Values in Japanese Mathematics Education from the Perspective of Open-ended Approach

Mathematics education community in Japan has continuously and extensively developed ‘mathematical thinking’ as an educational value. In this paper, the historical review was conducted on mathematical thinking in terms of its evaluation and educational method, textbook change, and research on treatment of diversifi ed mathematical thinking. Th is approach can provide methodologically an important perspective to grasp, clarify and make relative the values in mathematics education in diff erent times of each culture. Values here mean those attitudes which lay at the back of the intention, judgment, and selection of teaching-learning activity exhibited by primary teachers. As a result of this research, it is learnt that the theme in mathematics education research does refl ect values held by the primary mathematics teachers. Th ey, in turn, have held central ideas and value utilizing children’s diversifi ed mathematical thinking, letting them subjectively and extensively construct mathematical ideas in the lesson. Th e major characteristics of Japanese Mathematics education is the open-ended approach, which has been developed as an evaluation and educational method of mathematical thinking. Th is is available as translated version of “Th e Open-Ended Approach: A New Proposal for Teaching Mathematics” (Th e original version (Shimada) is in Japanese published in 1977).


Introduction
Th e Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom 1 takuba@hiroshima-u.ac.jp (Stigler & Hiebert, 1999) called in earnest for attention to be paid to the lesson study and Mathematics education in Japan, where the lesson study has been developed.In the same vein, a few other international eff orts sought to introduce Japanese Math-ematics education to the international community, and one of the typical ones is EARCOME 5 (East Asian Regional Conference on Mathematics Education).Such eff orts are rooted in the regional characteristic of Japan however, and they tend to be bound by Japanese perspective.Th e two mentioned initiatives however, start to raise fundamental questions over what introducing the Mathematics education in a particular country such as Japan means, what is fi rst of all the Mathematics education in Japan like, and what has been valued in Japanese Mathematics education by many people who have been involved in it.In return, these questions create the necessity of refl ecting and giving some answers.
In 2006, JASME (Japan Academic Society of Mathematics Education) held the symposium during the 22 nd annual conference with the theme of Cultural Aspects of Mathematics Education in Japan with a focus on Mathematical Th inking.It aimed at grasping and describing 'mathematical thinking' as an educational principle, which mathematics education community in Japan has continuously and extensively valued and developed, and exploring the future direction of it through the refl ection on its characteristics.Th e symposium confi rmed that the whole clothe of mathematics education in Japan has developed coherently with mathematical thinking being as it were its warp and social and historical needs as its weft (Baba, 2006).
In this paper, the persisting values of mathematics education community in Japan are refl ected from teachers' perspective as an example of openended approach.Here the values mean "those of primary teachers at the back of their intention, judgment, and selection of teaching-learning activity" (Baba et al., 2013).Th e open-ended approach is a good example in order to relativise and refl ect characteristics of Japanese mathematics education, because it retains those characteristics developing around mathematical thinking and diverse ideas, and it can be also referred by international researchers since it is available as translated version "Th e Open-ended Approach: A New Proposal for Teaching Mathematics" (Becker & Shimada) published in 1997.
Th e open-ended approach is typically exemplifi ed as the developmental work with Open-ended approach in Mathematics Education -New Proposal of Lesson Improvement (Shimada, 1977) and From Problem to Problem -Extensive Treatment of Problems for Improvement of Mathematics Lesson (Takeuchi & Sawada, 1984).Th is extensive treatment of mathematical problems is seen as an extension of Open-ended approach, and thus it is included in it.

Emergence of Mathematical Th inking as the Objective in the Course of Study
Th e term "mathematical thinking", which is a translation of "suugakutekina-kangaekata", fi rst appeared in 1958 in the objective of the course of study for primary education.Th e course of study was developed in Japan aft er the WWII and was intended to be the national curriculum in Japan.Its objectives at the time were as follows: 1. To enable students to understand basic concepts and principles about numbers and quantities, and geometrical fi gures, and let them develop more advanced mathematical thinking and how to treat it.2. To enable students to acquire basic knowledge and fundamental skills about numbers and quantities, and geometrical fi gures, and let them use those eff ectively and effi ciently according to the purpose.3. To enable students to understand the signifi cance of using mathematical terms and symbols, and let them use expression and think simply and clearly quantitative events and relations using the terms and symbols.
4. To enable students to extend the abilities to set up a appropriate plan and to think logically regarding quantitative events and relations, and let them treat things more self-dependently and rationally. 5. To enable students to develop attitudes towards a proactive mathematical thinking and how to treat it in daily life.(Underlined by the authors.)Th e phrase "mathematical thinking and how to treat it" in this objective is commonly referred as "mathematical thinking" to mean all components related to this mathematical thinking and treatment.From the above, it is expected to develop the acquired fundamental concepts and basic skills to the more advanced level and to grow the attitudes to apply them extensively to daily life situation.Historically speaking, the mathematical idea2 as philosophical stance in national textbook Jinjo-shogakusanjutsu used since 1935 preceded the mention of mathematical thinking (Ueda, 2006).So in this sense, there was a continuing aspiration of Japanese mathematics education community despite of temporal mutation during the WWII.
Th e community at the time tried to uplift the lowering standards of mathematics education when the term 'mathematical thinking' emerged, aft er the critical refl ection over the life unit learning which placed mathematics education as skills-based subject (Nakashima, 1981).In other words, the community aimed at raising effi ciency by teachers' clarifying and extending the basic ideas and principles through mathematical thinking.Th rough development of mathematical thinking abilities, students would have been able to fi nd out new ideas subjectively and use appropriately and effi ciently mathematical facts and relations, express and think of them in a concise and clear way, and treat them in-dependently and rationally.Despite these intentions, the meaning of mathematical thinking at the time was not clear enough to the majority of teachers expected to teach it.
Just before the emergence of the concept of mathematical thinking, there was a preceding idea, called the central concepts.Th e term fi rst appeared in the course of study for the senior high school in 1956.Th e characteristics of this course of study were the integration of Analysis I, Analysis II and Geometry into Mathematics I, Mathematics II and Mathematics III as mathematical subjects.At that time, central concepts exemplifi ed mathematical thinking as central ideas to bridge all the content of each mathematical subject although they were shown separately in terms of the algebraic and geometrical contents.For example, the central ideas for Mathematics I were described as follows: a. Expressing the concepts in symbols b.Extending concepts and laws c.Systematizing knowledge by deductive reasoning d.Grasping relation of correspondence and dependence e. Finding out invariance of equation and geometrical fi gures f. (Identifying) Relations between analytical and geometrical methods.Th e central ideas had an intention to integrate algebra and geometry in mathematics as one subject and to extract mathematical methods and activities common to both of them.Th ey are not exactly the same as mathematical thinking, which has become an objective of primary mathematics education, but it certainly had an infl uence on its introduction.When the course of study was revised in 1968 to introduce the idea of modern mathematics movement, it further emphasized the mathematical thinking we have been talking about.
Th e table 1 shows the name of sessions and the number of presentations in the session during the annual conference by the Japan Society of Mathematics Education (JSME).When the course of study was revised in 1968, the sessions on the newly introduced topics such as sets, function and probability and statistics were created in addition to the existing ones such as number and calculation, quantity and measurement, geometrical fi gures.Th e session of mathematical thinking was created only 6 years later in 1973.In other words, discussion over mathematical thinking started aft er discussion over the above contents had reached a certain level.

Eff orts analyzing and defi ning the mathematical thinking
Around the time of setting the session at the JSME in 1973, the analysis on concepts of mathematical thinking had already started.Katagiri of Tokyo Metropolitan Institute of Education ushered in the concept into his analysis Mathematical Th inking and its Teaching (Katagiri et al.)  a) the attitudinal aspects of mathematical thinking b) the process aspects of mathematical thinking such as generalization and analogy, and c) the contents related mathematical thinking such as unitary amount and relative amount.
In 1981 Nakashima published Mathematical Th inking at Primary and Secondary Mathematics Education, and stated that mathematical thinking consisted of abilities and attitudes to work autonomously and have an ability to apply these creatively through an activity appropriate to mathematics education.He clarifi ed that to develop mathematical thinking, one had to pay attention to this autonomous and creative process within an activity.
In 1988, Katagiri reorganized the above categorization of mathematical thinking into mathematical thinking related to methods and contents.Th rough these publications, interpretation of mathematical thinking has been gradually clarifi ed in Japanese context.As we have seen so far, Katagiri and Nakajima have been the most famous researchers that contributed to analytical research on mathematical thinking in Japan.
As shown in the following section, the research on evaluation and concretization of mathematical thinking has been developed simultaneously, while the above type of analysis continued.Both of these approaches -the analytical research and the concretization -we see as the diff erent sides of the same coin, and they have been infl uencing and referring to each other and deepening as a whole the fi eld of enquiry related to mathematical thinking.

Evaluating and developing the mathematical thinking through the Open-ended approach
For six years between 1971 and 1977, Mathematics education researchers in NIER (National Institute of Education Research): university professors, primary and secondary school teachers, formed an interest group and developed the research project, whose theme was to develop evaluation method of mathematical thinking, through the Grant-in-Aid by the Ministry of Education, Culture and Sports.Th is group, consisting of about 30 members, scrutinized the objectives of primary mathematics education carefully and stated that "mathematical thinking has been fl owing at the bottom of mathematics education in Japan since mathematical ideas in Jinjo-shogakusanjutsu (the national textbook during the pre-war period, Grade 1 of which was published by the Ministry in 1935) aiming to develop mathematical and scientifi c thinking, and the course of study in primary and junior secondary schools has already clearly stated it in 1958 and in senior secondary school in 1956.… In short, to be able to develop mathematical thinking can be regarded as the ultimate goal of mathematics education" (Hasihmoto, 1976: 21-22).
Th is interest group further conducted the survey questioning a wide array of stakeholders, from mathematicians, mathematics education researchers, to mathematics teachers across Japan, regarding some behavioral examples of primary and secondary students attaining the objectives of mathematical thinking.Th e group summarized the fi nd-ings from the answers about mathematical thinking as containing the following items (Hashimoto, 1976: 22), that are not necessarily independent from each other: 1. Being able to fi nd out relations that underline the situation within a problem and begin to construct it mathematically.2. Being able to solve non-routine problems which cannot be solved by common procedures.3. Being able to develop something new.4. Being able to fulfi ll one's own ideas in the group.5. General objectives (under the current course of study).Following these fi ndings, the researchers repeated the process of developing the Open-ended problems for evaluation and trialed them in classroom.Th ey had hypothesis that attainment level of mathematical thinking can be assessed through such incomplete Open-ended problems.Th ey used these problems in the lessons and let students fi nd out as many relations as possible and describe mathematically those relations.Evaluation is done by analyzing the relations in terms of quantity and quality, which is sophistication level of their description (Sawada & Hashimoto, 1972: 65).
Th e notable point for this project is that it focuses not only on evaluation method but also on effective teaching strategies to realize development of mathematical thinking.Th is basic stance of the group infl uenced the direction of the research.
Th e research theme for the fi rst year following the research project was "development research on evaluation method in mathematics education" but it was changed for the second and third year into "development of evaluation method in mathematics education and analysis of impact of various factors".Analysis of the factors is made possible by the fact that data collection at the classroom level had been done intensively from the beginning of this de-velopment work.In fact, the work done in the project paid attention to the students' group discussions during the lesson and tried to evaluate the change of this group discussion for the second year (Sawada & Hashimoto, 1972).
Th e experience and knowledge gained through the research project, and which have been accumulated through the data collection regarding students' responses, prompted the group to shift from "development work of evaluation method for mathematical thinking" to "development of teaching strategies for mathematical thinking." Even after this, students' responses in the lessons had been continuously collected in parallel to sophistication of evaluation method.And gradually they become a new standard of teaching strategy.
Th e report for fi fth year stated that the objective was "Th is year it aimed at trialing a few incomplete open-ended problems in the lesson during the second semester and confi rming through statistical survey if this form of teaching can promote the attainment of the above objectives, and showed also the following results from teachers' observation and students' remarks" (Shimada, 1976, 29-30): a. Th e middle and low achievers with less activity have become more active in expressing their ideas.(It is the same as the previous year).b.Especially the middle achievers in the daily activity have made most remarkable progress in elementary and junior secondary schools.
(It is the same as the previous year).c.Some of the high achievers in senior high schools have performed less than previous year, since they become too careful not to make a mistake.(It is the same as the previous year).d.In the previous year, it was reported in elementary schools that there were a few students who showed interests in mathematical properties aft er fi nding them within an openended problem, but in this year there were many examples which showed students took interest in these properties.
In the same year, the following ideas about teachers' work were found to be the case.
e.It may not be possible to say that being incomplete makes the problem eff ective.Rather, a problem posed to students should not only be incomplete, but it should also have a certain direction towards a solution, and something that is produced by students while they work on it, should be mathematically signifi cant.f.Th e open-ended problem approach is eff ective both at the introduction and at the summary of the lesson.When there is a good problem at the introduction, the lesson development becomes interesting.When it is given at the end or while summarizing the lesson, it is useful to review various aspects learned.
As for the summary and future issues of the research, the two points were listed as follows.
Th e fi rst point is that the two terms of the year during which the research project took place were too short to confi rm eff ectiveness of the teaching approach based on open-ended problems.Changes that were expected would be more visible only after a longer time has been spent in dedicating time to mathematical thinking and open-ended problem solving in the classrooms.In this sense, it was recommended to plan the activity from the beginning of school year in the following year.
Th e second point is that the problems used in the lessons were diverse not only in results but also in the processes and contexts they represented.Consequently, they had given diverse results, which could not always be correlated or compared with each other.
And you can see in the above point, diversities were noted in the process of research on evaluation method of mathematical thinking, and they demanded the necessity of systematizing and theorizing them as mathematical activities."Problem posing with diversity" was used as the evaluation method on devel-opment of mathematical thinking.In other words, mathematization of phenomenon was placed in the center of mathematical thinking, and it was assumed that possibility of such mathematization was not only one but also several, and so the signifi cance of "being diverse" was to be re-considered.It is notable that the research group located this as the future issue.
Th is research resulted in perceiving mathematical activity as coming and going between real and mathematical world and locating it in the phases of the Open-ended approach, which extensively utilizes the incomplete problems (Figure 1).And Takeuchi employed theory of scientifi c knowledge growth by Popper and approached this issue from the perspective of the nature of mathematical activity (Takeuchi, 1976: 11-12).Th is consideration played an important role in shift ing the research from the Open-ended approach to "extensive treatment of problems" (Takeuchi, 1984: 9-23). 3 In this way, the developmental research of the Open-ended approach has continued as mutual interaction between theory and practice and the treatment of diversity in mathematical learning has become systematized.

Diversifi ed ideas in mathematics textbooks
Lesson development like the one described above, and which used diversity of mathematical ideas has made an impact also on lesson structure.Th e textbooks published by one textbook company were compared by focusing on the area of plane fi gure (parallelogram) in the fi ft h grade.Th e textbooks from 1965 to 1975 (Figure 2) didn't have diversifi ed ideas, but they has already started development of evaluation method of mathematical thinking using incomplete problems.Th e textbooks in 1980 (Fig- ure 3) and that in 1985 (Figure 4) contained more than one idea.Th ey are diff erent ways of "cutting into pieces and pasting them together" and "moving trapezium and matching the corresponded areas".We must remember that the latter took place aft er the Open-ended approach was proposed as teaching method in 1977.Th is leads to current textbook (Figure 5).Adoption of diversifi ed ideas in the textbook produces the practical issues on how to treat them during the lesson.

Research on how to treat and summarize diversifi ed ideas
The diversifi cation of ideas as the ones shown above which made their way into the Japanese textbooks have infl uenced the developmental research on how children treat and summarize different mathematical ideas during the learning process."One objective of the problem solving through diversifi ed ideas is to ensure acquisition of the basic knowledge and skills and the understanding of mathematical thinking which can be encountered in the process of learning through presentation of those ideas, and to aim at the development of individual student's holistic growth including cognitive understand, emotional development and explaining skills through the whole class participation" (Koto, 1992, 19).Koto further stated that diversifi ed ideas should lead to development of mathematical thinking.
Koto (1992，1998) classifi ed diversifi ed ideas, which can be observed during mathematics lesson, in terms of teaching aims and quality, and proposed the instruction fl ow utilizing them as follows: Independent diversity: Paying attention to validity of each idea Prioritized diversity: Paying attention to efficiency of each idea Integrated diversity: Paying attention to commonality of each idea Structured diversity: Paying attention to mutual relations between ideas.
The research on how to treat and summarize the diversifi ed ideas is regarded as one of the necessary items for the lesson study in Japan in the spe-  EARCOME in 2010(Wada, 2010).This shows the signifi cance of research impact by Koto and others on the lesson development in Japan.

Summary
Engagement by Japanese mathematics education community regarding the open-ended approach can be summarized chronologically in the Figure 6.

Figure 6 .
Figure 6.Flow of Mathematics Education in Japan from the Perspective of Open-ended Approach