A Comparison Of The Instructional Content On Division Of Fractions In Turkish And Singaporean Textbooks

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International Journal of Mathematical Education in
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A comparison of the instructional content on
division of fractions in Turkish and Singaporean
textbooks
Suphi Önder Bütüner
To cite this article: Suphi Önder Bütüner (2019): A comparison of the instructional content on
division of fractions in Turkish and Singaporean textbooks, International Journal of Mathematical
Education in Science and Technology, DOI: 10.1080/0020739X.2019.1644681
To link to this article: https://doi.org/10.1080/0020739X.2019.1644681
Published online: 28 Jul 2019.
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INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY
https://doi.org/10.1080/0020739X.2019.1644681
A comparison of the instructional content on division of
fractions in Turkish and Singaporean textbooks
Suphi Önder Bütüner
Faculty of Education, Mathematics and Science Education, Yozgat Bozok University, Yozgat, Turkey
ABSTRACT
This study has compared Turkish and Singaporean textbooks with
respect to the instructional content of the unit on division in frac-
tions, which is a challenging topic for many students. Two Turkish
textbooks were compared against two Singaporean books, My Pals
Are Here and Targeting Mathematics. In Turkish and Singaporean
mathematics books, the contents of the unit on division in fractions
are based on students’ conceptual understanding and the develop-
ment of operational fluency. When compared to Singaporean text-
books, Turkish books included more solution methods. However, the
solutions in Turkish books displayed no relationship between visual
and symbolic representation. Both Turkish and Singaporean books
used only area models (circle, rectangle) in teaching the topic of
division in fractions, and included measurement and equal sharing
problems with step by step solution explanations. In sum, it may
be claimed that Singaporean books caused students to use fewer
solution strategies but offered more opportunities for grasping the
relationship between division and multiplication in fractions.
ARTICLE HISTORY
Received 12 December 2018
KEYWORDS
Division of fractions;
mathematics textbook;
instructional content
1. Introduction
Textbooks are a basic source that guide teachers as they plan their lessons (Li, 2000). The
Trends in International Mathematics and Science Study (TIMSS) has shown that, almost
everywhere around the world, most teachers make use of textbooks as they decide how to
present a given topic in their classes (Beaton et al., 1996). Textbooks influence what and
how teachers teach, and what homework or activities they subject their students to (Hirsch
et al., 2005; Weiss et al., 2003). If a topic is not included within the scope of a textbook, it is
not possible to present it in the classroom (Schmidt & McKnight, 1997). The way topics are
presented in textbooks is important as it activates ‘the pedagogic approaches and various
opportunities required for student learning’ (Stein, Remillard, & Smith, 2007). Textbooks
help identify curricular goals given in a formal curriculum guide (in other words, intended
curriculum). It also determines what will be taught in the classroom and the scope of what
will be learned (in other words, implemented curriculum). Therefore, textbooks comprise
a bridge between the ‘intended curriculum’ and ‘implemented curriculum’. An analysis of
textbooks gives a clearer picture of what needs to be taught and learned in the classroom
CONTACT Suphi Önder Bütüner s.onder.butuner@bozok.edu.tr
© 2019 Informa UK Limited, trading as Taylor & Francis Group

2 S. Ö. BÜTÜNER
than an intended curriculum (Flanders, 1994). Similarly, analysing a textbook instead of
an implemented curriculum is a more accessible way of documenting how instruction for
a large population should be sustained over a long period of time (Li, 2000).
Researchers, on the other hand, have contradictory views about what may be learned
by analysing mathematics textbooks. Some researchers claim that textbook analysis in
international comparative studies may explain the differences between student perfor-
mance (Fuson, Stigler, & Bartsch, 1988). Others, on the other hand, argue that textbooks
have very little effect on teaching and student learning (Freeman & Porter, 1989). There
are also researchers who believe that teachers use textbooks as a learning tool. Indeed,
teachers who are not knowledgeable in their subject area tend to rely more on text-
books. Naturally, textbooks can prove to be crucial for teachers by helping decision-making
(Collopy, 2003; Kaufman, 1997). The role of textbooks in instruction depends on how
students and teachers interact with them in the classroom. At the same time, examining
textbooks from different countries may reveal the similarities and differences in mathe-
matics learning opportunities offered to students around the globe. Such analyses may
show the performance expected from students in different countries; the level of prior-
ity placed by a given country on the conceptual understanding or operational fluency in a
set of textbooks; and how the treatment of mathematical content differs among countries
(Li, 2000).
In recent years, there has been an increase in textbook comparison studies. These
studies have mostly used textbooks from countries such as China, Korea, Japan, Tai-
wan, Singapore, and Finland, which score high on international exams like TIMSS or
PISA. In previous studies, researchers compared the number of topics, how topics are
introduced and developed, when topics are introduced, or characteristics of the prob-
lems presented. Studies generally focused either on content or problem sets (Hong &
Choi, 2014). The mathematics textbooks used in Turkey have been compared with the
books used in the top five countries in international exams (only books from Singapore)
with respect to design qualities (Erbaş, Alacacı, & Bulut, 2012) and problem type (Özer
& Sezer, 2014). Therefore, the present study compared Turkish and Singaporean text-
books, with respect to the instructional content of the challenging topic of division in
fractions.
This study compared Turkish and Singaporean mathematics textbooks regarding the
contents on division in fractions; identified the strengths and weaknesses of textbooks;
and aimed to reveal the learning opportunities in textbooks presented to students. In
addition, in line with the findings, the study makes recommendations for the curricu-
lum designers working at Turkish and Singaporean Education Ministries. The recom-
mendations for improving textbooks will guide curriculum developers in overcoming
the deficiencies of textbooks. Different textbooks can offer different learning oppor-
tunities to students and help explain the differences between students’ success levels
(Reys, Reys, & Chavez, 2004; Zhu & Fan, 2006). Therefore, the results of this study
may give an idea about the performances of Turkish and Singaporean students, who will
take international exams in future years, in divisions of fractions. The following section
focuses on book comparison studies to date about fractions and comparative studies
involving Turkish and Singaporean textbooks, thus emphasizing the need for compar-
ing Turkish and Singaporean textbooks and the importance of the topic of division of
fractions.

INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 3
1.1. Why the topic of division in fractions?
The effective teaching of operations with fractions is essential because the understand-
ing of concepts such as rate, percentage, inclination or decimals depends on the effective
learning of fractions (Son, 2011). Fractions are also closely related the algebra learning
area (Brown & Quinn, 2007; Siegler et al., 2010). Dividing with fractions is a topic related
with multiplying and subtracting with fractions as well. It is related to multiplication in
fractions because the solution of divisions in fractions largely relies on inverse multiply-
ing algorithm. It is also related to subtraction in fractions because dividing with fractions
means measurement-repeated subtraction (Van de Walle & Karp, 2010).
Division in fractions is a complicated and difficult topic for both teachers and
students. The topic is conceptually rich and difficult as its meaning requires it to be
explained through its connections with other mathematical knowledge, various represen-
tations, or real world contexts (Li, 2008). For example, one difficulty that the students have
is related to the meaning of division in fractions. The problem ‘Three bars of chocolate get
equally distributed among five children. How much chocolate does each child get?’ was
answered correctly by only 66% of 12-year-old children, and 63% of 13-year-old children.
Furthermore, even fewer children were able to state that 3 ÷ 5 may also be represented
as 3/5 (Gregg & Gregg, 2007). Kerslake (1986) used a number of alternative models for
fractions in an attempt to discover which models children relate to most. Similarly many
children were not able to state that 3 ÷ 4 may also be represented as 3/4. Therefore, stu-
dents should understand and feel comfortable with the example here written as 3 ÷ 4, 3/4.
This would give students the awareness that the dividend may be smaller than the divider.
When solving divisions in fractions, two algorithms may be used. The first algorithm is
the reverse and multiply algorithm, which is used commonly in schools. Students tend to
memorize this algorithm as teachers often teach it as a rote procedure (Sharp & Adams,
2002). It was also found that teachers and teacher candidates could not explain what
the reverse-multiply algorithm used in division in fractions means or how it is related
with the division operation (Borko et al., 1992; Isık & Kar, 2012; Li & Kulm, 2008). Stu-
dents should understand the underlying meaning and justifications for ideas and be able
to make connections among topics (for example, fractions-decimal fractions-percentages
or multiplication-division, etc.) because students cannot make sense of the results they
have obtained when they follow a procedure they do not understand. Also, algorithms
whose underlying meaning is not understood may be confused with one another (Van
de Walle & Karp, 2010). Indeed, even some college students may make mistakes simi-
lar to (a/b) + (c/d) = (a + c)/(b + d); (a/b) ÷ (c/d) = (ad ÷ bc)/bd (Li, Chen, & An,
2009). Other mistakes by students when dividing fractions include reaching the equation
1 ÷ (1/2) = (1/2) ÷ 1 = 2 by thinking ‘Reversing the dividend instead of the divider’ or
‘reversing both the dividend and the divider before multiplying the numerators and the
denominators’, ‘the division operation carries the quality of change’ or ‘the dividend is
always greater than the divider’. These mistakes may be attributed to the teaching of the
algorithms as a rule and not encouraging students to learn the underlying meaning (Tirosh,
2000). Divisions in fractions may also be solved by using the ‘common denominator strat-
egy’. This strategy is based on repeated-measurement subtraction. As students in early
school years carry out divisions with whole numbers with the thought of repeated subtrac-
tion (How many threes exist in 12?), this strategy is recommended to be used in the solution

4 S. Ö. BÜTÜNER
of divisions in fractions (Sharp & Adams, 2002). However, this strategy should not be used
mechanically. For example, textbooks should explain why the operation (3/4) ÷ (1/8) is
solved as follows: (6/8) ÷ (1/8) = (6 ÷ 1)/(8 ÷ 8) = 6. When the common denominator
strategy is taught as a rule, students will not be able to explain why they need to equalize
the denominators of fractions and why they divide the numerator of the divider fraction
by the numerator of the dividend. Fractions may also be divided by using visual models.
However, it is not enough to obtain the accurate result with visuals. The visuals should be
used to focus on the meanings of the reverse and multiply algorithm as well as the com-
mon denominator strategy and why these solution methods are preferred. Relationships
should be established between the solution methods. In light of these, it may be stated that
it is important to examine how the topic of divisions in fractions is presented in Turkish
and Singaporean textbooks, and which visual models (length, area, clusters) and solution
strategies are included in them.
The division operation has two meanings. Problems where group size is unknown are
known as equal sharing or disintegration problems. For example, here is an equal sharing
problem: when two and a half litres of orange juice is equally distributed among five peo-
ple, how many lt of juice does each person get? When group size is unknown but the size of
group pairs is known, these problems are known as measurement or repeated subtraction
problems. For instance, the measurement meaning of division is emphasized in the follow-
ing problem: ‘Two and a half liters of orange juice will be poured into half liter jugs. How
many jugs will be needed?’. Students, and even teachers, are known to have problems in
the topic of division in fractions (Carpenter et al., 1988; Isıksal & Çakıroğlu, 2008; Leung
& Carbone, 2013; Simon, 1993). When faced with equal sharing problems, students often
think that the divider needs to be a whole number, and the divider and the division need
to be smaller than the dividend. Student misconceptions about the sum of the division
always needing to be smaller may be attributed to their early elementary school years and
the operations they made with the elements of the natural numbers set (Tirosh, 2000). It
is therefore important that classes include both measurement and equal sharing problems.
Therefore, the textbooks were examined with respect to including equal distribution and
measurement-repeated subtraction problems solved step by step.
1.2. Book comparison studies on fractions
In order to reveal the importance of the study, the contents of book comparison
studies to date involving fractions, as well as the contents of studies comparing Turkish
and Singaporean textbooks were analysed. Table 1 shows the countries whose textbooks
have been analysed in book comparison studies, what topics were compared, and the coun-
tries whose textbooks have been compared with Turkish mathematics textbooks and from
which perspective.
As can be seen from Table 1, many book comparison studies involving fractions com-
pared textbooks used in the U.S.A. with those used in the top five countries in the TIMSS (Li
et al., 2009; Son, 2005, 2012; Son & Senk, 2010; Sun, 2011; Sun & Kulm, 2010). These studies
focused on the comparison of problem type in textbooks (Özer & Sezer, 2014; Zhu & Fan,
2006), contents and problem types (Alajmi, 2012; Charalambous, Delaney, Hsu, & Mesa,
2010; Kar, Güler, Şen, & Özdemir, 2018; Li et al., 2009; Son, 2012; Son & Senk, 2010) and
design characteristics (Özer & Sezer, 2014). More precisely, most book comparison studies

Table 1. An overview of book comparison studies involving fractions.
Authors Year Countries Unit
Son 2005 Korea–U.S.A. Multiplication and Division in Fractions
Li et al. 2009 China–Japan–U.S.A. Division in Fractions
Son and Senk 2010 Korea–U.S.A. Multiplication and Division in Fractions
Sun and Kulm 2010 China–U.S.A. Fractions
Yang, Reys, and Wu 2010 Singapore–Taiwan–U.S.A. Fractions
Charalambous et al. 2010 Taiwan–Cyprus–Ireland Additions and Subtractions in Fractions
Sun 2011 China–U.S.A. Division in Fractions
Alajmi 2012 Japan–Kuwait–U.S.A. Fractions
Erbaş et al. 2012 Singapore–Turkey–U.S.A. Design characteristics
Son 2012 Korea–U.S.A. Additions and Subtractions in Fractions
Özer and Sezer 2014 Singapore–Turkey–U.S.A. Problem Type
Cady, Collins, and Hodges 2015 U.S. textbooks Fractions
Vula, Kingji-Kastrati, and Podvorica 2015 Kosova–Albania Fractions
Watanabe et al. 2017 Japan, Korea, Taiwan Fractions and Operations in Fractions
Kar et al. 2018 U.S.A.–Turkey Multiplication in Fractions
Yang 2018 Finland–Taiwan Fractions
have analysed how the topics were taught and why type of problems were used. Consid-
ering the mathematics textbooks used in Turkey, the literature holds comparative studies
with American textbooks (Kar et al., 2018; Kar & Işık, 2015) and American-Singaporean
textbooks (Erbaş et al., 2012; Özer & Sezer, 2014). Turkish textbooks have been compared
with their Singaporean counterparts with respect to only design characteristics (Erbaş
et al., 2012) and problem type (Özer & Sezer, 2014). Differently from these previous stud-
ies, the present study compared Turkish and Singaporean textbooks, with respect to the
instructional content pertaining to the division of fractions, which is a challenge for many
students.
1.2.1. Book comparison studies on division in fractions
The literature includes studies that have compared textbooks with respect to how they teach
division in fractions (Li et al., 2009; Son, 2005; Son & Senk, 2010; Sun, 2011). However,
all of these studies have compared American books with those from far eastern countries
which take top place in international exams. Son (2005), conducted a content and problem
analysis on American and Korean mathematics textbooks. These textbooks aimed to make
students understand the topic of multiplication and division of fractions conceptually, thus
delaying their operational development. They also taught the meaning of multiplication
and division operations in fractions through paper folding exercises and area models. On
the other hand, Korean textbooks taught the meanings and algorithms of multiplication
and division operations in fractions simultaneously. While both groups of textbooks intro-
duced division in fractions with division in whole numbers, the American textbook used
the common denominator strategy and the reverse-multiply algorithm, but the Korean
textbook only provided the latter.
Li et al. (2009) conducted content and problem analysis on Chinese, Japanese and
U.S. mathematics textbooks. The Chinese textbooks treated division in fractions as a
separate unit from multiplication in fractions, while the Japanese textbooks treated the
two topics together in one single unit. Both Chinese and Japanese books presented the
reverse-multiply algorithm by using real life problems and different solution strategies.
Once again, both books explained the use of the reverse-multiply algorithm in division

6 S. Ö. BÜTÜNER
in fractions through visual, verbal and symbolic representations. Different from Chinese
books, Japanese textbooks used a diagram combining a number axis and area model in
division in fractions. Other than this solution, Japanese books offered verbal and sym-
bolic representation as other solutions. Similar to Japanese books, Chinese books also
emphasized that division in fractions is the opposite of multiplication in fractions. Japanese
textbooks used two different strategies in the solution of (4/5) ÷ 2 thus showing how divi-
sion in fractions is the opposite of multiplication. As can be seen, there was no direct
information in Chinese and Japanese books regarding how division in fractions is to be
undertaken. While visual representation and verbal explanations had an important role
in illuminating the reverse-multiply algorithm in Chinese textbooks, symbolic representa-
tion were used to show that different calculations may be used to reach the same outcome.
In Japanese books, numeric expressions and verbal explanations were used to explain the
division in fractions algorithm. Among these textbooks, the highest number of visual
representations was used in American books. In all American textbooks, the division in
fractions operation was associated with division in whole numbers (For 1:1/4, how many
1/4’s exist in 1). Only one American book explained how the reverse-multiply algorithm
worked by using visual and verbal representations. As in Chinese and Japanese textbooks,
no American textbook explained how the reverse-multiply algorithm worked.
Son and Senk (2010) conducted content and problem analysis on U.S.A. and Korean
mathematics textbooks. Both American and Korean textbooks presented division in frac-
tions in the second term of grades five and six. While American textbooks allotted one
class hour to the topic in both grades five and six, Korean books allotted four class hours
in grade five and six hours in grade six. American textbooks included problems requiring
the group number. American books used the common denominator strategy to teach divi-
sion in fractions in grade five and solve (3 ÷ (3/5) = (15/5) ÷ (3/5) = 15 ÷ 3 = 5), and
moved on to the reverse-multiply algorithm in grade six. Different from American text-
books, Korean textbooks included both equal sharing problems and those requiring the
group number. Korean textbooks presented equal sharing problems in grade five and prob-
lems requiring the group number in grade six. In Korean textbooks, problems involving
division in fractions were solved through the common denominator strategy and reverse-
multiply algorithm. While American books presented the reverse-multiply algorithm only
as a rule, Singaporean books explained the relationship between the common denominator
strategy and reverse-multiply algorithm. Korean textbooks were based on the simultane-
ous development of conceptual understanding and operational fluency whereas American
books emphasized conceptual understanding and delayed operational fluency. Korean
textbooks also mentioned a third meaning of the division in fractions operation: the reverse
operational meaning.
2. Methods
2.1. Sample of textbooks
In this study, Turkish and Singaporean textbooks were compared regarding the instruc-
tional content of the unit on division in fractions, which is a hard to grasp topic for
many students. In Turkey, the Ministry of Education determines the books to be stud-
ied at secondary schools. Teachers in Turkey use the textbooks distributed to schools by

Table 2. Selected textbooks from Singapore and
Turkey.
Country Selected textbooks
Singapore Kheong et al. (2018a)
Kheong et al. (2018b).
Ming (2018a)
Ming (2018b)
Turkey Commission (2016)
Güven (2014)
the Ministry. Indeed, the Ministry warns teachers not to use publications and materials
under the names ‘supplementary book’, ‘test book’ or ‘holiday exercises’. Therefore, the
main sources used in classes in Turkey are the textbooks published by the Ministry of
Education. In this study, the unit on division in fractions was examined in the textbook
published for the first time in 2016 by the National Education Publishing House (TB1)
(Commission, 2016) and Güven’s (TB2) (Güven, 2014) textbook which was used in Turk-
ish secondary schools during the 2017–2018 school year. These books were abbreviated as
TB1 and TB2.
In the Singaporean education system, elementary education consists of a four-year
‘Foundation Stage from grade 1 through 4 and a two-year Orientation Stage during grades
5 and 6’. The general goal of elementary education is to teach students good English, native
language and mathematics education. Currently, Singaporean elementary schools use four
mathematics textbooks: My pals are here!, New Syllabus Primary Mathematics (Kheong,
Soon, & Ramakrishnan, 2018a, 2018b), Shaping Maths and Targeting Mathematics (Ming,
2018a, 2018b). Approximately 60% of Singaporean secondary schools use the book My
pals are here! This book was first published in 2001 and is based on cognitive development
theories, metacognition theories and constructivism (Gregg & Gregg, 2007). In this study,
division in fractions was examined in the third edition of the book My pals are here! (MP)
published in 2018 and the first edition of the book Targeting Mathematics (TM) published
also in 2018. These books were abbreviated as MP and TM. The books examined in the
study are shown in Table 2.
2.2. Analysis of content
Studies generally focused either on content or problem sets (Hong & Choi, 2014). Son and
Senk (2010) investigated when and how the meaning of multiplication and division in frac-
tions is presented in American and Korean textbooks, and what kind of solution strategies
are used in these books. Similarly, Kar et al. (2018) studied when and how the meaning of
multiplication in fractions is presented in American and Turkish textbooks as well as the
types of solution strategies included in them. Li et al. (2009) compared Chinese, Japanese
and American textbooks on macro and micro levels concerning their content about divi-
sion in fractions. When comparing the books on the macro level, the authors focused on
the grade levels when division in fractions was presented, how the content was organized
(for instance, a separate section after multiplication in fractions), order of content, and the
page of numbers on this content. Following these, they compared how the meaning of the

8 S. Ö. BÜTÜNER
division in fractions operation is given, how the division algorithm is formed and what
kind of solution strategies were included in the books.
Parallel to the other studies cited in the literature, this study conducted content analysis
on division in fractions in mathematics textbooks used in Singapore and Turkey. In this
context, this study first analysed the grade levels and the extent of the topic of divisions in
fractions in the books, and then compared the order of topics. Then, the analysis focused
on how division with fractions was presented, how the algorithm was developed, and the
kinds of models that were used. Finally, the distribution of equal sharing and repeated-
measurement subtraction problems solved step-by-step in the books was analysed. More
precisely, it sought answers to the following research questions:
• What grade level(s) do the books focusing on division in fractions belong?
• What is the number of pages and percentage allocated to the unit on division in fractions
in the books?
• What is the order of topics in the books regarding division in fractions?
• When and how do the books introduce division in fractions?
• How do the books present the meaning of concepts, and what models and solution
strategies have been used?
• What is the distribution of measurement and separation into parts problems that are
solved through step by step explanations in the books?
Singaporean textbooks being currently used in Singaporean schools were gathered from
the relevant publishers. Turkish textbooks were obtained from the official website of
the Turkish Ministry of Education. Turkish mathematics books have been used in sec-
ondary schools during 2017–2018 academic year and the previous years, while Singapore
mathematics books have been used in secondary schools during 2018–2019 academic
year. The data analysis was conducted by three different academicians. First of all, in
these books, the number of pages containing the topic of division in the fractions were
determined and then the ratio of that number to the number of pages in each text-
book was calculated. Afterwards, three different researchers identified and noted how
this topic was introduced, what kind of models (length, area, cluster models), prob-
lems (measurement or fair share) and strategies in solving problems (inverse algorithm,
common denominator strategy, etc.) were used in these textbooks. They then compared
these notes and reached a consensus on a model of the steps that the textbooks fol-
lowed in teaching the division of fractions. Thus, the similarities and differences in the
teaching of the division of fractions in the Turkish and Singaporean textbooks were
determined.
3. Findings
3.1. Findings about the grade levels and extent of the topic of division in fractions
Differently from Singaporean textbooks, books TB1 and TB2 used in Turkey include a
unit on division in fractions only in the 6th grade. In Singapore, division in fractions
is presented in the first term of grades 5 and 6. The unit on division in fractions is
allocated 24 pages in TB1 (8.5%) and 10 pages in TB2 (6.9%). Singaporean books allot

more number of pages to division in fractions. In the book coded MP, division in frac-
tions is allocated 35 pages (19.6%), while in book TM it is allocated 25 pages (14.8%).
In books TB1 and TB2, division in fractions is treated in the unit about operations in
fractions.
3.2. Findings about order of topics in the books regarding division in fractions
The contents of this unit are in the following order: ‘ordering in fractions, addition, subtrac-
tion, multiplication and division in fractions’. In Singaporean books MP and TM, different
from the Turkish textbooks, the units on addition, subtraction and multiplication in frac-
tions do not go just before division in fractions. These topics that are related to division
in fractions are taught in the first term of grade 5. The books TB1 and TB2 offer no con-
tent on the relationship between division in whole numbers and fractions. Singaporean
books, on the other hand, mention the relationship between division in whole numbers and
fractions (4 ÷ 5 = (4/5)) in the first term of grade 5. In TB 1, the unit on division in frac-
tions is taught in the following order:a ÷ (1/b), (1/a) ÷ b, a ÷ (b/c), (a/b) ÷ c, (a/b) ÷
(1/c), (a/b) ÷ (c/d), while in TB2 it is taught in the following order:a ÷ (1/b), (1/a) ÷
b, a ÷ (b/c), (a/b) ÷ c, (1/a) ÷ (1/b), (a/b) ÷ (1/c), (a/b) ÷ (c/d). The unit finishes with
division in fractions problems and solutions. Different from Turkish books, Singa-
porean textbooks start the unit on division in fractions by associating division in
whole numbers with fractions (For instance; 3 ÷ 5 = (3/5)). Following this, the book
TM presents division in fractions with problem solutions which necessitate the follow-
ing operations: (1/a) ÷ b, (a/b) ÷ c, a ÷ (1/b), a ÷ (b/c), (1/a) ÷ (1/b), (a/b) ÷ (c/d),
while MP has adopted the following order: (a/b) ÷ c, (1/a) ÷ b, (a/b) ÷ c, a ÷ (1/b), a ÷
(b/c), (a/b) ÷ (1/c), (a/b) ÷ (c/d). Figure 1 presents an example of the association
between division in whole numbers and fractions. Similar examples can be seen in book
TM as well (TM 5A, 2018, pp. 46-47-48-49).
3.3. Findings regarding the teaching of the division of fractions in Turkish and
Singaporean textbooks
3.3.1. Dividing a fraction by a whole number
In Turkish and Singaporean mathematics books, the contents of the unit on division in
fractions are based on students’ conceptual understanding and the development of opera-
tional fluency. However, the importance that the books place on conceptual understanding
and operational fluency varies. These differences are shown through a detailed analysis of
the contents of these books regarding the unit on division in fractions.
In TB1, area models were used in the teaching of the division of a simple fraction into
a positive whole number, to solve separation into pieces and equal sharing problems. The
problem was solved, verbal explanations on the visual solution were offered, and then the
symbolic form was used in the solution of the problem. The solution in the symbolic form
used the common denominator strategy. The aim was to show the students an alternative
solution. Therefore, this book depends on the simultaneous development of conceptual
understanding and operational fluency in students. However, though there is no relation-
ship between the solution reached through symbolic representation and the one reached
through visual representation in TB1, the relationship between division and multiplication

10 S. Ö. BÜTÜNER
Figure 1. Relating fractions and division (MP, 5A, 2018, p. 54).
in fractions has not been explained. Similarly, the book does not use the reverse-multiply
algorithm when dividing fractions and does not explain what this algorithm means or why
it is used. As mentioned above, TB1 does not offer any content to explain the relationship of
division of whole numbers and fractions, but the solution obtained by using the common
denominator strategy in the problem ‘when half a tray of pastry is divided equally among
3 people, how much pastry does each person have?’ (Figure 2) was written not as 1 ÷ 6 but
as 1/6.
In TB2, three different solutions are used as a simple fraction was divided into a pos-
itive whole number. The problem is first solved by using area models and then via the
common denominator strategy and reverse-multiply algorithm. Similar to the book TB1,
problem solution is first made on the visual, verbal explanations are then given about the
visual solution, then followed by problem solution in the symbolic form (common denom-
inator strategy, reverse-multiply algorithm). Therefore, as in book TB1, this book also
depends on the simultaneous development of conceptual understanding in students and
operational fluency. However, TB2 does not include explanations showing the relationship
between visual representation and the reverse-multiply algorithm or questions that require
the students to make inferences about this relationship. It is stated that in direct division
of fractions, the first fraction is written as is and the second fraction is written by reversing
the numerator and the denominator, and is multiplied with the first fraction. The solution
methods of (3/4) ÷ 3 in TB2 are shown in Figure 3.
Similar to Turkish textbooks, Singaporean textbooks also use area models in teaching
the division of a simple fraction into a positive whole number and introduce divisions in
fractions by considering the simultaneous development of conceptual understanding and
operational fluency. However, the common denominator strategy is not used in solving

Figure 2. Dividing a fraction by a whole number [When half a tray of pie is distributed equally among
3 people, how much pie does each one get? Verbal solution: the figure shows that each person gets 1/6
of a tray. If an operation is undertaken, 1/2 must be divided into 3] (TB1. 6, 2016, p. 163).
Figure 3. Dividing a fraction by a whole number [Let us complete (3/4) ÷ 3 by modeling. Method 1:
we should first model 3/4. We should divide the model into 3 identical parts. Each part is 3/12 of the
figure. Method 2: let us solve (3/4) ÷ 3 by equalizing their denominators; Method 3: let us reverse the
dividing number and multiply it with the dividend] (TB2. 6, 2014, p. 101).

12 S. Ö. BÜTÜNER
Figure 4. Dividing a fraction by a whole number (MP. 6A, 2018, p. 35).
problems that require division in fractions. In the book MP, the problem was solved on
the visual, and then an association was made between division and multiplication in frac-
tions, and the origins of the reverse-multiply algorithm was shown by making use of the
changing characteristic of multiplication. This book uses ((a/b) × (c/d) = (c/d) × (a/b))
area models to show that multiplication in fractions has a changing characteristic. Simi-
lar to the book MP, the book TM uses the area model to explain the relationship between
division and multiplication in fractions. However, different from MP, TM states that divi-
sion in fractions may use the reverse-multiply algorithm by directly giving the equation
(a/b) × (c/d) = (c/d) × (a/b) but not proving this equation through area models. Also,
at the end of each unit, an example is offered about how the reverse-multiply algorithm can
be used. Figures 4 and 5 present sample problem and solution styles taken from books MP
and TM.
Singaporean textbooks presented the topic of dividing a fraction into a whole number
by basing it on the simultaneous development of conceptual understanding and opera-
tional fluency. The equal sharing problem in MP (Figure 4) states that a half cake is to be
shared equally among three kids and asks what fraction of the cake each child will get.
First the problem situation is modeled, and the model is used to show that each child will
get one third of the half cake. The book shows one third of the half cake symbolically as
follows: (1/2) ÷ 3 = (1/3) × (1/2). The the equation (1/2) × (1/3) = (1/3) × (1/2) is
used to write: (1/2) ÷ 3, (1/2) × (1/3) = (1/6). The book MP used in the first term of
grade 5 uses area models to show that one third of a half yields the same result as half
of one third (MP, 2018, p. 74). Another problem in the MP is that 3 children will share
3/4 of a pie equally and what fraction of the pie does each child get?. This problem is
solved primarily on the area model. Then MP provides symbolic expressions of division of
fractions(1/2) ÷ 3, (1/2) × (1/3) = (1/6) directly. Since 3/4 of the pie is divided equally
among 3 children, each child gets 1/3 of 3/4 of the pie. This verbal expression in the book
is written as (3/4) ÷ 3 = (1/3) of (3/4) = (1/3) × (3/4) = (1/4) [MP, 6A, p. 34].

Figure 5. Dividing a fraction by a whole number (TM. 6A, 2018, p. 30).
In the equal sharing problem in the book TM (Figure 5), the students are asked what
fraction of a pizza each child would get if the four fifths of a pizza is equally shared among
four children. The problem situation is first modelled, and it is shown through the model
that each child will get one fourth of four fifths of a pizza, namely one fifth of it. Symbol-
ically, this is written as (1/4) × (4/5). Then the equation (4/5) × (1/4) = (1/4) × (4/5)
is used as(4/5) ÷ 4, (4/5) × (1/4) for simplification and the result is 1/5. The book also
explains how the division operation is to be made by using the reverse-multiply algorithm
with the example (3/4) ÷ 2 = (3/4) × (1/2) (Figure 6). Another problem in the TM is that
a length of wire 3/4 m long is cut into 5 equal pieces. What is the length of each piece of
wire?. This problem is solved primarily on the model. The visual solution is then expressed
numerically. Since 3/4 of the wire is cut into 5 equal pieces, length of the each piece is
1/5 of 3/4 of the wire. This verbal expression in the book is written as (3/4) ÷ 5 = (1/5)
of (3/4) = (1/5) × (3/4) = (3/20) [TM, 6A, p. 31]. Seen from this perspective, Singa-
pore textbooks attempt to explain why the reverse-multiply algorithm is used in division
in fractions and what it means.
3.3.2. Dividing a whole number by a proper fraction
Regarding the content about the division of a positive whole number into fractions, the
books TB1 and TB2 use the area model to explain how many fractions consist in the whole
number. For example, in book TB1, the problem ‘how many quarter cheese sandwiches
can be made from three loaves of bread?’ (Figure 7) is interpreted as ‘how many 1/4 s

14 S. Ö. BÜTÜNER
Figure 6. The reverse-multiply algorithm in the division of a fraction into a positive whole number (TM.
6A, 2018, p. 32).
Figure 7. Dividing a whole number by a proper fraction [How many quarter sandwiches can be made
from 3 loaves of bread? Verbal solution: the ﬁgure shows that 3 loaves of bread yield 12 quarter sand-
wiches. So 3 wholes contain 12 quarters. The solution is also possible through an operation by dividing
3 by 1/4] (TB1, 2016, p. 161).
exist in three loaves of bread?’, and solved with the help of a visual. Right underneath
the visual representation in the book, there are verbal explanations about the solution. In
book TB1, the result was obtained through this question: How many quarters exist in 3
wholes? Even though the problem is solved by thinking about how many fractions exist
in a whole number, the connection between division and multiplication in fractions (the
reverse-multiply algorithm) has not been established. The second method used in the book
for problem solution is the common denominator strategy. However, the common denom-
inator strategy is given as a rule, and its relationship with the reverse-multiply algorithm is
not established. In addition, it may be stated that no connection exists between the visual
solution used and the symbolic solution.
In TB2, similar to TB1, the problem ‘Each step Ceyhun takes is 1/2 m. How many steps
will it take for Ceyhun to walk a 4 m road?’ (Figure 8) is solved by using a visual and

Figure 8. Dividing a whole number by a proper fraction [Method 1: In 4 wholes, there are eight 1/2’s.
In other words, 4 ÷ (1/2) = 8; Method 2: let us solve 4 ÷ (1/2) by equalizing denominators; Method 3:
let us reverse the divider and multiply it with the dividend] (TB2, 2014, p. 98).
focusing on how many halves exist in 4 wholes. Right under the visual are verbal explana-
tions regarding the solution. Different from TB1, TB2 uses the reverse-multiply algorithm
as a solution method but does not explain what this algorithm means and why it is used
by referring to a visual representation. In TB1, the problem is solved by using the com-
mon denominator strategy. However, no relationship is established between the common
denominator strategy and reverse-multiply algorithm.
Similar to TB1 and TB2, the Singaporean books MP and TM use the area model to
explain how to divide a positive whole number into a fraction. Different from the books
TM, TB1 and TB2, the book MP establishes a relationship between the operation of divid-
ing a positive whole number into a fraction and multiplication. For instance, when solving
the problem how many quarters exist in number 2, the operation is first modelled to write
2 ÷ (1/4), 2 × 4. In sum, the textbook states that since there are 4 quarters in a whole, there
should be 2 × 4 quarters in 2 wholes.
The book MP, when solving the question 6 ÷ (3/5) given in Figure 9, the area model
is used to show that 10 three fifths exist in 6 wholes. Considering there are 5 three fifths
in 3 wholes, there needs to be 5/3 three fifths in one whole. This leads to the solution
that in 6 wholes, there are 6 × (3/5) three fifths. As a result, the book MP uses the area
model to show that 6 ÷ (3/5) equals 6 × (5/3). Following some simplification, the sum
((2 × 5)/1) = (10/1) = 10 is reached. It can be seen that the other textbooks do not clearly
form a connection between division and multiplication. In the Singaporean book TM, the
area model is used to solve the problem situation in Figure 10 to show that 6 one third exists
in 2 wholes. The problem situation is written as 2 ÷ (1/3) = 6 in symbolic form. At the end

16 S. Ö. BÜTÜNER
Figure 9. Dividing a whole number by a proper fraction (MP 6A, 2018, p. 41).
Figure 10. Dividing a whole number by a proper fraction (TM 6A, 2018, p. 33).
of the topic, the book TM explains how to undertake the division via the reverse-multiply
algorithm by offering the example 2 ÷ (2/3) = 2 × (3/2) = 3 [TM, 6A, p. 35].
3.3.3. Dividing a proper fraction by a proper fraction
The division of two simple fractions into one another is treated in all books as finding how
many of the second fraction exists in the first one. In book TB2, division of two simple
fractions questions are first solved on visuals by using area models, and then the common
denominator strategy. Different from the book TB2, TB1 offers verbal explanations about
the relationship of the division of two simple fractions and the multiplication operation.

Figure 11. Dividing a proper fraction by a proper fraction [When a whole is divided into its quarters, 4
equal quarters are obtained. Therefore, 3/4 has (3/4) × 4 quarters] (TB1, 2016, p. 174).
However, the relationship between division and multiplication not being established in the
sections on the division of a fraction into a whole number and of a whole number into a
fraction emerges as a deficiency in book TB1. Murat has eaten a portion of the chocolate
bar that he has. He still has three fourths of it. Book TB1 solves the problem ‘He distributes
the remaining chocolate bar equally among his friends, giving each one a quarter of a whole
chocolate bar. How many friends receive chocolate from Murat?’ as follows:
Figure 11 shows that the solution was first reached by using an area model, followed by
the use of the common denominator strategy. Finally, it was both symbolically and ver-
bally shown that just as there are four quarters in a whole, there are (3/4) × 4 quarters
in three fourths. Therefore, while TB2 offers no content about a relationship between divi-
sion in fractions and multiplication, TB1 offers a delayed relationship between division and
multiplication in fractions at the end of the unit on division in fractions.
As book MP explains why the reverse-multiply algorithm is used when a whole num-
ber is divided into a fraction and a fraction into a whole number, as well as what this
algorithm means, the reverse-multiply algorithm is not directly mentioned in the solu-
tion of (5/8) ÷ (1/4). The solution is reached by starting from how many 1/4 exists in 5/8
(Figure 12). After the solution is reached by using a visual, it is written in the symbolic

18 S. Ö. BÜTÜNER
Figure 12. The modeling and solution of (5/8) ÷ (1/4) by using an area model (MP 6A, 2018, p. 46).
Figure 13. The modeling and solution of (4/5) ÷ (1/10) by using an area model (TM 6A, 2018, p. 37).
Figure 14. Simple division in fractions algorithm (TM 6A, 2018, p. 40).
form as (5/8) ÷ (1/4) = (5/8) × (4/1), simplification was done as the result 2(1/2) was
reached. Therefore, In book MP, the topics of dividing a whole number into a fraction
and dividing a fraction into a fraction were presented by basing them on the simultaneous
development of conceptual understanding and operational fluency.
In book TM, the area model was used to solve the questions in the division of two simple
fractions section. By the end of this section, the division of fractions by using the reverse-
multiply algorithm was exemplified with (4/5) ÷ (1/10) = (4/5) × (10/1) (TM 6A, 2018,
p. 40). The book does not directly mention why the reverse-multiply algorithm was used,

Figure 15. Hands-on activity for solving (1/2) ÷ (1/4) by paper folding (MP 6A, 2018, p. 45).
Figure 16. Hands-on activity on fraction disk use (TM 6A, 2018, p. 39).
what this algorithm means, why the reverse-multiply algorithm was used in the solution of
(4/5) ÷ (1/10) and what this algorithm means. In book TM, the question (4/5) ÷ (1/10)
was solved by using the visual representations in Figure 13, and the solution 8 was reached
by considering how many (1/10)s exist in 4/5. At the end of the topic, the rule to be used
when dividing a simple fraction into another simple fraction was given with the example
(4/5) ÷ (1/10) = (4/5) × (10/1) (Figure 14). While book TM based the topic of dividing
a fraction into a whole number on the simultaneous development of conceptual under-
standing and operational fluency, it based the topics of dividing a whole number into a

20 S. Ö. BÜTÜNER
Table 3. The number of measurement and equal sharing problems that have been solved step-by-step
in the textbooks under the heading division in fractions.
Problem type
Text books Shr Msr Problems in textbooks
MP 7 9 Devi had 2/3 of a pizza. She cut it into equal pieces. Each piece was 1/6
of the pizza. How many equal pieces did Devi cut it into? (MP, p. 45,
(a/b) ÷ (c/d) type measurement problem)
TM 10 11 Share 1/4 of a pizza equally between 2 children. What fraction of the pizza
does each child get? (TM, p. 29, (a/b) ÷ c type equal sharing problem).
TB1 5 12 If a half chocolate bar is shared equally between two children, what
fraction of the whole bar does each child get? (TB1, p. 170, (a/b) ÷ c
type equal sharing problem).
TB2 2 10 How many 3/2 lt bottles are needed to pour 18 lt of water? (TB2, p.
100–101, c ÷ (a/b) type measurement-repeated subtraction problem).
fraction and dividing a fraction into a fraction on the development of conceptual under-
standing, and delayed the development of operational fluency. However, both Singaporean
books introduced divisions in fractions by considering the simultaneous development of
conceptual understanding and operational fluency.
In book MP, the topics of dividing a whole number into a fraction and dividing a fraction
into a fraction were presented by basing them on the simultaneous development of con-
ceptual understanding and operational fluency. While book TM based the topic of dividing
a fraction into a whole number on the simultaneous development of conceptual under-
standing and operational fluency, it based the topics of dividing a whole number into a
fraction and dividing a fraction into a fraction on the development of conceptual under-
standing, and delayed the development of operational fluency. However, both Singaporean
books introduced divisions in fractions by considering the simultaneous development of
conceptual understanding and operational fluency. In addition, different from the Turk-
ish textbooks, both Singaporean books include hands-on activities at the end of each unit.
Book MP includes paper folding practices and activities whose solution requires the use of
area models (Figure 15). On the other hand, book TM encourages students to solve divi-
sion in fractions questions with fraction disks (Figure 16). This allows students to consider
the relationship between division and multiplication.
3.4. Findings about the distribution of measurement and equal sharing problems
that have been solved step-by-step in the textbooks
Table 3 includes sample problems and the number of measurement and equal sharing prob-
lems included in the textbooks under the topic of division in fractions and solved step by
step.
When the books were examined with respect to the measurement and separation into
pieces problems whose solutions are given with step by step explanations, it was found
that both Turkish and Singaporean books include problem structures in line with both
meanings of the division operation. TB1 offers step by step solutions to 5 sharing problems
and 12 measurement-repeated subtraction problems, while TB2 offers step by step solu-
tions to 2 sharing and 10 measurement-repeated subtraction problems. In book MP, there
are step by step solutions to 7 sharing problems and 9 measurement-repeated subtraction

problems, while in TM there are 10 sharing problems and 11 measurement-repeated sub-
traction problems. In all Turkish and Singaporean books, all (a/b) ÷ (c/d) type problems
with step by step solutions (the division of a simple fraction into a simple fraction) are
measurement-repeated subtraction problems (finding the group number).
4. Discussion and conclusion
Two Turkish textbooks and two Singaporean textbooks, My Pals Are Here and Targeting
Mathematics, were analysed. The Turkish textbooks used in the study were coded as TB1
and TB2, and the Singaporean books as MP and TM. The first difference between the math-
ematics textbooks used in Singapore and Turkey was the grade level in which the unit on
division in fractions is presented. In Singapore, division in fractions is presented in the
first terms of grades 5 and 6. In Turkey, on the other hand, the division operation is present
only in the grade 6 textbook. Previous comparative studies focusing on various mathemat-
ical topics have shown that countries that are highly successful in international exams start
the instruction of mathematics topics earlier than others (Ding & Li, 2010; Hong & Choi,
2014). For instance, Japanese, Korean and Taiwanese books include the division of frac-
tions into whole numbers in grade 5. Multiplication in fractions is presented in Taiwanese
books in grade 4, and in Japanese and Korean textbooks in grade 5 (Watanabe, Lo, & Son,
2017). On the other hand, in grade 6 books that were coded as MP and TM, the percentage
of pages allocated to division in fractions is 19.6 and 14.8 respectively, while in the Turk-
ish books coded as TB1 and TB2 allocated 8.5 and 6.9 percent respectively. This may lead
to the conclusion that Singaporean books attach more importance to the teaching of divi-
sion in fractions. Li et al. (2009) found that the percentage of pages allocated to division in
fractions in Japanese and Chinese books was higher than that in American books.
The second difference between Singaporean and Turkish textbooks is the location and
the contents of the units on division in fractions in the books. In books TB1 and TB2, the
units on addition, subtraction and multiplication of fractions, which are directly related to
division in fractions, are presented right before this unit. The books MP and TM, on the
other hand, the units on addition, subtraction and multiplication in fractions are intro-
duced in the first term of grade 5. Li et al. (2009) state that similar to Singaporean textbooks,
Chinese books treat the topic of division in fractions in a separate unit from multiplication
in fractions, while Japanese books combine multiplication and division in fractions in one
single unit. Different from Korean and Taiwanese books, Japanese books discuss how to
divide a fraction by a whole number in grade 5 before the topic of multiplication in frac-
tions. On the other hand, Korean and Taiwanese books wait till the end of multiplication
in fractions to present division in fractions (Watanabe et al., 2017).
While books MP and TM start the unit on division in fractions by establishing the rela-
tionship between division in whole numbers and fractions, books TB1 and TB2 start the
unit on division in fractions by showing how to divide positive whole numbers into a sim-
ple fraction. Although TB1 does not include any content about the relationship between
division in whole numbers and fractions, it is a significant deficiency that the result of
(1/2) ÷ 3, which is 1 ÷ 6 and obtained by using the common denominator strategy, is
given as 1/6. In order for students to make sense of the division of a fraction, the estab-
lishment of the relationship between division in whole numbers and fractions is essential
(Gregg & Gregg, 2007; Van de Walle & Karp, 2010). Son (2005) states that both Korean and

22 S. Ö. BÜTÜNER
American textbooks introduce division in fractions by using division in whole numbers.
Other than this difference in the order of the topic of division in fractions in Singaporean
and Turkish books, all four books proceed as follows: contents of division in fractions, the
division of a positive whole number into a simple fraction, the division of a simple frac-
tion into a positive whole number, the division of a simple fraction into another simple
fraction, division problems and solutions. Li et al. (2009), state that a similar instructional
order exists in Chinese, Japanese and American books as well.
The contents in Turkish and Singaporean mathematics books on division of fractions
are based on the development of students’ conceptual understanding and operational flu-
ency. All textbooks teach division in fractions by using area models. Researchers point to
the importance of collective use of area, length and set models in the teaching of fractions
(Clarke, Roche, & Mitchell, 2008; Siebert & Gaskin, 2006). For instance, Japanese books
use a diagram combining a number axis and area model in division of fractions. Korean
and Taiwanese books, on the other hand, use a bar diagram in solving divisions in frac-
tions (Watanabe et al., 2017). Therefore, as Turkish and Singaporean textbooks only use
part whole models, the solution of problems involving division in fractions need to entail
length and set models as well. Turkish and Singaporean textbooks vary with respect to
the explanations they offer on the relationship between visual and symbolic representa-
tions and the relationship between division and multiplication in fractions. For example,
book TB1 solves problems involving the division of a simple fraction into a positive whole
number and the division of a positive whole number into a simple fraction by using area
models and then the common denominator strategy. Book TB2 solves similar problems
primarily by using area models and then resorting to the common denominator strat-
egy and reverse-multiply algorithm. Therefore, both books are based on the simultaneous
development of conceptual understanding and operational fluency. Similarly, Son and Senk
(2010) state that Korean textbooks aim their content on division in fractions to simultane-
ously develop students’ conceptual understanding and operational fluency. Even though
book TB1 attempts to ensure students’ conceptual understanding by using area models
when solving problems about dividing a simple fraction into a positive whole number and
a positive whole number into a simple fraction, it offers no explanation about the rela-
tionship between this solution and the reverse-multiply algorithm. In TB1, the common
denominator strategy is used as an alternative solution to problems involving the divi-
sion of a simple fraction into a positive whole number and a positive whole number into
a simple fraction, but the relevance of this strategy with the reverse-multiply algorithm is
not mentioned. Similarly, although book TB2 uses the reverse-multiply algorithm when
solving division in fractions problems, the meaning of the algorithm or its history is not
explained. As a result, a serious deficiency in Turkish textbooks is the lack of a relationship
between solutions based on visual representation and those in symbolic form (Figures 3
and 7). However, it is claimed that students who move between visual, verbal and sym-
bolic representations may understand the topic better (Siegler & Pyke, 2013). Another
noteworthy deficiency is the lack of establishing a relationship between division and multi-
plication in fractions (the reverse-multiply strategy) when solving problems involving the
division of a simple fraction into a positive whole number and that of a positive whole
number into a simple fraction. It is essential for mathematics instruction to relate math-
ematical concepts with one another (for instance, fraction-decimal fraction-percentage
or the multiplication-division operation, etc.) and to explain what a mathematical rule

or algorithm means and how they emerged because when students follow a procedure
they do not understand, they have difficulty evaluating whether their results are mean-
ingful. Also, algorithms whose underlying meaning is not understood run the risk of
being confused with one another (Van de Walle & Karp, 2010). For example, (1/2) ÷ 3
was solved in TB1 by first using the area model and then the common denominator
strategy, and written as (1/2) ÷ (6/2) = 1 ÷ 6 = (1/6). The book TB1 does not include
any content on the relationship between the common denominator strategy and reverse-
multiply algorithm. Beyond this, the textbook does not explain the relationship between
the division and fraction but writes it as 1 ÷ 6, (1/6). However, the Turkish textbook could
include (1/2) ÷ 3, ((1 × 1)/(2 × 1)) ÷ ((3 × 2)/2 × 1) = ((1 × 1)/(3 × 2)), to reach the
equation (1 × 1)/(3 × 2) = (1/2) × (1/3). This would reveal the relationship between
division in fractions and the reverse-multiply algorithm. Also, prior to giving the equation
1 ÷ 6 = (1/6) the problem of equally distributing a cake to 6 people could be used to make
students learn the relationship between division and fractions, and represent it in the sym-
bolic form. Similarly, Son and Senk (2010) emphasize that American textbooks give the
reverse-multiply algorithm as a rule and do not explain the relationship between the com-
mon denominator strategy and the reverse-multiply algorithm, while Korean textbooks do
mention the relationship between the common denominator strategy and reverse-multiply
algorithm.
Singaporean textbooks did not use the common denominator strategy in the solution
of problems regarding the division of a simple fraction into a positive whole number and a
positive whole number into a simple fraction. Singaporean books MP and TM explain the
relationship between division and multiplication in fractions at the beginning of the unit
about division in fractions. Area models are used, followed by the explanation of the rela-
tionship between division and multiplication in fractions, when solving problems involving
the division of a simple fraction into a positive whole number and the division of a positive
whole number into a simple fraction in book MP, and when solving problems involving
the division of a whole number into a simple fraction in TM (Figures 4 and 5). Also,
they include fewer solution strategies than their Turkish counterparts when solving prob-
lems involving division in fractions. Son and Senk (2010) state that the unit on division in
fractions in American textbooks is designed for students’ conceptual learning, while their
acquisition of operational fluency is delayed. Book MP presents an equal sharing problem
in which half a cake is shared equally between three children. Considering that each child
would get one-third of the half cake, the equivalence between (1/2) ÷ 3 and (1/3) × (1/2)
was established. Following this, the equation (1/2) × (1/3) = (1/3) × (1/2) was used to
obtain (1/2) ÷ 3 = (1/2) × (1/3) (MP-6A, 2018, p. 35). Similarly, in book TM, one fourth
of a pizza is distributed equally between two children. When solving the problem what
fraction of the pizza each child will get, it is concluded that each one would get half of
the one-fourth of the pizza, and it is stated that (1/4) ÷ 2 equals (1/2) × (1/4). Following
this, the equation (1/4) × (1/2) = (1/2) × (1/4) was used to obtain (1/4) ÷ 2 = (1/4) ×
(1/2) (TM-6A, 2018, p. 29). Therefore, both Singaporean books introduced divisions in
fractions by considering the simultaneous development of conceptual understanding and
operational fluency.
The division of two simple fractions into one another is treated in all books as the oper-
ation of determining how many of the second fraction exists within the first fraction. In
book TB1, the solution is found by using area models supported by verbal explanations,

24 S. Ö. BÜTÜNER
therefore trying to establish the relationship between division and multiplication in frac-
tions (the reverse-multiply algorithm), albeit delayed. Book TB1 has the deficiency of not
mentioning the relationship between division and multiplication when dividing a fraction
into a positive whole number and a positive whole number into a fraction. In book TB2,
problems involving the division of a simple fraction into another simple fraction are solved
by using area models followed by the common denominator strategy. In TB2, the relation-
ship between division and multiplication in fractions, and the one between the common
denominator strategy and the reverse-multiply algorithm are not established. In the Singa-
porean book MP, problems involving the division of a simple fraction into another simple
fraction are first solved by using area models, and then by simplifying in the symbolic form
((a/b) ÷ (c/d) = (a/b) × (d/c)). On the other hand, the Singaporean book TM does not
continue the solution via area models with a solution based on the symbolic form such as
((a/b) ÷ (c/d) = (a/b) × (d/c)), and instead gives the result as (a/b) ÷ (c/d) = e. How-
ever, while TM includes a rule about the direct use of the reverse-multiply algorithm at the
end of the unit, MP does not give such a direct rule. In books MP and TM, as the relation-
ship between division and multiplication in fractions (the reverse-multiply algorithm) is
explained, there was no mention of this in the solution of the division of a simple fraction
into another simple one.
As a result, it may be argued that Singaporean books encouraged students to use fewer
solution strategies but offered them more opportunities to grasp the relationship between
division and multiplication in fractions. At the same time, the hands-on activities in Singa-
porean books may also help students to learn the topic of division in fractions conceptually.
Even though the use of area models in Turkish books makes room for students’ conceptual
learning, the delayed presentation of the relationship between division and multiplication
in fractions at the end of the unit only in book TB1 (the division of a simple fraction
into another simple fraction) and the presentation of the common denominator strat-
egy and the reverse-multiply algorithm only as rules may cause students to use these
rules mechanically. Li et al. (2009) and Sun (2011) state that Chinese and Japanese text-
books solve division in fractions problems with various solution strategies; establish links
between solutions made via visual, verbal and symbolic representations; and explain the
relationship between division and multiplication in fractions (reverse-multiply strategy).
Therefore, Turkish and Singaporean textbooks may use different solution strategies when
solving division in fractions problems and explain the relationship between division and
multiplication in fractions (the reverse-multiply strategy). Turkish textbooks should par-
ticularly pay attention to solving problems via visual, verbal and symbolic representations
and make sure that these representations are interrelated. Strengthening the ability to move
between different representations improves student learning and helps them retain their
learning (Van de Walle & Karp, 2010).
When the books are examined in relation to the measurement and separation into pieces
problems whose step-by-step solutions are given, both Turkish and Singaporean books
can be observed to include problem structures in line with both meanings of the divi-
sion operation. While there was a similar number of problems in Singaporean books in
line with the equal sharing and measurement meanings of the division operation, Turkish
books largely include problems in line with its measurement meaning. It is also a deficiency
that in all Turkish and Singaporean books, all (a/b) ÷ (c/d) type problems whose step-
by-step solutions are given (the division of a simple fraction into another simple fraction)

are measurement-repeated subtraction (finding the group number) problems. When faced
with equal sharing problems, students often believe that the divider needs to be a whole
number, and the divider and the division need to be smaller than the dividend. The student
misconception that the sum of the division should always be smaller stems from their early
elementary school experience of doing operations with the elements of the natural num-
bers set (Tirosh, 2000). Therefore, it is important to include both measurement and equal
sharing problems in classes in a balanced way. The findings have shown that Singaporean
students may display a better performance than Turkish students in problems based on
division in fractions.
5. Limitations, implications and future research
Findings from the study suggest that the Singaporean textbooks offer students with more
learning opportunities in learning divisions in fractions. Caution needs to be paid to the
generalizability of the results obtained in this study as it compared four textbooks from two
countries regarding their division in fractions contents. Also, considering the role of teach-
ers in the instructional process (Wijaya, Van den Heuvel-Panhuizen, & Doorman, 2015;
Yang, 2018), having full content in the textbook is not adequate on its own for effective
mathematics instruction as there are many other factors that affect student achievement
in mathematics (such as parents’ education level, the number of educational resources at
home, the role of the teacher, etc.). A textbook, well-designed or not, should come to life
in the hands of a well-equipped teacher. Such a teacher can identify the deficiencies in
textbooks and enrich classes with content that allows students to more easily learn the
underlying meaning of mathematical concepts via questioning. Such a teacher can also
bring to the classroom original problems that are not present in the textbook but serve to
measure conceptual understanding. S/he may use solution strategies that do not feature in
the textbooks. Therefore, caution is needed when generalizing the results from the study.
The implications of the findings may be discussed from two perspectives. The expla-
nation of the relationship between division and multiplication in fractions (the reverse-
multiply algorithm) at the beginning of the unit with real life examples in Singaporean
textbooks may help Singaporean students to grasp the relationship between these two
mathematical concepts. Of the Turkish books, only one (TB1) gives the relationship
between division and multiplication in fractions (reverse-multiply algorithm) in a delayed
way at the end of the unit. This may limit students’ experiences with the relationship
between the two concepts. The fact that Singaporean books represent the solutions reached
with area models in symbolic form as well may help students to move from visual repre-
sentation to symbolic representation. The lack of a relationship shown in Turkish books
between the solution based on area models and solution in the symbolic form (the com-
mon denominator strategy) (For example, Figures 2 and 3) may make it hard for students
to learn the link between visual and symbolic representation. However, the general defi-
ciency in all books regarding content that allows for two-way movement between visual,
symbolic and verbal representations may cause students difficulty in moving between rep-
resentations. The presence of many more solution strategies in Turkish textbooks than in
Singaporean textbooks, may enable Turkish students to use more than one strategy in the
solution of division in fraction problems. However, even though the use of area models
in Turkish books may give students the opportunity to learn the topic conceptually, the

26 S. Ö. BÜTÜNER
failure to explain the relationship between division and multiplication in fractions and the
solution methods given only as rules (the common denominator strategy, reverse-multiply
algorithm) may make students use these rules mechanically. This study first analysed the
grade levels and the extent of the topic of divisions in fractions in the books, and then
compared the order of topics. Following this, the analysis focused on how division with
fractions was presented, how the algorithm was developed, and the kinds of models that
were used. Finally, the distribution of equal sharing and repeated-measurement subtrac-
tion problems, whose step-by-step solutions were given in the books, was analysed. Future
studies may analyse fraction problems in Turkish and Singaporean textbooks by using the
categories of number of steps, type of answer, context and cognitive expectation. Besides,
future research may investigate how Turkish and Singaporean teachers use textbooks in
their courses, whether they make use of sources other than the textbook in their classes,
how they teach the meaning of the concept of division in fractions, whether they include
problems and solution strategies other than those in the textbooks, and what effects this
has on student learning.
Disclosure statement
No potential conflict of interest was reported by the author.
ORCID
Suphi Önder Bütüner http://orcid.org/0000-0001-7083-6549
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