**Suggested Citation:**"Appendix E: A Demonstration Lesson: Function Thinking at Sixth 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 E: A Demonstration Lesson: Function Thinking at Sixth 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 E: A Demonstration Lesson: Function Thinking at Sixth 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 E: A Demonstration Lesson: Function Thinking at Sixth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix E: A Demonstration Lesson: Function Thinking at Sixth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix E: A Demonstration Lesson: Function Thinking at Sixth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix E: A Demonstration Lesson: Function Thinking at Sixth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix E: A Demonstration Lesson: Function Thinking at Sixth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix E: A Demonstration Lesson: Function Thinking at Sixth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix E: A Demonstration Lesson: Function Thinking at Sixth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix E: A Demonstration Lesson: Function Thinking at Sixth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix E: A Demonstration Lesson: Function Thinking at Sixth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix E: A Demonstration Lesson: Function Thinking at Sixth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix E: A Demonstration Lesson: Function Thinking at Sixth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix E: A Demonstration Lesson: Function Thinking at Sixth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix E: A Demonstration Lesson: Function Thinking at Sixth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix E: A Demonstration Lesson: Function Thinking at Sixth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix E: A Demonstration Lesson: Function Thinking at Sixth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix E: A Demonstration Lesson: Function Thinking at Sixth Grade." National Research Council. 2002.

**Suggested Citation:**"Appendix E: A Demonstration Lesson: Function Thinking at Sixth Grade." National Research Council. 2002.

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Lesson Plan Lesson Transcript Shunji Kurosawa, Irk Teacher, Tokyo Gak~gei University, Setagaya Elementary School Postlesson Discusssion Transcript Takashi Nakamura, Professor, University of Yamanahz Postlesson Discussion: Questions and Responses 158 161 168 175

LESSON PLAN A DEMONSTRATION LESSON: FUNCTION THINKING AT SIXTH GRADE August 3, 1999 Setagaya Elementary School Sixth Gracle Instructor: Shunji Kurosawa BACKGROUND The Name and the Situation {7} Two changing variables Starting at the third grade, the field of quantity-related mathematics is brought into the curriculum, dealing with equa- tions, graphs, statistics, and functions. This particular lesson will be focusing on functions but not the functions defined in a math textbook. The lesson's idea is based on the "function-like thinking process" that deals with comparing two variables, one increasing and the other in relation to the first, finding the relation- ship, and expressing the realizations of the relationship to solve the problem. In the fourth grade, a lesson called "examining change" is introduced, and students are asked to compare two variables to examine the relationship. The relationship is described in graph and equation forms at the fifth-grade level. Now in the sixth grade, the com- parison of direct and inverse proportions is use(1 to (1evelop the "function-like thinking process." {2 ~ The situation This lesson is at the stage a step before studying proportions. So the objective is to solve a problem by applying the method airea(ly Earned, like the one in "examin- ing change." That is, to solve a problem by fin(ling a variable number within a given situation, compare it with another variable number that is dependent on the first one, and to find the rules of the relationships between the two. This kind of thinking process wall be used as the evaluation of the lesson. The Purpose The development of "function thinking" is the basic foundation for understanding functions and a goal that should be taught at an early stage. This is due to the fact that it develops a scientific mind by comparing the unknown to a known to construct an explanation. Yet it does create boredom just to experience each step of the basis of functions in order, no matter how important it may be. Function thinking begins to develop only when there is an unknown anti a (1esire to want to know the unknown. The aim here is to create a situation for students to be curious enough to fin(1 their own question within the given subject and evaluate the thinking process while solving the prob- lem. There is an unfortunate fact that this sort of thinking process is not commonly taught. Even if it was being taught, it often jumps to the stage of learning rules anti practicing, with two variables already provi(le(l. The truth is that the first step is being skippe(1 most of the time. In to(lay's lesson, students will see where each variable comes from while observing the growth of an abstract image. The Development {7} Objectives (a) Fin(1 a variable anti raise a question about what is going to happen. (b) Find another variable that is in relation to the first one, realize how they are related, and find an answer to develop the thinking process of functions. (c) In a(l(lition, encourage students to fin(1 their own rules (lescribing the changes to achieve (levelopmenta1 thinking. APPE N AX E

{2} The Lesson Plan The stages of learning and students' activities ~ · ~ Cautious points ~ · and evaluation points ~ · I. Observe the image shown below. · Each of the squares is moving away from the one in the center. · The number of squares is increasing by four. · It is like a virus.* · Begin with students' responses to the image. Listen to comments such as, "Like fireworks," "An explosion," '~here are nine squares," '~hey are expanding," etc. · Make sure to value comments that describe the change in the image, such as, '~he number is growing," and "away from the one in the center." after ~ min 2. Make it into a question. · If the amount of squares increases by four from the one in the center every minute, how many squares wall there be after ten minutes? · Let them think of the variable (time) that is related to the image. (?? at the beginning, ?? a minute later) · Develop the class according to their comments. 3. Each wall come up with their own solutions and answers. · Comment on each other's solutions and the reason behind the equations they built. 4. Argue the solutions. · The answer is 41 squares. · The reasonis: 4xX+ ~ or Xx4+ I. · Create own situations with changing variables. · Were they able to explain the reasons well by using graphs and equations? · Go over each problem, and plan to use them in the upcoming class. *Editor's note: The teacher stated as an assumption that we may not know how all viruses grow and that it may be the case that some grow at a constant rate. APPE N DIX E

(3) Evaluation Were students able to consider the two changing variables well enough to create a problem? APPE N DIX

LESSON TRANSCRIPT A Demonstration Lesson: Function Thinking in the Sixth Gracle August 3, 2000 Tokyo Gakugei University, Setagaya Elementary School Teacher: Shunji Kurosawa Kurosawa-senseii began the lesson by posting a yellow sheet of paper with a single black dot on the blackboard. He then posted another sheet with five black dots. This was followed by a sheet with nine black dots. And finally, there was a sheet with thirteen black dots. One student mentioned that it seemed to be "growing." Kurosawa-sensei asked students, "What is the subject of 'growing'? ITv~at is growing?" ~ Sensei means teacher APPE N DIX E

Students came up with a variety of responses. The line is getting longer. The Triangular area between the lines is getting larger. The size of the cross is getting bigger. The number of black dots is growing. After approximately 10 minutes of discussion, Kurosawa-sensei posed the same · ~ question again... Teacher: I'll ask the question one more time. You have come up with some answers seeing only this one. (Do you see other things) by looking at this ones Is there something else that is changing Is there anything else that is changing rapidly Arita-san, you can keep it, I will get it later. Yes, Ota-kun. Ota: The number of the dots will increase four pieces at a time. Teacher: Okay. I'll note that point. · · ~ "Writing} (Ota) Increase 4 at a time. Teacher: Increasing by four pieces. Yes. Student: Ota-kun also said that, if the dots, which increased by four pieces, were connected by lines, Then it would make a square and the squares would be increasing. APPE N DIX E

Teacher: Oh, you have said something interesting. You can see the square. Write it down please. YoshiJa is sharp. The squares have increased. "Writing} (YoshiJa) The square is increasing. Teacher: Okay, the squares are increasing rapidly. I see.. Yeses ... Do YOU know anything that increases rapidly like this in the worlds fIs , , v , , . there anything) in your everyday lifer Student: Debt. Teacher: Whets Student: I said debt. Teacher: Debt! You Debt. Is there anything that /'pohn!/' increases rapidly in your daily lifer Students: There's nothing, Kurosawa-sensei! Teacher: Is there really nothings Students: Increased This is tough. Whets Teacher: Nothings Whets Waters Waters Student: If the water falls. . . Teacher: I see, the water falls /' pohn/' then it will increase, I see. I see. Water. Water. Like water goes "pohn/' if stone falls into the water /'pohn./' Teacher: What else Arethere any other images Student: A balloon. Teacher: Whets A balloons Student: I mean the air makes it bigger. Teacher: (Blowing) Like this A balloon, as you blow "bwah/' into it more and more "bwah/' is that what you means Teacher: It will expand and in the end BANG! I see. I see. I came up with a much better idea when I saw this. I came up with cell division. Student: No such form exists. Teacher: You think sod What is it thence The thing which will increase. Bacteria! Umeki: It can increase in a fixed pattern. Teacher: Wait a minute. Umeki said a great thing. Say it again. Umeki: The black dots will increase in a fixed pattern. Teacher: Since the dots will increase in a fixed pattern, then you want to say it looks like whets Umeki: A virus. Teacher: A virus Students: ... It is not constant. Teacher: Isn't a virus constants Students: It's not constant. APPE N DIX E

Teacher: Is that sod We never know. Natural science behaves by rules, the virus may increase constantly. Virus. I understand. "Pohn!/' There's one (black dot). This is the beginning. This IwonJer how we should Jo this. This is after one second. Students: Teacher: It's fast Aheroneminute. Then this diagram is aher one minute. "Aher one minute." Aher one minute. So this is the diagram after the second minute. Therefore, aher thatone minute later, one minute later the virus The black dots became a virus. It's like the movie, My Neighbor Totoro, (about) that ghost who had a virus. Kobayashi, how much will the virus be increased aher 5 minutes, if it increases every one and two minutes It is easy to answer for five minutes or six minutes. How many minutes would you like to tryst One hour. Forty-three minutes. Ten minutes. Thirty-five minutes. An appropriate number is better. Appropriate! What is the most appropriate number of minutes A number that is easy to calculate. That is correct. That is right. Umeki. There is no number that is hard to calculate There is no number that is hard to calculate. Students: Student: Teacher: Student: Teacher: Umeki: Teacher: Umeki, speak. Umeki please make a problem. Please give us your virus problem. Please, go ahead. Umeki: Well, a virus will 48 minutes later. "Writing} (Umeki) A virus aher 48 minutes, how many are there Teacher: Then, from now. Okays Is there anything about the problem that you don't understands Is it hard for you to calculate 48 minutes Is it all rights Yes, what is its Students: ... Teacher: Just a moment. Did YOU try to say the answers Just a minute. Is this okay , . . . as a questions 48 minutes later. Then, I will give you 3 or 4 minutes from now, to calculate how much the virus will increase aher 48 minutes. Start now. The students worked independently on the problem for about eight minutes. During this time, Kurosawa-sensei walked among the students. He watched them work and occasionally stopped to talk with individuals. When he felt enough students were ready, he called the class together to discuss the problem. APPE N AX E

Kurosawa-sensei surveyed the students to find out their answers. Most students answered 193; a few had IS9. Kurosawa-sensei began the class discussion, concentrat- ing on the solutions that led to IS9 as the answer. Then, let's start with someone who has the answer 189. Kawamura-san, please. Kawamura: It increases by 4 pieces 47 times, so that's 47 times 4. And there is the dot in the center, but that does not increase 4 pieces at a time, so you only need to add 1. So it is 47 times 4 plus 1. Student: Teacher: |Writing} You can speak aher Kawamura-san is done. (Kawamura) 47 times, it increases by 4. 47 times 4, plus 1 in the center. 47 times 4 plus 1 is 189. Kawamura: The dots surrounding the four increase 4 at a time, But the first time is not included, and it increases by 4 pieces 47 times which is 1 88, And the Jot in the middle is added to it. Does anyone have an objection to my answers Yoshida: Kawamura says that four pieces increased 47 times, This means that since four pieces will increase 47 times aher one minute, Then I think 5, instead of 1, should be added for the first minute. "Writing} (YoshiJa) That is what happens one minute later therefore, must add five. Yoshida: Does anybody have any questions regarding my answers Teacher: Kawamura-san Kawamura: ... Teacher: You should say your opinion. YoshiJa ...you should name someone... Yoshida: Dobashi-kun. Dobashi: I agree with YoshiJa's opinion. Teacher: Ikeda-kun, did you understands Student: Yes. Teacher: Kawamura-san, are you okay arson Kawamura: Yes. Teacher: It Joes not become 193 after adding 5. Student: It will. 47 times 4 plus 5. Teacher: I see, I see. I understand. You mean that 47 times 4 plus 5. Is that what you means Student: 4 times 48. Teacher: Please speak. Student: It is not 47 times 4 plus 5. Well, it's okay if it's 48 times 4, plus 1. "Writing} (Umeki) 48 times 4 plus 1 is correct Teacher: I see. Then, can anyone explain Nakahara's answers Okay, not very many people. So, Nakai-kun please. APPE N DIX E

Tatsumi: Teacher: Nakai: Since four pieces will increase every minute, so four pieces will increase in one minute, 48 times 4 plus 1. "Writing} (Nakai) Every minute it increases by 4. 4 times 48 plus 1. Teacher: ... Nakai: Does anybody disagree with my opinions Teacher: Nakai's answer is the same as this one. But, it is slightly different. Is that okays 48 times 4 plus 1 and 4 times 48 plus 1. Can anybody explain the difference between the twos Student: ... Teacher: Someone has a different method. Wait a minute. There are others. Then, please explain it. Ogawa: I did 1 plus 4 times 48 "Writing} (Ogawa) Another way to describe it is 1 plus 4 times 48. Teacher: Say it again. Ogawa: 1 plus 4 times 48. Teacher: Oh, this way. What did you says Okay, I will let you explain what the differences and similarities are recardina these three methods v v It is easier to understand if it is marked as A here, and marked as B here. Tatsumi-kun, please explain Umeki-kun's way. A is- P~ease come up Tatsumi: You mean I should draw a pictured Teacher: I Jon't know, please ask. Tatsumi: A is, A is on the number line it assumes that 48 is 1, and seeks what is... over 4... Teacher: What you're saying is, 48 times 42 Tatsumi: 4 times 48 means 4 multiplied by 48, but what it means is that every 1 minute it increases by 4, and it is not increasing by 48 every minute, So I think it is better to say 4 times 48. Teacher: So A is betters I mean, B is betters Tatsumi: Between A and B. I thought B was better. But between B and C, there is a 1 at the very beginning, and I thought that was even better. Teacher: Then C is betters Tatsumi: Yes Teacher: Okay, Tatsumi-kun says that this one says 48 multiplied 4 times, and that one says 4 multiplied 48 times. So if it is increasing by 4, then B is the better choice. That is his opinion. OK Anyone disagreed APPE N DIX E

B is betters Tatsumi: 48 is a little big, so Teacher: OK, let's use this. Let's think of 3 minutes elapsed time. Yes, please. Tatsumi: ... 48 is so every minute, this part, not there, here increases by 1, by 4. Here, next to the [center] Jot, aher the first minute there is one Jot, and in 2 minutes there are 2, and aher 3 minutes there are 3, but then, over here there's 3 more, and 3 more here, and 3 here. So if you multiply 3 times 4, and then add 1 for the one in the center, you will get the answer. Teacher: How's thatch Now there are some of you who understand. Do you really understands So aher 48 minutes, there will be 48 more lined up here. Understands Aher 1 minute, there's one, after 2 minutes there are 2, and aher 3 minutes, 3 and so aher 48 minutes there will be 48. Please, go ahead. Student: I think that's okay, but the virus increase takes place uniformly 4 at a time, not 4 here at once, then 4 more there, so Teacher: Hmm. So we can think of the increase this way... or as 4 times 3. This is the difference, rights Do you see the differences As long as you see the difference, that's fine. But judging from the manner of the increase, the opinion is that B is better...or I mean C, because it adds the 1. Now, we're starting to run out of time. Hmm. Lastly, is there anyone who can produce a formula that will allow for the calculation of the increase no matter how many minutes have elapsed "Writing} In order to know the number of the viruses after an unspecified number of moments have elapsed Teacher: What if you want to figure out the number of viruses aher an unspecified number of elapsed minutes The class continued for only a few more minutes. Kurosawa-sensei ended class by asking students to consider "other scenarios that increase" for the next time they met. The sixth-grade lesson was followed by a postlesson discussion with the classroom teacher and the Japanese mathematics educators who were in attendance. Participants from the Congress observed the discussion. APPE N DIX E

POSTLESSON DISCUSSION TRANSCRIPT: Function Thinking at Sixth Gracle August 3, 2000 Tokyo Gakugei University, Setagaya Elementary School Facilitator: Takashi Nakamura Teacher: Shunji Kurosawa Takashi Nakamura from the University of Yamanashi facilitated the postlesson discus- sion. Nakamura-sensei began the session by having Kurosawa-sensei give his goals for the lesson. Facilitator: Let's start our postlesson discussion now. As I have explained, the content of our postlesson discussion is the subject of this meeting. First I want to note that we Jo not have much time. For about five minutes I suggest the teacher who conducted the classroom lesson tell us the goals of the lesson. Then, we will ask him questions and target the issues to discuss. First of all, I would like Kurosawa-sensei to talk about the goals of the lesson and the classroom teaching today. Teacher: Today, I had three goals. The first goal was to run the class in such a way that students come up with the math questions by themselves. To meet that goal, I showed them this chart and used the students' words. I wanted to set up the lesson to allow the students to come up with the questions. I would like your comments. The second goal was to teach students the concept of a function, which an important subject in math education. As you know, the concept of a function is important to understand measur- able changes of a subject. We need to understand two variables and find out the "rule of the change/' between them. And by using the "rule of change,/' we can solve math problems. This is how I think about teaching a function, step by step. And for the third goal, I wanted to teach students how to read and under- stand the equation that represents how things change. Now, I want to evaluate whether I accomplished these three goals in the class. . IS Regarding the first goal to let the students generate the questions and whether or not we achieved it... i welcome your opinion. Please tell me about it later. Today's math question was, "How many black dots will you have aher 48 minutes2" APPE N DIX E

I think students came up with the right questions on their own. That is because they used words such as "rapidly/' and "virus." Unfortu- nately, I introduced words like, "Aher certain minutes." I believe my students did indeed come up with the math questions on their own. The second point was the concept of a function. The concept of function involves finding how things change- then using examples associated with the change, and finding the "rule of change /' My students used the word "rapidly./' Regarding the use of the word "rapidly/' I would have preferred such terms as "spreading" or "increasing." But they were able to find many words to indicate change. For example, students knew the lines and area increased. However, as I've explained, I didn't Jo enough to help students with changes involving independent and dependent variables. Still, aspects such as shape and number of squares were used effectively to solve the fourth problem. The third goal was to help students understand the equation; I tried to get them to appreciate the difference between 48 x 4 and 4 x 48. Although they had some problems, they understood the meaning of not only4 x48 but also48 x4. For me, it is acceptable that one student understood, and the students around him understood by learning from him. Therefore, I think they learned how to read the equation appropriately. Additionally, I had hoped they would come up with an expression for dividing this into four parts. It produces a formula like this: 3 minutes plus 1/4, parentheses. (laughter) I wondered if these kids could come up with this concept. Whoops! That's wrong. It's actually multiplied by this, rights (laughter) I thought there might be a chance that these kids could have come up with this idea, but it did not happen. For them to Jo that, I should have taught them how to think about the concept of 1/4 or how to divide the element at the line. I thought it would be fun to see if they could come up with it. Anyway, I thought they more or less understood how to comprehend the equations. Teacher: The participants were given the opportunity to ask Kurosawa-sensei clarifying ques- tions about his goals. Nakamura-sensei took seven questions from participants then asked Kurosawa-sensei to address each of them. A summary of the questions and responses can be found following this transcript. The general discussion started with Kurosawa-sensei's first goal: To have students come up with the mathematical question. APPE N DIX E

And, overall, by focusing on the problem of the number of dots Then the students learned the relationship of variables and the concept of functions. They were taught different ways to view the problem and express it in an equation. For all these reasons, I agree with the way he conducted the class if we have to complete the lesson in 45 minutes. Facilitator: Let's consider other types of issues. By exploring the area of a triangle or length of the lines, at what point Jo students internalize the problems Is there any guarantee that will happen in the classy Or is that the responsibility of the teacher to have them learn it at homed Participant: This is related to what was discussed before. It is not important to let students think about what the teacher is thinking about. The important thing is to understand what students are thinking about. The opinion expressed by the teacher at the beginning was using his words he was bringing out students' awareness. Through Risremarksitwasevidentthat the teacher was drawing the students closer to his way of thinking- and the students started wondering what the teacher was thinking and wanted. Facilitator: I agree with your opinion. Yes, please. Participant: I also agree. It took 25 minutes to produce the math problem. While many math problems were expressed- Kurosawa-sensei tried to lead his students towards one problem. I think the students clearly saw his intent. Then, they knew the important problem was to find the total number of Jots. Facilitator: Finally we are getting a variety of views. (laugh) Yes, please. Participant: It took 25 minutes to develop the math problem. Nevertheless, the time it took is less important. The important thing is what you were able to accomplish within the time. If you listened carefully, they said the picture has plane symmetry or line symmetry. Students came up with a number of concepts that are associated with mathematics. Usually we see many nonessential things that are not related to mathematics. Still, depending on the content of the class, like today's, we should Jeter- mine the right amount of time and work hard on it. Although I Jo not disagree students used the word "spreading" repeat- edly as Kurosawa-sensei told us before. They did not use the word "increases." . . . APPE N DIX E

And, overall, by focusing on the problem of the number of dots Then the students learned the relationship of variables and the concept of functions. They were taught different ways to view the problem and express it in an equation. For all these reasons, I agree with the way he conducted the class if we have to complete the lesson in 45 minutes. Facilitator: Let's consider other types of issues. By exploring the area of a triangle or length of the lines, at what point Jo students internalize the problems Is there any guarantee that will happen in the classy Or is that the responsibility of the teacher to have them learn it at homed Participant: This is related to what was discussed before. It is not important to let students think about what the teacher is thinking about. The important thing is to understand what students are thinking about. The opinion expressed by the teacher at the beginning was using his words he was bringing out students' awareness. Through Risremarksitwasevidentthat the teacher was drawing the students closer to his way of thinking- and the students started wondering what the teacher was thinking and wanted. Facilitator: I agree with your opinion. Yes, please. Participant: I also agree. It took 25 minutes to produce the math problem. While many math problems were expressed- Kurosawa-sensei tried to lead his students towards one problem. I think the students clearly saw his intent. Then, they knew the important problem was to find the total number of Jots. Facilitator: Finally we are getting a variety of views. (laugh) Yes, please. Participant: It took 25 minutes to develop the math problem. Nevertheless, the time it took is less important. The important thing is what you were able to accomplish within the time. If you listened carefully, they said the picture has plane symmetry or line symmetry. Students came up with a number of concepts that are associated with mathematics. Usually we see many nonessential things that are not related to mathematics. Still, depending on the content of the class, like today's, we should Jeter- mine the right amount of time and work hard on it. Although I Jo not disagree students used the word "spreading" repeat- edly as Kurosawa-sensei told us before. They did not use the word "increases." . . . APPE N DIX E

You said, "I finally used the words 'increasing rapidly' because I couldn't wait any more for them to come up with those words." I have one more thing. Aher students mentioned the problem on the number of black Jots, you said, "Is there anything else changing rapidly2/' I wondered if you said that by mistake. How should the teacher conduct the class to draw out those words from the students I cannot offer a solution. (laugh) Facilitator: Yes, please. Participant: Regarding the concept of a function, I find the process of finding the variables of the problem to be important. In this sense, the beginning of the class was very interesting. However, it was the teacher who said, "This changes with time." What should we Jo to let students come up with that points In your presentation you clearly state, "The first chart, the second chart, the third chart./' Students can point out what they see. Time cannot be seen, but it is being shown the first chart, the second chart, and so on. Therefore, I think it should be possible to relate aher one minute, the first chart, then the second, and so on. However, its difficult because the children relate the first sheet (with the single black dot) to the first minute. Therefore, I think you should start with the second sheet with five dots. Facilitator: The second goal was how to develop the concept of a function in students' mind. What Jo you thinks Should we start from the sheet with a single Jot or 5 Jots Participant: That is not what I have in mind. A problem existed much earlier. The teacher used the word "virus" and that focused the subject at the , · ~ beginning. I think students produced mental images of the virus increasing over time. Nevertheless, were they all able to understand and visualize the problems Would they have gotten it without being told - "Today's subject is about the virus" and, 'What will happen in 48 minutes2" If they all achieved the concept, then the class was effective. At this point, Nakamura-sensei asked Kurosawa-sensei to respond to the comments that had been raised. Facilitator: We have discussed the equation on the boa rd. Now, can we address the function and the presentation of the problems Teacher: To tell you the truth, I have had a dilemma regarding the presentation of functions. APPE N DIX E

We were able to find a variety of variables, but we needed to choose one. The process of focusing on the most important variable is one subject. The next subject is defining the dependent variable in the problem. The challenge is effectively addressing these two subjects. Today's class demonstrated the difficulty but let me explain I did not randomly lead the students towards the problem of the number of dots. The line, distance between dots, size of the "x/', and area all rapidly increasing were related to the number of dots. I felt that the students realized the importance of the number of dots, and thus, I started to focus on it. I do feel the process of narrowing down to a single problem is an impor- tant subject. Next, if we can identify the important variable, then we can focus on the dependent variable. At that point, another subject that I must carefully consider is how I phrase my question. Once we identify the variable, should I say, "What is the cause of this changed/' When I use the term, "dependent variable,/' I didn't think students would understand. "What will change with it2/' does not sound clear. "To determine this value, what needs to be determined first2/' sounds too formal. So, after finding one variable, I think it is very difficult to ask the right question to find the dependent variable. As you've mentioned, aher defining one variable, I should consider carefully how to treat the dependent variable. To accomplish what we need, I think what is important is how you present the material. So, at the beginning, I showed this set of dots, but I wondered what kind of variables the students could identify. I worried that they would not find one. Therefore, I prepared something like this. And, at this stage, I used sound effects like "pohn/' and "wahn./' By using these sounds, I wanted to evoke images of something changing, like cells. I was looking for an example familiar to them. I wanted to draw upon their personal experiences. They may have seen cells increasing under a microscope in science class. I was looking for an example of something increasing, and the girl over there suggested mice. She said mice multiply. I wonder why she didn't say it earlier I had been struggling to find a good visual example of something increasing. APPE N DIX E

Therefore I expected the class to depend on what students have seen or experienced. Facilitator: What Jo you thinks Any opinions Yes. Participant: The students understood the concept. Then, it is the teacher's responsibility to wrap up. A function here is an addition of four dots. The students placed four dots, and four more dots, and four more dots of the square there. Therefore by observing the black Jots, we see the square is increasing in size. . They understood the change made by each set of four dots. So, instead of using the sound effect "Pohn! Pohn!/' on a daily basis- We should use the right words regularly so that we can treat functions appropriately... After this comment, Nakamura-sensei opened the discussion for final thoughts about the lesson as a whole. Facilitator: We only have about eight minutes. Sakai-san pointed out the presentation of the math problem and the concept of a function as discussed earlier. Do you have any opinions, insights, or thoughts about the lesson as a wholes Participant: In the latter portionhow confidently cilcl students come up with those two answers, and for what reasons Did they think so because the rest of the students got the same answers Is it correct for them to think that they all got the same answer so it should be right2 Teacher: Can we show the charts Is that rights Participant: The chart is acceptable... · · ~ A. I l 1. I I Facilitator: Therefore, you are saying that the students were able to formulate equa- tions from their ideas But, how they were able to arrive at these equations was never openly discussed. It might have been written in their notebooks. But when it is discussed in class, even the students who did not initially understand will benefit. Therefore, we use charts and graphs. I think displaying numbers like one, five, nine is a good idea. It remains on the board as a visual aid for the children · ~ Nakamura-sensei allowed a few minutes for visitors watching the discussion to ask questions and then concluded the session. APPE N DIX E

POSTLESSON DISCUSSION: QUESTIONS AND RESPONSES Function Thinking at Sixth Gracle The postlesson discussion between the observers and the teacher Shunji Kurosawa immediately followed the lesson on Function Thinking. Takashi Nakamura from the University of Yamanashi facilitated the discussion. Nakamura-sensei began the session by having Kurosawa-sensei give his goals for the lesson and describe the degree to which he felt he had achieved them. Nakamura-sensei took seven questions from the Japanese observers, then asked Kurosawa-sensei to address each of them. Nakamura- sensei grouped the questions as he presented them to Kurosawa-sensei for a response. The questions and the responses are summarized below, in the order of the responses. Observer: You intended to solve the problem by breaking it into several separate items, rights Observer: When you increased the number of the balls at the beginning, you pasted them one by one. Was that intentional As a result you did not place four balls on the sheet. Was that intentional Observer: Could you tell us about your method for presenting the math problems For example how to post the sheet, how to provide the sheet, how to present the problems Kurowasa-sensei: I had a dilemma about the method. If I were to post the sheet immedi- atelY I wondered if students could see how it increased. For fourth ~ , grade, it would be okay to post one, then another, and so on so that they could see it increasing. But for sixth graders, I wanted just to show the figure and let them discover how the black Jots increased. I was not sure they understood it or not when I saw their reactions today. Therefore, I placed the black dots as increasing from one to five and to nine. Going from five, six, seven, eight, nine, one at a time rather than r. . ... .. I one, five, nine was not Intentional. Observer: Someone mentioned the number line. Why didn't you use it on the boa rem Observer: You did not use graphs or charts. I want to know why. Kurowasa-sensei: In my class, I let my students judge whether they do multiplication using the number line or not. It is a definition. Therefore, there was no need to use the number line. I used the phrase "four times 48,/' and the phrase should be enough for them. And regarding the chart, where it explains the equation, I thought I might need to follow the chart to indicate that it increases four at a time. However, they are sixth graders. They understood that it increases four at a time without filling out the chart. When they fill out the chart, they can see the law of corresponding figures and numbers visually. But, I APPE N DIX E

wanted them to come up with these two equations and compare them today. Therefore, I did not ask them to make a chart. Observer: In your opinion, what Jo you think was the most important aspect of this lessons Kurowasa-sensei: I had a hard time deciding which goal should be number one. When someone asked me why, I would answer that I do not know whether the one I chose will be important in the future or not. I think it is important to grasp the variables in the equation. And I think it is fun seeing a lot of variables. Another important thing was how to read the equation. I wanted some students to grasp it. Finally one of my students, Tatsumi, got it and everyone got it too. I was happy. Observer: What percentage of students actually participated in the teacher's prepared lessons Although I do not have an accurate figure, about 12 or 13 students participated in the process again and again. What percent of the 33 students participated in the process in classy Kurowasa-sensei By what standard can we consider whether the students participated or note Is it a question about whether they came up with answers like 193 or 189 or not. I would say it is 100% because all the students got answers as far as I saw. Nevertheless, if you ask me what percent of my three goals were achieved, because I have to give you a figure, I would say 1 00 percent. APPE N DIX E