©Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach. Tokyo: CRICED, University of Tsukuba

Fraction for Teachers
Knowing What before Planning How to Teach

Masami Isoda, PhD/Prof. Director of CRICED, University of Tsukuba, JapanFirst Edition for the SMASE-INSET project by JICA, Japan and ME, Nigeria
Electronically Published from CRICED, University of Tsukuba, May 8, 2013
Revised September 21, 2019

2nd Edition prepared for SEAMEO School Network program

Mathematics Education to Develop Students Agency:
The Case of Fractions
October 23-December 4, 2021.
https://www.criced.tsukuba.ac.jp/seameo_online-program-fractions.html

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Preface for First Edition

Education is the work to prepare for the future. Developing children who learn mathematics byand for themselves is one of the major issues on mathematics education reforms in the world (Seesuch as Isoda & Katagiri, 2012). After the comparative study of mathematics classroom such as
TIMSS video study in 90s, Japanese lesson study is the world-shared methodology as for thetools for professional development because the study indirectly demonstrated the quality ofJapanese mathematics teaching and it is established by the lesson study. However, people often
misunderstand the lesson study as for the talking about the class rather than studying subject
matter. They enjoy the classroom observation likely listening to the music or watching thetheatre. However, through listening to the music, and even if we enjoy talking about actors, wecannot prepare the good player ourselves. In Japanese lesson study, most efforts are done for the
preparation of the class. The misunderstanding originated due to the limitation of the content
guidebook to refer in English. On this reason, I have developed several resources which show thetheory for the purpose to improve mathematics education with researches in the world.
For the workshop of SMASE-INSET project under Japan International Cooperation Agency(JICA), Japan and Federal Ministry of Education (FME), Nigeria, this booklet includes theessential theory for enabling teachers to plan the class for developing children who learn
mathematics by and for themselves. It focused on the innovation of elementary schoolmathematics based on the content which is well written in the textbooks in each country andknown by teachers. The workshop done in Nigeria was based on the author's experience inCentral and South America, South East Asia and Pacific as well as in Japan.
May 7, 2013
Masami Isoda, PhD
CRICED, University of Tsukuba, Japan
Copyright for First edition:
Author allows anyone (including SMASE-INSET) to use this booklet when he/she acknowledges it in his/her
reference completely: Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach.
Tokyo:CRICED,
UniversityofTsukubawhich
downloadedfrom
http://mathinfo.criced.tsukuba.ac.jp/museum/dbook_site/
Copyright for Second edition:
Second edition which divided the Chapter 6 to Chapter 6 and 7 and re-brushed up English was prepared for
SEAMEO School Network. Author allows anyone to use this booklet when he/she acknowledges it in his/her
reference completely: Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to
Teach. (2nd edition) Tokyo: CRICED, University of Tsukuba which downloaded fromhttps://www.criced.tsukuba.ac.jp/online-contents.html
Pictures of the English Edition of Japanese-Mathematics Textbook are extracted from
'Study with Your Friends MATHEMATICS for Elementary School (Gakko-Tosho;2005)'. When user extracts the pictures from the booklet, he/she needs thepermission
fromGakko-Tosho:
Katsuaki
Serizawa(e-mail:katsuaki.serizawa@gakuto.co.jp), GAKKO TOSHO CO., LTD. 3-10-36 Higashi-jujo Kita,
Tokyo, 114-0001, Japan. https://support.gakuto.co.jp/mathematics-textbook/2
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Acknowledgment
Author Masami Isoda, PhD, acknowledges Gakko-Tosho which allowed the author to use
the textbooks 'Study with Your Friends MATHEMATICS for Elementary School (2005)
for this booklet. The content of this booklet was adopted for the workshop in SMASEINSET by the author. Author acknowledges the following colleagues who contributed to
the workshop in Abuja, Nigeria in May, 2013.
National Trainers
Olasinde Kudrat, Hadizat Garba, Zainab Muhammad Shuaibu, Abubakar
IbrahimState Trainers
Ahmed Eneji, Mohammed Kabir Falalu,Monday Kurah
Japanese Staffs of JICA SMASE-INSET
Kazuhiro Okamoto, Tsuyoshi Ito, Hiroko Miura, Masako Tsuzuki, Hiromi Miyao
JICA Nigeria
Tatsuo Seki
Additionally, the author deeply acknowledges a number of colleagues who has beenworking with or supporting the author to develop the materials in the world.
Maitree Inprasitha (Thailand), Peter Gould (Australia), Catherine Lewis (USA),
Raimundo Olfos (Chile), Tenoch Cediro (Mexico), Ui Hock Cheah (Malaysia), Suhaidah
Tahir (Malaysia), Kaye Stacey (Australia), Max Stephens (Australia), Abraham Arcavi(Israel), Fadja (Indonesia) and all colleagues of APEC Lesson Study project:
http://www.criced.tsukuba.ac.jp/math/apec/
Akinori Ito (JICA, PNG project)3
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About the Author
Affiliation and Address:
CRICED/Graduate School of Education, University of Tsukuba, 305-8572, Japan.
Website:http://math-info.criced.tsukuba.ac.jp/
Title:PhD (Education), Waseda University;
Honorary PhD (Mathematics Education), Khon Kaen University, Thailand;
Associate Professor, University of Tsukuba (1998-).
Education:
Master Program of Education, University of Tsukuba (1984).
Academic Membership:
Advisory Board Member, History and Pedagogy of Mathematics (2002-);
Project Overseer (Director), APEC Lesson Study Project in Mathematics (2006-);
Council Member, World Association of Lesson Study (2011-);
Chief Editor, Journal of Japan Society of Mathematical Education (2010-).
Awarded by:
Minister of Education (MEXT), Japan, as for the Best Educational Software in
the Year 2005;
President of the Japan Publishers Association as for the Most Beautiful Book of
the Year 2010 in the area of natural science;
Honorary PhD (Mathematics Education) by Khon Kaen University, Thailand as
for his contribution to Khon Kaen University and Thailand, 2011.

Book for Acquisitions Program of Chilean Authors, Selected by National Committeeof Culture and Art, Government of Chile. 2012.
President of the University of Tsukuba, as for the Best Faculty Member of theYear 2012. 2013.
Further CV and Publications:Japanese Full
http://www.trios.tsukuba.ac.jp/Profiles/0006/0000997/profile.html
EnglishPart
http://www.trios.tsukuba.ac.jp/Profiles/0006/0000997/prof_e.html

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Table of Content

Preface for First Edition ...................................................................................................... 2
About the Author ................................................................................................................. 4
Table of Content .................................................................................................................. 5
Chapter 1: What is fraction? ............................................................................................... 7
Questions for professional development 1 ..................................................................... 10
Major Reference and Further readings 1 ...................................................................... 11
Chapter 2: Dividing Fraction and Quantity Fraction ....................................................... 12
Questions for professional development 2 ..................................................................... 15
Dividing Fraction ........................................................................................................... 16
Exercise: ......................................................................................................................... 17
Operational (measuring) Fraction ................................................................................. 17
Euclidian Algorithm....................................................................................................... 18
Exercise: ......................................................................................................................... 18
Quantity Fraction .......................................................................................................... 18
Unit Fraction for measuring unit .................................................................................. 19
Major Reference and Further Readings 2 ..................................................................... 20
Chapter 3: Addition of Fraction with Different Denominators ........................................ 21
Exercise .......................................................................................................................... 21
A Sample of Arguments: ................................................................................................ 22
Questions for professional development 3 ..................................................................... 25
Major Reference and Further readings 3 ...................................................................... 26
Chapter 4: When does Fraction become Number? ............................................................ 27
Equivalent Fraction ....................................................................................................... 27
Number Line .................................................................................................................. 27
Fraction is Expression: Fraction as Quotient ............................................................... 28
Exercise. Let’s read and answer. ................................................................................... 29
Questions for professional development 4 ..................................................................... 31

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Major Reference and Further readings 4 ...................................................................... 32
Chapter 5. Fraction in relation to ratio and proportion ................................................... 33
Questions for professional development 5 ..................................................................... 34
Fraction as Ratio ............................................................................................................ 39
Exercise .......................................................................................................................... 39
Rule of Three: two by two matrix table ......................................................................... 40
Tape and Number-Line diagram: Proportional number line ........................................ 40
Exercise .......................................................................................................................... 41
Multiplication as for the Origin of Proportionality ....................................................... 41
How to draw the proportional number line ................................................................... 43
Major Reference and Further readings 5 ...................................................................... 44
Chapter 6. Multiplication and Division of Fraction (1) .................................................... 45
Number line ................................................................................................................... 45
Multiplication ................................................................................................................. 46
How to find the expression from the situated problem ................................................. 47
Exercise .......................................................................................................................... 48
Fraction × Whole Number ............................................................................................. 48
Exercise .......................................................................................................................... 49
Fraction × Fraction ........................................................................................................ 49
Area Diagram ................................................................................................................. 49
Exercise .......................................................................................................................... 50
Major Reference and Further Readings 6 ..................................................................... 52
Chapter 7. Multiplication and Division of Fraction (2) .................................................... 53
Rules of Multiplication and Division ............................................................................. 53
Exercise: ......................................................................................................................... 53
Exercise: Fraction divided by Fractions ........................................................................ 55
Major Reference and Further Readings 7 ..................................................................... 56
Summary ........................................................................................................................... 57
Further Readings .............................................................................................................. 59

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Chapter 1: What is fraction?

The professor asked the teachers:
There is 2m tape. Where is the position of ⅔ m? Show it by ↓
0m

1m

2m
Isoda, M. (1993)

Possible Answers:
Som
0m

Any
1m

2m

0m

↑

1m

2m

↑

What do you want to do next?
I would liketo ask why?

Isoda&Katagiri (2012)

Yes, we would like todiscuss!

Professor: You are good teachers, aren't you? Because you already have the custom to
ask why to others and discuss. It means that you usually engage in a similar activity by
yourself. You already have the mind set for learning how to learn by yourself!

Discussion 1: Why do you think so?
1

Som: It is larger than 1m because 2/3 means:32

Ano: No, the whole is divided by 3, then shaded the 2 third of 3 parts
Som: Yeah, I should draw like that.

Any: Wow, 2/3 is less than 1. It is like this:Som & Ano: ..continue..(talking about part-whole)7
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Any: ...continue... (talking about part-whole)
Prof: I could not understand well all of your explanations and diagrams. What is theoriginal question?
Any: Where is the 2/3m?
Prof: Yes, then, the denomination ‘meter’ is missing in all of your explanations. We
should write the meter on the number line. Can you explain 2/3m, again?
Any: It is 2/3 of 1m

0m

1m

2m

Som: It is 2/3 of 2m

Prof: Now we can discuss what differences are there between these two ideas.
The denomination of quantity is important. When we explain something with thesituation, we usually omit some words which are already known in the situation. In thiscase, the unit of meter ‘m’ itself is written in the task; however the meter was missing
from their explanation and diagrams. In their discussion, if the denomination ofquantity is missing, we are not sure which part you are explaining.
People who participated in the discussion; however, not sure whichanswer is appropriate.
Then, Professor asked.

What is Fraction?

Discussion 2: What is Fraction?
Ano: It is a part of the whole.
Prof: It is not clear. Could you explain it more exactly?
Ano: When the whole is divided into parts, fraction is the number of the parts in thewhole.
Prof: Still are missing some important terms. Can anyone support?
Oth: When the whole is divided into ‘m equally parts’, ‘fraction n/m’ is ‘n pieces of theparts in the whole’.
Any: The whole for 2/3m is 1m.
Som: The whole for 2/3m is 2m.

What is the whole, here?
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Discussion 3: Which one should be the whole?
Any: 1m
Som: 2m
Ano: 1m or 2m, which one shall we chose for the whole, in this task?
Any: The whole for 2/3m is 1m
Som: The whole for 2/3m is 2m.
Oth: How can we discuss?

In mathematics, generally applicable idea is strong.
For checking it, we have to think 'For Example, ...."

Discussion 4: For Example, if..............
Prof: For example, if I change the task from 2/3m to 1/2m what will happen?
Any: The 1/2m is 0.5m.
Som: The 1/2 of 2m is 1m.
Ano: No, what you are saying is that 1/2m is 1m. However, 1/2m is 0.5m, isn't it?
Pro: If I change the original question ‘2/3m’ to ‘1/1m’ what will happen?
Oth: If Som's idea, 1/1m is the 1/1 of 2m, thus 2m. 1/1m =2m. It is strange.
Prof: Yes, we can generalize Any's idea and not generalize Som's idea. When we say
2/3m, the whole is denominated by the 2/3 'meters': The unit of meters is 1m in any
time.

What did you learn from this class? Let's write
your own resume based on what you learned.

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Questions for professional development 1
Q1. Why did the author choose this story as for the introductory chapter?
Q2. Professor asked several questions in the class. Which question is most important inthis class? Why do you think so?
Q3. Do you think the definition of fraction in the class is appropriate? Why do you
think so?
Q4. Explain the class using the term of appreciation.
Q5. If you conduct this task in your class or your teacher training program, what is yourobjective?
Q6. In your class, do you ask your students or teachers 'what do you want to do next?'.
If you do ask, when do you ask it? If you do not ask, why?
Q7. Please explain the professor's questioning and values from the viewpoint of thefollowing.
Three Major Objectives for Education as for Future Preparation
1. Human Character Formation ∋ {Developing Mindset, Attitude, Value} appreciation
2. Learning How to Learn ∋ {Knowing how to develop and reconstruct} reflection
3. Knowledge and Skills ∋ {Understanding and Proficiency} acquisition
Dizon, D., Ahmad, J., Isoda, M. (2017)

Q8. What is the explanation in mathematics at elementary school level? Using thediscussion in class, please explain what it is from the following three perspectives.
Explanation of:
1. Meaning, such as the base for the reasoning using different representation;
2. Significance or Objective, why I would like to think such a way;
3. Procedure, how I did.
Isoda, M.(2008, 2009)

Q9. In the argumentation, for progressive dialectic, the way of discussion below is
known as meaningful: 'If your saying is true, what will happen?'In the class, Professor has used this dialectic method. Where did he use it?
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Major Reference and Further readings 1
Isoda, M. (1993). The logic for Argumentation in Mathematics Classroom. (written in
Japanese:礒田正美(1993).算数授業における説得の論理を探る.北海道教育大学教科教育学
研究図書編集委員会編.教科と子どもとことば : 言語で探る教科教育.東京：東京書籍)
English translation: Isoda, M. (2008). Getting Others’ Perspectives through theHermeneutic Effort: Paper presented at TSG 26, ICME11 on

http://tsg.icme11.org/document/get/803

Isoda, M. (2008). Teaching Mathematics through Argumentation for Junior High School.

(written in Japanese: 礒田正美(2008). 思考・判断・表現による『学び直し』を求める数学の授業改善 : 新学習指導要領が求める対話:アーギュメンテーションによる学び方学習.東京：明治図書出版) Original theory for dialectic communication to develop mathematics
knowledge and thinking was in English: Isoda, M. (2007). Developing MathematicalThinking in Classroom, APEC Project report on

http://www.criced.tsukuba.ac.jp/math/apec/apec2007/progress_report/specialists_session/
Masami_Isoda.pdf
Isoda, M. (2009). Teaching Mathematics through Argumentation for Elementary School(written in Japanese: 礒田正美(2009). 思考・判断・表現による「学び直し」を求める算数の授業改善 : 新学習指導要領が求める言語活動 : アーギュメンテーションの実現. 東京：明治図書出版) Original theory for dialectic communication to develop mathematics
knowledge and thinking was in English: Isoda, M. (2007). Developing MathematicalThinking in Classroom, APEC Project Report on

http://www.criced.tsukuba.ac.jp/math/apec/apec2007/progress_report/specialists_session/
Masami_Isoda.pdf
Isoda, M. & Katagiri,S. (2012). Mathematical Thinking: How to Develop it in theclassroom. NJ: World Scientific. The part of 'What do you want to do next' can be
downloaded:

http://www.worldscientific.com/doi/suppl/10.1142/8163/suppl_file/8163_chap01.pdf

Dizon, D., Ahmad, J., Isoda, M. (2017). SEAMEO Basic Education Standards (SEABES): Common Core Regional Learning Standards (CCRLS) in Mathematics andScience. Penang, Malaysia: SEAMEO RECSAM

https://www.researchgate.net/publication/335465253_SEAMEO_Basic_Education_Standards_SEABES_Common_Core_Regional_Learning_Standards_CCRLS_in_Mathematics_and_Science11
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Chapter 2: Dividing Fraction and Quantity Fraction

On the Fourth APEC-Tsukuba Lesson Study Conference, Gould. P.(2010), Inspector of
NSW, Australia, lectured the roots of misconception of fraction in the case of Australian
students with Australian textbook. He mentioned that the standard fraction notation
itself encourages a ‘count’ interpretation of the regional ‘parts of a whole’ model (Figure1). He quoted Hackenberg (2007) who asked the students to draw seven-fifths of acandy bar if the drawing of the rectangle on their paper represented a candy bar.
Although both girls correctly created seven-fifths of the rectangle, when asked about thesize of the pieces in the bars they had drawn, the girls maintained that the pieces weresevenths. Depending on his report, the major root of misconception is just counting of
the parts without considering the whole. On his lecture, he also mentioned Japanesebetter approach.

Figure 1 Circle Model and Answer of Grade 5 student (Gould, P. 2010)

On the Fifth APEC-Tsukuba Lesson
Study Conference, Lewis, C. (2011)lectured about her research forprofessional development which
compared three groups: Lesson Studywith Japanese Resource for fractions,Lesson Study without it, and Ordinal
professional development programs. In
her lecture, she mentioned two points:
Firstly, Japanese approach for
introducing fractions is better approachfor understanding the meaning of
Figure 2 The effect of Lesson Study with Japanesefractions. Her recommended Japanese
Resource (Lewis, 2011)
Approach is shown on the Figure 3a&3b.
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Figure 3a. Study with your friends: Mathematics for elementary school, Gakkotosho (2005, Grade 4, vol.2,
pp.65-66; 2011, Grade 3, vol.2, pp88-89)

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Figure 3b. Study with your friends: Mathematics for elementary school, Gakkotosho (2005, Grade 4, vol.2,
pp.65-66; 2011, Grade 3, vol.2, pp88-89)

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Secondly, only the case using Japanese Resource for fractions has improved teachers'
fraction knowledge, significantly.
Both lectures recommended Japanese approach. One of their recommended approaches
is shown in Figure 3a and 3b which uses the remainder for measuring the unit length
1m.

Questions for professional development 2
Q10. Let's analyze the ideas of Som and Ano in Chapter 1 from the view point of Gould,
P. (2009).
Q11. Let's view Figure 3a and 3b. What is the common idea between Figure 3 and thesuggestion of Professor in Chapter 1?
Q12. In division, there are two meanings as follows. Let's analyze the dividing activity
in Figure 3a and 3b from the viewpoint of two different meanings.
Partitive division

Quotative Division

12 candies are divided by 4
children, equally. How many
candies one child can receive?

12 candies are distributed by 4
candies for one child. How many
children can receive candies?

Figure 4. Gakko Tosho Grade 3 (Vol2. p4,p8,
2005; vol1, p60, p64, 2011)

Both activities for dividing are different for children however
both activities include the repeated subtraction of the sameamount from the total amount (Only see the left-hand-part ofevery picture, in Figure 4 left and right). Partitive division
establishes the equally partition at first. Quotative division
quotes the same amount recursively until we cannot quote, notsure the number of partitions at the beginning.
.
Q13. Please do the activity in the Figure 3 by yourself and explain the difference withyour traditional approaches for teaching fraction based on Pizza model such as on
the figure 1.

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In mathematics, fraction n/m as for rational number is defined when n and m are
integers (or rational number) and m is not equal 0.1 If n or m are irrational or complex
numbers, the fraction is not rational number. In mathematics, the concept is usually
explained by the definition and the equivalent properties with exemplars. However, inschool mathematics, especially for the elementary school, the meaning is explainedusing the different representations, model or situation with action (See Chapter 1,
Q7,1). If we only teach number and calculation, children usually explain the ways of
calculation as a rule (See Chapter 1, Q7, 3) when we ask them to explain ‘Why?’. If
readers are hoping children to explain the meaning, you have to assist children to use
the further representations and models with situation. To meet these demands, Q12 on
this chapter has asked you to engage in the same activity by yourself. When you use the1m tape and you recognize the length of blackboard is 125cm, you can well prepare theactivity on Figure 3.
Fraction is used in various contexts/situations with different meanings even if it isclearly defined mathematically. First part of this booklet explains how Japanesedistinguish those meanings.2 Second part explains how Japanese teach four operationsof fraction.

Dividing Fraction
Many people believe that fraction is Dividing Fraction. It is thepart-whole relationship which means that fraction n/m is dividing
the whole into m equal parts and selecting n parts from them. It isdeeply related with the activity of partitive division. 2/3 means thepart-whole relationship which shows the whole is equally divided
into three parts and selecting two parts. If the pizza is divided into
three parts and take two parts, it means 2/3 of the pizza.
Paper folding, Origami, of a square paper includes the possible action for dividing theobject equally, such as half and quarter. On this action, children usually think that
dividing fraction cannot be larger than the one whole. However, as shown in Chapter 1
where people lost the denomination, children usually lose the process dividing the whole
into equal parts. As Gould mentioned, the misconception for dividing fraction isoriginated from giving children worksheets to shade the object already equally divided
through counting. On these worksheets, children are not needed to recognize the whole
and to think how to divide it at first, because they just count the number of parts on an
equally divided object. At the same time, on the ill-solution in Figure 1, it is arelationship to see the confusion of the fraction as ratio which allows to think the partpart relationship instead of the part-whole relationship described in Chapter 5. The task

1 Fraction, rational number, as a number system will be discussed at later chapters.
2 Internationally, there are various technical terms to explain the meaning of fraction on situations such as

Behr, M., et al. (1983) and Charalamous, C. Y & Pitta-Pantazi, D. (2007). However, it is un-usual todistinguish dividing fraction and operational fraction. Japanese technical terms are more precise toestablish fraction in relation to what students learned on multiplication and division (see Isoda&Olfos,
2021, p.91).

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itself expected to think part-whole relationship, thus, the major problem is related with
their losing of whole for equally-dividing the whole on the context. For developing better
understanding, we have to begin the activity to set a whole as one unit by children andask them to divide it into the number of equal parts. After dividing into equal parts,
children chose the necessary number of parts from the whole.
Any length of tape can be divided equally if
we use the parallel lines with sameinterval. Necessity of this diagram isrelated with dividing the whole to the equalparts.

Exercise:
Draw 1/3 of Pizza on your notebook andcompare the size of it with other’s drawing.

Operational (measuring) Fraction
Operational Fraction (measuring fraction)
can be seen as a kind of dividing fraction
however it tries to measure the whole as a unit by the remaining part. From theviewpoint of division, it can be seen the activity of quotative division. In the following,
three times of the remaining part is 1m. It means that 1/3 of 1m is the unit for counting
however the whole, that should be divided, is 1m. Additionally, the original length islonger than the whole 1m. Mixed fractions and improper fractions have already existed
even though children do not study them.

It is strange! We cannot always measure the 1m using the remaining
part. What shall we do when we get another remaining when wemeasure the whole by the part? The operation does not work!

Yes, you usually give me the good question. Euclid (BC 3C) approved
the way to find the unit (Greatest Common Divisor) for themeasurement. It works with whole numbers (Integers) and called
Euclidian Algorithm.

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Euclidian Algorithm
For example, Greatest Common Divisor of 18 and 42 is: 42=6x7, 18=3x6, 3 and 7 areprime numbers, then 6 is GCD of 18 and 42. It will be found using the remainders as
follows.
42=2x18+6

18=3x6→42 divided by 18, the remainder is 6.
182
181

18 divided by the remainder 6.
61
63
62

6

18

42

182

181

6
61
62

18

The rectangulardiagram is the betterway to demonstrate
Euclidian Algorithm.

63
42

Exercise:
Let's find the GCD of (41, 18) by the rectangular diagram.

Quantity Fraction
Quantity fraction is the fraction with denomination. 1/3 of 1m is 1/3m in quantity
fraction. The dialectic in Chapter 1 is done between quantity fraction (2/3m) anddividing fraction (2/3 of 2m). If we lose the denomination, we cannot distinguish
quantity and dividing fraction like the communication in Chapter 1. It means that wecannot arrange the position of fraction on the number line if we do not have thedenomination of the quantity. Quantity fraction allows us to compare the size of fraction
in relation to the unit quantity. Dividing fractions are not easy to compare on thenumber line because the size of the whole unit is not clear and it looks always less thanone.
In Figure 3a and 3b, all fractions are denominated with meter. All of them are quantity
fraction based on the unit meter. In Japanese textbook, fraction is introduced by thequantity fraction with operational fraction and dividing fraction. It is better than the
traditional approach introduced by dividing fraction for recognizing what is a unit
(whole). Because it allows us to extend the fraction larger than the whole unit, andenables us to compare the size of fraction on the number line.

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Unit Fraction for measuring unit
Unit fraction is the fraction in which numerator is one such as 1/3. Unit fraction is theunit for measuring up to improper fraction. Any fraction is represented as follows: (thespecific number) x (the unit fraction). It is the necessary base to extend proper fraction
to improper fraction. Thus, in fractions, there are two types of unit. Firstly, the whole isthe unit. Secondly, unit fraction. The unit fraction is the unit for counting thenumerator.
The idea of unit fraction on base ten place value system is related with decimals. For
example, 1 mm as unit quantity for length is usually introduced as dividing 1 cm by 10,
equally. The relationship between 1cm and 1 mm is a base to introduce decimals such as
10 mm is 1 cm and 1 mm is 0.1 cm. 1 cm and 1 mm scales are given on the ruler andtape measure, it is the base for number line which begins from 0 to +∞. On this context,
operational fraction and unit fraction are the bases for extension of numbers.
In world known Japanese approach, fraction is didactically explained using various
technical terms and introduced based on quantity and operational fraction, theremainder as measurement, for preparing the unit fraction (Figure 3b). If children
understand the quantity fraction and the unit fraction, they can easily represent thefraction on the number line such as in Chapter 1, and easier to think four operations offractions.

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Major Reference and Further Readings 2Behr, M., Lesh, R., Post, T. and Silver, E.: 1983, Rational number concepts. in R. Leshand M. Landau (eds.), Acquisition of Mathematics Concepts and Processes, Academic
Press, New York, pp. 91–125.
Charalabous, C. Y., Pitta-Pantaze, D. (2007). Drawing a theoretical model to study
student’s understanding of fractions. Educational Studies in Mathematics. 64, 239-316.
Gould. P.(2010). Fraction notation and textbooks in Australia. 4th APEC-Tsukuba
Conference. Sited on http://www.criced.tsukuba.ac.jp/math/apec/apec2009/index_en.php
Lewis, C. (2011). Lesson Study with Japanese Curriculum Materials: A Randomized
Controlled Trial. Sited on
http://www.criced.tsukuba.ac.jp/math/apec/apec2011/index_en.php
Masami Isoda et al. (2010). Mathematics Education Theories for Lesson Study: Problem
Solving Approach and the Curriculum through Extension and Integration, Journal ofJapan Society of Mathematical Education, 92(11). 1-158.
Shin Hitotsumatsu et al. (2005). Study with your friends: Mathematics for Elementary
School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda et al. (2010). Study with your friends: Mathematics for Elementary School(12 vols). Tokyo: Gakkotosyo.
Masami Isoda, Tenoch Esaú Cedillo Ávalos, et al. (2012). Matemáticas para la educación
normal (11vols). Estado de México : Pearson Educación de México.
Tenoch Cedillo, Masami Isoda, et al. (2013) Matemáticas para la educación normal :guía para el aprendizaje y enseñanza de la aritmética, Estado de México : Pearson
Educación de México.
Masami Isoda, Raimundo Olfos. Edited. (2021). Teaching Multiplication with Lesson
Study. Cham: Springer.

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Chapter 3: Addition of Fraction with Different Denominators

The professor asked the teachers:
In addition of the fractions, we teach the case of the same denominatorat first and then we introduce the case of different denominator.
1
2

1
3

In the case of + :1

1

We have two bottles with 2L and 3L of milk. How much L in total? Can
you imagine the student’s answer?
Possible Answers:

Isoda (1996)

Som

Any

Yes, we have children
who calculate as
follows:

Yes, we have. However
we have to teach:1 1 3 2 5
+ = + =
2 3 6 6 6

1 1
1+1
2
+ = 2+3 = 5
2 3
1How can you plan their argument?

; 3=ここに数式を入力します。

I would like to ask to
draw diagrams to
explain the meaning.

I would liketo ask why?

Professor: Yes, we would like to listen their ideas. Can you imagine their diagrams for
explanation? How do you teach to find the common denominator?

Exercise
Exercise1: Let's plan the process of dialectic discussion between two different ideas
through the drawing diagrams which support each of answers.
Exercise 2: When you plan, what are the expected knowledge for children before theclass and what knowledge do they have, actually, for those two ideas? Whatshall they learn before this class for enabling them such a discussion? Whatshall we teach before you plan your class?

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A Sample of Arguments:
Teacher: Good, now we have different answers. What shall we do?
C1: How did they get their answers?
C2: Yes.
Teacher: Then, we would like to ask: how do you get it?
1 1
2 3

C3: + =
1

1

1+1
2
= , I added numerators and denominators each.
2+3
5
3

2

5

C4: 2 + 3 = 6 + 6 = 6, I tried to find the common denominator.
Teacher: Do they explain in the same way?
C: (No, answer).
Teacher: Do you have any questions for them?
C1: I have a question for C3, why you added the numerators?
1 1

1+1

2

C3: As I explained, 2+3 = 2+3 = 5 , I added numerators and denominators.
2

1

1

1

1

C4: The total,5 is smaller than 2. If 3L of milk is added to 2L of milk is less than 2 L. It
is very much strange, isn't it?
C3: Yes, it's strange...then I cannot add denominators?
C5: No, it is not strange. I will explain the bottles like this:In the case of fractions, it is true.

+

C3: Aha, yes, it is!

=

Teacher: Then, you say in the case of fractions, it is possible to say that the total from1
1
1
L of milk and L of milk is less than L.
2
3
2

C: No! Yes!
Teacher: Yes or No? It looks C5 is less supported. Why?
C4: The milk bottle of C5, the size is not the same.
C5: Yes, the shape of the milk bottles are not usually the same.
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C4: ....
Teacher: Why did you use the common denominator, C4?
C4: Because we added when the denominators are the same.
We learned the addition of fractions in the case when the denominators are thesame.
C2: Yes, we know the addition of the same denominator. In that case, we added thenumerator. This task is not the same.
Teacher: Did you draw a diagram, like C5?
C4: No, I did not. I think that it is the rule.
C5: Oh, the rule? I could explain my idea by the diagram.
Please count in my diagram.
The way of C3 will be the right rule, not C4.
C6: I draw a similar diagram, not the same as C5:

It shows that the value is larger than the parts.

Teacher: Now, we have two drawings. Which diagram is reasonable for establishing theway of calculation?
1

1

3

2

5

C3: I do not think C6 is explaining C4: 2 + 3 = 6 + 6 = 6 .
1

1

5

1

1

3

2

It may explain the part of 2 + 3 = 6, however it does not explain 2 + 3 = 6 + 6.
1

Why you have to change the fraction on 6?
Teacher: It means that why C4 has to change the denominator, right?
C3: Is that a rule, also? If it is the rule, my own is better because we can explain the idea
by counting like C5.

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C6: If we have to explain it by counting, we have to share the unit at first. If we set the11unit for counting from the difference between 2 and 3:.

I could count six times of the unit.
1

Thus, the unit for counting is 6.

C5: Aha, I got it. However, why do you use the unit for counting by the difference? Is it
occasionally? Can you do that every time? Or, you already knew the difference infractions with different denominator?
C6: Not sure, however, if you consider every time, do you think your idea also works
every time?
Teacher: Yes, in mathematics, we develop generally applicable ideas. It must be
important point for discussion.
C5: Yes.
C4: Wow, did you add the denominators in the case that the denominators are the same?
C5: What?
1

1

C4: For example, how do you think in the case of 2 + 2.
1

1

1+1
= 1, right?
2

1

1

1+1

We already learned it is 2 + 2 =2

1

But if your idea is true, 2 + 2 = 2+2 = 4 = 2. It is strange.
C3,C5: Wow, yes, it is strange. Is it the reason, if the denominators are the same, we add
the numerators?
C4: Yes, now I understand why I did it. That is the objective why I changed thedenominators into common denominators! However, how can we produce thecommon denominator for the addition of fractions in the case of different
denominators. I am still not sure of the ways of calculation.
Teacher: Aha, for considering generally, we checked each proposed way by C3 and C4 on
the different denominators by the known case of the same denominator. Then, theway of C4 works and the way of C3 does not. However, the diagram by C5 still looks
fine. Why did it produce the inappropriate answer?

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C6: The size of the bottle should be the same. In this case, if we draw the diagram of a
1L milk bottle, it is fine.
Teacher: Yes, we need to write the size of quantity on the diagram. In this case, weshould draw 1L as for the whole bottle size in every diagram. We should use thesame size bottle for explaining the addition and subtraction of fractions. In the11diagram of C5, we counted different size of fractions. 2L is counted one and 3L is
counted one. We cannot count one, two because the size is not the same. The
1

diagram by C6 used 6 L for the same counting unit.
C4: Wow, this is the reason why I used the common denominator. Still not sure of theways.
Teacher: Then, from now, we would like to find the shorter and simple way of
calculation by considering the common denominator.

Questions for professional development 3
Q14. In the argument, you may see the same discussion which you read in Chapter 1
and it will be explained by the several terminologies as mentioned in Chapter 2.
Let's explain the argument by the terminologies. And explain your appreciation
about the terminologies for understanding the difficulty of fraction and what is thenecessary content for teaching.
Q15. In the argument, what kind of explanation did you find? Please read eachexplanation from the viewpoint of Chapter 1, Q8. Why do we need the diagram?
What C4 wished to say? For developing children who learn mathematics by/forthemselves, what type of argument you would like to establish in your classroom.
Q16. If you are not used to draw a diagram for explaining fractions by yourself, theargument might be difficult for you to understand. The difficulty that you
recognized is based on the unknown. It means that it is the chance for learning. At
first, let's discuss about which part do you feel a difficulty and then, talk about what
you shall learn.
Q17. When you feel a difficulty, your children also feel the same. Please ask yourchildren who knows addition of the same denominator and equivalent fraction, but
not yet learned the case of different denominators to read this discussion. And ask
them, how do they read? It is also a good chance to learn from children for knowing
what is the task for their learning and what is necessary for your preparations.
Q18. The argument itself was implemented in a class (see Isoda, 1996) and not unusual
in Japan. When we compared the classes, major difference is the teaching before theclass. What kinds of content shall we teach before this class?

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Major Reference and Further readings 3
Isoda, M. (1996). Problem Solving Approach with Dialectic. (written in Japanese:多様な考えを生み練り合う問題解決授業 : 意味とやり方のずれによる葛藤と納得の授業づくり. 東
京：明治図書出版) in English: Isoda, M. (2007). Developing Mathematical Thinking inClassroom, APEC Project report on

http://www.criced.tsukuba.ac.jp/math/apec/apec2007/progress_report/specialists_session/
Masami_Isoda.pdf
Masami Isoda et al. (2010). Mathematics Education Theories for Lesson Study: Problem
Solving Approach and the Curriculum through Extension and Integration, Journal ofJapan Society of Mathematical Education, 92(11). 1-158.
Shin Hitotsumatsu et al. (2005). Study with your friends: Mathematics for Elementary
School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda et al. (2010). Study with your friends: Mathematics for Elementary School(12 vols). Tokyo: Gakkotosyo.
Masami Isoda, Tenoch Esaú Cedillo Ávalos, et al. (2012). Matemáticas para la educación
normal (11vols). Estado de México : Pearson Educación de México.
Tenoch Cedillo, Masami Isoda, et al. (2013) Matemáticas para la educación normal :guía para el aprendizaje y enseñanza de la aritmética, Estado de México : Pearson
Educación de México.
Masami Isoda, Raimundo Olfos. Edited. (2021). Teaching Multiplication with Lesson
Study. Cham: Springer.

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Chapter 4: When does Fraction become Number?

Number system is the set with mathematical structures such as the relationship ofequivalence, greater/less and operations (addition and multiplication). Mathematically,
fraction n/m is an element of rational numbers, if n and m are integers and m≠ 0. If n or
m are irrational or complex, it becomes a part of a larger number set.
In school mathematics, when we extend numbers we usually discuss
existence/necessity/significance, equivalence/larger/smaller, and four arithmetic
operations. If fraction is completely learned, we can recognize the fraction as a part ofnumber set: It is a representation of rational numbers which are bridged with decimals.
In other words, fraction is not the number until it has been fully completed.
On this context, every teacher should know that the dividing fraction is not the number.
It represents the part-whole-relationship actions on the object such as half of a Pizza.
Quantity fraction can be arranged on the number line for quantity as well as othernumbers such as decimal numbers. Dividing fraction usually begins from the whole andit is not easy to represent on the number line as long as what is the whole which we
discussed on Chapter 1.

Equivalent Fraction
1
2
3
4
5
1
= 4 = 6 = 8 = 10... is the equivalent fraction of 2. At the lower grades the dividing
2

fraction, which is represented by paper folding activity, is learned for knowing the half
and the quarter, two quarters, and three quarters. It is a necessary activity for knowing
the procedure to get the half and the quarter. However, the paper folding activity for3
6

dividing fraction is usually missing , because it cannot be represented by the folding
procedure into half.

Number Line
Number line shows the magnitude/size of numbers up to real numbers as position which
is useful for comparison. Number line is scaled by using equally dividing of 0-1 span as
dividing fraction and by the unit fraction as operational fraction. As shown on the
Figure 3a in Chapter 2, we can divide any string into the number of equal parts. On thisbasis, we can draw the diagram of number lines which demonstrates the equivalence,larger and smaller (Figure 5). Through showing the position of fraction on the number
line, it supports to see the fraction as number as well as others.
The number line includes more than 1 on Figure 5 like a ray. If it is represented by asegment between 0 and 1 without any extra part more than 1, it is just a representationof dividing fraction which shows a part-whole relationship like folding a tape. Here, thewhole is 1 without considering the possibility of extension. As discussed in Chapter 2,
operational fraction is majored by a unit fraction (remaining part). The number of unit
fraction shows a numerator: then extended to improper and mixed fraction more than 1.

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2
3

Quantity fraction m implicates that the unit-quantity is 1m. If quantity is indicated on
the number line, the number line shows quantity.

Figure 5. Equivalent fraction on the number line. GakkoTosho(Grade 5, vol2, p24, 2005; Grade4,Vol2, p78, 2011)

Fraction is Expression: Fraction as Quotient
The answer of division is known as a number. When a fraction can be seen as a division
expression, it supports to see fraction as a part of number.
Fraction is related with division which includes both meanings of quotative and dividing
fraction on situations (see Chapter 2). In both situations, fraction represents the value of1

division such as 1 ÷ 3 = 3. We call it fraction as quotient. On this definition, fraction isconnected with a number as well as the whole number (See figure 6a and 6b) because
the answer of division should be a number on the perspective of the permanence of form.
1

In relation to two division situations, 1 ÷ 3 = 3 implicates equally dividing like dividing
fractions. 1 ÷ 3=0.3333333… in action related with operational fraction by focusing on
1
remainders. However, if we write 0. 3̅ (“―” comes to the top of 3) it is .
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Exercise. Let’s read and answer.

Figure 6a. Gakko Tosho Grade 5. (vol.2. p29, 2005; vol.1. p128, 2011)29
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Figure 6b. Gakko Tosho Grade 5. (vol.2. p30, 2005; vol.1. p129, 2011)30
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If children can see the fraction as quotient, now fraction is a part of number which can
1

be seen as the expression of division such as 3 = 1 ÷ 3. If fraction is given by theexpression of division, the meaning of equivalence fraction is also understandable
because it is not a mysterious property that only appeared on the fraction. In any
arithmetic operation, it has similar properties. When the equivalence of fraction isrecognized on the basis of the fraction as quotient, all four operations complete theproperties of equivalence on their expressions.
On mathematics, it is the discussion of equivalent class.

Questions for professional development 4
Q19. Let's find the equivalent property of addition and subtraction: Gakkotosho Grade1.

(pp82-83, pp92-93,2005; pp96-97, pp106-107, 2011)
Q20. In Chapter 2, Q11, there are two meanings of dividing activity. Which activity can
you see on the figure 6a and 6b and explain why.

( Gakkotosho Grade vol.2, p.92, 2005)
Q21. Let’s find the same product in the multiplication table.

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Major Reference and Further readings 4Masami Isoda et al. (2010). Mathematics Education Theories for Lesson Study: Problem
Solving Approach and the Curriculum through Extension and Integration, Journal of Japan
Society of Mathematical Education, 92(11). 1-158.
Shin Hitotsumatsu et al. (2005). Study with your friends: Mathematics for Elementary
School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda et al. (2010). Study with your friends: Mathematics for Elementary School (12vols). Tokyo: Gakkotosyo.
Masami Isoda (2012). Introductory Chapter: Problem Solving Approach to Develop
Mathematical Thinking. In Isoda, M. & Katagiri, S. edited (2012). Mathematical Thinking:
How to Develop it in the Classroom. Singapore: World Scientific. 1-28.
https://www.criced.tsukuba.ac.jp/math/apec/ICME12/Lesson_Study_set/MATHEMATICAL%20THINKING%20(c)World%20Scientific/MathematicalThinking_chap01(c)WorldScientific.pdf

https://doi.org/10.1142/9789814350853_0001

Masami Isoda, Tenoch Esaú Cedillo Ávalos, et al. (2012). Matemáticas para la educación
normal (11vols). Estado de México : Pearson Educación de México.
Tenoch Cedillo, Masami Isoda, et al. (2013) Matemáticas para la educación normal : guía
para el aprendizaje y enseñanza de la aritmética, Estado de México : Pearson Educación de
México.
Masami Isoda (2015). The Science of Lesson Study in the Problem Solving Approach. In
Inprasitha, M. et.al. edited. Lesson Study: Challenges in Mathematics Education. Singapore:
World Scientific. pp.81-108. https://doi.org/10.1142/9789812835420_0006
Masami Isoda (2015). Dialectic on the Problem Solving Approach: Illustrating Hermeneutics
as the Ground Theory for Lesson Study in Mathematics Education. In Sung, J. edit. Selected
Regular Lectures from the 12th International Congress on Mathematical Education. Cham:
Springer. pp.355-381. https://doi.org/10.1007/978-3-319-17187-6_21
Masami Isoda & Aki Murata edited (2011). Study with your friends: Mathematics for
Elementary School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda, Aki Murata, Aida Istino Yap edited (2015). Study with your friends:
Mathematics for Elementary School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda & Aki Murata edited (2021). Study with your friends: Mathematics for
Elementary School (12 vols). Tokyo: Gakkotosyo.
https://support.gakuto.co.jp/mathematicstextbook/
Masami Isoda, Raimundo Olfos. Edited. (2021).
Teaching Multiplication with Lesson Study.
Cham: Springer. DOI: 10.1007/978-3-030-28561-6

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Chapter 5. Fraction in relation to ratio and proportion

In relation to the fraction as quotient, fraction is used for representing the ratio ‘ : ’ as(Ratio) = (Comparing Quantity) ÷ (Base Quantity for the Unit of the Ratio). There areseveral contexts to use ratio. Japanese approach consistently uses Tape diagrams,
Number lines, Area diagrams and Tables, as representations for thinking.
Firstly, ratio (value of ratio) is the result of division. There are two contexts in relation
to the meaning of division. The following two types of dividing activity: partitive divisionand quotative division (Chapter 2, Q11).
Partitive Division

Quotative Division

12 candies are divided by 4
children, equally. How many
candies one child can receive?

12 candies are distributed by 4
candies for one child. How many
children can receive candies?

Ans. 3 candies for one childAns. 3 persons

When we see the underlined words in both tasks, the partitive division treats different
quantity/denomination such as candies and children, and quotative division treats thesame quantity/denomination.
On terminology, rate is the ratio for different quantities. Division by the different
quantity (partitive division) is the rate which is unknown quantity represented by thequantity per the different quantity such as (speed (km/h)) = (distance (km)) ÷ (duration(hour)). One person runs 10km/h and another person runs 8km/h. Even if two people run
together, they do not run at 18km/h. The quantity such as speed which cannot be added
is called connotative quantity. It is usually given specific name such as speed.
Division by the same quantity (quotative division) establishes the ratio of the same
quantity which produces the rate to show coefficient/multiple (how many times). If we
carefully read the task for quotative division situation, it is not exactly the same
quantity/denomination. Because, contextually, it can be represented by (number ofchildren) = (12 (candies)) ÷ (4 (candies for one child)). On the quantity expression, it is(𝐶𝑎𝑛𝑑𝑖𝑒𝑠)(

𝐶𝑎𝑛𝑑𝑖𝑒𝑠)𝑎 𝑐ℎ𝑖𝑙𝑑= (𝐶ℎ𝑖𝑙𝑑𝑟𝑒𝑛). Both numbers 12 and 4 are number of candies for students. Thus,

it is unnecessary to see them as different quantities.
There are countries that use the ratio symbol ‘ : ’ as the for division symbol ‘÷’. It meansathat a : b is the same as a÷b and b. To distinguish the ratio of different quantities(usually with special name as for new quantity) and the ratio of the same quantity (justin case, a<b and it is the narrow usage of WARIAI in Japanese) is reasonable. On theacontext of ratio a : b, b is called value of ratio.
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Questions for professional development 5
The following tasks show the various context of ratio. Let's answer the following
questions.
Q22.
1) Solve the following task. Provide the answer using percentage.
2) Explain the feature of the task using the following terms: Partitive division, Different
Quantities.

(Gakko Tosho, Grade 5, vol.2, pp57-58, 2005; pp85-86, 2011)34
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Q23.
1) Solve the following tasks and explain the base quantity for each task.
2) Explain the role of tape and number-line diagram.
3) Explain the situation using the quotative division and the same quantity.
1644) 16 ÷ 20 = 20 = 5 . Can we explain the task by the part-whole relationship?

(Gakko Tosho, Grade 5, vol.2, pp59, 2005; pp87, 2011)35
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Q24.
1) Solve the task using ratio.
2) If we consider the case of 6 mats for comparison, we do not need to use the idea ofratio.

(Gakko Tosho, Grade 6, vol.1, p74; Grade 5, vol.1, p19)

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Q25.
1) Let's solve the following tasks.
2) If you change the ratio to fraction, is it representing the part-whole relationship?

(Grade 6, vol2,p35,2005; p11, 2011)37
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Q26. Compare the following task with Q23. Can we see the fraction by the part-whole
relationship?

(Gakko Tosyo,Grade6, vol2,p27,2005; Grade 6, vol.1,p53,2011)38
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Fraction as Ratioab

On the context of ratio a : b, is called value of ratio. However, in some countries,
fraction is usually explained on the context of the part-whole relationship. Contextually,
it is correct, however, in the previous chapter, the author already explained that thisnarrow understanding of fraction has developed huge misconceptions of fraction itself.
nFraction m (m ≠ 0) in mathematics is a representation of rational numbers. It is thevalue of ratio n:m which cannot be defined by the part-whole relationship. For
understanding fraction, we should know the various meanings and must use suitable
representations for every meaning.
Indeed, on the context of usage on the ratio, we cannot generally explain the fraction by
the part-whole relationship. For example, speed is the quantity such as km per hour.
Distance is not the part of time duration. The ratio of boy and girl is 2:3. In fraction, it is22. It means that if there are three girls, the boys must be two: 3 × 3 = 2. Here, the boy is
3

not the part of the girl. The boy is the part of the whole human, not the girl. The ratio ofboy and human 2:5 can be seen as a part-whole relationship.
Thus, through the learning sequence, children relearn all meanings or contexts of using
fraction, and at the end of it. They can be re-presented/integrated by the idea of ratio.
Fraction as ratio is usually used in the context of multiplication. The ratio of boys and22girls is 2:3. The value of ratio is 3. 3 is used for knowing the number of boys when the2
3number of the girls is given: (the number of boys)= x (the number of girls). This is the
proportion. On the equation to represent the proportion, the ratio is used for thecoefficient. Until students well understand the proportionality3, we cannot say that thedividing fraction is the fraction as ratio.
Let’s think about the various representation of proportionality. Firstly, proportionality
appeared from each row of the multiplication table. However, on the multiplicationtable, children do not clearly think co-variable. It is difficult to distinguish the covariable with the term of multiples from the co-variable with the term ofdifference/increase at the introduction of rows because they are developing each row byadding multiplier based on accumulation.

Exercise
If there is proportionality between the length of ribbon and its price, please fill in the ? .

3 Here, proportionality means properties to correspond proportion such as if x value

become 2x, 3x, and so on, corresponding y value become 2y, 2y and so on.
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Rule of Three: two by two matrix table
In the situation of the proportionality, if three numbers are
given on the table, we can get the one in the remaining partof the table. Historically, this rule is called 'Rule of Three'which was the ancient’s method for ratio and proportional
reasoning. The idea itself is well known at the age of
Egyptians (BC17C). It has existed before the invention of
expressions and equations for algebra. Historically, the Rule of Three in 17th century isrepresented the arrangement of table with only four numbers.
In school mathematics, the rule of three was taught just as a rule in relation to ratio andproportion. Later, the rule of three is represented by the three different expressions orformula, however we should know that the original rule of three itself was used without
current expressions. It was used to be the number-arrangement on the paper to consider
the ratio and proportion. Currently, we distinguish it by different formula ofmultiplication and division depending on the arithmetic situations. However, rule ofthree is based on the ratio and we do not need to care to distinguish the multiplication
or division expressions depending on the situations. If we array numbers and consider
the relationship with idea of ratio and proportion by using multiplication and division, it
is enough. On this meaning, the person who explains the rule of three as the threedifferent types of formulas is not using the rule of three but thinking based on current
arithmetic representation oriented to algebra which should be represented by thedifferent expressions. Likely using rule of three in ancient era, table and arrow
representation for representing ratio must be easier for everyone and closer to theancients approach because rule of three does not need to distinguish the situations by
the three different types of expressions. On the algebraic representation of situation, if
given quantities on ratio x to y, the value of x/y is ratio and y is the base. The formulas,x/y=a, x=ya and y=x/a, correspond to three types of situations. On the rule of three, if
the situations are written on the two by two matrix table as one unknown on threeknown, we do need to distinguish the place of given numbers and multiples,unnecessary to memorize three formulas. If we use the rule of three meaningfully on ratio
and proportion, we do not need to distinguish these three types of formulas.

Tape and Number-Line diagram: Proportional number line
Proportionality or Rule of Three are represented on the tape and number-line diagram
as well as table. Japanese has been using these representations since 1960s.
If the rule of three on the table is just used for the rule, relatively, the tape and numberline diagram are better to represent the meaning of proportionality because it works torepresent co-variable with the images of the size of number on the line. Even if children
do not know the term of proportion it supports to establish the way to develop
expression and its calculation. We call it proportional number line. If we use the rulewith tape diagram instead of table, it shows meaningful magnitude.

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For explanation of meaning, we usually use various representations. Expressions can bederived from them. Proportional number line is also useful for studying fractions. The
diagram of operational fraction with the remainder in Chapter 2, Figure 3, is also seen
as a kind of proportional number line. Children draw table and diagram if their teacher
clearly taught how to draw and use them. Teachers have to learn how to draw the
proportional number lines.

(Gakko Tosho Grade 5, Vol.1, pp29-30,2005;pp30-31,2011)

Exercise
Let's get the price for 2.4m by the proportional number line.
For knowing how and what, please explain the way to use the proportional number line
by yourself.

(Gakko Tosho Grade 5, Vol.1, pp30, 2005;pp32,2011)

Multiplication as for the Origin of Proportionality
Origin of proportionality is multiplication table. At the introduction of multiplicationtable, every row can be seen by additive property and later, we re-see it for
multiplicative property. It is the origin of proportionality.

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For developing the proportionality, Gakko Tosho textbook (2021, Grade 3) introduces
proportional number line as follows after students learned multiplication table.

After students learned proportion in grade 5, it is drawn as follows (Isoda, Olfos 2021).

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How to draw the proportional number line
Firstly, draw the amount for unit which is known:
80 yen0

1mSecondly, draw the unknown for estimation:
? yen
80
0

1

2

One of the objectives to draw the
proportional number line issearching the way of calculation.
Its procedure for drawing is likethis. Arrows show the calculation
based on the proportionality.

m

2.4

Thirdly, add possibly known from estimation:
? yen
160
80
0

1

2

2.4

m

The price of 2.4m is not
sure. If I change theunit 'm' to 'cm,' it is 80
yen for 100cm. How
much in 240cm. Aha, 8
yen for 10cm: it means
0.1m!

If not clear, let's change the unit for measuring the unknown:
8
160
? yen
80
0

0.1

1

2

2.4

m

Finally, search the way of calculation under the proportionality:
× 24
8

÷ 10

80

0 0.1

÷ 10

1

× 24

160

? yen

2

2.4

m

80 × 2.4 =? The length corresponds the price. The re-measuring process is represented
by the arrows. The arrow sequence represents 80 ÷ 10 × 24. Aha!!

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Major Reference and Further readings 5
Masami Isoda et al. (2010). Mathematics Education Theories for Lesson Study: Problem
Solving Approach and the Curriculum through Extension and Integration, Journal of Japan
Society of Mathematical Education, 92(11). 1-158.
Shin Hitotsumatsu et al. (2005). Study with your friends: Mathematics for Elementary
School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda et al. (2010). Study with your friends: Mathematics for Elementary School (12vols). Tokyo: Gakkotosyo.
Masami Isoda (2012). Introductory Chapter: Problem Solving Approach to Develop
Mathematical Thinking. In Isoda, M. & Katagiri, S. edited (2012). Mathematical Thinking:
How to Develop it in the Classroom. Singapore: World Scientific. 1-28.
https://www.criced.tsukuba.ac.jp/math/apec/ICME12/Lesson_Study_set/MATHEMATICAL%20THINKING%20(c)World%20Scientific/MathematicalThinking_chap01(c)WorldScientific.pdf

https://doi.org/10.1142/9789814350853_0001

Masami Isoda, Tenoch Esaú Cedillo Ávalos, et al. (2012). Matemáticas para la educación
normal (11vols). Estado de México : Pearson Educación de México.
Tenoch Cedillo, Masami Isoda, et al. (2013) Matemáticas para la educación normal : guía
para el aprendizaje y enseñanza de la aritmética, Estado de México : Pearson Educación de
México.
Masami Isoda (2015). The Science of Lesson Study in the Problem Solving Approach. In
Inprasitha, M. et.al. edited. Lesson Study: Challenges in Mathematics Education. Singapore:
World Scientific. pp.81-108. https://doi.org/10.1142/9789812835420_0006
Masami Isoda (2015). Dialectic on the Problem Solving Approach: Illustrating Hermeneutics
as the Ground Theory for Lesson Study in Mathematics Education. In Sung, J. edit. Selected
Regular Lectures from the 12th International Congress on Mathematical Education. Cham:
Springer. pp.355-381. https://doi.org/10.1007/978-3-319-17187-6_21
Masami Isoda & Aki Murata edited (2011). Study with your friends: Mathematics for
Elementary School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda, Aki Murata, Aida Istino Yap edited (2015). Study with your friends:
Mathematics for Elementary School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda & Aki Murata edited (2021). Study with your friends: Mathematics for
Elementary School (12 vols). Tokyo: Gakkotosyo.
https://support.gakuto.co.jp/mathematics-textbook/
Masami Isoda, Raimundo Olfos. Edited. (2021).
Teaching Multiplication with Lesson Study.
Cham: Springer. DOI: 10.1007/978-3-030-28561-6

44
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Chapter 6. Multiplication and Division of Fraction (1)For developing the multiplication and division of fractions, this chapter mainly explains
the necessary knowledge needed to produce the idea for it.
Why did Professor
recommend us to
use the proportional
number line?

To explain the meaning of calculation, we mustchange the representation which explains theprocess in a visible and understandable way.

If we do not use it, we only teach skill as arule. Where shall we use it to explain themeaning? We have to teach how to draw.

The idea of proportional number line was originated from Rene Descartes(1637). Japanese Math-Educators such as Takeshi Ito invented it toestablish the Heuristic Teaching Approach for elementary schoolmathematics with the extension and integration curriculum sequence inthe 1960s. From the 1990s, it became the world famous approach as forthe representations to develop the competency for proportional reasoning.
Japanese textbooks such as Gakko Tosho established well teachingsequence for developing it.

Number line
In chapter 1 and 2, we recognize that shading activities, the parts for counting, is theroot for the misconceptions of fractions for missing the idea of whole as for a unit. For
developing children who learn mathematics for and by themselves (see Chapter 1),
children are necessary to draw and use appropriate diagrams for explaining their ideas.
On the previous chapters, appropriate diagrams need to show original unit withquantity fraction and the unit fraction for measuring its number, like measuring byusing the remainder as for operational fraction. The tape and number line withquantities are appropriate diagrams on this condition. For developing children who willdraw such a diagram, firstly, we have to develop children who draw the number line by
and for themselves.
Number line which shows the position of number on the line is introduced by taking thesame intervals by the unit of measurement recursively for comparing the size of
number. At this moment, it looks like a line of discrete numbers because it is given45
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by interval as for the scale on the line and there is only one interval but no number
between two numbers. On the process of extension of numbers, when students re-scale itby using smaller units or larger units, it begins to function as number lines which shows
the position of various numbers and used for extension of numbers. At the beginning,children learn the '0' is the starting point on
the line instead of ‘nothing.’ On the number
line, 0 shows the origin of position as formeasuring by the interval (unit). The
difference of positions shows the distance (thenumber of intervals: cardinal number).
If teacher does not teach the measuring by
the unit from the starting point 0, children
may confuse one on counting intervals as
scale number 0 on the left instead of thescale number 1 on the right. Children learn
the number line as for comparing the sizeand ordering of numbers, and recognize thenumber on the base ten system. Taking
interval is the preparations for themultiplication and division, too.

(Gakko Tosho, Grade 1, pp71-72, 2005; pp82-83,2011)
If your children do not know how to draw the number line, let's give them theopportunity to draw it by themselves. It is the activity of measurement by using
arbitrary unit.

Multiplication(Gakko Tosho, Grade 2, vol.2 pp2-3, 2005; pp6-7,2011)
In the case for lower elementary school mathematics, children study how to use daily
language mathematically on the situation. Definition of arithmetic operations is usually
done on the situation in daily context because we have to develop children to use fourarithmetic operations on their daily life.
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On this issue, multiplication is introduced at the following situations: the number ofdishes and the number of objects for each dish.
From the viewpoint of measurement, the situation is used to explain multiplication thatthe multiplication is measuring the amount of the quantity by the unit of quantity when
the unit for the amount is known by the quantity (Ministry of Education, Japan, 1960;
Freudenthal, 1983). For example, when the amount is 8 dishes and the unit for theamount is 2 cakes in each dish, the measured amount by the quantity is 8 x 2 =16(cakes).
This definition works for multiplication of decimal numbers and fractions as well as thesituation of repeated addition. For example, when the amount of steel is 2.4 m and theunit weight for the steel is 1.5 kg for one meter, the measured amount by the quantity is
1.5 x 2.4 = 3.6 (kg)4:
1.5kg

0

1.5

1.5x2 1.5x2.4

0

1

2

2.4

kg
m

In relation to how to calculate, multiplication is explained with the repeated addition,
however, multiplication of decimal numbers and fractions is not explained by therepeated addition but explained by multiplication table with distribution law on baseten place value system. For explaining the multiplication of decimal numbers andfractions, we can use the proportional number line which represents the meaning ofmultiplication by measuring.

How to find the expression from the situated problem
At the end of last chapter, you learned how to draw the proportional number line by thetask below:

4 This textbook is using Japanese notation: 1.5 (kg/m) x 2.4 (m) = 3.6 (kg). In English

notation, it should be 2.4 (m) x 1.5 (kg/m) when ‘2.4 x 1.5’ is read as ‘2.4 times 1.5’. In
English, ‘2.4 times’ implicates ‘multiplied by 2.4’. Thus, as long as you read ‘1.5 x 2.4’ as‘1.5 multiplied by 2.4,’ Japanese notation of multiplication is understandable. Indeed, ‘ax b’ can be read as ‘a multiplied by b’ even in English. English usage has inconsistency.
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The box (in blank) is 2.4 at the end of last chapter. The children who have not yet
learned the multiplication of decimal numbers and fractions cannot easily recognize that
this task is multiplication. On the other hand, if we put the whole number such as 2 into
the box, children who learned the multiplication of whole numbers could easily
understand that this task is multiplication because multiplication is introduced in daily
situations on the whole numbers.
On the context of extension of numbers and operations, Japanese teachers usually
prefer this problem posing strategy like this form and ask children to put any number
they want into the box and discuss how. Through putting into a simple number, children
recognize this task as multiplication and in the case of whole numbers, they alreadylearned, and in the case of fraction and decimal numbers, they did not yet learn. Whenthe class begins this way, children recognize this task as the task for multiplication andthey would like to inquire how to find the answer using what they already learned.

Exercise3
2

Let's draw the proportional number line when the box (blank) is 2 (m), 2.3 (m) or (m)on 1 .
In this exercise, for answering in the case of 2.3 m, we have to change the unit from 1m
to 0.1m as well as the case of 2.4m. In the case of fraction, we usually change the unit
1

1from 1m to the unit fraction: this case 2 m is the unit for measuring. If 1m is 80 yen, 2 m
1
2

3
2

is 80 ÷ 2 yen. If m is 80 ÷ 2 yen, m is 80 ÷ 2 × 3 yen. For considering like this,children need to draw a proportional number line and apply multiplication and divisionon the number line.

Fraction × Whole Number
Please explain the following:48
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If 1 piece is

7
7
7
7
7
7
28
m and addition of fractions is known, 4 pieces are × 4 = + + + = .
5
5
5
5
5
5
5

After the explanation, we should ask children as follows: Is it possible to find an easieror faster way? Then,

7×4
7
is recognized as a simple way for 5 × 4.
5

In mathematics, we usually produce shorter and simple ways. Seeking simplicity is abasic value of mathematics.

Exercise
In Gakko Tosho textbooks, the proportional number line changes from the tape diagramand number line to two number lines at Grade 5. Let's put the number in the box and4

answer: "We can cover an area of 5 m2 with 1dl paint. How many m2 can we cover withdl of the paint?"

Fraction × Fraction
If the box is a whole number, we already learned.
2

If the box is a fraction such as 3 dl, we can draw
the proportional number line:
If we develop the way of calculation as well as
multiplication of decimal numbers, it can be4

calculated as follows: 5 ÷ 3 × 2.
After the explanation, we should ask children as follows: Is it possible to find an easier
4

4×2

4

2

or faster way? Then, 5 ÷ 3 × 2 = 5×3 is recognized as the simple way for 5 × 3.
In mathematics, we usually produce shorter and simple ways.

Area Diagram
In school mathematics, area diagram is usually recommended for use in explaining(a+b)(c+d)=ac+ad+bc+bd. On the area diagram, multiplication is a two-dimensional idea
and it functions as for the model of commutativity. Some people strongly believe that thearea diagram is the best way for explaining multiplication and division because it
provides the wall painting/shading metaphor based on two dimensions. The
misconception will appear if students do not feel the necessity to draw the same size
diagram.
As long as teachers try to explain fraction as dividing fraction it might be true, however,shading activities of the area diagram itself is a major source of the misconceptions if
children recognize fraction only by dividing fraction (see Chapter 2).
What is necessary to develop in students by and for themselves is that students are able
to draw the area diagram by and themselves as for the tool for reasoning as well as the
proportional number lines.

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Historically, Euclid produced the theory under the dimension and Descartes overcame
the wall between dimensions by the proportional number line which defined
multiplication by the measurements.

Exercise
1) Let's solve the following task by three different methods.

Gakkotosho G6, vol.2, p3, 2011; G6,vol.1. pp34-35.
1) Let's compare three methods. Which one do you recommend? Why do we need more?
2) There are a number of students who get the answer using the area diagram such as
412× 3 = 15. Why does the area diagram produce such answer?
5

There are two types of area diagram. The first type is already shown in the textbook. It
shows the area of the wall itself for showing the painting. Another type of area diagram
represents the following situation:
4

There is a wall in which we use 5 dL of paint for painting 1 meter of wall. How muchliters do we need for painting 3 meters?
For this task, we draw the following two diagrams for the same meaning.

dL

dL

4
dL
5

4
dL
5

4
dL
5

4
dL
5

1
dL
5

1

+

+

0

5

1m

2m

3m

dL

1m

1m

1m

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In both diagrams, the denominations of quantities are necessary because children mightdevelop misunderstanding such as

412× 3 = 15 if there are no denominations.
5

Area diagram is fine to explain meaning. However, the children who still keep thedividing fraction, misunderstand the meaning of whole. Indeed, the left hand side of the4

above diagram can be read as 5 if the square is the whole even though it shows quantity.
Here, the key is a unit fraction

1dL (quantity fraction!) as well as the whole 1 L. If
5teachers ask students just shading without considering these two units, we are not surestudents understand well or not, even though teachers felt success to explain for
him/herself.

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Major Reference and Further Readings 6
Masami Isoda et al. (2010). Mathematics Education Theories for Lesson Study: Problem
Solving Approach and the Curriculum through Extension and Integration, Journal ofJapan Society of Mathematical Education, 92(11). 1-158.
Shin Hitotsumatsu et al. (2005). Study with your friends: Mathematics for Elementary
School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda et al. (2010). Study with your friends: Mathematics for Elementary School(12 vols). Tokyo: Gakkotosyo.
Masami Isoda (2012). Introductory Chapter: Problem Solving Approach to Develop
Mathematical Thinking. In Isoda, M. & Katagiri, S. edited (2012). MathematicalThinking: How to Develop it in the Classroom. Singapore: World Scientific. 1-28.
https://www.criced.tsukuba.ac.jp/math/apec/ICME12/Lesson_Study_set/MATHEMATICAL%20THINKING%20(c)World%20Scientific/MathematicalThinking_chap01(c)WorldScientific.pdf

https://doi.org/10.1142/9789814350853_0001

Masami Isoda, Tenoch Esaú Cedillo Ávalos, et al. (2012). Matemáticas para la educación
normal (11vols). Estado de México : Pearson Educación de México.
Tenoch Cedillo, Masami Isoda, et al. (2013) Matemáticas para la educación normal :guía para el aprendizaje y enseñanza de la aritmética, Estado de México : Pearson
Educación de México.
Masami Isoda (2015). The Science of Lesson Study in the Problem Solving Approach. In

Inprasitha, M. et.al. edited. Lesson Study: Challenges in Mathematics Education.
Singapore: World Scientific. pp.81-108. https://doi.org/10.1142/9789812835420_0006
Masami Isoda (2015). Dialectic on the Problem Solving Approach: Illustrating
Hermeneutics as the Ground Theory for Lesson Study in Mathematics Education. In

Sung, J. edit. Selected Regular Lectures from the 12th International Congress on
Mathematical Education. Cham: Springer. pp.355-381. https://doi.org/10.1007/978-3319-17187-6_21
Masami Isoda & Aki Murata edited (2011). Study with your friends: Mathematics for
Elementary School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda, Aki Murata, Aida Istino Yap edited (2015). Study with your friends:
Mathematics for Elementary School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda & Aki Murata edited (2021). Study with your friends: Mathematics for
Elementary School (12 vols). Tokyo: Gakkotosyo.
https://support.gakuto.co.jp/mathematics-textbook/
Masami Isoda, Raimundo Olfos. Edited. (2021).
Teaching Multiplication with Lesson Study.
Cham: Springer. DOI: 10.1007/978-3-03028561-6

52
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Chapter 7. Multiplication and Division of Fraction (2)The reasoning for ratio and proportion isalready begun from the multiplication at
the second grade. Proportional number lines
and Rules of three on the table are therepresentation of the proportionality:
Children should study them before learning
ratio and proportion for multiplication anddivision of fraction by and for themselves byusing what they already learned.

Rules of Multiplication and Division
Rules of Multiplication and Rules of Division which appears the comparison of
expressions for multiplication and division can be seen as the representation of the
proportionality. Both rules are also useful for the extension of multiplication anddivision into decimals and fractions from the whole number.

Exercise:
The followings are the samples of the rules of multiplication and division:1) Let's find the rules of division.

Gakkotosho G4, vol.1, pp21-24 (2011)53
http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/©Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach. Tokyo: CRICED, University of Tsukuba2) Let's find the rules of multiplication:

Gakkotosho G4, vol.1, p89 (2011)
3) Let's explain those rules by using four number table for the Rule of three.
4) Rules of multiplication are easier ways to find the answer of multiplication of
fractions. Let's compare the following two approaches and explain how to use the rule ofmultiplication below (see page 49).

Gakkotosho, G6, vol.1, p36 (2011)54
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Exercise: Fraction divided by Fractions
1) Let's find the answer of the following task by the proportional number line, rule ofthree, rule of division, and area diagram.

Gakkotosho G6. vol.1, pp44-45 (2011).

2) Explain the following approaches to obtain the answers.
a) Rule of Divisionb) Proportional Number Linec) Area Diagram

Gakkotosho G6. vol.1, pp46-47 (2011).

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http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/©Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning How to Teach. Tokyo: CRICED, University of Tsukuba3) Let's compare the three approaches. Which approach uses the unit fraction forchanging the measurement scale? Which approach directly shows the way for themultiplication of inverse fraction? Which approach has the possibility to produce the8wrong answer 20?

Major Reference and Further Readings 7
Masami Isoda et al. (2010). Mathematics Education Theories for Lesson Study: Problem Solving
Approach and the Curriculum through Extension and Integration, Journal of Japan Society ofMathematical Education, 92(11). 1-158.
Shin Hitotsumatsu et al. (2005). Study with your friends: Mathematics for Elementary School (12vols). Tokyo: Gakkotosyo.
Masami Isoda et al. (2010). Study with your friends: Mathematics for Elementary School (12vols). Tokyo: Gakkotosyo.
Masami Isoda (2012). Introductory Chapter: Problem Solving Approach to Develop MathematicalThinking. In Isoda, M. & Katagiri, S. edited (2012). Mathematical Thinking: How to Develop it in
the Classroom. Singapore: World Scientific. 1-28.
https://www.criced.tsukuba.ac.jp/math/apec/ICME12/Lesson_Study_set/MATHEMATICAL%20THINKING%20(c)World%20Scientific/MathematicalThinking_chap01(c)WorldScientific.pdf

https://doi.org/10.1142/9789814350853_0001

Masami Isoda, Tenoch Esaú Cedillo Ávalos, et al. (2012). Matemáticas para la educación normal(11vols). Estado de México : Pearson Educación de México.
Tenoch Cedillo, Masami Isoda, et al. (2013) Matemáticas para la educación normal : guía para el
aprendizaje y enseñanza de la aritmética, Estado de México : Pearson Educación de México.
Masami Isoda (2015). The Science of Lesson Study in the Problem Solving Approach. In
Inprasitha, M. et.al. edited. Lesson Study: Challenges in Mathematics Education. Singapore:
World Scientific. pp.81-108. https://doi.org/10.1142/9789812835420_0006
Masami Isoda (2015). Dialectic on the Problem Solving Approach: Illustrating Hermeneutics as
the Ground Theory for Lesson Study in Mathematics Education. In Sung, J. edit. Selected
Regular Lectures from the 12th International Congress on Mathematical Education . Cham:
Springer. pp.355-381. https://doi.org/10.1007/978-3-319-17187-6_21
Masami Isoda & Aki Murata edited (2011). Study with your friends: Mathematics for Elementary
School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda, Aki Murata, Aida Istino Yap edited (2015). Study with your friends: Mathematics
for Elementary School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda & Aki Murata edited (2021). Study with your friends: Mathematics for Elementary
School (12 vols). Tokyo: Gakkotosyo.
https://support.gakuto.co.jp/mathematics-textbook/
Masami Isoda, Raimundo Olfos. Edited. (2021).
Teaching Multiplication with Lesson Study. Cham:
Springer. DOI: 10.1007/978-3-030-28561-6

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Summary
As discussed at the exercises of Chapter 1, this textbook is written for developingchildren who learn mathematics by and for themselves. For developing children, we
should set the appropriate task sequence from what they already learned. Difficulty offraction is related with various meanings of fraction and originated from many teachers
who believe that the meaning of fraction is only dividing fraction. This textbook
explained that it is the source of misconceptions. It demonstrated various meanings offractions with terminology and the necessity of teachers considering them forovercoming difficulties.
In Japan, the terminology to distinguish the conceptual difference of fraction and so on
were already known in the 1960s, at least, as a result of lesson study since 1873 withsurveys of various countries’ mathematics education. Terminology has been used toestablish curriculum and task sequence on the textbooks. The author learned it fromProf. Tatsuro Miwa on his lectures for undergraduate students in 1981.
Due to difference of curricula sequence and textbooks, teachers who read this textbook
may feel uncomfortable to use it. It might be normal as long as keeping the mindset toteach procedure by exercise with minimum explanation by teachers. However, if
teachers try to develop children who explain their mathematical ideas by using what
they already learned, the terminology might be significant. Indeed, Japanese teachers
who show excellent practice use it for clarifying what children already learned and how
they use it for thinking, and knowing necessity as for preparation of future learning.

Quantity
Fraction

Equivalence of fraction
on number line

Figure 7. Various meanings of fraction and their development57
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This textbook used Gakko Tosho textbooks, see QR codes in the reference, which havebeen written by teachers in Elementary School of the University of Tsukuba which has
been leading lesson study since 1873.
For using the terminology of this textbook, the task sequence to develop children who
learn mathematics by and for themselves will be considered by Figure 7. Please note, it
illustrates the necessary process for conceptual change under the extension andintegration up to the fraction as ratio as for the base of rational number.
As long as teachers believe fraction means dividing fraction, dividing fraction may
include operational fraction in a broad sense, however, in a narrow sense, the action for
dividing into equally corresponds to the action for partitive division and the action for
scaling by the unit fraction corresponds to the action for quotative division. On thisdifference, a narrow meaning of the dividing fraction should be distinguished with
operational fraction. To change dividing fraction as comparable fraction, denominationof quantities is necessary for fixing and showing the size of the whole. To recognize
fraction as number, equivalence of fraction on the number line and to see fraction as theanswer of division are necessary. To consider the multiplication and division of fraction,
it is necessary to treat the fraction under proportionality. In relation to proportionality,
fraction is ratio. In relation to the meaning of division, there are three types on ratio. At
the introductory stage, dividing fraction is a pair of numbers and not a number itself.
For extension of number, we should discuss existence, comparison, and operations. Aftercompletion of these three discussions, fraction becomes the representation of rational
number.
This book focused on fraction. Teaching of decimals can be considered analogically.

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Further Readings
Tenoch Cedillo, Masami Isoda, et al. (2013) Matemáticas para la educación normal : guía
para el aprendizaje y enseñanza de la aritmética, Estado de México : Pearson Educación de
México.
Masami Isoda y Raimundo Olfos, coodinadores ; [contribuyentes] Ubiratan D'Ambrósio [etal.] (2011). Enseñanza de la multiplicación : desde el estudio de clases japonés a las
propuestas iberoamericanas, Valparaíso [Chile] : Ediciones Universitarias de Valparaíso,
Pontificia Universidad Católica de Valparaíso
Masami Isoda, Raimundo Olfos (2009).El enfoque de resolución de problemas : en la
enseñanza de la matemática : a partir del estudio de clases. Valparaíso : Ediciones
Universitarias de Valparaíso.
Masami Isoda, Max Stephens, Yutaka Ohara, Takeshi Miyakawa (2007). Japanese Lesson
Study in Mathematics: Its Impact, Diversity and Potential for Educational Improvement.
NJ: World Scientific.
Masami Isoda, Shigeo Katagiri (2012). Mathematical Thinking: How to Develop it in theClassroom. NJ: World Scientific.
http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/
http://www.criced.tsukuba.ac.jp/en/seameo_index.html
http://www.criced.tsukuba.ac.jp/math/apec/
http://www.criced.tsukuba.ac.jp/renkei/msa/
https://www.criced.tsukuba.ac.jp/online-contents.html

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