Handbook of psychology volume 7 educational psychology
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- The Growth of Argument 359
- Conversational Structure as a Resource for Argument
- From Pretense to Proof
- Development of Representational Competence
- Knitting Possibilities: Counterfactual Reasoning
- The Growth of Argument 361
- The Skills of Argument
THE GROWTH OF ARGUMENT Arguments in mathematics aim to provide explanations of mathematical structures. Proof is often taken as emblematic of mathematical argument because it both explains and pro- vides grounds for certainty that are hard to match or even imagine in other disciplines, such as science or history. Although everyday folk psychology often associates proof with drudgery, for mathematicians proof is a form of dis- covery (e.g., de Villiers, 1998), and even “epiphany” (e.g., Benson, 1999). Yet conviction does not start with proof, so in this section we trace the ontogeny of forms of reasoning that seem to ground proof and proof-like forms of explanation. Our approach here is necessarily speculative because there is no compelling study of the long-term development of an epistemic appreciation for mathematical argument. More- over, the emblem of mathematical argument, proof, is often misunderstood as a series of conventional procedures for arriving at the empirically obvious, rather than as a form of explanation (Schoenfeld, 1988). International comparisons of students (e.g., Healy & Hoyles, 2000) confirm this impres- sion, and apparently many teachers hold similar views (Knuth, 2002; Martin & Harel, 1989). Nonetheless, several lines of research suggest fruitful avenues for generating an epistemology of mathematical argument that is more aligned with mathematical practice and more likely to expose progenitors from which this epistemol- ogy can be developed. (We are not discounting the growth of experimental knowledge in mathematics but are focusing on grounds for certainty here. We return to this point later.) In the sections that follow, we suggest that mathematical The Growth of Argument 359 argument evolves from everyday argument and represents an epistemic refinement of everyday reasoning. We propose that the evolution is grounded in the structure of everyday conversation, is sustained by the growth and development of an appreciation of pretense and possibility, and is honed through participation in communities of mathematical in- quiry that promote generalization and certainty. Conversational Structure as a Resource for Argument Contested claims are commonplace, of course, and perhaps there is no more common arena for resolving differing perspectives than conversation. Although we may well more readily recall debates and other specialized formats as sparring grounds, everyday conversation also provides many opportunities for developing “substantial” arguments (Toulmin, 1958). By substantial, Toulmin referred to argu- ments that expand and modify claims and propositions but that lead to conclusions not contained in the premises (unlike those of formal logic). For example, Ochs, Taylor, Rudolph, and Smith (1992) examined family conversations with young children (e.g., 4–6 years of age) around such mundane events as recall of “the time when” (e.g., mistaking chili peppers for pickles) or a daily episode, such as an employee’s reaction to time off. They suggested that dinnertime narratives engender many of the elements of sound argument in a manner that parallels scientific debate. First, narratives implicate a prob- lematic event, a tension in need of resolution, so that narra- tives often embody some form of contest, or at least contrast. Second, the problematic event often invites causal explana- tion during the course of the conversation. Moreover, these explanations may be challenged by conarrators or listeners, thus establishing a tacit anticipation of the need to ground claims. Challenges in everyday conversations can range from matters of fact (e.g., disputing what a character said) to mat- ters of ideology (e.g., disputing the intentions of one of the characters in the account). Finally, conarrators often respond to challenges by redrafting narratives to provide alternative explanations or to align outcomes more in keeping with a family’s worldview. By means of explicit parallels like these, Ochs et al. (1992) argued that theories and stories may be generated, critiqued, and revised in ways that are essentially similar (see Hall, 1999; Warren, Ballanger, Ogonowski, Rosebery, & Hudicourt-Barnes, 2001, regarding continuities between everyday and scientific discourses). Studies like those of Ochs et al. (1992) are emblematic of much of the work in conversation analysis, which suggests that the structure of everyday talk in many settings is an important resource for creating meaning (Drew & Heritage, 1992). For example, Rips (1998; Rips, Brem, & Bailenson, 1999) noted that everyday conversationalists typically make claims, ask for justification of others’ claims, attack claims, and attack the justifications offered in defense of a claim. The arrangement of these moves gives argumentation its charac- teristic shape. Judgments of the informal arguments so crafted depend not only on the logical structure of the argu- ment but also on consideration of possible alternative states of the claims and warrants suggested. Rips and Marcus (1976) suggested that reasoning about such suppositions, or possible states, requires bracketing uncertain states in mem- ory in order to segregate hypothetical states from what is currently believed to be true. In the next section we review evidence about the origins and constraints on this cognitive capacity to reason about the hypothetical.
Reasoning about hypothetical states implicates the develop- ment of a number of related skills that culminate in the capacity to reason about relations between possible states of the world, to treat aspects of them as if they were in the world, to objectify possibilities, and to coordinate these objects (e.g., conjectures, theories, etc.) with evidence. Both theory and evidence are socially sanctioned and thus cannot be properly regarded apart from participation in communities that encourage, support, and otherwise value these forms of reasoning. We focus first on the development of representa- tional competence, which appears to originate in pretend play, and then on corresponding competencies in conditional reasoning. We turn then from competence to dispositions to construct sound arguments that coordinate theory and evidence and, in mathematics, to prove. Because these dispo- sitions do not seem to arise as readily as the competencies that underlie them, we conclude with an examination of the characteristics of classroom practices that seem to support the development of generalization and grounds for certainty in early mathematics education.
One of the features of mathematical argument is that one must often reason about possible states of affairs, sometimes even in light of counterfactual evidence. As we have seen, this ca- pacity is supported by everyday conversational structure. However, such reasoning about possibility begins with repre- sentation. This representational capacity generally emerges towards the end of the second year and is evident in children’s pretend play. Leslie (1987) clarified the representational de- mands of pretending that a banana is a telephone, while knowing very well that whatever the transformation, the
360 Mathematical Learning banana remains a banana, after all. He suggested that pretense is founded in metarepresentational capacity to constitute (and distinguish) a secondary representation of one’s primary rep- resentation of objects and events. Metarepresentation expands dramatically during the preschool years. Consider, for example, DeLoache’s (1987, 1989, 1995) work on children’s understanding of scale mod- els of space. DeLoache encouraged preschoolers to observe while she hid small objects in a scale model of a living room. Then she brought them into the full-scale room and asked them to find similar objects in the analogous locations. DeLoache observed a dramatic increase in representational mapping between the model and the world between 2.5 and 3 years of age. Younger children did not seem to appreciate, for example, that an object hidden under the couch in the model could be used to find its correspondent in the room, even though they readily described these correspondences verbally. Yet slightly older children could readily employ the model as a representation, rather than as a world unto itself, suggesting that they could sustain a clear distinction between representation and world. Gentner’s (Gentner & Loewenstein, 2002; Gentner & Toupin, 1986) work on analogy also focuses on early devel- oping capacities to represent relational structures, so that one set of relations can stand in for another. For example, Kotovsky and Gentner (1996) presented triads of patterns to children ranging from 4 to 8 years of age. One of the patterns was relationally similar to an initially presented pattern (e.g., small circle, large circle, small circle matched to small square, large square, small square), and the third was not (e.g., large square, small square, small square). Although the 4-year-olds responded at chance levels, 6- and 8-year-olds preferred relational matches. These findings are consistent with a relational shift from early reliance on object-matching similarity to later capacity and preference for reasoning rela- tionally (Gentner, 1983). This kind of relational capacity undergirds conceptual metaphors important to mathematics, like those between collections of objects and sets in arith- metic, and forms the basis for the construction of mathemati- cal objects (Lakoff & Nunez, 2000). Moreover, Sfard (2000) pointed out that although discourse about everyday events and objects is a kind of first language game (in Wittgenstein’s sense), the playing field in mathematics is virtual, so that mathematical discourse is often about objects that have no counterpart in the world. Knitting Possibilities: Counterfactual Reasoning Collectively, research on the emergence of representational competence illuminates the impressive cognitive achievement of creating and deploying representational structures of actual, potential, and pretend states of the world. However, it is yet another cognitive milestone to act on these representations to knit relations among them, a capacity that relies on reasoning about relations among these hypothetical states. Children’s ability to engage in such hypothetical reasoning is often dis- counted, perhaps because the seminal work of Inhelder and Piaget (1958) stressed children’s, and even adults’, difficulties with the (mental) structures of logical entailment. However, these difficulties do not rule out the possibility that children may engage in forms of mental logic that provide resources for dealing with possible worlds, even though they may fall short of an appreciation of the interconnectedness of mental opera- tors dictated by formal logic. Studies of child logic document impressive accomplishments even among young children. For the current purpose of considering routes to mathematical argument, we focus on findings related to counterfactual reasoning—reasoning about possible states that run counter to knowledge or perception, yet are considered for the sake of the argument (Levi, 1996; Roese, 1997). This capacity is at the heart of deductive modes of thought that do not rely exclu- sively on empirical knowledge, yet can be traced to children’s capacity to coordinate separate representations of true and false states of affairs in pretend play (Amsel & Smalley, 2001). In one of the early studies of young children’s hypotheti- cal reasoning, Hawkins, Pea, Glick, and Scribner (1984) asked preschool children (4 and 5 years) to respond to syllo- gistic problems with three different types of initial premises: (a) congruent with children’s empirical experience (e.g., “Bears have big teeth”), (b) incongruent with children’s empirical experience (e.g., “Everything that can fly has wheels”), and (c) a fantasy statement outside of their experi- ence (e.g., “Every banga is purple”). Children responded to questions posed in the syllogistic form of modus ponens (“Pogs wear blue boots. Tom is a pog. Does Tom wear blue boots?”). They usually answered the congruent problems cor- rectly and the incongruent problems incorrectly. Further- more, children’s responses to incongruent problems were consistent with their experience, rather than the premises of the problem. This empirical bias was a consistent and strong trend. However, unexpectedly, when the fantasy expressions were presented first, children reasoned from premises, even if these premises contradicted their experiences. This finding suggested that the fantasy form supported children in orient- ing to the logical structure of the argument, rather than being distracted by its content. Subsequently, Dias and Harris (1988, 1990) presented young children (4-, 5- and 6-year olds) with syllogisms, some counterfactual, such as, “All cats bark. Rex is a cat. Does Rex bark?” When they were cued to treat statements as
The Growth of Argument 361 make-believe, or when they were encouraged to imagine the states depicted in the premises, children at all ages tended to reason from the premises as stated, rather than from their knowledge of the world. Scott, Baron-Cohen, and Leslie (1999) found similar advantages of pretense and imagination with another group of 5-year-old children as well as with older children who had learning disabilities. Harris and Leevers (2001) suggested that extraordinary conditions of pretense need not be invoked. They obtained clear evidence of counterfactual reasoning with preschool children who were simply prompted to think about the content of coun- terfactual premises or, as they put it, to adopt an analytic perspective. Further research of children’s understandings of the entailments of conditional clauses suggests that at or around age 8, many children interpret these clauses biconditionally. That is, they treat the relationship symmetrically (Kuhn, 1977; Taplin, Staudenmayer, & Taddonio, 1974), rather than treat- ing the first clause as a sufficient but not necessary condition for the consequent (e.g., treating “if anthrax, then bacteria” as symmetric). However, Jorgenson and Falmagne (1992) assessed 6-year-old children’s understanding of entailment in story formats and found that this form of narrative support produced comprehension of entailment more like that typi- cally shown by adults. O’Brien, Dias, Roazzi, and Braine (1998) suggested that the conflicting conclusions like these about conditional reasoning can be traced to the model of ma- terial implication (if P, then Q) based on formal logic. O’Brien and colleagues argued that it may be a mistake to evaluate conditional reasoning via the truth table of formal logic (espe- cially the requirement that a conditional is true whenever its antecedent is false). This perspective, they think, obscures the role of conditionals in ordinary reasoning. They proposed in- stead that a set of logic inference schemas governs conditional reasoning. Collectively, these schemas rely on supposing that the antecedent is true and then generating the truth of the con- sequent. They found that second- and fifth-grade children in both the United States and Brazil could judge the entailments of the premises of a variety of conditionals (e.g., P or Q, Not- P or Not-Q) in ways consistent with these schemas, rather than strict material implication. Even preschool children judged a series of counterfactual events, for example, those that would follow from a character pretending to be a dog, as consistent with a story. An interesting result was that they also excluded events that were suppositionally inconsistent with the story, for example, the same character talking on the phone even though those events were presumably more consistent with their experience (i.e., people, not dogs, use phones). Collectively, these studies of hypothetical reasoning point to an early developing competence for representing and comparing possible and actual states of the world, as well as for comparing possible states with other possible states. Moreover, these comparisons can be reasoned about in ways that generate sound deductions that share much, but do not overlap completely, with formal logic. These impressive competencies apparently arise from the early development of representational competence, especially in pretend play (Amsel & Smalley, 2001), as well as the structure of everyday conversation. However, despite these displays of early com- petence, other work suggests that the skills of argument are not well honed at any age, and are especially underdeveloped in early childhood.
Kuhn (1991) suggested that an argument demands not only generation of possibilities but also comparison and evalua- tion of them. These skills of argument demand a clear sepa- ration between beliefs and evidence, as well as development of the means for establishing systematic relations between them (Kuhn, 1989). Kuhn (2001) viewed this development as one of disposition to use competencies like those noted, a development related to people’s epistemologies: “what they take it to mean to know something” (Kuhn, 2001, p. 1). In studies with adults and adolescents (ninth graders) who attempted to develop sound arguments for the causes of unemployment, school failure, and criminal recidivism, most of those interviewed did not seem aware of the inherent uncertainty of their arguments in these ill-structured domains (Kuhn, 1991, 1992). Only 16% of participants generated evidence that would shed light on their theories, and only about one third were consistently able to generate counterar- guments to their positions. Kuhn, Amsel, and O’Loughlin (1988) found similar trends with people ranging in age from childhood (age 8) to adulthood who also attempted to gener- ate theories about everyday topics like the role of diet in catching colds. Participants again had difficulty generating and evaluating evidence and considering counterarguments. Apparently, these difficulties are not confined to compara- tively ill-structured problems. For example, in a study of the generality and specificity of expertise in scientific reasoning, Schunn and Anderson (1999) found that nearly a third of college undergraduate participants never supported their conjectures about a scientific theory with any mention of empirical evidence. Kuhn (2001) further suggested that argu- ments constructed in contexts ranging from science to social justice tend to overemphasize explanation and cause at the expense of evidence and, more important, that it is difficult for people at all ages to understand the complementary epis- temic virtues of each (understanding vs. truth). 362 Mathematical Learning Proof The difficulties that most people have in developing epistemic appreciations of fundamental components of formal or scien- tific argument suggest that comprehension and production of more specialized epistemic forms of argument, such as proof, might be somewhat difficult to learn. A number of studies con- firm this anticipation. For example, Edwards (1999) invited 10 first-year high school students to generate convincing argu- ments about the truth of simple statements in arithmetic, such as, “Even x odd makes even.” The modal justification was, “I tried it and it works” (Edwards, p. 494). When pressed for further justification, students resorted to additional examples. In a study of 60 high school students who were invited to gen- erate and test conjectures about kites, Koedinger (1998) noted that “almost all students seemed satisfied to stop after making one or a few conjectures from the example(s) they had drawn” (p. 327). Findings like these have prompted suggestions that “it is safer to assume little in the way of proof understanding of entering college students” (Sowder, 1994, p. 5). What makes proof hard? One source of difficulty seems to be instruction that emphasizes formalisms, such as two- column proofs, at the expense of explanation (Coe & Ruthven, 1994; Schoenfeld, 1988). Herbst (2002) went so far as to suggest that classroom practices like two-column proofs often bind students and their instructors in a pedagogical paradox because the inscription into columns embodies two contradictory demands. The format scripts students’ re- sponses so that a valid proof is generated. Yet this very emphasis on form obscures the rationale for the choice of the proposition to be proved: Why is it important to prove the proposition so carefully? What does the proof explain? Hoyles (1997; Healy & Hoyles, 2000) added that curricula are often organized in ways that de-emphasize deductive reasoning and scatter the elements of proof across the school year (see also Schoenfeld, 1988, 1994). In their analysis of university students’ conceptions of proof, Harel and Sowder (1998) found that many students seem to embrace ritual and symbolic forms that share surface characteristics with the symbolism of deductive logic. For example, many students, even those entering the university, appear to confuse demonstration and proof and therefore value a single case as definitive. Martin and Harel (1989) examined the judgments of a sample of preservice elemen- tary teachers enrolled in a second-year university mathemat- ics course. Over half judged a single example as providing a valid proof. Many did not accept a single countercase as invalidating a generalization, perhaps because they thought of mathematical generalization as a variation of the general- izations typical of prototypes of classes (e.g., Rosch, 1973). Outcomes like these are not confined to prospective teachers: Segal (2000) noted that 40% of entry-level university mathe- matics students also judged examples as valid proofs. Although many studies emphasize the logic of proof, oth- ers examine proof as a social practice, one in which accept- ability of proof is grounded in the norms of a community (e.g., Hanna, 1991, 1995). These social aspects of proof sug- gest a form of rationality governed by artifacts and conven- tions about evidence, rigor, and plausibility that interact with logic (Lakatos, 1976; Thurston, 1995). Segal (2000) pointed out that conviction (one’s personal belief) and validity (the acceptance of this belief by others) may not always be con- sistent. She found that for first-year mathematics students, these aspects of proof were often decoupled. This finding accords well with Hanna’s (1990) distinction between proofs that prove and those that explain, a distinction reminiscent of Kuhn’s (2001) contrast between explanation and evidence. Chazan (1993) explored the proof conceptions of 17 high school students from geometry classes that emphasized empirical investigation as well as deductive proof. Students had many opportunities during instruction to compare deduc- tion and induction over examples. One component of instruc- tion emphasized that measurement of examples may suffer from accuracy and precision limitations of measurement devices (such as the sum of the angles of triangles drawn on paper). A second component of instruction highlighted the risks of specific examples because one does not know if one’s example is special or general. Nevertheless, students did not readily appreciate the virtues of proof. One objection was that examples constituted a kind of proof by evidence, if one was careful to generate a wide range of them. Other students believed that deductive proofs did not provide safety from counterexamples, perhaps because proof was usually con- structed within a particular diagram. Harel (1998) suggested that many of these difficulties can be traced to fundamental epistemic distinctions that arose during the history of mathematics. In his view, students’ understanding of proof is often akin to that of the Greeks, who regarded axioms as corresponding to ideal states of the world (see also Kline, 1980). Hence, mathematical objects determine axioms, but in a more modern view, objects are determined by axioms. Moreover, in modern mathematics, axioms yield a structure that may be realized in different forms. Hence, students’ efforts to prove are governed by epis- temologies that have little in common with those of the math- ematicians teaching them, a difficulty that is both cultural and cognitive. Of course, the cultural-epistemic obstacles to proof are not intended to downplay cognitive skills that students might need to generate sound proofs (e.g., Koedinger, 1998). Nevertheless, it is difficult to conceive of
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