We observed (there were over 100 observers) a lesson on finding patterns in a number chart. The teacher began the lesson by posting a large paper showing a hundreds chart (a square with the numbers 1 to 100 in rows). He had a square magnetic frame that he could place anywhere on the chart creating a square with nine numbers inside. He asked the students to find the sum of the numbers as quickly as they could with a calculator (see example below).

11 12 13

21 22 23

31 32 33

At the same time, he asked Dr. Takahashi to figure it out without a calculator. Dr. Takahashi found his answers quickly and the students began to wonder if he was using some kind of trick. So the teacher posed the following problem: “Let’s come up with a strategy to calculate the total of the 9 numbers efficiently and explain the strategy.

Students first began solving the problem on their own. After several minutes, when the teacher noticed that many students were unable to think about a strategy, he called students who needed help getting started to the floor at the front of the room to give them hints. After several more minutes, he asked students to share their strategies.

The first strategy presented was (1+2+3) x 3 + (10+20+30) x 3. This student said that the ones digits in each column are always 1, 2 and 3 so since there are 3 columns, the sum of the ones digits is (1+2+3) x 3. Likewise, when the ones digits are removed, each column has 10, 20 and 30, so the sum of all the tens is (10+20+30) x 3. The teacher had arranged that the ones digits (the problem was on a large paper on the chalkboard) could be removed to illustrate this solution method. The kids remarked, “Our teacher is really prepared for this lesson!”Another student explained that by decomposing the numbers, you could make all the numbers in one column the same (e.g. move 10 from 31 to the 11 so all three numbers in the column are 21). Therefore, the problem could be solved by 21×3 + 22×3 + 23×3.

The teacher then asked the students that if by looking at these methods, they could come up with a better strategy. Many students quickly came up with the idea that by decomposing the numbers, all of the numbers could be made equal to the middle number so you could just multiply the middle number, 22, by 9. Next, the teacher asked if this idea could be used to find the sum of numbers in other parts of the chart. Most students were able to quickly find the solution to the next problem. Afterwards, several students shared what they learned aloud. (I would like to add that I really liked Mr. Tsuruoka’s teaching style. Whenever he asked a question, he would wait, even when many students had their hands up. He wanted all students to think and it was obvious that students were used to this type of inquiry-based, problem-solving lesson, and it was not just something he was doing for the sake of this public research lesson.)

During the post-lesson discussion, one teacher acted as the facilitator, another acted as recorder, and final comments were given by Professor Nakamura. During his comments, Prof. Nakamura emphasized the importance of careful observation and careful note taking during the research lesson observation. A good post-lesson discussion is the result of careful observation. He showed his notebook that had 7 pages of detailed notes that were color-coded with four different colored pens (see photo). The different colors were used to record different aspects of the lesson such as math expressions, student reactions. Another interesting aspect of how he observed the lesson is that he focused on one student, writing down what the student said and what he was thinking (aloud), misconceptions, etc. He said that while you are teaching, you cannot focus on how one student is understanding the lesson. This is the beauty of being a lesson study observer—you are able to focus on details that you would not normally be able to do in your classroom teaching. Afterwards, I asked him how he selects the one student he will observe. He told me that he listens carefully to what students are saying to themselves and chooses a student who is “thinking out loud.”

After the lesson, we had our final “enkai” (big party) with the teachers, professors, administrators, district superintendent, and others. Mr. Toki, my good friend and former principal of the Greenwich Japanese School (GJS) in Connecticut also came. (I had the privilege of working on a research lesson on the area of plane figures with Mr. Toki and several GJS teachers in 2003 and it was definitely one of the highlights of my career. Mr. He is now a principal in the Setagaya Ward.) I must say that this was one of the best enkais that we have had. Many people made speeches, there were many kampais (toasts), and I was even asked to say a few words. My final act in Japan was to put on my “indoor shoes” I had brought along to wear during school visits and ran outside and jumped up and down! (In Japan you must remove your shoes and wear “indoor shoes” or slippers. Now, all of the shoes I own are dirty!)

Well, what can I say? The 2007 LSIP is over but in many ways it is just beginning. We now need to take what we learned back to our schools and figure out what to do next. This will be very different for every one of us. In our morning reflection meeting, we talked about how important it is to keep in touch and for the lesson study communities in North America to network and support one another. Many questions were also raised such as, “How can we make lesson study work in North America without having the underlying support system for lesson study that exists in Japan?” In Japan, everyone—universities, professors, administrators, teachers, students, superintendents, ministry of education officials—value and support lesson study. In the U.S. and Canada, many of us are, as one LSIP participant remarked, like “underground freedom fighters.” Maybe part of the answer is, in the words of the famous New Orleans musician Dr. John, to just simply “keep on keeping on.” But now we can do that—continue lesson study, form new lesson study groups, improve upon what we have been doing—with new knowledge and renewed enthusiasm and commitment.

Lastly, sayonara to all the wonderful friends I have made from the U.S. and Canada. Y adios a mis Buenos amigos mexicanos, que gusto conocerles y compartir con ustedes en el idioma celestial! Let’s all keep in touch!

Adios,

Bill Jackson

Tokyo, July 4, 2007

P.S.

I am now off to Guatemala to spend a few weeks vacation with my wife and kids whom I miss very much and who have sacrificed and supported me on this trip. I will be thinking about you all from time to time as I lay on a lounge chair sipping some kind of tropical drink with an umbrella in it. God bless you all and enjoy your summer!

]]>We arrived around lunchtime and the kids were already prepared for us. Students at this school didn’t speak nearly as much English as their counterparts in Tokyo but they were very happy to see us. We ate lunch with the students, visited many different classes, and had a wonderful time. Afterwards, the principal, vice-principal, and the head of the school’s research committee discussed the lesson study goal. We also attended a “konaikenshu” (school based lesson study) meeting which was the seventh such meeting they’ve had this year (the year began in April). The next day (Tuesday), we observed a public research lesson.

Overview of lesson study at the school

Teachers at Ishida Elementary School were concerned about having good relationships with each other but they were not looking at their lessons with critical eyes. Japanese people tend to go along with the group instead of being individuals and, therefore, tend to have similar ideas. This is hindering their efforts to improve. The same problem exists with the students. Instead of having their own ideas, they are often content with going along with the group. To solve this problem, the school is working together to foster many different points of view among both students and teachers. Through lesson study, they are addressing these issues in order to improve the education at the school.

The school’s lesson study goal is “to foster solid understanding and learning in students.” To achieve the goal they want to “bring out multiple points of view, recognize different points of view, and come up with ideas” and “develop communication skills in students.” By communicating with each other, students can learn from each other and raise their level of understanding. To do this, however, they need to be able to express their own ideas and listen to the ideas of their friends. In order to foster this trait in students, teachers need to be able to communicate as well. In lesson study meetings, they encourage teachers to share and communicate their thoughts with each other, and design activities in ways that students come up with the tasks and ideas they want to study themselves. Lesson study is done around this goal in all subjects.

Since Ishida Elementary School is a designated research school by the Japanese Ministry of Education and the Yamanashi Board of Education, they receive grants to pursue this goal. One of the things the grant provides them is assistance from the education professors at Yamanashi University. The school is preparing for a lesson study open house on November 7, 2007 and many teachers will attend from throughout the prefecture to hear about the results of their research and view public research lesons around this goal.

Lesson study planning meeting and public research lesson

The lesson study planning meeting we attended and the lesson we observed involved a second grade lesson on ordinal numbers. The problem the teacher (Mr. Hayakawa) posed to the students was the following: “Children were lined up in a straight line. Yoshiko (a girl’s name) is the 6th person from the front of the line and the 7th person from the back of the line. How many children are there altogether?” The most common error students come up with in this problem is 6 + 7 = 13. Since being 6th from the front involves 6 people including Yoshiko and being 7th from the back involves 7 people including Yoshiko, the error of adding 6 and 7 involves counting Yoshiko twice. Mr. Hayakawa wanted to pose the problem in the context of students waiting on line to buy bread, which is something they did on a recent field trip. They ran out of buns and needed to make more for the students. The baker could see only the first 6 people and his assistant could see only the last 7 people. Both could see Yoshiko.

The main parts of the lesson were—understanding the problem, individual problem solving, sharing in groups, presentation and discussion of solutions, and reflection through journal writing. There was much discussion about the problem being posed, materials that were to be used, etc. The entire staff was there and part of the reason for the meeting was to prepare them for a new style of post-lesson discussion that was meant to foster communication among the observers (I will explain this later).

On the day of the public research lesson, there were at about 60 observers which included us, teachers from the school, the principal and vice-principal, pre-service teachers from Yamanashi University, and two university professors—Professor Nakamura and Professor Tabata. The teacher took a longer amount of time than anticipated posing the problem, so the lesson went over the time allotted. Also, students came up with some unexpected responses that the teacher was unsure about how to deal with. One of them was 5 + 6 = 11 (5 in front of Yoshiko and 6 behind her but no Yoshiko). Many students had the incorrect answer (6 + 7) but a correct representation with manipulatives or drawings. The post lesson discussion focused on these things and some others, including whether or not students were really communicating during the small group sharing time.

Since I have talked about other post-lesson discussions before on this blog, I would like to focus now on the way they conducted the discussion, not on the content of it. Each observer was given three post-it (sticky) notes—blue, yellow, and pink. On the blue note each observer was to write (only a couple of words) something positive about the lesson, the yellow note was to write a question you had about the lesson, and the pink note was to write something the teacher should consider for future teaching. In the hallway, they posted large copies of the lesson plan and after the lesson the observers (including us—they made large English copies too!) placed the notes on the lesson plan. Teachers were divided into small groups of about 8 people, each with its own large lesson plan to post the notes. Then they were told to organize the results and categorize them according whether they had to do with three sections of the lesson they were interested in getting feedback about—the posing of the problem, the small group communication piece, and the representations students used (this includes both expressions and drawings or use of manipulatives). After this, each group discussed the lesson and the reason why they wrote what they did on the notes for about a half an hour. The conversations in each small group of observers were very engaging. Afterwards, each group was asked to share what they discussed. Final comments were brought by Professor Tabata, and at the end Mr. Hayakawa, the teacher of the lesson, shared reflections about what he learned from the research lesson.

It is difficult to express in this blog how powerful this type of post-lesson discussion was. It really did achieve the goal of fostering communication. It also left me with a question: Can we really teach students to communicate and share their ideas if we as teachers don’t do so ourselves?

Bye Bye (This is the way many young people in Japan say goodbye these days. Try to imagine hearing it with a Japanese accent.)

Bill Jackson

Under the shadow of Mount Fuji, Kofu City, Japan

July 3, 2007

The role of the mathematics educator (Prof. Yukio Yoshikawa)

It is not enough for student teachers to simply learn about “instructional materials” (a broad term that encompasses textbooks, curricular materials, manipulatives, math problems, etc.). They must experience them to be able to “see them from the students’ perspective.” Professor Yoshikawa demonstrated this through an example of a very rich 8th grade math problem involving finding a missing angle between parallel lines. He explained about 10 different methods students often use to solve the problem, in order from very simple to quite complex.

Teachers must be prepared for the many different ideas that students come up with. By examining these different methods, students to begin to see patterns and relationships, look for “harmony in ideas,” and appreciate mathematical reasoning. All students will not become mathematicians, but by recognizing mathematical relationships, they can discover that mathematics is enjoyable and something that enriches their lives.

It is important that student teachers learn to go beyond the way they were taught in order to provide quality learning experiences for children. The most important quality a teacher needs the ability to understand how students think and have a passion not only for teaching, but also for learning.

Student teaching and research lessons (Prof. Takashi Nakamura)

Lesson study has three main purposes—(1) to solve educational problems by developing new curriculum and instructional approaches, (2) enable practitioners to examine and improve their own practice, and (3) stimulate a shared community of practice among teachers within a setting. In addition, for mathematics teaching, it is important for student teachers to learn how to teach a problem-solving based lesson. These lessons, typically involve four steps—understanding the problem and the task, attempting to solve the problem on their own, class discussion about the solutions, and reflection on individual solutions. The best way to do this is through lesson study.

Education majors at Yamanashi University spend six weeks in student teaching (3 weeks in June and 3 weeks in September-October) of their junior year. For both elementary (grades 1-6) and lower secondary (grades 7-9) school teachers, one of these experiences must take place in an elementary school and one in a lower secondary school. This is because teachers can be asked to teach or transferred to any of those grades during their career. In the final week of their student teaching experience, student teachers must conduct lesson study and teach a public research lesson.

One of the most important aspects of these research lessons is the post-lesson discussion. During this discussion, the teacher who taught the lesson reflects first. Next, there is a question and answer period followed by discussion and comments by the observers. The discussion is concluded by comments and suggestions by a university professor.

There are three foci for the post lesson discussion. The first focus is on materials. This includes the choice of numbers for the problem, the variety of students’ solutions, and the mathematical activity students were asked to engage in. The second focus is on points of instruction. This includes the clarity of the question, allocation of time, the choice of student presenters, and the use of journal writing. The third focus is on student understanding. This includes the how the students reacted to the activity, student thinking, interest, and satisfaction, and the knowledge and skills that they learned.

There are three results that the university hopes to accomplish by requiring lesson study for student teachers—(1) internalization of the teaching profession, (2) self-reflection and “understanding the characteristics of self,” and (3) the realization of the need to continuously study and improve. In addition, professors at Yamanashi University want to develop a “joy for teaching” in the student teachers.

Testimonial by a student teacher on her student teaching experience (Masami Kawakubo)

Miss Kawakubo was assigned to work with a 4th grade class for her three-week student teaching experience. Each day she arrived at school before 8:00 a.m. to prepare. She had to observe lessons in classrooms, teach lessons, and interact with students throughout the day. After school, she would meet with a teacher assigned as her instructional advisor to reflect on the day and engage in discussions about how to improve her teaching. She would go home at around 6:00 p.m., but even her evenings were occupied by writing in a “student teaching journal,” conducting kyozaikenkyu (materials and curriculum study), writing lesson plans, and preparing for next day’s lessons.

The first week of her student teaching was devoted to observing different lessons in many classrooms. On her 2nd week, she began to teach lessons. This required preparing a detailed lesson plan for every lesson which was later discussed, often several times, with her instructional advisor. (Student teachers typically teach about 20 lessons during the their student teaching.) In the 3rd week of her student teaching experience, she had to teach a research lesson.

The most difficult part of her student teaching was writing the student teaching journal. There are three main sections in the journal. In the first section, “understanding students,” she had to record the kinds of learning activities the students were engaged in, their states of learning, and their actions over the course of a three-week period. Since this had to be done for every one of the 40 students in the class, she had to conscientiously interact with each student on a daily basis. In the course of writing this section, she found herself writing more about some students than others. This made her realize that there were some students “she was not connected with.” In the second section, “reflections on instruction,” she had to write about the lessons she taught, reflections from the post-lesson discussions, and her thoughts about her teaching. This experience helped her to reflect on her lessons “carefully and deeply.” In the final the section, “reflections on classroom management,” she had to record and reflect on the kinds of activities the students were engaged in during the day and what kinds of interactions occurred among the students. These reflections were then discussed with her instructional advisor in order to improve her classroom instruction each day.

In the final week of her student teaching, Miss Kawakubo engaged in lesson study in mathematics. The most difficult part of this was conducting “kyozaikenkyu” (instructional material investigation). First, she planned a five-lesson unit on isosceles and equilateral triangles (all the lessons were taught and three were observed by other teachers and student teachers). To prepare the lessons, she investigated documents such as the Elementary School Mathematics Teaching Guide for the Course of Study (available in English at http://www.globaledresources.com), the teachers’ manuals for the textbook, and current research papers. Then, she wrote a research lesson plan for the final lesson in which she asked students to make isosceles triangles using geoboards. For this lesson, she consulted with instructional advisors, university professors, senior teachers, and other colleagues to get their suggestions and opinions. In the post-lesson discussion, she received many helpful suggestions and learned a lot, especially about the importance and difficulty of conducting kyozaikenkyu. She also realized that it is important to have colleagues she can talk to, get encouragement from and encourage, and “engage in heated discussions with.”

The three weeks that Miss Kawakubo spent student teaching were not easy and they took a toll on her both mentally and physically. On weekdays during her student teaching, she only slept about 2 to 4 hours per day. It was very beneficial, however, because it made her think about many things, try and make mistakes, and reflect on herself and her teaching. Even now, she still thinks about many things that she could have done better.

I would like to end this part of the blog by quoting what Miss Kawakubo wrote in her reflections about the experience. “Even though I felt so tired every day, when I saw the students’ faces every morning, I felt marvelous. The more seriously I took my responsibility with the students, the better they reacted to me. Through this experience, I realized that teaching is a worthwhile profession.”

Sayonara,

Bill Jackson

Kofu, July 2, 2007

There are six major math textbooks in Japan. These textbooks are very similar. According to Mr. Ogasawara, this is due to the fact that the other companies have copied Tokyo Shoseki’s textbooks because of their coherence and excellent sequencing. These textbooks were written based on lesson study findings. Textbook company executives attend large lesson study open houses in Japan and also consult many published lesson research lesson findings to learn what to incorporate into their textbooks. This is one reason why Tokyo Shoseki’s mathematics textbooks are the most widely used math textbooks in Japan.

Several years ago, a new subject called “integrated studies” was added to the Japanese curriculum. In addition, Saturday school was eliminated. Therefore, the number of class hours per year dedicated to mathematics teaching was reduced. Now, 1st graders spend 114 hours, 2nd graders 155 hours, and 3rd through 6th graders 150 hours in math classes per year respectively (1 hour = one 45 min. lesson). Japanese students generally have math class 3 to 4 times per week, depending on the grade level. As a result, textbook companies had to reduce the content of the math textbooks and move certain topics into higher grades. This has created somewhat of an outcry among Japanese educators and because of that, the Ministry of Education is revising the curriculum to possibly include more hours of mathematics in the near future.

Perhaps the first reaction one gets when looking at Japanese math textbooks is that they are thin, lightweight paperbacks with colorful cartoon illustrations. The total number of pages ranges from 120 to 210 per grade level. This is vastly different from the 500 to 600+ pages found in typical, heavy U.S. textbooks. (I couldn’t help thinking that if for some reason the number of hours devoted to math education were reduced in the U.S., we wouldn’t bother reducing the content.) This embodies the Japanese philosophy of teaching a few important math topics per year in depth instead of what is typically done in the U.S.—teaching many topics per year superficially. Japanese textbooks are also inexpensive and they are given to the children to keep. We saw many instances at the schools we have visited of kids writing in their textbook, something unheard of in the U.S. (unless they’re writing graffiti!)

Another characteristic of Japanese elementary math textbooks is that they are designed so teachers can teach the subject without having any special knowledge. Elementary school teachers in Japan typically teach every subject (except gym, art, music, etc.) so they are generalists, not specialists. Teacher’s guides for the textbook offer suggestions on how to teach each lesson based on the results of actual research lessons. Examples are given of questions that may be asked by students according to various stages of learning as well as helpful hints on how to deal with these questions and typical student responses and errors. They even include diagrams on how to organize the chalkboard.

Not only are the teacher’s guides useful, but the textbooks themselves are also very helpful for teachers. Oftentimes, teachers will teach lessons without using the textbook because the textbook pages include pictures of children solving the problems in several different ways. These illustrations are given to help both the teacher and the students. When a student is stuck and cannot think of a strategy, the teacher will oftentimes tell the student to look at the textbook to get ideas. In addition, several textbook pages often show pictures of what the teacher’s chalkboard should look like.

As noted above, Japanese textbooks have many pictures and colorful, cartoon illustrations. These illustrations, however, have a purpose and are not just for show as is often seen in U.S. elementary textbooks. In Tokyo Shoseki’s textbook, there is friendly cartoon space creature who gives helpful hints or states important mathematical ideas. This makes the textbooks very child friendly. Instead of statements like “solve the problem below” Japanese textbooks say “Let’s solve the problem below,” reflecting the philosophy of whole class, student-centered problem solving. (I was wondering if the kids might think that “let’s” is referring to them and the friendly space creature though!)

In the past, math teaching in Japan focused on implementing knowledge thoroughly by direct instruction (“cramming” was the word Mr. Ogasawara actually used.) In recent years, however, the focus has shifted to having students understand the process of solving a problem by thinking for themselves. Therefore, Japanese textbooks focus much on problem solving and having students find various solution methods as well as analyzing the merits of each method. An example that was shared was a page from the 6th grade textbook (English version available at http://www.globaledresources.com) that showed pictures of three students finding the area of a trapezoid by transforming it in different ways. One child is seen dividing the trapezoid into two triangles, another attaches a congruent trapezoid upside down to one side to create a parallelogram with twice the area of the trapezoid, and another child cuts the trapezoid at half of its height and rotates it to create a parallelogram with the same area as the trapezoid. Each of these methods can be used to derive the formula for the area of a trapezoid. The teacher’s guide also offers various other possibilities students might try. I think that this type of approach is largely absent in most U.S. math textbooks (in fact, all the ones I’ve ever seen!)

Textbooks are revised every three to four years in Japan, but before they begin the revision process, the textbook company surveys up to 2000 university scholars and teachers to ask for ideas on how to improve the textbooks. These ideas are then incorporated into the new version. This is very important because Japanese mathematics textbooks tend to improve slowly over time reflecting the philosophy of continually polishing and improving upon what has been learned. Different U.S. textbooks tend to offer radically different approaches and tend to “throw out the bay with the bath water” (at least that’s this particular blog writer’s opinion). It seems to me that we don’t share this idea that improving math teaching and learning is a long-term process based on the results of actual classroom research.

After Mr. Osawagara’s talk, Dr. Tad Watanabe gave the LSIP participants a workshop on geometry in elementary Japanese textbooks. In this workshop he showed us how to use set-squares and compasses to construct parallel and perpendicular lines and draw polygons. Most of us were commenting on how we had little experience in this area in elementary school and witness little of it still today. Japanese students are encouraged to use these tools to draw accurate rectangles, parallelograms, rhombuses and many other figures. How many 3rd graders do you know who can draw an accurate isosceles triangle using a compass and a ruler (or adults, for that matter)? Well, in Japan they do (and we also did, and, yes, it was a lot of fun)!

After the workshop, we were invited to a wonderful Bento Box lunch at Tokyo Shoseki’s company cafeteria. Afterwards, we had a tour of their one-of-a-kind textbook history museum where we saw school textbooks from the 18th century on up (and even earlier examples of wooden panels of math problems). It was interesting to find out that teaching mathematics through story problems has a very long history in Japan. These story problems often involved animals. Even very old textbooks have cartoon illustrations of animals for the problems. One textbook from the 1930’s had an illustration of 8 frogs who jumped into a pond but only 7 came out. The panic-stricken friends jumped back in and pulled their half-drowned friend to shore. Now that makes for an interesting math problem!

Domo (which basically can mean anything from hi to bye to thanks and just about anything else—good word to know if don’t know what to say!)

Bill Jackson

Tokyo, Japan

June 28, 2007

We were very well received by the principal, Mrs. Ichinose, who spoke to us about the philosophy of the school, which is based on three things—openness, collaboration, and challenge. She said that she has very good teachers and they work very hard toward that goal. (I would like to note that both teachers and principals are referred to by the same title “sensei” or “teacher,” and it seems like Mrs. Ichinose sees her position not so much as boss but as part of a collaborative effort.) She also emphasized the importance of not just parental involvement, but involvement of the entire community.

In the morning, we split into small groups and visited classes. A parent or community volunteer, who spoke English, led each small group. We visited many different classes including art, swimming, Japanese language, music, mathematics and English.

Although Narimasu is quite different from Takehaya, there was still that feeling of openness and the feeling that they want to educate the whole child. Japanese schooling attempts to develop students’ creative talents, not just academic skills. There was student art all over the walls and I must say it was quite good. Children are genuinely happy, playful and loved. Teachers go out of their way to be kind to the children and, just like Takehaya, teachers were generally not yelling at kids, and as bad behavior was seemingly ignored, it quickly dissipated.

One of the highlights of the day was eating lunch with the students. It is really something to see how the students serve and eat lunch, usually unsupervised. All students wait until everyone is served and then together say the blessing “itadakimas” (we gratefully receive this) and eat. Nothing is wasted and afterwards they all clean up. I ate lunch in a sixth grade class and the students were very happy to practice their English. Common questions I was asked are, “What is your favorite _________ (hobby, color, food, etc.)?” I really laughed a lot with the students at my table as they taught me Japanese and I taught them English!

In the afternoon, we all observed a research lesson. The Narimasu school is conducting a different kind of lesson study where three lessons are conducted at the same time in the same grade and subject in different classes. Through this, observers can observe the previous day’s lesson, the current day’s lesson, and the next day’s lesson all at once in three different classes. Observers switch between classes to observe. These classes have been downsized in order to have classes of a little more than 20 students per class instead of the typical 40 or more. The students are divided into the three classes heterogeneously based on a pre-test so all classes have the same mix of ability levels. We observed 4th grade mathematics lessons taught by three teachers—Mr. Suzuki, Ms. Adachi, and Mr. Koizumi. We divided into three different groups to move about every 10 minutes between the classes. For the sake of brevity, I will comment on Ms. Adachi’s class only.

The goal of Ms. Adachi’s lesson was for students to see the advantages of writing a unified expression for a division problem. The problem that she posed was, “If you share 8 dozen pencils among 6 people, how many pencils will each person get?” Students wrote down their solution methods and answers on magnetic white boards and put them on the board. The teacher then classified all of the solutions into 3 basic methods: (1) 12 x 8 ÷ 6 = 16, (2) 12 x 8 = 96, 96 ÷ 6 = 16, and (3) 8 ÷ 6 = 1 R2, 24 ÷ 6 = 4, 12 + 4 = 16. When students explained the solutions, Ms. Adachi told them that they have to tell what the numbers mean in each expression. Students concluded that methods 1 and 2 were similar but many were confused about solution method #3. Through discussion, students understood that this method divided 8 dozens among 6 people so each person gets one dozen. The R2 means that there are 2 dozen or 24 left. These had to be divided among 6 people so each person gets 4 pencils. So each person receives 12 (1 dozen) + 4 or 16 pencils. Then, the teacher had students analyze the methods based on 3 criteria: (1) Is it efficient? (2) Is it simple? and (3) Is it accurate? Students concluded that the best method is method #1, the unified expression, although one student insisted that method #2 was the best. The teacher said that they would discuss that the next day, summarized the lesson, and asked students to write in math journals.

After the lesson, the entire staff met for the two-hour post-lesson discussion (two hours seems to be typical). The discussion focused on the importance of problem solving, the use of hint cards, the meaning and importance of mathematical expressions, and the purpose of writing unified expressions. There was much debate and discussion about all three lessons. Dr. Takahashi and Dr. Fujii brought final comments. This discussion was a good example of typical lesson study discussions in Japan, but it was not nearly as critical or intense as the one we saw at the Takehaya school.

Afterwards we all went out for a post-lesson “enkai” at a local restaurant. It was very exciting and fun. They made a lot of jokes (even about the principal!), the Japanese teachers all sang a Japanese song, the American teachers sang “America the Beautiful,” and two Mexican teachers sang a rousing rendition of “Mexico Lindo.” Again, it reinforced the idea in my mind of how important these post-lesson parties are to build friendship and community, and to support and honor the teachers who put themselves out there by teaching a public lesson so that everyone could learn.

Mata-ne (see you later),

]]>We were all very impressed by both the freedom and the responsibility of the students. Children play (and even learn at times) unsupervised, they serve lunch to themselves, and they clean the school. The overwhelming feeling that you get is that children are truly happy and enjoying their elementary and middle school experience. In many ways, it was heartbreaking to reflect on what, in comparison, is the oppressive nature of the typical U.S. school. At this school, kids play freely and enjoy themselves, classes are engaging and fun, and no one has to scream at them or say “shhhh.”

This is not because the students are better behaved than our students, however. We noticed that oftentimes, negative behaviors were simply ignored. Teachers generally do not stop the flow of a lesson because one or few students are misbehaving or not paying attention. We asked Dr. Takahashi about this and he told us that teachers make a conscious effort to do this and. Whereas in the U.S., student behavior is often seen apart from academics, Japanese teachers see it as part of academics and through their many lesson study experiences, teachers often discuss student behaviors and how to best deal with them, especially at this particular school. The result is that there is a very light feeling to the school and even teachers seem to be genuinely happy in their interactions with students, each other, and administrators. The principal of the school, Dr. Fujii, does not have a top down management style but very much sees teacher professional development as a long-term process in which he is intimately involved.

This leads me to think about one research lesson in particular that was taught to 6th grade students by Mr. Yamada on division of fractions. In Japan, division of fractions is often taught in the context of word problems such as, “With ¾ dl of paint you can paint 2/5 square meters of boards, how much board can you paint with 1 dl of paint?” (see Tokyo Shoseki Mathematics for Elementary School 6A p. 17, available at http://www.globaledresources.com.) Mr. Yamada, however, gave students the problem 9/20 divided by 3/5 as well as the answer, ¾ based on what students had learned previously by changing the fractions to decimals and dividing to get 0.45/0.6=0.75=3/4. Then students thought about how to get this answer, ¾, directly by calculation of fractions.

Students quickly noticed that you can just divide the numerators and denominators to get the answer but were not sure if this worked in every case. One student suggested the problem 9/30 divided by 3/5 as one case. Although the teacher had planned on using ¾ divided by 2/5, he made the decision to go with this problem the student suggested and students tried to solve it to see if there was another way. Many students noticed that you could invert the dividend and multiply but they were unable to explain why you can do this very well. To make a long story short, this took the lesson in a very different direction and the teacher was unable to achieve the main goal of the lesson.

Afterwards, the teacher sat down to discuss the lesson with a panel of observers. (There were about 60 observers in all, including much of the school faculty and invited knowledgeable others.) what really shocked us was the candor of the discussion. Many teachers on the panel openly questioned the teacher’s decisions to not use a word problem, to accept the problem suggested by the student, and to not use either diagrams or manipulatives. The comments were very direct and it amounted to a severe two-hour grilling for this particular teacher where he was forced to question almost everything he did in the course of the lesson—very different from the general niceness we have witnessed in much U.S. lesson study discussions.

At the end, comments were made by the invited knowledgeable other—Dr. Nakamura of Tokyo Gagukei University. He said that often word problems are not necessary because in this case, painting boards is not really something kids can relate to. He said that these problems were used, however, so students could think about the proportional relationship, which is essential to understanding why we invert and multiply.

After the discussion, we all went out for an “enkai” (big party) with almost the entire school staff. We really learned the importance of these large parties because as everyone ate and drank together, we witnessed much caring, sharing and affirmation, especially for Mr. Yamada (who, by the way, as the teacher who taught the lesson did not have to pay). At one point, Mr. Yamada stood up and told everyone not to feel sorry for him because of the grilling he received at the post-lesson discussion. He said that this was not for him but for his students and that he appreciated his colleagues’ frankness because their intent was to help him to become a better teacher. Wow! It was very moving, to say the least.

There is so much more to say about how wonderful the visit to the Takehaya school was and all that we learned. But what sticks out the most to me are those words from Mr. Yamada. If we don’t open ourselves to such critique, how can we ever improve what we are doing? Thank you Mr. Yamada for teaching all of us a very valuable lesson.

Dewa Mata (bye),

]]>__Summary of Dr. Takahashi’s Presentation__:

Both the National Council of Teachers of Mathematics (NCTM) and the Japanese Course of Study (COS) emphasize teaching mathematics *through* problem solving. This means that important mathematical concepts and skills are presented and taught in a problem-solving context. This is very different than teaching about (or for) problem-solving, as is usually done in most U.S. curricula and textbooks which teach problem-solving as a separate skill. To develop and improve this type of mathematics teaching, Japanese teachers engage in lesson study.

In a lesson study cycle, teachers collaboratively plan, teach, and reflect on an actual classroom lesson. This should not be understood as just teaching one lesson, however. Teachers plan an entire unit and several lessons are taught prior to the research lesson. Also, they do not plan the lesson from scratch. Textbooks, teacher’s guides, previous lesson studies, and other materials are carefully consulted.

A lesson study group involves the planning team that plans and teaches the lesson as well as subgroup members who help observe and discuss the research lesson. There are also often participants from outside the school who also act as observers and discussants. While traditional professional development for teachers usually begins with an answer, lesson study begins with a question. Traditional professional development can be likened to a cooking show. Many people watch the show but few actually try the recipe. Even those who actually try it, often run into problems since they are working alone, may have to substitute hard-to-find ingredients, etc. Lesson study is more like a cooking group that tries the recipe and then collaboratively thinks about how to modify and improve upon it.

There are different types of lesson study. The most common type is school-based lesson study. In school-based lesson study, teachers from within one school work on an over-arching goal for 5 to 7 years. Its purpose is to achieve systematic improvement, consistent instruction, and common vision at the school. We will see this type of lesson study at Takehaya Elementary and Lower Secondary (middle) School in Tokyo, which is affiliated with Tokyo Gagukei University. This school provides professional development activities for teachers and student teachers, and often teachers from outside come to the school for several months to learn. Every month, the entire faculty participates in observing and discussing research lessons in different subject areas.

We will also see this type of lesson study at Narimasu Elementary School in Tokyo. This school is a designated research school. In Japan, each prefecture (and sometimes each city within the prefecture) has one of these schools, which receive government grants to investigate new directions in curriculum and instruction. This particular school is studying achievement level grouping. They give students a pre-test at the beginning of each unit and based on the results, divide them into 3 groups. This is different from ability level grouping which tracks students into ability level groups for the entire year.

Another type of lesson study is district-wide lesson study. In this type of lesson study, teachers from across schools plan research lessons in different subject area groups to achieve a district-wide goal. Its purpose is to develop communication and exchange ideas among schools, and improve teaching and learning in the district as a whole. University professors are usually invited to act as knowledgeable others for the research lessons they teach.

We will observe this type of lesson study at two different schools—Ishida Elementary School, in Yamanashi Prefecture (near Mt. Fuji), and Kyuden Elementary School in the Setagaya Ward of Tokyo. The latter school provides a half-day, district-wide professional development day every month where teachers from throughout the district come to the school to observe research lessons. Each month, a different subject area group teaches research lessons. Since there are so many observers (100 or more), the lesson are discussed by a selected panel of discussants while the others observe and learn from the discussion.

We had the afternoon free and many participants went shopping for souvenirs, electronics, and books, some went to the Kabuki theater, and some went exploring Tokyo. In the evening, we all shared a wonderful Kobe beef sukiaki dinner at a wonderful restaurant. The food is so good in Japan, I think the author of this blog may have to go on a diet after this trip!

Ja-ne (see you later),

Bill Jackson

Tokyo, June 24, 2007

]]>Today was our first full day in Japan and to take advantage of the effects of the jet lag (almost everyone was wide awake at around 2:00 or 3:00 AM), we went to the Tsukiji fish market at 5:00 AM on a chartered bus. Our very helpful (and funny) guide, Yoko-san, told us that this is the largest fish market in the world. Fortunately, we got there in time to see the tuna auction where buyers bid on huge blue fin tuna that fetch up to $20,000.00 a piece. We also saw so many different kinds of fish and sea creatures. We were all so impressed by how clean the fish market was, not a single fly!

Seeing so much fish made us all very hungry so afterwards, we went back to the hotel for breakfast where we could choose from a typical Japanese breakfast or an American breakfast. Many of us chose the Japanese breakfast, which consists of miso soup, rice, grilled fish, pickled vegetables, fresh fruit and other delicacies. I heard that the American-style breakfast was also very good.

At 9:00 AM, we all went on a sightseeing tour of Tokyo, beginning with the beautiful Asakusa Kannon Temple where we saw and learned about Japanese religious customs, and even witnessed a bride and groom coming to the shrine for a traditional Japanese-style wedding. Afterwards, we went to the Imperial Palace, which is where the Emperor of Japan, Akihito, and his wife live with 1000 servants. We were only allowed to view the impressive 280-acre compound from the outside (visitors are only allowed inside twice a year on special holidays), but we learned many interesting facts about the history and culture of the Japan.

After the sightseeing tour, we had lunch at a Kusheage restaurant where we ate skewers of battered, deep-fried fish, meat and vegetables. It was absolutely delicious! After lunch, we all had the afternoon and evening free. Many people went to museums, parks, shopping, or just walked around to take in the multitude of sights and sounds of Tokyo. All in all, it was a wonderful beginning to our trip! Now, we are really looking forward to our school visits where we will be observing classes, eating lunch with students, and observing and discussing research lessons.

Sayonara from Japan,

Bill Jackson

Tokyo, June 23, 2007

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