Math.Edu.

ICME 12. July 9, 2012

Univ.of Tsukuba

Dialectic on the Problem Solving Approach:
Hermeneutic Efforts for Designing Classroom
Mathematical Activity
Masami Isoda
Center for Research on International Cooperation in Educational Development,
University of Tsukuba, Japan.
Project Overseer (Representative), APEC Lesson Study Project
Chief Editor, Journal of Japan Society of Mathematical Education
JSME (since 1919)

 Hermeneutics is the ground theory for interpretation.
It’s support understanding activity.
 Hermeneutic efforts for designing classroom are thebasic activity to develop theory for teaching.

Math.Edu.

Content

Univ.of Tsukuba

Objective: Knowing the importance of thehermeneutic efforts (ISODA, 2001)
Setting
 Knowing Problem Solving ApproachKnowing Hermeneutics (Abraham, Isoda,
2007)

ExamplesInternet Communication (Isoda, McCrae,
Stacey 2007) for knowing the significancefor humanizing mathematics.
Fraction (Isoda, 1993) for knowing theunderstanding beyond the cognitive veiw.

2

Teaching Approaches which are developed throughLS. Challengers try to develop new ways of teaching.

Math.Edu.
Univ.of Tsukuba

The lesson study community to make an effort todevelop the children who learn…
a

b

c
vs

No, some ideas cannot generalizeabc

All ideas arefine.

Problem

Injection

Dialectic
Problem Solving
ApproachOpen ApproachMath.Edu.

Where is the 2/3 m from the 2m tape?

Univ.of Tsukuba

By Isoda, Inprasitha, Anake: Nohda 1981
0m

1m

2m

0m

 The caseof schoolteacher
who do notknow thelessonstudy withJ.
Textbook..

2m

Q4. Where is [fraction]m?

IET

44.0

IPS

55.4

81.7

KKU

83.6

20%

4.7

23.2

RU

0%

0m

33.2

40%

18.1

12.5

9.2

10.4

60%

80%

8.9

5.0 4.2

4.51.5

100%

2m

4

Let’s see the video

Math.Edu.
Univ.of Tsukuba

Why, the classroom became quiet.
 Problem solving approach distinguish task and problem.
 Why, some of them could get it and a few changed theirunderstanding?

How can you explain their understanding?
Because they try to understand what Minami saying.
They can re-construct Minami’s idea by themselves.
In general
It can be said that because they try to get the others
perspective.

We can get children’s perspective in Math Classroom.
We can get other teacher’s p. through the lesson study

5What is Lesson Study?

Math.Edu.
Univ.of Tsukuba

In Isoda, M.(2011)
Journal of Science and Mathematics Education in Southeast Asia, v34 n1 p2-25

Process/Lesson Study cycle. Various dimensions of Open Classroom.
Theme of Lesson Study. Lesson plan. Teachers' mind. Various
Outcomes. Sequential experience for sharing the heritage.
Additionally, developing children who learn by/for themselves
A US teacher said, I developed the eye to look at students andsubject matters "Kodomo wo miru me." Now, I am wellaware of my responsibility for my class. In the lesson study,
with other teachers, I preferred the more challenging
lessons such as with open-ended problems. When I foundthat students can challenge such difficult problems, I
recognized self-confidence in my lesson. (Lewis, C. 2006)

Teachers are usually re-invent, discover or re-understand objective
in relation to the children and subject matter through the lesson
study and listening from others.
On this meaning, Lesson study is the reproductive science.

6

What is mathematics and it’s learning?

Math.Edu.
Univ.of Tsukuba

 Pre-established harmony: Unknown/not proved
statement became the theorem in the system which is
known.
 Mathematics learning should be done based on what
learned before even if children have to extend theirideas.
Constructivism: Organism VS Environment

Solipsism

Mathematics is only existed in the mind?
Social Con.: Inter subjective to subjective

Materialism

Mathematics is only existed in the language?
Through the activity for getting other’s perspective we canbridge constructivism to social constructivism.

7

What is the theory of Education?
General vs. Local Theory in relation to the Classroom practice;
A Stereo Type images which does not have general meaning General Theory explains something.
 Observer observes, describes and say something but cannot propose.
 Epistemological Obstacle exists as obstacle but it does not tell how to beyond the obstacle
itself.

 Local Theory solves local situation.
 Local theory fixed on the situation is developed from ideas for challenges.
 Lesson study develops the local theory of teaching.
 Problem solving teaching approach tells how to beyond the obstacles.

 Local theory discuss necessary conditions for developing students but do notdiscuss sufficient conditions.

 General Theory in my research framework Hermeneutics
 Conceptual and Procedural Knowledge

 A theory for developing problem solving approach inclassroom Hermeneutic Efforts for developing mathematical knowledge with understanding.
 Meaning and procedure for planning the lesson based on the curriculum which
emended the children’s conceptual development.

Math.Edu.
Univ.of Tsukuba

Hermeneutics: General theory for Interpretation
General Theory:Implicit theory: the methodology of qualitative
study for scientific INTERPRETATION.
Explicit theory:
Modern hermeneutics: Gadamer, H.G. Hermeneutic
cycle; “we can not understand others”
Historical hermeneutics: Schleiermacher, D. F.
Hermeneutic cycle is the process for the subjective toobjective; “I can understand everything until
contradiction.” Until logical inconsistency, our interpretation, understanding, is true.

In Math Education:
Modern hermeneutics: Brown, T. (1997).
Historical hermeneutics: Jahnke, H.N. (1994)

Math.Edu.
Univ.of Tsukuba

Hermeneutic Effort:
Originated between the lines:L on Hebrew

Dead Sea Scrolls

ラメド

Math.Edu.
Univ.of Tsukuba

Math.Edu.

Hermeneutic Effort: local theory ofUniv.of Tsukuba

understanding for humanizing mathematics
education
Hermeneutic effort by Historians
First circle
ObjectScientist
TheorySecond circle
Object: past activity
S1:Historians
T1:Historical Interpretations
Jahnke, H.N. (1994)

Math.Edu.
Univ.of Tsukuba

Cultural Awareness(Lerman, 1994. ISODA, 2001.)
Result and Significance of the Projects
Enculturation through Cultural AwarenessStudents got cultural perspectives in mathematics
through their hermeneutic effort to another culture.

 Meet the Unknown.
 The Unknown Functions as the Mirror of Oneself.
 Cause the Enculturation

Hermeneutic Effort: local theory ofunderstanding for humanizing mathematics Education
Internet Project (Isoda, McCrae, Stacey 2007)
A Japanese Students said;“It was fun that we could talk with students
in far-off Australia. We could neither see
them nor hear them, but the three of themcertainly exist on the other side of theocean and were thinking about the samemathematical problems as we were. Just
imagining that makes me happy”
Imagine! The existence of others

Math.Edu.
Univ.of Tsukuba

Meet the Unknown

Math.Edu.
Univ.of Tsukuba

Australia

JapanMath.Edu.
Univ.of Tsukuba

Cultural Awareness(Lerman, 1994. ISODA, 2001.)
 Meet the Unknown.
 The Unknown Functions as the Mirror of Oneself.
 Cause for the Enculturation
Japanese Students met
the sentence type description in Algebra, and so on.

Australian Students met
the formal type description in Algebra, and so on.

Math.Edu.
Univ.of Tsukuba

Cultural Awareness(Lerman, 1994. ISODA, 2001.)
 Meet the Unknown.
 The Unknown Functions as the Mirror for Oneself.
 Cause for the Enculturation
Japanese Students met
the sentence type description in Algebra, and so on.
→ Our mathematics is more formal thanAustralian.
Australian Students met
the formal type description in Algebra, and so on.
→ Our mathematics is more basic than Japanese.

Math.Edu.
Univ.of Tsukuba

Australia

Cultural Awareness(Lerman, 1994. ISODA, 2001.)
 Meet the Unknown.
 The Unknown FunctionsJapan
as the Mirror of Oneself.
 Cause for the
Enculturation
Japanese
Students met
the sentence type description in Algebra, and so on.

Australian Students met
the formal type description in Algebra, and so on.
Australia

Math.Edu.
Univ.of Tsukuba

Isoda, M. (2001) defined Hermeneutic effort forhumanizing mathematics education by following
activities.

Understanding； one’s interpretation
Getting others’ perspectives (the assumptionof the positions of others)；the appropriate
interpretations of a text is only possible
through a subjective approach whereby weassume the writer’s (or speaker’s) position,
feelings and sympathetically attempt to put
ourselves into the position of another (writeror speaker).

Math.Edu.
Univ.of Tsukuba

Isoda, M. (2001) defined Hermeneutic effort forhumanizing mathematics education by following
activities.

Instruction from experience (selfunderstanding)； one obtains an instruction
about oneself with comparison of others’
perspectives.
The hermeneutic cycle: if we have someunderstanding, we apply it to new situations
and if it is applicable, it will become moreobjectively correct.
‘Hermeneutic cycle’=‘subjective to objective’

Math.Edu.
Univ.of Tsukuba

Math.Edu.
Univ.of Tsukuba

Case Study for Illustrating H.E.
Divisional/Partitive Fractions vs.
Quantitative Fractions
How difficult for getting other’s perspectives.
What kinds of arguments will be necessary.
Counter examples are not always counterexamples.

21

Making (creating) a 2/3 m pieceof tape from a 2 m piece of tape;

Math.Edu.
Univ.of Tsukuba

even if they already learned.
Partitive/divisional fractions; fraction inpartition, n parts from among m equallydivided parts of the whole
37 children

0m2m

Quantitative fractions; n parts from among mequally divided parts of a unit quantity (such as‘1 m’), where m<n is also possible such as ‘3/2
m’
1 children

0m

2m
22

Math.Edu.

Dialectic between two ideas
MatsuUra0m

Univ.of Tsukuba

MinamiYama

2m

0m
This argumentation is used as the good example
of Dialectic in the textbook of Educational
Philosophy at the Univ. of Kyoto,2mArgument by
MinamiG

Argument by
MatsuG

Sublation, Aufheben
23

Scene 1: Presentation of the Resultsof Individual Problem-Solving
0m

2m

0m

MatsuUra; 37 students
MinamiY (1):MatsuU G:
MatsuU G:
MatsuU G:
Suzuki (1):Math.Edu.
Univ.of Tsukuba

2mMinamiYama; 2 students

Basically, it’s half of this (MatsuUra’s tape).
No, it’s two thirds!
Wait, I get it; I’m in MinamiYama’s Group,
too.
When MatsuU G knew the MinamiY idea,
The waysome
the teacher
wrote
this isn’t what“get 2/3of themremembered
they hadof 2 m from
2m oftape”...
learnedbefore.
MinamiY idea functioned as a
Teacher,counter
this canexample.
be interpreted either way.

Matsuura Group: two parts of the three

Either Way

equal segments of the 2 m

Is Fine22 students

11 students

Undecided

Minamiyama

Group:

2/3m
2 students

4 students

24

Math.Edu.

Scene 2: Exchanges Regarding Each Side’s
Position (Each others’ Opinion)
MatsuUra;0m

2m

MinamiYama

Univ.of Tsukuba

0m

2m

Maybe the problem is coming from the way of question.
According to Matsuura Group, the answer is 2/3 of 2 m.
MinamiY G:
But it says “from 2 m.”
Either G:
So the reason for Matsuura’s answer of 2/3 is that since 1
or 1 m is taken from the 2 m tape.
MatsuU G:
You just divide it into three and take two of thosesegments.
Either G:

MinamiY G:

But it’s 2/3 from the 2 m tape
(Note: the “m” of “2/3 m” is missing in his explanation).

Either G understand both sides
25

Math.Edu.

Scene 2: Exchanges Regarding Each Side’s
Position (Each others’ Opinion)
MatsuUra;0m

2m

MinamiYama

Univ.of Tsukuba

0m

2mThis is a 2 m piece of tape, so with 2 m, you get 2/3 m,right? Usually when you have a fraction, the base number is 1. Since
it’s 2/3 m here, you have to get the base to 1. It says 2/3 m, right?
Since there’s an “m” on it, that means 2/3 of 1 m. So it’s 2/3 m from 2
m of tape, and Minamiyama first threw out this half (1 m), and I thinkyou use two of the three segmentsSuzukiof the remainingtape.
If sides andunderstandboth
Matsuura’s Group did this withoutexplain
the “m”correctin “2/3answer.
m”, I think it
would be just like Matsuura’s answer.

Suzuki (2):MinamiY G:
MatsuU G:
MinamiY(2):Now there’s the “m”, so wouldn’t Minamiyama be right?
Why is it that Matsuura’s right only if there is no “m” (in “2/3m”)?
I thought that 3/3 is equal to 1 m.

Teacher (1):MinamiY(3):One more time.
3/3 means 1 m, right?

MatsuU G do not understandSuzuki’s explanation .

(Note: the “m” of “3/3 m” is missing in his explanation).

26

Math.Edu.

Scene 3: Teacher Intervention 1
MatsuUra;
Teacher (2):0m

2mMinamiYama

Univ.of Tsukuba
0m

2m

3/3 m is 1 m, no doubt about it
MatsuU: They already(Note: he emphasized the “m”).
feel contraction but
MinamiY G:
Right!
reasoning from
MatsuU G:
No, absolutely not.
conclusion
Suzuki (3):Teacher, it is not related with the problem (Note: the originalquestion), isn’t it?
MatsuU G:
You take 2/3 from 2 m, right? So maybe Minamiyama’s 3/3 m is 2 m.
Suzuki try to share the ground of
MinamiY(4):(Pointing at the 2 m figure) This half is 1 m, and these two segments
reasoning.
are 2/3.
MatsuU G:
No mentioned 2/3 of “1 m” in the original problem.
MinamiY G:
It doesn’t say create “2 m” tape. It just says is “from 2 m of tape”.
MinamiY G:
Since the original problem doesn’t say to make this only from a 2 mtape, you can make it from 1 m as well.
Teacher (3):Minamiyama, if your answer is 2/3 m, then we would like to say thatthe base is 1 m. This is the reason why 3/3 m is 1 m, and the base is 1 m, according27
to what you are trying to say, right, Minamiyama?

Math.Edu.

Scene 3-4: Intervention-Sharing
MastuUra;0m

2m

MinamiYama

Univ.of Tsukuba
0m

2m

Teacher (4):Well, this is a problem, isn’t it?
MatsuU G develop their meaning
MatsuU G:
Since Minamiyama has left 1 m over, doesn’t that mean he reallyof MinamiY on their ground,remain 1 and 2/3?
MatsuU G:
So Minamiyama does not take 1, but 1/6.
Teacher (5):No. Minamiyama’s answer works when he’s only using this (1 m).
The remaining 1 m is irrelevant for him.

If the original problem involves making 2/3 of a 2 m tape,then Matsuura’s side is right, I mean, I think Matsuura’s argument iseasier to understand. Since you’re supposed to create 2/3 m from a 2
m piece of tape then it must be 2/3 m. So you ignore the 1 m, and this
3/3 m is also 1 m. Since you are going “from”, you’ve got to deal withboth “from” and “m”. If there wasn’t this “m”, and if “from” was“of”, then I would agree with Matsuura. (Repeating while reviewing
the figure) This 2/3 m means that the base is 1 m. If there wasn’t an“m”, then you could use any amount of “m” as the base, but since28
there is an “m”, then 1 m must be the base.

Suzuki (4):Math.Edu.

Scene 4: Sharing
MatsuUra;0m

Univ.of Tsukuba

2mMinamiYama

0m

2m

MatsuU G:
If the problem is “create 2/3 from a 2 m tape”,
or “create 2/3 m of a 2 m tape”?
MatsuU G & MinamiY G: The first one, “create 2/3 from a 2
m tape”, is Matsuura Group but what about the second one?
MatsuU G:
2 m might be the base, but since its 2/3 m, 1 m
might be the base, too.
Teacher (6): So the second one would be strange andcontradicting.
Shared the Problematic ‘from’ or ‘of’Matsuura Group: two parts of the three

Either Way

equal segments of the 2 m

Is Fine16 students

No student

Undecided

Minamiyama

Group:

2/3m
2 students

20 students

29

Scene 5: Teacher Intervention 2
In order to stir things up again, the teacher
asked the students to forget the originalquestion and whether or not “3/3 m is 1
m” temporarily. Some of Matsuura groupare still the opinion that “it is three equallydivided parts of 1 m or 2 m”. This opinion
indicates that some students are stillcaught up in the idea of divisional
fractions.
 The teacher asks “can we change
tracks?” and continued as follows.

Math.Edu.
Univ.of Tsukuba

30

Math.Edu.

Scene 6: Teacher Intervention 3

Univ.of Tsukuba

Teacher (7):
If we have 0.5 m, then do we indicate what the length is in cm?
MatsuU G &MinamiY G: Yes, it’s the same as 50 cm.
Teacher (8):Can we express this as a fraction? (Detailed discussion omitted) So is
it the same as 1/2 m, or is it different?
Some MU understand but
MinamiY G:
It’s the same.
some developed hard core.
MatsuU G:
Wow!
(Note: this is taken to mean that they are realizing their contradiction.)
MatsuU G:
It’s different.
Suzuki (5):If 0.5 m is the same as 1/2, then what is 1/2?
Teacher (9):And if I asked you to express 1/2 m as a decimal of m, what wouldthat be?
(Note: he added ‘m’.)
MatsuU G:
0.5. (Note: it still lost ‘m’.)
MatsuU G:
It might be 1/2 of 2 m.
MinamiY (5):0.5 m = 1/2 m and 1/2 m = 0.5 m are the same thing, all you’re doingis reversing the order. So I think you can say that 3/3 m is 1 m. But if the basechanges, I’m not sure if you can still say that 1/2 m = 0.5 m.
MatsuuraGroup:

twopartsof the three

EitherWayequal segments of the 2 m

Is Fine14 students

No student

Undecided

Minamiyama
2/3m

1 students

23 students

Group:

31

Scene 7: The Next day

Math.Edu.
Univ.of Tsukuba

After some feedback.

Teacher (10):The class seems to be in Minamiyama’s direction. Matsuura Group,
do you have anything to add to the discussion?
MatsuU G:
It says “from” a 2 m tape, right? If it said “from a 1 m tape”, or if it
didn’t say “from” ( “of 1m”), then Minamiyama would be right, but it does say“from”, so 2 m is the base.
MinamiY G:
2 m is larger than 1 m, right? So we can just forget 1 m of the 2 m forthe moment, and take 2/3 m from 1 m, for instance.
MinamiY G:
Just ignore where it says “from”.
Teacher (11):So that you are saying, just “create a 2/3 m tape” is the same as theoriginal question.
Suzuki (6):For instance, you have a blackboard and you have 2/3 of a
blackboard. We say this is 2/3. For instance, if you have a blackboard eraser, you
could say 2/3 of this blackboard eraser. Understand?
Teacher (12):I know what you’re trying to say. I really do understand.
32

Scene 7: The Next day

Math.Edu.
Univ.of Tsukuba

Suzuki (7):You can go with anything whatever. But it says 2/3 m. Since it has an“m” on it, that “m” must be the base. We studied that it was determined by the
distance from the Equator to the North Pole divided by some tens of millions, right?
Before they standardized it that way, “1 m” was not always equal, right? If you use
2 m as the base, you back then against the determination. Anyway, since m has 1 m
as the base. This is the difference between when you have a given base and when
you don’t.
Suzuki try to share the ground.
Teacher (13):Suzuki wants to say that since there is a unit affixed, the base isalready completely settled. So that’s why she feels she has to join the Minamiyama
Group.
Teacher (14):We haven’t heard from the Matsuura Group at all lately. Can we end
this discussion now, then? Since “m” is affixed, the base is “1 m”, but since we are
dealing with 2/3, we can change the base accordingly. It’s as simple as that, isn’t it?
Is this fine with everyone? (The class ends at this point.)

Although there are no longer any counterarguments from Matsuura Group, some of themembers remain unconvinced.
33

Math.Edu.
Univ.of Tsukuba

MatsuUra0m

MinamiYama

2m

0m

learned?
Either G understand both sides
MatsuU G do not understand.
Reasoningfromconclusion

MatsuU G
reason on
their
ground anddeveloping
hard core.

3/3 m =1m?

2m

lost ‘m’

Suzuki understand both sides
and explain correct answer.
Suzuki try to share the ground.

Shared the Problematic ‘from’ or ‘of’
0.5m ?

Teacher support Suzuki
Suzuki assert the invariant of 1m.
34

Types of Counter Example for
MatsuU0m

Math.Edu.
Univ.of Tsukuba

2mType A: MinamiY’s answerer itself

0m

2m

Type B: 3/3 m=1m (not functioning wellbecause it deny MatsuU)
Type C: 0.5m=50cm (Possibility ofGeneralization of MatsuU; some of them
functioning)
Suzuki’s equator is related with Type C.
Type B & C developed the hard core at the same time.

35

Why it’s fucntioned as counterexample?
0m

Math.Edu.
Univ.of Tsukuba

2mType A: MinamiY’s answerer itself

0m

2m

Type B: 3/3 m=1m (Even if feeling strange, notfunctioning well because it deny MatsuU)
Type B’: 0.5m=50cm (Possibility ofGeneralization of MatsuU; some of them
functioning). If we deny it, 1m is notinvariant (Suzuki’s last comment)
Type B & B’ developed the hard core, too.

36

Teacher’s roles

Math.Edu.
Univ.of TsukubaFirst, he gave opportunity of assertion fromboth sides (Scene 1),
Second, he allowed Suzuki who understands
both sides to explain,
Third, he gave a counter example 3/3m in
relation to the original question (Scene 3),
Fourth, he gave generalized counter example
1/2m (Scene 6), andFifth, he supported Suzuki who explained thenecessity to fix ‘1m’ as a unit for conclusion.
37

Why teacher have to support
Suzuki?

Math.Edu.
Univ.of Tsukuba

Why some children developed hard core?
Because they try to say their conclusion is true.
Because they do not well understand.
Because they thought emotionally but did not logically.

In the process, they applied

Type A

Type B

Type Cthe appropriate procedure of division

Kept

Kept

Kept

the appropriate meaning of the quantity

Kept

non → having non → non38

Case Study for Illustrating H.E.

Math.Edu.
Univ.of Tsukuba

Divisional (Partitive) Fractions vs.
Quantitative Fractions
How difficult for getting other’s perspectives.
What kinds of arguments will be necessary.
Counter example is not always counterexample.
Depending on the teachingsequence we can avoid thedeveloping hard core!
39

Math.Edu.

Conclusion

Univ.of Tsukuba

Objective: Knowing the importance of thehermeneutic efforts (ISODA, 2001)
Setting
 Knowing Problem Solving ApproachKnowing Hermeneutics (Abraham, Isoda,
2007)

ExamplesInternet Communication (Isoda, McCrae,
Stacey 2007) for knowing the significancefor humanizing mathematics.
Fraction (Isoda, 1993) for knowing theunderstanding beyond the cognitive veiw.

40