Available online at www.sciencedirect.com

Procedia Social and Behavioral Sciences 8 (2010) 17–27

International Conference on Mathematics Education Research 2010 (ICMER 2010)

Lesson Study: Problem Solving Approaches in Mathematics
Education as a Japanese Experience
Masami Isoda*
Center for Research on International Cooperation in Educational Development, University of Tsukuba

Abstract

Lesson study is a scientific activity for Japanese teachers on their meaning. In Japan, research topics for lesson study
usually shared through the regular revisions of curriculum and research movement by several societies (Isoda et al.;2007). As a result of teachers' challenges, several local theories of teaching were shared. 'Problem Solving
Approach' has been known as Japanese teaching approach (Stigler & Hiebert; 1999). It is a well known theory ofteaching for developing children who learn mathematics by/for themselves in Japan. It includes teaching about
learning how to learn. Another important products of lesson study are theories for curriculum, described in teacher's
guidebooks as a number of technical terms which are only used by teachers and math-educators for sharing
pedagogical content knowledge.
© 2010 Elsevier Ltd. All rights reserved.
Keywords: Japanese lesson study; Problem Solving Approach; Pedagogical Content Knowledge; History of lesson study

1.

Introduction

Lesson study is a scientific activity for teachers who try to develop their theories for developing and sharing goodpractices. Product of lesson study is not limited within what each participant learned from the class and the postclass reflective discussion. Each participant reproduces the class with their own developed theories of practice ineach of their contexts. On the personal meaning, their theories are just a kind of pedagogical content knowledge,
which is working as their local theory on teaching in each of their practices. In the case of Japan, the regularrevisions of curriculum supply national research topics for lesson study (Isoda et al.; 2007). As a result of teachers'
challenge on the same topic from a school to national level, local theories are integrated into a shared theory ofteaching. Teachers' journals and academicians who usually participate in lesson study support this theorization
through proposing necessary technical terms for improving practice.
'Problem Solving Approach' has been known as Japanese teaching approach, which was well described by Stigler &Hiebert (1999). It is one shared theory for developing children who learn mathematics by/for themselves in Japan. It
includes teaching about learning how to learn, which means how to develop mathematics by/for themselves. It is theconsequence of more than one hundred years of lesson study.

* Corresponding author.
E-mail address: isoda@criced.tsukuba.ac.jp

1877-0428 © 2010 Published by Elsevier Ltd.
doi:10.1016/j.sbspro.2010.12.003

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Masami Isoda / Procedia Social and Behavioral Sciences 8 (2010) 17–27

Another important product of lesson study is theories for curriculum, described in teacher's guidebooks as a numberof technical terms which are only used by teachers and math-educators for sharing pedagogical content knowledge.
For example, in US, Cognitive Guided Instruction Project showed us the evidence of the counting types for additionand subtraction in 1980s. In the case of Japan, counting types and the problem situations were categorized afterWorld War I and before World War II. In 1960s, there was a confrontation on the definition of multiplications
between fixing the limitations of dimensions (like Euclid) and going beyond the limitation (like Descartes).
Proportional number lines have been introduced to overcome the limitation. Japanese national curriculum clearlyenhanced the extension and integration as for the principle of curriculum sequence to develop children for learning
mathematics by/for themselves.
Both major achievements are well described at the Special Issues of Journal of Japan Society of Mathematical
Education for EARCOME 5 (Isoda & Nakamura; 2010). The origins of technical terms are different depending oneach term. For example, various ancient mathematics such as Chinese mathematics and Arabic mathematics used todescribe the methods of solutions depending on the problem types. Thus, historically, analyzing the problem type
itself used to be a mathematical activity. Distinguish of division between Partitive division (Tobunjyo) andQuotative division (Hoganjyo) related with Japanese Language. On the other hands, the history of problem solving
approach is comparatively easy to find the origin.
In this paper, the meaning of products of lesson study is explained and then, the brief history until problem solving
approach is illustrated. Using a case of school level lesson study project, the practical theory of teaching for the
Problem Solving Approach is explained. It is a sample of lesson study on the problem solving approach.
2.

What is lesson study and What are the products of lesson study

2.1

What is lesson study

There are various understandings of lesson study. In many of English articles, lesson study is understood asenhancing the school based approach for improvement of teaching as for professional development. In the case ofJapan, we have much more contexts of lesson study. Here, the Japanese lesson study is recognized with following
features.
Process/lesson study cycle: Plan (Preparations), Do (Observations) and See (Discussion and Reflection) activities
involving with other teachers.
Various Dimensions of Open Classroom: Personal (by master teacher), Whole School, Regional and National
lesson study but Systematic.
Theme of lesson study: Study Topics and Objective are different. Study Topics such as Developing MathematicalThinking, Learning for/by themselves in relation to development, reform or improvement. Objective is
specified at each class related with curriculum. In the case of Japan, the objective is often described by thesentence ‘Through A, students learn/understand/enable to do B’ because Japanese curriculum asked
teachers to teach learning how-to and achievement as for outcome.
Lesson Plan: A format is not fixed, usually developed/improved depending on a study topic of lesson study. Somecountries recommend a set of national lesson plans as a part of curriculum, but lesson study is implemented
for new challenges and pushes the new format of lesson plan and ways of teaching approach.
Teachers’ mind: Lesson study is conducted by teachers for developing students in a classroom and making each
student developing him/herself, not for researchers who just observe a classroom through their telescopes.
Even though researchers participate as for their research, if they do not understand teacher’s objectives fordeveloping children, and if they do not work together with them, it is just the activities as a social scientist
as an observer. In this sense, lesson study recommends that researchers are teachers who propose
improvement of class, as well as teachers are researchers who analyze children’s understanding.
Results: Lesson study usually considers achievement in relation to study topic and objective. At the same time, aimsof lesson study change depending on participants and are not always the same as seen in the following:
Model teaching approach, New ideas for traditional approach, Understanding objectives, What students

Masami Isoda / Procedia Social and Behavioral Sciences 8 (2010) 17–27

19learned before the class, What students learned and could not learn in the class, Teachers’ values, Students’values, Professional development, Ideas for the curriculum reform, Theory of mathematics education, andso on.
Sequential experience for sharing the heritage: Lesson study cycle continues beyond the generation. It’s usually
opened for newcomers and changes experienced bearers. On this context, similar experiences usually arerecognized as new experiences with challenges. That is the reason why lesson study develops the learning
community.
One of the most sharable products is a description of model approaches. The guidebooks for teaching contents andteaching approaches have been written by teachers. In these twenty years, videos have been used for sharing good
approaches by making them more visible (APEC lesson study project 2006: first announcement). In some countries,
a model approach sometimes means a teaching manual with the sequence of teacher’s questions and children’s
answers which are expected to be followed by every teacher such as hamburger shops. On the other hand, in the caseof lesson study, it is nature of that to work beyond a model because lesson study usually includes a proposal todevelop something new in their group based on their own theme of lesson. And usually, new challenges includes
some difficulty should be overcome. Thus, on the context of lesson study, a model approach means an illuminating
approach and major resources for adapting a model into each teacher’s classroom. And sometimes it means an
object of improvement for specific aims. On this meaning, lesson study is a reproductive science for teachers.
2.2

What are the products of lesson study

There are no limitations on the products of lesson study. On APEC Lesson Study project directed by Isoda, M. &Inprasitha, M. since 2006, it has been engaging in the lesson study to develop good practice on the theme forenhancing Mathematical Thinking (2007), Mathematical Communication (2008), and Assessment (2010). The
project aimed to spread lesson study movement for improving teaching practice. Math-educators have beendeveloping lesson study community with elementary school teachers in their economies. On this context, the project
asked the specialists the following questions (Figure 1) in relation to how lesson study is productive or influential inparticipating 19 economies on APEC project (Isoda & Inprasitha; 2008).
Useful for improvement of the quality of mathematics education.
100 %Influential to other subjects
93%
Used for developing innovative teaching approach.
93%
Used for curriculum improvement.
80%
Used for sharing model teaching approaches.
80%
Used for developing teachers
80%
Used for developing students
80%
Used for developing practical/local theories of Mathematics Education
53%
Figure 1: Survey results by the specialists from APEC economies (Isoda & Inprasitha; 2008)
Most of APEC economies have been in the stage for introducing lesson study. On this context, developing theories
of mathematics education are still unusual. On the other hand, in some economies such as Japan, teacher cannot
recognize the theory of mathematics education without considering their practice. There are researches which haveenhancing theories of mathematics education through the practical experiment. Gravemeijer,K. (2007) explained theprocess to develop the local theory of teaching (Figure 2). This is a case of Netherland to develop local theory basedon classroom practice on his research Mathematics in Context. Depending on the research context, the meaning oftheory for mathematics education is difference but Figure 2 by Gravemeijer is an illuminating case to show its
existence.

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Masami Isoda / Procedia Social and Behavioral Sciences 8 (2010) 17–27

Figure 2: Development of Local Teaching Theory by Gravemeijer, K. (2007)

3.

History of Japanese lesson study from the viewpoint of the theme

In the case of Japan, theories of teaching approaches and theories of subject matters or curriculum have been the
product of lesson study. This custom is deeply related with that each lesson study is done by the study theme such as
special topic of the lesson study. Following Figure 3 shows samples of shared study topics (revised, Nagasaki,
E.;2007).
Instead of lecture style, argumentation has been enhanced from the origin of lesson study at Tokyo Normal School(University of Tsukuba) in 1880s. The problem posing by children progressed at the attached school of Nara
Women's Higher Normal School (Nara Women's University) in 1920s. Open-ended problems began to be enhanced
before World War II. Beyond the occupation after WWII, the approach had been theorized again and again. For
developing mathematical thinking, the prototype of this approach was well-known. In 1980s, the approach began toshear. The junior high school textbook based on problem solving approach began to be published in 1980s. A
number of teaching guidebooks for explaining how to implement this approach have been published from 1980s. In
following sections, some extracts are explained from history.

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Masami Isoda / Procedia Social and Behavioral Sciences 8 (2010) 17–27

1880s
1910s
1930s

Topic of Lesson Study
Pestalozzi Method and Dialog Method (includingargumentation between teacher and students)
Mathematics for Life (including problem posing)

1950s

Curriculum Integration in Mathematics (including OpenEnded Problems)Core curriculum movement based on the social study

1960s
1970s

Mathematical Thinking (Japanese way of New Math.)
Open-Ended Approach and Problem Solving Approach

1980s

Problem Solving

Not only limited
mathematics.
Not
only
mathematics.
From 1900s

limitedUnder the occupation after
WWII.
Related with New MathFor
developing
Mathematical Thinking.
Related with US

Figure 3: Samples of Lesson Study topics.
3.1

An Origin of lesson study

Japanese lesson study originated in 1872 when the Education code was establishedand the Normal School (University of Tsukuba) and the Attached Elementary
School (Elementary School Attached to the University of Tsukuba) were
established at the same time (Isoda et al., 2007). It began from the observation ofteaching methods in whole classroom teaching which were firstly introduced inthose schools beyond the temple school culture on tutorial teaching methods.
People observed the ways of teaching for knowing how-to. Teachers’ Canon was
published by the Normal School in 1873 which already mentioned the etiquette forentering classroom for observation as for avoiding troubles during observations(Figure 4).
3.2

An Origin of Developing Students thinking and learning by/for themselves

Figure 4: Teacher’s Canon
Sometimes, general educators and educational management researchers enhance(1873)
the function of the professional development on the lesson study but do notconcern preparation of subject meter and teaching approaches for improvement. If it does not have the subject andteachers’ perspective for developing children, it is not satisfying the meaning of lesson study. The history of lesson
study has been described with a new theme and a new approach on lesson study for developing children because the
new theme and approach themselves are the aims of study and represent the reform, improvement, or focus of study
itself.
The first known lesson study guidebook for teachers in Japan which have these features is ‘Reform the Methods ofTeaching’ (1883: see Figure 5). The lesson study topic was Pestalozzi methodology of teaching approach for wholesubjects, but it was not same as the original version in German because it was imported from New York Oswego
Normal School and adapted in the Japanese way. In those days, lesson study had been introduced in Japan in a topdown way as well as establishment of the school system with an initiative of the government.
Another important feature of the first guidebook is the establishment of model teaching approach throughquestioning (‘Hatsumon’, as we call it today) for developing students who think by themselves. For enhancing a
dialogue style of classroom communication in whole classroom teaching, the model approach itself was describedthrough the dialogues such as ones of Plato and Confucius. The model dialogues in order to represent the process
within a limited number of pages at high cost of publication are a recommended process for enabling teachers toplan their lesson and did not develop for following the protocol to describe social phenomena by current researchers
on social science. Teachers’ guidebooks in Japan have been keeping the custom of the model dialog because it ismuch reproductive than the social-science-like protocol. From the viewpoint of teachers who are trying to reproduce

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Masami Isoda / Procedia Social and Behavioral Sciences 8 (2010) 17–27

his approach based on the model approach, model dialogue description style is reasonable because careful protocol
as for data only describes the past as the object of interpretation and does not aim for designing new practice.

Format of Lesson Plan

Necessity of Lesson Plan

Figure 5: ‘Reform the Methods of Teaching’ (1883)
3.3

Origins of Problem Solving Approach

In 1910s, Jingo Shimizu was a known case. He wrote a book ‘Teaching Elementary School Mathematics throughProblem Posing’ in Japanese (1924), which explained the innovative teaching approach including a fact that an
activity of learning mathematics begins from children’s problem posing (Figure 6). In that era, Japanese Teaching
Principle, ‘Learning by/for Themselves’ had been described by teachers and educators who wrote the teachers
guidebook for teaching. As seen this example, those samples could be seen as several origins of current problem
solving approach.

Masami Isoda / Procedia Social and Behavioral Sciences 8 (2010) 17–27

23

Figure 6: Problem Posing Approach by Jingo Shimizu (1924)
Japanese Problem Solving Approach, known as the process through ‘posing a problem’, ‘independent solving’,‘comparison and discussion’, and ‘summary and application’, was known in the US through the comparative study
on problem solving in the 80s by Tatsuro Miwa and Jerry Becker. It influenced the world through the TIMSS video
study in 90s (Stigler & Hiebert, 1999). Jerry Becker and Shigeru Shimada (1997) explained the approach from theview point of open ended problems. Shimada’s idea itself originated 1940’s. It is a good approach to introduce
lesson study in any school (Inprasihta, M.; 2006)
Problem Solving Approaches are one of the shared approaches in Japan and developing such a sharable approach
itself is one of the long-term results of lesson study. Lesson study is known in the world with Problem Solving
Approach. It may not have been spread if it were only explained by the lesson study cycle. The problem solving
approaches combined with lesson study has spread to the world from Japan through the comparative studies andteacher training programs for developing countries from 1980s, the Japan International Cooperation Agency’s
projects from 1993 (See, Isoda et al., 2007) and APEC projects from 2006.
4.

Local theory for Problem Solving Approach

Each Japanese elementary school usually sets a theme of lesson study project on school level through a year
depending on the demands of national reform movements, teachers and school district. Major themes of lesson study
projects at elementary schools are Japanese, Mathematics or general topics. General topics are usually related withcrossing curriculum topic such as Physical and Mental Health. More than 50 years, improvement of mathematics
teaching for better achievement of curriculum has been a major theme of lesson study (Isoda et al; 2007). Especially,in these days, the achievement of Japanese, Mathematics and Science on PISA has been lower due to the 20%

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Masami Isoda / Procedia Social and Behavioral Sciences 8 (2010) 17–27

reduced curriculum in 1999, Mathematics and Japanese are two major subjects in Elementary School lesson study
project.
On this context, more than 30 teaching guidebooks for elementary school mathematics are published, every year, as
for explaining theories of teaching. Here, for explaining designing strategy of class on Problem Solving Approachand showing the meaning from the improvement of children’ performance, the teachers’ guidebook titled‘Designing Problem Solving Class with the Basic Standards for Teaching Given by Check Sheets’ by Isoda et al.
(2009), is explained because it is currently known as one of the best-sellers in this area (the 1st printing was sold out
within two months). Many of guidebooks written by math-educators were adopted to develop check sheets. A
characteristic feature of this book is that it is written as the result of school level lesson study project and describedfor novice teachers who do not know well how to teach mathematics, even if they might have several years ofexperience. It also published in Spanish (Isoda & Olfos; 2009).
In Japan, problem solving approaches are shared to develop children’s ability to think and learn by themselves. For
knowing their achievement, there are two sets of national assessment problems. First type focuses on understanding
and skills and second type focuses on mathematical thinking including mathematical argumentation. Both tests
problems are developed on the national curriculum standards and the problems of the second type are deeply relatedwith problems solving approach itself.
4.1

Checking List for Implementing Problem Solving ApproachThis book by Isoda et al. (2009) has checking lists (Figure 7 as for a sample) as for explaining and sharing thisschool’s local theory of problem solving approach.
Problem Posing
1. The lesson sets tasks that can be solved in a variety of different ways byapplying previously learned knowledge, and presents the content to belearned.
2. The lesson planned with tasks (problem given by teacher) and problems(problematic from students), and promotes problem (problematic) awareness.
3. The teacher expected methods and solutions before.
Independent Solving
1. The children can recall and apply what they have already learned.
2. The children’s ideas are predicted before.
3. Inappropriate solutions are predicted, and advice and hints are prepared forthem before.
4. The teacher, walking around, observes and helps children to insure thatchildren use mathematical representation to solve the problems.
5. Notebook are written and taken in a manner such that they will be helpful forpresentation as well.
Comparison and Discussion
1. Steps (Validity, Compare, Similarity and Generalization or Selection) areplanned for comparative discussion.
2. The ideas to be taken up are presented in an order that is planned before.
3. The method for writing presentation sheets is planned in advance anddirections are provided.
4. In addition to develop the ability to explain, children are also fostered withthe ability to listen and the ability to question.
5. When ideas are brought together (generalized), it is important to experience
them by themselves.
6. The reorganization or integration of ideas proceeds smoothly from thepresentation and communication of children.
Summary
1. Activities are incorporated that let children experience for themselves the
merits of the ideas and procedures that are generalized.

Self-Evaluation
4 3 2 14 3 2 14 3 2 14 3 2 14 3 2 14 3 2 14 3 2 14 3 2 14 3 2 14 3 2 14 3 2 14 3 2 14 3 2 14 3 2 14 3 2 1

Masami Isoda / Procedia Social and Behavioral Sciences 8 (2010) 17–27

25

2. The summary matches the aims and problems (problematic) of this lesson.
4 3 2 1
3. It is recognized that both correct and incorrect answers (to the task) have4 3 2 1something good in the foundation of their ideas.
4. Children are made to experience the joy and wonder of learning.
4 3 2 1Figure 7: Lesson Planning Checklist: Self-Evaluation[4: Achieved; 1: Not Achieved] (Isoda, 2009; Isoda & Olfos, 2009)
Behind of this list, there is a local theory of problem solving approach. For example, the difference of problem (task)
and problematic (problem) is a key because problematic is necessary for children leaning by/for themselves, and it isalso related with the objective of the lesson. Without the problematic, any collect answerers are good answerers. In
various kinds of answerers, children can discuss which answerers are appropriate. On the other hand, when theschool began to use the checking lists on their project, most teachers did not understand the meaning of eachchecking list, because in the case of this school most of teachers do not know how to teach mathematics well, even
if they have a chance to see other teacher’s problem solving approach. After conducting school-level lesson study
project for a year and half, through having lesson study once a month in each grade, the teachers well understood themeaning of check lists and developed high achievement.
4.2

The Achievement of the Lesson Study Project on Math by the School using various checking lists

After the one and half year mathematics lesson study project in Ozone Elementary School through using checking
lists for mathematics, children’s achievement improved as follows (Isoda et al.; 2009).
In Figure 8, Children’s mathematical thinking ability which is a key for leaning by/for themselves is improved. Itshows that achievement of children in the 5th grade improved by 15 points in mathematical thinking test comparedwith the average of the whole prefecture. Figure 9 implies that the achievement of school-level mathematics lesson
study during one and half year is not only limited to the improvement of children’s mathematics achievement, butalso influenced positively other subjects such as Japanese, Science and Social Studies. It means that the lesson study
efforts on the teaching approach in mathematics through using checking lists may influence other subject ofteaching. Indeed, in Ozone Elementary School, a teacher teaches almost all subjects. Children’s awareness of
empowerment in mathematics led to improvement of their interests of learning and developed their wish to study.
ᅗ 䠍 䠊 ᆅSchool’sᇦ ᖹAcademicᆒ 䛛 䜙Abilities䜏 䛯Compared኱ ᭯ ᰿to theᑠ RegionalᏛ ᰯ 䛾
Ozone Elementary
AverageᏛ ຊthGraders䠒6 ᖺ⏕Graders䠑5 ᖺ⏕

A Year before
Implementationᐇ ᪋ ๓ ᖺ11stᖺYear
┠

th

th
䠐 4ᖺ
⏕
Graders

22nd ᖺ
Year┠

+5

+0

-5

-1 0

Difference from average in mathematical thinking problem
set (converted to 100 points)

Figure 8: Ozone Elementary School’s Academic Abilities Compared to the Regional Average

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Masami Isoda / Procedia Social and Behavioral Sciences 8 (2010) 17–27

ᅗ  ᆅᇦᖹᆒ࡜኱᭮᰿ᑠᏛᰯᏛຊẚ㍑

6th Grade
Mathematics
6th Grade
Japanese
2nd yearbefore6th Grade 4
Subjects
5th Grade
Mathematics
5th Grade
Japanese
5th Grade 4
Subjects

Numerical values are point differences from regional
average points

Figure 9: Ozone Elementary School’s Academic Abilities Compared to the Regional Average

Independent䛷䛝䛶䛔䜛Achieved
Solving

Problem Posing
Achieved

Not Achieved

Start
Time

Start
㛤ጞ᫬
Time

After
1.5
Years

After
1 .5 ᖺ ᚋ
1.5
Years

Start
㛤ጞ᫬
Time

16

After
1.51 .5 ᖺ ᚋ
YearsSummary
Not Achieved
Start
㛤ጞ᫬
Time

79
0%

2 0%

40%

80 %

2 0%

40 %

1 00 %

14

60%

䛷䛝䛶䛔䜛

80 %

Not Achieved

32

68

91
0%

2 0%

Figure 10: Improvements of Teacher Instruction by
Lesson Study with Checklist

100%䛷䛝䛶䛔䛺䛔

Achieved

After
1 .5 ᖺ ᚋ
1.5
Years

21

60 %

70

86

䛷 䛝䛶䛔䛺 䛔

84

Not Achieved

30

0%Comparison䛷䛝䛶䛔䜛and Discussion Achieved䛷䛝䛶䛔䛺䛔

40%

9

60%

80 %

1 00 %

Masami Isoda / Procedia Social and Behavioral Sciences 8 (2010) 17–27

27

The achievement is the result of lesson study through using the checklists in the school. For improvement ofclassroom teaching, it is important that teachers and children share objectives. For sharing objectives, behindtheories such as for Problem Solving Approach and the theory of curriculum had been learned in the process oflesson study. Ozone Elementary School developed Lesson Planning Checklist base on the theory of problem solving
approach (Isoda et al. ; 2009), Children Leaning How to Learn Checklist, and Lesson Plan Checklist, and also morechecklists such as the way or blackboard planning were used (Isoda; 2009) and improved on Isoda and Olfos (2009)
for Latin America. They provide opportunities for children to check by themselves for reflecting on what should be
improved.
Figure 10 is the result of the self-evaluations by teachers on the lesson planning checklist in order to verify theirinstruction method and the problem solving approach have been appropriate or not. Figure 10 compares theachievement at start time with that of 1.5 years later. At the beginning of this research (1.5 years before the lesson
study open school), teachers were not sure of the meanings of the words listed on the lesson planning checklist. By
taking on the challenge of this project throughout the entire school for one and a half years, the teachers gained
confidence in their instruction method. Through the improvement of teachers’ teaching practices through the school
lesson study project in only one and half years, teachers teaching methods are improved and then, children’sachievement are improved beyond mathematics. It was the result of collaborative lesson studies by Ozone
Elementary School teachers.
5.

Final Remarks

This paper illustrated the historical development of Japanese lesson study in the case of mathematics and explained
the theories of mathematics education have been developed on this history. For explaining a case of developed local
theories, this paper briefly illustrated the case of Ozone Elementary School, school based approach.
The achievement of Ozone Elementary School is just a case of the school level lesson study on Problem Solving
Approach. As a number of lesson study activities are done by schools and they are develop their local theories of
their Problem Solving Approach for their schools. Those theories are teachers' theories developed by teachers withsupport of researchers or supervisors to improve their daily practice. Shared Japanese Theory of Problem Solving
Approach and theory of curriculum are well described at the Special Issues of Journal of Japan Society ofMathematical Education for EARCOME 5 (Isoda & Nakamura; 2010).
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