©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

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 thepreparation 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.
http://mathTokyo:CRICED,UniversityofTsukuba
which
downloaded
frominfo.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 forSEAMEO 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 toTeach. (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 the permission from Gakko-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 tothe 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 inthe Year 2005;
President of the Japan Publishers Association as for the Most Beautiful Book ofthe Year 2010 in the area of natural science;
Honorary PhD (Mathematics Education) by Khon Kaen University, Thailand asfor 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 ................................................................................................................................. 2
About the Author ................................................................................................................. 4
Table of Content .................................................................................................................. 5
Chapter 1: What is fraction? ............................................................................................... 7
Questions for professional development 1 ..................................................................... 10Major Reference and Further readings 1 ...................................................................... 11
Chapter 2: Dividing Fraction and Quantity Fraction....................................................... 12
Questions for professional development 2 ..................................................................... 15
Dividing Fraction ........................................................................................................... 16
Exercise: ......................................................................................................................... 17Operational (measuring) Fraction ................................................................................. 17
Euclidian Algorithm....................................................................................................... 18
Exercise: ......................................................................................................................... 18
Quantity Fraction .......................................................................................................... 18Unit Fraction for measuring unit .................................................................................. 19
Major Reference and Further Readings 2 ..................................................................... 19
Chapter 3: Addition of Fraction with Different Denominators ........................................ 20
Exercise .......................................................................................................................... 20
Questions for professional development 3 ..................................................................... 24
Major Reference and Further readings 3 ...................................................................... 25
Chapter 4: When does Fraction become Number? ............................................................ 26
Equivalent Fraction ....................................................................................................... 26
Fraction is Expression ................................................................................................... 27
Fraction as Quotient ...................................................................................................... 27
Questions for professional development 4 ..................................................................... 30

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Major Reference and Further readings 4 ...................................................................... 31
Chapter 5. Fraction in relation to ratio and proportion ................................................... 32
Questions for professional development 5 ..................................................................... 33
Fraction as Ratio ............................................................................................................ 38
Exercise .......................................................................................................................... 38
Rule of Three .................................................................................................................. 39
Tape and Number-Line diagram: Proportional number line ........................................ 39
Exercise .......................................................................................................................... 40
How to draw the proportional number line ................................................................... 41
Major Reference and Further readings 5 ...................................................................... 42
Chapter 6. Multiplication and Division of Fraction (1) .................................................... 43
Number line ................................................................................................................... 43
Multiplication ................................................................................................................. 44
How to find the expression from the situated problem ................................................. 45
Exercise .......................................................................................................................... 46
Fraction × Whole Number ............................................................................................. 46
Exercise .......................................................................................................................... 47

Fraction × Fraction ........................................................................................................ 47
Area Diagram ................................................................................................................. 47

Exercise .......................................................................................................................... 48
Major Reference and Further Readings 6 ..................................................................... 50
Chapter 7. Multiplication and Division of Fraction (2) .................................................... 51
Rules of Multiplication and Division ............................................................................. 51
Exercise: ......................................................................................................................... 51
Exercise: Fraction divided by Fractions ........................................................................ 53
Major Reference and Further Readings 7 ..................................................................... 54
Further Readings .............................................................................................................. 55

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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:
Som0m

Any1m

2m0m

↑1m2m↑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 toask why to others and discuss. It means that you usually engage in a similar activity byyourself. You already have the mind set for learning how to learn by yourself!

Discussion 1: Why do you think so?

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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 partsSom: 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 1mSom: 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 anytime.

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 inJapanese: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_Scie
nce11

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