Jou nal o Educa o s, Teache s and T aine s JETT, Vol. 11 (1); ISSN: 1989-9572 68
ISSN 1989 - 9572
DOI: 10.47750/je .2020.11.01.007
Mohammad Reza Ma danbeigi5
Jou nal o Educa o s, Teache s and T aine s, Vol. 11 (1)
h ps://je .labos o .com/
Da e o ecep ion: 15 Ma ch 2020
Da e o e ision: 18 June 2020
Da e o accep ance: 14 Augus 2020
1,2,4,5Depa men o Ma hema ics, Science and Resea ch B anch, Islamic Azad Uni e si y, Teh an, I an.
3Depa men o S a is ics, Science and Resea ch B anch, Islamic Azad Uni e si y, Teh an, I an.
The Impac o Bybee and Synec ics Models on C ea i i y,
C ea i e P oblem-sol ing, and S uden s’ Pe o mance in
Geome y
Zah a Kalan a nia1
Ahmad Shah a ani semnani2
Mohammad Hassan Behzadi3
Mohsen Ros amy -MalKhali eh4
Zah a Kalan a nia, Ahmad Shah a ani semnani, Mohammad Hassan Behzadi, Mohsen
Ros amy -MalKhali eh, Mohammad Reza Ma danbeigi (2020). The Impac o Bybee and Synec ics
Models on C ea i i y, C ea i e P oblem-sol ing, and S uden s’ Pe o mance in Geome y. Jou nal o
Educa o s, Teache s and T aine s, Vol. 11(1). 68-78.
Jou nal o Educa o s, Teache s and T aine s, Vol. 11 (1)
ISSN 1989 – 9572
h ps://je .labos o .com/
Jou nal o Educa o s, Teache s and T aine s JETT, Vol. 11 (1); ISSN: 1989-9572 69
1,2,4,5Depa men o Ma hema ics, Science and Resea ch B anch, Islamic Azad Uni e si y, Teh an, I an.
3Depa men o S a is ics, Science and Resea ch B anch, Islamic Azad Uni e si y, Teh an, I an.
Email: Zah a_kalan a [email protected], Shah a ani. [email protected], behzadi@s biau.ac.i ,
os amy@s biau.ac.i , m ma danbeigi@s biau.ac.i
ABSTRACT
The p esen s udy aimed o in es iga e he e ec o Bybee and Synec ics on c ea i i y, c ea i e
p oblem-sol ing, and pe o mance o nin h-g ade s uden s in geome y. The esea ch me hod was
quasi-expe imen al wi h p e- es , pos - es , and con ol g oup. F om he en i e popula ion o he
nin h-g ade emale s uden s o public high schools in Teh an, h ee in ac class ooms we e selec ed by
he clus e sampling me hod, each consis ing o 30 s uden s. Then, wo classes we e andomly
assigned o wo expe imen al g oups and one con ol g oup. In addi ion, esea ch ins umen s
included Abedi’s c ea i i y es , Basadu ’s p oblem-sol ing c ea i e es , and a esea che -made
geome y es . In o de o collec da a, a i s , p e- es s o pe o mance, c ea i i y and c ea i e
solu ion we e pe o med on he subjec s. A e pe o ming he pa e ns in he g oups, pos - es s o
pe o mance, c ea i i y and c ea i e solu ion we e pe o med on he subjec s. Finally, desc ip i e ( he
mean and s anda d de ia ion) and in e en ial ANCOVA s a is ics we e used o analyze he da a by
SPSS so wa e. The indings indica ed ha using he pa e ns o Bybee and Synec ics on s uden s’
c ea i i y, c ea i e p oblem-sol ing, and pe o mance in geome y we e signi ican ly mo e in luen ial
compa ed o adi ional eaching me hods . The use o educa ional pa e ns app op ia e o he
educa ional con en will lead o he aining o c ea i e people.
Keywo ds: Bybee model, Synec ics model, C ea i i y, C ea i e p oblem-sol ing, Geome y
INTRODUCTION
Geome ic concep s a e he mos impo an issues when eaching ma hema ics. Di e en iews on heo ies such
as misconcep ions, concep ual e o s, men al diso de s, and he lack o in o ma ion a e possible a almos e e y
le el. When conside ing hese pe spec i es and he impo ance o eaching geome y, i is unde s andable how
o o e come he p oblems ha may a ise du ing he p ocess (Fy e e al., 2015). I is wo h no ing ha , in he
con en ional iew o ma hema ics eaching, he end objec i e o eaching concep s is o imp o e s uden s’
p oblem-sol ing abili ies. In his iew, p oblem-sol ing equa es wi h eaching he co ec answe (Ka imi,
Shah a ani, & Hagh e di, 2019).
To imp o e he s uden s’ awa eness o p oblem-sol ing s a egies, eache s a e equi ed o be in o med o
common p oblems ha impede quali ied ma hema ics educa ion. The eache s need o be skilled in using a wide
ange o s a egies, echniques, and ac i i ies in o de o help building he p e equisi e knowledge and s eng hen
he connec ion be ween s uden s and a pa icula concep . This may include discussing, elling s o ies, using
ole-playing and isual image y, encou aging sea ch pa e ns, and using eal-li e examples and analyses, along
wi h me apho s and explana ions (McLa en, 2010). Fu he mo e, addi ional concep s can be lea ned by he
de elopmen o concep s and hei ela ionships (Gallens ein, 2013). I seems ha some eaching models can
The Impac o Bybee and Synec ics Models on C ea i i y, C ea i e
P oblem-sol ing, and S uden s’ Pe o mance in Geome y
Zah a Kalan a nia1, Ahmad Shah a ani semnani2, Mohammad Hassan Behzadi3, Mohsen Ros amy
-MalKhali eh4, Mohammad Reza Ma danbeigi5
Jou nal o Educa o s, Teache s and T aine s JETT, Vol. 11 (1); ISSN: 1989-9572 70
play a ole when used in a speci ic con ex wi hin he class ooms. In he cu en s udy, Bybee and Synec ics
models we e used since i was assumed ha hese wo s a egies would imp o e c ea i i y, c ea i e p oblem-
sol ing, and pe o mance o nin h-g ade s uden s in geome y.
Piage ’s cogni i e de elopmen ini ially consis ed o explo a ion, in en ion, and disco e y s ages, and hen was
modi ied by Roge Bybee o cons uc i ism. This model o eaching enhances lea ning ou comes o s uden s
wi h di e en cogni i e le els (Walia, 2012). I was de eloped by Bybee (2009) and called he “5E aining
cycle model” because each s ep begins wi h he le e “E”. An unbiased eaching me hod has se e al bene i s. I
is lea ne -cen e ed, includes mo e meaning ul lea ning ac i i ies, and p e en s he me e e en ion o
in o ma ion. In addi ion, his me hod abso bs and adap s in o ma ion h ough p oblem-sol ing and in o ma ion
acquisi ion, and inally, encou ages lea ne s o p omo e hei ex acu icula ac i i ies in o de o ha e c i ical
hinking and e hics (Sucia i, Vincen isia, & Ismiya in, 2015). In he majo i y o he s udies, he 5E Model was
ex ensi ely used in in e na ional ma hema ics educa ion (Bybee, 2009).
The Synec ics model was he o he model ha was used in he cu en s udy. Low concep ual unde s anding is
conside ed as one o he majo p oblems in ma hema ics, in gene al, and in geome y, in pa icula , which is
p obably because he s uden s’ pe cep ions o his subjec a e un ela ed o e e yday li e. Acco dingly, using a
Synec ics model o each geome ic concep s is one possible solu ion o his p oblem. Va ious s udies explo ed
he use o a Synec ics model o cla i y and de elop concep s (Heid, 2008).
The model was pos ula ed by Go don e al. in 1961. The p ocess o synec ics p obably implies ha i can join
di e se and seemingly i ial elemen s o c ea e new ideas abou a concep . I is a c ea i e p ocess ha wo ks
elying on one’s men al capaci y o disco e and uni e subjec s in di e en and un ela ed ways (Go don, 1961).
Among all ac o s, Synec ics seems o play wi h un ela ed esponses. Fu he , his ole is used o p oduce ene gy
in o de o sol e p oblems and s imula e new pe spec i es and p oblems (Blisse & Mc G a h, 1996).
I is wo h men ioning ha he c ea i e p oblem-sol ing p ocess is de ined as he applica ion o c ea i i y o he
eal wo ld, indus y, o ganiza ion, o he social con ex o p oblem-sol ing, as well as p esen ing and
implemen ing solu ions o complex issues. The c ea i e p oblem-sol ing model in ol es a h ee-s ep p ocess o
p oblem inding, p oblem-sol ing, and solu ion execu ion (Basado , G aen, & G een, 1982). Choosing and
implemen ing an app op ia e app oach o he goals o each lesson equi e a a ie y o esea ch and a
comp ehensi e e iew o he bene i s and limi a ions o adi ional and ac i e eaching models in geome y.
Conside ing he impo ance o Bybee and Synec ics models in geome y, he cu en s udy sough o e alua e
he e ec s o hese models on c ea i i y, c ea i e p oblem-sol ing, and he pe o mance o nin h-g ade s uden s
in geome y.
THEORETICAL AND EMPIRICAL BACKGROUND
Resea che s ha e shown ha ma h ins uc ion is e ec i e when s uden s a e ac i ely in ol ed in he lea ning
p ocess, hus ma h eache s should use in e ac i e g oup ac i i ies o encou age s uden s ins ead o using an
explana o y eaching app oach. The 5E eaching model is one app oach h ough which s uden s ac i ely
pa icipa e in he lea ning p ocess (Runisah Hemen, & Dahlan, 2016). The s eps o he 5E lea ning cycle model
a e as ollows.
1. Engage: A his poin , s uden s a e s imula ed by dis up ing hei cogni i e balance o d awing hei a en ion
o eal-li e e en s.
2. Explo e: A his s age, he s uden ’s expe iences a e used o obse ing, collec ing da a, p edic ing es
ques ions, and co ec ing he hypo heses in o de o be able o answe he ques ions posed in he p e ious s ep.
3. Explain: A his phase, he s uden s p o ide hei own obse a ions and da a ha gi e a scien i ic explana ion
o hei ob ained esul s.
4. Elabo a e: A his s age, o he issues a ise o s uden s ega ding using hei new knowledge, p o iding
solu ions, making decisions, and eaching a ional esul s.
5. E alua e: E alua ion is needed o de e mine whe he s uden s ha e a co ec unde s anding o he concep s o
a e able o ex end hei lea ning o o he a eas (Acisli, Yalem, & Tu gu , 2011).
A bulk o esea ch has ocused on his a ea. Fo ins ance, Demi cioglu and Caga ay (2014) a emp ed o
examine he e ec o labo a o y s udies based on Bybee’s model on he s uden s’ comp ehension o chemis y.
Thei indings showed ha using his pa e n was signi ican ly mo e e ec i e on s uden s’ comp ehension
compa ed o adi ional eaching models.
Jou nal o Educa o s, Teache s and T aine s JETT, Vol. 11 (1); ISSN: 1989-9572 71
In hei s udy on he applica ion o he lea ning cycle model (5E) lea ning wi h cha a ia ions o s uden s’
c ea i i y, Saci a e al. (2015) demons a ed ha using he Bybee model would imp o e s uden s’ c ea i i y.
Simila ly, Teze and Cumhu (2017) ocused on he 5E eaching model and ma hema ical modeling wi hou
al e ing he main s uc u e o he model and ound ha using his model had a posi i e impac on academic
achie emen and p oblem-sol ing skills, as well as he inc eased mo i a ion o s uden s o ma hema ics. The
hy hmic pa e n is designed o enhance s uden s’ c ea i i y and c ea i i y in p oblem-sol ing. In his model,
h ee ypes o me apho s (i.e., di ec analogy, pe sonal analogy, and in ense con lic ) o m he basis o he
sequence o synec ics ac i i ies. S uden s de elop an unde s anding o geome ic concep s and build on hei
new knowledge based on pas li e expe iences.
In addi ion, Go don de ined synec ics o me apho ical hinking based on ou ideas. C ea i i y is impo an in
e e yday li e, no jus o a and imp o isa ion. In o he wo ds, i is pa o ou li es in ou wo k and leisu e.
Fu he , he c ea i e p ocess is no e y mys e ious and no limi ed o speci ic people who a e amilia wi h he
basics o c ea i i y. Based on ano he idea, c ea i e inno a ion is simila in all a eas o empi ical sciences,
ma hema ics, and humani ies. Finally, indi idual and g oup c ea i e hinking o inno a ion is e y simila o one
ano he . Mo e p ecisely, indi iduals and g oups c ea e hei ideas and p oduc s in simila ways (Joyce e al.,
2013). The s eps o his pa e n a e desc ibed as ollows.
1. P oblem Desc ip ion: A his poin , he eache encou ages he s uden s o desc ibe he new si ua ion.
2. Di ec Analogy: A di ec analogy is a simple compa ison o wo exis en s o concep s. In his compa ison, no
all aspec s a e necessa ily iden ical. A s aigh o wa d compa ison is simply made by pushing o wa d he
condi ions o a gi en subjec o a eal si ua ion ha ing a p oblem wi h ano he si ua ion in o de o p esen a new
heo y o a p oblem.
3. Pe sonal Analogy: In he pe sonal analogy, s uden s mus place hemsel es in o he elemen s o eel he
a ibu es o o he elemen s. Simula ion is he basis o pe sonal analogy. This eplica ion may occu wi h a
pe son, plan , animal, o inanima e being.
4. Compac Con lic : I desc ibes wo con adic o y wo ds o he same subjec , gi ing he deepes and wides
insigh .
5. Di ec Analogy: A his s age, s uden s selec and p esen ano he di ec analogy based on in ense con lic s. I
should be no ed ha he e is no wo d on he main issue a his poin .
6. Assignmen Re iew: A his poin , he eache encou ages he s uden s o e u n o he main ask o ques ion
and use he la es analogy wi h all he expe iences o heo izing (Shabani & Hassan, 2016).
In he esea ch en i led “Using In eg a ed Analogy in Physics Educa ion o Building Concep o Rep esen a ion:
The Way o be G ea In en o ”, A i iyan i and Wahyuningish (2015) ound ha he inno a i e model and he
analogies se e as use ul ools o eaching, lea ning, and unde s anding abs ac concep s.
Ahmad Khan and Mahmoud (2017) also examined he ole o synec ics in s uden s’ comp ehension o
geome ic concep s and concluded ha hei comp ehension and lea ning skills signi ican ly inc eased when
hey we e ins uc ed acco ding o his model. Based on he e iew o he ela ed li e a u e and o ou bes
knowledge, no simila s udy has been conduc ed in he con ex like I an in he big a ea o geome y using he
abo e-men ioned models.
METHOD
Based on i s pu pose, he p esen s udy used a quasi-expe imen al design and sough o p o ide p ac ical and
usable indings in he ield o eaching me hods. In his s udy, a p e es -pos es wi h he con ol g oup was used
in his ega d. The s udy popula ion included all nin h-g ade emale s uden s o he public high schools o
Teh an, I an. In addi ion, h ee in ac classes o nin h-g ade emale s uden s we e selec ed by a clus e sampling
echnique and each class consis ed o 30 s uden s. The wo classes we e andomly selec ed as Bybee and
Synec ics expe imen al g oups, as well as he con ol g oup. Fu he , a esea che -made es was de eloped
based on s uden s’ p io knowledge o geome ical concep s in o de o in es iga e he scien i ic le el o all
s uden s in all g oups. The expe imen al and con ol g oups we e also es ed on c ea i i y p e- es s and c ea i e
p oblem-sol ing. Subsequen ly, he con en o he lessons was augh by Bybee and Synec ics pa e ns and he
adi ional me hod o expe imen al and con ol g oups in six 90-minu e sessions, espec i ely. A e comple ing
Jou nal o Educa o s, Teache s and T aine s JETT, Vol. 11 (1); ISSN: 1989-9572 72
he sessions and ac i i ies, c ea i i y and p oblem-sol ing es s and esea che -made pe o mance es s we e
pe o med on all h ee g oups and he indings we e compa ed acco dingly.
Pa icipan s
Gi en ha geome y opics a e essen ially aised in he nin h g ade and a conside able po ion o his book is
de o ed o geome y, he nin h g ade was selec ed o his esea ch. F om he popula ion o he nin h g ade
emale s uden s o Teh an seconda y schools, he sample o his s udy was andomly selec ed om a emale
public school in Teh an. Th ee nin h g ade class ooms we e selec ed by he clus e sampling me hod, each
con aining 30 emale s uden s (16 yea s old). The s uden s in he expe imen al g oups comp ised o 6 g oups o
5 and one ep esen a i e we e selec ed o each g oup. The cu iculum based on Bybee and Synec ics models
p epa ed by he examine s was ailo ed and e iewed p io o he ea men .
Ins umen s
Abedi’s c ea i i y es was used o de e mine he ex en o he s uden s’ c ea i i y p omo ion. Abedi adop ed his
es based on To ance’s heo y o c ea i i y in 1984. This es had 60 i ems on a 3-poin Like - ype scale which
consis s o ou sub ypes o luidi y, expansion, ingenui y, and lexibili y. The o al c ea i i y sco e o each
subjec was be ween 60 and 180. The eliabili y o Abedi’s c ea i i y es was achie ed h ough he es - e es
me hod om he seconda y school s uden s o Teh an in ou es sec ions. The eliabili y coe icien o he luid
ac ion, ini ia i e, lexibili y, and ex ension was 0.85, 0.82, 0.84, and 0.80, espec i ely. The co ela ion
coe icien be ween he o al sco e o To ance’s es and his es was calcula ed o be 0.63 (Abedi, 1993). To
assess he eliabili y o he es , C onbach’s alpha in each sec ion was calcula ed as 0.83, 0.86, and 0.89 o
con ol, Bybee, and Synec ics, espec i ely.
Fu he mo e, he c ea i e p oblem-sol ing es o Basadu was used o de e mine he amoun o c ea i e
solu ion. The c ea i e solu ion o he p oblem began wi h Wallace’s wo k in 1926. Howe e , one o he mos
p es igious c ea i e p oblem-sol ing models belongs o Basadu . This es had 16 i ems on a 5-poin Like - ype
scale anging om ne e , e y ew, some imes, o en, o e y o en. Ques ions 1, 2, 3, 4, 5, 6, 7, 9, 10, 13, 14,
and 15 a e sco ed di ec ly, meaning ha he choice o “ne e ”, “ e y ew”, “some imes”, “o en”, and “ e y
o en” we e sco ed 1 o 5, espec i ely. Howe e , ques ions “8, 11, 12, and 16” we e sco ed e e sely. The
minimum and maximum sco es we e 16 and 80. The alidi y o his es was es ima ed using he concu en
alidi y o his es wi h To ance’s e bal c ea i i y es . The indings showed ha he subscales o lexibili y
(R = 0.603, P <0.1%), luidi y (R = 0.596, P <0.1%), and ini ia i e (R = 0.464, P <0.1) signi ican ly co ela ed
wi h Basadu ’s c ea i e p oblem-sol ing es , indica ing he accep able alidi y o he subjec . Mo eo e ,
C onbach’s alpha o he whole es was 0.884, ep esen ing ha his es was eliable. The e o e, his ensu es
ha all i ems a e app op ia e and he e is no need o elimina e some i ems (Za e e al., 2014). Finally,
C onbach’s alpha was calcula ed o con ol, Bybee, and Synec ics sec ions as 0.80, 0.81, and 0.77 in o de o
assess he eliabili y o he es .
The geome y esea che -made es was used o e alua e s uden s’ lea ning le els. Fo he p e- es , 10 ques ions
we e selec ed, which we e ela ed o he p e equisi es o he opics o he nin h-G ade P e- o med Geome y
Ques ions. These included i e, h ee, and 2 easy, in e media e, and di icul ques ions, espec i ely. The pos -
es consis ed o nine ques ions o he augh opics, including h ee, ou , and wo easy, in e media e, and
di icul ques ions, espec i ely. Ma hema ics p o esso s discussed and app o ed he ex en o which he es
ques ions ep esen ed he con en and pu pose o he esea ch. Each es sco e was ou o 20. The in e nal
consis ency es (spli -hal es ) was used o e alua e he eliabili y o he es s in his s udy and i s alue was
ob ained in he ollowing g oups. The alue o eliabili y o he p e es o con ol, Synec ics, and Bybee g oups
was 0.70, 0.78, and 0.70, espec i ely. The alue o eliabili y o he pos es o con ol, Synec ics, and Bybee
g oups was 0.70, 0.71, and 0.72, espec i ely.
Examples o opics
The ollowing a e some examples o geome y opics in acco dance wi h Bybee and Synec ics models.
Py amid and cone olume acco ding o he Bybee model (5E)
In he i s s ep, he esea che ied o mo i a e he s uden s by de ining exci ing e en s ha we e ela ed o he
p ope ies o he py amids. Then, s uden s we e asked ques ions abou hei p io knowledge. The shapes we e

Jou nal o Educa o s, Teache s and T aine s JETT, Vol. 11 (1); ISSN: 1989-9572 73
p o ided wi h he de ini ions o he py amid and cone. Nex , he s uden s we e asked o de ine he cylinde and
s a e he cylinde olume calcula ion o mula ha hey had lea ned p e iously. The esponses we e w i en on
he boa d o expose s uden s o he iewpoin s, and hen hey we e asked o indica e he a ea o a ious shapes.
The ques ion “wha is he ela ionship be ween cone olume and cylinde olume? ” was aised by he
esea che . In he second phase (Explo e), each g oup indi idually p oduced pape cylinde s and cones. Du ing
he ques ion and answe p ocess, a numbe o s uden s modi ied hei cylinde s, and his g oup began es ing
again. Meanwhile, he esea che ac ed as a acili a o and encou aged lea ne s o expe ience, es , and he
de ailed e iew o he p oblem.
In he hi d s ep (Explain), he esea che asked he ep esen a i e o each g oup o sha e hei expe iences and
indings and he indings we e e iewed acco dingly. In he ou h s ep (Elabo a e), a ques ion was asked (How
can he olume o he cone be calcula ed?) Then, he g oups we e eques ed o examine he ques ions
indi idually. Simila ly, he esea che allowed he s uden s o e alua e hei lea ning and eamwo k p og ess in
he i h s age (E alua e). Likewise, g oups we e asked o epo he indings a e examining he ques ions.
Finally, assignmen s a di e en le els o lea ning we e iden i ied o he s uden s.
Sphe e olume and sphe e a ea acco ding o he Synec ics model
In he i s s ep (p oblem desc ip ion), s uden s we e encou aged o p o ide in o ma ion on he subjec o he
lesson and o desc ibe he exis ing si ua ion o wha is p esen . The s uden s we e hen di ided in o six g oups o
i e and ep esen a i es we e selec ed o each g oup. In addi ion, equipmen was p o ided o he g oups,
including he colo ed pape , scisso s, glue, a 7-cm diame e ball, a 7-cm diame e cylinde , and colo ed wa e . In
he second s ep (di ec analogy), he s uden s we e asked o ind sphe ical pa e ns in he book o class oom and
o desc ibe he u ili y and bene i s o hese objec s. Some examples a e as ollows.
Ball: Easie o mo e, has no angle, and his speeds up he ball.
Sky Objec s: Spend he leas amoun o ene gy o achie e maximum s abili y. Acco dingly, hey a e all
sphe ical. In he hi d s age (in ensi e con lic ), each g oup was asked o conside wo aces o si ua ions o he
sphe e ha we e in con lic . In he ou h s ep (di ec analogy), he s uden s we e eques ed o p o ide mo e
examples and his ime hey could use con as ing wo ds o exp ess newe examples. The mos beau i ul ex ac
was as ollows.
Ea h: Due o he sphe ical na u e o he Ea h, he e a e always di e en hou s o he day a all imes
(Day/nigh ).
In he i h s ep (pe sonal analogy), s uden s we e asked o conside he ollowing ques ion.
Suppose you a e a sphe ical adius ha needs o be pain ed in wo s ages o become a billia d ball. In he i s
s age, you a e h own in o a cylind ical chambe wi h a adius cm and a heigh o 2 cm, illed wi h he desi ed
colo . “Wha ac ion o he colo does ge ou o he cylind ical shed?”
In s ep six ( e-assignmen e iew), he g oups we e asked o w i e down hei indings, and he ep esen a i es
o each g oup we e eques ed o announce he iews. Then, he iews o he g oups we e w i en on he boa d.
The second ask was o s ick he s icke s in ci cles o he adius o he manu ac u e . How many s icke s do you
need o co e you en i e su ace? Finally,, he esea che summa ized and comple ed he con en by showing
anima ions ela ed o each sec ion and hen assigned homewo k o each o he s uden s a di e en le els o
lea ning. In Figu es 1, 2, and 3, s uden s’ solu ions in he h ee g oups a e compa ed o one p oblem.S uden s’
solu ions in he Bybee expe imen al g oup (Figu e 1) show c ea i i y in esponses demons a ing deep lea ning
and s ong men al skills. Acco ding o he gained expe iences in he class, s uden s used a simple and sho
answe a he han a de ailed answe . The conical olume was 1/3 o he cylinde olume equal o he egula
heigh .
Jou nal o Educa o s, Teache s and T aine s JETT, Vol. 11 (1); ISSN: 1989-9572 74
Fig.1: S uden s’ solu ions in he Bybee g oup
The solu ion o he Synec ics g oup (Figu e 2) demons a es he c ea i i y in w i ing and expanding he isibili y
o lea ne s’ analy ics capabili y. By simpli ying he o mula o his p oblem, he cylinde heigh was calcula ed
by equa ing he olume o he cone and cylinde .
Fig.2: S uden s’ solu ions in he Synec ics g oup
The con ol g oup’s solu ion (Figu e 3) ep esen s a poo unde s anding o geome ic olumes, a
misunde s anding o he ela ionship be ween he a iables, and inaccu acy in easoning and concep ualiza ion.
Despi e he knowledge o olume calcula ion o mulas, he s a egy was no p ope ly used, and s uden s
conside ed he ele a ion o he liquid le el in he cone a he han he heigh o he wa e in he cylinde .
Fig.3: S uden s’ solu ions in he con ol g oup
RESULTS
In Tables 1 o 3, he mean and s anda d de ia ion we e used o es desc ip i e s a is ics. The indings showed
ha he mean sco es in he p e- es s we e he same and he g oups we e scien i ically simila while he mean
sco es in he pos - es sco es o Bybee and Synec ics g oups in pe o mance es s, c ea i i y, and c ea i e
solu ion we e mo e compa ed o he con ol g oup. Addi ionally, he highes amoun o de ia ion in c ea i i y,
pe o mance, and c ea i e sol ing es s was ela ed o he p e- es o he Bybee g oup.
In he con ol g oup, s uden s had un esol ed p oblems. Mo e p ecisely, hey had di icul ies in d awing,
calcula ions, and a gumen s due o he lack o an accu a e pic u e o geome ic su aces. In cong uen iangles,
hey some imes did no ecognize he sides o angles o did no conside he midline angle in he “ wo sides
Jou nal o Educa o s, Teache s and T aine s JETT, Vol. 11 (1); ISSN: 1989-9572 75
be ween” and “ wo angles be ween” posi ions. In addi ion,, hey we e less con used on ques ions ega ding
which iangles we e aised indi idually. Due o he lack o basic in o ma ion, he s uden s used he o mulas o
a ea and olume in e changeably o w o e he o mulas mo e o less. Some imes, hey did no unde s and he
hypo hesis and induc ion co ec ly. In o he wo ds, he s uden s did no ha e he igh s a egy using he
cohesi e s a es. Using he cong uen o m, s uden s did no ha e he igh s a egy. A i s , hey w o e all o he
da a and hen decided on which cong uen o m o use acco dingly.
Table 1: Desc ip i e S a is ics o he C ea i i y Tes
Cha ac e is ics
N
Mean (SD)
P e es con ol
30
76.90 (11.77)
Pos es con ol
30
77.43 (11.74)
P e es Bybee
30
77.33 (14.92)
Pos es Bybee
30
88.33 (12.16)
P e es synec ics
30
80.10 (12.39)
Pos es synec ics
30
89.20 (11.93)
No e. N: Numbe ; SD: S anda d de ia ion.
Table 2: Desc ip i e S a is ics o C ea i e Sol ing Tes
Cha ac e is ics
N
Mean (SD)
P e es con ol
30
53.43 (7.48)
Pos es con ol
30
54.13 (8.43)
P e es Bybee
30
54.43 (8.89)
Pos es Bybee
30
62.16 (7.74)
P e es synec ics
30
56.20 (6.35)
Pos es synec ics
30
61.40 (6.78)
No e. N: Numbe ; SD: S anda d de ia ion.
Table 3: Desc ip i e S a is ics o Pe o mance Tes
Cha ac e is ics
N
Mean (SD)
P e es con ol
30
13.44 (3.50)
Pos es con ol
30
13.86 (2.69)
P e es Bybee
30
13.45 (4.63)
Pos es Bybee
30
16.61 (2.86)
P e es synec ics
30
13.19 (4.19)
Pos es synec ics
30
15.92 (2.47)
No e. N: Numbe ; SD: S anda d de ia ion
Kolmogo o -Smi no es was used o e alua e he no mali y o he da a and he indings showed ha he sco es
in all g oups we e no mal (P>0.05) con i ming he use o pa ame ic es s.
Table 4: ANCOVA Resul s o he E ec o Bybee Model on C ea i i y, C ea i e Sol ing, and
Pe o mance
Tes s o Be ween-Subjec s E ec s
Dependen a iable: VAR00004
Sou ce
Type III Sum o Squa es
d
Mean Squa e
F
Sig.
Co ec ed model
84499.420a
56
1508.918
62.573
0.000
In e cep
5438.781
1
5438.781
225.540
0.000
G oup
379.429
1
379.429
15.735
0.000
VAR00003
7350.216
55
133.640
5.542
0.000
E o
795.777
33
24.114
To al
364575.000
90
Co ec ed o al
85295.197
89
a. R Squa ed = 0.991 (Adjus ed R Squa ed = 0.975)
No e. ANCOVA: Analysis o co a iance.
Jou nal o Educa o s, Teache s and T aine s JETT, Vol. 11 (1); ISSN: 1989-9572 76
Acco ding o da a in Table 4, he e is a signi ican di e ence be ween he p e es and pos es sco es o
c ea i i y, c ea i e sol ing, and pe o mance (P = 0.000). These indings suppo he posi i e e ec o using
Bybee in pa icipan s’ c ea i i y, c ea i e sol ing, and pe o mance.
Table 5: ANCOVA Resul s o he E ec o he Synec ics Model on C ea i i y, C ea i e Sol ing,
and Pe o mance
Tes s o Be ween-Subjec s E ec s
Dependen a iable: VAR00002
Sou ce
Type III Sum o Squa es
d
Mean Squa e
F
Sig.
Co ec ed model
87386.114a
65
1344.402
86.680
0.000
In e cep
2139.170
1
2139.170
137.922
0.000
G oup
78.000
1
78.000
5.029
0.034
VAR00001
6701.113
64
104.705
6.751
0.000
E o
372.240
24
15.510
To al
365286.188
90
Co ec ed o al
87758.353
89
a. R Squa ed = 0.996 (Adjus ed R Squa ed = 0.984)
No e. ANCOVA: Analysis o co a iance.
Based on he ob ained da a in Table 5, he e is a signi ican di e ence be ween he p e es and pos es sco es o
c ea i i y, c ea i e sol ing, and pe o mance (P = 0.034). Conside ing ha he mean sco e o he pos es was
signi ican ly highe , synec ics had a posi i e e ec on s uden s’ c ea i i y, c ea i e sol ing, and pe o mance.
Table 6: ANCOVA Resul s o he E ec o he T adi ional Model on C ea i i y, C ea i e Sol ing, and
Pe o mance
Tes s o Be ween-Subjec s E ec s
Dependen a iable: VAR00006
Sou ce
Type III Sum o Squa es
d
Mean Squa e
F
Sig.
Co ec ed model
68139.494a
58
1174.819
191.342
0.000
In e cep
1298.458
1
1298.458
211.479
0.000
G oup
0.245
1
0.245
0.040
0.843
VAR00005
7528.678
57
132.082
21.512
0.000
E o
190.336
31
6.140
To al
279838.375
90
Co ec ed o al
68329.831
89
a. R Squa ed = 0.997 (Adjus ed R Squa ed = 0.992)
No e. ANCOVA: Analysis o co a iance.
As indica ed in Table 6, he le el o signi icance is 0.83, implying ha he di e ence be ween he p e es and
pos es sco es o c ea i i y, c ea i e sol ing, and pe o mance was no signi ican . The e o e, he sco es in he
pos es demons a ed no signi ican change in he con ol g oup. In o he wo ds, Bybee and Synec ics me hods
pe o med be e compa ed o he adi ional one, and he use o hese wo me hods had posi i e impac s on
s uden s’ pe o mance and c ea i e p oblem-sol ing in geome y.
DISCUSSION AND CONCLUSION
In gene al, using adi ional me hods in geome y eaching p e en s s uden s om he p ac ical obse a ion o
he con en hus hey p e e o eso o inapp op ia e me hods such as memo izing he de ini ions o he
geome y, which impedes hei le el o geome ic hinking. The applied eaching me hods a e di ec ly ela ed o
he g ow h o s uden s’ c ea i i y and c ea i i y can be enhanced h ough ins uc ion among he s uden s. Using
hese me hods while eaching, s uden s explo e he subjec ma e wi hou limi a ions and a e in ol ed in he
solu ion p ocess, which inc eases he possibili y o c ea i i y in s uden s.
The p esen s udy a emp ed o in es iga e he e ec s o Bybee and Synec ics me hods on he c ea i i y, c ea i e
p oblem-sol ing, and pe o mance o nin h-g ade s uden s in geome y. To his end, he p e es sco es o
c ea i i y, c ea i e p oblem-sol ing, and pe o mance o all h ee g oups we e iden i ied and he indings
e ealed ha expe imen al g oups and he con ol g oup we e a he same le el. A e aining based on Bybee