Handbook of psychology volume 7 educational psychology

The Growth of Argument

363

why students might acquire the skills of proof if they do not

see its epistemic point.

Reprise of Pretense to Proof

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