**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

*Studying Classroom Teaching as a Medium for Professional Development: Proceedings of a U.S.-Japan Workshop*. Washington, DC: The National Academies Press. doi: 10.17226/10289.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

*Studying Classroom Teaching as a Medium for Professional Development: Proceedings of a U.S.-Japan Workshop*. Washington, DC: The National Academies Press. doi: 10.17226/10289.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

*Studying Classroom Teaching as a Medium for Professional Development: Proceedings of a U.S.-Japan Workshop*. Washington, DC: The National Academies Press. doi: 10.17226/10289.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix F: A Study Lesson: Large Numbers at Fourth Grade." National Research Council. 2002.

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Background Lesson Plan Hiroshi Nakano Lesson Plan Transcript Blackboard , Teacher, Tokyo Gak~gei University, Setagaya Elementary School Postlesson Discussion Transcript 178 ~3 191 205 207

BACKGROUND Japanese Counting System The Japanese counting system is similar to the system used in the United States. The Japanese numbers are written like U.S. numbers and, for numbers ranging from 1-9999, are read with a very similar number-naming scheme. For each place value, there is an indicator of how many in that place. For example, a number like 32 would be read as san jug ni which translates literally as three Jim (i.e., tens) two. This is somewhat like "thirty two" where the word "thirty" is used to refer to three tens, and "two" indicates there are two units or ones. The number 532 is go hyaku san jug ni, or five hyaku ~un(lre(ls), three jug (tens), two much like our "five hundred thirty two." This similarity continues through the thousands. So a number like 6532 would be read as rokku sen go hyaku san jug ni, which is six sen (thousands), five hyaku ~un(lre(ls), three jug (tens), two. The American and Japanese number- naming systems, however, (liverge after the number 9999. The U.S. number system goes in blocks of three for unit names: hun(lre(ls, thousands, millions, anti so on. The Japanese number system, however, uses unit names in blocks of four. Whereas an American would read 16532 as "sixteen thousand, five hun(lre(l, thirty two," specifying how many thousands there are (i.e., sixteen), the Japanese specify how many ten thousand there are. The Japanese wor(1 for "ten thousand" is "man". Thus, 16532 is read as ichi man rokku sen go hyaku san jug nit This translates as one man (ten thousand), six sen (thousan(ls), five hyakyu ~un(lre(ls), three jug (tens), two. Just as in English, rather than saying "thousand thousand" we introduce the word "million". In Japanese, rather than saying "man man" (ten thousand ten thousan(ls), the new wor(1 "ok~," meaning "hun(lre(1 million" is use(l. The naming of the next four place values follows the same naming pattern as above, except with the unit oku instead of man. Number Place Value Name (Japanese) Place Value Name (English) ., . Icnl ones 10 jou ten 100 hyaku hundred 1000 sen thousand 10,000 ichi (one) man ten thousand 100,000 jou (ten) man hundred thousand 1,000,000 hyaku thunclrecl) man one million (new place value name) 10,000,000 sen (thousancl) man ten million 100,000,000 ichi (one) oku (new place value name) one hundred million 1,000,000,000 jou (ten) oku one billion (new place value name) 10,000,000,000 hyaku (hundred} oku ten billion 100,000,000,000 sen (thousancl} oku one hundred billion 1,000,000,000,000 ichi (one} cho one trillion (new place value name) APPE N DIX F

Large Numbers at Gracle Four: Problems Coverec! in the Lesson The mathematics objective in the fourth grade was to have students understand and work with large numbers, beginning in this lesson with a particular focus on the unit, okay. In the lesson, students solved four problems that dealt with the quantity or measure of one okay objects. Using a "hint" provided by the teacher, students tried to determine how much one okay would be. Problem 1. What is the height of a stack of 1-ok?z (1 hundred miBion) yen if you use 1-man yen bills? Hint: If you stack 1-man yen bills for lOO-man yen, the height of the bills will be ~ centimeter. Answer: lOO-man centimeters high. On the board: APPE N DIX F

Problem 2. How many classrooms would 1-ok?z (~1 hundred miBion) liters of water fig? Hint: lO-man liters would fill half a classroom. Answer: 500 classrooms. On the board: APPE N DIX

Problem 3. How many Sato-san's (a student named Sato) w'll it take to make 1-ok?z (~1 hundred miBion) kilograms? Hint: 500 Sato-sans weigh I-man kilograms. Answer: 500-man people. On the board: - It 1-man Fig 1 layout Imll 1-oku~ (~700O,000) 500 people 1 1~.111~11. 500-~n peddle APPE N DIX F

Problem 4. How long is a row of 1-oku Daraemon comic books? Hint: If you line up I-man Daraemon comic books in a row, it wall be 150 meters long. Answer: lOO-man meters. On the board: APPE N DIX

LESSON PLAN December 7, 1999 (Tuesday), 5th period Setagaya Elementary School (a primary school attached to Tokyo Gakugei University Education Department} Fourth Grade (class #id 40 Students (M. 20, F. 20} Instructor: Hiroshi Nakano 1. Name of the Unit: Large Numbers 2. The Goal of the Unit: Understanding the structure, how to read, and how to write the large whole numbers that reach up to "Okay (100 million)" and "Cho (~1 trillion)" and deepening the understanding of the decimal positional notation system.) Interest, Desire, and Attitude A. Try to find a better way of thinking (solving) and commonality between two different solutions by presenting own ideas to the class and listening to others' ideas. B. Show an interest in large numbers in everyday life and try to use or try to investigate them. Mathematical Thinking C. Using previously learned knowledge of the number system of the numbers up to one thousand, a student can think about the system of large numbers that reach up to "okay (100 million)" and "cho (~1 trillion)." D. Understanding the relative size of large numbers based on the system of the unit used for numbering. Expression and Manipulation E. Be able to write and read the numbers up to "cho (~1 trillion)." Knowledge and Understanding F. Understand the numbers up to "cho (~1 trillion)", and that the system is based on a decimal positional notation system. 3. About the Unit: Up to now, my students learned numbers up to ~ man (ten thousand) by counting numbers when they were second grade students. During the third grade, they learned the numbers up to 1000 man (10 million) based on previ- ously learned knowledge of the number ~ man (10 thousand). In this unit, the ~ (Translator's note) The numeration system for place value in English and Japanese is different. In English, different numerations are used every three place value. For example, thousand, million, billion, and trillion. In Japanese, different numerations are used every four place values. For example, "Man (10 thousand)," "Oku (100 million)," and "Cho (1 trillion)." APPE N DIX F

study of the size of whole numbers comes to the final stage, and the study range of the large numbers reaches to "oku (100 million)" and "Cho (1 trillion)." As the study of the size of whole numbers approaches this stage, the most important study points of this unit become the students' understanding of the decimal positional notation system, the Japanese numeration system, and the relative size of large numbers. The decimal positional notation system has the following main points: · When numbers reach 10, it is a new unit. · The quantity of each unit is determined by the place of the numbers. · The number "O" is used to show that there is no value in the unit (place). The students were already exposed to these main points when they were in first though third grade. In this unit, the students wall learn that all whole numbers are expressed using the numbers O though 9 based on the previously learned principles mentioned above. Japanese numerations were invented for showing a large number using the small number of units. For example, the unit goes up "ichi (one)," "Jim (ten)," "hyaku hundred)," and "sen (thousand)." Moreover, the cycle of these units is used to show larger numbers combined with the other units "man (10 thou- san(l)", "oku (100 million)", anti "cho (1 trillion)" that are used for every four place values. In this unit the students wall understand the merit of the Japanese numerations as well as recognize the necessity for a unit for large numbers like "oku (100 million)" and "cho (1 trillion)." In a(l(lition, to fostering the students' ability to look at large numbers relatively by thinking about how many ~ man (10 thousand), the need to create large numbers like "oku (100 million)" and "cho (1 trillion)" is an important concept in this unit. The ability to look at numbers relatively means "1 oku (100 million) can be shown as ~ man (10 thousand if we look at the number base(1 on 1-Man unit." Therefore such ability is very important in order to foster a rich sense of numbers among the students. The ability to look at numbers relatively is important in order to deepen the understanding of the decimal system and eventually becomes useful for estimating numbers anti calculating with (leci- mals. Moreover, the image of large numbers such as "oku (100 million)" and "cho (1 trillion)" wall be enriched by looking at numbers relatively. 4. Gui(lelines for Instruction (6 lessons): The first lesson (this lesson): (A) Based on the previously learned knowI- edge of the number system up to 1000 man (10 million), the student wall learn that there are units like "oku (100 million)" and "cho (1 trillion)" above the numbers they previously stu(lie(1 and learn about the mechanism of those large numbers. (B) Think about the size of ~ oku (100 million). The second lesson: Un(lerstan(ling how to write and rea(1 the numbers more than ~ oku (100 million). APPE N AX F

The third lesson: Thinking about how to express whole numbers when the numbers were multiplied by 10, 100, and I/10. The fourth lesson: Recognizing the merits of the decimal positional nota- tion system through understanding that any size of whole numbers can be expressed using the numbers O though 9. The fifth lesson: Calculating addition and subtraction of large numbers. The sixth lesson: Doing problems. 5. About the Learning of This Lesson: The focal point of this lesson is learning about new units like "oku (100 million)" and "cho (1 trillion)." However these units are often used on TV and in books; therefore, the names of units are not foreign concepts to the students. Particularly the unit "oku" is often used in "3 oku yen lottery" in everyday life. Thus the unit "oku" is not really new to the students. Although the students know the word "ok~," ~ suspect that they do not know the fact that the unit is nested in the highly developed Japanese numeration system, and they have not understood the actual quantity of the unit. Therefore ~ considered these circum- stances to develop this lesson. When presented with a number more than ~ oku (100 million), students respond as follows: "we cannot read more than 1000 man (10 million) (because they have not learned it)." Then the teachers often used the response as a problem that the whole class needs to solve. This was a common way to start this lesson. However, as ~ described above, the problem like "we cannot read more than 1000 man (10 million)" is not a good whole class problem. It may be used individually, but it will be not natural to use it as a whole class problem. Therefore, at the introduction of this lesson, ~ decided to check the stu- dents' previously learned knowledge. In other words, ~ decided to check the students' knowledge of the numeration system up to 1000 man (10 million). Sen Hyaku Juu Ichi Sen Hyaku Juu Ichi I Man I Man I Man I Man l l l l By presenting this kind of chart, the students (even the students who do not know the unit "oku") can notice the cycle of "ichi (one)," "jug (ten)," "hyaku (100)," and "sen (1000)," and the new cycle begins after the number reached "sea," and "ichi"comes next. In addition, the students can notice that after "sen man" a new unit wall be needed. In this way, the knowledge that the students learned previously can be used for learning this lesson. By checking the level of understanding of previously learned knowledge and making sure that the students notice the system of the numbers, then we can teach that the new unit that comes after sen man is "oku" and the units "ichi," "jug," "beaks," and "see" form a cycle. ~ believe that this kind of introduction to this lesson wall provide for the different kinds of student needs. The students who did not know the new unit "oku" will learn the necessity of a new unit in order to make numbers more APPE N DIX F

than 1000 man. And the students who already knew the unit "oku" will have a chance to deepen their understanding of the Japanese numeration system and recognize the merit of the system. As ~ mentioned before, many students today are exposed to the unit "oku." However, ~ have doubts that the students actually have a sense of the quantity of ~ oku. (100 million). Therefore, after the students have recognized that the unit "oku" comes after "1000 man" and that the unit is nested in part of the Japanese numeration system, ~ decided to prepare an activity that makes the students actively think about the quantity of the number ~ oku. For example: · How high is a stack of ~ man yen bills if we have ~ oku yen? If we imagine this classroom as a cup to ladle out ~ oku liter of water, how many cups would it be? · How many children do we need in order to have a weight of ~ oku kilogram? These are the kinds of activities that ~ plan to have. Even these quantities that we wall talk about during the activities are often difficult to imagine in reality so we need to imagine in our head. (For example, the quantity of ~ oku liter of water is about 500 classroom "cups".) However, because you cannot actually count, it is very hard to imagine how large population numbers and budget numbers are that are presented as examples in the textbooks. Moreover, just practicing reading and writing such numbers represented in the textbooks wall not help this situation. Therefore, those numbers may become meaningless numbers that the students wall see just on their desks. Even if the students cannot see the actual size of the numbers in front of their eyes, ~ believe that preparing some activities to help them to see the quantity of ~ oku in various ways will help (levelop the students' interest in the number ~ oku. In addition, ~ believe that helping the students see the number ~ oku in more concrete ways as ~ proposed above wall help foster students' understanding of the relative size of numbers. For example, the height of the stack of ~ man yen bills wall be ~ centimeter for 100 man yen. The height of the stack of ~ man yen bills wall be 100 centimeter for ~ okay yen. In this case, the students can imagine the relationship of the two numbers, 100 man and ~ okay (100 times of 100 man is ~ okay) using the example of the thickness of bills. ~ would like to plan this lesson, incorporating the mentioned activities, to support the students discovering the relative size of numbers on their own. 6. This Lesson: (~) The Goal of This Lesson: · Based on the previously learned knowledge of the principles of the Japanese numeration system up to 1000 man (10 million), the students think about what kind of unit names might be logical to use. And the students will learn that the new units such as "okay" and "cho" also follow the principles of the Japanese numeration system. APPE N AX F

· Understanding the relative size of the number ~ okay using various ~ okay things and using the knowledge to solve a problem (the length of ~ okay comic books). · Developing interest in thinking about the quantity of ~ okay. Learning Activities2 Main Hatsumon (question) and students' anticipated reaction. * Evaluation + Points to remember I. Checking the students' previously learned knowledge (concepts) T *How can YOU read the number · ICE," ~ I`, ·~,J~1 ~ 5-a 2-Man 5-Sen identify the unit/position per? 1 2 1 5 1 2 1 5 T 7 T ° 1 6 Hyaku Juu Ichi Sen Hyaku Juu Ichi 1 Man I Man I Man l l l l e anything from this be two cycles of -sen. a~-hyak?~-sen cycle does hing added but the a~-hyak?~-sen cycle has unit. write when we multiply by TO? down a "O" at the end of 5706. clarify the unit/position per? hildren + Tell the students that the rule for adding "O" at the end of a number means multiplying the number by 10 was previously learned knowledge. + Tell the students that they learned the concepts when they were in third gra(le. ~ · ~ AV Y. ~" ~ ~ V ~ 12525706? C: Sen 2-Hyaku 7-Hyaku 6. T: Why(lon'tw' of each num} C: 1 Sen Man T: Di(1 you noti chart? C: The units ha, ichi-juu-hyak, C: The first ichi not have any secon(1 ichi-j: man for each T: What can we this number C: We can write number -1252 T: Why(lon'tw~ of each num] 2 T. Teacher; C, c APPE N AX F

1 1 2 1 5 1 2 1 5 1 7 1 0 T 6 1 0 1 Sen Hyaku Juu Ichi Sen Hyaku Juu Ichi Man Man Man Man 2. Think about the new unit T: Let's think about how we can read the number "I" in the 125257060 based on what we know about the numbers up to 1000 man. C: ~ think the nest cycle of ichi-ju~ hyaku-sen starts so ~ think it has ichi. C: ~ think we wail have a different unit name instead of man. T: The next position of the see-man is read as ichi-oku. * Are the students actively thinking about the new large unit based on what they learne(1 before (the number up to 10 million)? (Listen to what students say.) The teacher will help the students think that bigger numbers also follow a similar pattern. Next position of ichi-oku is juu-ok~, then hyaku-ok~, then sen-oku. Finally help the students to think about the unit of "cho (1 trillion)." Let the students know that the number 125257060 is the number that represents the population of Japan two years ago. 3. To know what quantity the number + ~ oku represents T: The height of a stack of ~ man yen bills for 100 man yen is ~ centimeter. What is the height of a stack of ~ oku yen, if you use ~ man yen bills? C: About ~ meter. T: If we have 10 man (100 thousand) liters of water, we can fill half of this classroom. If we have ~ oku liter of water, how many classrooms can we fill? In addition, if a child's weight is 20 kilograms, ~ man kilograms would be500 children. How many children (lo we nee(1 for ~ oku kilograms. Ask if any of the students hear(1 the unit "oku" in everyday life anti relate the conversation to the height of the stack of bills. Ask the students to estimate the answers. Tell the students the answer 500 classrooms anti 500 man children. APPE N AX F

T: Doraemon comic books have sold I-oku up to now. If we lined up ~ man (10 thousand) books, the row wall be 150 meters long. How long would the row be if we lined up ~ oku books? + Ask students to think about the answer based on what we discovered through working on the problems before. 4. Think about the length of the book on their own C: In the case of the bill problem, it was * ~ centimeter for 100 man yen and 100 centimeters for ~ oku yen. So ~ oku is 100 times bigger than 100 man. Therefore, we can find the answer by calculating 150 x 100 x 100. C: In the case of the children's weight problem, ~ man kilograms is equal to 500 children, and ~ oku kilogram is equal to 500 man children. So, ~ oku is ~ man times of ~ man. Thus, we can find the answer by 150 x 10000. C: ~ man comic books equals 150 meters. 10 man books is 10 times it so 1500 meters. 100 man books is 10 times so it is 15000 meters. Then ~ continue to do the calculation by multiplying by 10 in each step. Ask students to write down the reasons for their answer. Are the students solving the problem using the relative quantity of ~ oku? (Look at students' notebooks.) 5. Presenting the solution T: Please explain the solution carefully. T: The length of the comic book wall be 1500 kilometers. APPE N DIX F When the numbers are gathered and form 10, a new unit is formed. So when the position goes up one, then the number was multiplied by 10. Make sure all students understand this concept. + Make sure the students un(lerstan the relative quantity of ~ oku by helping the students to un(lerstan each solution method. * Di(1 the students (1evelop an interest in the number ~ oku? (Look at students' notebooks.)

6. Summarizing today's lesson T: Please write down your thoughts about the number ~ oku in your notebook. C: ~ understood how large the number ~ oku is. C: ~ would like to think about the size of ~ oku using other things. 7. Evaluation · Did the students understand the necessity for new units based on the numeration system of the numbers up to 1000 man? · Did the students figure out the length of ~ oku comic books using the acquired knowledge of relative quantity for the number ~ oku? · Did the students develop interest in the number ~ oku? APPE N AX

LESSON PLAN TRANSCRIPT A Stucly Lesson: Large Numbers in the Fourth Gracle December 7, ]999 Tokyo Gakugei University, Setagaya Elementary School Teacher: Hiroshi Nakano Nakano-sensei3 began the lesson by reviewing what students had learned in earlier grades about place value and large numbers up to seven places. The students briefly discussed how the Japanese place value system is grouped in sets of four. This is different than the U.S. system in which a new unit name is introduced every three places: hundredths, thousandths, millions, etc. A description of the Japanese counting system is included at the beginning of this appendix. After about eight minutes, Nakano-sensei was ready to move to the main subject of the lesson, I-oku or 100,000,000. On the blackboard was written: ; ~ ,7 0 ~ ~ Teacher: Well, I would like to write the name of units just like we did before: ichi (ones), juu (tens), hyaku "hundreds), sen (thousands). Just like Matsumoto-kun said before, this is a (cycle of) units that goes around one more time. Ichi (ones), juu (tens), hyaku (hundreds), sen (thousands), and these units have the unit man (ten-thousands). 3 sensei means teacher APPE N DIX F

S'70 6 ~ S ~ S S Teacher: Now, you can see that the number shikeJ one position so a number sticks out further than the units we studied previously. The next step is to think about how we can read the number. It is going up like ichi (ones), juu (tens), hyaku "hundreds), sen (thou- sands)- lchi (ones), juu (tens), hyaku (hundreds), sen (thousands) - like what Matsumoto-kun said. Tanaka-san was saying the same thing. What Jo you think the next unit will beg Students: Yes, yes. Teacher: What is your guess Yes, please. Student: Ichi (ones). Teacher: Is that rights Students: Yes, yes. Student: It is "ichi (one)-something/' number. Teacher: That's right. I think maybe the unit one-something comes next. Well, this figure became strange. Students: Slaughters Teacher: This becomes ichi (one)-something. If this is ichi (one)-something, does that mean you can't put (the label) man (ten-thousands) herein Yes, yes. Whys We used man (ten-thousands) before. Yes, Otsuka-kun. Students: Teacher: Student: Teacher: APPE N DIX F

Otsuka: The unit man (ten-thousands) is used for the last cycle of numbers so- Teacher: Oh, I see. We used it here. Otsuka: So the next unit will use a different name. It will be next... aher 9 sen man (9000 man), so it will not be man (ten- thousands). Teacher: So it is not man (ten-thousands). It is not man. Yes, Shirai-kun. Shirai: In addition to what Otsuka-kun said, if you put the unit man again here (where the 1 is)- Then it will become ichi-man man (ten-thousand man). And the position of the unit goes back to the man position. So it is not man. Teacher: I see, I think it is okay to call it ichi-man man (ten-thousand man) Student: That sounds like a name of a food. Students: Slaughters Teacher: Ichi man of ichi man and ichi-man man. I think it sounds good. But So we need to have a different name for the unit that replaces man, Jon't wed I think you already have heard this many times before... that is... Students: Oku thundred-m il l ions) . Teacher: When I asked you a couple of days ago, there were few who had never heard the word oku. You must have heard the word somewhere before, haven't your Bitt the word eke veneers in this sih~otion . ~ - i- err -a Well, this is the first time for us to learn oku formally and- because there were two students who did not know oku when I asked before this is called ichi-oku (1 hundred-million). If we write ichi-oku (,1 hundred-million) using numbers, it becomeslike this and we have One, two, three, four, five, six, seven, eight. Eight zeros. Students: Wow, wow... There are a lot of them. Teacher: This is something we are going to learn in the fourth grade. Nakano-sensei spent a few more minutes discussing the new unit, okay, then shifted to the problems on which the students would be working. Teacher: Well, today, we would like to study the number 1 oku {,1 hundred-million) further. So we will have a quiz competition called, "How much is 1 oku (,1 hun- JreJ-million) quiz competition." Students: Yea! Writing: How much is 1-oku {,1 hundred-million) quiz competition. Teacher: Oh, somebody said something really good. APPE N DIX F

Somebody was talking about 1 -oku {,1 hundred-million) yen. I have a 1 -man yen bill here. Students: WowI want to have it. I want to have it Teacher: Well this 1 -man yen bill. Students: I want it. I want to have it Teacher: What would be the height of a stack of 1 -oku {,1 hundred-million) yen if we used 1 -man yen bills This is the first question. Student: Yes. Teacher: Yes, Shirai-kun please. Shirai: I think it reaches up to the height of the blackboard. Teacher: I see. It will be from the bottom to the top of this blackboard. Yes, Matsumoto-kun. Matsumoto: About 2 meters. Teacher: The height 2 meters is a little bit taller than me so about this tall. Okay, I will give you a hint. I wish I can take out 1 OO-man yen right now, but I don't have that kind of money. Students: Slaughters Student: If that's the case, you can sell your house. Students: Slaughters Students: You can't Jo that so easily. Where could he live if he sold his housed Teacher: A stack of 1 OO-man yen. If you stack 1 -man yen bills for 1 OO-man yen, the height of the bills will be 1 centimeter. Students: I got itI got it Student: Why does it become 1 centimeters Teacher: I wish I can take out real bills for 1 OO-man yen as evidence. If I stack 1 -man yen like this, the height will be 1 centimeter. So now, how about 1-oku (,1 hundred-million) yens Student: I can't see the board. Teacher: Yes, please. Student: I think it is about 10 centimeters. Teacher: About 10 centimeters. So it is about this much. Okay, Keita-kun. Keita: 1 meter. Teacher: 1 meter. I will tell you the answer. The answer is- For the height of 1 -oku (,1 hundred-million) yen is- 1 00 centimeters. APPE N DIX F

m:-~ yak . Blat COCOA=) f - ~ i~ JO Students: Yes, yes !!! Teacher: It is 1 meter. The height of 1 -oku (1 hundred-million) yen is 1 meter. The height will become about this much. Are you okay with twist You can talk about the 3-oku (3 hundred-million) yen lottery later. Have you noticed something by looking at twist When you move from 1 OO-man yen to 1 -oku (1 hundred-million) yen 1 00 times. I heard somebody say something good. 1 00 times. That's right, isn't its The height is 100 times as big. That means the value is also 100 times as big as, isn't its You said something really good. Shintaro-kun, you are very sharp. So the value is also 100 times bigger. Okay, now Okay, we will go on to the second question. Yea! It is similar to a mathematical principle we learned before. If the answer was 100 times the original one, the number that is multiplied is also 100 times. Student: Teacher: Student: Teacher: Shintaro: Teacher: Students: Student: APPE N DIX F

Teacher: I see. It is one of the principles of calculation. Student: Yes, we did it long time ago. l~i~-~ ye I J ` ~ 1 IFS UPON f - ~ i~ Jo4~m JO ~ ~ F. "F ( F ~= ~ ~ ) 1 fin The class moved on to the next problem: How many classrooms would 1-oka~ (1 hundred million) liters of water fill? The students were given the hint that lO-man (1 hun(lre(1 thousand liters would fill half a classroom. Students suggested possible answers. After four minutes, Nakano-sensei stated that 500 classrooms filled with water would be approximately ~ okay liters about 1000 times the given hint. APPE N DIX

Teacher: I see. It is one of the principles of calculation. Student: Yes, we did it long time ago. l~i~-~ ye I J ` ~ 1 IFS UPON f - ~ i~ Jo4~m JO ~ ~ F. "F ( F ~= ~ ~ ) 1 fin The class moved on to the next problem: How many classrooms would 1-oka~ (1 hundred million) liters of water fill? The students were given the hint that lO-man (1 hun(lre(1 thousand liters would fill half a classroom. Students suggested possible answers. After four minutes, Nakano-sensei stated that 500 classrooms filled with water would be approximately ~ okay liters about 1000 times the given hint. APPE N DIX

l ]~ ~~> 10° Bare 1-~ =_~- Nakano-sensei recorded the answer on the board, then moved on to the final question. Teacher: Students: Teacher: Students: Teacher: Okay, now we will go on to the last problem. Yea! Excuse me for changing the subject but, when I was listening to the radio the other day, (laughter) the radio said that it has been 30 years since Doraemon was born. It said that the Doraemon comic book series reached sales of 1 -oku (,1 hundred-million) books. Students: REALLY!! Teacher: It is said that 1 -oku {,1 hundred-million) copies of Doraemon books were sold in the last 30 years. Why Jon't we solve a problem on Doraemon for the celebration of it's 30th anniversary Students: Yea! Student: Even if it is a Doraemon problem, it is a mathematics problem. APPE N DIX

Student: I don't know what will happen. Teacher: Well, a person lined up the Doraemon books like this: 1, 2,... Lined up... If you lined up the books like this for 1 -okut1 hundred-million) books, how long will the row of books beg Student: It might be 1-okut1 hundred-million) centimeters Teacher: How about your What did you says Student: It must be about 1-okut1 hundred-million) centimeters. Teacher: I see. It is 1 -okut1 hundred-million) centimeters. How far can we go if we said 1 -oku centimeters Student: I think it is about the size of this classroom. Teacher: You mean if you line the books this ways I see. Well, I will give you a hint, because if you think about the problem as it is, you may not go forward. Actually, a person actually lined up some books Students: Wow Teacher: It said that the person lined up 1-man books. Student: Wow, for 1-man books Teacher: If 1 -man books are lined up for 1-man books it is 150 meters. 1~o~ J- let 10m Students: Wow Teacher: It is about the length of this school playground. So it is from here to about Mt. Donguri. Could you wait for a seconds Well, how about 1-okut1 hundred-million) books What kind of length IS that going to beg Could Hashimoto-kun wait for a seconds Because everybody needs to listen a little bit more. If you use these things, if you use these things, you can figure out what the length for 1 -okut1 hundred-million) books would be, can't your In your notebook, please write down the expression. What would the expression beg APPE N DIX F .

And write down the reasons for the expression. While thinking about these things, please write out how long the row of oku Doraemon books will be. The students worked independently while Nakano-sensei walked among them, looking at their work and stopping to talk briefly with some about their solutions. The observers from the lesson study team also moved among the students, watching how they approached the problem and taking notes. After approximately six minutes, Nakano-sensei called the class together. Teacher: Who could solve this problems Well, first... I would like you to tell me the expression. Well, I want somebody to tell me the expression. How about Shirai-kun~ Shirai: Well, 150 times 10 man. Teacher: 150 times 10 man. One, two, three, four, five (counting zeros). So this is good. And what is the answers Shirai: The answer is Well Teacher: Okay. How many zeros aher 152 Shirai: There are six zeros. Teacher: Six. One, two, three, four, five, six. Student: Ichi (ones), juu (tens), hyaku (hundreds), sen (thousands), man (ten- thousands), juu-man (ten man), Teacher: Ichi (ones), juu (tens), hyaku (hundreds), sen (thousands), man (ten- thousands) Jyu-man (ten man), hyaku-man "hundred man), see-man (thousand man). 1500 man. Is it okays So the answer is Student: The answer is 1 500-man meters. /~x /~ = I A ~ fly -~0 Teacher: Yes. Yes, this is one way to solve this problem. Is there any other way to solve this problems APPE N DIX F

Teacher: Other way... Well, who should I asks How about Matsumoto-kun~ Matsumoto: I Jid it a different way, but my answer is a little bit different. Teacher: I see your answer is a little different. Matsumoto: My answer is different from Shirai-kun's. Teacher: Yes. Matsumoto: 150timesparenthesisl okut,1 hundred-million)dividedbyl man... Teacher: 150 times 1 oku {,1 hundred-million)... One, two, three, four, five, six, seven, eight, (counting zeros) and divided by... Matsumoto: 1 man. Teacher: 1 man. And thence Matsumoto: Equals...150 man. fame Student: Yes, the answer became the same. Teacher: The answers Student: Okay! Teacher: Are there any other expressions Or something you want to adds Yes, well Excuse me. Who are your Hayashi-kun. Hayashi: The answer is Teacher: Could you come to the blackboards I think it is better to copy your note- book. Hayashi-kun's solution is kind of difficult to understand. Hayashi-kun's Then... This one became... 1 50-man meters... And this became 1500 kilometers. APPE N DIX F

! t~~ ~ 1~ - fly ~ fat I- Teacher: Okay. Are there any other expressions Yeses 4 - A = into One other student, Maho-san, shared her solution with the class. Nakano-sensei wrote her equation on the board ~ . ~ fork /0~o = /~ _ _ ~ Teacher: Well, now Matsumoto-kun, Hayashi-kun, and Maho-san got 1 50 man, and only Shirai-kun's was 1500 man. Well, what is going one Could Shirai-kun explain in front of the class how you thought about your solutions Okay, everybody. Please put your pencils down and listen to your friend's Shirai: explanation. From here, these zeros are increasing one by one. And this one is also times 10 so... APPE N DIX F

150 times 10 man equals 1500 man. This is how I thought about it. Do you have any questions Teacher: If you Jon't have any questions, Joes that mean it's okay with your Did you understand Shirai-kun's explanations Okay, you can ask. Student: Could you explain it one more timed Teacher: Could you explain it slowly Shirai: From here, the number of zeros increased one by one so Teacher: Wait a second, so this one has So this one has two zeros (times 1001. Shirai & Teacher: Three zeros. Shirai & Teacher: Four zeros. Teacher: So you thought that this must have 5 zeros. That's what you thought. Oh, I see. Then Then- 150 times 1 0-man equals Students: Teacher: Shirai: 1 500 man. Teacher: Yes, that is wonderful, isn't its I see you used the pattern that these zeros increase one by one. I see. So Jo you think this is a right answers Student: I am trying to read the teacher's face (expression). Teacher: How about Tanabe-san~ Tanabe: But if you look at these, the zeros are decreasing one by one. But 1 -man kilograms and 1 -man books are both 1 man so I think it should be 10,000 times, just like this. Teacher: Well, could you explain a little mores Tanabe: This is 100 man and this one is 10 man and this one is 1 man. And these · · , zeros are Increasing one lay one. And 1-man kilograms and 1-man books both have 1 man in it, so I think the number 10,000 times should also be the same. Shirai: Teacher: ~ see. If we Jon't say anything addressing what Shirai-kun said, then this lesson will end with Shirai-kun's solution. Shirai-kun. Well, just like Maho-san said The units used, books and kilograms, are fine because the number is the same. What do you think about twist Shirai-kun, were you convinced with Maho-san's explanations Well... The units used for weight and length are different. So I think the 1 man times becomes 10 man times. I see, it becomes 1 man times to 10 man times. I see. Teacher: Student: This is a tricky problem that doesn't have an easy answer. APPE N DIX F

Although the zeros increase by one, it shouldn't be that easy like 10 man because it's a continuation of the increase. Student: I think what Shirai-kun saiJ the pattern of the increasing zeros was just speculation. So I think, the reasons...because of the pattern of the increasing zeros or because the units are different...are not right. Student: When we started this lesson, nobody told us there is a pattern like that and the increasing pattern is not a rule. So it is his own convenient reason. Teacher: So what Jo you think, Shirai-kun~ Some said, "Your own convenient reason" or something like that or it's too much to say "because I see the pattern here./' However, the reason that somebody gave "The numbers are the same even though the units are different./' What Jo you think about this statements Teacher: Well, we are going over the time limit by a lot but We will end this lesson with tentative agreement that the answer might be lOmantimes. Then during tomorrow's lesson, we will look at Matsumoto-kun, Hayashi- kun, and Maho-san's answers, and think about how long a row will be if we had 1 -oku Doraemon books. Nakano-sensei ended the class by asking students to write down their thoughts about the lesson in their notebooks and to turn them in. APPE N AX F

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POSTLESSON DISCUSSION TRANSCRIPT December 7, ]999 Monitor: Let's start the postlesson discussion with the teacher's (who taught the lesson) self- reflection about the lesson. Teacher: In teaching this unit ("Large Numbers"), ~ wanted to make this lesson help the stu- dents foster the image of the size of large numbers like oku (100 million)" and "cho (1 trillion)." By preparing the lesson in this way, ~ thought that the students can be inter- ested in a number like "oku", and they will have a desire to investigate it on their own. However, the actual lesson did not go as well as ~ had thought. By having the students work on several problems to help their thinking about the relative size of the number ~ oku (100 million) using a quiz show format, ~ thought they could use the newly acquired concept (the relative size of numbers) and be able to solve the Doraemon problem (comic book problem). However, in the actual lesson, the students could not see the merit of using the concept of the relative size of the number ~ oku. Particularly the concept that ~ oku is ~ man times ~ man. Monitor: Please ask questions if you have any. A . When did the students learn that, "if you multiply a number by 10 then the number will have "O" at the end of the number (in the one's position)," that is necessary for this lesson? Teacher: In our school's curriculum, this unit comes after "Multiplication of Large Numbers." noticed that some textbooks use the opposite order. In the unit of "Multiplication of Large Numbers" the students learned the multiplication of numbers like "200 x 30" and they learned the concept of adding "O" at the end of the number. B: ~ notice that you did not write down the concept of adding "O" when a number is multiplied by 10 on the blackboard. Do you have any reason for not doing that? Teacher: ~ thought ~ would write down the concept when one of the students mentioned some- thing like "because the number was multiplied by 1000 we need to add three zeros at the end of the number." Ito: Is your definition of "image of ~ oku" similar to the definition "1 oku is a very large number?" APPE N DIX F

Teacher: ~ was thinking about the definition and the relative size of the number ~ oku which is something like ~ oku is ~ man times ~ man. Monitor: C: If you have any opinions please tell us. I question whether the students deepened their understanding of the concepts of the relative size of large numbers from today's lesson. For example, when the students were working on the Draemon comic book problem, the students did not seem to realize the size difference of 10 man times ~ man and ~ man times ~ man. ~ thought the students were thinking as if one of the numbers has one more "O" than others. ~ thought they are not really conceptualizing the size of these large numbers. Therefore, the conversation between the teacher and the students was more focused on the "trickiness" of the numbers rather than the actual size. ~ thought the teacher should have used more visually convincing materials to show the size to the students. For example, the teacher could use something like a number line. D: When ~ teach this lesson, ~ try to convince the students how large ~ oku is by creating materials that wall help the students to visualize the size of the number. Thus, ~ use a 1- millimeter grid for the lesson. ~ use a ~ millimeter by ~ millimeter square and have students think about how large the figure wall be if we have 1-oku (100 million) squares. If we do it this way, ~ oku becomes a ~ meter by 10 meter rectangle. So while ~ was watching the lesson, ~ thought that if the students can not conceptualize the size of ~ okay, it is very difficult for the students to know the relative size of ~ oku. ~ know that you cannot bring actual I-man yen bills to show the height differences between lOO-man yen and I-oku yen, but the teacher needed to show the height of both cases so that the students feel how much these numbers are different. C: ~ understand your suggestion, but ~ thought showing actual number size differences and letting the students experience it is the content of the third grade curriculum. ~ think in the fourth grade it is much important to acquire the concept of the relative size of the large numbers, and the focus of the lesson should be on that. Ito: The concept of relative size of the large numbers is a really important concept the students need to learn in the fourth grade. However, some aspects of learning large numbers overlap in the thir(1 anti fourth gra(les. Therefore, based on the numeration of whole numbers, the students need to see how large the numbers are by using different types of quantities (i.e., length, volume, weight, etch. If the students do not have an image for how large the large numbers with different units are, ~ think it is a very dangerous situation. The scenario to have the students acquire such an image of large APPE N AX F

numbers should not occur by the teacher telling them. Instead, it is better for the students to think about the image. For example, the teacher can ask the students "please think about the number by using something you can imagine." In addition, when we use large numbers to show the size of large objects, we usually used a larger measurement unit in order to show its quantity. For example, we use the measurement unit kilometers when we talk about a very long distance instead of meters. When we teach this lesson, we should also think that we need to be consistent, just like teaching units of measurements. When we talk about how far the sun is from the earth, even if we used the unit of kilometers it will be a very large number. ~ hope that we can make the students understand that the numbers that they are learning now are very large numbers. F·~e UJ11: ~ thought if the teacher went over the relationship of numbers such as "Coke liter equals 500 classrooms" and "Coke kilogram equals 500-man children" then the students could gain a better image of the number ~ oku. They should have a chance to talk about something like "if we have 500 classrooms, we can not fit all the classrooms in our school property." If the teacher and the students could have such conversations and had gone over those problems more carefully, ~ thought the students could feel that the number oku is a very large number. Shimizu: ~ think the students usually do not have a good concept of large numbers like these. think there is a tendency for students usually to think that ~ oku and 10 oku are not so different in size. ~ thought today's lesson had a good exercise for the students in esti- mating the size of the large numbers by changing the yardstick of the unit (e.g., looking at I-oku kilogram by 20 kilogram of a child). The lesson went from length, volume, weight, and back to length when the students worked on the Draemaon (comic book) problem. ~ thought that if the teacher related the two length problems it might be more effective for the students' understanding. Ito: ~ thought the lesson went too fast. ~ thought the teacher could go over the concept if the number is multiplied by 1000, then the resulting number has three zeros added. E: Up to the Draemon problem, the explanation of each problem was something like "because ~ centimeter became 100 centimeters is why it is 100 times; therefore, ~ oku is 100 man's 100 times." However, when the students needed to work on the Draemon problem, the students needed to think about ~ oku is ~ man times ~ man first, then multiply the 150 meter by ~ man. In other words, the students needed to know how many times ~ man is in ~ oku in order to solve this problem. Therefore, the teacher nee(le(1 to gui(le the students to think ~ oku equals how many times for specifie numbers. APPE N DIX F

F·~e UJll: ~ noticed the same thing. Up to the Doraemon problem all the explanations for each problem were: "1 centimeter became 100 centimeter so it is 100 times" and "A half of classroom became 500 classrooms so it is 1000 times." ~ thought the explanation of each problem should be: "Because from 100 man to ~ oku is 100 times so ~ centimeter is also multiplied by 100 to become 100 centimeters." Ito: Because the first three problems were in a quiz show format, ~ guess the response to the problems were "right" or "wrong," but ~ think the situation of "1 centimeter became 100 centimeters so it is 100 times" should have been gone over carefully by identifing how many zeros we needed to add and thinking about the relation of 100 man and ~ oku. Teacher: ~ thought my students could understand 100 times by looking at the relationship between ~ centimeter and 100 centimeters. F·~e UJll: The explanation used in the lesson was that way, but actually the 100 times should be found in the relation between 100 man and ~ oku. B: ~ have a comment on the students' answers for those problems. When the students were working on the second problem that used liters, one of the students said "~-oku liters equals 12 classrooms" although the students worked on the first problem, ~ oku is 100 times 100 man. ~ thought the student's response was clearly a wrong guess. The teacher mentioned that he wanted to do the first couple of problems in a quiz show style, but does he really think that it is okay for the students to guess the answers. ~ think guessing without thinking is not good especially after the teacher provided a hint for solving the problem. Teacher: In my mind it was more like a guessing quiz show style until the second problem that used liters. ~ wanted to discuss the problem as a mathematics problem so some of the students start to think about how they can find the answer by calculating. C: If ~ were a student in the classroom, ~ would like to know the size of 1-oku liter by relating to the I-liter container that the teacher showed in the classroom. The teacher gave a half of the volume of the classroom as a hint to think about the problem, but in my mind ~ felt like the hint did not come from any relation to the container. ~ thought the same way for the weight problem. ~ thought the students wanted to know the relation- ship between ~ kilogram and 1-oku kilograms, and the hint of a chil(l's weight was introduce as a surprise. APPE N AX F

Teacher: Because the focus of the lesson was to have the students foster their ability to look at the relative size of the numbers, the numbers that ~ prepared for those hints were artificially chosen numbers. ~ understand what you said, but ~ think there is a value for looking at the relative size of the numbers something like "1 oku equals ~ man ~ man times," even if the hints were somewhat artificially chosen ones. F·~e UJll: ~ think the students can easily imagine the size of 1-oku liters if they are told it is about the size of "500 classrooms" rather than telling them it is "1 oku times 1-liter container." In today's lesson, the teacher was helping the students to understand the relative size of numbers by talking about an absolute size comparison of "~-oku liters equals the size of 500 classrooms" and relative size comparison of "I- oku liters equals 1000 times lO-man litters." ~ thought the teacher was covering these two aspects of the relative size of numbers using "length," "volume," and "weight." B: ~ think it is hard to express a large number like ~ oku if we don't use an absolute comparison. ~ think even you said 1-oku liter. It is hard to express the size without using the size of"pools" and "classrooms." Monitor: Often people express a large number using the height of the Tokyo Tower. Ito: Even in this case ~ thought the teacher could have asked the students, 'what should we use to show how large I-oku liters is? Should we use the size of this classroom which everyone knows very well?" ~ wanted to have a better flow for the lesson by having the teacher include a conversation like this. A. . In this lesson, "1 oku is 100 man times 100 man" was talked about as one of the quiz style problems. ~ think the teacher needed to go over these explanations carefully, using a chart (the chart shows each position (unit) of the number). Using the chart the teacher could explain, "because the numbers moved two positions on the chart, the number was multiplied by 100." ~ think such clear explanations could help the students be able to solve the problem such as the Draemon problem. Although the students had already learned some parts of concept of the decimal positional notation system using the chart in the third grade, ~ thought the content that was introduced in today's lesson was a little bit difficult for the students to learn. ~ thought the content that was taught in today's lesson should have moved to the third lesson of this unit. Learning about the concept that a zero is added because the number was multiplied by 10 should come after the students fully understand the concept of the decimal positioning notation system. APPE N DIX F

Monitor: Finally, ~ wouic! like to ask the advisers to comment on the lesson and close this · ~ c .lscusslon. Shimizu: The lesson style that we often see these clays is the problem solving style (mondai kaiketsu-gata) that goes through the steps of posing a problem, students solve the problem on their own, student presentation and discussion, etc. However, ~ think some lessons can be different from the problem solving style just like tociay's lesson. The focal point of the discussion tociay was "sense (feeling) of quantity (of large numbers)." In tociay's lesson the teacher was trying to foster the students' sense of large numbers. Therefore, ~ thought the teacher needled to go over the clecimal positional notation system using the chart that was mentioned by someone before. In aciclition, the students learned four problems tociay, but ~ thought the teacher needled to go over the problems more carefully and unify the relationship in them. ~ thought the teacher needled to use a visual that shows ~ oku as a center of the focus and use some arrows to inclicate "1000 times 100 man" and "1 man times ~ man." If the teacher hac! a chance to use a diagram, that might help the students' unclerstancling of the relation between the numbers. ·~e FUJll: Because the problems were introclucec! in a quiz show style, ~ thought the lesson cirew the students' attention well. ~ thought the students were thrilled waiting to fief! out the sizes of the number 1-oku which were presented in length, volume, and weight. ~ thought these experiences helnec! foster students' conception of the quantity of large numbers. The differences between what students learn in the thirc! ant! fourth gracle are that thircI-gracle students learn the number as clecimal numbers and the fourth-gracle students learn the numeration system (where different numerations are used every four positions). In other words, the second one shows that "1 man times ~ man" becomes a new unit "ok~," and if we multiply the number by ~ man again, we will get another new unit. ~ thought tociay's lesson was carefully planed to cover the concept well. The numbers "1 man times ~ man" show the characteristic of the number ~ oku well. Anc! the teacher mentioned, this relation in the problem of kilograms en c! Doraemon. How- ever, as some people mentioned the explanation of the relation of the numbers shouic! not be "100 times ~ centimeter equals 100 centimeters," it shouic! be "100 times 100 man equals ~ oku." If the explanation hac! been this way, ~ thought the students couic! have solved the Doraemon problem. Ito: In this unit, the focus of the lesson is usually on learning about the concept of the numeration system, and the lessons tenc! to be not so interesting. ~ think the teacher's effort in preparing a thought out lesson that is more attractive for the students can be praised highly. ~ think these kincis of lessons need to be more fun. However, this unit introduces not only the numeration system but also the clecimal positional notation system. The merit of the clecimal positional notation system is that the position of a number moves up one, which means the number was multipliec! by 10. ~ thought that if APPE N AX F

the teacher could go over this in the decimal positional notation system where the number was multiplied by 100 and 1000, the lesson could have been better. ~ noticed that there was a student who wanted to show 150 man meters using kilome- ters. ~ thought the student's effort was important. Moreover, ~ hope that the teacher also spends some time to help visualize the size of the number used in kilometers. In many cases, students do not know how far apart Tokyo is from Osaka. ~ think using such numbers for the two cities could provide a clearer image of large numbers. If the teacher could present something that the students could use to 'feel' how large a number is, ~ thought the lesson would have been better. APPE N DIX F

the teacher could go over this in the decimal positional notation system where the number was multiplied by 100 and 1000, the lesson could have been better. ~ noticed that there was a student who wanted to show 150 man meters using kilome- ters. ~ thought the student's effort was important. Moreover, ~ hope that the teacher also spends some time to help visualize the size of the number used in kilometers. In many cases, students do not know how far apart Tokyo is from Osaka. ~ think using such numbers for the two cities could provide a clearer image of large numbers. If the teacher could present something that the students could use to 'feel' how large a number is, ~ thought the lesson would have been better. APPE N DIX F