a study of reflective thinking: patterns in interpretive discussion

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A STUDY OF REFLECTIVE THINKING: PATTERNS IN INTERPRETIVE DISCUSSION

Sophie Haroutunian-Gordon The School of Education and Social Policy

Northwestern University

INTRODUCTION A cardinal preoccupation of the current educational reform movement is that of

“teaching people to think.” This is a preoccupation that John Dewey popularized,’ and many since his time have urged its importance.2 Yet, what we should do to teach people to think remains in question.” In what follows, I address the question from a limited but, I hope, helpful perspective. I use the term “thinking” in reference not to every kind of thinking but to that thinking which takes place as a group of people converse about the meaning of some text, that is, engage in what I call ”interpretive discu~sion.”~ Interpretive discussions occur in many classrooms, in all subjects, and at all grade levels, and so are of particular interest to educators. The thinking that one sees taking place in such conversations shares features with Dewey‘s notion of “reflective thinking.”j The exploration that follows illuminates, I believe, some features of this style of thinking. It also has implications for the problem of ”teaching people to think” - a problem that will receive further definition as the discussion proceeds.

In How We Think, Dewey gives the example of a ditch that must be crossed, although there is no bridge at hand.6 From my perspective, the situation he describes

1. John Dewey, How We Think: A Restatement of the Relation of Reflective Thinking 10 thc Educative Process (Boston: Heath, 19931, particularly chap. 7. 2. Gareth 8 . Matthews, Philosophy and the Young Child (Cambridge: Harvard University Press, 1980) and Mortimer J. Adler, Reforming Education: The Opening of the American Mind [New York: Macmillan, 1988). More recently, the authors whose work is represented in Re-Thinking Reason: New Perspeclives ~n Critical Thinking, ed. K.S. Walters (Albany: State University of New York Press, 1994) and Reason and Values: New Essays in Philosophy of Education, ed. J.P. Portelli and Sharon Bailin [Calgary, Alberta: Detselig Enterprises Ltd., 1993) argue the importance of educating to reason and make judgments from a variety of perspectives. A somewhat broader discussion of thinking as it takes place in dialogue may be found in Nicholas C. Burbules, Dialogue in Teaching: Theory and Practice (New York: Teachers College Press, 1993). The discussion of the present paper focuses upon the ways in which students engage in and develop patterns of thinking in one type of dialogue situation. 3 . The debates on issues pertaining to this question may be explored by perusing several volumes in the Philosophy of Education Research Library, including the following: J.E. McPeck, Teaching Critical Thinking: Dialogue and Dialectic [New York: Routledge, 1990); V.A. Howard, ed., Vurieties of Thinking: Essays from Harvard’s Philosophy of Education Research Center (New York: Routledge, 1990); and Harvey Siegel, Educating Reason: Rationality, Critical Thinking, andEducation (New York: Routledge, 1988). For more recent commentary on the debates, see also footnote #2. 4. Sophie Haroutunian-Gordon, Turning the Soul: Teaching Through Conversation in the High School (Chicago: University of Chicago Press, 1991). 5. Dewey, How W e Think, chap. 7 . 6 . Ibid., 107-15.

EDUCATIONAL THEORY / Winter 1998 / Volume 48 / Number 1 0 1998 Board of Trustecs / University of Illinois

34 E D U C A T I O N A L T H E O R Y WINTER 1998 / VOLUME 48 / NUMBER 1

constitutes a “text” that must be reflected upon in order to be understood. For Dewey, reflective thinking involves resolving an indeterminate, problematic situation. The procedure for resolution has been called “following the scientific method.”’ In my view, “following” involves what George Mead describes as the creation of meaning:

Thinking always implies a symbol which will call out the same response in another that it calls out in the thinkcr. Such a symbol is a universal of discourse; it is universal in its character. We always assume that the symbol we use is one which will call out in the other person the same response, provided it is a part of his mechanism of conduct. A person who is saying something is saying to himself what he says to others; otherwise, he does not know what he is talking about!

According to Mead, thinking involves using symbols, The symbols are ”universal” because they call out of each member of the symbol-using community the same response when they are used. To think the phrase, ”This is a chair,” calls out of each thinker in the community a limited and well-defined set of responses - responses that each member has learned from others in the community by watching them respond under the conditions in which the phrase “this is a chair” is uttered. Thinking reflectively, or following the scientific method, then involves discovering and, in some instances, creating meaning: As one speaks with others in the community, one discovers the community’s patterns of response to what one says; in some instances, participants in the conversation come to new, agreed-upon patterns of response to what is said (they create meaning). Through these exchanges, conclusions are drawn and actions are determined.

In what follows, we will see how meaning is created as well as discovered as certain patterns of responding to symbols are fol l~wed.~ The patterns we shall study can be seen when a group of students has what 1 and others call an “interpretive discussion.”l” A prototypical interpretive discussion occurs in a classroom among a group of fifteen to eighteen discussants and a leader, all seated in a circle. The aim

7. See, for example, Ernest Nagel, The Structure of Science in the Logic of Scientific Explanation, 2d. ed. (Indianapolis: Hackett Publishing Co., 1979).

8. George H. Mead, Mind, Self, and Society, ed. C.W. Morris [Chicago: University of Chicago Press, 19341, 147.

9. One might say, with Wittgenstein, that the word “thinking” refers to a set of activities or language gamejsj that goes on in a community of language users under certain conditions. The game is played by manipulating symbols - symbols whose meaning is established by members of the community and which, when used in the community, draw out of each language user similar responses. Furthermore, the participants presuppose that conditions being equal, others in the community will respond to the symbols as do thcy themselves. The activity of thinking, as defined here, then, involves participating in a community that follows accepted rules and thcreby uses and modifies terms for thought.

10. As will become clear, the “patterns” to which I refer are not what Courtney Cazden, Classroom Discourse, The Languages of Teaching and Learning (Portsmouth, N.H.: Heineman, 1988) calls discourse patterns. The patterns described in what follows are patterns in the content of what is said in discussion, not patterns in the style, structure, or function of the discourse. See Great Books Foundation, Leader Training Manual, for use of the phrase, “interpretive discussion.”

SOPHIE HAROUTUNIAN-GORDON is Professor, School of Education and Social Policy and Director of the Science in Education Program at Northwestern University, 11 7 Annenberg Hall, 21 15 North Campus Drive, Evanston, IL 60208-2610. Her primary areas of scholarship are philosophy of education, philosophy of psychology, aesthetics, teacher education, and dialogue and text analysis.

HAROUTUNIAN-GORDON A Study in Reflective Thinking 35

of a discussion is to cultivate questions about the meaning of some text - questions that the group genuinely wishes to resolve. The “text” may be a set of data, a literary work, a film, an artifact, a painting, a philosophical work - indeed, any object that has enough ambiguity to permit questioning about its meaning. The text must also be complex enough to permit resolution to be pursued, so that the conversation both develops and addresses questions about the meaning of the text that all participants have read or examined. Unlike more traditional classroom conversations, the aim is not to impart to discussants the “correct” interpretation (namely, the teacher’s interpretation), although the concern for correctness is ever-present. The authority and arbitrator of the dispute is sought in the text itself, not the leader or teacher; evidence is culled from work, and arguments are formed to explain the evidence. The dispute is resolved by the argument that best accounts for the evidence, not an authority’s pronouncement. Discussants and leader work to define the question that the group addresses and to pursue resolution by identifying and interpreting aspects of the text that seem relevant to it. The parts of the text that are “covered” in the discussion are those that have bearing upon the questions that arise in the conversation, not ones the leader believes “everyone should understand,” as one might say.

To say that discussants are “thinking” in an interpretive discussion means that they are using terms they share to create an understanding of what the text means - an understanding that arises through their conversation and depends upon the rules they follow in using words, as well as upon the text before them.” The participants in the group are said to be “thinking” because they are saying things to one another about the meaning of the conditions in the text - things to which others in the conversation listen, and about which they may agree or take issue.12 They are, then, establishing points of agreement and disagreement [issues) about responses to the text, thereby creating questions about its meaning. The responding that one sees taking place is based upon each participant’s assumption that others in the conversation can share his or her view of the text, provided that they are shown the nature and basis of that interpretation.

Analysis of transcripts of interpretive discussions reveals that the remarks people make are anything but random. Indeed, as shall be argued, the creation of shared meaning seems to proceed by forming questions -by uncovering what people believe the text to be saying and, in so doing, creating questions that presuppose the existence of certain conditions. In addition, the questions are created as people follow certain patterns that may be identified. The creation of meaning, as described here, is the similar to the well-known exchange in Plato’s Meno, where a slave develops a growing recognition of the height and length required to double the area of a square

1 1. Do not assume that the meaning of the text is radically indeterminate due to thereader’spreconceptions, contra Harold Bloom, The Anxiety of Influence: A Theory of Poetry (New York: Oxford Universlty Press, 1973) and Stanley Fish, Is there a Text in this Class! The Authority of Interpretive Communities [Cambridge: Harvard University Press, 1980). I hold, with Hans-Georg Gadamer, that the reader’s preconceptions need to be explored, but that the nature of the interpretation is constrained by the features of the text and the rules governing its interpretation.

12. Of course, images and ideas may indeed arise “in the heads” of the participants, but the word “thinking” is not used here in reference to these mental events.


-a “recollection“ that seems to be drawn out as Socrates questions him.13 My view is that as the slave speaks with Socrates, talking about the problem by following particular patterns, the outcome of each of which is accepted or rejected, the two create both questions and a shared understanding of the problem‘s features. I use the word “create“ because the questions and points of agreement are reached through the dialogue; they didnot exist before the dialogue began. Likewise, as discussants follow the patterns and assent to or reject particular statements about a text, they create questions and points of agreement, and discover implications following from these.I4

In summary, the use of the term ”thinking” here presupposes the manipulation of symbols whose meaning is or becomes shared by those in the linguistic community. Given the definition, one may ask: How, then, is the meaning created and discovered? That is, how, by following patterned practices for usinglanguage that the discussants accept, are points of agreement and disagreement reached, so that conditions in the text are identified and questions opened - questions that, when addressed, modify ideas about features of the text? This is a more precise formulation of the opening question, How does reflective thinking (as it occurs in interpretive discussion) proceed? The remainder of the paper addresses this question. The outcome of this analysis has implications for classroom practices that help reflective thinking to occur and for teacher education, and some of these implications will be briefly discussed in the concluding section of the paper.

DISCOVERING AND CREATING MEANING THROUGH CONVERSATION

To understand how thinking proceeds in interpretive discussion, it helps to focus upon a text. In what follows, we consider three texts. Two of them are works of art. The first is Plato’s Protagoras, which comes from the so-called ”early” period of Plato’s dialogues in which Socrates appears in conversation with pe0p1e.l~ The second text is the courante from French Suite #5 by J.S. Bach, BWV 816, first published in Bach’s notebook for his wife, Anna Magdalena, in 1722. Bach‘s music, in which rich conversation between voices takes place, is like Plato’s Protagoras in that both present examples of dialogue that may be studied so as to address our question.

Both, however, are works of art, so that our use of them raises an issue: Do spontaneous conversations about the meaning of texts, as might occur in classrooms, discover and create meaning in ways similar to those observed in the artistic

13. Plato, Menu, 82c-8ha.

14. A thoughtful reader of an earlier draft of this essay noted that Mead, in M i n d , Self, and Society, maintained that awareness or consciousness is not rcquired for situations to draw forth a meaningful response (pp. 77-8 1). Consciousness of conditions, however, can arise as images, based upon previous experience, allowing us to identify aspects of our situations (pp. 344.46). From Mead’s perspective, one might say that the creation of meaning, which occurs as one acts and observes the responses of others to one’s actions, opciis the possibility for the discovery of further meaning- meaning that is entailed, at least logically, by that which has been created. The discoveries are made as imagcs form andareuscd to interpret situations.

15. The ”carly” period includes those dialogues composed before Plato’s first trip to Syracuse, Sicily, in 388 B.C.

HAKOUTUNIAN-GOKDON A Study in Reflective Thinking 37

examples? This question is explored later in this essay, where we examine an actual classroom conversation that took place at Northwestern University. We begin with the artworks because they suggest ideas about patterns of discovering and creating meaning that seem to be followed in nonfiction conversations as well. The patterns are cleaner and easier to see in the artistic pieces, Furthermore, the courante from Bach’s French Suite #5 reveals features of the patterns that are less obvious yet still present in the literary example, Plato‘s Protagoras. So, these works function as analogies - analogies that suggest a way in which reflective thinking about the meaning of texts seems to proceed.

We begin, then, with the Protagoras, and I use this example to introduce the idea that reflective thinking about meaning seems to follow the pattern of a “cluster” of related questions that evolve as the thinking moves along.

THE ANALYSIS THE CLUSTER OF QUESTIONS

The features of a cluster of questions can be grasped by considering the following excerpt taken from Plato’s Protugorus, which opens with the following conversation between Socrates and an unnamed friend:

Friend: Where have you come from, Socrates? No doubt from 309

pursuit of the captivating Alcibiades. Certainly when I saw him only a day or two ago, he seemed to he still a handsome man, but

between ourselves Socrates, “man” is the word. He’s actually growing a beard.

Socrutes: What of i t? Aren’t you an enthusiast for Homer, who says

that the most charming age is that of the youth with his first beard,

just the age Alcihiades now? Friend: Well, what’s thc news? Have you just left the young man, and how is he

disposed toward you? Socrates: Very well, I think, particularly today, since he came to my assistance and spoke up for me at some length. For as you guessed,

I have only just left him. But I will tell you a surprising thing.

Although he was present, I had no thought for him, and often forgot him altogether.

Friend: Why, what can have happened between you and him to make such a difference? You surely can‘t have met someone

more handsome -not in Athens at least?

Socrutes: Yes, much more. Friend: Really? An Athenian or a foreigner? SocraLes: A foreigner. Friend: Where from?

Socrates: Abdera. Frieiid: And this stranger struck you as such a handsome person that you put

him above the son of Clinias in that respect?

Socrutes: Yes. Must not perfect wisdom take the palm for handsomeness?

b

C


Friend: You mean you have just been meeting with some wise man? Socrates: Say rather the wisest man now Iwing, if you agree that that description fits Protagoras. Friend: Whnt! Protngoras is in Athens? Socrates: And has been for two days. Friend: And you have just now come from seeing him? Socrates: Yes, we had a long talk together. Friend: Then lose no time in telling me about your conversation, if you arc free. Sit down here; the slave will make room for you. Socrates: Certainly I shall, and be grateful to you for listening. Friend: And I to you for your story.16

d

310

This introductory exchange may seem perfunctory, but reflection upon it could lead one to the following question: Why does Socrates wish to tell the story of his meeting with Protagoras to the unnamed friend? The lines of the dialogue suggest several possible answers. For example:

1. Does Socrates wish to tell the story because he wants the chance to brag about his intellectual victory over Protagoras? If so, is this why he refers to Protagoras as ”the wisest man now living” (309d)? 2. If Socrates wishes to brag, why does he indicate that he does not think Protagoras is wise by saying, ”the wisest man now living, if you agree that that description fits Protagoras” (309d)? 3. Does Socrates want to tell the story to the friend in order to figure out whether he got the better of Protagoras in the argument? If so, is this why he says that Protagoras has “perfect wisdom” (309c)? Is it why he tells the friend that he and Protagoras had “a long talk together” (3 lOa)? 4. Does Socrates want to tell the story in order to figure out the insights that were reached in the discussion with Protagoras? If so, is this why he says that the friend will do him a “favor” by listening to the story (310a)? 5. Finally, does Socrates wish to tell the story so as to deepen the friend’s admiration for him? If so, is it because the friend is already impressed with Socrates’ wit (309a) and his “captivation” of Alicibiades (309c)? If Socrates wishes to deepen the friend’s admiration, why does Plato not dignify the friend by giving him a name? The forgoing constitutes a “cluster” of questions. That is, it consists, first, of a

“Basic Question” (BQ) - a question about the meaning of the text that cannot be resolved definitively but can be explored given what the text presents. It is a question about whose answer there is genuine doubt, and let us call it an ”Interpretive Question,” as it is a question about the meaning of the text. Second, there are

16. Quotations from Plato’s dialogues are taken from the translations collected in the edltion by Huntington and Cairns, Collected Dialoglies (Princeton, N.J.: Princeton University Press, 1961).

H AROUTUNIAN-GORDON A Study in Reflective Thinking 39

questions about the text’s meaning which, again, cannot be resolved definitively (in other words, interpretive questions) but which seem to have implication for resolving the BQ. So, the question “Why does Socrates wish to tell the story of his conversation with Protagoras to the unnamed friend?” is a BQ, as there are more interpretive questions that, if pursued, may have implication for resolving the issue.

Now, the cluster of questions above is itself a work of fiction. It arose after reading the dialogue, writing down genuine questions - real questions for me - about its meaning, and then seeing how the questions related to one another.I8 In looking at the questions, it occurred to me that Socrates’ desire to tell the story to the unnamed friend is not clearly explained, and curiosity about this fact intensified. Some of the questions I had written pointed to passages in the text that seemed to have implications for resolving the problem, and those were then included in the cluster. Further questions occurred as the text was reviewed with the BQ in mind. While the cluster itself is a fiction, the questions evolved in a natural way as I began to draw out the meaning of the text by questioning it and pursuing resolution.

So much for the definition of the phrase “cluster of questions.’’ Now, let us turn to the twofold hypothesis to be explored below. My claim is that in his dialogues, Plato presents conversations between characters, often including Socrates, in which one or more clusters of questions evolve over the course of the conversation. Furthermore, these questions arise as the participants follow certain identifiable patterns. In Protagoras, the question that engages Socrates and Protagoras is: What does Protagoras teach his students (319b)? This is an interpretive question because the text - the things Protagoras says -provides evidence for pursuing its resolution yet does not answer it definitively. Since the text before the characters is the statement of their own beliefs, the dialogue consists of people raising questions about the meaning of what others have said, questions that build upon agreements reached about certain points. My hypothesis is that these questions are neither random nor meandering, although it may appear otherwise. Rather, resolving these questions has implications for addressing the initial dilemma, and so, constitute follow-up questions. They also arise as participants follow certain recognizable patterns. CONVERSATION IN A FICTIONAL EXAMPLE: A CLUSTER OF QUESTIONS’9

To see that the conversation in Protagoras evolves through the formation of a cluster of questions, consider further the conversation between Socrates and Protagoras. Socrates begins their exchange by asking Protagoras what he hopes to

17. For example, if one argues, in rcsponse to Question #3, that Socrates is trying to figure out whether hc got the better of Protagoras in the argument, one might then go on to argue that given his goal, Socrates wishes to recount his conversation in order to resolve his question Question #3 and that the others listed follow-up the BQ because there is at least one way of resolving it that has implications for the resolution of the BQ.

18. As Gadamer distinguishes between an “open” and “distorted” question - one that has to be asked, given the presuppositions: If one accepts x and y, then the question of z is raised; Hans-Georg Gadamer, Truth and Method, 2d. ed. [New York: Crossroads Publishing Company, 1985), 327.

19. I call this dialogue “fictional” because I assume that it is not a verbatum transcript of an actual conversation between Socrates and Protagoras, or at least not entirely so.


teach his pupils (318a). After some questioning by Socrates, Protagoras declares that he ”makes men good citizens’’ (3 19a). Socrates’ question, taken together with Protagoras’ response, functions as the BQ: The question of what Protagoras hopes to teach his pupils and his answer are explored in depth through the ensuing conversation. The question is addressed and clarified as its meaning is created through the remainder of the dialogue.

In response to Protagoras’ claim, there is a second exchange: Socrates presents some observations from which he concludes that men cannot be taught to be good citizens, as virtue cannot be taught (320b). To this, Protagoras responds with a long allegorical tale and an argument, from which he concludes that virtue can be taught and that he is “rather better than anyone else at helping a man to acquire a good and noble character” (328b). Again, we may speak of this exchange as having two phases: Socrates’ argument to the conclusion that virtue cannot be taught and Protagoras’ story and argument to the conclusion that it can be taught. This second exchange feels like a follow-up question because it has implications for resolving some questions raised by the first exchange: Socrates began by asking Protagoras what he claims to teach people, to which Protagoras responds that he teaches them to be good citizens (BQ): Socrates responds by arguing that virtue cannot be taught, to which Protagoras responds with an argument to the effect that it can be taught (Follow-up Question #l). So, when Socrates responds to Protagoras by arguing from examples to the conclusion that virtue cannot be taught, and Protagoras counters with an allegory and argument from which he concludes that the opposite is true, the issue posed initially deepens: Each speaker has presented further evidence for exploring the question of whether Protagoras can rightfully claim to teach his pupils to be “good citizens.” As the reader examines these arguments, the meaning of the initial exchange between Socrates and Protagoras becomes clearer and more intense, and the issue of whether virtue can be taught becomes more pressing.

Now, the second exchange between Socrates and Protagoras does not fully resolve the BQ - the issue of what Protagoras teaches his pupils. Indeed, when Protagoras finishes ”his long and magnificent display of eloquence” (328d), which terminates their second exchange, Socrates says,

Now thcn, Protagoras, there is just one small question left, your answer to which will give me all I want. You say that virtue is teachable, and there IS no one I would believe sooner than you. But there is one thing which took me by surprise in your speech, and I should like you to fill this gap in my mind. You said that Zeus bestowed on mcn justice and respect for their fellows, and again at several points in your discourse justice and self-control and holiness and the rest were mentioned as if together they made LIP one thing, virtue. This is the point I want you to state for me with more precision. Is virtue a single whole, and are justice and self-control and holiness parts of it, or are these latter all names for one and the same thing (329b-329d)!

To Socrates’ question of whether virtue is one thing for which there are different names or a single whole with different parts, Protagoras responds that the latter is the case (329d). This exchange may be thought of as a second Follow-up Question to the issue of what Protagoras teaches people. For in exploring Protagoras’ answer, Socrates shows that it is contradictory and, hence, that Protagoras‘ claim is based upon fallacious reasoning. Since it is not clear what virtue is- whether it is one thing with different parts or not - then it is not possible to say whether it can be taught

H AROUTUNIAN-GORDON A Study in Reflective Thinking 41

and, hence, whether Protagoras is right in saying that he teaches people to be good citizens.

Socrates attempts to resolve the issue of whether virtue is one thing or many by drawing an analogy to the face (329d), asking Protagoras whether he believes that different virtues are like different parts of the face. Here, then, is a third Follow-up Question. For again, if this question is resolved, it may be possible to say what virtue is and hence whether it can be taught. Protagoras’ response to the question saddles him with a conclusion he cannot accept (332a), so Socrates tries another tack, this time asking Protagoras whether he recognizes the existence of folly (332a) -what may be called Follow-up Question #4. For again, if the inquiry is resolved, the possibility of defining virtue and determining its teachability exists.

Once again, Protagoras’ response dissolves in contradiction (333b). In trying to determine which horn of the dilemma to embrace, Socrates poses yet another question (Follow-up #5): Are good things those that are beneficial? Protagoras resists this definition of “good,” and the conversation breaks down until he assumes the role of the questioner and asks Socrates to consider the definition of virtue with reference to apoemby Simonides (339b): Is the poet consistent or inconsistent? asks Protagoras (339c). It is Protagoras, then, who poses Follow-up Question#6. For the poem is about the meaning of the term ”good,” so that interpretation of it may cast light on the definition of the term “virtue.” “Consistent,” is Socrates’ response (339c), to which the rejoinder again is disagreement (339d) and the debate continues (to 347a).

After this discussion, Socrates asks, Do the terms “wisdom,” “temperance,” ”courage,” “justice,” and “holiness,” stand for ”a single reality” or does each refer to a particular entity underlying it, ”a reality with its own separate function, each different from the other” (349b)Z This is a reiteration of an earlier question, but with added specificity andin a new context, so that we may call it Follow-up Question #7. Protagoras’ response is that “all these are parts of virtue, and that four of them resemble each other fairly closely, but courage is very different from all the rest. The proof of what I say is that you can find many men who are quite unjust, unholy, intemperate, and ignorant, yet outstandingly courageous” (349d). An attempt to respond to this claim brings Socrates to another question: Is pleasure, qua pleasure, a good thing and pain, qua pain, bad (351c)? Here, then, is Follow-up Question #8: If pleasure, qua pleasure, is good, and pain, qua pain, bad, then it follows that in choosing the pleasurable, one must always choose the good, so that virtue will constitute knowing what is pleasurable (good); in other words, virtue will be knowledge. The two investigate the answer to Socrates’ question through discussion, the conclusion of which is that all the virtues are, in fact, one, namely knowledge (360d-360e). Socrates himself acknowledges the relation between Fol- low-up Question #8 and the basic question with which they began:

It seems to ine that the present outcome of our talk is pointing at us, like a human adversary, the finger of accusation andscorn. If it hadavoiceit would say, “What anabsurdpair you are, Socrates and Protagoras. One of you, having said at the beginning that virtue is not teachable, now is bent upon contradicting himself by trying to demonstrate that everything is knowledge - justice, temperance, and courage alike - which is the best way to prove that virtue is teachable. If virtue were something other than knowledge, as Protagoras tried to provc, obviously it could not be taught” (36 1 b) .

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So what has happened in the conversation between Socrates and Protagoras? They began with a question - What Protagoras hopes to teach people - which functioned as the Basic Question in the conversation. The eight Follow-up Ques- tions, then, were not random, and pursuing their resolution shed light upon the initial issue. Indeed, in addressing the follow-up questions, Protagoras and Socrates appear to have reversed their initial responses to the BQ.

My position, then, is that Plato’s Protagoras presents a conversation that shares many features with the interpretive discussion defined above, although the “text” is the contents of people’s remarks rather than a document. The aim in both Socrates’ conversation with Protagoras and the interpretive discussion is to resolve a question about meaning that the discussants have. The Protagoras provides evidence that the conversation is not random; the questions that arise, spontaneously it seems, form a cluster of questions.

The analysis to this point, however, raises the following question: If conversation about the meaning of texts proceeds by cultivating a sequence of questions, including a BQ and Follow-up Questions, then how do Follow-up Questions evolve in the conversation so that the meaning of the basic question is created? May one identify patterns that discussants seem to follow such that the cluster of questions, with its coherence, is created? It is to this matter that we now turn, first with reference to a musical example, then back to the Protagoras, and, finally, to an example of a classroom Conversation.

CREATING THE CLUSTER OF QUESTIONS IN CONVERSATION BY FOLLOWING PATTERNS To begin, it helps to use an example that shows the patterns more simply than

does the Protagoras. With the patterns clearly in view, one may return to Protagoras and see them there with greater ease. Let us look, then, at Figure 1, a musical example, the courante from the Bach French Suite #5, written in the key of G major.‘” The courante, a baroque dance, in triple meter (three beats per bar), is divided into two sections (A and B), each of which has sixteenmeasures. One might say that this dance is a kind of brief conversation in which Section B constitutes the response to section A. For section A begins in the key of G and ends in the dominant of that key -D major -while section B begins in the dominant and returns to G major at its conclusion.

Let us consider in detail section A of Bach’s courante. M1, the first phrase or musical idea, is set forth immediately by the upper voice (measures 1 and 2) and has the following features: it is in the key of G major; it begins on G with an eighth note followed by a descending sequence of sixteenth notes (measure l), followed by a sequence of eighth notes that ascends back to G, from whence it began. It is the “return” to G that makes the secondpart of M1 feel as if it were “answering” the first. The “answer” is not random: it is there because the first phrase of M1 is what it is, and because Bach is writing in a certain style.21 While he might have “answered” the

20. J.S. Bach, French Suite #5, reprinted by permission of G. Henle Verlag, Munich. 21. To speak of the “baroque style” is to speak of rules of conversation that Bach followed. As Gadamer, (Truth and Method) and Ludwig Wittgenstein have shown in detail, conversations are governed by rules, so that not just anything can be said and still permit the conversation to continue - or continue with the same character.


Figure 1

A1 (cont.)

I M2 (reversed) - - - M1 (reversed)

A2 (cont.)

10 hE(varied)

I

A2 (cont.)

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Figure 1 (continued)

HAKOUTUNIAN-GORDON A Study in Reflective Thinking 45

first part of M1 differently, there are many things he could not have written (such as an A-flat minor chord) and still have maintained the conversation he began.

M2, the second musical idea or motive, as it first appears in the upper voice (measure 3) , begins again on G and again with a sequence of sixteenth notes. M2 feels like an ”answer” to M1 because it begins on G (where the second phrase of M1 ended). Here then is the first pattern that seems to be followed in conversations of many sorts: A new sequence begins where one offered previously left o f f and returns to about the same startingpoint (Pattern #l) . Not only do such juxtaposed sequences appear to ”answer” one another, but their presence in the conversation gives it the character of saying more about, or developing an idea.

A second pattern is that of repeating a sequence or some portion thereof in a new context. We see this first in measure 2, where Bach repeats the first part of M1 in the lower voice. The repetition begins an octave lower on the keyboard than did M1 originally. Likewise, the first two of the four-sixteenth note groups that make up the sequence appear no less than five times in the course of Al. The location of the sequence varies: its starts twice on G, three times on D, and once on A. By changing the context in which the sequence or portion of the sequence is repeated, its meaning is elaborated, that is, the number of ideas to which it is related increases. We have, then, Pattern #2: Repeating a sequence or portion thereof in a different context.

Looking again at M1 andM2 as they appear in measures 1 through 4, one sees that M2 varies Ml’s opening sixteenth-note sequence and its eighth-note response by reversing the direction of the note sequences, all the while staying within the octave range established by M1. Instead of descending, as occurred in M1, the sixteenth-note sequence in M2 has an upward thrust (measure 3) ; instead of ascending as it did in M1, the eight-note sequence in M2 has a downward motion (measure 4). We will see further instances in which new ideas are created by reversing a sequence or a portion of a sequence that has appeared previously. Here, then is Pattern #3 by which new ideas are developed in conversations: Reversing the direction of a sequence or portion thereof.

Now we notice a fourth pattern, as the eighth-note answer portion of M2 (first appearing in measure 4) is extended from two notes to six. Interestingly enough, the extension of the answer portion in M2 (upper voice, measure 10) reminds us of the eighth-note answer portion of M1 (upper voice, measure 2), which is constructed upon intervals of a third. That is, the first four notes of the six note sequence in measure 2 are achieved by moving two steps and creating the distance of a third from the previous note. In the sequence in measure 10 (upper voice), however, the second and third notes are a third apart, but the fourth note is the distance of a fourth from the previous one, the fifth drops down a half step so its distance from the previous note is one half step, and the interval between the fifth and sixth notes is a sixth. It is as if Bach is drawing out the meaning of the original answer portion in measure #2 in order to construct his variation of M2 in measure 10. And so we have Pattern #4: Drawing out the elements of a sequence or part thereof.

Bach continues to follow Pattern #4 as the lower voice takes up M2 in measure 9. Interestingly enough, the two parts of M2 are reversed here, part two appearing in

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measure 9 and part one in measure 10 (Pattern #2). In measure 9, then, the intervals between the first four notes are thirds, while the distance between the fourth and fifth notes is a fifth, andbetween the fifth and sixth notes, an eighth, or octave. Notice that the amount of expansion varies as Bach follows Pattern #4: The thirds may become fourths, fifths, sixths, or even eighths; at times, they inay diminish and then reach further. The point is that variation is achieved by expanding a sequence so that its limits or boundaries appear to be stretched out.

The reverse of Pattern #4 occurs when the details of a sequence are filled in so that its structure becomes clearer - what we shall call Pattern #5: Filling in the details of the elements in a sequence. We see an example of this in measures 13 and 14, where the detail of the second part of M1 is drawn out by the upper voice. Here, the eighth notes have become sixteenth notes, and so there are twelve notes rather than six in the upper voice in each of these two bars. By starting on the first note of the upper voice, measure 13, and observing every other note in the bar, one sees almost the sequence that first appears in bar two, upper voice, with the direction reversed so that it moves down rather than up. By adding the second, fourth, sixth, eighth, tenth, and twelfth notes in bar thirteen, Bach embellishes the sequence in bar two, as though its structure - the intervals and the direction of the motive -have been underscored rather than drawn out or extended by the added detail, as is the case when Pattern #4 is followed. The same effect may be observed in the upper voice of measure 14, where again the sequence presented in measure 2 forms the basic structure of the measure and added notes accentuate that structure.22

The foregoing analysis suggests that the meaning of M1 evolves as the composer follows certain patterns:

Pattern #I: Starting a new sequence where a previous one left off;

Pattern #2: Repeating a sequence in a different context;

Pattern #3: Reversing the direction of a sequence or of one of its elements; Pattern #4: Drawing out the elements of a sequence, as occurs when intervals of a third become intervals of a fourth, fifth, or sixth; Pattern#5: Filling in the outline of a sequence so that its pattern becomes clearer.

Through further study of the example, the reader may observe that the patterns appear to be followed repeatedly. And it is by following these patterns that Bach seems to create the new musical phrases that explore and deepen the meaning of his original idea, M 1.

My claim is not that Bach’s courante displays all of the patterns that one may find people following in a conversation where they work together to create meaning. Indeed, there may be additional ones, and the description of those identified here might be revised. However, the description to this point enables me to argue that the discussants in Plato’s Protagoras repeat and vary ideas in ways analogous to Bach’s,

22. One might argue that to draw out logical implications is not to create meaning but to reveal meaning that is already there. In either case, however, new relations between elements become visible, and it is the new appearance of relations that changes the meaning of what has been said.


although their conversation is verbal andnot musical. To see that this is the case, we must look at the Protagoras once again.

After Socrates poses his initial question to Protagoras - “What effect will [Protagoras’] teaching have upon [his pupils]?” (3 1Sa) - Protagoras says they will “make progress toward a better state” (3 18a). As indicated above, this exchange poses the Basic Question of the conversation, for the remainder of the discussion has implications for resolving it. Socrates responds by saying, “What shall [they] get better at and where shall [they] make progress?” (31Sc). He then gives some examples: If one asked the question of Zeuxippus, he would say ”In painting”; if one asked the question of Orthagoras, he would say, “In playing the flute” (3 1 Sc, 3 1 Sd). Having given these examples, Socrates puts the question to Protagoras again. This time, Protagoras responds by contrasting himself with other Sophists, who

plunge their students into special studies [such as arithmetic, astronomy, geometry].. , .But from me [the pupilj will learn only what he has come to learn. What is that subject? The proper care of his personal affairs, so that he may best manage his own household, and also the state’s affairs, so as to become a real power in the city, both as speaker and man of action (318e).

The first point to notice is that each of Socrates’ rejoinders begins from the point at which Protagoras’ end: When Protagoras says that pupils under his tutelage will “make progress toward a better state,” Socrates responds, for example, by asking, “What shall I get better at?” Second, he follows his question by giving examples of the kind of response he is seeking, thereby filling in the detail of the question, “What shallIget better at?” Immediately, then, we see evidence of Patterns #1 and#5, above.

The second exchange between Socrates and Protagoras begins where the first leaves off (again, evidence of Pattern #l): After confirming with Protagoras that he seems to be claiming to make men good citizens (3 19a), Socrates says that it is up to him to say why he believes that such a thing cannot be taught by one person to another (319b). He then argues that while the state would consult specialists in a building project (architects, for example) they would listen to one of any occupation if the topic was something that had to do with the governing of the country. Once again, Socrates enumerates examples to make his point: “The man who gets up to advise them may be a builder or equally well be a blacksmith or a shoemaker, merchant or ship owner, rich or poor” (319d). Again, one might argue that he is following Pattern #5, filling in the detail of his point by giving examples. Similarly, Socrates continues, Pericles gave his two sons the best education he could, but he did not train them in his “own special kind of wisdom,” nor could he train his ward Clinias, whom he finally gave over for instruction to Ariphron. The latter “gave Clinias back; he could make nothing of him” (320b). By describing specific instances, Socrates is filling in the detail of his claim that one person cannot teach another to be virtuous. Or one may put it this way: By following Pattern #5, Socrates creates follow-up question number one - Can one teach others to be good citizens? - and simultaneously argues that one cannot.

An additional point may be made here: As was true in their first exchange, which created the BQ, Socrates argues to his conclusion (that virtue cannot be taught) by citing examples. One might argue that Socrates is repeating the same pattern

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(argument by example) in a new context, much as variations of M1 and M2 repeat their elements in different contexts. Hence, from this perspective, Socrates seems to be following Pattern #2 in creating his response to Protagoras. Furthermore, the meaning of the examples is not quite the same in Socrates‘ two exchanges with Protagoras, just as the meaning of the elements in M1 and M2 vary in different contexts. In the first exchange, Socrates uses the examples to provide analogies: Just as Zeuxippus and Orthagoras would say that their instruction makes their pupils better painters and flute players, so Protagoras ought to say what, specifically, his instruction would enable his students to do. In the second exchange, Socrates uses the examples to help demonstrate a truth: If virtue could be taught, the state would call upon trained experts (people trained to be virtuous) for advice about issues of governance, and virtuous fathers would impart their virtuous characters to their sons. Since neither is the case, virtue cannot be taught. So, in Socrates’ second use of examples, they become not an illustration of how to proceed but the basis upon which the conclusion is drawn. Notice, too, the sense in which Socrates’ use of examples in his first and second exchanges with Protagoras is reversed, much as the elements of M1 and M2 are reversed in some instances: In the first case, Zeuxippus and Orthagoras are used as examples of teachers who are able to impart their expertise (in painting and flute playing) to their students; in the second instance, the examples given (Pericles and Ariphron) are unable to impart that in which they are expert, namely their virtue. Here, then, is evidence that Socrates is following Pattern # 3 . 2 3

The analysis in the preceding paragraph, which argues that Socrates is following not one but three of the patterns identified previously, raises the issue of whether such a thing is possible. The answer appears to be “yes,” which is why it is sometimes difficult to determine the origins of statements that arise in conversations. At times the shift from one pattern to another seems to occur rapidly, as shall be seen. This is not to say that discussants are aware of the shift or of intentionally following the patterns. Perhaps the following and shifting from pattern to pattern so as to create ideas is a skill acquired through participation in conversation.

Let us turn now to the second Follow-up Question: Whether virtue is one thing for which there are several names or a single whole for which there are several parts. When this question arises, Socrates seems again to follow Pattern #1. for he says to Protagoras, “You said that Zeus bestowed on men justice and respect for their fellows, and again at several points in your discourse justice and self-control and holiness and the rest were mentioned as if together they made up one thing, virtue” (329c). Socrates seems to mean that he has heard two things in Protagoras’ remarks: that Zeus bestowed several different virtues on man and that justice, self-control, and holiness were different names for one thing, called virtue. Which, asks Socrates, is Protagoras claiming? In posing this question, Socrates takes up an issue that has been opened by Protagoras’ statements, much as M2 takes up a musical idea that has been

23. Even if one argues that Socrates’ arguments are invalid, his dialogue with Protagoras seems to follow the patterns I have indicated.


posed by M1. Protagoras responds by saying: “Well, that is easy to answer.. . .Virtue is one, and the qualities you ask about are part of it” (329d). This response answers the question and returns the conversation almost to the point at which it was before Socrates posed the second Follow-up Question.

But Socrates is not satisfied that the issue has been resolved, and so he asks, “DO you mean.. .as the parts of the face are parts - mouth, nose, eyes, and ears - or like the parts of a piece of gold, which do not differ from one another or from the whole except in size?” This third follow-up question seems to arise as Socrates again presents examples that function as analogies and aim to show Protagoras, in greater detail, what characteristics his response ought to have. One might say, then, that Socrates creates this third follow-up question by following Pattern #2. He generates examples in a new context. Furthermore, these examples are given to illustrate the meaning of previous question in more detail (Pattern #5).

The fourth Follow-up Question, Does Protagoras recognize the existence of folly? (332a), arises as Protagoras has reached the point of what Socrates sees as frustration:

Socrates: And this is how you suppose justice to be related to holiness, that therc really is only a slight resemblance hetween them?

Protagoras: Not quite that, but not on the other hand in the way that you seem to believe Socrutes: Well, said I, this line of argument doesn’t seem to be agreeable to you, so lets drop it and look a t something else that you said. You recognize the existence of folly [331e-332a)?

Here, Socrates declares that he is coming to the new question by taking up something Protagoras said previously (Pattern #l). However, it is not clear what Protagoras has said that brings Socrates to this question. Protagoras has not stated that he believes in the existence of folly. So why does Socrates begin at this point? One possibility is that Socrates is drawing out the implication of a previous remark to arrive at his question. At 326e Protagoras says, “The state sets up laws.. .and compels the citizens to be ruled and be ruled in accordance with them. Whoever strays outside the lines is punished.” Now, if the state has set up laws and punishes those who violate them, then the possibility of violating laws must exist. If violating laws is equivalent to folly, then Protagoras’ statement may have led Socrates to pose the question of whether Protagoras recognizes the existence of folly. In so doing, then, he may have been following Pattern #4.

Socrates’ question of Protagoras is like Bach’s activity, when, in measure 10, upper voice, he draws out the second part of M2 as it appears in measure 4. The two eighth notes set one third apart become six eighth notes in measure 10, the first three of which are one third apart, then comes an interval of a fourth, then down a half step so that the fifth note is a fifth apart from the first, and the last is a sixth apart from the previous note, returning almost to the G from which the sequence began. The whole feeling of measure 10, upper voice, is that of expanding the relations between the two eighth notes in measure 2. The possibility for this expansion is limited by the octave range that is established by the structure of scales employed in Bach’s music. Likewise, one has the sense that Socrates has arrived at his question, “DO you

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recognize the existence of folly?” by drawing out the implications, by examples, of something said previously by Protagoras.

As further evidence that the patterns previously identified in Bach’s courante are followed by discussants in Protagoras, consider now Protagoras’ use of comparison in the foregoing passages. In his first exchange with Socrates, where the BQ is created, Protagoras compares himself with other sophists and points out the differences between them. The differences are the reverse of one another: the other sophists ”treat their pupils badly” and “plunge [them] into special studies again, ” while Protagoras follows the opposite procedures, he says (318d-318e). In his second exchange, with Socrates, Protagoras tells a story to illustrate why he believes virtue can be taught - a story that compares the activities of Epimetheus to Prometheus (320d-322a). In these instances, comparison is used for two different purposes: in the first case, the aim of the comparison is to clarify Protagoras’ qualities as a teacher; in the second case, comparison is used not to clarify but to account for man’s talents and characteristics. Hence, both Socrates in his use of examples and Protagoras in his use of comparison seem to follow Pattern #2. Furthermore, in Protagoras’ seconduse of comparison, there is attention to filling out the details of the comparison so that it becomes clear what Epimetheus and Prometheus each contributed (Pattern #5).

There is one portion of Protagoras in which one can see Socrates developing questions and creating meaning by following four of the five patterns previously identified. This is the section in which he “puts on his own magnificent display of eloquence’, - an action for which he chastised Protagoras previously (334d). Socrates’ oration (342 b-347b) nearly rivals Protagoras’ (320d-328d) in length, and takes up sequences previously initiated by Protagoras, including that of analyzing a poem ( Protagoras at 339b; Socrates 342b-347b). One might argue, then, that by engaging in his oration, Socrates reverses his previous admonishments and practice of short question-and-answer exchange (Pattern #3). Furthermore, Socrates takes up the analysis of the poem where previous commentaryby Protagoras left it off, returns to that point (Pattern #l), and sets the poem, as well as elements in the analysis, such as the use of examples (345a), in a new context (Pattern #2). The “new context” is the Spartan model of education, with a focus on Pitticus, who sympathized with the model and is mentioned in the poem.

In his long oration, Socrates analyzes the lines of the poem in much greater detail than had been done previously in the conversation, thereby drawing out their features to a much greater extent (Pattern #4). So, he ends by interpreting the line, “I praise and love all willingly Who do no baseness,“ as Simonides’ condemnation of Pitticus on the grounds that he has made an “utterly false statement about something of the highest import” (346e-347a). In Socrates’ view, that false statement is the one Simonides quotes from Pitticus, ”To be noble is hard” (339e). According to Socrates, Simonides’ view is that to be noble is impossible, not hard, as only the gods can be noble. Man can only seek to become noble, as man has not already achieved perfection. The analysis of the poem by Simonides, which Protagoras introduced to defend his claim that one ought to become an authority on poetry (339a), is taken up

HAROUTUNIAN-GORDON A Study in Reflective Thinking 5 1

and elaborated by Socrates until he demonstrates that Protagoras is not an authority on poetry. Socrates appears to beat Protagoras at his own game of oratory, which clearly has implications for addressing the basic question of what Protagoras teaches people.

If one looks at the speeches of Protagoras and Socrates in relation to one another, it appears that the remarks of each permit the other to develop new Follow-up Questions and thereby create new understanding of the Basic Question (What does Protagoras teach people?). Over and over, one may see each beginning where the other left off, as though not completely satisfied that the previous exchange had resolved the issue on the table (Pattern #l). We watch them taking up topics, rhetorical devices, and questions offered by the other in a previous context, reversing the use of these in order to reach the opposite conclusion or new conclusions and questions, filling in the outlines of patterns so that details become clearer, and even drawing out the sequences in new ways, By following these patterns, Socrates and Protagoras appear to move the discussion as one sees it unfold. For it is by following the patterns that the discussants raise questions and objections, draw inferences, and arrive at particular conclusions and generalization^.^^

Now, the conversations in the two works of art considered here, Bach’s and Plato’s, seem to proceedvery naturally. That is, the topics that arise, the patterns that are followed, and the variations in the patterns that one sees seem entirely consistent with one another and form unified wholes in which the meaning of an initial question is both discovered and created. But since Plato’s Protagoras and Bach’s courante are works of art, one may ask, Is the course of conversation observed in each the consequence of artistic genius and therefore not observable in natural discourse? In other words, would natural conversation about the meaning of texts, as occurs in classrooms among real people, also have the character of an evolving cluster of questions? If so, would the questions and conclusions develop through the patterns outlined above? Before turning to this topic, it needs to be clear why so doing is of critical importance.

If it turns out that actual classroom conversations about the meaning of texts proceed by cultivating clusters of questions following the patterns identified, then we will be in a position to describe discussions of this sort in ways that reveal their order and structure. We will gain perspective on how they “cover” the material of texts and what we can expect to happen when conversations are allowed to pursue their natural courses. We will, then, be in a position to reflect upon a number of standard classroom practices, including those of lesson planning, discussion organi- zation and management, and, indeed, classroom management more generally. Furthermore, the analysis will have implications for teacher preparation practices, as well .

24. In other words, my claim is that thc patterns described here are the mechanism by which reflective thinking is accomplished. They arc the means by which questions, conclusions, and other steps in acts of reasoning are achievcd.


AN EXAMPLE FROM CLASSROOM CONVERSATION

The example I wish to consider comes from a class that I conducted with a mathematician, David Tartakoff, in the Fall Quarter 1992, at Northwestern Univer- sity, entitled “Conversation in Mathematics and Music: A Study of Teaching.”25 The texts for the course were mathematical, philosophical, and musical.26 The class consisted of conversations about the meaning of the various texts and about the questions themselves so that these became clarified and their resolutions pursued.

For the second meeting of the class, the students had prepared by reading four descriptions of Euclid’s proof of the infinity of prime numbers, including one by the mathematician, G.H. Hardy. Much of the conversation focused upon the meaning of “direct” and “indirect” forms of proof, which may be summarized as follows:

1. The “indirect“ or “nonconstructive” proof, which proceeds by a reductio ad absurdurn argument: Suppose that there are only finitely many prime numbers, p,, p2.. .p,; then form Q = 1 + p, x pr. ..p,. Either Q is prime (a new prime number, for it is larger than any of the original ones) or it is not. If it is not a new prime number, take its prime factorization - the prime numbers which, if multiplied together, would give us Q. These factors must all be new, since the original primes all left nonzero remainders [actually, remainders of 1) when divided into Q. 2. The more constructive or “direct” proofs, which offer a procedure for finding new prime numbers: Suppose that some primes, say A, B, and C, are given. According to Euclid’s argument, there must be another prime, different from those. This can be shown by taking the product of A, B and C and adding 1 and calling the result Q: Q = A x B x C + 1; as above, either Q is prime (and new) or is not and has a new prime factorization.*’

To open the discussion, David and I asked the following question: If Hardy believes that there could be another proof of the same result that is as beautiful as [his version of] Euclid’s, what qualities would it need to have? We had come to this question, in part, because the topic of mathematical “beauty“ had been introduced in the previous discussion. As the conversation proceeded, the following questions arose, in this order:

1. Why is finding Q powerful?

25. Further discussion of the course and the experiment conducted therein may be found in Sophie Haroutunian-Gordon and David Tartaltoff, ”On the Learning of Mathcmatics Through Conversation,” For the Learning o f M a t h e m a t i c s 16, no. 2 (June 199hJ, 2-10,

26. The texts included: Richard Courant, and Herbert Robbins, W h a t is Mathematics{ An Elementary Approach to Ideas and Methods (Oxford: Oxford University Press, 1947); Albert Einstein, Autohiographi- cal Nores, trans. P.A. Schilpp (La Salle, Ill.: Opcn Court Publishers, 19911; Eduard Hanslick, On the Musical ly Beaut i ful , trans., Geoffrey Payzant [Indianapolis: Hackctt Publishing Company, 19861; G.H. Hardy, A Mnfhernu~icicui’s Apology (Cambridge: Cambridge University Press, 1992, Canto Books); T.W. Surette, Music a n d l i f e (Boston: E.C. Schirmer Music Company, 1957); andLudwig Wittgenstein, Remarks on the Foundntions of Morhenmtics, ed. G.H. von Wright et al. (Cambridge: MIT Press, 1967).

27. The difference hetween “indirect” and “constructive” proofs was taken up by the students, as shall be seen. Hence, attcmpts at clarification are omitted here.

HAKOUTUNIAN-GORDON A Study in Reflective Thinking 53

2. How might Euclid have worked? Did he first “know” that the list of primes was infinitely long and then discover a proof for this fact, or in seeking a proof for an intuition, did he find it and thereby establish the fact? 3. What is beautiful about the use of Q? (Sources of its beauty.) 4. What are the similarities and differences between direct and indirect (or “constructive” and “nonconstructive”) proofs? 5. Is Q necessarily a prime number in the direct, constructive proof? 6. Again, what are the differences between the direct and indirect forms of proof? (This time, the question is reiterated in the context of an additional question, namely, does one generate all the prime numbers in a sequence by carrying out the direct form of the proof, or will it skip some?) 7. What do specific examples tell us about the answer to the previous question and to those preceding?

Now, do these questions, which were not imposed by the instructors but arose naturally in the course of the conversation, constitute follow-up questions to the one posed initially (If Hardy believes that there could be another proof of the same result that is as beautiful as Euclid’s, what qualities would it need to have?) If so, do these questions arise as people follow the patterns outlined above? The answer to both questions appears to be “yes.” To begin with, Question 1, Why is finding Q powerful? seems to arise as one student, George, draws out of the text one element for additional scrutiny, namely, Q, and begins to identify its features. George seems to be following Pattern #4. He observes that Q is powerful because it “brings up a contradiction,” as he puts it. If George is correct, one might ask, Would Hardy argue that any proof as beautiful as Euclid’s needs the simplicity and definitiveness of the reductio ad absurdurn form of argument? George’s question is a follow-up to the one posed initially, what I shall call the Basic Question (BQ), because its resolution has implications for resolving the BQ.

As the discussants reflect with George upon his observationlZX they begin to follow another pattern that brings them to Question 2, Did Euclid discover a proof for a fact that he knew already - that there are infinitely many primes - or did he discover the infinity of primes because he had found a convincing proof? The question here is not fully clear. But interestingly enough, Lisa, another student, seems to come to the question by reversing George’s observation (Pattern #3) . One can say that Q proves the infinity of prime numbers because it reveals a contradiction. Can one also say that the existence of the infinity of prime numbers - a truth - enables one to discover a proof of its existence? By reversing George’s observation that Q is powerful because it reveals a contradiction and thereby proves a truth, Lisa raises a question about how Euclid actually worked. However, Lisa’s question is not yet clear, nor is it clear how resolving it would have implications for determining the criteria of mathematical beauty that a proof must meet (BQ). As shall be seen, Lisa’s question was not pursued.

28. The contradiction arises because whether Q is prime or not, its factors will includc ncw primc numbers, so that Q will contradict the initial assumption that there is a finite number of prime numbers.

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Question 3, What is beautiful about the use of Q? presents a modification of the first follow-up question and begins to offer a resolution: Q stands in a particular relation to the rest of the proof, and this relation is both powerful and beautiful. What, then, are the sources of its beauty? It appears that in exploring the power of Q - how it proves the infinity of prime numbers- the speaker comes to repeat a sequence (the BQ, which asks about the criteria of beauty that a proof must meet, according to Hardy) in a new context, namely, that of Q. Here, the speaker seems to be following Pattern #2. If one could resolve the issue of what makes Q beautiful, then one might gain insight into the criteria that proofs acceptable to Hardy would need to meet. So, Question 3 follows up the BQ.

Question 4, What are the similarities and differences between direct and indirect forms of proof? like the previous one, reiterates an earlier question in a new context, perhaps reflecting a deepened, clarified confusion about the differences between the two forms of the proof. The idea suggested now is that the indirect, or nonconstructive form, which proceeds by contradicting an initial assumption (reductio ad absurdum), is ”neater” than the constructive proof, as one student puts it. Here, then, is a newly identified difference that arises as Pattern #2 is followed. But, is the constructive form of proof necessarily less beautiful, even if it is messier? In responding to this question, a new one is opened, as excerpt 7 from the classroom transcript reveals:

Lisa: “Okay, so you guys are saying that equation will not get a prime?” Sam: ”No, it will get a prime. All I said is that it won‘t give you.. .it doesn’t give you all the primes. But it does allow you to keep computing primes as long as you want.” Sue: “You will end up with all the primes. If there’s one you don’t know it will eventually pop up.” Sam: “Oh, I see, okay.’’ Sue: ”All you really need to start with is the definition [Q = A x B x C + 11 You try two and find out if it’s a prime number. You start like he did and it gives you three. You know? And then if you went with two and three it gives you seven. If you went with two, three and seven.. . .I mean, eventually you’d find out five is a prime number.”

When Lisa says, “so you guys are saying that equation [Q = A x B x C + 11 will not get a prime,” she points to Question 5, Is Q always a prime number? When Sam says, “No, it willgetaprime.All1saidis that it ... doesn’tgiveyouall theprimes,” heseems to be making the claim that the procedure for generating a constructive proof will not give all the prime numbers in a sequence. Question 5 and the response to it appear to have arisen as the group filled in details about the features of Q so that its nature became clearer. The group might be said, then, to be following Pattern #5. For the scrutiny of Q reveals features of it that were not at first apparent, namely, the possibility that the procedure for generating prime numbers may skip some of the primes in a sequence. The examination of Q, then, raises anew the issue of the difference between the direct and indirect forms of proof. In order to test out Sam’s


claim and uncover differences between the indirect and direct forms of the proof (Question 6), the group turned to consider particular examples in which prime numbers were plugged into the equation [Q = A x B x C + 1). The turn to the examples happened naturally: the instructors had not planned the move in advance. The Transcript continues as follows (Excerpt 8):

[DT is at the chalkboard, working out an example. He is trying the product of primes 5 x 7 and then adding 1, which yields 361 Mort: “Well, thirty six isn’t a prime number, right?” [Several people;] “No.” Lisa: “So what are its factors?” Mike: “This proof doesn’t seem quite so beautiful now that you [DT)] are up there. ” [Laughter.] Sum: “It’s so much trial and error. And like you were saying ... I mean ...y ou have.. .you have to presuppose, it seems to me, a little bit more. Maybe I’m wrong about that, but it seems you have to presuppose a little more with this proof [than the indirect form of proof, which is based upon the reductio ad absurdum] because you have to actually get down with the numbers. ”

When Mort begins by observing that 36 is not a prime number, he appears to set to rest the first issue, namely, is Q always a prime number? Lisa immediately moves on to ask, “So what are its factors?” thereby testing the claim that either Q is a new prime number or it contains as a factors new primes. Lisa’s question is an attempt to draw out the features of Q in the particular case before the group. She might, then, be said to be following Pattern #5. Mike, in observing the features of Q as they are revealed in this case, observes that the constructive approach to the proof “doesn’t seem quite so beautiful.” In so doing, he raises in the context of these particular examples the issue of what makes mathematics beautiful. Sam responds by saying, “It [the procedure of the constructive proof] is so much trial and error.. .you have to really get down with the numbers.” Notice that while Lisa, Mike, and Sam might all be said to be drawing out and scrutinizing features of Q in aparticular instance, Sam, and perhaps Mike, seem to do so in the context of the BQ, that is, what makes a mathematical proof beautiful. They, then, might be said to be following Pattern #2. The conversation continues in Excerpt 9:

Sue: “I think also we’re kind of hung up on finding all of them [all the prime numbers] and I don‘t think this was a big deal with this proof. I don‘t even think anybody cared about finding all of them. I think they cared about knowing there were an infinite number of them. And in that respect if we use this [the constructive, direct proof] to say we want to find them all, well yeah, it’s not going to be very pretty, cause that’s not the way it was intended.” Sum: “Except my concept of infinity is ... is damaged if I can’t find all of them. I mean, I know that’s not necessary [laughter] but if I leave out seven I have a problem.” Sue: “And we know there’s always one more, too.”

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Sam: “I know, I know, I know. My mind tells me it works, it’s just ... not as beautiful, that’s all.” Mike: “You’ll spend the rest of your life computing all the rest of them.” Sam: “Yeah right, hoping 1/11 get to seven.” Mort: ”I don’t think that this proof is intended to give you all prime numbers.” Fay: ”Yeah, I don‘t think so either.” Linda: “I think this proof is nice because you have freedom of choice about what primes you start out with and everyone can start out with different ones and there’s room for individual freedom.’’ Sam: “The freedom of primes.’’

In the preceding excerpt, we see discussants continuing to struggle with the issue of whether the constructive proof is beautiful. In so doing, they seem to be looking at their intuitive reactions to the particular examples and examining assumptions about what makes a proof beautiful. When Sue argues that the constructive procedure was “not intended” to generate all the primes, hence, ”it’s not going to be very pretty” if judged from that perspective, she may be suggesting that beauty should be judged in relation to intended function. Sam expresses dissatisfaction, it seems, when he says that “his concept of infinity is damaged” if the procedure does not allow him to find all the prime numbers in a sequence. When Sue points out that the proof will always generate a new prime, both Mike and Sam resist: Sam adds to Mike’s remark by saying, “Yeah, right, hoping 1’11 get to seven,” suggesting that he measures the constructive form of proof against the criteria of order and completeness. It is not until Linda remarks that the proof gives you “freedom of choice about what primes you start with” that Sam shows evidence of modifying his criteria of beauty. His iteration, “freedom of primes,” seems to suggest that he is entertaining a new criterion, namely, freedom of choice, measured against which the constructive form of the proof may not fare so badly.

Notice that the preceding exchange was introduced by the questions (1) Is Q always aprime number? and (2) Will the procedure for generating prime numbers give you all the prime numbers in a sequence or will it skip some? These questions seem to have arisen as discussants followed the patterns outlined above. Furthermore, the question of whether the constructive proof is beautiful arises as the discussants examine the elements of Q that study of specific examples allows them to draw out. They thereby fill in aspects of the pattern previously not apparent and so may be said to be following Pattern #5. The identification of a new criterion allows them to consider the proof from a new perspective - to return to the issue of its beauty and reiterate the question in a new context.

The foregoing analysis suggests, then, that the questions that arise in a classroom conversation about the meaning of texts may seem random but may in fact have the form of a cluster of questions. That is, a Basic Question, which may be the question posed initially, may be defined and pursued as discussants raise questions about aspects of the text. The resolutions of these questions may have implications for resolving the BQ, and the discussion may proceed by raising these questions and


drawing out the implications of the resolutions. In short, the evidence presented here suggests that real conversations, as well as ones found in literary and musical works, may not be at all random but may proceed by cultivating patterned clusters of questions as defined above.

CONCLUSION

In the foregoing sections we have considered the following issue: How does reflective thinking, as it occurs in interpretive discussion, proceed? The hypothesis proposed is that questions evolve in the form of a cluster, and that the questions evolve as the discussants follow patterns that include: (1) Starting a new sequence where a previous one left off and returning t o about the same starting point; (2) Repeating a sequence in a different context; (3) Reversing the direction of a sequence or one o f i ts elements; (4) Drawing out t he elements of a sequence; and (5) Filling in the outline o f a sequence so that i ts patterns become clearer. We have seen evidence to support the hypothesis in a musical example, a literary one, and an actual classroom conversation. Study of these examples revealed the patterns that seem to be followed so that meaning is created.

The claim here is not that all the patterns followed in conversation about the meaning of texts have been identified, or that every interpretive discussion evolves as a cluster of questions. The latter is an empirical claim, the basis for which three cases would hardly suffice. The former claim - that the patterns followed in discussion are as indicated - will no doubt be revised as further examples are scrutinized. However, given the diversity of the three cases considered, the claim that patterns like the ones identified may be followed in many instances has some plausibility.

Let us, then, assume that the analysis suggests features that may be found in other conversations about the meaning of texts. What, then, are the implications for classroom practices, teacher preparation for class, and teacher education? A fuller answer to these questions is receiving detailed attention elsewhere.29 But let me briefly sketch arguments for three claims here. First, if it is the case that conversation about the meaning of texts proceeds by forming interpretations and questions in the ways described, thereby creating meaning, then one may argue that such conversations should be allowed to flow freely without the preoccupation of "covering" certain (or all the) material in the text. Teachers leading such discussions should recognize that while some parts of a text may not be discussed, those that do receive attention will beponderedmeaningfully, as the study of them will be related to issues students care to resolve. As it is frequently the case that the time available does not permit all aspects of a work to be discussed, the issue of which parts to cover arises in many traditional classroom situations. The traditional resolution - the teacher decides which parts to scrutinize - may distance the students from the text if their concerns are unrelated to these passages (or, indeed, if they have no concerns with which to address the text). In light of the foregoing analysis of examples, it seems

29, Sophie Haroutunian-Gordon, Preparing to Turn the Soul: Teacher Education Today, unpublished manuscript, 1997.

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reasonable to suggest that interpretive discussions, if allowed to proceed according to the patterns discussants naturally seem to follow, will cover material in the text in ways that illuminate its meaning for those participating. Furthermore, such discussions may to be more meaningful than those in which the meaning of the text is dictated by the teacher.

Second, if it turns out to be true that texts are covered most effectively if the conversation about meaning is allowed to proceed as it does naturally in an interpretive discussion, then instead of planning in advance what portions of the text to cover and how to look at these, teachers should prepare for class by developing clusters of questions about the meaning of the texts for themselves - questions of genuine interest to them and ones for which they are unsure of the answers. They should spend time perusing the text, writing down questions that occur to them, relating the questions to one another so as to identify issues of deeper interest and passages in the text that help to explore these issues. Such a process is time- consuming. Once completed, the teacher might never pose the cherished questions, especially if the students wish to put their own on the table. However, the reward of such preparation is at least two-fold: it allows one to pursue reflection upon issues of genuine interest to oneself - an important source of teacher renewal - and it breeds deep familiarity with the text, which enables the teacher to listen to students with an open, free mind so as to help cultivate their questions and interpretations. Such preparation can enable students to relate to the texts in powerful and productive ways. It shares little with the traditional lesson planning in which teachers determine, in advance, procedures by which students will reach an understanding that the teacher already has - an understanding that may or may not have meaning to the students.

Finally, it may be that in preparing teachers to lead discussion, we should be helping them to clarify their questions and to listen so as to assist them in doing the same with their students. Were we to turn our attention to such activities, there might be much less preoccupation with lesson planning and classroom management issues than is frequently the case in teacher preparation programs. Rather, our focus, like the teacher’s, would be upon helping the formation of clear questions about which there is genuine doubt. Teachers and students alike need serious help with the project of forming questions. It is not an activity to which our educational institu- tions have traditionally given sustained and focused attention. If it turns out that interpretive discussion helps students to engage in reflective thinking about the meaning of texts and to study the texts in meaningful ways, as the foregoing analysis suggests, and if, with Dewey, we wish to teach students to think reflectively, it behooves us to focus teacher education on teachers’ questions, the aim being to help them make these as clear and reflective of their genuine points of doubt as possible.

I wish to thank the following persons, representing philosophy, music, mathematics, and education, for enormously helpful comments on one or more versions of this essay: Eric Bredo, Nick Burhules, Karen Fuson, Bennett Reimer, Harvey Siegel, David Tartaltoff, and anonymous reviewers.

a study of reflective thinking: patterns in interpretive discussion

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