Handbook of psychology volume 7 educational psychology
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- Mathematical Argument Emerges in Classrooms That Support It
- Teacher Orchestration of Mathematical Conversation
- The Growth of Argument 365
- Pathways to Proof
The Growth of Argument 363 why students might acquire the skills of proof if they do not see its epistemic point.
The literature paints a somewhat paradoxical portrait of the development of mathematical argument, especially the epis- temology of proof. On the one hand, mathematical argument utilizes everyday competencies, like those involved in resolving contested claims in conversation and those under- lying the generation and management of relations among possible states of the world. On the other hand, mathematical argument invokes a disposition to separate conjectures from evidence and to establish rigorous relations between them— all propensities that appear problematic for people at any age. Moreover, the emphasis on structure and certainty in mathe- matics appears to demand an epistemological shift away from things in the world to structures governed by axioms that may not correspond directly to any personal experience, except perhaps by metaphoric extension (e.g., Lakoff & Nunes, 1997). To these cognitive burdens we can also safely assume that the practices from which this specialized form of argument springs are hidden, both from students and even (within subfields of mathematics) from mathematicians themselves (e.g., Thurstone, 1995). Despite this paradox, or perhaps because of it, emerging research suggests a synthesis where the everyday and the mathematical can meet, so that mathematical argument can be supported by—yet differenti- ated from—everyday reasoning. In the next section we explore these possibilities. Mathematical Argument Emerges in Classrooms That Support It As the previous summary illustrates, research generally paints a dim portrait of dispositions to create sound argu- ments, even in realms less specialized than mathematics. Nonetheless an emerging body of research suggests a con- versational pathway toward developing mathematical argu- ment in classrooms. The premise is that classroom discourse can be formatted and orchestrated in ways that make the grounds of mathematical argument visible and explicit even to young children, partly because everyday discourse offers a structure for negotiating and making explicit contested claims and potential resolutions (e.g., Wells, 1999), and partly because classrooms can be designed so that “norms” (e.g., Barker & Wright, 1954) of participant interaction can include mathematically fruitful ideas such as the value of generalization. Rather than treating acceptance or disagree- ment solely as internal states of mind, these are externalized as discursive activities (van Eemeren et al., 1996). A related claim is that classrooms can be designed as venues for initi- ating students in the “register” (Halliday, 1978; Pimm, 1987) or “Discourse” (Gee, 1997, in press) of a discipline like mathematics. Dialogue, then, is a potential foundation for supporting argument, and studies outside of mathematics suggest that sound arguments can be developed in dialogic interaction. For example, Kuhn, Shaw, and Felton (1997) asked adoles- cents and young adults to create arguments for or against capital punishment. Compared to a control condition limited to repeated (twice) elicitation of their views, a group en- gaged in dyadic interactions (one session per week for five weeks) was much more likely to create arguments that ad- dressed the desirability of capital punishment within a framework of alternatives. Students in this dyadic group also were more likely to develop a personal stance about their arguments. The development of argument in the engaged group was not primarily related to hearing about the posi- tions of others, but rather to the need to articulate one’s own position, which apparently instigated voicing of new forms of argument. Moreover, criteria by which one might judge the desirability of capital punishment were elaborated and made more explicit by those participating in the dyadic conversations. Studies of argument in classrooms where it is explicitly promoted are also encouraging. For example, Anderson, Chinn, Chang, Waggoner, and Yi (1997) examined the logical integrity of the arguments developed by fourth-grade chil- dren who participated in discussions about dilemmas faced by characters in a story. The discussions were regulated by norms of turn taking (students spoke one at a time and avoided interrupting each other), attentive listening, and the expectation of respectful challenge. The teacher’s role was to facilitate student interaction but not to evaluate contributions. Anderson et al. (1997) analyzed the microstructure of the resulting classroom talk. They found that children’s argu- ments generally conformed to modus ponens (if p, then q) if unstated but shared premises of children were taken into account. This context of shared understandings, generated from collective experiences and everyday knowledge, resolved referential ambiguities and thus constituted a kind of sound, conversational logic. However, “only a handful of children were consistently sensitive to the possibility of backing arguments with appeals to general principles” (Anderson et al., 1997, p. 162). Yet, such an emphasis on the general is an important epistemic component of argument in mathematics, which suggests that mathematics classrooms may need to be more than incubators of dialogue and the gen- eral norms that support conversational exchange.
364 Mathematical Learning Mathematical Norms Cobb and his colleagues have conducted a series of teaching experiments in elementary school classrooms that examine the role of conversational norms more explicitly attuned to mathematical justification, such as those governing what counts as an acceptable mathematical explanation (e.g., Cobb, Wood, Yackel, & McNeal, 1992; Cobb, Yackel, & Wood, 1988; Yackel & Cobb, 1996). Cobb and his colleagues suggested that mathematical norms constitute an encapsula- tion of what counts as evidence, and a proliferation of norms suggests that students in a class are undertaking a progressive refinement and elaboration of mathematical meaning. In this research several conversational gambits appear reliably to frame the emergence of mathematically fruitful norms. One is discussion of what constitutes a mathematical difference, prompted by teachers who ask if anyone has solved a problem in a different way. Yackel and Cobb (1996) described interactions among students and teachers solving number sentences like 78 Ϫ 53 ϭ ____. During the course of this interaction the teacher accepted strategies that involved recomposition or decomposition of numbers as different, but simple restatements of a particular strategy were not ac- cepted as different (e.g., similar counts with fingers vs. teddy bears). The need to contribute to this kind of collective activity prompted students to reflect about how their strategy was similar to or different from those described by class- mates, a step toward generalization. Moreover, McClain and Cobb (2001) found that negotiation of norms such as what counted as a mathematical difference among first-graders also spawned other norms such as what counted as a sophis- ticated solution. This cascade of norms appeared to have more general epistemological consequences, orienting chil- dren toward mathematics as pattern as they discovered rela- tionships among numbers. Hershkowitz and Schwarz (1999) tracked the arguments made by sixth-grade students in small group and collective discussions of solution strategies and also noted steps toward mathematics as pattern via discussion of mathematical differ- ence. They observed that pedagogy in the sixth-grade class they studied was oriented toward “purifying” students’ in- vented strategies by suppressing surface-level differences among those proposed. The resulting distillation focused student attention on meaningful differences in mathematical structures. Here again a negotiation of what counted as a mathematical difference inspired the growth of mathematical thinking. Krummheuer (1998) suggested that mathematical norms such as difference operate by formatting mathematical conversation, meaning that they frame the interactions among participants. Krummheuer (1995) proposed that formatting is consequential for learning because similarly formatted argu- ments invite cognitive recognition of similarity between approaches taken in these arguments, thus setting the stage for the distillation or purification noted previously. For exam- ple, Krummheuer (1995) documented how two second-grade boys initially disagreed about the similarity of their solution methods to the problem of 8 ϫ 4, but later found that although one subtracted four from a previous result (9 ϫ 4) and another eight from a previous result (10 ϫ 4), they were really talking about the “same way.” This realization initiated discovery of what made them the same—a quality that, in turn, was staged by the norm of what counted as different. Teacher Orchestration of Mathematical Conversation The work of the teacher to establish norms is by no means clear-cut because privileging certain forms of explanation may compete with other goals, such as including all students. Hence, part of the work of the teacher is to find ways to orchestrate discussions that make norms explicit while also de- veloping means to make a norm work collectively (McClain & Cobb, 2001). In her study of argumentation in a second-grade classroom, Wood (1999) illustrated the important role played by teachers in formatting participation itself. She traced how a second-grade teacher apprenticed students to the discourse of mathematical disagreement, differentiating this kind of dis- agreement from everyday, personal contest. Children appren- ticed in problem-solving contexts well within their grasp, so that when they later disagreed about the meaning of place value (one student counted by tens from 49 and another disagreed, contending that counts had to start at decades, as in 50, 60, etc.), the resulting argument centered around mathe- matical, not personal, claims. Wood cautioned that what might seem like fairly effortless ability to orchestrate arguments about mathematical difference relies instead on prior spade- work by the teacher. In this instance, much of that spadework revolved around formatting disagreement. Other classroom studies indicate that teachers assist mathematical argument by explicit support of suppositional reasoning. For example, Lehrer, Jacobson, et al. (1998) conducted a longitudinal study of second-grade mathematics teachers who increasingly encouraged students to investigate the implications of counter- factual propositions (e.g., “What would happen if it were true?”).
The work of the teacher to develop norms and format argument is part of a more general endeavor to understand how teachers assist student thinking about mathematics dialogically. Henningsen and Stein (1997) found that student engagement in classroom mathematics was associated with a sustained press for justification, explanations, or meaning through teacher questioning, comments, and feedback.
The Growth of Argument 365 Spillane and Zeuli (1999) noted that despite endorsing math- ematics reform, teachers nevertheless had difficulty orienting conversation in the classroom toward significant mathemati- cal principles and concepts. O’Connor and Michaels (1996) suggested that teacher orchestration of classroom conversations “provides a site for aligning students with each other and with the content of the academic work while simultaneously socializing them into particular ways of speaking and thinking” (p. 65). The conver- sational mechanisms by which teachers orchestrate mathemat- ically productive arguments include “revoicing” student utterances so that teachers repeat, expand, rephrase, or animate these parts of conversation in ways that increase their scope or precision or that juxtapose temporally discrete claims for con- sideration (O’Connor & Michaels, 1993, 1996). For example, a student may explain how she solved a perimeter problem by saying that she counted all around the hexagonal shape. In response, her teacher might rephrase the student’s utterance by substituting “perimeter” for her expression “all around.” In this instance, the teacher is substituting a mathematical term, “perimeter,” for a more familiar, but imprecise construction, “all around,” thereby transforming the student’s utterance spoken in everyday language into mathematical reference (Forman, Larreamendy-Joerns, Stein, & Brown, 1998). Revoicing encompasses more complex goals than substi- tution of mathematical vocabulary for everyday words or even expanding the range of a mathematical concept. Some revoicing appears to be aimed at communicating respect for ideas and at the larger epistemic agenda of helping students identify aspects of mathematical activity, such as the need to “know for sure” or the idea that a case might be a window to a more general pattern (Strom, Kemeny, Lehrer, & Forman, 2001). For example, in a study of second graders who were learning about geometric transformations by designing quilts, Jacobson and Lehrer (2000) examined differences in how teachers revoiced children’s comments about an instruc- tional video that depicted various kinds of geometric trans- formations in the context of designing a quilt. They found an association between teacher revoicing and student achieve- ment. In classes where teachers revoiced student comments in ways that invited conjectures about the causes of observed patterns or that drew attention to central concepts, students’ knowledge of transformational geometry exceeded that of counterparts whose teachers merely paraphrased or repeated student utterances. Pathways to Proof In classroom cultures characterized by cycles of conjec- ture and revision in light of evidence, student reasoning can become quite sophisticated and can form an important underlying foundation for the development of proof (Reid, 2002). For example, Lampert (2001; Lampert, Rittenhouse, & Crumbaugh, 1996) described a classroom argument about a claim made by one student that 13.3 was one fourth of 55. Other students claimed, and the class accepted, 27.5 as one half of 55. Another student noted that 13.3 ϩ 13. 3 ϭ 26.6, with the tacit premise that one fourth and one fourth is one half, and hence refuted the first claim. Lampert (2001) noted that the logical form of this proof also served to generate an orientation toward student authority and justification, so that the teacher (Lampert) was not the sole or even chief au- thority on mathematical truth. Ball and Bass (2000) docu- mented a similar process with third-grade students who worked from contested claims to commonly accepted knowl- edge by processes of conjecturing, generating cases, and “confronting the very nature and challenge of mathematical proof” (p. 196). Although generating conjectures and exploring their ram- ifications is an important precursor to proof, ironically it is grasping the limitations of this form of argument that moti- vates an important development toward proof as necessity. In classrooms like those taught by Lampert and by Ball, the need for proof emerges as an adjunct to sound argument. For example, a pair of third graders working on a conjecture that an odd number plus an odd number is an even number gener- ated many cases consistent with the conjecture. Yet they were not satisfied because, as one of them said: “You can’t prove that Betsy’s conjecture always works. Because um, there’s, um like, numbers go on and on forever and that means odd numbers and even numbers go on forever, so you couldn’t prove that all of them aren’t” (Ball & Bass, 2000, p. 196). Children’s recognition of the limits of case-based induc- tion has also been observed in other classrooms where teach- ers orchestrate discussions and develop classroom cultures consistent with mathematical practices. For example, Lehrer, Jacobson, et al. (1998) observed a class of second-grade stu- dents exploring transformational geometry who developed the conjecture that there would always be some transfor- mation or composition of transformations that could be ap- plied to an asymmetric cell (a core unit) that would result in a symmetric design. The class searched vigorously for a single countercase among all the asymmetric core units designed by the children in this class and could not generate any refutation. Nevertheless, a subset of the class remained unconvinced and continued to insist that that they could not “be really sure.” Their rationale, like that of the third grader described earlier, focused on the need to exhaustively test all possible cases, a need that could not be met because “we’d have to test all the core squares in the world that are asymmetric” (Lehrer, Jacobson, et al., 1998, p. 183). They went on to note that this criterion could not possibly be met due to its infinite size 366 Mathematical Learning and also because “people are probably making some right now” (p. 183). Hence, in classrooms like these, the need for proof arises as children recognize the limitations of the gener- alization of their argument. Of course, such need arises only when norms valuing generalization and its rationale are established. When children have the opportunity to participate regu- larly in these kinds of classroom cultures, there is good evi- dence that their appreciation of mathematical generalization and the epistemology of proof take root (e.g., Kaput, 1999). For example, Maher and Martino (1996) traced the develop- ment of one child’s reasoning over a five-year span (Grade 1 through 5) as she participated in classrooms of literate math- ematical practice. A trace of conceptual change was obtained by asking Stephanie to figure out how many different towers four or five cubes tall can be made if one selects among red and blue cubes. In the third grade Stephanie attempted to generate cases of combination and eliminate duplicates. Her justification for claiming that she had found all possible tow- ers was that she could not generate any new ones. By the spring of the fourth grade, Stephanie was no longer content with mere generation and instead constituted an empirical proof by developing a means for exhaustively searching all possibilities. In another longitudinal study (Grades 2–3), Lehrer and his colleagues followed students in the same second-grade class that had discovered the limits of case-based generalization into and over the course of the third grade. These students’ mathematical experiences continued in a classroom empha- sizing conjecture, justification, and generalization. Over the course of the third-grade year, researchers recorded many instances of student-generated proof in the context of class- room discussion. At the end of the third grade, all children in the class were interviewed about their preferences for justifi- cations of mathematical conjectures to determine whether proof genres sustained in classroom dialogues would guide the thinking of individual students (Strom & Lehrer, 1999). Four conjectures were presented in the interview, two of which were false and two of which were true. Justifications for true conjectures included single cases, multiple cases, simple restatement of the conjecture in symbolic notation, abstrac- tion of single cases (notation without generalization, as in using an abstract pattern of dots to represent the commutative property of a case), and valid generalizations, in the form of visual proofs (e.g., the rotational invariance of an arbitrary rectangle for commutative property of multiplication). The range of justification types was designed to distinguish be- tween case-based and deductive generalizations on the one hand and the form of proof (the restatement of the conjecture in symbolic notation) from its substance on the other. A similar format of justifications was employed for false conjectures, such as, “When you take half of an even number, you get an even number.” Here, however, we also included a single coun- terexample. Students rank-ordered their preferences. For the false conjecture, over half (55%) of the students selected the counterexample as the best justification and the single case as the worst. For the true conjectures, the majority chose the visual proof as best and either the single case or simple trans- lation of the statement into symbolic notation as worst. Strom and Lehrer (1999) also observed processes of proof generation for these 21 students, asking students to prove that two times any number is an even number. Two of the 21 stu- dents rejected the claim immediately, citing counterexamples with fractions (we had intended whole numbers as a tacit premise). Three other students cited the problem of proof by induction, generated several cases, noted that they were “pretty sure” that the conjecture was true, and then decided that they could not prove it because, as one put it, “because the numbers never stop. . . . I couldn’t ever really prove that” (p. 31). Other students (n ϭ 3) followed a similar line of rea- soning, suggesting that they had “proved it to myself, but not for others” (p. 32). Five students solved the problem of in- duction, either by drawing on definition to deduce the truth of the conjecture or by describing how the patterns they noticed from exploring several cases constituted a pattern that could be applied to all numbers. For example, two of these five stu- dents verified the conjecture for the numbers 1 through 10 and then stated that for numbers greater than 10 “any number that ends in an even number is even” (p. 32). Then each student showed how this implied that the pattern of even numbers they had verified for 1 through 10 extended to all numbers— “The rest of the numbers just have a different number at the beginning” (p. 32). The remaining students generated several cases, searched for and failed to find counterexamples, and then declared that they saw the pattern and so believed the conjecture true. In summary, students who had repeated op- portunity to construct generalizations and proofs during the course of classroom instruction were sensitive to the role of counterexamples in refutation, and nearly all appreciated the limitations of relying on cases (unless one could exhaustively search the set). Generation of proof without dialogic assis- tance was considerably more difficult, but in fact, many were capable of constructing valid proofs, albeit with methods con- siderably more limited than those at the disposal of partici- pants well versed in the discipline. In well-constituted classrooms, young students can suc- ceed at these forms of reasoning with appropriate assistance. However, work with adults illustrates how difficult it can be to acculturate students to proof-based argument. Simon and Blume (1996) conducted a study of prospective mathematics |
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