In
te
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ation
a
l Jou
rn
al
o
f E
v
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u
a
t
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on
a
n
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R
e
se
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in
Ed
u
c
ation
(
IJERE
)
V
o
l.7,
N
o.3,
S
eptem
b
er
2
01
8
,
pp. 229~
2
3
5
IS
S
N
: 2252-
88
22
D
O
I
: 10.
1
1
591
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.v7.i
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142
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tp:
//i
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.
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.
p
hp/IJ
ERE
/
Activities Written by Prospectiv
e Primary Teachers on Realistic
Mathematics Education
Hatice
Kub
r
a Guler
Dep
a
rt
em
ent
o
f
E
le
m
e
n
t
ary M
a
them
ati
c
s
Edu
cati
on, Du
zce Un
iversi
ty
Art
i
cl
e In
fo
ABSTRACT
A
r
tic
le hist
o
r
y
:
Re
ce
i
v
e
d
Ju
l
1
1,
201
8
Re
vise
d A
ug
7
, 2018
Ac
ce
p
t
ed
Au
g
2
0
,
2
018
It
i
s
im
p
o
rt
ant
fo
r
st
uden
t
s
to
c
o
n
n
ect
w
i
t
h
real
l
if
e
i
n
m
ath
e
m
a
tics
c
ou
rses
.
The
ai
m
of
t
hi
s
study
is
t
o
invest
iga
t
e
the
act
i
v
it
i
e
s
which
ar
e
writt
e
n
by
pro
s
p
ectiv
e
primary
sch
o
o
l
t
each
ers
in
t
h
e
c
on
t
e
xt
o
f
Reali
s
tic
M
athematics
Ed
ucat
ion.
T
his
re
s
ear
ch
i
s
a
cas
e
stu
d
y
of
w
hich
d
ata
were
c
ol
lect
ed
by
qualitative
methods.
T
he
p
artici
p
ants
o
f
the
resear
ch
i
ncl
u
ded
8
1
p
ro
sp
ectiv
e
pri
m
ary
t
eachers
wh
o
are
jun
i
o
r
s
t
u
d
e
n
t
s
at
ten
d
ed
p
ri
m
a
ry
t
each
er
dep
a
rtm
e
nt
.
It
w
as
a
s
k
ed
t
h
e
m
t
o
w
rite
a
n
act
iv
it
y
su
itab
l
e
f
o
r
prim
ary
m
a
th
ematics
curricu
lu
m
b
y
c
on
si
de
ring
R
ealis
t
i
c
Mathema
t
i
c
s
Edu
cati
on
pri
n
ci
p
l
es
.
Th
ey
w
e
r
e
g
i
v
e
n
on
e
mo
nt
h
t
o
w
rite
t
he
acti
v
ities
.
T
h
e
acti
v
ities
writt
en
b
y
pros
p
ect
iv
e
t
eachers
a
n
alyzed
b
y
d
e
sc
rip
t
i
v
e
m
e
th
od
s
i
n
t
e
r
m
s
o
f
learn
i
n
g
d
om
ains,
ty
pes
of
c
on
tex
t
a
n
d
p
rin
c
ip
les
of
R
eal
isti
c
Mathema
tics
Ed
ucat
ion.
T
otally
3
6
acti
v
iti
e
s
were
w
ri
tt
en
b
y
p
r
os
pect
ive
t
e
a
c
hers.
B
ased
on
t
he
a
nalysis,
it was
observed
t
hat no
t
all
th
e
con
t
ext
s
were
e
xactly
r
eal
life
con
t
ex
ts.
It
w
as
d
eterm
i
ned
t
h
a
t
p
ros
p
ect
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teacher
c
ou
ld
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o
t
di
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gu
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s
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rou
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word
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ro
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l
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s
f
r
o
m
r
eal
life
pro
b
l
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m
s
.
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can
b
e
sai
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t
hat
t
h
eo
retical
knowledge
of
R
eal
istic
M
athe
matic
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E
d
u
cation
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i
nad
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quat
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f
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w
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iti
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v
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osp
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im
ary
teach
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ed
t
o
pract
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n
w
riti
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c
t
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ie
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r
e
g
a
r
d
i
n
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th
e
me
nt
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d
th
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o
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y
.
K
eyw
ord
:
P
r
ospe
ct
ive P
r
ima
r
y T
eac
hers
R
e
a
lis
tic
M
a
t
h
e
m
a
tics
Ed
uca
t
i
on (
R
ME)
Tea
c
h
i
n
g
A
nd
Le
arni
ng
A
c
ti
vi
ty
Co
pyri
gh
t © 2
018 In
stit
u
t
e
of Advanced
En
gi
neeri
n
g
an
d
Scien
ce.
All
rights
res
e
rv
ed.
Corres
pon
d
i
n
g
Au
th
or:
Hat
i
c
e Ku
b
r
a Gü
l
e
r
,
Depa
rtem
ent o
f
E
lem
e
n
t
ary
Ma
them
at
ics E
duc
a
t
i
o
n,
Duzc
e U
n
i
v
e
r
s
i
t
y
,
K
o
n
u
ral
p
C
a
m
pus 8
1
6
2
0
D
uz
ce
/
T
ur
key
Em
ail:
hat
i
c
e
g
uler@
d
uz
ce
.ed
u
.
t
r
1.
I
N
TR
OD
U
C
TI
O
N
Rec
e
n
t
l
y,
t
he
o
p
i
ni
ons
a
r
e
s
ha
red
an
d
re
se
arc
h
es
a
re
c
arri
e
d
o
u
t
rega
r
d
i
ng
t
h
e
fa
c
t
t
ha
t
ed
uca
tio
n
sho
u
l
d
b
e
prac
ti
c
a
l
a
n
d
r
e
l
ate
d
t
o
r
eal
l
i
f
e
[1
].
T
he
m
ent
i
o
n
e
d
s
itua
t
io
n
g
o
e
s
for
m
a
them
atic
s;
b
eca
use
of
t
ha
t
ma
them
at
i
c
s
is
a
bs
t
r
act
i
o
n
o
f
t
he
r
ea
l
li
fe.
Ther
efore
,
i
t
i
s
i
m
p
orta
nt
f
or
s
t
ude
n
t
s
t
o
c
on
nec
t
w
i
t
h
r
eal
l
i
f
e
i
n
ma
them
at
i
c
s
cour
ses
[2]
,
[
3].
Real
is
t
i
c
Ma
them
at
ics
Ed
uc
at
ion
(
R
M
E)
i
s
a
u
n
i
que
t
he
or
y
u
s
ed
i
n
ma
them
at
i
c
s
e
duca
t
i
o
n,
w
h
i
c
h
w
as
p
ut
f
or
w
a
rd
b
y
H
a
n
s
F
re
uden
t
ha
l
t
o
r
ela
t
e
ma
t
h
e
m
ati
c
s
wi
th
r
eal
l
i
f
e.
I
n
th
i
s
t
he
ory,
i
t
is
w
orked
on t
h
e
c
i
rcum
stance
s
in re
a
l li
fe,
or
possib
l
e
ha
pp
eni
n
g in r
ea
l
life
c
ond
iti
o
n
s.
C
ons
t
r
uc
t
i
n
g
m
a
t
hem
a
t
i
c
a
l
kn
ow
le
d
g
e
w
a
s
ca
lle
d
m
a
t
h
e
m
atiza
t
i
o
n
b
y
F
re
uden
t
ha
l
w
ho
de
vel
o
pe
d
RME.
M
a
t
he
m
a
tiza
t
i
o
n
is
d
i
v
ide
d
i
n
t
o
t
w
o
b
y
h
or
i
z
o
n
t
a
l
a
nd
ve
rt
ic
al
m
athe
ma
ti
za
tio
n.
A
t
h
o
rizo
n
t
a
l
ma
them
at
i
z
at
i
on,
s
t
ude
n
t
s
st
ar
t
ac
t
i
vi
t
y
w
it
h
a
r
e
a
l
c
on
te
x
t
p
r
ob
l
e
ms
a
nd
th
e
n
t
h
e
y
u
s
e
in
fo
rma
l
l
angu
ag
e
an
d
symb
ols
to
d
efine prob
lem
s
.
A
t
t
h
i
s
le
ve
l,
e
very
s
tu
de
n
t
c
a
n
d
i
sc
ov
e
r
a
d
if
fe
ren
t
s
ol
u
tio
n
w
a
y.
A
s
for
ve
rt
ical
ma
them
at
i
z
at
i
on,
a
t
th
is
l
e
v
e
l
,
st
u
d
e
n
t
s
s
t
a
r
t
t
o
w
o
r
k
w
i
t
h
ma
t
hem
a
t
i
c
a
l
sym
b
o
l
s
a
nd
t
h
ey
can
r
e
ach
f
or
mulas
by
e
s
t
a
b
lish
i
n
g
r
ela
t
ions
h
i
p
am
ong
the
c
onc
e
p
ts
[
4]
.
F
r
e
ude
nt
ha
l
d
e
fi
n
e
d
ho
ri
zo
nt
al
m
at
h
e
mat
i
zat
i
o
n
a
s
trans
f
or
min
g
r
e
a
l
c
o
n
t
e
x
ts
t
o
sym
b
ols
w
h
il
e
vert
ica
l
m
at
hem
a
t
i
z
a
t
i
on
a
s
m
ov
in
g
in
s
ym
bo
ls
/
m
at
he
m
a
tics
w
o
rld.
K
n
o
w
l
edge
i
s
c
o
nst
r
ucte
d
w
h
ile
s
tud
y
i
n
g
on
r
e
al
c
o
n
t
e
x
t
prob
l
e
ms
a
n
d
t
hen
t
h
e
co
ns
t
r
ucte
d
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2252-
8822
IJERE
V
ol
.
7,
N
o.
3,
S
eptem
b
er
20
18 :
2
2
9
– 235
23
0
k
n
o
w
l
e
dg
e
b
eco
me
a
m
at
h
e
mat
i
cal
f
o
r
m
a
f
t
e
r
mat
h
emat
iza
t
i
o
n
.
A
f
t
er
t
hat,
i
t
ca
n
be
u
se
d
i
n
e
xe
rci
s
e
an
d
furt
her
a
c
tiv
it
i
e
s to c
o
n
str
u
c
t
n
ew
k
n
o
w
l
e
d
g
e
.
I
n
t
he
p
roc
e
s
s
of
c
o
n
struc
tin
g
m
a
them
atica
l
k
n
o
w
l
e
d
ge
,
th
ere
ar
e
t
h
r
ee
ke
y
pri
n
c
i
p
l
e
s
.
Gra
v
em
eijer
[4]
st
a
t
e
d
t
ha
t
the
y
w
ere
didact
ic
p
h
e
nom
e
n
o
l
og
y,
g
u
i
d
e
d
d
i
sco
v
er
y
a
nd
inc
l
usi
o
n
o
f
m
odel
s
.
The
m
a
in
que
st
ion
is
h
o
w
t
he
s
u
ita
ble
d
i
da
c
tic
al
s
t
r
uc
tures
of
m
at
he
m
a
ti
c
s
s
u
b
j
e
c
t
s
c
a
n
b
e
e
s
t
a
b
l
i
s
h
e
d
f
o
r
t
h
e
r
e
l
e
v
a
n
t
age
gro
u
p
.
Th
is
i
s
n
o
t
o
n
l
y
a
bo
ut
t
he
d
e
f
in
iti
o
n
o
f
t
h
e
m
a
them
a
t
i
c
al
c
o
n
cept
s
but
a
l
s
o
est
a
bl
i
s
hing
t
h
e
rela
tio
ns
hip
be
tw
e
e
n
rea
l
c
o
n
te
x
t
a
n
d
m
a
t
h
e
ma
tic
al
w
or
ld.
A
c
c
o
r
di
ng
t
o
d
id
a
c
t
i
cal
p
h
e
no
me
n
o
lo
gy
,
t
h
e
ma
them
at
i
c
al
a
c
tiv
i
ties
s
hou
l
d
s
tart
w
i
t
h
t
h
e
k
now
led
g
e
w
h
ic
h
i
s
me
ani
ngfu
l
f
o
r
s
tu
den
t
s.
T
hi
s
i
s
s
u
ppo
rt
ed
t
h
e
l
e
arn
i
ng
p
ro
ce
ss.
D
i
d
acti
c
a
l
p
h
e
no
men
o
l
og
y
expl
a
i
n
s
h
o
w
a
m
at
he
m
a
ti
c
a
l
c
once
p
t
i
s
c
on
st
r
u
c
t
ed
b
y
ana
l
yz
i
n
g
the
m
.
A
c
cordin
g
to
gui
de
d
di
sc
ove
r
y
,
stu
d
en
t
s
s
ho
u
l
d
have
t
he
o
p
por
tu
n
i
ty
t
o
try
a
m
e
th
o
d
o
r
stud
y
s
i
mi
lar
to
t
he
i
nve
n
tion
o
f
m
a
t
hem
a
ti
c
s
.
G
u
i
d
e
d
d
i
s
co
ve
ry
c
a
n
b
e
imp
l
em
en
ted
t
h
ro
u
gh
i
n
form
al
kn
ow
le
d
g
e.
I
nform
a
l
k
now
l
e
dge
a
n
d
s
tra
t
e
g
ie
s
can
b
e
u
s
ed
a
s
a
w
ay
t
o
form
al
s
tra
t
eg
ies.
C
ons
i
d
er
ing
inc
l
usio
n
of
m
ode
ls,
s
t
u
d
e
n
ts
s
h
oul
d
be
a
b
l
e
t
o
c
re
ate
t
h
e
i
r
o
w
n
m
ode
ls
w
he
n
so
l
v
i
n
g
pro
b
lem
s
.
F
i
rst
l
y,
stude
n
t
s
de
ve
l
op
a
m
ode
l
t
h
at
t
he
y
a
r
e
fa
m
ili
a
r
w
it
h
t
h
e
i
r
ow
n
in
formal
k
n
o
w
l
e
dge
.
The
m
o
del
w
h
ic
h
w
a
s
deve
l
ope
d
a
f
te
r
the
ge
ne
ral
i
z
a
ti
o
n
a
nd
f
o
rm
aliza
t
i
on
proce
s
s
b
e
ca
me
a
s
t
a
nd-a
l
one
s
tru
c
ture
g
rad
u
a
l
l
y
.
T
h
is
proce
s
s
is
c
a
l
l
e
d
“
t
he
m
ode
l
tr
ans
i
t
i
on
pro
c
e
s
s”
.
A
f
te
r
th
i
s
t
r
a
n
s
i
t
i
on,
t
he
m
ode
l
c
a
n
b
e
use
d
a
s
a
form
al
ma
them
at
i
c
al
m
ode
l
.
I
n
R
M
E,
c
onstr
uct
i
ng
for
m
a
l
m
ode
l
is
t
he
u
l
t
i
m
a
t
e
goa
l.
S
o,
i
t
shou
ld
n
o
t
b
e
s
t
a
r
te
d
w
ith
f
orm
a
l
know
l
e
dge,
w
h
ic
h is t
he
ul
tim
at
e
goal o
f
t
he mathe
m
a
t
ics
teac
hin
g
[
4].
I
n
R
M
E
,
con
t
e
x
t
prob
lem
s
a
re
t
he
b
as
is
f
or
t
he
m
a
t
he
ma
tic
al
p
r
o
ce
ss.
T
h
e
c
on
t
e
xt
p
rob
l
ems
l
e
ad
stude
n
t
s
to
d
isc
over
t
h
e
k
n
o
w
le
dge
a
nd
e
na
ble
t
h
em
t
o
r
eac
h
for
m
a
l
m
a
them
a
tica
l
m
ode
ls
[
5].
The
pro
b
le
m
sho
u
l
d
a
ll
ow
t
he
h
orizo
n
t
a
l
a
nd
verti
c
a
l
m
athem
a
tiza
t
io
n
pr
oces
s
as
n
e
e
de
d
f
o
r
e
x
plo
r
ation.
T
he
t
e
a
c
he
rs’
ma
in
que
st
ion
sh
o
u
l
d
be
“
ho
w
can
I
d
is
co
v
e
r
th
is
?
”
.
First
o
f
a
l
l,
s
tude
n
t
s
a
r
e
g
i
ve
n
pr
ob
lem
s
t
ha
t
t
h
ey
c
a
n
deve
l
op
s
p
ec
ific
m
eth
ods
f
or
t
he
s
i
t
ua
ti
on.
I
f
the
stu
d
e
n
t
ca
n
cre
a
te
n
e
w
m
ethods
t
o
so
lve
pr
ob
lem
,
t
he
ir
me
tho
d
s
bec
o
m
e
m
odels.
In
o
t
h
er
w
ords,
mode
ls
a
re
d
e
v
e
l
o
p
ed
w
i
t
h
t
h
e
a
c
t
i
vi
ti
es
o
f
t
h
e
st
ud
en
t
s
.
An
ot
h
e
r
criter
i
on
i
s
t
ha
t
the
model
a
l
l
o
w
s
s
t
ude
n
t
s
to
do ver
t
ic
a
l
m
ath
em
ati
c
s [6].
G
r
avem
eijer
[7]
disc
usse
d
th
e
r
o
l
e
o
f
m
ode
ls
d
e
v
e
l
o
p
e
d
d
uri
ng
t
h
e
f
o
r
mal
mat
h
emati
c
a
l
p
ro
cess
i
n
h
i
s
wo
rk
.
Th
e
u
n
d
e
rlin
e
d
phil
o
soph
y
o
f
h
i
s
s
tu
dy
w
a
s
t
h
a
t
ma
t
h
e
m
a
t
i
z
a
t
i
o
n
w
a
s
a
w
o
r
k
d
o
n
e
b
y
s
t
u
d
e
n
t
s
.
I
n
th
i
s
p
a
p
e
r
,
it
w
a
s
m
e
n
t
i
one
d
how
t
he
m
o
d
e
l
s
w
a
s
e
x
p
l
a
i
ne
d
more
a
n
d
ho
w
the
m
o
del
s
w
er
e
deve
l
o
pe
d
over
ti
m
e
.
A
course
w
hic
h
d
ea
lt
w
ith
m
en
ta
l
ca
l
c
ul
a
tio
n
s
t
r
a
te
g
i
es
a
n
d
a
d
d
i
t
i
o
n
u
p
t
o
1
0
0
w
a
s
d
e
s
i
g
n
e
d
.
A
n
a
l
y
s
e
s
w
e
r
e
r
e
v
e
a
l
e
d
t
h
r
e
e
r
e
l
a
t
e
d
p
r
o
c
e
s
s
.
F
i
r
s
t
o
n
e
w
a
s
t
h
a
t
t
h
e
m
o
d
els
were
i
nitiall
y
p
roduc
t
s
o
f
inf
o
rm
al
mat
h
e
m
a
t
i
c
a
l
a
ct
ivit
i
e
s,
b
u
t
l
at
er
t
h
e
y
we
re
d
e
v
el
op
ed
b
y
f
o
rma
l
m
a
th
em
atica
l
a
ct
iv
itie
s.
T
he
s
e
c
o
nd
on
e
in
vol
ve
d
t
h
e
c
ons
truc
t
i
o
n
o
f
new
ma
them
a
t
i
c
a
l
t
ru
t
h
s
w
ith
f
orm
a
l
re
l
a
t
i
o
n
s
by
t
ra
nsf
o
rmin
g
m
ode
l
of
t
o
mode
l
for
.
T
hi
r
d
ly,
i
t
w
as
s
e
e
n
t
h
a
t
t
h
e
re
w
e
r
e
m
o
re
t
han
o
n
e
m
ode
l
t
h
a
t
c
o
u
l
d
be
c
rea
t
e
d
a
t
the
end
of a
ct
i
v
i
tie
s.
S
o
me
r
esea
rche
rs
w
ere
show
e
d
t
ha
t
usi
ng
RME
th
eory
i
n
m
a
t
h
em
at
i
c
classe
s
impr
o
v
e
d
stu
d
e
n
ts’
ma
them
at
i
c
s
u
c
cesses
[8]-[1
2
]
a
nd
i
t
e
na
ble
s
f
or
t
hem
to
i
n
t
e
r
n
al
ize
t
h
e
conce
p
ts
d
ee
pl
y
[
1
3]-[1
6
]
.
Kwon
[1
7]
u
se
d
the
me
nt
i
o
ned
t
h
e
o
r
y
f
or
u
nd
er
g
r
adua
te
s
tu
den
t
s
in
d
i
fere
n
tia
l
e
q
ua
nt
ion
s
c
ou
r
s
es.
I
t
w
as
f
o
u
nde
d
at
t
he
e
n
d
o
f
the
s
t
udy
th
a
t
s
t
u
den
t
s’
a
bi
t
ili
t
y
o
f
m
odel
l
i
n
g
i
m
p
r
o
v
e
d
.
G
r
a
v
e
m
e
i
j
e
r
a
n
d
D
o
o
r
m
a
n
[
5
]
di
sc
usse
d
us
i
n
g
RME
t
h
eor
y
f
or
s
t
u
den
t
s
i
n
o
rder
t
o
re
ach
f
or
ma
l
m
a
t
h
em
ati
c
sl
m
od
e
l
s.
T
h
e
y
p
r
ep
ar
ed
a
ca
l
c
u
l
us
c
our
se
u
sin
g
R
ME
a
nd
foc
u
se
d
on
mode
ll
in
g.
T
h
e
a
i
m
o
f
the
i
r
st
udy
w
a
s
rea
c
hi
n
g
m
ode
l
for
stage
.
V
a
n
de
n
H
e
uv
e
l
-
P
a
nhuiz
e
n
[
10]
w
orke
d
on
RME
a
n
d
d
e
fi
ne
d
i
t
a
s
a
m
odel
w
h
ic
h
e
n
ab
le
d
for
stude
n
t
s
to
i
m
p
ro
ve
t
he
ir
m
athem
a
t
i
ca
l
t
h
in
kin
g
.
H
e
s
ug
ges
t
e
d
l
i
n
e
gr
ap
h
ic
a
s
a
s
t
ron
g
m
ode
l
f
o
r
perc
enta
ge
s
s
u
b
j
ec
t
in m
athem
a
tics
.
Mo
de
ls
a
re
i
m
por
tan
t
i
n
t
h
e
m
a
them
at
ics
le
arni
ng
a
n
d
tea
c
h
in
g
p
roc
e
ss
b
o
t
h
ea
rl
i
e
r
a
n
d
up
per
a
g
es.
Tre
ffe
rs
[
18]
s
ug
ges
t
e
d
t
ha
t
t
h
e
em
pty
n
u
mbe
r
l
in
e
as
a
n
e
ffe
c
t
ive
m
o
de
l
in
o
rder
t
o
te
a
c
h
number
li
ne
i
n
ea
rly
a
g
es.
A
s
T
re
ffe
rs,
A
l
t
u
n
[19]
s
t
u
d
i
e
d
o
n
tea
c
h
i
ng
nu
m
b
er
li
ne
b
y
the
hel
p
o
f
RM
E.
I
n
h
i
s
s
t
ud
y
,
it
w
a
s
sug
g
es
t
e
d tha
t
“
apple
st
a
i
r
-
mode
l”
c
o
u
ld
b
e
use
d
for
tea
ch
i
ng n
um
b
e
r
l
i
ne
i
n ea
rly a
g
e
s
.
C
ons
ide
r
i
ng
t
h
at
s
t
u
de
nts
a
r
e
at
c
o
n
cre
t
e
ope
rat
i
o
n
a
l
s
ta
g
e
,
RM
E
ac
t
i
v
i
t
i
es
a
p
p
l
i
c
ab
le
f
or
p
rim
a
ry
sch
o
o
l
s
sh
o
u
l
d
b
e
de
v
e
lope
d
for
s
t
ude
n
t
s
t
o
l
e
a
rn
m
athem
a
tic
s
w
e
ll.
Z
ulka
rd
i
[2
0]
a
rrange
d
a
pr
ojec
t
t
a
k
i
n
g
fo
ur
y
ea
rs
a
n
d
t
ra
i
n
ed
p
ros
p
ect
i
v
e
teac
he
r
on
RME.
I
n
t
h
i
s
c
om
p
re
h
e
n
s
i
v
e
re
se
arc
h
,
it
w
a
s
f
ound
t
h
a
t
pros
pec
t
i
v
e
m
a
the
m
a
tics
tea
c
her
s
l
e
a
rn
t
p
h
i
l
os
op
h
y
o
f
R
M
E
a
n
d
t
he
y
c
o
u
l
d
w
r
ite
a
c
t
i
v
i
t
i
e
s
on
R
M
E.
A
s
seen,
no
t
on
l
y
i
m
por
tan
t
i
ss
u
e
i
s
u
s
i
n
g
RM
E
act
iv
i
t
ie
s
i
n
c
lass
es
b
ut
a
ls
o
it
is
i
m
porta
nt
f
or
t
eac
her
t
o
d
e
s
ig
n
an en
v
ir
onm
en
t
su
i
t
a
b
l
e
f
or
R
ME
a
n
d
t
o
w
r
ite
a
ctivie
s ac
co
rdi
n
g
to t
h
e
m
e
nt
i
one
d
t
h
e
o
r
y
. It
i
s
r
eall
y
d
iff
i
c
u
lt
for
te
ac
he
rs
t
o
find
a
c
t
i
v
it
ies
sui
t
a
b
l
e
f
or
R
ME
;
i
t
i
s
be
ca
use
t
h
e
r
e
a
r
e
f
e
w
R
M
E
a
c
t
i
v
i
t
i
e
s
i
n
t
e
x
t
b
o
o
k
s
,
lite
rat
u
re
a
n
d
e
tc
.
A
l
so,
te
ac
he
r
cou
l
dn’t
u
s
e
a
n
a
c
tiv
i
t
y
n
e
ar
a
t
ha
nd
be
cau
se
o
f
tha
t
t
he
s
t
u
den
t
s’
p
rior
kn
owle
d
g
e,
ar
ea of resi
d
enc
e
a
n
d
etc
.
A
s see
n
,
t
h
e ea
rl
i
e
r st
ud
i
e
s
are
g
e
n
e
ra
l
l
y
me
nti
o
n
e
d
ho
w RM
E coul
d b
e
use
d
i
n classes
and w
h
e
t
her i
t
w
as rea
l
l
y im
pro
v
e
d
s
tu
de
nt
s m
a
t
h
e
ma
t
i
c
s
ac
h
i
ev
ement
s
.
Th
e
r
e
f
o
r
e
,
t
h
e
a
i
m
o
f
th
i
s
s
tu
d
y
i
s
t
o
i
nve
sti
g
ate
the
act
iv
i
ties
w
h
ic
h
are
w
r
it
te
n
b
y
pr
ospe
c
t
i
v
e
prim
ary
sc
ho
o
l
t
ea
c
h
ers
in
t
he
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ERE
I
S
S
N
:
2252-
88
22
Wri
t
t
e
n by
Pro
s
pec
tive
Pr
im
a
r
y T
e
ac
h
e
rs on
Re
alist
i
c M
a
th
e
m
atic
s
Ed
uca
t
ion
(
H
at
ice
Ku
bra
G
u
ler)
23
1
c
o
n
t
e
x
t
of R
M
E
.
The
m
a
in
ques
tio
n
of
t
he
r
esea
r
c
h
is
w
he
t
h
e
r
t
h
eo
ret
i
c
al
k
no
wl
ed
g
e
o
f
R
M
E
i
s
a
d
e
qu
at
e
f
o
r
wr
i
t
i
n
g R
M
E a
c
tiv
it
ies or
n
ot.
2.
RESEARCH METHOD
Th
is
r
esea
r
c
h
i
s
a
case
st
ud
y
of
w
hic
h
d
a
t
a
w
e
r
e
c
ollec
t
e
d
b
y
q
ua
l
i
t
a
tive
me
t
h
ods.
T
h
e
par
tic
ipa
n
ts
of
t
he
r
ese
a
r
c
h
i
n
c
l
ude
d
81
p
r
ospe
c
t
i
v
e
pr
i
m
ar
y
tea
c
he
r
s
w
ho
w
e
r
e
j
u
n
i
o
r
st
ude
n
t
s
at
t
e
nde
d
pr
im
ar
y
t
e
ache
r
de
par
t
m
e
nt.
T
h
e
par
tic
i
p
an
ts
t
o
o
k
t
he
m
at
h
e
m
a
tics
e
duca
t
io
n
c
o
u
rse
a
n
d
t
h
e
y
h
a
d
t
h
e
t
h
e
o
r
e
t
i
cal
k
nowl
e
dg
e
r
e
gar
d
ing
R
M
E.
T
he
t
heor
y
a
nd
i
t
s
pr
i
n
c
i
p
l
es
w
er
e
exp
l
a
i
ne
d
by
u
s
in
g
t
w
o
d
i
ffe
r
en
t
R
M
E
ac
t
i
v
i
t
i
e
s
n
am
ed
“
S
nake
s”
a
nd
“
What
d
i
d
n’
t
t
h
e
ki
n
g
k
now
?”.
T
he
a
im
o
f
the
“
s
nak
e
s
”
pr
ob
lem
w
a
s
f
o
r
st
ude
n
t
s
to
c
o
n
s
tr
uc
t
ge
ome
t
r
i
c
sets
w
hile
t
he
a
im
o
f
t
h
e
“
w
hat
di
d
n
’
t
k
i
ng
kn
ow
?
”
w
a
s
t
o
b
e
c
onstr
uc
ted
geom
etr
i
c
ser
i
es.
A
n
d
a
l
s
o
i
n
t
h
e
c
our
ses,
B
lo
om’
s
t
axo
n
o
m
y
w
as
c
om
par
e
d
w
i
th
R
M
E
i
n
te
r
m
s
o
f
h
o
r
iz
o
n
ta
l
a
v
e
r
t
i
c
a
l
m
a
them
at
iza
t
i
on.
T
he
f
o
l
lo
w
i
n
g
F
i
g
ur
e
1
w
a
s
give
n
stu
d
e
n
ts
a
nd
i
t
w
a
s
d
i
s
c
u
s
s
e
d
i
n
th
e
cl
ass
in
t
e
r
ms
o
f
Bl
oo
m’s
t
a
x
ono
my
a
n
d
R
M
E.
F
i
gur
e
1.
H
or
i
z
on
t
a
l
and
ver
t
i
c
a
l
m
a
t
hem
a
t
i
z
a
ti
o
n
It
w
a
s
e
x
p
la
i
n
ed
s
tu
de
nt
s
th
a
t
horiz
o
n
t
a
l
m
a
them
at
izat
i
on
star
t
e
d
w
i
t
h
a
p
p
l
i
c
a
t
i
o
n
,
a
s
s
e
e
n
f
r
o
m
F
i
g
u
r
e
1
.
A
f
te
r
cons
tr
uc
ti
n
g
k
n
o
w
l
e
d
ge
,
ve
r
tica
l
m
a
t
hem
a
ti
z
a
t
i
o
n
c
o
nt
i
n
ues
a
s
B
loom
’
s
t
a
x
o
n
o
moy.
I
n
t
h
e
fol
l
o
wi
ng
c
ou
rs
es,
t
h
e
p
r
i
c
in
ip
l
e
s
o
f
acti
v
e
le
arn
i
n
g
was
ex
pl
a
ine
d
a
nd
re
l
a
te
d
wi
t
h
R
M
E
.
Ac
tivi
t
i
e
s
w
hic
h
w
e
r
e
car
r
i
ed
out
d
ur
i
n
g
the
c
our
se
w
er
e
inv
e
st
iga
t
e
d
i
n
ter
m
s
o
f
the pr
inc
i
p
l
e
s
o
f
act
ive
le
arni
ng a
n
d R
M
E
.
The
pa
r
tic
ipa
n
ts
w
e
r
e
pr
ovid
e
d
a
n
o
p
t
i
o
n
to
s
et
u
p
gr
o
u
p
s
of
m
a
xi
mu
m
t
h
re
e
p
e
opl
e
i
f
t
h
e
y
wa
nt
ed
to.
The
y
g
r
o
u
p
ed
12
gr
ou
ps
c
on
s
i
s
tin
g
of
t
hr
e
e
s
tu
de
n
t
s
and
21
gr
ou
ps
c
ons
isti
n
g
o
f
t
w
o
st
u
d
en
ts.
Thr
e
e
st
ude
n
t
s
prefer
r
e
d
to
w
ri
te
a
c
tiv
it
ies
a
l
one.
It
w
as
a
ske
d
e
a
c
h
g
ro
u
p
t
o
write
a
n
ac
ti
v
ity
s
u
ita
b
l
e
for
pri
m
a
r
y
m
a
them
at
ics
c
u
rricu
lum
by
c
ons
ide
r
i
n
g
R
M
E
pri
n
c
i
p
l
es.
T
h
ey
w
er
e
g
ive
n
one
m
on
t
h
t
o
w
r
i
t
e
the
ac
t
i
v
i
tie
s.
The
act
i
v
i
ties
wr
i
tte
n
b
y
p
ro
spec
tive
t
e
ac
he
rs
a
na
l
y
z
e
d
b
y
desc
r
i
pti
v
e
me
t
hod
s
i
n
t
e
r
ms
o
f
l
e
a
r
ning
d
o
m
ai
ns
(
n
u
m
ber
s
,
geo
m
etry,
m
e
a
s
urem
ent,
a
n
d
,
da
t
a
a
n
d
s
ta
ti
stic
s)
,
t
y
p
e
s
o
f
c
onte
x
t
(
s
cie
n
ti
fi
c,
pub
l
i
c,
p
e
r
s
ona
l,
a
n
d
oc
cu
pa
ti
o
n
a
l
)
a
nd
pr
i
n
c
i
p
l
es
o
f
R
M
E
(
g
ui
de
d
r
e
in
ve
n
t
i
o
n,
d
ida
c
t
i
c
al
p
h
e
no
meno
l
o
gy
,
an
d
i
n
clu
s
i
on
of
m
ode
l
s
)
.
I
n
add
i
t
i
o
n
,
i
t
w
a
s
d
isc
u
sse
d
w
i
th
p
r
o
s
p
ec
ti
ve
t
e
a
c
hers
o
n
a
c
t
iv
itie
s
wr
itte
n
by
t
he
m
in
a
w
ho
le
cl
ass discussion.
I
n
t
he
a
na
lyse
s
w
h
i
c
h
w
e
r
e
d
o
n
e
ac
cor
d
i
ng
t
o
t
he
p
r
i
nc
ip
les
of
R
M
E,
t
he
f
oll
o
w
i
n
g
c
r
i
t
e
r
i
a
we
re
c
onsidere
d
:
1)
D
i
d
t
he
p
r
o
bl
e
m
s
/
ac
t
i
vi
t
i
e
s
w
hich
w
er
e
w
r
it
te
n
b
y
p
r
o
spe
c
t
i
v
e
te
ac
h
e
rs
h
a
v
e
a
fam
i
l
i
ar
c
o
n
te
x
t
for
st
ude
n
t
s
?
2)
D
id
t
he pro
blem
s/a
c
t
i
vi
t
i
e
s
w
hi
c
h
w
e
r
e writte
n
b
y
p
r
o
s
p
ec
ti
v
e tea
c
h
er
s
ha
ve
t
he
c
ha
r
a
c
t
e
r
is
tics of
the
rea
l
co
n
te
x
t
?
3)
D
id
t
he
p
r
o
b
l
em
s/
a
c
t
ivi
t
i
e
s
whic
h
were
w
ritte
n
by
pr
ospe
ct
i
v
e
te
a
c
her
s
e
na
b
l
e
stude
n
t
s
t
o
de
v
e
l
o
p
d
i
ffe
r
e
nt
p
r
o
b
l
e
m
s
o
l
vi
n
g
s
tr
ate
g
i
e
s
and
to
d
o
e
xpla
n
at
ion
s
b
y
us
in
g
inf
o
r
m
al
l
an
gua
ge
?
4)
D
i
d
t
he
s
o
l
u
tio
n
of
t
he
p
r
o
blem
s
ena
b
l
e
t
o
use
p
hys
ica
l
m
ode
ls
?
5)
A
t
the
e
nd
of
t
he
a
ct
i
v
it
ie
s,
c
ou
ld
i
t
be
c
on
st
r
u
c
t
e
d
a
s
tr
u
ct
u
r
e
whi
c
h
co
ul
d
us
e
solv
i
n
g
a
not
h
e
r
pr
o
b
lem
o
r
c
ould
it
be
m
ade
a
gene
r
a
liza
tio
n
?
The
f
i
r
s
t
t
h
r
e
e
c
r
iter
i
a
ar
e
r
e
ga
r
d
in
g
“
g
ui
d
e
d
r
e
i
n
ven
t
i
o
n
”
a
n
d
“
dida
c
t
ic
al
phe
n
o
m
e
n
o
l
o
gy”.
T
h
e
f
o
u
r
th
a
nd
f
i
f
t
h
ones
ar
e
r
e
gardi
n
g
“
i
n
c
l
u
sio
n
of
m
odels”.
3.
RESU
L
T
S
A
ND ANALY
S
IS
To
tal
l
y
36
a
c
tiv
i
t
i
e
s
w
e
r
e
w
r
i
t
t
en
b
y
pr
os
pect
i
v
e
t
e
ac
he
r
s
.
O
n
e
o
f
t
h
e
m
w
as
e
xcl
u
d
e
d
fr
om
t
he
a
n
al
ys
i
s
b
e
c
a
u
se
it
wa
s
rela
te
d
to
a
ri
t
h
me
tic
s
e
q
uence
;
w
h
i
c
h
w
asn’
t
a
pr
i
m
a
r
y
m
a
them
atic
s
su
b
j
ec
t.
O
ut
o
f
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2252-
8822
IJERE
V
ol
.
7,
N
o.
3,
S
eptem
b
er
20
18 :
2
2
9
– 235
23
2
35
a
c
t
i
v
i
t
i
e
s
,
16
o
f
t
hem
w
e
re
r
e
g
ardi
ng
n
u
m
bers,
16
o
f
t
he
m
w
e
r
e
r
ega
r
ding
me
asure
m
ent,
a
nd
3
of
t
he
m
w
e
r
e
r
egar
di
n
g
ge
om
etry l
e
a
rni
n
g dom
ai
n.
Ther
e
w
ere
no
t
a
n
y
a
c
t
iv
it
ies r
e
gar
d
ing
da
ta and
s
ta
t
i
s
t
ic
s le
ar
nin
g
doma
i
n
.
T
he
s
ub
jec
t
s
o
f
t
he
a
cti
v
i
t
es
a
r
e
m
easurem
ent
o
f
a
re
a,
fr
acti
ons,
decim
a
l
nu
m
b
er
s,
p
att
e
rns,
d
iv
i
s
i
on,
symm
etry,
pe
rime
t
e
r of re
c
tangle,
m
ea
sur
e
ment o
f l
i
q
u
i
ds a
nd et
c.
A
l
t
h
o
u
gh
s
o
m
e
o
f
the
a
c
t
i
v
it
ies
ha
d
the
ga
me
f
eature
s
,
all
c
o
nt
e
x
t
s
w
er
e
persona
l.
E
spe
c
ia
l
l
y
th
e
con
t
e
x
t
s
o
f
t
h
e
a
c
t
i
vi
tie
s
rega
rd
in
g
num
b
e
rs
l
ear
n
i
n
g
d
om
ain
w
e
re
q
ui
te
c
l
o
se
d
t
o
r
ou
ti
ne
p
ro
b
l
em
s
in
the
t
ex
t
b
o
oks.
A
s
f
or
R
ME
p
rinc
ip
les,
it
ca
n
be
a
na
l
y
se
d
doma
i
n
b
y
d
o
m
a
i
n.
T
he
f
ami
l
i
a
rity
o
f
th
e
c
o
nt
ex
ts
w
as
gi
ve
n in Ta
b
le
1
.
Tab
l
e
1. F
a
m
il
i
a
rity
o
f t
h
e
co
n
t
ex
ts
Le
a
r
ning Doma
ins
No
t
f
a
milia
r
Pa
r
tia
ll
y
f
a
m
ili
a
r
F
ami
l
i
a
r
T
o
ta
l
Nu
m
b
e
r
s
3
0
13
1
6
M
e
a
s
u
r
e
m
ent
0
3
13
1
6
G
e
o
m
e
t
r
y
0
0
3
3
Tot
a
l
3
3
29
3
5
A
s
s
e
e
n
f
r
o
m
T
a
b
l
e
1
,
2
9
o
f
t
h
e
c
o
n
t
e
x
t
s
w
e
r
e
f
a
m
i
l
i
a
r
f
o
r
p
r
i
m
ar
y
st
ude
n
t
s.
F
rom
the
fa
mi
liar
ac
t
i
v
i
ti
e
s
1
3
o
f
t
hem
w
e
re
r
ega
r
d
i
n
g
n
um
bers,
13
o
f
t
h
e
m
w
ere
r
ega
r
ding
m
e
a
s
ur
en
t
an
d
3
of
t
he
m
w
e
r
e
rega
rdi
ng
ge
o
m
etry.
F
r
om
T
ab
le
1
it
c
a
n
b
e
sa
id
t
ha
t
t
h
e
a
c
t
i
v
i
t
i
es
w
rit
t
en
by
pros
pec
t
i
v
e
t
e
ac
he
rs
w
er
e
fa
mil
i
a
r
f
or
p
r
i
ma
ry
s
tu
de
nt
s
.
B
ut
n
o
t
a
l
l
t
he
f
a
m
i
liar
co
nte
x
t
s
we
re
r
eal
l
ife
s
itua
t
i
o
ns.
It
c
an
b
e
se
en
f
r
om
Table 2.
Ta
b
l
e 2.
N
um
ber
of
r
ea
l
conte
x
t pr
oble
m
s
Le
a
r
ning
D
o
m
a
ins
N
o
t
r
e
a
l
c
ont
e
x
t
pro
b
lem
Pa
rtia
ll
y
r
e
a
l
c
on
text
pro
b
le
m
R
eal
c
o
n
t
ex
t
pr
o
b
le
m
To
t
a
l
Nu
m
b
e
r
s
5
3
8
16
M
e
a
s
u
r
e
m
ent
2
8
6
16
G
e
o
m
e
t
r
y
1
1
1
3
Tot
a
l
8
12
15
3
5
A
s
s
ee
n
from
T
ab
le
2
,
15
o
f
t
he
w
ri
tten
a
c
t
i
v
it
ies
w
e
r
e
r
e
a
l
c
o
nte
x
t
prob
l
e
m.
A
nd
o
nly
8
o
f
1
6
con
t
e
x
t
s
r
ega
r
di
n
g
n
um
ber
s
w
e
r
e
rea
l
c
on
te
xt
p
r
oble
m
s
w
h
i
l
e
6
o
f
1
6
c
on
te
xt
s
rega
rdi
n
g
m
e
a
s
urem
ent
w
e
r
e
r
e
a
l
s
i
t
u
a
t
i
o
n
s
.
A
s
f
o
r
g
e
o
m
e
t
r
y
,
o
n
l
y
o
n
e
o
f
t
h
e
m
h
a
d
c
h
a
r
a
c
t
e
ristic
o
f
real
p
roblem
.
From
t
hi
s
re
s
u
lt
it
c
a
n
be
s
a
i
d tha
t
pro
spe
c
tive
tea
c
h
e
r
s
had
difficu
l
t
y
in
w
rit
i
n
g
rea
l con
t
e
x
t
pr
ob
le
ms rega
r
di
ng
geom
etr
y
s
u
b
je
cts.
“Part
i
al
l
y
r
eal
c
o
n
t
e
x
t
p
robl
e
m
”
me
an
s
t
h
a
t
t
h
e
r
outi
n
e
p
r
o
b
l
e
m
s
w
h
i
c
h
were
f
o
u
n
d
i
n
t
e
xtb
ook
s
w
e
r
e
w
ritte
n
i
n
c
o
n
te
x
t
s.
F
rom
T
a
ble
2,
i
t
ca
n
be
s
ai
d
t
h
at
n
ea
rly
h
a
l
f
o
f
th
e
con
t
ext
s
w
ere
re
a
l
c
o
n
t
ext
pro
b
lem
s
.
A
s
f
or
s
tra
t
eg
ies
w
h
i
c
h
w
e
re
p
rob
l
em
s
ne
ede
d
a
nd
e
n
a
b
l
e
d
st
u
d
en
ts
t
o
d
o
e
x
p
l
a
na
tio
ns
i
n
i
n
form
al
l
a
ngu
a
g
e, t
he
r
es
ul
t
s
ca
n
b
e se
en
f
ro
m Ta
b
l
e 3
.
Tab
l
e
3.
N
umbe
r of the
u
se
d strate
gi
e
s
Le
a
r
ning
D
o
m
a
ins
Only
one
r
outi
n
e
stra
t
e
gy
M
o
re
t
h
a
n
on
e
but
routi
n
e
strat
e
gi
e
s
Non-
r
out
i
n
e
stra
te
gi
e
s
To
t
a
l
Nu
m
b
e
r
s
8
1
7
16
M
e
a
s
u
r
e
m
ent
6
4
6
16
G
e
o
m
e
t
r
y
1
1
1
3
Tot
a
l
15
6
1
4
35
To
tal
l
y
1
4
o
f
t
h
e
a
c
t
i
v
i
tie
s
ena
b
l
e
d
stu
d
e
n
ts
t
o
e
x
pl
a
i
n
t
h
e
i
r
so
lu
tio
n
w
a
ys
i
n
f
orm
a
lly
a
n
d
t
he
ir
pro
b
lem
so
lv
i
n
g
s
t
ra
t
e
gi
e
s
c
ou
ld
b
e
n
on-r
o
u
tine
s
t
rate
g
i
e
s
.
15
of
t
h
e
a
c
t
iv
i
t
i
e
s
we
re
r
o
u
t
i
n
e
p
r
obl
e
m
s
a
nd
the
y
d
id
n’t
ha
ve
t
he
c
har
a
c
h
te
r
i
st
ics
w
h
ic
h
e
n
a
b
l
e
d
s
t
u
d
en
ts
t
o
e
xp
la
in
t
he
ir
s
o
l
ut
i
o
n
i
n
form
all
y
;
i
t
w
as
bec
a
u
s
e
t
h
e
y
onl
y
re
quire
d
alg
hor
itm
ic
o
p
e
ra
t
i
ons.
F
r
om
T
a
b
l
e
3
,
i
t
c
a
n
b
e
sa
i
d
t
hat
l
e
ss
t
han
ha
lf
o
f
t
h
e
con
t
e
x
t
s
re
q
u
i
r
e
d no
n-
rou
tine
strate
gie
s
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IJERE
I
S
S
N
:
2252-
88
22
Writte
n
by Pro
s
pec
t
i
v
e P
r
im
a
r
y
Te
achers on
Real
ist
i
c
Ma
t
h
e
m
at
i
c
s Ed
uca
t
io
n (H
at
ice
Ku
bra
G
u
ler)
23
3
Tab
l
e
4.
E
nab
ling
t
o
use
p
hysi
c
a
l
m
od
e
l
s
Le
a
r
ning
D
o
m
a
ins
N
o
t
to e
na
bl
e
O
pt
iona
l
T
o
e
na
bl
e
T
o
ta
l
Nu
m
b
e
r
s
6
1
9
16
M
e
a
s
u
r
e
m
ent
2
4
10
1
6
G
e
o
m
e
t
r
y
1
0
2
3
Tot
a
l
9
5
21
3
5
F
r
om
T
able
4
,
it
c
a
n
b
e
see
n
how
m
any
pr
o
b
lem
s
’
sol
u
t
i
o
n
s
ena
b
l
e
t
o
u
se
p
hy
si
cal
m
o
d
e
l
s.
9
o
f
th
e
w
r
itte
n
a
c
t
i
v
it
i
e
s
d
i
dn’
t
req
u
i
re
t
o
u
s
e
ph
ys
i
c
a
l
m
o
d
e
l
s
w
h
i
l
e,
2
1
of
t
he
m
requ
ire
d
t
o
use
.
O
nly
in
5
a
c
tivi
t
i
e
s,
ph
ysic
a
l
m
ode
l
usi
ng
w
a
s
opti
o
nal.
T
he
p
r
o
b
l
em
s
c
o
u
l
d
solve
w
i
t
h
c
o
n
c
re
t
e
m
ate
r
ials
o
r
w
i
t
hou
t.
B
oth
of
them
w
er
e
p
o
s
s
i
b
l
e
.
Bu
t
m
o
re
t
h
a
n
ha
l
f
o
f
them
e
na
ble
d
t
o
use
c
o
n
c
r
ete
ma
t
e
rial
s
to
s
ol
ve
p
r
o
b
l
e
m
.
B
u
t
it
w
a
s
r
eally
i
n
t
e
r
e
s
t
i
ng
t
h
at
o
n
l
y
1
1
a
ct
iv
it
ies
ga
ve
t
he
o
ppo
r
t
u
ni
t
y
s
t
ude
n
t
s
t
o
m
o
v
e
fro
m
phys
ica
l
m
o
d
el
t
o
ma
them
at
i
c
al
m
odel
an
d t
o
m
ake
gene
ral
i
z
a
t
i
o
n.
I
t c
a
n
b
e
se
e
n f
rom
Tablo 5.
Whe
n
t
he
w
ritte
n
act
i
v
i
ties
are
e
x
am
i
n
e
d
,
it
w
as
s
een
t
hat
m
o
s
t
o
f
the
m
s
up
por
ted
t
h
e
use
of
conc
re
te
m
ater
i
a
ls
e
ve
n
th
ou
gh
th
ey
w
ere
no
t
re
al
c
on
te
xt
p
ro
bl
em
s.
M
a
ny
of
t
he
m
w
e
re
like
ga
m
e
s
and
bec
a
u
s
e
o
f
t
ha
t
r
eason
it
ca
n
be
s
ai
d
tha
t
t
h
e
y
requ
ire
d
u
si
ng
p
hys
ica
l
m
ode
ls.
Wh
ile
t
hey
r
e
q
u
i
red
p
hys
ica
l
mode
l
s
,
they
d
idn’
t
give
o
pp
o
r
tu
ni
ty
t
o
mov
e
m
athem
a
tica
l
m
ode
l
s.
Tab
l
e
5.
E
nab
ling
t
o
m
ake
gen
e
r
a
liza
t
i
o
n
Le
a
r
ning
D
o
m
a
ins
N
o
t
to e
na
bl
e
O
pt
iona
l
T
o
e
na
bl
e
T
o
ta
l
Nu
m
b
e
r
s
9
3
4
16
M
e
a
s
u
r
e
m
ent
6
4
6
16
G
e
o
m
e
t
r
y
1
1
1
3
Tot
a
l
16
8
1
1
35
As
s
e
e
n
T
a
b
l
e
5
,
1
6
o
f
t
h
e
w
ri
tt
e
n
a
c
t
i
v
i
t
i
e
s
d
i
dn
’t
g
iv
e
oppo
r
t
uni
ty
s
t
ude
n
t
s
t
o
ma
k
e ge
n
e
rilaz
a
tio
ns
and de
vel
o
p
ma
t
h
e
m
a
t
ic
a
l
m
ode
ls.
In
8
o
f t
h
e
w
r
it
t
e
n
a
c
t
i
vi
t
i
es
,
ma
ki
ng
gene
r
a
liza
tio
n
de
pe
n
d
ed o
n
te
a
c
he
rs.
H
o
w
t
h
e
te
ach
e
r
g
u
i
de
d
t
h
e
stude
n
t
s
d
u
ri
n
g
t
he
a
c
t
i
v
it
y
w
a
s
a
n
in
di
c
a
ti
on
t
h
a
t
t
h
e
y
w
o
u
l
d
d
ev
e
l
op
o
r
n
ot
deve
l
o
p
the
i
r
m
a
the
m
a
tica
l
m
odel
s
.
If
t
he
t
eac
her
ca
rrie
d
o
ut
t
he
a
c
tiv
i
tie
s
as
i
n
a
tra
d
it
io
na
l
a
n
d
te
a
c
he
r-
ce
nt
e
r
e
d
c
lass,
i
t
w
a
s
t
a
ke
n from
the
p
o
ssi
b
i
l
i
t
i
es o
f
gener
a
l
i
zat
i
on o
f
t
h
e
stu
de
nt
s.
I
n
t
he
i
nterv
i
ew
s,
p
ar
t
i
ci
pa
nt
s
sta
t
e
d
t
ha
t
t
h
e
y
h
a
d
d
i
ffi
c
u
l
t
y
i
n
c
a
rr
yin
g
o
u
t
t
he
p
rinc
i
p
le
o
f
“
i
ncl
u
s
i
on
of
m
ode
ls”
.
The
data
w
hi
ch
p
resen
t
e
d
i
n
ta
b
l
es
c
o
n
f
i
r
me
d
th
ei
r
st
a
t
emen
t
s
.
Th
e
d
a
t
a
w
sh
o
w
ed
tha
t
t
he
y
did
n
’t
h
a
v
e
di
ffic
ul
t
y
i
n
c
onsi
d
er
ing
“
g
u
i
ded
rei
nve
n
t
i
on
”
a
n
d
“d
i
d
acti
c
a
l
p
h
e
n
o
m
en
ol
ogy
”
wh
en
com
p
are
d
w
it
h
“
i
nc
lus
i
on o
f
m
ode
l
s
”
.
A
l
so, an
i
nte
r
est
i
n
g
da
t
um
w
a
s
t
ha
t
a
n
a
c
tiv
it
y
w
a
s qu
i
t
e
s
i
m
i
l
ar
w
ith
Tr
effers’
[
1
8]
a
nd Altun’s [
1
9]
ac
t
i
v
i
ti
e
s
w
h
i
c
h
w
ere
“
e
m
pty
numbe
r
l
i
n
e
”
a
nd
“
a
pp
le
s
tai
r
s”.
T
heir
a
cti
v
i
t
y
w
a
s
r
e
late
d
to
n
umbe
r
l
i
n
e
a
nd
the
y
use
d ap
p
l
e st
a
i
r
s
to c
ons
truc
t
n
u
mbe
r
l
i
n
e.
P
rospect
ive
t
e
ac
hers’ draw
in
gs a
bo
u
t
t
he pr
oble
m
can
b
e se
en
in Fig
ure
2.
F
i
gur
e 2.
A
ppl
e
st
a
i
r
s
prob
l
e
m
w
r
i
tte
n
by p
r
ospec
tive
tea
c
h
ers
As
s
een
f
rom
the
Fi
g
u
re
2
,
p
r
ospec
t
i
v
e
t
e
ac
he
rs
d
r
e
w
an
a
pple
s
ta
irs.
T
h
e
n
t
h
e
y
til
ted
it
a
n
d
a
f
ter
tha
t
t
he
y
simu
l
a
te
d
it
to
a
n
u
m
be
r
li
ne
.
In
t
hi
s
w
a
y,
t
he
y
aim
e
d
for
prim
ary
stu
d
e
n
t
s
t
o
con
s
truc
t
num
ber
li
n
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2252-
8822
IJERE
V
ol
.
7,
N
o.
3,
S
eptem
b
er
20
18 :
2
2
9
– 235
23
4
kn
ow
le
d
g
e
by
us
in
g
R
M
E.
W
he
n
ha
d
a
n
i
n
t
e
r
v
i
ew
w
i
t
h
pros
pec
t
i
v
e
t
e
a
c
h
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r
s
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h
o
w
r
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t
e
t
h
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a
c
t
i
v
i
t
y
,
t
h
e
y
st
a
t
ed
t
ha
t
t
h
e
y
h
a
d
n
e
v
er
s
ee
n
seen
T
reffe
r
’
s
a
rticle
;
wh
i
l
e
t
hey
saw
A
ltu
n’s
a
r
ti
c
l
e
af
t
e
r
the
y
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d
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r
e
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red
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c
t
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t
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A
nd t
h
e
i
r
ac
t
i
vi
t
y
h
ad
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l
l
t
hree
pri
nci
p
les of R
M
E.
4.
DISC
USSION
A
N
D
I
M
P
L
I
CATIO
NS
Gr
avem
eijer
[4]
sta
t
e
d
t
ha
t
t
h
ere
wer
e
t
hre
e
k
e
y
p
ri
n
c
ip
l
e
s
i
n
t
h
e
proc
es
s
of
c
on
st
r
u
c
t
i
n
g
ma
them
at
i
c
al
k
n
o
w
l
e
d
ge
a
nd
t
hey
w
e
re
d
i
d
actic
p
he
nom
e
n
o
l
og
y,
g
u
i
d
ed
d
i
s
cov
e
ry
a
nd
in
c
l
u
s
i
o
n
of
m
o
d
e
ls.
Ear
l
ier
r
e
sea
r
ches
[
4]-
[
7],
[11]
e
mpha
size
d
t
h
e
imp
o
r
t
a
n
ce
o
f
“
inc
l
u
s
ion
of
m
o
d
e
l
s”
i
n
R
M
E.
W
h
e
n
in
ves
tiga
t
in
g
fr
om
t
he
v
iew
p
o
i
n
t
o
f
t
he
p
r
i
n
c
i
p
le
s
of
R
eal
i
s
ti
c
Ma
them
at
i
c
s
Educa
t
io
n,
a
c
t
i
v
it
i
e
s
w
h
ic
h
w
e
r
e
w
r
itte
n
i
n
the
c
urrent stu
dy mode
ra
tel
y
ha
d
t
he
f
e
a
t
ures o
f gui
de
d
re
in
ve
n
t
i
o
n an
d
d
i
dac
tic
al phe
nom
en
ol
o
g
y
,
alt
h
ou
gh
t
h
e
y
d
i
d
n
ot
h
a
v
e
the
fe
at
ures
o
f
i
n
cl
us
ion
of
m
odel
s
a
d
e
q
u
a
tel
y
.
T
h
is
s
itua
t
ion
w
a
s
s
t
a
t
ed
b
y
pros
pec
t
i
v
e
te
a
c
her
i
n
t
he
i
n
t
e
r
v
i
ew
.
The
y
h
a
d
d
i
f
f
i
c
u
lt
y
in
g
i
v
i
n
g
o
pp
u
r
tu
ni
t
i
e
s
s
t
ude
nt
s
to
t
ra
nsfor
m
t
h
e
in
form
al
m
ode
ls
i
nt
o
f
o
rm
al
one
s.
F
rom
th
i
s
r
e
s
u
l
t,
i
t
c
a
n
b
e
s
ugg
e
s
t
e
d
t
h
at
p
rosp
ecti
v
es
p
ri
ma
ry
t
e
ach
e
r
s
n
e
ed
ma
t
he
ma
ti
c
a
l
mo
d
e
lli
ng
a
ct
ivit
i
e
s.
Ma
the
m
a
tiza
tion
pro
c
e
s
s
ha
s
t
w
o
s
t
e
p
s
:
h
oriz
on
ta
l
a
n
d
ver
tic
al
[4]
.
H
oriz
on
ta
l
m
a
t
h
em
atiza
tio
n
st
a
r
ts
w
i
t
h
a
p
r
ob
lem
(an
a
p
pl
i
c
a
t
io
n
)
a
n
d
i
t
fi
nis
h
es
w
i
t
h
c
on
struc
tin
g
the
kn
ow
le
d
g
e
i
n
ot
her
w
o
r
d
s
w
i
t
h
in
form
al
m
ode
l.
A
ft
e
r
t
hat
v
e
r
i
c
a
l
m
a
them
atiza
t
i
o
n
s
t
a
r
ts
a
n
d
f
i
n
ishe
s
w
ith
f
orm
a
l
m
ode
l.
B
eca
u
s
e
to
b
e
com
p
le
te
d
t
h
e
RME
ac
tiv
it
y
an
d
m
a
the
m
atiza
t
i
o
n
pro
c
ess,
s
tude
n
t
s
sh
ou
l
d
m
ak
e
a
g
e
n
e
ri
l
aza
ti
on
a
nd
con
s
truc
t
t
h
e
ma
them
at
i
c
al
m
ode
ls
w
h
i
c
h
c
a
n
u
se
f
or
s
o
l
vin
g
o
t
h
er
p
ro
blem
s.
A
c
c
ord
i
n
g
t
o
pri
n
c
i
p
l
es
w
hi
ch
w
e
r
e
s
t
a
te
d
by
G
r
a
v
em
eijer
[4
]
,
t
he
act
i
v
itie
s
w
e
ren’t
su
ita
b
l
e
f
or
c
om
pl
e
t
i
n
g
ma
them
at
i
z
ati
o
n
proc
ess
in
t
he
curr
ent
re
seac
h.
F
rom
thi
s
r
esu
l
t
,
i
t
ca
n
b
e
s
u
gges
t
e
d
t
h
a
t
pr
o
sp
ec
tiv
es
p
ri
ma
ry
t
ea
ch
ers
n
e
e
d
t
o
do
a
n
d
prepa
r
e ac
tiv
i
t
i
e
s
b
o
t
h
h
o
riz
o
nta
l
a
nd ve
rt
ic
al ma
t
he
ma
tiz
a
tio
n
; it i
s
beca
use th
is
s
it
ua
tion d
i
rec
tly a
ffe
c
ts
t
ha
t
stude
n
t
s
use
a
n
d con
s
truc
t
form
al m
odels.
O
n
e
gr
ou
p
w
r
ote
an
a
c
tiv
i
t
y
qu
i
t
e
s
i
mi
lar
to
T
r
e
ffer’
s
[
18]
a
nd
A
l
t
un’s
[
1
9
]
p
rob
l
ems
wi
t
hou
t
me
etin
g
t
h
e
m
.
Th
is
g
ro
u
p
r
ea
l
l
y
m
o
t
i
va
t
e
d
a
nd
the
y
w
e
r
e
inter
e
s
t
e
d
in
m
a
t
he
ma
t
i
c
s
t
e
a
c
hi
n
g
.
This
s
i
t
ua
ti
on
show
e
d
t
ha
t
if pros
p
e
c
t
i
ve
pri
m
a
ry
t
e
a
c
h
ers moti
va
te
d,
t
he
y c
o
u
ld
p
repa
re
a
c
v
it
ies su
i
t
a
b
le
f
or RM
E
.
Zu
lkar
d
i
[
2
0
]
sta
t
e
d
t
ha
t
t
h
e
pros
pect
i
v
e
tea
c
h
ers
in
h
i
s
s
t
u
d
y
l
ea
rn
t
RME’s
p
h
i
lo
so
phy
an
d
co
ul
d
w
r
ite
a
ct
iv
it
ies
rega
rdi
n
g
it.
B
ut
i
n
th
e
c
u
rr
ent
stu
dy,
i
t
c
a
n
b
e
sa
i
d
t
hat
a
l
t
h
ou
g
h
p
ros
p
ecti
v
e
tea
c
h
er
s
l
e
arn
t
the
m
e
nti
o
ned
the
o
r
y
,
t
h
ey
h
a
d
d
i
f
fic
u
l
t
ie
s
o
n
w
ri
tin
g
a
c
t
i
v
iti
e
s
o
n
i
t
.
When
t
he
w
ritte
n
ac
tiv
i
t
i
e
s
w
e
r
e
e
x
ami
n
ed
,
ev
en
i
f
t
h
e
a
c
t
ivi
t
y
h
a
d
t
h
e
rea
l
c
o
n
t
e
x
t
p
ro
bl
em
c
h
a
rac
t
e
r
is
ti
cs,
t
e
ache
r
s
co
ul
d
ta
ke
it
as
a
r
ou
t
i
ne
pro
b
lem
.
S
o,
w
hile
a
c
t
i
v
it
ie
s
are
c
a
rr
yi
n
g
o
u
t
,
tea
c
h
ers
h
a
ve
t
o
o
m
a
n
y
ta
sks.
T
he
y
s
h
ou
l
d
b
e
a
n
e
ff
ecti
v
e
gu
ide
for
s
t
ude
nt
s,
o
nly
i
n
t
h
i
s
w
a
y
a
c
t
i
v
i
tie
s
co
u
l
d
be
car
rie
d
o
u
t
a
s
a
RME
ac
tiv
i
t
y.
O
therw
i
se,
t
h
e
a
c
ti
v
i
t
y
is
c
o
nde
mne
d
t
o be
a
r
outine
pro
b
lem
.
S
o, tea
cher
s
sho
u
ld
know
and
u
s
e
t
h
e
pric
inc
i
ple
s
of R
M
E
w
e
l
l
.
5.
CONCL
U
S
ION
The
a
i
m
of th
i
s
re
sear
ch was
t
o i
n
ve
s
t
i
g
ate
whe
t
he
r
t
h
e
o
r
e
t
i
cal
k
now
l
e
d
g
e of RM
E
w
a
s
a
de
q
u
a
t
e for
w
r
itin
g
R
M
E
a
c
ti
v
ities
or
n
o
t
.
To
ta
ll
y
36
a
cti
v
i
t
ies
w
e
re
w
ritt
en
b
y
p
r
os
p
ect
iv
e
t
e
ache
r
s
.
O
n
e
o
f
t
h
em
w
a
s
e
x
cl
ud
ed
f
r
o
m
t
h
e
an
al
y
s
i
s
b
e
c
a
u
se
i
t
wa
sn
’t
a
s
u
b
j
e
c
t
o
f
p
r
i
m
a
ry
m
a
t
h
e
m
a
tic
s.
A
ll
the
w
r
it
ten
co
n
t
e
x
ts
w
e
r
e
p
e
rson
al
.
It
w
a
s
a
sk
ed
p
a
r
ti
cip
a
nt
s
to
s
t
a
rt
p
re
p
a
ri
ng
acti
v
iti
es
w
it
h
a
c
o
nt
e
x
t pr
ob
lem
.
H
ow
ever
, base
d on
t
he
ana
l
ys
is,
i
t
w
a
s
o
bserve
d
tha
t
n
o
t
a
ll
the
c
o
n
t
e
x
ts
w
e
r
e
exac
tl
y
re
al
l
ife
co
nte
x
ts.
It
w
as
d
et
er
mine
d
tha
t
pros
pec
t
i
v
e
te
ac
her
cou
l
d
n
o
t
dist
i
ngu
ish
r
o
u
t
i
n
e
w
o
rd
p
ro
blem
s
f
r
o
m
rea
l
life
pro
b
l
em
s.
T
he
acti
v
it
ies
w
r
itte
n
b
y
p
ro
spec
ti
ve
t
ea
c
h
er
s
w
e
re
f
am
ili
a
r
f
or
p
rima
ry
s
tu
de
n
t
s.
N
ea
rly
ha
lf
o
f
th
e
c
o
n
t
e
x
t
s
w
e
r
e
r
eal
con
t
e
x
t
pr
ob
le
m
s
.
Less
tha
n
h
al
f
of
t
he
c
o
n
t
ex
ts
r
equ
i
red
no
n-r
ou
tine
s
t
ra
t
e
g
i
e
s
.
G
e
ner
a
l
l
y
r
o
u
t
ine
str
a
teg
i
es
w
e
r
e
e
no
ug
h t
o
s
ol
ve
t
he
p
ro
blem
s. Bu
t
th
i
s
is
n’t e
n
ou
g
h
f
or R
M
E. In
RM
E, i
t
i
s
e
xp
e
c
t
e
d
st
ud
en
t
s
t
o
ex
pl
ain
situa
t
io
ns i
nform
a
lly a
n
d
use
n
o
n
-
r
ou
t
i
ne
pr
o
b
l
em
sol
v
i
ng
st
r
a
t
eg
i
e
s.
H
a
lf
o
f
t
h
e
w
r
i
tte
n
a
c
t
iv
itie
s
e
n
ab
le
d to
u
se
c
onc
re
te
m
ateri
a
l
s t
o
so
l
ve
p
r
o
ble
m
.
Ma
ny o
f
t
he
m
w
e
r
e
like
ga
me
s
an
d
beca
use
of
t
hat
rea
s
on
i
t
c
a
n
b
e
sai
d
t
ha
t
th
e
y
r
e
q
u
i
red
usin
g
phys
ica
l
m
odel
s
.
Wh
il
e
the
y
requ
ire
d
ph
y
s
i
cal m
ode
ls,
the
y
di
d
n’t g
i
ve
o
pp
ort
uni
t
y
to
m
ove
ma
them
ati
c
a
l
m
ode
ls.
Co
n
s
i
d
erin
g
t
h
e
v
i
e
w
poi
nt
o
f
l
e
arni
ng
domai
n
s
,
i
t
wa
s
se
en
t
h
a
t
t
h
ey
w
rote
c
on
te
xt
s
inc
l
ud
in
g
nume
r
ic
o
pera
tio
ns.
A
c
t
i
v
itie
s
for
c
once
p
tua
l
l
e
a
rni
n
g
w
e
re
n
o
t
w
rit
t
e
n
.
F
o
r
e
x
am
pl
e
,
g
eo
m
e
try
subje
c
t
s
gi
ve
stude
n
t
s
op
p
u
rtu
n
ities
us
i
ng
c
oncre
te
m
a
t
erial
s
w
he
n
c
o
mpa
r
ed
w
ith
o
the
r
d
o
m
ai
n
s
l
ike
nu
mb
ers,
a
ri
t
h
me
t
i
c
a
n
d
e
t
c.
I
t
wa
s
exp
e
ct
e
d
f
o
r
p
ro
sp
ec
ti
v
e
t
e
ach
e
r
s
t
o
w
ri
t
e
m
o
r
e
t
ha
n
one
a
c
t
ivi
t
y
o
n
ge
om
etry
s
u
b
j
ec
ts.
Bu
t
the
y
o
nl
y
w
r
ot
e
3
ac
t
i
v
i
t
i
e
s
rega
rdin
g
th
a
t
i
ssue
;
i
n
ad
di
t
i
o
n
o
n
l
y
o
n
e
o
f
th
em
h
ad
r
ea
l
c
o
nt
ext
pro
b
lem
c
h
ara
c
teris
tic
.
A
c
cordi
n
g
to
a
l
l
t
he
f
in
d
i
n
g
s
o
f
th
i
s
s
t
u
dy,
i
t
c
a
n
b
e
sai
d
t
ha
t
t
h
eore
t
i
c
a
l
k
n
ow
le
dge
o
f
R
e
al
ist
i
c
Ma
them
at
ics
E
duca
tio
n
w
a
s
i
n
ade
q
ua
t
e
f
or
w
riti
ng
RME
a
c
ti
v
ities
.
A
l
t
h
ou
g
h
s
om
e
of
t
he
w
ri
tten
ac
t
i
v
i
t
i
e
s
Evaluation Warning : The document was created with Spire.PDF for Python.
IJERE
I
S
S
N
:
2252-
88
22
Writte
n
by Pro
s
pec
t
i
v
e P
r
im
a
r
y
Te
achers on
Real
ist
i
c
Ma
t
h
e
m
at
i
c
s Ed
uca
t
io
n (H
at
ice
Ku
bra
G
u
ler)
23
5
w
e
r
e
s
uita
b
l
e
for
the
t
h
e
o
ry,
pros
pec
t
i
v
e
pr
im
ar
y
teac
hers
n
e
e
d
e
d
t
o
pr
ac
tice
o
n
w
rit
i
ng
acti
v
i
t
ies
r
e
g
a
r
d
i
ng
the m
e
nt
i
o
ned
the
o
r
y
.
REFE
RENCES
[1]
Org
a
nis
a
ti
on
f
or
E
con
o
mi
c
Co-operat
i
o
n
a
nd
D
evel
opm
en
t
(OE
C
D)
.
“
PISA 20
15
a
s
s
e
ssment
and
analytica
l
fr
am
e
w
o
r
k: s
c
ie
n
ce,
rea
d
i
n
g
,
math
ema
t
i
c
an
d financi
a
l
li
ter
a
cy
”.
O
ECD
pu
blis
hing,
2
0
16.
[2]
N
at
ion
a
l
Cou
n
c
i
l
of
T
eachers
of
M
athe
m
a
tics
(NCT
M
)
.
“
P
r
inci
ples
a
n
d
standa
rd
s
for s
c
hool
m
a
th
ema
t
i
c
s
”.
Re
sto
n
, Va
.
:
NC
TM,
20
00
.
[3]
Cot
t
on, T. "M
a
th
emat
ics teach
ing i
n
t
he real wo
r
ld
."
Is
sues in
math
ematics teaching
.
Routl
e
dge,
P
p.
39-5
3
,
2
0
02.
[4]
Gravemei
jer,
K
.
“
D
evelo
p
in
g real
is
ti
c math
e
m
a
t
ics
edu
c
at
ion
,”
F
reud
en
t
h
al
In
s
t
i
t
u
te,
Utrech, 1
99
4.
[5]
Grav
emei
jer,
K
.
and
D
o
o
r
m
a
n,
M
.
"Co
n
t
e
xt
p
ro
b
l
em
s
i
n
r
eali
s
tic
m
a
th
em
atics
e
d
u
c
at
io
n:
A
calc
u
l
u
s
cours
e
a
s
a
n
exam
p
l
e."
Educational
s
t
udi
e
s
in mat
h
ema
tics
3
9(1
-
3).
P
p
.
1
11
-12
9
,
1
99
9.
[6]
Grav
emei
jer,
K
.
&
Doo
r
m
a
n,
M
.
"Co
n
t
e
xt
p
ro
blems
i
n
r
ealist
i
c
mat
h
em
ati
c
s
educat
io
n:
A
cal
cu
l
u
s
cou
r
s
e
a
s
an
exam
p
l
e,"
Ed
uca
tio
nal Stu
d
i
e
s
in M
a
t
h
em
a
t
i
c
s,
v
ol/is
sue: 39(1-3). Pp
.
111-129
,
1
9
99.
[7]
Grav
emei
jer,
K
.
"
H
ow
e
m
e
rg
e
n
t
m
o
de
ls
m
ay
f
oster
th
e
consti
t
uti
o
n
of
f
orm
a
l
m
a
t
h
em
ati
c
s,
"
Mathematic
al
T
h
i
n
kin
g
and
L
e
ar
ni
ng,
v
o
l/issu
e:
1
(2),
155
-17
7
,
199
9.
[8]
Çiling
i
r,
E
.
“
T
h
e ef
fect o
f
th
e re
al
is
ti
c ma
t
h
em
atic edu
c
ati
o
n
(
r
me
)
a
p
p
r
oa
ch o
n
vis
u
a
l
ma
th
lit
e
ra
cy
s
e
lf
e
ffi
cacy
per
cepti
on
s an
d
pr
ob
lem
s
o
l
v
i
n
g
achi
evement
of 4th g
r
ad
e
studen
t
s
”
,
Unp
ublished
master
t
hesi
s,
Ç
ukurova
Unive
r
sit
y
:
Ad
ana,
201
5.
[9]
Özkaya,
A.
“
The effect
s of realistic mathematic
s
educati
o
n on st
udent
’
s
achivemen
ts, a
ttitudes
and sel
f
repor
t
s
in
5t
h gra
d
es
mathe
m
at
ics lesso
n
”,
U
np
ub
lis
h
ed
m
ast
e
r th
esis,
G
azi U
ni
versity:
Ankara,
2
016
.
[10
]
V
an
D
ijk,
I.
M
.
A.
W
.
,
e
t
al.
“S
t
r
at
eg
i
c
l
earnin
g
i
n
p
r
im
a
ry
m
athem
a
t
i
cs
e
du
cati
on:
E
ff
ects
of
a
n
exp
e
rim
e
nt
al
program i
n
mod
elling,”
Edu
c
atio
na
l
R
e
sear
ch and
Eva
l
u
a
tion
,
vol
/i
ss
ue:
9(2).
P
p
.
16
1-18
7,
2
0
03.
[11
]
V
an
d
en
H
eu
vel
-
P
a
nhu
izen
,
M
.
“
The d
i
dacti
cal
u
se of
m
o
dels
i
n
real
isti
c
m
a
th
e
m
atics
edu
cati
on:
A
n
ex
am
p
l
e
f
r
om
a lo
ng
it
udinal
t
r
a
j
ecto
r
y o
n
p
ercent
a
g
e
,”
E
duca
tion
a
l
S
t
udies
in
Mathematics,
v
o
l/issue
:
54
(
1).
Pp
.
9
-
35
,
20
03
.
[12]
Kurt,
E.
S
.
Ger
ç
ekçi mat
e
ma
tik e
ğ
i
t
i
m
inin
u
z
un
luk ö
l
çme
konu
su
nd
a b
a
ş
ar
ı
ve
kal
ı
c
ı
l
ı
ğ
a e
t
kisi
.
Unpu
blish
e
d
mas
t
er
the
s
is,
On
do
ku
z
Ma
yıs Un
iv
e
r
si
ty
:
Sa
msu
n
, 20
1
5
.
[13]
F
a
uzan
,
A
.
Appl
yi
ng realis
t
i
c
mathematics educat
ion (
R
ME
)
in
t
e
achi
ng
g
e
om
etry
in
End
o
n
e
sia
n
pr
im
a
r
y s
c
hoo
ls
.
Do
ctoral
dis
s
e
rtat
io
n
,
U
ni
versit
y
of Twe
n
t
e
:
E
nsc
h
e
d
e
, 20
0
2
.
[14]
Bo
no
tto
,
C
.
Ho
w
in
fo
rma
l
o
u
t
-o
f-sc
ho
ol
m
a
t
h
e
ma
tic
s
c
a
n
h
e
l
p
stu
d
e
nts
m
a
ke
s
en
se
o
f
f
o
rm
a
l
i
n
-
school
m
a
th
e
m
atics:
t
he
cas
e
o
f
m
u
l
tip
lyi
n
g
b
y
d
eci
m
a
l
nu
m
b
ers.
M
a
thema
t
i
c
al T
h
inki
n
g
&
L
e
ar
n
i
ng
,
7
(
4
)
.
Pp
.
31
3-3
4
4
,
20
05
.
[15]
Tunalı,
Ö
.
K.
Aç
ı
kavr
am
ı
n
ı
n g
e
rçekç
i m
a
t
e
ma
ti
k
ö
ğ
r
e
timi
ve yap
ı
land
ı
rm
ac
ı
k
u
ra
ma
gö
re
ö
ğ
retimi
n
i
n
kar
ş
ı
la
ş
t
ı
r
ı
lma
s
ı
.
Un
pu
bl
ish
e
d m
a
ste
r
the
s
i
s
, Ulu
d
a
ğ Unive
r
s
i
ty
:
Bu
rsa
, 20
1
0
.
[16]
Uça,
S
.
Ö
ğ
ren
c
il
eri
n
on
dal
ı
k
k
e
s
i
rleri
anla
m
land
ı
rm
a
s
ı
n
d
a
ge
rçekçi
mat
e
ma
tik
e
ğ
it
i
m
i ku
l
l
an
ı
m
ı
:
bi
r
ta
sa
r
ı
ar
a
ş
t
ı
rmas
ı
.
U
n
p
ubl
is
hed
m
a
st
e
r
t
h
e
sis,
A
dn
an
M
en
deres
U
n
i
v
ers
i
ty
:
Aydın
,
2
014.
[17
]
K
w
on,
P
.
“
H
o
p
e
,
d
ef
ens
e
m
echa
n
is
m
s
,
an
d
ad
ju
s
t
m
e
n
t
:
Im
p
lic
ati
o
n
s
f
o
r
f
als
e
hope
a
n
d
d
efen
si
v
e
h
op
eless
n
es
s,
”
J
o
urna
l o
f
Pe
rso
n
a
l
i
t
y
,
v
o
l/is
su
e
:
7
0(2
)
.
P
p
.
2
0
7
-231,
2
0
0
2
.
[18
]
T
reff
ers,
A
.
"Meet
in
g
i
n
n
u
m
e
racy
a
t
p
r
im
ary
sch
ool
,"
Edu
c
at
io
na
l S
t
u
d
ie
s in
Mathematics,
v
o
l
/
i
s
s
ue
:
2
2
(
4)
.
P
p
.
33
3-3
52,
1
9
9
1
.
[19
]
A
ltu
n,
M
.
“
S
ayı do
ğrus
un
un
ö
ğ
r
etim
i
n
d
e
y
en
i
bir
yakl
aşım
,
”
İ
lkö
ğ
retim On
line,
vol/is
sue: 1(2).
P
p.
3
3-39, 2002.
[2
0]
Z
ulka
rdi
,
Z
.
Devel
op
in
g a l
e
arn
i
n
g
en
viro
nment
o
n
r
e
alistic
m
a
th
ema
t
i
c
s edu
c
at
ion
for Indon
esi
a
n
stu
d
en
t
teachers
.
D
oc
t
o
ral
Dis
s
ert
a
ti
on
, Un
i
v
e
si
ty
o
f
T
w
en
te:
Ens
c
hed
e
,
2
002
.
B
I
BL
IOGRAPHY OF AUTHO
R
A
ssis
t
.
P
r
of.
D
r
.
Hatice
Küb
r
a
Gü
ler
i
s
c
o
m
pl
e
t
ed
h
er
B
A
D
e
g
r
ee
a
t
Do
kuz
E
y
l
ul
U
n
i
v
e
rsity,
Bu
ca
F
a
cul
t
y
of
E
d
u
cati
on,
Elemant
ary
M
a
t
h
em
at
ics
Ed
ucati
o
n
Pro
gram
a
n
d
c
o
m
pl
eted
h
er
M
A
D
e
gree
at
G
azi
U
ni
versi
t
y
on
th
e
sam
e
p
rogram
.
S
h
e
to
ok
her
P
h
D
D
eg
rees
a
t
Ul
udag
U
n
iv
ersi
ty,
E
l
e
m
ent
a
ry E
du
ca
tio
n
Pro
gram
. S
h
e h
a
d b
een at
S
i
en
a
Un
i
v
e
rsit
y for s
i
x m
onths
as
a
P
h
D
stu
d
ent
by
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he
h
elp
o
f
E
RAS
M
US
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t
u
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t
E
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c
hang
e
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r
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ram
.
H
e
r
res
earch
i
nt
e
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es
ts
a
re
m
a
th
em
ati
c
s
t
eacher
t
rain
in
g,
m
athem
a
t
i
cal
l
i
t
eracy,
ef
f
ecti
v
e
mat
h
em
ati
c
s
teachi
n
g
,
g
eo
m
e
try
t
eachin
g an
d m
a
t
h
em
atical ab
s
t
r
acti
on.
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