Handbook of psychology volume 7 educational psychology

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.

From Pretense to Proof

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.

Development of Representational Competence

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.

The Skills of Argument

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