International Education Studies; Vol. 10, No. 7; 2017 ISSN 1913-9020 E-ISSN 1913-9039

Published by Canadian Center of Science and Education

48

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: teguhwibowo743@gmail.com

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.

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can still be cate to find anothe

nal Education Stu

52

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side is equalual, so the lenbecause the r

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53

ed to in a matte

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ensory and cre

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are

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ength that fill

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Code

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side length of

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by psg sl

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sh

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atical Imagina

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form of sensoblems that infcture above thatal length and wand width. But

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th is 5 units

th is 5 units and to

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re 5. S3’s begin

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precise, so that er number. Theof small square

nal Education Stu

54

otal length is 20 un

and large square is

nd large square is

the previous

ch as possible

Subject 3 (S3)

thinking aloudway”, then repe

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d calculates hoimes. This indwidth of the imf this subject.

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subject crossee method of dee is 20 cm. Me

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d for 70 seconeats reading thsubject said q

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ed number 10 ceciding the sideans that the le

s pjg psg kc 5

pjg psg kc 5 & pjg

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psg spt sl

s pjg psg kc 9

l 36

psg bs 12

hs pd gb

pjg psg kc 9 & bs

pjg ss kn tl = kr tl

v

s pjg psg kc 6

l 24

s psg bs 8

hs pd gb

pjg psg kc 6 & bs

m bbr pjg psg idn d

ksl mnm pjg psg sb

nds. S3 reiterahe matter to unquietly. Subjesubject repea

uare in the prot is still tryingion is 10 cm a

g

bject. These imlength of tota

, and then the for each squar

of square side t

cm and 12 cm de length of smength side of l

Vol. 10, No. 7;

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l

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the equal lenation of the sub

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ults are much?he order of thes in the matter.13.34 cm. Repr

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by the studenty to construct istudent exper

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ustrated in the d

ngth, so that bject.

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a form of studeas that can hength side of smof small squar

?” the authors e question. Sub. Subject then resentations w

Fig

r solution that th side of squaes “the importathe student’s ption, so it buio on”, this repalize the solutihen the subjecs (Wibowo & ideas from difrience in solvn a general form

ove, S3 is capThe structure pdiagram below

Internation

the length sid

re 6. S3’s findi

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said. “No” rbject seeks solsaid “the lengt

were made by su

gure 7. S3’s fin

fills these imaare as much asant one is the lprevious expeild imaginativepresentation ision by double ct is able to geAs’ari, 2014).

fferent perspecving the probm that fill the s

pable to producprocess of sen

w.

nal Education Stu

55

de of large s

ing the length

e mathematicathe problem (Wcm in order tlarge square is

replied subjectlutions again tth side multiplubject looks li

nding the lengt

ages (10 cm as possible. Whlength side muerience in solve image (Van A

s a form of resize the lengt

eneralize it is a. Currie & Ravctives of an exblem. Generalsolution.

ce sensory, crensory, creative

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square is 6.67

side of square

al imagination.Wibowo & Aso obtain a tots 6.67 cm.

t, then subjectto fill the lenglied by two”, tike the image b

th other side

and 13.34 cm).hen the author ultiplied by 2, ving the probAlphen, 2011)creative matheth of each sidea form of recrvenscroft (200xperience. Genlization is bu

eative and recrand recreative

7 cm. This is

. Imagination s’ari, 2014). Real length of sq

t is reading thgth side of howthat is 5 cm x 2below.

. However, thiasked “can yomultiplied by

blem. The exp). In Figure 7 aematical image, and then it wreative mathem2) says, recreaneralization is uilt by subjec

reative matheme mathematical

Vol. 10, No. 7;

evident from

is associated elevant idea hequare 20 cm.

he matter agaw many square2 = 10 cm and

is solution doeou find as muc2, and so on”.

perience makeabove subject gination of stuwill obtain thematical imaginative imaginati

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matical imaginl imagination o

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org

Diagram 3

3’s term code p

Term

Problem

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“there are four c

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“congruent”

Draw a square a

Count the numb

Repeat reading t

Deciding wide o

Perception

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Looking the len

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Difficulty in fin

Cross out numb

Deciding the len

Relevant ideas

The total length

The length of la

Creative mathem

“is it that much”

3. S3’s process

process of mat

hink aloud

congruent squares

the matter

side the length of e

as in the problem

ber of square on th

the matter

of all square is 10

matical imaginatio

ngth of each square

the length and wid

nding the length of

er 10 cm and 12 c

ngth of small squa

h of small square is

arge square 20 : 3,

matical imaginatio

”

Internation

of sensory, cr

thematical ima

s & three congruen

each square”

he matter

cm and length of

on

e

dth of its total

f each square

cm

are is 5 cm

s 20 cm

gained 6.67 cm

on

nal Education Stu

56

eative and recr

agination

nt squares “

f all square is 12 cm

udies

reative mathem

C

M

M

“a

M

“c

“K

M

M

M

m M

Ps

IM

M

M

K

M

M

Id

Pj

Pj

IM

“a

matical imagin

ode

Msl

Mbc dg sr kr

a 4 psg kgn & 3 ps

Mgl bc sl

cr pjg ss mg msg p

Kgn”

Mgb psg spt sl

Mht by psg sl

Mgl bc sl

Mms lb psg 10 & pj

sp

MS

Mcr pjg msg psg

Mgn pjg & lb tl

Ksl mnm pjg msg p

Mct ak 10 & 12

Mms pjg psg kc 5

d rlv

jg tl ss psg kc 20

jg psg bs 20:3, dpr

MK

ap sd sby”

Vol. 10, No. 7;

nation

sg kgn”

psg”

pjg psg 12

psg

r 6,67

2017

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57

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

6,67 cm x 2 = 13,34 cm

5 x 2 = 10

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.

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