ZDM Mathematics Education (2007) 39:503–514 DOI 10.1007/s11858-007-0052-1
ORIGINAL ARTICLE
Toward the problem-centered classroom: trends in mathematical problem solving in Japan Keiko Hino
Accepted: 8 July 2007 / Published online: 28 July 2007 FIZ Karlsruhe 2007
Abstract In this paper, I summarize the influence of mathematical problem solving on mathematics education in Japan. During the 1980–1990s, many studies had been conducted under the title of problem solving, and, therefore, even until now, the curriculum, textbook, evaluation and teaching have been changing. Considering these, it is possible to identify several influences. They include that mathematical problem solving helped to (1) enable the deepening and widening of our knowledge of the students’ processes of thinking and learning mathematics, (2) stimulate our efforts to develop materials and effective ways of organizing lessons with problem solving, and (3) provide a powerful means of assessing students’ thinking and attitude. Before 1980, we had a history of both research and practice, based on the importance of mathematical thinking. This culture of mathematical thinking in Japanese mathematics education is the foundation of these influences. Keywords Mathematical problem solving Mathematical thinking Research and practice Assessment
recommendation, ‘‘Problem solving should be the focus of school mathematics in the 1980s,’’ was received by Japanese mathematics educators as a strong message. At that time, we had been conducting both research and practice based on the idea of mathematical thinking. During the 1980s and 1990s, much research was conducted under the title of mathematical problem solving. When we look at the outcomes of these studies, as well as the progress of the curriculum, textbook, evaluation, and teaching, several influences can be identified. In this paper, I summarize the influence of mathematical problem solving on Japanese mathematics education from three perspectives: research, practice, and assessment. After a brief report on the introduction of mathematical problem solving in Japan, I will focus on its influence on advancing our effort to study and improve teaching practice in the classroom. Together with the illustration of several events and examples, it is observed that the community of Japanese mathematics educators has been investigating the methods of realizing a problem-centered classroom. It will also be pointed out that the culture of mathematical thinking in Japan lies behind, as well as provides the basis for considering such a direction.
1 Introduction The purpose of this paper is to report the influence of mathematical problem solving on mathematics education in Japan. Here, ‘‘mathematical problem solving’’ refers to the idea that, starting in the 1970s, was advocated strongly in the Agenda for Action by NCTM in 1980. Their first K. Hino (&) Department of Education, Utsunomiya University, 350 Mine-machi, Utsunomiya, Tochigi 321-8505, Japan e-mail:
[email protected]
2 Introduction to problem solving in mathematics education in Japan In Japan, we have national curriculum standards called the Course of Study, which is prescribed by the Ministry of Education. After World War II, the Course of Study has been revised almost every 10 years. In each revision, its goals and standards reflected the interests and needs of the Japanese society, and its education requirements. Figure 1 shows the emphasis placed on the Course of Study in each
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504 Fig. 1 Progress of the Course of Study after World War II
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period for these 60 years (Japan Society of Mathematical Education Research Section, 2000). The progress of the Course of Study indicates that we have administered mathematics education based on our own philosophy and at the same time by reflecting the worldwide reform and movement. Owing to the prosperity of progressive education, in the period the Course of Study emphasized experiences in daily life, we can see problem solving as one strand in elementary school curriculum. In the 1951 version, problem solving was stated as a goal of school mathematics. It was mainly related to solving problems involving addition, subtraction, multiplication, and division as well as proportion (Nagasaki, 1990). However, this was the only period in which problem solving was included in the goal statement.1 In the Course of Studies after 1958, fostering the students’ mathematical thinking was emphasized as the goal of mathematics education. In the 1958 Course of Study, at the elementary school level, we can find goal statements such as ‘‘To make children understand basic concepts and principles with respect to numbers, quantities, and geometrical figures; enable them to create advanced mathematical thinking and ways of handling the problem situation by using mathematics,’’ and ‘‘develop their attitudes toward utilizing mathematical thinking and ways of handling the problem situation by using mathematics willingly in their daily life.’’ Mathematical thinking continued to be included in the Course of Study, which emphasized modernization of mathematics, and even until now it has been a major goal of school mathematics in Japan. When the Agenda for Action was released, Japanese mathematics educators were in the midst of implementing a new Course of Study that emphasized the development of individuality and basics in mathematics. This Course of Study was intended to change tracks from the emphasis on mathematics modernization. Reduction of class hours also forced the selection and integration of content into the basic matters. The principle adopted by the mathematics educators was to maintain the essential idea of incorporating modernized mathematics into the curriculum, i.e., fostering the students’ creativity and mathematical thinking. The interest in improving the method of teaching and 1 Although problem solving was not in the goal statement, it was mentioned in the instruction manual of the current Course of Study.
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learning was increased rapidly, because it was thought to be fundamental to the realization of the spirit of modernization of mathematics. Mathematical problem solving was one of their central research topics. Mathematical problem solving was conceived as an important factor in reexamining mathematics education in Japan (Shimizu, 1990). The results of the First and Second International Mathematics Studies also compelled them to pay greater attention to mathematical problem solving since they revealed the Japanese students’ weakness in the realm of mathematical thinking, compared to computational skills (Mase, 1982). Due to the emphasis on mathematical thinking in the community of Japanese mathematics educators in those days, introduction to mathematical problem solving also brought about a state of confusion among the educators. They were trying to elucidate the relationship between mathematical problem solving and mathematical thinking, and to try and differentiate between the two (e.g., Koto, 1983; Katagiri et al., 1985). Through these examinations it was found that both have far more commonalities than differences. However, it appears that mathematical thinking was considered relevant in creating and forming mathematical concepts, rules and algorithms, and so on, while problem solving was considered relevant to using and applying such concepts and algorithms flexibly and effectively. After stating that the major role of problem solving is to enable students to understand mathematical content, Nagasaki (1990) distinguished three approaches to problem solving: (1) problem solving as the goal of instruction in mathematics education, (2) as the process of instruction, and (3) as the content of instruction. In the first approach, the goal of instruction is to develop student’s problem solving ability or ability to think and foster their attitude and habit toward thinking. Here, every context of learning can be considered as problem solving. In the second approach, the purpose of mathematics instruction is to teach students the processes of creating mathematical concepts through solving problems. This second approach was included in the instruction, to aim at the student’s acquisition of thinking relating to content and method essential to mathematics. He described that both (1) and (2) were observed in the long history of mathematics education in Japan. The approach of (3) can also be seen in history, but it was emphasized more after 1980. In this approach, it is considered important for students to acquire the method of
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problem solving. Here, procedures, stages, or strategies as the method of problem solving were studied and devised as the content of instruction. It can be said that teaching about problem solving (Schroeder & Lester, 1989) attracted the attention of mathematics educators as a fresh perspective in the introduction to mathematical problem solving.
3 Studying, thinking, and learning process of students Yamazaki (1995) summarized the progress of research on mathematical problem solving in Japan from 1980 to 1995. By collecting a wide range of data from books, both research- and practice-based journals, and conference proceedings, he observed that the number of literature and presentations increased after 1980. The number of studies that appeared in, and presented at, research-based journals and conferences increased after 1985, and then remained fairly constant. He concluded that during this period, research on mathematical problem solving was one of the important themes among university researchers and schoolteachers. In the last 10 years, the number of research papers that contain ‘‘problem solving’’ in their titles appears to be decreasing. However, there have also been studies that do not explicitly state ‘‘problem solving,’’ but deal with it in substance, such as studies on social interaction in the learning of mathematics, on the process of mathematical modeling, and studies on the process of learning specific content of school mathematics. This tendency can be interpreted to mean that while there are studies that are an extension of the previous ones, studies on problem solving have been conducted in multiple areas of study, based on various research interests (Shimizu, 2002; Nunokawa, 2002). 3.1 Analysis of students’ processes of solving problems Yamazaki further analyzed the title and abstract of each study and summarized them. According to him, many studies attempted to clarify the process of mathematical problem solving by analyzing the problem solving process used by students. They often focused on the student’s use of problem solving strategies and mathematical thinking. Some were also interested in the characteristics of mathematics when students were in the process of mathematical problem solving. These studies often adopted approaches from cognitive psychology and provided detailed descriptions of the actual thinking processes of the students. A summary of research papers and presentations in journals and annual conferences by the Japan Society of Mathematical Education during 1991–2000 was provided by Ito (2001). He categorized the research content into four
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areas: behaviors in the process of problem solving, strategies in problem solving, structures and settings of the problem, and various abilities related to problem solving. In line with Yamazaki’s analysis, we can gauge the researchers’ interest in what goes on in the student’s mind when they engage in solving mathematical problems. Focuses in the analyses and interpretations were varied, including representation, goal, motivation, elaboration, perplex, control/monitoring, and reflection. Fine-grained analyses of the transformations of the student’s representations and the use of strategies in the course of problem solving were also observed. Both the progress of the analysis of the students’ problem solving processes and the tasks for future study have been pointed out. In the summary presented above, Ito noted the tendency of the atomization of research topics, and called for an integration of research findings obtained thus far. Okubo (2002) also pointed out the necessity of considering a global perspective that goes beyond the analysis of each stage of problem solving, and looks at problem solving throughout the lesson or in the entire teaching unit. As these studies were mainly conducted in the laboratory setting, integration between research and practice was also suggested (Sakitani, 2002).2 Pursuing the point of contact with teaching practice, some researchers considered the classroom situation as the target of their study. Besides conducting the teaching experiment on problem solving strategies, effects of the intervention on facilitating the student’s use of strategies was also investigated (e.g., Shimizu & Yamada, 2003). The role of the teacher in a classroom where problem solving is emphasized is one of their concerns. Another direction is the use of these findings in studying the mechanism of acquiring a certain level of understanding of specific content of mathematics through classroom teaching (Nunokawa, 2005). He proposed this direction as ‘‘a clinical approach to learning processes.’’ For example, Nunokawa (2007) analyzed the behavior of target children in a third grade classroom when the teacher dealt with proportion problems in the class. He was interested in learning how each child can, or cannot, construct a sub-unit and use it for solving the problem. In the analysis, he used research findings of the use of problem solving strategy, i.e., drawing figures. He focused on what figures each child drew in her/his notebook and how each child interacted with her/his figures. One difference between two children 2
As described in ‘‘Development of teaching materials and effective lesson organization with problem solving’’, in Japan we have the tradition of practice-based research by schoolteachers. Their active involvement in such research activities seems to realize a closer relationship between research and practice. However, Sakitani interviewed five school teachers and pointed out concretely the need of research findings that are useful enough for their teaching practice.
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who could construct and use a sub-unit and two who could not was that only the former children were able to modify the figures they once drew. Nunokawa argued that these differences were related to their disposition of looking back at the problem situation when they got stuck in solving the problem. Utilizing research findings on the students’ processes of problem solving, Nunokawa is investigating the possibility of changing the understanding of individual students by respecting their own thoughts. The perspective of the problem solving process is also used for elucidating and overcoming the students’ difficulties in the learning of content in school mathematics (Shimizu, 2005). For example, Shimizu (1995, 2005) approached the students’ difficulties in understanding the reason for division by a fraction from the viewpoint of ‘‘meta-thinking’’. Students tend to learn the topic as a rule without understanding the reasons. He explored students’ thinking about division of fractions when they were asked to answer the written tests, which included a correct but unfamiliar procedure (‘‘Yoshiko’s method’’): the numerators and denominators are divided (e.g., 8/15‚2/5 = (8‚2)/ (15‚5) = 4/3). More than 70% of the sixth grade students, and about 65% of the seventh grade students, judged Yoshiko’s method as an incorrect procedure. Many students insisted that Yoshiko’s method was incorrect because it was different from the algorithm they knew. Clinical interviews and instructional interventions further showed that it was very difficult to change their judgment. All this indicated the rigidity in their argument and suggested their rule-oriented attitude toward mathematics learning, or, in other words, their weakness in meta-thinking. He proposed the need for teaching the division of fractions by enhancing the students’ meta-cognitive activities such as stressing the properties of division and the relationship between multiplication and division. Since overcoming the students’ difficulty by making references to real world situations is more widely recognized in Japan, the point he made suggests that there are multiple possibilities to exploit the difficulties. 3.2 Attention to problem solving in the real world It should be noted that recently in Japan, there have been an increased number of studies on problem solving in the real world. We have a long history of studies on world problems. Their focus has mainly been the area after the problem was formulated from the real world situation. The process of formulating a mathematical problem itself had not been sufficiently studied. A similar tendency can be observed in the history of research on mathematical thinking, in which the research focus was more on the process of creation and extension within the world of mathematics. Analysis of the process of solving real world problems and the use of real world situations as a genuine opportunity for fostering the
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students’ problem solving ability were considered important research topics. The results of the Japanese students’ low values on mathematics, by an international assessment of mathematics (e.g., Mullis et al., 2000, 2004), have been spurring us in our efforts in these areas. Mathematical modeling is a growing area of research. A major content of the research includes the processes contained in problem solving in the real world, knowledge, skill, and thinking that relate to the process; importance of teaching mathematical modeling; development of teaching materials; investigation of real world situations; and development of instructional stages. Here again, one of the tasks is the implementation of the processes of mathematical modeling at the classroom level. We do not have sufficient information on selecting tasks and organizing a lesson or a series of lessons, especially because our students are not very familiar with mathematical modeling in the first place. As one exception, Ikeda (2004) developed a number of teaching objectives and tasks to foster the students’ skills in mathematical modeling. The following three stages were set: (1) understanding the meanings for setting up assumptions; (2) acquiring several important ideas that are required in constructing and analyzing a mathematical model; and (3) solving real world problems by applying a variety of ideas. He implemented lessons in the ‘‘optional course’’3 for students in a lower secondary school, attached to a university. The series of lessons were implemented once a week and each lesson took 100 min. It was not easy to implement lessons like mathematical modeling in public schools because of the time constraints. One reasonable choice would be the use of the optional course, as done by Ikeda. Ikeda found it effective for the teacher to make students understand the meaning of setting up assumptions, before teaching the particular ideas that are required for constructing and analyzing a mathematical model. In order to deal with mathematical modeling in the classrooms, in which there are students with a wide variety of backgrounds and knowledge base, much care should be bestowed upon developing problems and ways of teaching in order to develop settings that can lead the students to comprehend the meanings of mathematical modeling. 3.3 Cross-cultural research in Japan and the US on problem solving Research on mathematical problem solving in Japan was conducted collaboratively with researchers in other countries, such as the US, France, and China. These studies 3
Mathematics as an optional course in the ninth grade was adopted in the Course of Study revised in 1989. In accordance with the students’ characteristics, it was intended that various learning activities should be designed and dealt with while implementing lessons in the course.
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provided valuable opportunities for Japanese mathematics educators to reflect on their curriculum and teaching, by making invisible rules visible. Here, I describe some results obtained from the Japan–US collaborative research on mathematical problem solving (Miwa, 1992). The research was conducted in 1988–1990 by 5 US researchers and 14 Japanese researchers. One of their focuses was to compare the solution processes of students in Japan and the US by using five common problems. The results showed several differences between the two countries. One was that Japanese students used multiplication and division more often than their US counterparts. Another difference was found in their use of mathematical expressions. Japanese students tended to write mathematical expressions even though this did not always lead them to the correct answers. These differences reflected the differences in emphasis in the curriculum and classroom teaching (Miwa, 1992). Another focus was to compare classroom practices of problem solving using a common topic. Six Japanese and five US teachers implemented lessons in the fifth–eighth grades using the Telephone Line Problem: ‘‘We connect two houses by a direct telephone line. We put just one telephone line between each pair of houses. How many telephone lines are there for twenty houses?’’ Miwa analyzed the lesson plans and classroom practices in the two countries in detail. He pointed out that in both countries, teachers provided time to the students to think out solutions freely by themselves and in groups. However, Japanese teachers compared and refined the students’ solutions and provided a summary at the end of the lesson, while the U.S. teachers did not necessarily spend time for these activities (Fujii et al. (1998) examined this as the difference between the mathematically convincing and personally convincing views of teaching). He also pointed out the emphasis on mathematical expressions for communicating the students’ ways of thinking by Japanese teachers. These teachers tended to advise their students to make their approaches clear by using mathematical expressions and also used mathematical expressions to attract their attention to the generalization of the results and approaches. Miwa stated the importance of considering classroom culture in interpreting the differences across countries. Although this research focused on mathematical problem solving, we can observe a similarity between the findings obtained by this research and the results of recent international comparisons of classroom practices (e.g., Stigler & Hiebert, 1999; Hiebert et al., 2003).
4 Development of teaching materials and effective lesson organization with problem solving Mathematical problem solving has also attracted the attention of schoolteachers. In the analysis by Yamazaki
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mentioned above, he pointed out that in the periodicals for teachers in elementary and lower secondary schools, special issues on mathematical problem solving appeared regularly during the 1980s. Moreover, in Japan, we have a tradition of practice-based research by teachers. They present the results of their investigations in both regional and national meetings. In the annual meetings of the Japan Society of Mathematical Education, the section of ‘‘problem solving’’ has been included since 1955 in the session of elementary school mathematics. According to Nagasaki & Senuma (1986), the number of studies presented in this section, when compared to the total number of research presentations, exceeded 10% in 1985. Subsequently, the number of studies presented in this section has not been small and has not decreased (Yamazaki, 1995). Until today, many practice-based studies have been presented by expanding their focuses into diverse topics, including those of fostering the students’ mathematical thinking and creativity, and capitalizing on individual differences among students in the classroom. At the lower secondary school level, many studies on problem solving were also related to developing tasks and organizing lessons in the ‘‘problem situation learning’’.4 These practice-based studies can be divided into two categories. The first is the development of teaching materials that aim at fostering the students’ ability in problem solving, and the other is the attempt at the effective organization of lessons. Here, I discuss studies on teaching by an open-ended approach as one of the former category. Concerning the latter, after mentioning several studies, I describe a case of lesson study by a group of teachers, because it provides an image of how teachers struggle in improving their lessons with respect to the style of problem solving. 4.1 Teaching by open-ended approach Mathematics teaching by using open-ended problems is one of the representative methods of promoting a student’s mathematical problem solving ability in Japan (Yamazaki, 4
‘‘Problem situation learning’’ was introduced in the Course of Study revised in 1989 at the lower secondary school level. The purpose of ‘‘problem situation learning’’ is to stimulate the students’ spontaneous learning and to foster their views and ways of thinking mathematically by setting up appropriate problem situations by, e.g., integrating the learning content in different areas, relating mathematics to the events in everyday and social life, and by emphasizing activities such as concrete manipulation, observation, or experimentation. Problem situation learning was intended to make more room for cultivating motivation and attitude than the acquisition and consolidation of knowledge and skills. It was planned to be included in the eighth and ninth grades, in a total teaching plan with an appropriate allocation and implementation. In the Course of Study revised in 1998, both problem situation learning and mathematics as the optional course were extended to all grades at the lower secondary school level.
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1995). An ‘‘open-ended problem’’ is a problem that is formulated so as to have multiple correct answers. The origin of the open-ended approach was the research on evaluation conducted by Shimada and others at the beginning of the 1970s. Their goal was to develop a method of evaluating the students’ achievement of the objectives of higher-order thinking in mathematics education. Higher-order thinking implies that the students can mathematize a problem situation and deal with it. In doing so, it is important to ‘‘bring forth an (important) aspect of the problem into student’s favored way of thinking by mobilizing their repertories of learned mathematics, reinterpreting it to deal with the situation mathematically, and then applying their preferred technique’’ (Becker & Shimada, 1997, pp. 2–3). Therefore, they needed to develop a problem situation that could be analyzed by students from different perspectives. It was here that they developed open-ended problems. The result of their investigation showed the need for teaching aimed at achieving the higher objectives of mathematics education, which also led their attention toward the development of teaching materials and ways of organizing lessons using open-ended problems. Teachers in the research team began using the approach in their mathematics classrooms. Later, teaching materials as well, as ways of teaching, became more variable, and as indicated by the production of many books (e.g., Nohda, 1983; Takeuchi & Sawada, 1984; Koto, 1992), the idea of openness in teaching and evaluation has since then developed and expanded in various ways through collaboration between university researchers and school teachers (Nohda, 2000). We can find open-ended problems in textbooks that are used in elementary and lower secondary schools. Even though the number of such problems is not large, they are spread across all the grade levels. They include the problems of finding patterns and relationships from tables or from geometrical figures, creating the students’ own designs or figures that satisfy certain conditions and making their own problems based on the original problem. Figure 2 shows an example. Many of these open-ended problems are used for introductory and enrichment activities. Their aims are to arouse the students’ interest and to foster their mathematical views and thinking. For the open-ended problems in lower secondary schools, teachers also deal with them in their lessons of problem situation learning.
Fig. 2 An example of openended problems in the textbook
4.2 Investigation of effective lesson organization The TIMSS video study identified the lesson patterns as cultural scripts for teaching in Germany, Japan, and the US (Stigler & Hiebert, 1999). They identified the Japanese pattern of teaching a lesson as a series of five activities: reviewing the previous lesson; presenting the problem for the day; students working individually or in groups discussing solution methods; and highlighting and summarizing the major points (p. 79). Here, a distinct feature of the Japanese lesson pattern, compared with the other two countries, was that presenting a problem set the stage for students to work on developing solution procedures. In contrast, in the US and in Germany, students work on problems after the teacher demonstrates how to solve the problem (U.S.) or after the teacher directs students to develop procedures for solving the problem (Germany). This pattern, or the motto of Japanese teaching, has been called ‘‘structured problem solving’’ by Stigler & Hiebert. Shimizu (1999) said that one of the origins of such a cultural script is the traditional use of the framework for planning and implementing lessons. He described a common framework for lesson plans as having segments such as ‘‘posing a problem,’’ ‘‘students’ problem solving on their own,’’ ‘‘whole-class discussion,’’ and ‘‘summing up.’’ The origins of such frameworks would be worth studying. Here, I would only like to mention that investigation into lesson organization had already begun in the 1960s and that introduction to mathematical problem solving further promoted research and practice concerning effective organization of the lesson with respect to the style of problem solving. As described before, in the 1960s and 1970s, we had been stressing on the students’ mathematical thinking. Japanese mathematics educators were making an effort to develop ways of making students discover new ideas and construct knowledge on their own in their learning. Different approaches and proposals were developed with reference to leading works such as those by Polya and Poincare. The tradition of lesson study contributed to the active involvement of the teachers. One such study was conducted by teachers in a public elementary school (Sugita elementary school, 1964). They investigated lesson organization and teacher’s questioning that are useful for fostering children’s thinking. They were interested in finding out how to let children discover mathematical
The Problem of Calendar, 9th grade “When you choose and enclose three numbers by a circle, what rules or patterns can you find in the three numbers? How about when you choose and enclose four numbers?” (Sawada, Sakai, et al., 2005)
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content, operation, and thinking in the classroom. Their proposed segments of a lesson were as follows: grasp the problem, make a plan of learning and investigation, find a preliminary conclusion, verify the conclusion, and summarize the content of that day’s lesson. Here, we can see the exact type of segments of structured problem solving. After 1980, the interest in developing useful lesson organization for fostering mathematical thinking continued. The educators thought that letting students experience the process of problem solving would be an excellent way to use and recognize mathematical thinking. For example, Katagiri (1988) examined mathematical thinking used in each stage of problem solving and developed a list of questions that fostered the students’ mathematical thinking in the classroom. Many of these practice-based studies targeted ordinary lessons. In other words, in teaching mathematical content, it was attempted that students create mathematical ideas and knowledge by themselves, by experiencing the process of problem solving. Here, they can be said to have taken the approach of problem solving as a process of instruction, as proposed by Nagasaki. This trend and active involvement by teachers continue till today, especially by teachers in elementary schools. 4.3 Role of teacher in structured problem solving: notes from a lesson study
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how they summarize the lessons; and how they introduce, negotiate, or utilize mathematical norms during the lessons (Hino, 2006; Shimizu, 2006; Sekiguchi, 2006). These analyses also show that the teacher plays a major role in the lesson with problem solving. Here, I describe a case of a lesson study, which illustrates the points teachers discuss, about the role of a teacher, and how they improve their lessons by incorporating them. The lesson study was conduced in 2006 by a group of teachers working in either elementary or lower secondary schools in a city and several people outside the schools, including superintendents of the city. The theme of the group was to encourage the students’ ability to express and communicate their thinking with their friends and the teacher, while creating mathematics in the classroom. Three preparatory research lessons were conducted in sixth grade classrooms in three elementary schools. Each lesson was followed by a debriefing session and by the improvement of the lesson plan based on the suggestions given during the discussion. The final research lesson was open to all the teachers in the group, superintendents, and other experts in teaching mathematics. Next, I describe the points of their discussion on a preparatory research lesson, and how the points were incorporated in the final lesson to make an improvement possible. 4.3.1 Outline of a research lesson
In Japan, teaching mathematics in the style of structured problem solving is generally appreciated by teachers. In lesson studies, almost all the research lessons are conducted in such a style. However, the problem of deal letter has been pointed out (e.g., Zenkoku Sansu Jyugo Kenkyu Kai, 2006). On the other hand, there still exist many everyday lessons that are conduced by direct instruction and demonstration by the teacher, with no room for students to think about the problem for themselves. Morii (2003) expressed the teacher’s confusion about the concept of teaching a lesson with problem solving. He distinguished four types of lessons with problem solving using a 2D model (teacher- or student-centered; and whole class or individualized learning) and argued for the need of studentcentered lessons in both types of learning. These issues indicate that it is not always easy to effectively teach mathematics through structured problem solving, and this is why we have been investigating the characteristics of a good lesson with problem solving. Further investigation into the lesson segments that are suitable for a specific goal of the lesson is required. The role of the teacher in the classroom is considered as another big issue. In Lerner’s Perspective Study, analyses of lessons taught by three experienced mathematics teachers in Japan were performed from different points of view: i.e., how those teachers support students during the seat work;
The goal of the preparatory research lesson was to make students devise ways of finding the volume of one composite solid in the shape of a step (see Fig. 3). The lesson was related to the application of the formulas for volumes of cube and rectangular parallelepiped that was learned in the previous lessons. An outline of the lesson is as follows. Reviewing the previous lesson and presenting a preliminary problem for the day. After the teacher stated the goal of the lesson, he inquired the children about the solids, the volumes of which they had learned. The children answered, ‘‘cube and rectangular parallelepiped.’’ They also expressed the formulas for these volumes. Then, the teacher showed them a picture of the solid on a blackboard
Fig. 3 A composite solid dealt with in the lesson
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(Fig. 3) and said that they should try it for that day’s lesson. In the picture, the solid was presented without numerical values (lengths). Working on the problem individually and presenting the ideas. The teacher asked the children to work on the problem on their own. Worksheets were distributed to the children to allow them to record their work. He instructed them to write their ideas (not the answer) and to try different ideas if they finished one. For the next 7 min, the children worked individually on the problem. The teacher moved around the room, answering questions and providing hints. After individual problem solving, the children presented their ideas in front of the class and shared them with their classmates. The presentations were made individually by those children who wanted to do it. It resulted in six different ideas of transformation of the original solid into the rectangular parallelepipeds (see Fig. 4). Figure 4 shows the representations of the six ideas from the front. Then, the teacher asked all the children in the class to raise their hands by asking them which ideas they found on their own. Presenting the problem for the day. Here, the teacher introduced the lengths of the sides of the solid (see Fig. 3). Working on the problem individually and presenting solutions. The next 10 min were again spent with the children working individually on the problem. This time, they were asked to choose one idea and to work out the solution. They were also asked to try another idea when they finished one. The teacher again moved around the room, answering questions and providing hints. At the same time, he let four children write their solutions on the blackboard (see Fig. 5). Their solutions were based on the first four ideas in Fig. 4. In the presentation, the four students explained their solutions in front of the class. Each explanation was followed by a short comment by the teacher, e.g., checking
Idea 1
Idea 2
Idea 3
Idea 4
Idea 5
Idea 6
the mathematical expressions or the answer. There were no questions or comments by the children. Comparing the solutions and thinking of a convenient method. Then, the teacher asked the children, ‘‘Let’s compare these four solutions. Which do you think is the most convenient?’’ The children became noisy, but no one responded. Therefore, the teacher asked them to raise their hands if they agreed with any solution. Most children agreed with Solution B. Summarizing and children’s writing of comments on their learning. The period was almost over, and the teacher was rushed to say, ‘‘It is important to think about the method to follow according to the type of solid’’. The remaining time was used by the children to write their comments on the lesson in their notebooks. 4.3.2 Reflection on the lesson In preparing this research lesson, the teachers thought about the different techniques in providing students opportunities to think and express their thinking: – To provide the children with the preliminary problem in which the lengths of the sides of the solid were deleted. By doing so, they attempted to turn the children’s attention to the ideas of transforming the unfamiliar solid into rectangular parallelepipeds that they had learned. – To devise the children’s worksheet by dividing a sheet of paper into four parts to enable them to think and write multiple ways of transforming the solid into ones they are familiar with. – To allot time for preparing the children’s self-report at the end of the lesson. An attempt was made to get the children to reflect on what they did and thought in the lesson, and, furthermore, to enable the teacher to evaluate his teaching. It appears that these attempts were partially effective. All the children in the class were able to develop their own idea(s). In their presentations, the children were active in raising their hands and explaining their work, although their presentations were not always skillful. When the four children presented their work on the blackboard, all of them wrote mathematical expressions to clearly express their methods of calculation to their friends.
Solution A
Solution B
8 * 4 * 3 = 96
8 * 10 * 7 = 560
Solution C
Solution D
8 * 4 * 7 = 224
8 * 4 * 3 = 96 8 * 4 * (7-3) = 128
8 * 10 * 4 = 320
8 * 6 * 3 = 144
8 * 6 * 4 = 192
96 + 320 = 416
560 – 144 = 416
224 + 192 = 416
8 * 6 * (7-3) =192 96 + 128 + 192 = 416
Fig. 4 Six ideas of transformation presented by the children
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Fig. 5 Four solutions presented by the children
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However, the lesson also provided an opportunity to the group of teachers to discuss the effectiveness of the role of the teacher in achieving the lesson goal and in enhancing the children’s abilities in thinking and expressing their thinking. One issue raised was whether the children were motivated enough to tackle the task. The meaningfulness and necessity for the children to solve the problem for the day are often considered important issues. In this lesson, even though the teacher reviewed the content of the previous lesson, concern was raised about whether the children were somewhat forced into solving the task given by the teacher. Another issue was the lack of time for validating and comparing different solutions presented by the children. This issue is also commonly addressed in lesson studies. The teacher asked the children to present their solutions on the blackboard. However, each presentation was followed by only a short comment from the teacher. Moreover, the teacher just let the children raise their hands for the most convenient solution, and did not ask them their reasons or initiate a discussion. A similar observation was made in the preliminary problem. The teacher did not delve into the validity of each idea, although some children did try to assess if it was possible to transform the solid into a rectangular parallelepiped using the ideas of 5 and 6. Another issue was the lack of time for summing up the lesson. At this stage, the lesson was very close to the end. Here, the teacher asked the children to write comments on their notebooks. A question was also raised whether this was enough for them to reflect on their learning of the lesson. 4.3.3 Revision of the lesson plan These issues were incorporated when the teachers revised the lesson plan. Here, I show the points made in the lesson plan (see Table 1) and give the outline of only the comparison and summary stages in the final lesson. In the final research lesson, after the presentations of four solutions by the children (they were similar to those in Fig. 5), the teacher pointed out three major ideas of transforming the solid into a rectangular parallelepiped in
their solutions. They separated the solid into right and left sides, upper and lower parts, and reduced the larger rectangular parallelepiped into the smaller one. The teacher then put a summary paper on the blackboard and concluded the lesson by saying ‘‘We came up with different ideas today. They are ‘dividing and then bringing together’ and ‘reducing a part from the whole.’ By applying these ideas, we can calculate the volume of such a solid using the formulas for volumes we have already learned.’’ After the summary, the teacher also asked the children to solve a similar problem, and write a comment on their learning of that day’s lesson. In solving the problem, each child was requested to choose the method, which she/he thought was better than the rest. After the implementation of this final research lesson, the teachers had time for another discussion. I will not explain the details any further. Instead, I will only stress that teachers need to play a vital role in a lesson with problem solving (Lester, 1994). Providing the students with time to work on the problem themselves is not enough. We need to ask ourselves such questions as, ‘‘Is the problem situation enough for the students to increase their interest in pursuing and tackling the task?’’ ‘‘In what way should different solutions be managed so that the students can be led to the mathematical content we want them to learn?’’ and ‘‘To what extent should we guarantee opportunities for all the students to actually engage in the activity in each stage of the lesson, and to reflect on their activity?’’ All these questions need to be addressed because it is not the style of structured problem solving per se but the teacher’s careful management of the students’ thinking in such a style that has an influence on the development of their problem solving abilities and mathematical thinking.
5 Problem solving as a way of assessing students’ mathematical thinking Evaluating the students’ achievement with respect to the goals of mathematics teaching is one of the important areas
Table 1 Issues and revisions made in the final research lesson Issues
Revision
Meaningfulness for the children to solve the problem for the day
Showing a black box and a solid made with building blocks
Time for comparing different solutions and methods of summarizing the lesson
Reduced time for dealing with the preliminary problem into just exchanging different ideas
Letting them estimate before calculation
Devising steps toward lesson summary More support for expressing their thinking
Preparing physical models prepared from expanded polystyrene and cutters Revising the worksheet for the children Letting other children explain the solutions written on the blackboard
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of research in mathematics education in Japan. As described earlier, the origin of the open-ended approach was in the research on the development of methods for evaluating the students’ achievements of the objectives of higher-order thinking in mathematics education. The results of international assessments also affected our curriculum. As described, the students’ low valuing of mathematics and their low scores in the questionnaire items on beliefs and attitudes toward mathematics were always big concerns in the community of mathematics educators in Japan. In the Course of Study revised in 1998, the word ‘‘joy of mathematical activity’’ was stated as a goal of school mathematics because of these negative results obtained by international studies. These results have been investigated further by several types of assessment by the Ministry of Education (e.g., Ministry of Education, Culture, Sports, Science and Technology, 2007). Research on the assessment has become active by the accumulation of our knowledge of mathematical problem solving. We have learned that the processes of problem solving provide fruitful information on the student’s use of their knowledge and skill. Furthermore, they provide us information on their mathematical views and ways of thinking mathematically. Thus, in developing test items, we came to examine the possibility of assessing the students’ mathematical thinking and their attitudes toward mathematics. Studies and practices have also been conducted in the classroom. In Japan, the importance of formative evaluation is commonly shared by the teachers. They often observe the students during the lessons and analyze their worksheets after the lessons (Nagasaki & Becker, 1993). Researchers further investigated the method of obtaining valuable information on their thinking and interests in, and attitudes toward, mathematics and giving useful feedback to students in the classroom (e.g., Shigematsu, 1995; Yabe, 2000; Ninomiya, 2005). These studies aim at enhancing the students’ ability in meta-cognition and self-evaluation. In their lessons, the teacher asked the students to write about their thinking processes and reflections on the problem dealt with in the lesson. While collecting information, they also devised ways in which the students could express their thinking willingly and purposefully. Finally, I discuss our system of evaluation at the national level. We have been examining the standards for evaluating the students’ mathematical learning. Currently, we have a system of evaluation based on criterion-referenced evaluation. Each student is compared with a pre-set standard for acceptable achievement based on four criteria: interest, motivation and attitude; mathematical thinking; skill of expressing and handling; and knowledge and comprehension (National Institute for Educational Policy Research, 2002). These criteria were established because
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there was a tendency to focus on knowledge and skills when evaluating the students’ achievement.5 Introduction to these criteria and the criterion-referenced evaluation was a big driving force in expanding the studies in these areas. To conduct a criterion-referenced evaluation, it is inevitable to analyze the instructional objectives, make a plan of teaching, and a method of evaluation in every lesson. Although all these processes are hard to work out, in developing a lesson plan, we explain the connection between the instructional objectives in a teaching unit and the goals of each lesson on one hand, and the method of evaluation on the other hand. For example, the teachers in the lesson study group described in ‘‘Development of teaching materials and effective lesson organization with problem solving’’ developed the lesson plans containing such information. Figure 6 shows their framework of construction of the teaching unit ‘‘Volumes’’ and the method of evaluation. This figure only shows their plan on the final research lesson that was described earlier as an example. They set up eight specific objectives that covered all the four criteria of evaluation in teaching the unit of ‘‘Volumes’’. From Fig. 6, we can see that the teacher intended to specifically achieve the criteria of mathematical thinking, knowledge, and comprehension in the objectives, and tried to judge the children’s acceptable achievement based on the written and oral information of whether they transformed the original solid into rectangular parallelepipeds, and whether they showed an understanding of the meaning of volumes and others.
6 Final remarks In this paper, I summarized the trend of mathematical problem solving from three perspectives: research, practice, and assessment. As I looked at the tendency and problems we tackle today, in the sections ‘‘Studying, thinking, and learning process of students’’, ‘‘Development of teaching materials and effective lesson organization with problem solving’’ and ‘‘Problem solving as a way of assessing students’ mathematical thinking’’, the three perspectives cannot be regarded as being independent of each other. Nevertheless, I observed that mathematical problem solving has a definite influence on advancing our efforts in studying and improving teaching practices in the classroom. It drew our attention to the students’ processes of solving problems, 5
In Japan, we do not have standardized examinations at the national level. Therefore, these four standards have not been developed for the purpose of preparing examination items. Instead, they were developed as a way of writing school reports to parents, etc. At the same time, in the assessment conducted by the Ministry of Education mentioned earlier in this section, the standards have been used. Overall, these educational policies have an influence on classroom practice.
Toward the problem-centered classroom Fig. 6 A framework of construction of teaching unit and method of evaluation
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Specific
Number
Construction of the
Connection with the criteria
Method of evaluation
themes in
of class
teaching unit and
of evaluation
in the learning
the
hours for
goals in each lesson
inter
thin
teaching
the theme
est
king
unit Volumes
skill
kno
activities
wled ge
4
Lesson 4) To find the
of cube
volume of a composite
child can find ideas to
and
solid by applying the
transform the
rectangular
formulas for cube and
composite solid into
parallelepi
rectangular
rectangular
ped
parallelepiped.
parallelepiped.
To understand the meaning of “solid,” etc.
Knowledge — If the child shows correct use of “solid” and “volume” and the correct writing of “cm3.”
and, further, their processes of thinking mathematically and the learning of mathematics in the classroom. It promoted studies and practices by stimulating our interest and the need to develop teaching materials and effective ways of organizing lessons for the purpose of enhancing the students’ ability in problem solving. Looking into the processes of problem solving also provided a way of assessing the thinking and attitude of the students. I also observed that the culture of mathematical thinking lies behind these influences. Mathematical problem solving was conceived as an important perspective, partly because we had been investigating teaching that aimed at developing the students’ abilities in thinking mathematically. After a period of intense activity of research and practice on mathematical problem solving, together with an active discussion relating to mathematical thinking, it seems that we have settled down. This means that in some way we have incorporated the perspective of mathematical problem solving into our view of mathematics teaching and learning. By recognizing the problem solving approach as a powerful way of learning mathematics, we continue to investigate conditions for, and roles of, the teacher in realizing a classroom in which the students are actively engaged in the activity of solving problems and developing mathematics. References Becker, J. P., & Shimada, S. (Eds.). (1997). The open-ended approach: a new proposal for teaching mathematics. Reston, Virginia: NCTM. (Original work published 1977). Fujii, T., Kumagai, K., Shimizu, Y., & Sugiyama, Y. (1998). A crosscultural study of classroom practices based on a common topic. Tsukuba Journal of Educational Study in Mathematics, 17, 185– 194. Hiebert J., et al. (2003). Teaching mathematics in seven countries: results from the TIMSS 1999 video study (NCES 2003-013), US Department of Education. National Center for Education Statistics, Washington DC.
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