Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.

Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 157

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

OCR for page 157

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

OCR for page 157

{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

OCR for page 157

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

OCR for page 157

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

OCR for page 157

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

OCR for page 157

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

OCR for page 157

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 that—one
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

OCR for page 157

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

OCR for page 157

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

OCR for page 157

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

OCR for page 157

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

OCR for page 157

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

OCR for page 157

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

OCR for page 157

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

OCR for page 157

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

OCR for page 157

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

OCR for page 157

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

OCR for page 157

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

OCR for page 157

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