2013. Vol.4, No.8A, 19-24
Published Online August 2013 in SciRes (http://www.scirp.org/journal/ce) http://dx.doi.org/10.4236/ce.2013.48A005
Copyright © 2013 SciRes. 19
How Student Teachers Use Proportional Number Line to Teach
Multiplication and Division of Fraction: Professional Learning
in Context of Lesson Study and Open Approach
Tipparat Noparit1,2, Jensamut Saengpun2*
1Centre of Excellence in Mathematics , CHE, Bangkok, Thailand
2Programme in Mathematics Education, Faculty of Education, Chiang Mai University, Chiang Mai, Thailand
Email: firstname.lastname@example.org, *email@example.com
Received July 7th, 2013; revised A ugust 7th, 2013; accepted August 14th, 2013
Copyright © 2013 Tipparat Noparit, Jensamut Saengpun. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original wor k is properly cited.
The objective of this study is to examine how student teachers use proportional number line in teaching
multiplication and division of fractions in context of lesson study and open approach. Teaching experi-
ment was employed to conduct this case study qualitative research with two mathematics student teachers
who participated in the study as case studies. This research was carried out in two sixth grade mathemat-
ics classrooms from two schools project innovated by lesson study and open approach in the second se-
mester of 2012 academic year. Research data included lesson plans on multiplication and division of frac-
tions, classroom observation with video tape recordings, students’ written works and interviewing with
the student teachers. The results showed that the student teachers use proportional number line in three
ways. Firstly, they use it for asking students to interpret the problems using proportional table aiming at
writing equation of the problem correctly in the step of posing open-ended problem situation. Secondly,
they use it as a tool for giving student think about how to calculate the answer by themselves. Thirdly,
they use it for connecting and checking the various way of thinking about calculation of fractions. The
student teachers conceived that learning to teach calculation of fraction with proportional number line is
beneficial to their own professional learning in teaching through open approach and student learning and
thinking proportionally thought to be the most complex of the elementary mathematics curriculum.
Keywords: Proportional Number Line; Multiplication and Division of Fraction; Lesson Study and Open
Abu Ja’far Muhammad ibn Musa al-Khwarizmi, known for
short as “al-Khwarizmi” introduced to the Arabic world in
Baghdad around 800 BC the Hindu notion of fraction, and
showed how operations on these fractions were more or less
mechanical, as distinct from the sophisticated relational thought
required to interpret and operate with ratios in Euclid’s sense
(Crossley & Henry, 1990). What al-Khwarizmi saw was a ma-
jor intellectual achievement, many students—perhaps most—
see as the first significant cabalistic mystery of mathematics:
operations on fractions (Davis, 2003). The mechanical opera-
tions on fractions seemed to have the effect of turning teachers’
and students’ away from the beautiful ideas of proportion and
ratio to mechanical manipulation of odd pairs of whole num-
bers, written as “a/b”. Fractions have been said to be the most
intricate numbers to deal with in arithmetic and division has
been thought to be most complex of mathematical operations.
What does happen in school age learning on fraction?
It is hardly surprising that many adolescent students who can
apply numerical approaches meaningfully in addition context,
not able to apply such approaches to the multiplicative struc-
tures associated with proportional reasoning (e.g., Karplus,
Pulos, & Stage, 1983). Indeed many of errors that students
show in multiplication and division on fraction problems illus-
trate that they apply additive or subtractive thinking processes
rather than multiplicative processes (Karplus et al., 1983). So,
exposing students to routine multiplication and division prob-
lems of fraction alone has not been effective in helping students
to develop deeper understanding of more complex problems
needed proportional thinking. This is in part because students
need to understand fractions as well as multiplicative concepts
(Lo & Watanabe, 1997). How then to teach operation of frac-
tion for making sense for them?
The teaching and learning of fractions is problematic (Pearn
& Stephens, 2004). Pearn and Stephens (2004) found that this
topic was a major problem for students because they did not
understand the part-whole relationships described in fraction
notation. Teaching this topic should use multiple representa-
tions of fractions such as discrete and continuous quantities and
the number line. Many students often struggle with how to
think about calculating the fraction and interpreting fraction
operation problems. In authors’ experience, we have much
more problems in teaching fraction, especially in teaching the
operations on fraction of which the causes can be characterized
T. NOPARIT, J. SAENGPUN
in two parts; the first is teachers’ lacking knowledge for teach-
ing fraction with understanding of proportional relationship and
the second is lacking linkage between fractions and proportion
in mathematics textbooks.
Japanese mathematics textbooks introduced how to think
about calculation of fraction with proportional number line in
order to develop the way students learn to calculate fraction by/
for themselves (Isoda, 2010). As Kishimoto (2010) defined a
proportional number line as use of two number lines to show
the multiplicative relationship between two quantities, and can
also be referred to as a line segment diagram, a multiple line
diagram, a double number line, or a functional scale. The pro-
portional line first came to be widely used in mathematics edu-
cation when the Japanese educational guidelines announced in
1958 assigned the “concept of ratios” as the implication behind
the multiplication and division of decimals and fraction (Ki-
shimoto, 2010). Typical Japanese mathematics textbook (e.g.
Gakkoh Tosho, 1995) first used tape diagram represent the
quantities at first grade and used tape diagram as a graphic
representation of multiplication in fifth grade on unit about
decimal multiplication, and the use of proportional number line
to calculate multiplication and division problem in sixth grade
since 1971. This approach is the consequence of the hundred
years of lesson study, Japanese teacher professional develop-
Thailand has adopted lesson study to teacher education by
Center for Research in Mathematics Education, Khon Kaen
University since 2004. Inprasitha (2010) took open approach as
a teaching approach incorporating with lesson study process in
order to provide teacher who come to work collaboratively with
construct task and problem situation to teach student learning
and thinking by/for themselves. In Thailand, lesson study and
open approach is then becoming an innovation for Thai teacher
professional development that helps teacher recognize this as-
pect of students’ mathematics learning. The Open Approach as
“problem solving approach” used in Japan is one shared theory
for developing children who learn mathematics by/for them-
selves (Isoda, 2010). It includes teaching about learning how to
learn. The students often gain opportunities to learn mathemat-
ics with understanding and meaningful. Inprasitha (2010) has
conceived that the “Open Approach” is a teaching approach
used in cooperated with lesson study to design learning units
and lesson plans. The open approach is consisted of 4 steps as
follows: 1) posing open-ended problem situations, 2) students’
self-learning, 3) whole-class discussion and comparison, and 4)
summarization through connecting students’ mathematical
ideas emerged in the classroom.
One feature of lesson study in Thailand is usage of Japanese
elementary mathematics textbooks (e.g. Gakkoh Tosho, 1995)
to design unit and problem situation for teaching in real math-
ematics classroom. A group of student teacher from faculty of
education majoring in mathematics has participated in lesson
study and open approach project. They worked with in-service
teachers in lesson study process for one academic year and they
usually use the Japanese textbook as tool for their planning the
lesson. Although they have trained for teaching through open
approach, they often struggle with the concept of calculation on
fraction with proportional number line embedded in their text-
book. With their effort to beyond the struggle, in seminar class
for student teacher they took this topic to discuss with their peer
and supervisors to think critically on this subject matter and
how teaching should be. In this case, the authors concerned that
the teachers’ learning for teaching mathematics in their profes-
sional development is a crucial role to enhance and support stu-
dents’ learning (Ball et al., 2008; Ma, 1999).
This study aims to provide detailed insight into student
teacher’s learning by examining how they use proportional
number line to teach multiplication and division of fractions.
Specifically, we concerned what knowledge student teachers
gained from using Japanese mathematics textbooks plentiful
with proportional number line and also analyzed the student
learning process and outcomes from their teaching through each
step of open approach.
Context and Methodology
This study is a case study qualitative research. The study was
employed teaching experiment (Cobb and Steffe, 1983) for
conducting the research. Two case studies of mathematics stu-
dent from program in mathematics education, Chiang Mai Uni-
versity, who taught in sixth grade in Chum Chon Ban Bua Krok
Noi school and Ban Mae Sa school, the elementary schools in
Chiang Mai province, which participated in the “Project for
mathematics teacher professional development innovated by
lesson study and open approach in northern educational service
areas” since 2009 academic year. The project was conducted by
the center for research in mathematics education, Khon Kaen
University and the mathematics education program, faculty of
education, Chiang Mai University.
The teaching experiment was carried out in two sixth grade
mathematics classroom which is each class taught by each stu-
dent teacher. The first class includes 41 students aged 11 - 12
year and another class with 26 students the same age. Each
student teacher taught two chapters on “Multiplication and Di-
vision of Fraction” according to the sixth grade Japanese ele-
mentary mathematics textbook volume 2 (Gakkoh Tosho,
1995). The first chapter includes “Calculations of Fraction ×
Whole Number” and “Calculations of Fraction ÷ Whole Num-
ber”. Another chapter includes “Calculations of Fraction ×
Fraction” and “Calculations of Fraction ÷ Fraction”.
Data were collected during December 2012-January 2013 in
the second semester of 2012 school year and included daily
videotaped recording of whole unit consecutive lessons on
“Multiplication and Division of Fraction” in their mathematics
classroom teaching made by two cameras. During each class-
room teaching, one camera focused primary on interaction be-
tween teacher and student, especially in whole-class discussion.
The second camera focused on students’ group working.
Moreover, documentation consists of lesson plans; students’
written works; daily field notes that summarized classroom
events and student’s ways of thinking. The data from the class-
room video recordings of each class were transcribed into pro-
tocol and analyzed the way student teachers use proportional
number line in teaching calculation of fractions by video analy-
sis. After teaching the unit, the authors interviewed the student
teacher after they already taught the unit with the series ques-
tions in theme of “What you have learned for teaching calcula-
tion of fraction with proportional number line in context of
lesson study and open approach?” The interview was recorded,
and the transcripts were analyzed according with the video
Copyright © 2013 SciRes.
T. NOPARIT, J. SAENGPUN
Teaching Mul tip l ic ati o n an d Di vi si on of Fractions
with Whole Numbers
Through the analysis of the unit lesson plans on multiplica-
tion and division of fraction with whole number, the goal of
this learning unit was to explore mathematical ideas, methods
and learning processes on multiplication and division of frac-
tion with whole number. Then teachers aimed student to write
an expression to represent the given problem situation. Students
would be learned how to calculate and solve the problem situa-
tions through using unit fraction, commutative law, division
rule, drawing, showing the calculation of fraction with tape
diagram in order to conclude the concept of multiplication and
division of fraction with whole number. The major learning tool
of learning unit is “tape diagram” cooperated with “unit frac-
tion” used for think about how to calculate the answer.
In their classroom teaching through open approach on first
lesson of the unit, in posing problem session, student teachers
posed problem situation according to the textbook as show in
Problem Situation for Calculation of “Fraction × Whole
“On this fence, 1 dl of green paint is enough for an area of
2. How many m2 of this fence can be painted by 3 dl of
1) Color in the figure.
2) Write an equation and think about how to calculate the
The teachers attempted to convey their students in class to
look for the quantitative relationship between area of painting
(m2) and quantity of color (dl) which is represented by fence
picture and a number line (Figure 1). The teachers used the
illustration to interpret the problem situation and then to write
equation for calculation of fraction correctly. Moreover, it lead
student to think about how to calculate with using unit fraction
as show in Figure 2.
In session of students’ self-learning, most students colored
the space representing the answer along with the number line.
Some of them used repeated addition 222
late the problem. Some students concerned the meaning of unit
fraction; they looked for one piece of color painting that is rep-
resenting 25 and counted it and multiplied it by 3. One stu-
dent in the classroom used proportional number line as the Fig-
ure 3 shown. He wrote a sentence near the fi gure that “2 multi-
ply 3 is 6”. Teacher noticed and took a note this way of think-
In whole-class discussion and comparison session, teacher
selected and sequenced students’ written works for presentation.
The ideas for discussing sequenced through repeated addition
tiplication of fraction with whole number.
to using proportional number line. Teacher ex-
tended the idea of using proportional number from the record to
the whole class and asked for discussion (Figure 4).
In summing up the lesson session, teacher attempted to con-
nect the student’ ideas from repeated addition, using propor-
tional number line, using unit fraction and multiplication rule
respectively. In last session, teacher conveyed student to notice
each calculation ideas and then concluded the principle of mul-
Figure 1. n of problem with picture and a number line (Gakkho Representatio
Tosho, 2005: p. 3).
Figure 2. ith unit fraction (Gakkho Tosho, 1995: p. 4). Calculation w
Figure 3. of proportional num ber line.
Figure 4. tending proportional number line student used.
From the classroom analysis above, we found that propor-
nal number line emerge from one student. Teacher used the
idea of using number line as a tool for extending and connect-
ing with other ideas emerged in the classroom. Moreover, the
lesson study team reflected after classroom teaching on the
issue about connecting using unit fraction and proportional
Copyright © 2013 SciRes. 21
T. NOPARIT, J. SAENGPUN
number line during student solving problem. Then, teacher
decided to concentrate with designing the lesson plan with
proportional number line appropriately in next lesson.
In teaching first lesson on “Fraction ÷ Whole Number”, the
ion for Calculation of “Fra
Nnce, 2 l of paint is enough for an area of
udent teachers aimed students to write equation Fraction ÷
Whole Number from analyzing and interpreting problem situa-
tion with proportional number line. Although the Japanese ma-
thematics textbook does not approach proportional number line
in this topic, but teachers still use it as a tool for thinking about
how to calculate the fraction because the students have familiar
with and conceived proportional number line to solve the prob-
lem. The figure below is the illustration of conception of calcu-
lation “Fraction ÷ Whole Number” that the student teachers
generated with themselves and explained it in the les- son plan
(see in Figure 5).
Problem Situatction ÷ Whole
On this fe56 m.
e painted with 1 l of paint?
Teaching Mul tip l ic ati o n an d Di vi si on of Fractions
ysis of the unit lesson plans on multiplica-
room teaching through open approach on first
tion of “Fraction × Frac-
ti an cover an area of
How many m2 can be painted with each 1 l of paint
1) Write an equation.
2) How many m2 can b
Find the answer by coloring in the figure.
Through the anal
n and division of fractions with fractions, we found that the
goal of this learning unit was to explore mathematical ideas,
methods and learning processes on multiplication and division
of fractions with fractions. Student will be able to analyze prob-
lem situation and then write expression to represent given
problem situation easily. Students would be learned how to
calculate and solve the problem situations through using crucial
learning tools including using unit fraction, proportional table
and proportional number line with multiplication and division
rule. Student could be used previous concept of multiplication
and division of fraction with whole number helping in calculat-
ing in this unit. In addition, student could be inferred the prin-
ciple of multiplication and division of fractions with fractions
In their class
sson of the unit, the problem situation was posed according to
the textbook as show in Figure 6.
Problem Situation for Calcula
We c45 m with1 l of blue paint
1) How many m2 can wever with 13 l of paint? Write
ane f equation then check the area by using tigure on the right.
2) How many m2 can we cover with h
23 l of paint? Write
an he area that can be covered with
3) Check t23 l of paint
by te the area that can be covered
using the figure on the right.
4) Think about how to calcula
ith 23 l of paint.
The teacher posed problem 1) first and then asked students to
interpret the problem with the figure and coloring the area with
13 l of paint. The teacher conveyed students to notice the
nge in the figure and present the change discussed in form
“proportional table”. The teacher wrote the table on the black-
board as show in Figure 7.
From Figure 7, in order to
Figure 5. number line for Calculation of Fraction ÷
Figure 6. ation with picture and a number line
orrectly, the teacher asked student to discuss the proportional
(Gakkho Tosho, 2005: p. 13).
table and looked for the area that could be calculated by using
multiplication rule. This teaching scene was very important to
support student thinking with ratio. Most of student was able to
address correctly that the equation for problem 1) is 41
leaded student to think about how to calculate the area in prob-
- 4), stu-
lem 1) with proportional number line by themselves.
During students’ self-learning in solving problem 2)
thinking about how to calculate 42 two
ough proportional number line and using the
pr rough unit fraction.
write the equation of the problem
1) Thinking thr
2) Thinking th
Copyright © 2013 SciRes.
T. NOPARIT, J. SAENGPUN
Copyright © 2013 SciRes. 23
often used proportional number line to check and verify the
answer from the other ways of calculation.
The following illustration is some example of learning out-
come of students in using proportional number line to calculate
and solve problem on “Fraction ÷ Fraction”. It showed the
learning process that student able to think proportionally when
solving the problem of fraction.
What the Student Teachers Have Learn from Lesson
Study and Open Approach Context
According to analysis of interviewing transcripts, the authors
found that both of the teacher student recognized that Japanese
mathematics textbook used in the project is very important and
has a crucial role in teaching multiplication and division of
fraction in mathematics classroom taught by open approach.
They accepted that learning calculation of fraction is the most
difficulty topic in elementary mathematics especially with in-
terpreting the problems. They told that Japanese mathematics
textbooks help teacher and student learn to interpret the prob-
lem with using proportional number line. Although in the be-
ginning step of open approach; posing open-ended problem
situation, it functions as a tool only for interpretation of prob-
lem, but when student comes to solving the problem by them, it
takes a role as tool for actual leaning by and for themselves.
They conceived that the proportional line helps student write
equation represented the problem correctly and easily. More-
over, it help student to think proportionally which is normally
hard to understand. They accepted that before they have par-
ticipated the lesson and open approach project, more and more
topic in mathematics they learn by rote learning. When they
became to be a student teacher, they recognized that the devel-
opment of students’ learning depend on teacher knowledge
about how to teach student to construct leaning tool by them-
selves. Additionally, working with other teachers in their lesson
study team encourages them to develop their knowledge for
Figure 7. oport io n a l t a b l e.
The students recognized unit fraction of this picture that is
The use of pr
2. They easily addressed that there are 4 × 2 set of is
2. Then the area is 42
In w parisession, teacher
as hole-class discussion andcomon s
ked students presented their ideas about how to calculate
42content of discussion and comparison in this lesson
both aimed at checking how to
e, proportional table came to
use proportional number line
and unit fraction and finding the common principle for calcula-
tion of Fraction × Fraction according to students’ ideas. In ses-
sion of lesson summing up, the teacher asked students notice
and concluded about common principle for calculation of Frac-
tion × Fraction that they gained.
From classroom analysis abov Discussion
This study provided an opportunity to understand how stu-
dent learn to teach calculation of fraction with using propor-
tional number line in context of lesson study and open approach.
As we known, teaching calculation of fraction especially multi-
a mediating tool for making the correct equation from the
problem situation and it leaded into making the proportional
number line in order to think about how to calculate “Fraction ×
Fraction” objectively and easily. Additionally, another teacher
Example of student’s us in g p roportional number line to calculate 23
T. NOPARIT, J. SAENGPUN
plication anf fraction is very difficult. Teacher often
g the problems with using proportional table
r line is beneficial to their own learning and students’
by Centre of Excellence in Mathe-
matics, the Comm (CHE), Si Ayut-
thaya Rd., Bangko
D. L., Thames, M. H. & Phelps, G. (2008). Content knowledge for
teaching: What makeeacher Education, 59,
d division o
orithm of calculation of fraction directly without
ng. From the results, we learn important thing in Ball,
approaches of calculation of fraction in Japanese mathematics
textbook used in the project. Fortunately, the student teachers
who were participated in lesson study and open approach pro-
ject were the people who have more curiosity to learn new
teaching approach. They use amount of time to read and under-
stand the textbook and discuss together how to design task and
how to teach.
The results showed that the student teachers use proportional
number line in three ways. Firstly, they use it for asking stu-
ming at write equation of the problem correctly in the step of
posing open-ended problem situation. Secondly, they use it as a
tool for giving student to think about how to calculate the an-
swer by themselves. Thirdly, they use it for connecting and
checking the various way of thinking about calculation of frac-
The results indicated that the student teachers conceived that
learning to teach calculation of fraction with proportional
arning. For student teachers, it appears that uncovering learn-
ing goal with supportive learning tools is very important and
needed to know and improve continually (Murray, 1996). For
students, they should be provided with opportunities to develop
an understanding of fraction concepts with proportional number
line before the formal introduction of algorithms. While many
students have experienced great difficulty in solving problems
that involve fractions (Davis, 1993, 2003; Copp et al., 1983),
these students demonstrated proportional thinking and under-
standing of division of fractions, thought to be the most com-
plex of the elementary mathematics curriculum (Ma, 1999).
The implication of the study here is that when a new genera-
tion of student teachers are given the time and the opportunity
to explore and develop their mathematical knowledge
aching in a supportive professional development process such
as lesson study and open approach, they become empowered to
think about teaching mathematics in order to develop profes-
sional learning simultaneously with develop students’ learning
by /for themselves.
This work was supported
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