ERIC EJ1146478: The Stages of Student Mathematical Imagination in Solving Mathematical Problems

2017·1.2 MB·English

by  ERIC

International Education Studies; Vol. 10, No. 7; 2017
ISSN 1913-9020 E-ISSN 1913-9039
Published by Canadian Center of Science and Education
The Stages of Student Mathematical Imagination in Solving
Mathematical Problems
Teguh Wibowo1, Akbar Sutawidjaja1, Abdur Rahman As’ari1 & I Made Sulandra1
1 Department of Mathematics Education, State University of Malang, Indonesia
Correspondence: Teguh Wibowo, Department of Mathematics Education, State University of Malang, Indonesia.
E-mail: [email protected]

Received: February 8, 2017 Accepted: March 12, 2017 Online Published: June 27, 2017
doi:10.5539/ies.v10n7p48 URL: https://doi.org/10.5539/ies.v10n7p48

Abstract
This research is a qualitative study that aimed to describe the stages of students mathematical imagination in
solving mathematical problems. There are three kinds of mathematical imagination in solving mathematical
problems, namely sensory mathematical imagination, creative mathematical imagination and recreative
mathematical imagination. Students can produce one kind of mathematical imagination or other kinds of
mathematical imagination. Problem sheet is used as a supporting instrument to find out the stages of students
mathematical imagination in solving problems. Three students are used as research subjects in whom students
were able to produce their mathematical imagination in solving mathematical problems. The results showed that
there are three stages of students mathematical imagination in solving mathematical problems, the first stage is
sensory mathematical imagination, the second stage is creative mathematical imagination, and the last stage is
recreative mathematical imagination.
Keywords: sensory mathematical imagination, creative mathematical imagination, recreative mathematical
imagination
1. Introduction
Research Nemirovsky and Ferrara (2008) on mathematical imagination shows that mathematical imagination
and cognition representative of students involves gestures activity (hand gesture, speech, and other activities of
sensory motor) on learning is very important in developing the creativity and innovation of students in solving
mathematical problems. Similarly the research that done by Swirski (2010), Samli (2011), Kotsopoulos and
Cordy (2009), Van Alphen (2011), they shows these studies support the involvement of imagination in the
learning process. Wilke (2010) and Chapman (2008) emphasize the importance of imagination in the learning
process and helping students in solving problems.
Imagination is not only the capacity to form the image, but the capacity to think in a certain way. Students should
be encouraged to think for themselves by emphasizing their imagination. Imagination can be a main focus in
effective learning (Wilke, 2010). In addition, the imagination has been implicated as the key to mathematical
creativity in generating and manipulating images (Abrahamson, 2006). In the process of mathematics learning or
mathematical problem solving, the ability to imagine has no limits and constraints (Carroll et al., 2010). Egan
and Steiner’s perspective claim that children between the ages of 5 to 14 years learn best through imagination,
because this is a natural mode and strongest when they are involved with knowledge (Van Alphen, 2011). Both
perspectives are stressed in order that involve student imagination ability in building knowledge or solve a
problem.
There are three types of students’ mathematical imagination in solving mathematical problems, namely sensory
mathematical imagination, creative mathematical imagination and recreative mathematical imagination (Wibowo
& As’ari, 2014). Sensory mathematical imagination can be seen through the emergence of students perception
based stimuli problem in the matter. Creative mathematical imagination manifested through the emergence of
relevant or irrelevant ideas in solving the problem. Recreative mathematical imagination manifested through the
emergence of generalizing the idea of solving mathematical problem. This study was to determine the stage of
students’ mathematical imagination in solving mathematical problems. So the aim of this study is to determine
stages of students’ mathematical imagination in solving mathematical problems.
48
ies.ccsenet.org Internationnal Education Stuudies Vol. 10, No. 7;2017
2. Researcch Methodoloogy
This studyy is descriptivve qualitative research. Resppondents in thhis study were students of class VIII SMMPIT
Logaritmaa Kebumen, Ceentral Java, Inndonesia, whicch amounted too 12 students. Then, we takke three studennts to
become ouur research subbject. Subjects were selectedd based on studdents ability too solve problemms in this studyy and
were able to appear maathematical immagination. Too see mathemaatical imaginaation that emeergence studennts in
solving prroblem viewedd through a visual representtation of the sttudent in solving problem (Wibowo & As’ari,
2014).
The mainn instrument iin this study is the researrchers themsellves, because researchers wwho plan, deesign,
implementt, collect data, analyze the daata, draw concclusions and mmake a report (MMoleong, 2004). The researcchers
also used the test (probllem sheet) andd unstructured interviews to collect the daata. Problem ssheet consists oof an
item the mmatter related too solving mathhematical probblems. Here prooblem sheet ussed in this studdy.

The imagee is consisted oof four congruuent squares annd three congruuent squares. FFind the lengthh of square sidde (as
much as) ppossible on eacch square in orrder to fill the image!
The probleem in above emmphasizes howw students cann find the lengtth of each squaare side in ordeer to fill the immage.
The lengthh of square sidde here are a coouple of numbbers that fill thhe length side oof small squarre and large sqquare.
If studentss find one the llength side of square is likelly to determinee the length off other square side. The results of
students viisual representtation can be sseen as how thee stages of maathematical imaagination in soolving the probblem.
Visualizatiion is the act oof bringing thee image to the screen of the mmind and the iimagination annd visualization are
generally taken as synoonyms (Kotsoppoulos & Cordy, 2009). Thhe informationn processing thheory said thaat the
visual reprresentation off knowledge generally talks about mental imagery (Sollso et al., 20088). In other wwords,
mental imaagery (imaginaation) can be sseen through a visual represeentation of knoowledge.
To collectt the data, we use problem sheet, recordiing and intervview. The colllected data in this research is in
descriptivee data (verball data) then annalyzed in indductive way. TThe collected data analyzedd using qualittative
analysis teechnique (it iincludes interaactive analysis technique mmodel) improvved by Cresswwell (2014) wwhich
included rreducing the ddata, presenting the data andd making conclusion. To deecide the accuuration of dataa, the
researcherrs do triangulaation method ((compare the ddata by studennt think aloud result, problemm sheet, field note
and interviiew).
3. Result && Discussion
Below aree descriptions oof each subjecct in solving thhe problem. Too concise the wwriting each suubject were wrritten
as S1, S2 aand S3. S1 is aas subject 1, S22 as subject 2, S3 as subject 3 and P as a reesearcher.
3.1 Analyssis of Mathemaatical Imagination Stage inn Subject 1 (S1))
S1 started to work on thee problems witth reading the question and tthink aloud forr 80 seconds. SS1 then movinng the
fingers over the matter, it indicates subbject tried to eexamine and uunderstand the question. Thiss activity lasteed for
120 seconnds, but subjecct has not beeen able to undderstand the qquestion. It is seen S1 repeeats to readingg the
question foor getting the iinformation, coorresponding tthe statement oof following thhe subject.
S1 : Thee length is not kknown.
Subjects wwere informed that the lengthh of each squarre on the matteer is not knowwn. From this innformation theen S1
see imagess more carefully in order to oobtain the infoormation:
S1 : Thee length of fourr small square sides is equal to the length oof three large ssquare sides.
49
ies.ccsenet.org Internationnal Education Stuudies Vol. 10, No. 7;2017
Subjects can know that tthe length of ffour small square sides is eqqual to the lenggth of three larrge square sidees by
drawing sqquare containeed in the matterr. It shows S1 has been able to understand the intent of thhe question.
Furthermoore S1 trying to find the soluution of the prroblem. S1 mooving the fingeers on the tablle, it shows suubject
uses his thhinking processs to find the soolution of the pproblem. This activity lastedd for 70 secondds, and then suubject
says:
S1: The lenngth of all is 112.
The subjecct inferred the length side off all squares is12 cm. Althouugh the subjectt can determinne the length side of
all squaress are 12 cm, bbut the subjectt looks still diffficulty in deteermining the llength side of each square. AAfter
contemplaate a moment, ssubjects have pperception thaat the length sidde of small squuare is 3 cm annd large squaree is 4
cm. This iss consistent wiith statement oof subject:
S1: The lenngth of small ssquare is 3 cmm and large squuare is 4 cm.
This idea eemerges on baased subject peerception after received stimuulus of the prooblem in questiion. Stimulus iis the
order in thhe matter to deetermine the llength side of each square thhat fill the immage. These stiimuli are arisinng to
subject peerception which then emergeence the idea tto determine tthe length of eeach side of square. Van Allphen
(2011) saiid that imaginnation appearss from percepption through the senses. TThis idea is a form of studdents’
mathematiical imaginatioon in facing oof such problemms or so-calleed sensory matthematical imaagination. Wibbowo
and As’arri (2014) said,, sensory matthematical imaagination can be seen throuugh the emerrgence of studdents’
perceptionns due to stimuulus of the probblem. This is reinforced by thhe answers ofsubject that deemonstrates this.

Figuree 1. S1’s drawiing of square wwith the lengthh

As confirmmed why subjeect determines the length sidde of small squuare is 3 cm annd large squaree is 4 cm, subject is
not able too disclose the rreason. This reeinforces the nootion that senssory mathemattical imaginatioon is closely liinked
to perceptiion of studentt. The basic off sensory imaggination is perrception, such as the experience of presennce a
stimulus (CCurrie and Raavenscroft, 20002). It is reinfoorcement by Feerrara (2006), said imagination in mathemmatics
learning iss as student mmotoric and peerception mainn activity. To rreinforce this ccase, the reseaarcher conducted a
interview wwith subject:
P : Whyy is 3 cm and 44 cm, instead oof 3 cm and 5 cm?
S1 : Sileent.
Subject is not able to dissclose the reasoons of choosinng the length side of square iis 3 cm and 4 ccm not others.This
indicates tthat the subjecct’s perceptionn determines tthe length sidee of square 3 cm and 4 cmm by looking aat the
relationshiip between thee total length (112 cm) with thhe length of eacch square.
Descriptioon in above shoows S1 is ablee to appear a seensory mathemmatical imaginnation in solvinng the problemm, but
can not appear another tyype of mathemmatical imaginaation. Structurre of sensory mmathematical immagination proocess
of S1 can bbe illustrated iin the diagram below.
50
ies.ccsenet.org Internationnal Education Stuudies Vol. 10, No. 7;2017

Diagram 1. SS1’s process off sensory mathhematical imaggination

Table 1. S1’s term code pprocess of matthematical imaagination
Term Code
Problem Msl
Moving the ffingers Mgr jr tgg
Repeating reaading the matter Mgl bc sl
Digging information in the queestion Mgg inf dl sl
The length iss unknown Pjg bl dikk
Seeing picturres of the matter mmore carefully Mlt gb sll lb sks
Length of 4 ssides of small squaare = length of 3 ssides of large squaare Pjg 4 ss ppsg kc = pjg 3 ss ppsg bs
Subject able to understand of iintent the matter Sb mmh mk sl
Do up the lenngth of square sir?? Ap tsr pjg psg?
Please Slk
Finding soluttions Mcr sls
Moving the ffinger Mgr jr tgg
The length siide of all 12 Pjg ss smm 12
Seeing relatioonship between thhe total length (12 cm) with the lenggth of each squaree Mlt hub ppjg tl 12 dg pjg msg psg
The length siide of small squaree is 3 cm and largge square is 4 cm Pjg ss psg kc 3 & bs 4
Perception Psp
Sensory mathhematical imaginaation IMS
Subject has nnot been able to mmake a more distannt relationship bettween the total lenngth with the lenggth of Sb bl mbbt hub pjg tl dg pjg msg
each square psg
Finished Sls

3.2 Analyssis of Mathemaatical Imagination Stage in SSubject 2 (S2)
S2 begins working on thhe problems byy reading and thinking aloudd for 90 seconnds. S2 repeats reading the mmatter
to understaand the probleem by pointingg the fingers aat the problemm lasted for 150 seconds. Thhen S2 said, “TThere
are three laarge squares annd four small ssquares”. Baseed on the pictuure on the mattter, subject cann mention that there
are three oof small squarees and four of llarge squares. Then subject ddraws it on thee answer sheet as shown beloow:
51
ies.ccsenet.org Internationnal Education Stuudies Vol. 10, No. 7;2017

FFigure 2. S2’sdraw beginninng square

Subject mooves his fingerrs to trace squuare image andd decides the leength of small square is 5 unnit, so the lenggth of
all squaress is 20 unit. Nuumber that cann be crossed bbeside number 9 is number 55 (Figure 2 aboove). This ideaa is a
subject peerception that appears due tto stimulus off the problemms that can be regarded as a form of sennsory
mathematiical imaginatioon.
Then subject said, “the total of right side and left side is equal represented 20: 3”. Subjectt analyzes the total
length of lleft square sidee and right squuare side is equual, so the length of one squuare in the righht side is 20: 3. But
it is becomming cognitivee disagreemennt on subject, because the rresult is in decimal form. TTo further simpplify,
subject finnds another ideea that the soluution is not in ddecimal form.
Subject deecides the lenggth of small squuare is 6 unit, so that the totaal length of smmall square is 224 unit. This leength
is equal too the total lengtth of large squuare, the result of the length of each large ssquare is 8 uniit derived fromm 24 :
3. Subjectt concludes that the length oof each side oof large squaree is 8 unit and 6 unit for ssmall square. IIt the
representaation of subjectt can be seen bbelow:

Figure 33. Deciding ofSmall and Larrge Square Do S2

From above, subject iss able to appeear relevant iddeas in solvinng the problem which is aa form of creeative
mathematiical imaginatioon. Wibowo and As’ari (22014) said creeative mathemmatical imaginnation can be seen
through apppears of releevant or irreleevant ideas inn solving mathhematical probblems. Imaginnation of this type
appears ass a form of stuudent creativityy in solving thhe problems. CCreative imagiination is the eemergence of ideas
with unexpected ways iin solving prooblems (Curriee & Ravenscrooft, 2002). Eggan (2005) saidd that, throughh the
imaginatioon the studentts can make mmore extensivee and creative in their thinkking way. Apppearing of creeative
mathematiical imaginatioon through divvergent thinkinng was by subjject. This opinnion is same ass Saiber and Tuurner
(2009), it ssaid divergent thinking can iinfluence the suubject imaginaation ability.
After findiing the length of square sidee is 6 unit and 8 unit, subjectt draws square as the matter by decided thaat the
side lengthh of small squuare is 9 unit. TThus obtained the total length of small squuare is 36 unitt, the length eqquals
with the tootal length of large square. RRetrieved the llength of one side of large ssquare is 36: 33, which is 12 unit.
The repressentation of subbject can be seeen below:

Figurre 4. S2’s decidding of other sside the length
This idea iis the previouss step, which ccan still be categorized as crreative mathemmatical imaginaation of studennts in
solving thee problem. Subbject was ablee to find anotheer number to ffill the length sside of square that corresponnding
52
ies.ccsenet.org Internationnal Education Stuudies Vol. 10, No. 7;2017
to the pictture. But this hhas not filled wwhat is referreed to in a matteer to find the side length off square as much as
possible inn order to fill thhe image.
Subject stiill have troublle to find howw many is posssible square leength that fill the image. It is accordance with
what is dissclosed by subjject:
P : Howw many is posssible square length?
S2 : I doo not know sir.
From the aabove, subjectt has already bbeen able to finnd several pairrs of the lengthh sides of smaall square and large
square, buut subject is stiill difficulty inn generalizingg the solutions. In this case subject can prroduce sensoryy and
creative mmathematical imagination tto solving thee problems. SSubject has nnot been ablee to issue another
mathematiical imaginatiion in solvingg this probleem. The struccture of senssory and creaative mathemaatical
imaginatioon of subject caan be illustrateed in the diagraam below.

Diagramm 2. S2’s processs appear of seensory and creative mathemaatical imaginattion

Table 2. S22’s term code oof mathematiccal imaginationn process
Term Codee
Problem Msl
Repeat readiing the matter Mgl bbc sl
Fingers poinnt to the problem Jr tg mnj sl
Identifying aa square lot in queestion Mdf bby psg sl
There are 3 llarge squares, therre are 4 small square Psg bbs 3 psg kc 4
Its length is up Pjg tssh
Draw a squaare as in the probleem Mgb psg spt sl
Moving the fingers trace for immages Mgr jjr tg mnl gb
53
ies.ccsenet.org Internationnal Education Stuudies Vol. 10, No. 7;2017
Deciding a ssmall square lengtth is 5 units Mmss pjg psg kc 5
Perception Psp
Obtained a ssmall square lengtth is 5 units and tootal length is 20 unnits Dpr ppjg psg kc 5 & pjgg tl 20
Sensory matthematical imaginnation IMS
Total right side = total left sidde Ss knn tl = ss kr tl
Comma Ka
Draw a squaare as the matter Mgb psg spt sl
Deciding a ssmall square lengtth is 9 units Mmss pjg psg kc 9
Total length is 36 units Pjg tll 36
Large squaree length is 12 unitts Pjg ppsg bs 12
Results Reprresentation in the image Rps hhs pd gb
Retrieved thhe length of small square is 9 units aand large square iss 12 units Dpr ppjg psg kc 9 & bs 12
Analyzing tootal length of righht side = left side Mgs pjg ss kn tl = kr tll
Relevant ideeas Id rlvv
Deciding a ssmall square lengtth is 6 units Mmss pjg psg kc 6
Total length is 24 units Pjg tll 24
The length oof large square sidde is 8 units Pjg ss psg bs 8
Result Repreesentation in the immage Rps hhs pd gb
Obtained thee length of small ssquare is 6 units annd large square is 8 units Dpr ppjg psg kc 6 & bs 8
Creative mathematical imaginnation IMK
Finding a feww square length thhat is identical to tthe previous Mnmm bbr pjg psg idn ddg sbl
Still having trouble to find squuare length as much as possible Ms kksl mnm pjg psg sbby
Finished Sls

3.3 Analyssis of Mathemaatical Imagination Stage in SSubject 3 (S3)
S3 begin wworking on thhe problems byy reading and thinking aloudd for 70 seconnds. S3 reiteraated “there are four
congruent squares and thhree congruentt squares anywway”, then repeeats reading thhe matter to unnderstand the intent
of the queestion. “Find tthe possible side length of each square” subject said qquietly. Subject is still tryinng to
understandd the intent off the question. Subject then mutters “conggruent?”, then subject repeaats the phrase “find
the possiblle side length””.
After a feew moments, subject drawss a square andd calculates hoow many squuare in the prooblem. But suubject
re-reads thhe questions aand repeated thhem several tiimes. This inddicates subjectt is still tryingg to understandd the
intent of thhe question. Subject then deecided all the wwidth of the immage in questiion is 10 cm aand all the lenggth is
12 cm. Thiis is the evidennt from the reppresentation off this subject.

Figurre 5. S3’s beginnnings percepttion that doingg

This repreesentation is a form of sensory mathemattical imaginatiion on the subbject. These immaginations apppear
with stimuulus of the prooblems that infflicts to the peerception of suubject on side length of totaal square and wwidth
square. Shhown in the piccture above thaat 10 cm and 112 cm was maade by subject,, and then the subject crossed out
it. After deeciding the tottal length and wwidth of squarre, subject searrches a lengthfor each squarre with the outtlines
from the tootal of length aand width. Butt subject is still trouble to finnd the length oof square side tto be equal.
Subject knnows that the ssteps are less pprecise, so that subject crosseed number 10 ccm and 12 cm on the answerr (see
Figure 5 aabove) and replace it by otheer number. Thee method of deeciding the sidde length of smmall square is 5 cm
in order too obtain a totall side length oof small squaree is 20 cm. Meeans that the leength side of llarge square is 20:3
54
ies.ccsenet.org Internationnal Education Stuudies Vol. 10, No. 7;2017
to obtain the equal lenngth, so that the length sidde of large ssquare is 6.677 cm. This is evident fromm the
representaation of the subbject.

Figurre 6. S3’s findiing the length side of square

This repreesentation is aa form of studdents’ creativee mathematicaal imagination.. Imagination is associated with
appearing of relevant ideeas that can heelp in solving tthe problem (WWibowo & Ass’ari, 2014). Reelevant idea heere is
the appearring of the lenngth side of smmall square 5 cm in order to obtain a total length of sqquare 20 cm. Thus
obtained thhe length side of small squarre is 5 cm and large square iss 6.67 cm.
“The resuults are much??” the authors said. “No” rreplied subjectt, then subjectt is reading thhe matter agaain to
confirm thhe order of thee question. Subbject seeks sollutions again tto fill the lenggth side of howw many squaree that
fill imagess in the matter.. Subject then said “the lengtth side multipllied by two”, tthat is 5 cm x 22 = 10 cm and6.67
cm x 2 = 113.34 cm. Reprresentations wwere made by suubject looks liike the image bbelow.

Figgure 7. S3’s finnding the lengtth other side

Subject caan find anotherr solution that fills these imaages (10 cm aand 13.34 cm).. However, thiis solution doees not
say that it fills the lengtth side of squaare as much ass possible. Whhen the author asked “can yoou find as mucch as
possible?””, subject arguees “the importaant one is the llength side muultiplied by 2, multiplied by 2, and so on”.This
idea appeaars based on the student’s previous expeerience in solvving the probblem. The expperience makes the
student ennriches a percepption, so it buiild imaginativee image (Van AAlphen, 2011)). In Figure 7 aabove subject write
“multipliedd by 2 and soo on”, this reppresentation iss a form of recreative matheematical imaggination of stuudent.
Subject is able to generaalize the solutiion by double size the lengtth of each sidee, and then it wwill obtain thee new
length for each side. Whhen the subjecct is able to geeneralize it is aa form of recrreative mathemmatical imagination
produced bby the students (Wibowo & As’ari, 2014).. Currie & Ravvenscroft (2002) says, recreaative imaginatiion is
the abilityy to construct iideas from diffferent perspecctives of an exxperience. Genneralization is obtained baseed on
previous student experrience in solvving the probblem. Generallization is buuilt by subjecct through various
perspectivves resulting inn a general formm that fill the ssolution.
From the ddescription aboove, S3 is cappable to producce sensory, creeative and recrreative mathemmatical imagination
to solving the problem. TThe structure pprocess of sennsory, creative and recreativee mathematicall imagination oof S3
can be illuustrated in the ddiagram beloww.
55
ies.ccsenet.org Internationnal Education Stuudies Vol. 10, No. 7;2017

Diagram 33. S3’s process of sensory, creative and recrreative mathemmatical imaginnation

Table 3. S33’s term code pprocess of matthematical imaagination
Term Code
Problem MMsl
Reading with thhink aloud MMbc dg sr kr
“there are four ccongruent squaress & three congruennt squares “ “aa 4 psg kgn & 3 pssg kgn”
Repeat reading tthe matter MMgl bc sl
“Find possible sside the length of eeach square” “ccr pjg ss mg msg ppsg”
“congruent” “KKgn”
Draw a square aas in the problem MMgb psg spt sl
Count the numbber of square on thhe matter MMht by psg sl
Repeat reading tthe matter MMgl bc sl
Deciding wide oof all square is 10 cm and length off all square is 12 cmm MMms lb psg 10 & ppjjg psg 12
Perception Pssp
Sensory mathemmatical imaginatioon IMMS
Looking the lenngth of each squaree MMcr pjg msg psg
Elaborating on tthe length and widdth of its total MMgn pjg & lb tl
Difficulty in finnding the length off each square KKsl mnm pjg msg ppsg
Cross out number 10 cm and 12 ccm MMct ak 10 & 12
Deciding the lenngth of small squaare is 5 cm MMms pjg psg kc 5
Relevant ideas Idd rlv
The total lengthh of small square iss 20 cm Pjjg tl ss psg kc 20
The length of laarge square 20 : 3, gained 6.67 cm Pjjg psg bs 20:3, dprr 6,67
Creative mathemmatical imaginatioon IMMK
“is it that much”” “aap sd sby”
56
ies.ccsenet.org International Education Studies Vol. 10, No. 7; 2017
Not as many as that Bl sby
Reading the matter again Mbc sl kbl
Clarifying the command in the matter Mpj prt sl
Finding solutions of the length side of square as much as possible Mcr sls pjg ss psg sby
The Length side multiplied by 2 Pjg ss dk 2
5 cm x 2 = 10 cm 5 x 2 = 10
6,67 cm x 2 = 13,34 cm 6,67 x 2 = 13,34
Another solutions Sls ln
“How often?” “Brp sby”
“ the important one is the length side multiplied by 2, multiplied by 2 and so on” “yg ptg pjg ss dk 2 dst”
Each of length side multiplied by 2, so it acquired the new length Stp pjg ss dk 2, dpr pjg ss br
Generalization Gnr
Recreative mathematical imagination IMR
Finished Sls

The description in above indicates that the stage of students mathematical imagination in solving mathematical
problem consist of, the first is one stage of mathematical imagination, the second is two stage of mathematical
imagination, and the third is three stage of mathematical imagination. S1 can show sensory mathematical
imagination, but has not been able to show creative and recreative mathematical imagination. It can be said S1
only through one stage of mathematical imagination. Students who can show creative mathematical imagination,
they would show sensory mathematical imagination first. S2 is able to show creative mathematical imagination
by issuing sensory mathematical imagination first, but S2 has not been able to appear recreative mathematical
imagination. S2 through two stages of mathematical imagination in solving mathematical problem, they are
sensory and creative mathematical imagination. Similarly, students who can show recreative mathematical
imagination, they should show sensory and creative mathematical imagination first. S3 can show recreative
mathematical imagination by issuing sensory and creative mathematical imagination first. S3 through three
stages of mathematical imagination in solving mathematical problem, they are sensory, creative and recreative
mathematical imagination.
4. Conclusion
From the description in above, it can be concluded that the stage of mathematical imagination which through the
students in solving mathematical problems proceeded with showing sensory mathematical imagination (first
stage), then creative mathematical imagination (second stage) and then recreative mathematical imagination
(third stage). S1 is only through one stage of mathematical imagination in solving mathematical problems, which
is sensory mathematical imagination stage. S2 can be through two stages of mathematical imagination in solving
mathematical problems, they are sensory mathematical imagination and creative mathematical imagination.
Students who can show creative mathematical imagination (second stage), they would show sensory
mathematical imagination first (first stage). While S3 can be through three stages of mathematical imagination in
solving mathematical problems, sensory mathematical imagination at first, then creative mathematical
imagination, and recreative mathematical imagination in the end. Students who can show recreative
mathematical imagination, they would show sensory and creative mathematical imagination first. But it does not
work on other way, students can show sensory mathematical imagination but they can not necessarily issue a
creative and recreative mathematical imagination. That`s also students who can appear sensory and creative
mathematical imagination, but they can not necessarily issue a recreative mathematical imagination.
Acknowledgments
We thank the many colleagues who have given advice on this paper. We appreciate to educational institution
Logaritma Junior High School, the teachers and students who helped in this study. We also thank a few people
who have read and make suggestions for this paper.
References
Abrahamson, D. (2006). The Three M’s: Imagination, Embodiment, and Mathematics. Paper presented at the
annual meeting of the Jean Piaget Society, June, 2006, Baltimore, MD.
Carroll, M., Goldman, S., Britos, L., Koh, J., Adam, R., & Hornstein, M. (2010). Destination, Imagination and
the Fires Within: Design Thinking in a Middle School Classroom. International Journal of Art & Design
Education, 29(1), 37-53. https://doi.org/10.1111/j.1476-8070.2010.01632.x
57