er mathematicaongruent or frac
“How does aning the objectdescribe the upt of the obj
re expected to
etation of the
d in this topic f
oncept of fractl numbers anSilver, 1983;
ions ordinary wd melon. As can
udies
up congruent p
1/3 and ¼ withbut the interp
e value ¼ of th
learning fracticonceptions stu
ions meaningalso must be
othier, Vance, &e and real woral ideas that actional equal s
n understandint of which isunderstandingject which is
be useful for:
students in un
for further rese
tions is not a snd integers, so
Kieren, 1976
will be more mn be seen in Fi
part
hout mentioninpretation of thehe red, the valu
ions for the nexudents in learn
if he can estae able to estab& Bezuk, 2006rld situations. are important say (Fosnot &
ng of elementagiven to fracof elementarygiven to fra
nderstanding t
earch.
imple conceptometimes mak; Streefland, 1
meaningful whigure 2 below:
Vol. 9, No. 7;
ng the color inese students diue ¼ of the ye
xt. If this conding fractions.
ablish relationblish a relation6). RepresentaIn addition toto be built byDolk, 2002).
ary school studctions with finy school studenactions with L
the sub conce
t; the uniqueneking it difficu1991) and mak
hen preceded b
2016
n the d not ellow
dition
ships nship ations o the y the
dents niting nts in Limit
pt of
ess of ult to kes it
y the
www.ccsen
Figure 2 abcut again iwith melon
Then proc
Fractional fractions s
Furthermocircular orThis can b
net.org/ies
F
bove, supposeinto each of thn.
eed to introduc
Figure 3
beam is shapstarting at 1, 1/
ore shading par square, so thebe seen in Figu
Figure 2. Introd
e an apple cut ihe two parts to
ce the concept
. Introduce the
ped props piec/2, 1/3, 1/4, 1/5
aper that statee precise pleat
ure 4 below.
Internation
duce the conce
in two equal paogether again,
t of fractions u
e concept of fr
ces beam or a5 and so on.
s the area. Suts covering eac
nal Education Stu
147
ept of fractions
arts having a fthen the value
using fractional
actions using f
a circle divide
uch as fractionch other. The
udies
s using concre
fractional valuee of the pieces
l block as in F
fractional beam
d into several
nal 1/2 can bepart that is fol
ete objects
e ½ and two eq into ¼ and so
igure 3.
ms (Husna, 20
l sections. The
e demonstratedlded open and
Vol. 9, No. 7;
qual parts hado on. So is the
12)
e size of the b
d by folding p shaded as des
2016
been case
beam
paper sired.
www.ccsen
Demonstra
2.2 Fractio
Accordingexpressed quantity, iamong oth2/3 can be(square, reand orienta
Regardlessnumbers st
Fractions
net.org/ies
ation for fracti
Figure
on Meaning
g to Musser, Brelative amou
it states the nuhers, can be a le shown or exectangle, paralation or perspe
s of size (smtated 2/3.
Figure 6. Fra
based on the
Figure 4. Sta
ions , , a
e 5. Stating the
Burger, and Punt (relative amumber of fraclong, wide are
xpressed as thellelogram-paraective (in view
mall, medium,
actions based o
quantity of st
Internation
ating the conce
and can be
concept of fra
Peterson (2006mount), and (2ctional part ofa, or volume. e area shadedallelograms, tr
w of shading or
large), shape
on the quantity
tating the relat
nal Education Stu
148
ept of fractiona
seen as in Figu
actional ,
6), the fraction2) as a numbef a number ofFor example for colored, re
riangles, circler coloring parts
e, composition
y to stating the
tive amount c
udies
al (Sukayati
ure 5 below.
, and (S
n has two meer that has thef equal parts.fractional 2/3,egardless of sies, or other shs: a flat, vertic
n, or orientati
e relative amou
an be express
i, 2003)
Sukayati, 2003
eanings, nameemblem. As tQuantities whas the relative
ize (small, mehapes), arrangal, oblique, cu
on, all of the
unt (Muhsetyo
sed as a symb
Vol. 9, No. 7;
3)
ely (1) as quathe meaning ohich are expree amount, fractdium, large),
gement or locaurved
e following m
, 2014)
bol or numera
2016
antity of the essed, ional build ation,
model
al (or
www.ccsenet.org/ies International Education Studies Vol. 9, No. 7; 2016
149
name), to indicate a relationship “many parts of the whole part of the same” (a part-to-whole relationship).
If p and q are numbers chopped and q ≠ 0, then the fraction or p/q, showed p dividing by q equal parts
(equivalent), p is called the numerator (numerator) and q is called the denominator (the denominator).
Several studies have been conducted to investigate the difficulties in understanding and learning the material fractions. Vale (2007) found that students will be more likely to make mistakes on a fractional operation if only a fraction of learning material focuses on memorizing formulas and procedures for operating without sufficient attention to the meaning of fractions. Part-whole relationship as a sub identifies key ideas and overall understanding of fractions which will be interpretation in fractions (Kieren, 1988). In addition, the complexity of the characteristics and stages of understanding the concept of fractions requires that made it can not be understood in a relatively short period (Yusof and Malone, 2003). Further research conducted Clarke, et al. (2007) found that the methods and learning strategies that are less precise can also contribute to the misconceptions students.
According to Bell, Castello, and Kucheman (1983) the concept of fractions can be interpreted into seven ways, namely:
a. Part group congruent part
Fractions as part group congruent part. Fraction is interpreted as an area divided into b congruent parts and
pay attention to a part.
b. Part group non congruent part
Fractions as part group non congruent part. Fraction is interpreted as an area divided into sections that are not
congruent b and pay attention to a part. c. Part whole congruent part
Fractions as part whole congruent part. Fraction is interpreted as a set consisting of b objects are congruent
and noticed an object d. Part whole non congruent part
Fractions as part whole not congruent part. Fraction is interpreted as a set consisting of b objects that are not
congruent and noticed a object. e. Part group comparison
Fraction as part group comparison. Fraction is interpreted as a relative comparison of two areas A and B. The
number of areas that are congruent to A is A, while the B as b. f. Part whole comparison
Fraction as part whole comparison. Fraction is interpreted as the ratio of the bunch is the comparison of the
relative number of objects in a set A with a lot of objects in a set B g. Number line
Fractions as the number line. Fraction is interpreted as a point on the number line. In this case, a unit on the
number line is divided into b equal parts, then point to note-a
The ability of representation is one of the standard math learning process should be developed and owned by the students. These process standards should not be submitted separately to the mathematical material. Unfortunately,
www.ccsen
the repres(Depkdiknmathematiother ideas
2.3 Limit I
The term iunderstandinteractionexperienceattempt tothe objecti
3. ResearcThis reseaNegeri 2 Mbeen learninstrumentdata.
Given the making noerror in co
Based on tand why reInterviews
Selected ustudent anthe form o
4. FindingResults of Merjosari into fragmset of obje
Results ofresults of t
net.org/ies
sentation is ofnas, 2005: 51ical concepts as, in recognizin
Intervention
intervention inding the intervn that is designe (structured l change behavives of the lear
ch Methods arch includes Merjosari Maln about fractiots are Student
subject matterotes, also recoonverting an ob
the results of stelated to a gives were used in
unstructured innd the answer wof unstructured
gs and Discussf this research grade 5 is use
ments that repreects which are
f interpretationthe students’ an
ften taught an). In fact, the
and relationshing the relation
n general is an vention by Samned to achieve learning expervior, thoughts rning experien
exploratory relang academic ons. The mainTask Sheet m
r of the researcorded with audbject into fragm
tudent work inen problem andthis study can
terviews becauwill be the star
d interviews.
sion was elected of
ed as the subjeesent fractionsgiven.
n fractional asnswers, as sho
Internation
nd learned ase representatiips in communnship between t
attempt to chamani and Hariythe formation
rience). So intand feelings oce fractions.
esearch with qyear 2014/20
n instrument imaterial fractio
ch through thediovisual. Thements. So that
n completing thd the results asbe categorized
use researcherrting point of
f five collectedect of the reses as a ratio and
s part group cown in Figure 7
Tran
Figure 7. Qu
nal Education Stu
150
if it stood aon is expecte
nicating mathemthe concepts o
ange behavior,yanto (2011) aof character wtervention inof a person to
qualitative app015 in the oddin this study
ons and compl
e student assignmatter obtainthe students w
he task to explowell as other pd as unstructur
rs do not yet knthe developme
d students whoearch is basedd parts that are
congruent part7 below
nslate version
uestion answer
udies
lone withouted to supportmatics, argum
of mathematics
, thoughts andan action to dewith the implem
solving mathdevelop the s
proach conducsemester, whis the researceteness, interv
nment sheet. Wned from studewill serve as res
ore the subjectpossibilities thared interviews
now what the ent of the ques
o were taken fon a wrong ane congruent an
t of the first su
1 of S1
any connectiothe students’
ments, and an uns (NCTM, 200
feelings (Marevelop an atmomentation of a h problems astudents’ know
cted on 32 stuere the subjec
chers themselvview, audio-vis
When it was obents representisearch subject
t of an intervieat arise from th.
answer will bestions that will
from 32 studennswer in transnd incongruen
ubject (S1) can
Vol. 9, No. 7;
on in mathem’ understandinnderstanding o0, p. 206).
rkam, 2003). Wosphere of learstructured lear
at a fractionwledge in achie
udents grade Vct has been andves and supposual, and trans
bserved directing encounteres.
ew about what,he impact ques
e obtained froml be followed u
nts of SD Negsforming the ont of the whole
n be seen from
2016
matics ng of of the
While rning rning is an eving
V SD d has orting script
ly by ed an
, how stions.
m the up in
geri 2 object e or a
m the
www.ccsen
Based on t
Researche
S1
Researche
S1 Interpretat
From this group conga fractiona1/3 look oarrows (re
Because SLimited in
Researche
S1 Researche
S1
Researche
S1
Researche
From inter
net.org/ies
the results writ
r : Is the fra
: It seems
r : How do y
: Figure 7tion of the resu
diagram 1 S1gruent part, beal value of bluof the groove bd arrow groov
S1 the answerntervention to S
r : How ma
: 4, sir. r : How m
: 3 blue an
r : Yes, well
(Limite
: Blue is 3
r : Very cle
rviews research
tten by S1, the
actional value
correct, sir.
you determine
7, its value 3/1 ults of interview
1 finish the anecause S1 prov
ue (B) to pink blue arrow, a f
ve that settleme
Diagram 1. I
r is not in accS1 the followin
any part entirel
any parts the b
nd pink there i
l, if the value o
ed intervention
3/4 and 1/4 is p
ver.
hers using Lim
Internation
e researchers co
that you wrote
e the value of s
blue to pink, pws with S1 can
nswer is not invide initial ans(P) is 3/1 visibfractional valuent should be).
Interpretation
cordance withng
ly?
blue ones? Ho
is none.
of the blue sha
n)
pink.
mited intervent
nal Education Stu
151
onducted inter
e is correct?
such fractions
pink to blue 1/3n be seen in di
n accordance wswer as followble on furrow
ue blue to all i.
of the results o
h the settlemen
ow many parts
ards how? And
tion by S1 can
udies
rviews with S1
in figure 7?
/3, blue againsiagram 1.
with the desirews the number
green arrow,is ¾ and pink
of interviews w
nt, the researc
of pink ones?
d what is pink f
be seen in dia
1 as follows
t all 3/4 and 1/
ed answer to part of the wha fractional vato all is ¼ vis
with S1
chers conducte
fractional valu
agram 2 below
Vol. 9, No. 7;
/4 pink agains
a fraction as phole was 4 (N)alue pink to blsible on furrow
ed interviews
ue?
:
2016
t all.
parts with
lue is w red
with
www.ccsen
Once the ithe questio
Next resulFigure 8.
Based on t
Researche
S2
Researche
S2
Interpretat
net.org/ies
D
intervention ison no one can
lts interpret fr
the results writ
r : Is the fra
: Yes, sir.
r : How do y
: Fraction
fractiona
tion of the resu
Diagram 2. Th
s Limited to Ssee the plot on
ractions as par
tten by S2, the
actional value
you determine
ns value in figu
al value is 4/2,
ults of interview
Internation
he results of int
1 by the resean diagram 2 on
rt group non c
Tran
Figure 8. Qu
e researchers co
that you wrote
e the value of s
ure 8, No 4 is b
while for the y
ws with S2 can
nal Education Stu
152
terviews S1 w
archers, the resn the right side
congruent par
nslate version
uestion answer
onducted inter
e is correct?
such fractions
blue, the yellow
yellowis 2/4
n be seen in di
udies
with limited inte
sults are consi.
rt of the secon
2 of S2
rviews with S2
in figure 8?
w are no 2. Th
4.
iagram 3.
ervention
istent with the
nd subject (S2)
2 as follows
en the blue
Vol. 9, No. 7;
e desired answ
) can be seen
2016
wer of
from
www.ccsen
From the yellow (Y)grooves bl
Because Swith interv
Researche
S2
Researche
S2
Researche
Interventio
From the rappears thand yellow
net.org/ies
diagram 3 loo) is 4/2 indicatlue arrows.
S2 wrong answvention limited
r : How ma
: 6, sir.
r : Yes. Wel
: owh yes,
r : Very goo
ons conducted
D
results of the Lat the S2 begin
w is 2/6.
Diagram 3. I
oks S2 interprted by green a
wer is not in ad to S2 as follo
any of the value
ll if so, how is
, it’s 4/6. 2/6 fo
od.
by researchers
Diagram 4. Th
Limited interven to understan
Internation
Interpretation
ret many partsarrows groove
accordance witows
e whole?
blue fractiona
for a fractional
s of the S2 sho
he results of int
entions conducnd the value of
nal Education Stu
153
of the results o
s of the wholewhile the yell
th the fraction
al value of the w
l value that is y
own in diagram
terviews S2 w
cted by researcf fractions as p
udies
of interviews w
e 6 (N) with alow to blue fra
nal value, the r
whole? (Limit
yellow .... hehe
m 4 below:
with limited inte
chers of the S2part group non
with S2
a fractional vaactional value
researchers con
ted intervention
e
ervention
2 in question n congruent pa
Vol. 9, No. 7;
alue of blue (Bis 2/4 indicate
nducted interv
n)
no 2 in diagramrt to the blue i
2016
B) to ed by
views
m 4, it is 4/6
www.ccsen
Then the nanswer to
Based on t
Researche
S3
Researche
S3
Interpretat
In the diagblack (Bk)
Because Sresearcher
Researche
S3
Researche
S3
Researche
S3
Researche
Interventio
net.org/ies
next result is tquestion no 3
the results writ
r : Is the fra
: Yes, sir.
r : How do y
: its orang
value of f
tion of the resu
gram 5 visible) and blue (B)
S3 wrong ansrs conducted in
r : 2/7 the fr
: Hmmmm
r : Yes, so h
: Oh yes, c
: Yes, Plea
: For the o
r : That’s ri
ons conducted
the interpretatbelow
tten by S3, the
actional value
you determine
ge color was in
f fractions 2/7.
ults of interview
Diagram 5
e S3 interpret ois 2/7 indicate
swer is not innterviews with
fractional valu
m seems to be o
how many frac
can I fix it, sir
ase
orange value f
ight.
by researchers
Internation
tion of fraction
Tran
Figure 9. Qu
e researchers co
that you wrote
e the fractional
n front of black
ws with S3 can
5. Interpretatio
overall the numed by the flow
n accordance Limited interv
e for the color
orange, sir.
ctional value of
?
fractions is 2/7
s to S3 shown
nal Education Stu
154
ns as part who
nslate version
uestion answer
onducted inter
e is correct?
l value?
k and blue whi
n be seen in di
on of the result
mber of partsarrows in blue
with the fracvention to S3 a
r orange or fra
f her black and
7, 4/7 black an
in diagram 6 b
udies
ole congruent
3 of S3
rviews with S3
ile the total of
iagram 5.
ts of interview
is 7 (N) withe, green and pu
ctional valueas follows
actional value
d blue? (Limit
nd blue colors 1
below:
part of S3 can
3 as follows
f number 7, the
ws S3
a fractional vaurple.
corresponding
overall?
ted intervention
1/7
Vol. 9, No. 7;
n be seen from
en the
alue of orange
g to the color
n)
2016
m the
e (O),
r, the
www.ccsen
From the r6, it appea2/7, black
Then procbelow:
Based on t
Researche
S4
Researche
S4
Interpretat
net.org/ies
D
results of the Lars that the S3 is 4/7 and blue
ceed with the i
the results writ
r : How abo
: Is it wro
r : Try to an
: 2/4 Oran
tion of the resu
Diagram 6. Th
Limited intervebegan to undee is 1/7.
nterpretation o
tten by S4, res
out the image 1
ng, sir?
nswer how mu
nge against bla
ults of interview
Internation
he results of int
entions conducerstand the val
of fractions as
Tran
Figure 10. Qu
searchers condu
10?
ch the value of
ack, black aga
ws with the S4
nal Education Stu
155
terviews S3 w
cted by researcue of fractions
s part whole n
nslate version
uestion answer
ucted interview
f the pieces?
ainst the 4/2.
4 can be seen i
udies
with limited inte
chers of the S3s as part whol
ot congruent p
r 4 of S4
ws with S4 the
in diagram 7.
ervention
3 in question nle congruent p
part from S4 s
e following
Vol. 9, No. 7;
no 3 in the diaart to the oran
shown in Figu
2016
gram nge is
re 10
www.ccsenet.org/ies International Education Studies Vol. 9, No. 7; 2016
156
Diagram 7. Interpretation of the results of interviews S4
In the diagram 7 visible S4 interprets many parts of the whole 6 (N) with a fractional value of the orange (O) to black (Bk) is 2/4 indicated by the flow arrows in blue and fractional value of the black to orange is 4/2 indicated by a green arrow groove.
Because S4 wrong answer is not in accordance with the fractional value corresponding to the color, the researchers conducted interviews with intervention Limited to the following S4
Researcher : What is the sum of that images a collection?
S4 : 6, sir.
Researcher : Exactly. Then how many images are in black? And how many pictures are orange?
(Limited intervention)
S4 : The black is ones 4 and the orange is 2. The black fractions 4/6 and 2/6 orange, right sir?
Researcher : great.
Interventions conducted by researchers to S4 shown in the diagram below 8:
Diagram 8. The results of interviews S4 with limited intervention
From the results of the Limited interventions conducted by researchers of the S4 at no 4 on the diagram 8, it appears that the S4 begin to understand the value of fractions as part whole non congruent part to black is 4/6 and orange is 2/6.
While the next picture interpret fractions as the part group comparition of the S5 are shown in Figure 11 below:
Fractions
NBk = 4 NO = 2
N = 6
2/64/6
Fractions
NBk = 4 NO = 2
N = 6
4/22/4
www.ccsen
Based on t
Researche
S5
Interpretat
In the diagand the fra
Because Swith interv
Researche
S5
Researche
S5
Researche
Interventio
net.org/ies
the results writ
r : How is F
: Yes, sir.
tion of the resu
gram 9 visible actional value o
S5 wrong answvention Limite
r : Do you k
: No, I don
r : The ratio
(Limite
: owh, its
r : Exactly.
ons conducted
tten by S5, the
Figure 11? Is c
Orange value
ults of interview
Diagram 9
S5 interprets tof brown (Br)
wer is not in aed to S5 the fol
know the part g
n’t sir.
o is a compari
ed intervention
value is 2/4 or
by researchers
Internation
Tran
Figure 11. Qu
e researchers co
correct?
is 2/6 and 4/6
ws with S5 can
9. Interpretatio
the fractions vais 4/6 indicate
accordance witllowing
group compar
ison of the over
n)
r ½ right sir?
s of the S5 sho
nal Education Stu
157
nslate version
uestion answer
onducted inter
6 is brown.
n be seen in di
on of the result
alue of orangeed by green arr
th the fraction
rison?
rall, so how m
own in the diag
udies
r 5 of S5
rviews S5 with
iagram 9:
ts of interview
e (O) is 2/6 indrows groove.
nal value, the r
much is the ima
gram 10 below
h the following
ws S5
dicated by the f
researchers con
age 11 fraction
w:
Vol. 9, No. 7;
g
flow arrows in
nducted interv
nal value?
2016
n blue
views
www.ccsen
From the r
10, it appe
is 2/4. The next p
Based on t
Researche
S3
Interpretat
In the diagand the fra
Because Swith interv
Researche
S3
net.org/ies
D
results of the L
ears that S5 be
picture interpre
the results writ
r : What is y
: fractiona
total num
tion of the resu
gram 11 looks actional value o
S3 wrong answvention Limite
r : Do you k
: It’s a dis
Diagram 10. Th
Limited interve
egin to underst
et fractions as
tten by S3, the
your answer in
al value in Fig
mber of 7, then
Diagram 1
ults of interview
S3 interpret thof purple (Pu)
wer is not in aed to S3 as foll
know about the
stribution, sir!
Internation
he results of in
entions conduc
tand the value
the part whole
Tran
Figure 12. Qu
e researchers co
n figure 12?
gure 12, its gre
n the value of f
11. Interpretati
ws with S3 can
he fractional vis 3/7 indicate
accordance witlows
e ratio?
nal Education Stu
158
nterviews S5 w
cted by researc
of the part gro
e comparison o
nslate version
uestion answer
onducted inter
een color was o
fractions is 3/7
on of the resul
n be seen in th
value of greened by green arr
th the fraction
udies
with limited int
chers of the S5
oup compariti
of S3 are show
r 6 of S3
rviews with S3
on top of a pur
7.
lts of interview
he diagram 11.
(G) is 3/7 indirows groove.
nal value, the r
tervention
5 in question n
ion based on th
wn in Figure 12
3 as follows
rple color and
ws S3
icated by the f
researchers con
Vol. 9, No. 7;
no 5 on the dia
he orange to b
2 below:
d a
flow arrows in
nducted interv
2016
gram
rown
blue
views
www.ccsenet.org/ies International Education Studies Vol. 9, No. 7; 2016
159
Researcher : True or with other words the ratio is of = comparisons.
And what if the fractional value based on the ratio? (Limited intervention)
S3 : Oh yes, sir. 3/4 of is the value
Researcher : You are very smart!
Interventions conducted by researchers to S3 shown in the diagram 12 below:
Diagram 12. The results of interviews S3 with limited intervention
From the results of the limited interventions conducted by researchers of the S3 in question no 6 on the diagram 12, it appears that the S3 began to understand the value of a fraction as the part whole comparition based on the green to purple is 3/4.
The last picture interpret fractions as the number line shown in the students’ answers below:
Fractional as the number line
Figures 7 and 8 above error interprets the fractional value of the primary colors (first/previous) is above as the basis for determining the value of the fraction. Whereas in Figure 9 and 10 above error interprets the fractional value of the second color as the basis for determining the value of such fractions by looking at other parts. Later Figures 11 and 12 above interprets the fractional value in the second color as the basis for determining the value of a fraction without seeing the whole.
5. Conclusions Completion of the process of student answers appeared that students cannot understand fractions as ratios and parts are congruent and incongruent of the whole or a set of objects which was given, it is seen from the inability of students to write answers to a fractional value of the three categories in the discussion. Students interpreted the fractional value soch as
1) The dominant colors are located on top of the other colors to see part of the overall
2) Colors are asked as part of other colors regardless of the whole.
3) The combination of No. 1 and 2.
After the intervention of the settlement of the problem of students, then the problem can be solved with the desired questions.
References Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational number concepts. In M. Landau (Ed.),
Acquisition of mathematics concepts and processes (pp. 91-126). Hillsdale, NJ: Erlbaum.
Bell, A. W., Castello, J., & Kucheman, D. E. (1983). A Review of Research in Mathematics Education (Part A). England: MFERNELSON.
Bruner, J.S. (1966). Toward a Theory of Instruction. Cambridge Mass.: Harvard University Press.
Cathcart, W. G., Pothier, Y., Vance, J. H., & Bezuk, N. S. (2006). Learning mathematics in elementary and middle schools: A learner-centred approach (4th ed., Multimedia ed.). Upper Saddle River: Pearson Prentice Hall.
Fractions
NG = 3 NPu = 4
3/4
www.ccsenet.org/ies International Education Studies Vol. 9, No. 7; 2016
160
Clarke, D. M., Roche, A., & Mitchell, A. (2007). Year six fraction understanding: A part of the whole story. In J. Watson, & K. Beswick (Eds), Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 207-216). Hobart, Tasmania: University of Tasmania.
Depdiknas. (2005). Pedoman Penulisan Buku Ajar. Jakarta: Pusat Perbukuan.
Fosnot, T. F., & Dolk, M. (2002). Young Mathematicians at Work: Constructing Fractions, Decimals, and Percents. Portsmouth: Heinemann.
Goldin, G., & Shteingold, N. (2008). Systems of representations and the development of mathematical concepts. In A. Cuoco & S. R. Curcio (Eds.), The roles of representation in school mathematics (pp. 1-23). Reston: The National Council of Teachers of Mathematics.
Hadi, S. (2005). Pendidikan Matematika Realistik dan Implikasinya. Banjarmasin: Tulip.
Harries, T., & Sutherland. (2009). The Representation of Mathematical Concepts in Primary Mathematics Textbooks: A Focus On Multiplication. Retrieved from http://math.unipa.it/~grim/Jharries.PDF
Husna, N. (2012). Mari Belajar Pecahan dengan Menggunakan Balok Pecahan. Retrieved from https://nikmatulhusna13.files.wordpress.com/2012/12/balok-pecahan.pdf
Ismail, S. (2009). Deskripsi Sub Konsep Bilangan Pecahan (Fraction) untuk menghindari Miskonsepsi pada Pecahan. Matsains, 6(2).
Kieren, T. (1976). On the mathematical, cognitive and instructional foundations of the rational numbers. In R. A. Lesh (Ed.), Number and measurement: Papers from a research workshop (pp. 101-144). Athens, GA: ERIC/SMEAC.
Kieren, T. E. (1988). Personal Knowledge of Rational Number-Its Intuitive and Formal Development. In J. Heibert, & M. Behr (Eds.), Number Concepts and Operations in the Middle Grades, NCTM (pp.162-181). Lawrence Erlbaum Associates, Reston, VA.
Markam, S. (2003). Pengantar Psikologis Klinis. Jakarta: Universitas Indonesia Press.
Muhsetyo, G. (2014). Membandingkan Pecahan dengan menggunakan bahan Manifulatif Strip dan menggunakan Benchmark. J-TEQIP Jurnal Peningkatan Kualitas Guru Malang 05/2014, Tahun V, 1(Teaching School Mathematics), 1-8. Retrieved from https://www.researchgate.net/profile/Gatot_Muhsetyo/publications
Musser, G. L., Burger, W. F., & Peterson, B. E. (2006). Mathematics for Elementary Teachers: A Contemporary Approch. Hobogen, NJ: John Wiley & Sons.
NCTM. (2000). Principles and standards for school mathematics. Reston, Va: National Council of Teachers of Mathematics.
Samani, M., & Hariyanto. (2011). Konsep dan Model Pendidikan Karakter. Bandung: PT Remaja Rosdakarya.
Streefland, L. (1991). Fractions in realistic mathematics education: A paradigm of developmental research. Dordrecht: Kluwer Academic Publications. http://dx.doi.org/10.1007/978-94-011-3168-1.
Sukayati. (2003). Pecahan. Depdiknas PPPGM Yogyakarta.
Vale, C. (2007). Using number sense when adding fraction. Prime Number, 22(2), 5-10. Retrieved from http://search.informit.com.au/documentSummary;dn=006254399365786;res=IELHSS
Yusof, J., & Malone, J. (2003). Mathematical errors infractions:A case of Bruneian primary 5 pupils. In L.Bragg, C. Champbell, G. Herbert, & J. Mousley (Eds.), Proceedings of the 26th Annual Conference of the Mathematics Education Research Group of Australasia (Vol. 2, pp. 650-657). Geelong: Deakin University.
Copyrights Copyright for this article is retained by the author(s), with first publication rights granted to the journal.
This is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).