© Yasuhiro
Sekiguchi
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This proposal discusses relationships between
argumentation and mathematical proof from the cultural
perspective. Recently, some researchers suggest that
learning of mathematical proof or proving should be located
in the context of argumentative activity, that is, in the
exchanges of conjectures, explanations, justifications, and
refutations among students. I discuss this idea from a
cultural perspective, using Japanese culture as a case, and
indicate that careful consideration on cultures of the
society and classroom is necessary to actualize the learning
of mathematical proof in argumentative activities.
For this purpose I use the concept of
communication as an overall framework. This is because I
consider that (1) cultural aspects are well-reflected in the
style of communication, (2) argumentation is a type of
verbal communication, and (3) mathematical proof is an
important component of communication in the mathematical
community.
In the following, I first describe communication
styles in Japanese culture, comparing with those of the
Western culture. Then, I discuss argumentation and
mathematical proof in Japanese schools, focusing on how they
are related to general styles of communication in Japanese
culture.
Communication and Argumentation in
Japanese Culture
Barnlund (1975) pointed out that Japanese traditional
culture does not always place the highest value on verbal
communication in the communicative activity. The goal of
communication in public is a harmony ("wa") among the
participants. Difference between opinions among the
participants is conceived of as a threat to the harmony.
Therefore, people tend to avoid explicit expression of
disagreement in public. The harmony is often symbolized by
uniformity or homogeneity in appearance, behaviors,
expressions, and so on, within a community. The community
emphasizes following social obligations ("gimu," "giri,"
"tatemae") of the community. Cooperation rather than
competition is highly valued within a community. Therefore,
a person who disregards the community's obligations
sometimes receives rather emotional reactions--e. g.,
accusation, isolation, or expulsion--than rational ones. It
is well-known that even in academic conferences, Japanese do
not openly argue with each other very much. Expressing
direct opposition is considered impolite: Opposition is
usually indirectly or euphemistically expressed. This
communication style of Japanese may be called the "group"
model (Moeran, 1984).
Individuals do not always agree with each other,
of course, in any culture. They need to have opportunities
to express their own opinions and negotiate them. The group
model does not describe those opportunities. As Moeran
points out, there is a complementary model--a "social
exchange" model--of Japanese communication, where
individuals exchange their spontaneous opinions and feelings
(Moeran, 1984). In informal opportunities like private talk
with close friends or conversation in a drinking party of
coworkers, people express rather openly their natural
opinions and feelings ("ninjo," "honne"), and negotiate
them.
For the process of exchanging opinions in the
Western cultures, Toulmin (1958) described a pattern of
argument ("Toulmin model"), consisting of four components:
claims, grounds, warrants, and backing. These are
"armaments" of the Western style argumentation. As Lakoff
and Johnson (1980) suggest, a "war" metaphor underlies this
argumentation:
"Many of the things we do in arguing are
partially structured by the concept of war. Though there
is no physical battle, there is a verbal battle, and the
structure of argument--attack, defense, counterattack,
etc.--reflects this." (Lakoff & Johnson, 1980, p. 4)
In contrast, in Japan, exchanging talks in either public
or private is usually referred to as "hanashi-ai": The word
means mutual conversation or consultation, and does not
signify a war. Because people try to avoid direct
confrontation, they try to put their opinions ambiguously so
that they can withdraw or change them easily when others
indicate opposition. As a result, people in "hanashi-ai" do
not usually bring up such full logical defense devices like
"grounds," "warrants," and "backing." Even in those
situations where the social exchange model is working,
people tend to avoid bringing up logical armaments because
they feel that arguing logically is impersonal
("katakurushii"). In ordinary life, logic ("ronri") is often
equated with "rikutsu." The latter is often used
derogatorily. Arguments that emphasize "rikutsu" are
considered superficial and not reaching the audience's
hearts. Therefore, even in the social exchange model,
logical argumentation is not preferred.
Argumentation and Proof in
Japanese Classrooms
In Japanese schools, the classroom processes contain both
formal and informal opportunities, in the sense of the
general Japanese culture. Classroom lessons usually contain
processes of exchanging opinions in a whole-class or
small-groups. These processes are again called "hanashi-ai"
as in the adult societies, and teachers play an important
role for managing "hanashi-ai" in the class.
As mentioned already, confronting someone's
argument in public is not encouraged in Japanese culture. In
school, children are not totally socialized to the adult
culture, however. They sometimes directly express opposition
or disagreement in classroom talks, and may endanger the
classroom harmony. The classroom teacher plays an important
role here. The teacher expresses respects to individual
children's ideas, whether they are valid or not. The teacher
tries to use a conflict between children's claims as a good
opportunity to deepen children's understanding of the issue
in question. That is, the teacher handles the conflict not
just as a problem between the involved children but instead
frames a problem of the whole class from it: The conflict is
shared among the classroom participants, and becomes "our"
problem (cf. Lewis, 1995, pp. 125-130). The teacher
encourages the whole class to think about it and give
suggestions. All the class members are supposed to work
together towards resolution of the problem, so that the
reached resolution produces a recovery of the harmony in the
classroom community.
Japanese teachers pose a rather challenging
problem in the opening of lesson. They encourage children to
present their own ideas for solving the problem. In the
lesson, the teacher asks children to do "hanashi-ai" in
small groups, or a whole class. Because the problem is
difficult, children often make wrong conjectures and ideas,
and procedural mistakes. Also, since the problem is often
open-ended, children may produce several different
solutions. The teacher encourages them to compare their
ideas and solutions with each other. At those occasions,
counterexamples may be found, and counterarguments may
occur. The teacher intentionally uses such opportunities to
stimulate children's thinking. Japanese traditional
discipline (or moral) places great emphasis on reflecting
("hansei") on one's own mistakes and appreciating
contributions from others, so that it encourages cooperation
among children (cf. Lewis, 1995). Though "hanashi-ai" may
eventually conclude which solution is better, correct,
efficient, elegant, or whatever, competition among children
is generally discouraged. Therefore, in principle, no winner
and no loser exist in "hanashi-ai," unlike the Western-style
argumentation.
Mathematics lessons in Japanese schools
emphasize "wakaru" (understanding) of mathematical ideas.
Memorizing formulas and mastering skills are not considered
the central theme of learning. In school mathematics, we
emphasize the importance of asking questions "why?" in
thinking: "Why" questions encourage asking to search the
"origin" (causes or basic premises) of the phenomenon in
focus and to describe a (causal or logical) path
("sujimichi") leading from the origin to the phenomenon.
Answer to the "why?" is termed "wake" or "riyu" (reasons).
The activities of finding and explaining "wake" or "riyu"
are considered essential for learning of mathematical proof
in Japan.
Mathematical proof is called "shoumei" in
Japanese. In junior high school, explaining "wake" or "riyu"
is often simply called "setsumei." Activities of doing
"setsumei" are commonly held before introducing the concept
of mathematical proof "shoumei." The terms "wake," "riyu,"
and "setsumei" are all commonly used in students' everyday
life. In contrast, the term "shoumei" rarely appears in
everyday life; therefore, it has to be explicitly introduced
and instructed in school. In Japanese schools, the term
"shoumei" is first introduced to the students in geometry
lessons of the eighth grade mathematics. In lesson,
"shoumei" of a mathematical claim is usually defined either
as an act of showing logically that the claim is true, or as
a written document of that act. And, "shoumei" is conceived
of as a special kind of "setsumei," characteristic of
mathematics.
The instruction of mathematical proof has been
traditionally conceived of in the above mentioned group
model of Japanese communication. "Shoumei" has to deduce the
stated claim by following the accepted premises. This
corresponds well to the idea of "following the social
obligations of the community." Therefore, the group model of
Japanese public communication seems to fit well the process
of showing proofs. The instruction of mathematical proof and
the structure of "hanashi-ai" in Japanese classrooms seem
more consistent with Japanese traditional communication
styles than Toulmin model.
The above conclusion suggests that it would be culturally
difficult for Japanese to fully actualize the argumentation
of Toulmin model in the classroom, and to teach mathematical
proving as argumentative activity. The idea that learning of
mathematical proof should be located in the context of
argumentative activity seems highly influenced from the
Western cultures. Should Japanese educators adopt that
idea?
References
Barnlund D. C. (1975) Public and private self
in Japan and the United States: Communicative styles of two
cultures. Tokyo: The Simul Press.
Lakoff G., Johnson M. (1980) Metaphors we
live by. Chicago: The University of Chicago Press.
Lewis C. C. (1995) Educating hearts and minds:
Reflections on Japanese preschool and elementary
education. Cambridge, UK: Cambridge University
Press.
Moeran B. (1984). Individual, group and seishin:
Japan's internal cultural debate. Man 19,
252-266.
Toulmin E. S. (1958) The uses of argument.
Cambridge. UK: Cambridge University Press.
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