©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 asTIMSS 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
whichdownloaded
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 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 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 thepermission
fromGakko-Tosho:
KatsuakiSerizawa(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 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 ........................................................................................................... 16Exercise: ......................................................................................................................... 17
Operational (measuring) Fraction ................................................................................. 17
Euclidian Algorithm....................................................................................................... 18Exercise: ......................................................................................................................... 18
Quantity Fraction .......................................................................................................... 18
Unit Fraction for measuring unit .................................................................................. 19
Major Reference and Further Readings 2 ..................................................................... 19
Chapter 3: Addition of Fraction with Different Denominators ........................................ 20
Exercise .......................................................................................................................... 20
A Sample of Arguments: ................................................................................................ 21
Questions for professional development 3 ..................................................................... 24
Major Reference and Further readings 3 ...................................................................... 25
Chapter 4: When does Fraction become Number? ............................................................ 26
Equivalent Fraction ....................................................................................................... 26
Number Line .................................................................................................................. 26
Fraction is Expression: Fraction as Quotient ............................................................... 27
Exercise. Let’s read and answer. ................................................................................... 28
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: two by two matrix table ......................................................................... 39
Tape and Number-Line diagram: Proportional number line ........................................ 39
Exercise .......................................................................................................................... 40Multiplication as for the Origin of Proportionality ....................................................... 40
How to draw the proportional number line ................................................................... 42
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 ........................................................................................................ 47Area 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
Summary ........................................................................................................................... 55
Further Readings .............................................................................................................. 57

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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 linesand 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 ofexpressions for multiplication and division can be seen as the representation of theproportionality. 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)51

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 offractions. 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)52

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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.

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

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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 thewrong answer

8?
20Major 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: Mathematicsfor 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 development55

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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 deMéxico.
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 laenseñanza 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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