European Journal of Education Studies
ISSN: 2501 - 1111
ISSN-L: 2501 - 1111
Available on-line at: www.oapub.org/edu
Volume 3 │ Issue 1 │ 2017
doi: 10.5281/zenodo.221098
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING
OF THE INTEGERS IN THE MIDDLE SCHOOLS
Mevhibe Kobak Demir1i, Nursen Azizoğlu2, Hülya Gür3
Research Assistant, ”alıkesir University, Faculty of Necatibey Education, ”alıkesir, Turkey
1
Assist. Prof. Dr., ”alıkesir University, Faculty of Necatibey Education, ”alıkesir, Turkey
2
Prof. Dr., ”alıkesir University, Faculty of Necatibey Education, ”alıkesir, Turkey
3
Abstract:
This study aims to determine the difficulties that middle school mathematics teachers
have in teaching integers, the analogies they use to solve these difficulties, and the
efficiency of these analogies. The study was conducted with 10 mathematics teachers, 6
females and 4 males, who were working in middle schools affiliated with the Ministry
of National Education (MNE), selected using the convenience sampling method. The
study used the case study design among the qualitative research methods. The data
were collected using open-ended questions and analyzed using descriptive and content
analysis. The findings were presented under three categories which are the difficulties
they have in teaching integers, the curriculum on the subject of integers, and the analogies
found in textbooks or used by teachers and their efficiency. Recommendations were made
based on these findings. Centralized Early Childhood Development Policy of the early
childhood education.
Keywords: analogy, difficulty, teaching integers, curriculum, mathematics text books
1. Introduction
The new world of constant and rapidly increasing information combined with
technological developments produced a social order based on knowledge and
information, and this social order expects different skills from individuals today (MEB,
2009). The increased necessity of using and understanding mathematics in daily life and
the negative results obtained on national and international exams (Altun, 2009) has
necessitated Turkey's education system to make changes to the framework of its
i
Correspondence: email mevhibekobak@balikesir.edu.tr
Copyright © The Author(s). All Rights Reserved.
© 2015 – 2017 Open Access Publishing Group
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Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
expectations from students; and accordingly, curriculum have been reorganized based
on this constructive approach. The constructive approach defines learning as an active
process where individuals construct their own interpretation and explanation of new
information they learned from their environment based on their previous knowledge
and experience (Driver and Bell, 1986). According to constructivism, learning new
information depends on prior information, because new information is constructed
upon the previously acquired information; and the prior information is accepted as a
starting point for teaching new information ”ıyıklı, Veznedaroğlu, 5ztepe and Onur,
2008). In this regard, the necessity of connecting the new and prior information for a
more meaningful and permanent learning experience becomes prominent. A way to
facilitate these connections is analogies based on the constructive approach (Glynn and
Takahashi,
Pittman,
Şahin, G(rdal and ”erkem,
Kaya and Durmuş,
2011).
An analogy is the process of understanding the unknown by comparing and
connecting the known and unknown information based on the conditions of the known
G(nay ”ilaloğlu,
. “ccording to another source, an analogy can be defined as a
mapping mechanism that helps students construct the new information based on their
previous knowledge (Parida and Goswami, 2000). In analogies, which are the bridges
between the known prior information and unknown new information Kesercioğlu,
Yılmaz, Huyug(zel Çavaş and Çavaş,
, the known concept is called the source
analog , and the concept being taught unknown is called the target concept “kkuş,
2006). The aim of the analogy is to facilitate the understanding of the target by finding
appropriate similarities between the source and the target G(lçiçek, ”ağı and Moğol,
2003). Glynn (2007) highlights that the target should be identified, the source should be
organized according to the target, the similarities between the target and the source
should be determined, these similarities should be compared, the circumstances under
which the analogy does not work should be identified, and conclusions should be
drawn about the target concept.
Studies show that analogies contribute to solidifying abstract subjects (Günay
”ilaloğlu,
“ykutlu and Şen,
Heywood,
, constructing information
“kkuş,
, eliminating misconceptions “tav, Erdem, Yılmaz and G(c(m.,
,
drawing students' attention to the course (Duit, 1991), learning and developing
scientific ideas and concepts, making comparisons with the real world, increasing
students' motivation and taking students' previous knowledge into consideration
(Dagher, 1995), and improving students' scientific thinking and creativity (Aykutlu and
Şen
. G(nay ”ilaloğlu
stated that analogies also enable students to grasp
certain information and summarize the subjects comprehensibly. However, although
analogies are useful in the learning and teaching processes as mentioned above, they
lead to negative results and inefficiency when used incorrectly.
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Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
Palmquist (1996) stated that a good analogy should include the following
features:
Structural Richness: An analogy should show a variety of relationships between
itself and the other ideas or concepts being compared to in terms of meaning.
be applicable for the concept and not cause misunderstandings.
Applicability: An analogy should include the structure of the relationships and
Appropriateness: An analogy should be appropriate for the target group.
Comprehensiveness: An analogy should be understood by the target group in the
same way.
“kkuş
expressed that teachers should be sure that they make appropriate
and similar connections between the target and analog concepts, and the analog should
help teachers map the similarities and differences of these connections for analogies to
be useful. Kaptan and Arslan (2002) stated the issues that teachers should consider
when using analogies in learning and teaching activities as follows:
Teachers should identify how and for which subject they will use an analogy and
be able to draw students' attention to the analogy by making a plan according to
this identification.
analogies and use visual materials when required.
and connected to the students' prior information from their daily lives.
Teachers should direct and give opportunity to students to create their own
Teachers should ensure that the analogy they use is closely related to the subject
Teachers should ensure that the analogy they use is appropriate for the students'
cognitive and comprehension level.
These issues emphasize the importance of using analogies correctly and
appropriately by teachers to be efficient in teaching.
Analogies are strong strategies in mathematical research (Krieger, 2003). They
play a significant role in mathematical thinking Saygılı,
. One of the subjects in
which students are known to have problems of understanding is the concept of integers
and the mathematical operations with them ”ahadır and 5zdemir; 2013). Studies show
that the natural numbers that previously exist in students' minds can help with learning
positive integers, but negative numbers cannot be learned by observing the physical
world since nonpositive objects or object groups do not exist (Davidson, 1992; Mc
Corkle,
Şeng(l and Dereli,
. Işıksal ”ostan
stated that the biggest
problem with negative numbers is the inability to interpret and comprehend these
numbers. Although students do not have difficulty in placing negative numbers on the
number line, they have difficulty in comparing their superiority. It can be said that
these difficulties and mistakes in comparing negative numbers are based on the
misconception that the features of positive numbers can also be generalized for negative
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Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
numbers. Another difficulty is the calculation errors due to incorrect use of the addition
and minus signs since students cannot understand whether these signs reflect the
numbers' direction or the operation itself. In multiplication and division, students try to
use the generalizations in natural numbers and cannot grasp the conceptual
information behind these operations, which leads them to make calculation errors.
Teachers have an important role in the elimination of these misconceptions that
students have in conceptualizing negative numbers, interpreting the operations made
with these numbers, and the calculation errors they make due to this misconception
Işıksal ”ostan,
. ”ased on this, teachers’ use of analogies in eliminating the
misconceptions about integers and concretizing the relevant concepts is considered to
be effective.
This study aims to determine the difficulties that middle school mathematics
teachers have in teaching integers, the analogies they use to solve these difficulties, and
the efficiency of these analogies. The answers of the following questions were sought in
line with the aim of this study:
What difficulties do mathematics teachers have in teaching integers?
Which analogies are used in the mathematics curriculum and textbooks in the
subject of integers?
Which analogies do mathematics teachers use in teaching integers?
How efficient are the analogies used by the teachers in the subject of integers?
2. Method
This study used the case study design among the qualitative research methods to
answer the study questions and obtain in-depth information Yıldırım and Şimşek,
2008).
2.1 Sample of the study
The study was conducted with 10 mathematics teachers, 6 females and 4 males, who
were graduated from elementary mathematics teaching and working in middle schools
affiliated with the Ministry of National Education (MNE), selected using the
convenience sampling method. Three teachers are older than 30, the others' age is
changed between 25-30 years. Furthermore, two teachers of the participations have an
experience between 5 and 10 years, the others have an experience between 2 and 5
years.
2.2 Instruments and Data Analysis
The data were collected using a questionnaire that included 8 open-ended questions
questioning the difficulties that teachers have when teaching the learning objectives
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Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
related to the integers topic which is included in the mathematics curriculum applied
in the 2013-2014 academic year and the textbooks approved by the MNE for middle
schools. Additionally, the questionnaire asked for the analogies that the teachers use to
resolve these difficulties. This draft questionnaire prepared by the researchers was
finalized according to the corrections recommended by three field experts. In this
questionnaire, firstly, the concept of analogy was defined, then the following examples
of the analogies used in mathematics were shown and finally teachers were asked to
specify the analogies they use in teaching the topic of integers.
As we take from one side of the scales, we take the same quantity of load from the other
side so as not to tip the scales. Similarly, we divide both of the sides of the equation by 60.
A sphere is like an orange.
An acute angle is similar to the angle that occurs when you open a pair of scissors.
15x2: (10x2)+ (5x2):: 14x3 : ______
The coding system developed by Serin-Ergin (2009) to analyze the source-target
match was used to determine the efficiency of the analogies that teachers developed
different from the curriculum and textbooks. Accordingly,
Full Association (FA): This means fully and correctly expressing the similarities
and differences of the source-target relationship.
similarities and differences of the source-target relationship.
concept other than the target.
Partial Association (PA): This means correctly but incompletely expressing the
Incorrect Association (IA): This means associating the source with a different
No Association (NA): This means that the source-target relationship is not
associated with the analogy, i.e. the answer includes statements such as not
related I couldn't associate. or the question was not answered at all.
Association Including Misconception (AIM): This means expressing the sourcetarget relationship in a way that causes misconception.
In this study, the valid, partially valid will be used instead of full association and
partial association. Also, invalid will be used for the incorrect association, no
association and association including misconception.
The data were analyzed upon adding the new themes generated after an indepth analysis of the themes and categories determined by reviewing the relevant
literature, and the results were comprehensibly interpreted, including direct quotations
as a way to reflect the themes. The teachers' names were not given in these direct
statements to protect their identity; instead, they were coded with numbers in
parentheses.
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Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
The mathematics curriculum and textbooks for the sixth and seventh grades
were the other sources of data. The documents were analyzed using the content and
descriptive analysis methods Yıldırım and Şimşek,
.
3. Findings
3.1 The Mathematics Teachers' Opinions on the Difficulties They Have in Teaching
Integers
Table 1 shows the themes and subthemes generated from the middle school
mathematics teachers' opinions on the difficulties they have in teaching integers.
Table 1: Themes and Subthemes Regarding the Middle School Mathematics Teachers' Opinions
on the Difficulties They Have in Teaching Integers
Themes
Subthemes
Interpreting integers and showing them on
the number line
Not having difficulty
Inability to comprehend negative integers
Inability to show negative integers on the number line
Determining and explaining the absolute
value of a integer number
Not having difficulty
Inability to extract negative integers from absolute value
Comparing and putting integers in order
Inability to comprehend that -1 is the biggest negative
integer number
Not paying attention to the signs before integers
Inability to put negative integers in order
Adding and subtracting integers
Not having difficulty
Having difficulty in adding integers with different signs
Comprehending
that
subtraction
in
integers means to add the opposite sign of
the minuend
Memorizing the rule
Inability to understand the reasons
Inability to understand the concept of the negative of
negative integers
Using the features of addition as strategies
for a fluent operation
Having difficulty in understanding the order of operations
Having difficulty in distributing the negative sign before
the parentheses into them
Multiplying and dividing integers
Not having difficulty
Inability to comprehend that the division and
multiplication of numbers with the same sign will always
result in a positive sign
Miscomprehending multiplication
Indicating the repeated multiplication of
integers as exponential numbers
Not having difficulty
Perceiving repetitive multiplication as addition
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Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
Having difficulty in exponentiating according to the
location of the parentheses when exponentiating the
numbers
Inability to identify the sign of the result of the even and
odd exponents of negative numbers
Table 1 indicates, based on the teachers' opinions on the curriculum addition of
Interpreting integers and showing them on the number line , that students have difficulty
in comprehending negative numbers and showing them on the number line. A teacher
considered that he did not have difficulty on this issue and said I don't have difficulty in
the comprehension of integers and showing them on the number line. Because students start the
sixth grade as already having learned the natural numbers, they can transfer their information
on natural numbers into integers. T
The statements of some teachers who consider that they have difficulty in
teaching integers are presented below:
Negative numbers are difficult to understand. Students ignore the minus sign before the
number as they think they are positive numbers. (T1)
They place negative numbers on the number line as -1, -2, -3, -4... from left to right
towards zero. Namely, they cannot show negative numbers on the number line. (T2)
I don't generally have difficulty in positive integers, but students have difficulty in
placing negative numbers on the number line. (T6)
These statements reveal that students can use their knowledge about natural
numbers on positive integers but have difficulty in comprehending negative numbers
and showing them on the number line.
T5, who thought that he did not have difficulty with the curriculum addition of
Determining and explaining the absolute value of an integer number, said There is no
problem when the absolute value of integers are associated with problems in daily life and are
well-exemplified. I did not have difficulty in this subject. The statements of some teachers
who emphasized the difficulty in extracting negative integer number from the absolute
value are presented below:
They cannot comprehend that the absolute value changes the sign; they extract negative
numbers as negative. T
They learn to extract negative numbers from the absolute value as positive, but they
begin to extract positive numbers as negative. T
It can be concluded from the teachers' statements in Table 1 that all teachers have
difficulty with the curriculum addition of Comparing and putting integers in order. It is
seen, according to the subthemes, that the difficulties mostly focus on the inability to
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Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
comprehend that -1 is the biggest negative integer number, ignoring the signs before the
numbers, and the inability to put negative numbers in order. Some statements of
teachers on the difficulties teachers have are as follows:
It takes time for them to understand that -1 is the biggest negative integer number and
the others are smaller. They make the operations with the information they memorized.
(T9)
Students generally do not pay attention to the sign before integers when comparing
them. (T5)
Students think that negative integers are getting bigger as they get away from zero.
(T2)
The subthemes generated from the opinions on the addition and subtraction of
integers are not having difficulty and having difficulty in the addition of integers with
different signs. The teachers who did not have difficulty stated that this is due to the
information that students acquired about addition and subtraction in natural numbers.
The statement of one teacher who did not have difficulty is presented below:
Actually, the students use their knowledge about addition and subtraction in natural
numbers when adding and subtracting integers, only the number needs to be highlighted
when adding and subtracting two negative numbers. T
Some statements of the teachers who have difficulty with the addition of integers
with different signs are as follows:
They do addition without seeing the negative number when adding positive and
negative numbers at once. (T4)
There is no problem when adding two positive integers. But they have difficulty in
adding negative and positive numbers or two negative numbers. (T8)
According to the Table 1 for the curriculum addition of Comprehending that
subtraction in integers means to add the opposite sign of the minuend, students memorize
the rule and cannot understand the reasons of the addition or the concept of the
negative of negative integer number. The difficulty with this addition is the inability to
add the integers with different signs. The statements of the teachers who have difficulty
in the said curriculum addition are as follows:
Students can only memorize this addition as a rule. (T1)
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Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
Comprehending subtraction is an important addition, but they cannot understand why
they are doing it. (T6)
They cannot make the connection of negative of the negative. They make calculation
errors when they distribute the minus before a negative number. (T7)
The subthemes generated from the opinions of the teachers on the difficulties
they have with the curriculum addition of Using the features of addition as strategies for a
fluent operation are having difficulty in understanding the order of operations and
distributing the negative sign before the parentheses into the parentheses. All teachers
had difficulty with this addition, and some of their statements are as follows:
They often do multiplication, division, and even addition or subtraction before doing the
operations in the parentheses. (T4)
Students have difficulty in giving priority to the operations in parentheses when solving
the problems. T
They forget to distribute the minus sign before the parentheses into the parentheses.
(T3)
Another problem that teachers have in teaching integers concerns multiplication
and division. While T , who stated that he did not have difficulty, said No problems
occur when they understand multiplication, the subthemes generated from the opinions of
the teachers who had difficulties are the inability to comprehend that the division and
multiplication of numbers with the same sign will always result in a positive sign and
miscomprehending multiplication. Some of the teachers' statements under the said
subthemes are presented below:
They make incorrect generalizations about the fact that the multiplication and division
of numbers with the same signs will be positive. (T10)
They have difficulty in understanding multiplication. They even do addition
sometimes. (T5)
The last learning objective indicated in the mathematics curriculum about
integers is Indicating the repetitive multiplication of integers with themselves as exponential
numbers. The difficulties with this addition are perceiving repetitive multiplications as
additions, having difficulty in exponentiating according to the location of the
parentheses when exponentiating the numbers, and the inability to identify the sign of
the result of the even and odd powers of negative numbers. Some statements of the
teachers on this learning objective are presented below:
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Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
Students perceive repetitive multiplication of exponential numbers as repetitive
additions. (T8)
They are confused whether to take the power in or out of parentheses when the exponent
is in parentheses. Especially for negative numbers, this confusion causes calculation
errors since the exponent's being in or out of parentheses affect positivity or negativity.
(T2)
They can exponentiate positive numbers. They have difficulty in identifying whether the
result is negative or positive. (T6)
3.2 The Analogies Used for Integers in the Middle School Mathematics Curriculum
and Textbooks
Table 2 shows the analogies used for integers in the mathematics curriculum applied
from the 5th to 8th grades and in the mathematics textbooks advised by the MNE for
the 2013-2014 educational year.
Table 2: Analogies Used for Integers in the Mathematics Curriculum and Textbooks
Objectives
Analogies in the
curriculum
Interpreting
integers and
showing them on the
number line
Stating the floors in an
elevator
Air temperatures below and
above zero
Sea level
Thermometer
Elevator
Steps forwards and backwards
Profit-loss relationship
Determining and explaining
the absolute value of a
integer number
Associating
with
real
situations using elevators,
thermometers,
or
bank
accounts
The distance between ants moving in
opposite directions from a starting point
The relationship between the object and
the image in a mirror
Comparing the depths that a diver dives
Comparing and
integers in order
-
Air temperature
Debit-credit relationship
Profit-loss relationship
Associating the tools such
as
an
elevator
or
thermometer with vertical
and horizontal number line.
The backwards and forwards moves of a
seismic search and rescue ship produced in
Turkey for a Norwegian firm is associated
with addition and subtraction in integers
Addition and subtraction are also
associated
with
debit
and
credit
relationship by modeling with counting
pieces
Addition and subtraction in integers are
explained through air temperatures in
Ankara and the foothills of a mountain
Modeling
pieces
---
Adding
integers
and
Comprehending
subtraction in
putting
subtracting
that
integers
with
counting
European Journal of Education Studies - Volume 3 │ Issue 1 │ 2017
Analogies in the textbooks
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Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
means to add the opposite
sign of the minuend
Using the features of
addition as strategies for a
fluent operation
-
---
Multiplying
integers
dividing
-
Multiplication in integers is associated
with finding the depth a diver dives in 3
minutes by giving the depth that diver
dives in two minutes
Determining the scores they obtain from a
test (multiplication)
The profits and losses of a company is used
(multiplication and division in integers)
The change occurring in the temperature of
a substance (multiplication)
Likening division to multiplication and
transferring the features of multiplication
into division (finding that the answer of
the operation 18:6=? is three through the
operation of 6x?=18)
Indicating
the
repeated
multiplication of integers as
exponential numbers
-
Likening to the number pattern of marbles
being put in boxes for a competition in an
electronic appliances store
Likening to finding the number pattern
through modeling of a rectangular prism
consisting of unit cubes
Associating the reproduction of bacteria
with the exponents of integers
and
Integers are addressed in two sections in different grades in the curriculum
prepared based on the constructive approach. The comprehension, comparison,
addition, and subtraction of integers are addressed in the 6th grade while the
multiplication and division of integers are addressed in the 7th grade. Thus, students
should understand the concept of integers in the 7th grade. While it is recommended in
the mathematics curriculum that in the 6th grade integers be associated with elevators,
air temperature, thermometers, and bank accounts, no analogies were found in the
addition of multiplication, division and exponentiation of integers in the 7th grade. This
may be to prevent miscomprehension about the concept of integers and make students
meaningfully construct the concept of integers.
According to the Table 2 the textbooks included analogies created by associating
integers with sea level, thermometers, elevators, steps forwards and backwards, profitloss or debit-credit relationships, air temperature, and number patterns. These analogies
are similar to the analogies recommended in the curriculum, and differently from the
curriculum, textbooks include analogies for multiplication, division, and exponentiation
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Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
of integers. For the curriculum learning objectives above, the analogies of number
patterns and the field are different from the other learning objectives. Another highlight
is that the curriculum and the textbooks do not include analogies for the learning
objectives of subtraction and addition with the opposite sign of the minuend in integers.
3.3 The Analogies Used by Mathematics Teachers in Teaching Integers
Table 3 shows the analogies used by mathematics teachers in teaching integers and the
efficiency of these analogies. In the Table 3, the analogies used by the teachers but
similar in the curriculum and textbooks are showed as T-C. Also, the coding system
developed by Serin-Ergin (2009) to analyze the source-target match was used to
determine the efficiency of the analogies that teachers developed different from the
curriculum and textbooks. According to the this coding system, Full Association is
called as a valid and partially association is called a partially association.
Table 3: The Analogies Used by Mathematics Teachers in Teaching Integers and the Efficiency
of These Analogies
Curriculum
objective
Learning
Interpreting
integers and
showing them on the number
line
Determining and explaining
the absolute value of a integer
number
Comparing
integers
and
ordering
The Analogies Used
Efficiency
Floors in a shopping center (T1)
T-C
Temperature (T9)
T-C
Depth or altitude according to the sea level (T7)
T-C
Credit (+) and debit (-) relationships (T8)
T-C
Thermometers (T2, T6)
The minus reflect the contrasts, therefore the number gets
smaller as it gets bigger (T3)
T-C
Partially
Valid
Elevators (T4, T6)
T-C
I choose a certain point in the class and state that it is
ahead of or behind where I stand (T5)
T-C
The distance of fish at various depths under the sea level
(T1, T7)
T-C
I liken the absolute value to a washing machine. If we put
dirty clothes (-), we take them as clean (+). If we put clean
clothes (-), we take them as clean again (+). (T2, T3, T4,
T10)
Valid
I determine the starting point on the number line as home
and associate the absolute values of integers with the
distance between home and any negative and positive
numbers (T6)
T-C
Credit>debit (T1).
T-C
I associate integers with air temperature, asking if we
chill more at -1°C or at -10°C and which one is colder (T2,
T4, T10)
T-C
The minus reflect the contrasts, therefore the number gets
smaller as it gets bigger (T3)
Partially
Valid
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Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
I categorize integers as negative- and positive-signed and
then associate them with debit and credit relationships
(T5, T8, T9)
T-C
Going up from one floor to another in a building (T1)
Valid
The + sign represents the credits. I have a credit of +5 TL
from a friend and +10 TL from another friend; then I have
a total credit of +15 TL. And I explain negative numbers as
debts. (T2, T5, T7)
T-C
In the operation of 4+ (-8), the minus and plus cannot get
along, so the sign in the middle will be minus.
In the operation of 4+ (+8), the same signs get along, so the
sign in the middle will be plus (those who get along will
be + and who do not get along will be -) (T3)
Valid
Blowing up the + and - balloons (T4)
Valid
Using counting pieces for + and - integers (T6)
T-C
I associate negative numbers with holes in the soil and
positive numbers with filling these holes (T10)
Partially
Valid
I use the concept of zero pair (T2, T9)
Valid
In the operation of +4-(-8), the minus and minus get along,
so the sign in the middle will be + (Two signs cannot be
side by side) (T3)
Valid
I use the method of blowing up the + and - balloons (T4)
Valid
I ask the students to think of subtraction as if it was
addition and put the sign of the integer number with
bigger number value to the beginning of the operation
(T6, T7)
Valid
Comprehending
that
subtraction in integers means
to add the opposite sign of the
minuend
I explain by modeling with integers (T2)
T-C
Valid
Using the features of addition
as strategies for a fluent
operation
The parentheses are our apartment, and the exponent
concerns everything in the building as if it is out of the
apartment. But it concerns only the flat with that number
if it is in the apartment. It does not intervene the sign. (T2)
Valid
I tell the students to consider themselves as doctors to
make them pay attention to the signs' distribution. I tell
them to pay attention to the parentheses just as they
should be careful not to hurt the patient’s heart in a
surgery if they were doctors.
Valid
Associating the operation of (+10)-(-2)=+10+2=+12 with the
operation of (+5)-(-1)=+5+1=+6 (T6)
Valid
I use the example of my friend's friend is my friend or
my enemy's enemy is my friend
by associating friends with positive numbers and enemies
with negative numbers (T1, T2, T4)
Valid
Adding integers
(this curriculum addition was
analyzed in two sections)
Subtracting integers
(this curriculum addition was
analyzed in two sections)
Multiplying
integers
and
dividing
I use the example of a car moving (T6)
European Journal of Education Studies - Volume 3 │ Issue 1 │ 2017
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Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
Indicating
the
repeated
multiplication of integers as
exponential numbers
Each number is as powerful as its exponent and
multiplied as much (T3)
Valid
Associating with a unicellular's reproduction (T4)
T-C
I try to draw their attention to which number is multiplied
and how many times it is writted (T5, T7)
T-C
** For example T5 means teachers.
It was observed that the teachers mostly used the analogies in the textbooks and
the curriculum when teaching integers. The analogies used were focused on
temperature, credit-debit relationships, thermometers, elevators, moves forwards or
backwards, counting pieces, modeling with zero pairs, and using mathematical
patterns. The majority of the analogies created and used by the teachers except for the
analogies in the curriculum and textbooks were found to be sufficient. However, the
analogy of the minus reflect the contrasts, therefore the number gets smaller as it gets
bigger used by teachers to show integers on the number line and compare them was
partially valid. The target and source relationship for the analogy was expressed
correctly but incompletely, which led to a partial association. Students may have
difficulty in perceiving the biggest negative number, -1, if they cannot comprehend the
number value in this analogy. In addition, an unclear teaching method can lead to
alternative concepts and misconception when comparing positive and negative
numbers.
Another partially valid analogy created by the teachers is associating the holes in
the soil with negative numbers and filling these holes with positive numbers when
adding integers, which was used by the teacher coded as T10. This analogy was fully
associated with the addition of integers with different signs (positive and negative),
however, it is considered to lead to misconception and calculation errors in the addition
of integers with the same sign (positive and positive or negative and negative) since
students will always be focused on filling the holes. For example, in the addition of two
negative integers such as the operation of (-2)+(-3)=?, students can perceive addition of
3 holes to 2 holes as filling the holes, and cannot find the correct result if they do not
pay attention to the second number. Therefore, explaining the similarities and
differences in the target and source relationship in analogies considering all aspects of
the subject will prevent misunderstanding. In this regard, it is important that teachers
have sufficient field information and knowledge in creating analogies, and that they use
these analogies correctly and appropriately.
4. Conclusion and Discussion
This study analyzed the difficulties that middle school mathematics teachers have in
teaching integers, the analogies they use to eliminate these difficulties, and the
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Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
efficiency of these analogies under three titles which are the difficulties they have in
teaching integers, the analogies used for integers in the curriculum and textbooks, and the
analogies used by the teachers and their efficiency.
Although the teachers stated that students have no difficulties in learning
positive integers, they state that students have difficulty in comprehending and
comparing negative integers and showing them on the number line. Students also have
difficulty in the operations with negative integers; they have problems in the addition
of numbers with different signs and the multiplication and division of negative
numbers. In addition, they cannot understand the concept of the negative of a negative
integer number and make calculation errors in the result being positive or negative
according to the parentheses and the exponent when exponentiating negative integers.
These errors may mainly be because students have difficulty in interpreting and
solidifying the concept of negative integers. If the concept of negative integers is not
sufficiently comprehended in the 6th grade, then it negatively affects the subject of
operations with integers and becomes a serious problem for future learning.
The literature indicates that the subject of integers, taught in middle school for
the first time, is one of the subjects that is difficult to learn due to its abstract structure
Şeng(l and Kör(kç(,
. Işıksal ”ostan
stated that while the students have no
difficulty with positive integers, they have difficulty in interpreting and comprehending
negative numbers. They can associate positive integers with real objects, but some
features of negative numbers conflict with the perception of counting numbers
(Linchevski and Williams, 1999). This is because natural numbers that previously exist
in students' minds can help with learning positive integers, but negative numbers
cannot be learned by observing the physical world since nonpositive objects or object
groups do not exist Davidson,
Mc Corkle,
Şeng(l and Dereli,
.
According to Piaget's (1952) cognitive developmental periods, concrete
operations correspond to primary school and the abstract operations correspond to
middle school periods; and problems occur in comprehending and carrying out
operations with integers, an abstract subject, when passing from concrete operations to
abstract operations (Dereli, 2008). Studies show that even if the students do not have
difficulty in placing negative numbers in the number line, they have difficulty in
comparing their superiority, which is similar to the findings of this study. In addition,
students cannot fully comprehend whether the addition and minus signs reflect the
numbers' direction or the operation itself and make calculation errors due to incorrect
use of these signs; they tend to subtract the number with the smaller number value
from the number with the bigger absolute value Işıksal ”ostan,
“vcu and
Durmaz, 2011). The teachers that participated in this study stated that students have
problems in identifying the sign of the result in the multiplication or division of
integers. Avcu and Durmaz (2011) found similar results in their study with students in
European Journal of Education Studies - Volume 3 │ Issue 1 │ 2017
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Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
the 6th and 7th grades. Another finding of the present study is that students cannot
fully understand the order of operations and make calculations ignoring the fact that
the sign before the numbers change the sign of the result. The study of Avcu and
Durmaz (2011) supports the findings of the present study.
Regarding the sixth grade curriculum learning objective of Using the features of
addition as strategies for a fluent operation teachers stated that students have
difficulty in understanding the order of operations and make errors in distributing the
negative sign before the parentheses.
Integers are addressed in two sections in different grades in the curriculum
prepared based on the constructive approach; the comprehension, comparison,
addition, and subtraction of integers are addressed in the 6th grade while the
multiplication and division of integers are addressed in the 7th grade. Thus, it was
aimed that students acquire the concept of integers in the 7th grade. While it is
recommended in the mathematics curriculum that in the 6th grade, integers be
associated with elevators, air temperature, thermometers, and bank accounts, any
analogies were found in the additions about the multiplication, division, and
exponentiation of integers in the 7th grade.
When the mathematics textbooks approved by the MNE for the 6th and 7th
grades in the 2013-2014 academic year were analyzed it is observed that, the textbooks
address the comprehension, comparison, and expression of the absolute values of
integers in the 6th grade and the operations with integers in the 7th grade, while the
curriculum recommend giving only the objectives that "multiplying an dividing integers"
and "indicating the repeated multiplication of integers as exponential numbers . This conflict
between the curriculum and the textbooks can negatively affect teaching integers. The
textbooks included analogies created by associating integers with sea level,
thermometers, elevators, steps forwards and backwards, profit-loss or debit-credit
relationships, air temperature, and number patterns. These analogies are similar to the
analogies recommended in the curriculum, and differently from the curriculum, the
textbooks include analogies for multiplication, division, and exponentiation of integers.
It can be highlighted that particularly for the curriculum additions above, number
pattern and the field analogies have been focused, differently from the other additions.
Another highlight is that the curriculum and textbooks do not include analogies for the
additions of subtraction and addition with the opposite sign of the minuend in integers.
Considering the difficulties that teachers have in teaching integers, it can be said
that these analogies stipulated in the curriculum will be insufficient. Duit (1991) stated
that the analogies are used too infrequently and generally at a very simple level in
textbooks. The common and non-common points between the target and the source is
not clearly specified in the analogies used in the curriculum and textbooks. However,
analogies are strong tools if how and why the common and non-common features and
European Journal of Education Studies - Volume 3 │ Issue 1 │ 2017
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Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
similarities are formed is explained mapping Saygılı,
Kanalmaz, 2010).
Explanation of the concept of integers and the operations with integers, which students
have problems with interpreting by associating with something concrete in daily life,
guides students in the formation of mathematical operations (Kilhamn, 2008; Hayes and
Stacey,
Şeng(l and Kör(kç(,
Şeng(l and Dereli,
. Therefore it is
important to enrich the curriculum and textbooks with more analogies and to specify
the common and non-common features so as not to cause misunderstanding when
using the given analogies in teaching to help teachers who are accustomed to the
traditional approach adapt to the new education and training environment.
It was observed that the teachers who participated in this study mostly used the
analogies in the textbooks and the curriculum when teaching integers. The analogies
used were focused on temperature, credit-debit relationships, thermometers, elevators,
moves forwards or backwards, counting pieces, modeling with zero pairs, and using
mathematical patterns. The teachers mostly used the given analogies rather than
creating analogies. This may be because teachers are incompetent in creating analogies.
The findings show that the majority of the teachers used the given analogies
rather than creating analogies themselves. Treagust et al. (1990) indicated that teachers
do not have sufficient knowledge in analogies and thus use the analogies in textbooks;
however, this situation conflicts with constructive teaching. Yet, they may have to
create analogies for the subject to counter against the difficulties not predicted in the
curriculum or textbooks. Therefore, teachers should be provided with in-service
training to create correct analogies. An incorrect analogy made by teachers, who have a
significant place in the teaching and learning process, may lead to misconceptions that
are hard to get rid of (Treagust, Harrison and Venville, 1996). Teachers have important
roles in eliminating misconceptions and the calculation errors students make due to
such a misconception Işıl and ”ostan,
. ”ased on this information, the awareness
of teachers should be raised on creating, using, and adapting analogies into their
teaching to enable them to use efficiently the correct analogies at the right times.
The teachers mostly used the given analogies rather than creating analogies. This
may be because teachers are incompetent in creating analogies. In this regard, teachers
should be provided with the required training and the in-service training should be
expanded. In addition, the faculties of education have great responsibilities in training
the pre-service teachers, who will raise the next generations, during their
undergraduate education in terms of creating analogies and how to adapt them into
their teaching. The training on analogies should not be neglected considering their
benefits in education and training.
European Journal of Education Studies - Volume 3 │ Issue 1 │ 2017
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Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
Acknowledgments
A part of this study was presented at the 13th Mathematical Symposiums as an oral
presentation.
References
1. “kkuş H.
. Kimyasal tepkimelerin dengeye ulaşmasının öğretiminde
kullanılabilecek bir analoji meslek seçimi analojisi, Kazım Karabekir Eğitim
Fakültesi Dergisi, 14, 19-30.
2. “ltun S. “.
. İlköğretim öğrencilerinin akademik başarısızlıklarına ilişkin
veli, öğretmen ve öğrenci gör(şlerinin incelenmesi, İlköğretim Online, 8(2), 567586.
Retrieved
January
18,
2014,
http://ilkogretim-
from
online.org.tr/vol8say2/v8s2m24.pdf
3. “tav E., Erdem E., Yılmaz “. & G(c(m ”.
. Enzimler konusunun anlamlı
öğrenilmesinde analojiler oluşturmanın etkisi, Hacettepe Üniversitesi Eğitim
Fakültesi Dergisi, 27, 21-29.
4. “vcu T. & Durmaz ”.
. Tamsayılarla ilgili işlemlerde ilköğretim d(zeyinde
yapılan hatalar ve karşılaşılan zorluklar, 2nd International Conference on New
Trends in Education and Their Implications (ICONTE), Antalya.
5. “ykutlu I. & Şen “. İ.
. Fizik öğretmen adaylarının analoji kullanımına
ilişkin gör(şleri ve elektrik akımı konusundaki analojileri, Hacettepe Üniversitesi
Eğitim Fakültesi Dergisi, 41, 48-59.
6. ”ahadır E. & 5zdemir “. Ş.
. Tam sayılar konusunun canlandırma tekniği
ile öğretiminin öğrenci başarısına ve hatırlama d(zeyine etkisi, International
Journal Social Science Research, 2(2), 114-136.
7. ”ıyıklı, C., Veznedaroğlu, L., 5ztepe, ”. & Onur, “.
. Yapılandırmacılığı nasıl
uygulamalıyız? “nkara ODTÜ Yayıncılık.
8. Dagher Z. R. (1995). Analysis of analogies used by sciences teachers, Journal of
Research in Science Teaching, 32(3), 259-270.
9. Davidson P. M. (1992). Precursors of non positive integer concepts. Biennial Meeting
of the Society for Research in Child Development, Baltimore, MD.
10. Dereli M. (2008). Tam sayılar konusunun karikat(rle öğretiminin öğrencilerin
matematik başarılarına etkisi, Yayınlanmamış y(ksek lisans tezi, Marmara
Üniversitesi, İstanbul.
11. Driver, R. & ”ell, ”.
. Students’ Thinking and the Learning of Science “
Constructivist View. School Science Review, 67, 443 – 456.
12. Duit, R. (1991). On the role of analogies and metaphors in learning science.
Science Education, 75(6), 649-672.
European Journal of Education Studies - Volume 3 │ Issue 1 │ 2017
62
Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
13. Glynn, S. M. (2007). Methods and strategies: Teaching with analogies. Science and
Children, 44(8), 52-55.
14. Glynn, S. M. & Takahashi, T. (1998). Learning from analogy-enhanced science
text. Journal of Science Teaching, 35(10): 1129-1149.
15. G(lçiçek Ç., ”ağı N. & Moğol S.
. 5ğrencilerin atom yapısı-g(neş sistemi
pedogojik benzeştirme analoji modelini analiz yeterlilikleri, Milli Eğitim Dergisi,
159, 74-84.
16. G(nay ”ilaloğlu R.
. Erken çocukluk döneminde fen öğretiminde analoji
tekniği, Çukurova Üniversitesi Eğitim Fakültesi Dergisi, 2(30), 72-77.
17. Hayes, B., & Stacey, K. (1990). Teaching negative number using integer tiles.
Unpublished doctoral thesis, University of Melbourne, USA.
18. Heywood, D. (2002). The place of analogies in science education. Cambridge
Journal of Education, 32(2), 233-247
19. Işıksal ”ostan M.
. Negatif sayılara ilişkin zorluklar, kavram yanılgıları ve
bu yanılgıların giderilmesine yönelik öneriler. ”ingölbali E. & 5zmantar M. F.
(Ed.) İlköğretimde karşılaşılan matematiksel zorluklar ve çözüm önerileri içinde (ss.
155-
“nkara Pegem “kademi Yayınları.
20. Kanalmaz T.
. İlköğretim . sınıf matematik dersi ölçme öğrenme alanında
analoji yöntemine dayalı öğretimin öğrencilerin akademik başarılarına etkisi,
Yayınlanmamış y(ksek lisans tezi, Gazi Üniversitesi, Ankara.
21. Kaptan F. & “rslan ”.
. Fen öğretiminde soru-cevap tekniği ile analoji
tekniğinin karşılaştırılması, V. Ulusal Fen ”ilimleri ve Matematik eğitimi kongresi,
Ankara.
22. Kaya S. & Durmuş “.
. ”ilişim teknolojileri öğretimi için geliştirilen örnek
analojilerin incelenmesi, “hi Evran Üniversitesi Eğitim Fakültesi Dergisi, 12 (2), 235254.
23. Kesercioğlu T., Yılmaz H., Huyug(zel-Çavaş P. & Çavaş ”.
. İlköğretim fen
bilgisi öğretiminde analojilerin kullanımı "örnek uygulamalar", Ege Eğitim
Dergisi, 5, 35-44.
24. Kilhamn, C. (2008). Making sense of negative numbers through metaphorical
reasoning. Retrieved from www.mai. liu.se/SMDF/madif6/Kilhamn.pdf.
25. Kriger, M. H. (2003). Doing mathematics. New York: World Scientific.
26. Linchevski L. & Williams J. (1999). Using intuition from everyday life in 'filling'
the gap in children's extension of their number concept to include the negative
numbers, Educational Studies in Mathematics, 39(1-3), 131-147.
European Journal of Education Studies - Volume 3 │ Issue 1 │ 2017
63
Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
27. McCorkle, K. (2001). Relational and instrumental learning when teaching the
addition and substraction of positive and negative integers. Unpublished master
thesis. California State University, Dominques Hills.
28. MEB (2009). İlköğretim matematik dersi 6-8. sınıflar öğretim programı ve kılavuzu.
“nkara ME” Yayınları.
29. Palmquist, R. (1996). The Search for an Internet Metaphor: A Comparison of
Literatures.
American
Society
of
Information
Science
Conference,
http://www.asis.org/annual-96/ElectronicProceedings/ palmquist.html.
30. Parida B.K. & Goswami M. (2000). Using Analogy as a Tool in Science Education,
School Science Quarterly Journal of Science Education, 38 (4). Retrieved December,
21, 2013 http://www.ncert.nic.in/journalnew/sschap10.htm.
31. Piaget, J. (1952). The child’s conception of number. Humanities press, New York.
32. Pittman K. M. (1999). Student generated analogies: Another way of knowing?
Journal of Research in Science Teaching, 36(1), 1-22.
33. Saygılı S.
. “naloji ile öğretim yönteminin . Sınıf öğrencilerinin matematik
başarılarına ve yaratıcı d(ş(nmelerine etkisi. Yayınlanmamış y(ksek lisans tezi,
Çanakkale On Sekiz Mart Üniversitesi, Çanakkale.
34. Serin Ergin Ö. (2009). 5ğrenci ve öğretmenlerin
ilişkili
analojilerdeki
benzerlik
ve
. sınıf kimya konuları ile
farklılıkları
belirleme
d(zeyleri,
Yayınlanmamış y(ksek lisans tezi, ”alıkesir Üniversitesi, ”alıkesir.
35. Şahin, F., G(rdal, “. & ”erkem, M. L. (2000). Fizyolojik kavramların anlamlı
öğrenilmesi ile ilgili bir araştırma. IV. Fen ”ilimleri Eğitimi Kongresi, Ankara,
”ildiriler Kitabı,
-23.
36. Şeng(l S. & Kör(kc( E.
. Tam sayılar konusunun görsel materyal ile
öğretiminin altıncı sınıf öğrencilerinin matematik başarıları ve kalıcılık
düzeylerine etkisi, International Online Journal of Educational Sciences, 4(2), 489508.
37. Şeng(l S. & Dereli M.
. Tam sayılar konusunun karikat(rle öğretiminin .
sınıf öğrencilerinin matematik tutumuna etkisi, Kuram ve Uygulamada Eğitim
Bilimleri, 13 (4), 2509-2534.
38. Treagust D. F., Duit R., Joslın P., & Lındauer I.
. “ naturalistic study
of science teachers' use of analogies as part of their regular teaching.
Paper
Presented at the Annual Meeting of the American Educational Research Association,
Boston.
39. Treagust, D.F., Harrison A.G., & Venville G.J., (1996). Using an analogical
teaching approach to engender conceptual change. International Journal of Science
Education, 18 (2), 213-229.
European Journal of Education Studies - Volume 3 │ Issue 1 │ 2017
64
Mevhibe Kobak Demir, Nursen “zizoğlu, H(lya G(r
USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
40. Yıldırım, “. & Şimşek, H.
. Sosyal bilimlerde nitel araştırma yöntemleri.
“nkara Seçkin Yayıncılık.
.
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USING ANALOGIES TO OVERCOME DIFFICULTIES IN TEACHING OF THE INTEGERS
IN THE MIDDLE SCHOOLS
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