La lettre de la Preuve

ISSN 1292-8763

 
Argumentation and Mathematical Proof in Japan

This paper is a response to some issues addressed by Nicholas Balacheff about relationships between argumentation and mathematical proof in the cultural context. We discuss these issues from a Japanese perspective, and focus on the ways Japanese culture affects argumentation, mathematical proof, and their relationships.
  For this purpose we take the concept of communication as an overall framework. This is because it encompasses argumentation, mathematical proof, and their relationships, that is, we consider that

(1) argumentation is a type of verbal communication,

(2) mathematical proof is an important component of communication in the mathematical community, and

(3) communication is one of the main topics of cultural studies.

In the following, we first describe communication styles in Japanese culture, comparing with those of the Western culture. Then, we 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 and Western Cultures

In Western culture, the goal of communication is considered to be a valid conclusion. Discussion is a tool to explore the problem concerned. Expressing one's own opinion and confronting others are conceived of as deepening the understanding of each other. Here, better understanding of differences between opinions is considered facilitating good human relation.
  Barnlund (1975, 1989) 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. Barnlund illustrated Japanese style of communication in decision-making processes, comparing with that of the United States:

"In the United States problems are sharply defined, causes of difficulty identified, alternative proposals offered and challenged, decisions hammered out through a process of argument and compromise. In Japan, decision-making follows a different course. The discussion may proceed at some length without any clear specification of the problem. Participants proceed cautiously, attempting to decipher the opinions of others without asking them directly. Various points of view are intimated, but so expressed that they can later be qualified or retracted if they encounter resistance. The leader in the American case alternatively challenges and crystallizes the views expressed. He presses to effect a decision in the allotted time. In the Japanese case the leader sensitively listens for or nourishes whatever themes seem to be drawing unanimous support. At any suggestion of a serious difference of opinion the meeting may be postponed. Perhaps at another time group members may be more of the same mind. If not, the matter can and should be delayed until everyone is comfortable with its disposition. Differences appear to be emphasized and encouraged in the United States as a way of stimulating a wider variety of solutions. Differences appear to be minimized or suppressed in Japan in the interest of preserving the harmony of the group." (Barnlund, 1975, pp. 136-137)

The above communication style of Japanese may be called the "group" model, and it originated from Confucianism of ancient China (Moeran, 1984, 1989/1993).
  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, 1989/1993). 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, Lakoff and Johnson (1980) suggest that a "war" metaphor underlies argumentation in the Western culture:

"This metaphor [ARGUMENT IS WAR] is reflected in our everyday language by a wide variety of expressions: ...
  It is important to see that we don't just talk about arguments in terms of war. We can actually win or lose arguments. We see the person we are arguing with as an opponent. We attack his positions and we defend our own. We gain and lose ground. We plan and use strategies. If we find a position indefensible, we can abandon it and take a new line of attack. 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. It is in this sense that the ARGUMENT IS WAR metaphor is one that we live by in this culture; it structures the actions we perform in arguing." (Lakoff & Johnson, 1980, p. 4)

This holds true in a single argument also. Toulmin (1958) and Toulmin, Rieke, & Janik (1984) described a pattern of argument ("Toulmin model"), consisting of four components: claims, grounds, warrants, and backing. These are to responses to (real or hypothetical) challengers' questions. A "claim" is a statement which clarifies a topic of the discussion, and a position the arguer tries to defend about the topic. "Grounds" are the data or information which the claim is based on, responding to questions like "What have you got to go on?" "Warrants" are to justify the relevance of the grounds to the claim, taking the form of rules, principles, standards, and the like, responding to questions like "How do you get there?" "Backing" is to assure that the warrants are reliable, and that they are applicable to the present context, responding to challenges to the warrants. Thus, the argument structure also reflects the Western style argumentation, as van Eemeren, Grootendorst, Jackson, and Jacobs (1997) point out

"..., the argument structure [of Toulmin model] is really the product of an interaction with each part of the argument defined in terms of some specified interactional function--as answers to particular questions or challenges to the initial claim." (p. 217)

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 (Nakayama, 1989). 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.

Proof and Argumentation in Japanese Classrooms

Mathematical proof is called "shoumei" in Japanese. In the following, we first describe where the concept of "shoumei" is located in mathematics teaching in Japan, and discuss how Japanese culture affects its instruction. Then we discuss the argumentation in mathematics classrooms in Japan, and how it is related to Japanese culture.
  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.

Mathematical Proof in Classrooms

Mathematics lessons in Japanese schools emphasize "wakaru" (understanding) of mathematical ideas (e. g., Stigler & Hiebert, 1999). Memorizing formulas and mastering skills are not considered to be 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 (cf. Kumagai, 1998). These include descriptions about problem solving processes (e.g., "Write an equation to represent the problem situation") and justification of procedures or steps employed in those processes (e. g., "Why did you do so?").
  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.

Argumentation in Mathematics Classrooms

As mentioned already, confronting someone's argument in public is not encouraged in Japanese culture: Opposition is usually indirectly or euphemistically expressed. 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 wrong 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 (e. g., Stigler & Hiebert, 1999). 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.

Concluding Remarks

We indicated that the instruction of mathematical proof and the structure of "hanashi-ai" in Japanese classrooms are more consistent with Japanese traditional communication styles than Toulmin model. One may wonder if the instruction of mathematical proof and structures of discussion in the mathematics classroom of Western countries are consistent with Toulmin model. That seems not necessarily to be the case. For example, as Schoenfeld (1988) and Gregg (1995) pointed out, the instruction of mathematical proof in the United States seems not to encourage students' argumentative activity. Though there are some attempts that succeeded in producing Toulmin-type argumentation in mathematics classrooms (Fawcett, 1938/1995; Krummheuer, 1995), they are exceptional. We are afraid that this gap between the classroom instruction of mathematical proof and the general communication styles may further intensify the isolation of the former from the social life in the United States.

References

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