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CHAPTER 15

Mathematical Learning

RICHARD LEHRER AND RICHARD LESH

357

MATHEMATICAL LEARNING

357

THE GROWTH OF ARGUMENT

358

Conversational Structure as a Resource for Argument

359

From Pretense to Proof

359

Mathematical Argument Emerges in Classrooms

That Support It

363

Reprise of Mathematical Argument

367

INSCRIPTIONS TRANSFORM MATHEMATICAL THINKING

AND LEARNING

367

Disciplinary Practices of Inscription and Notation

368

The Development of Inscriptions as Tools for Thought

369

Inscriptions as Mediators of Mathematical Activity

and Reasoning

369

GEOMETRY AND MEASUREMENT

373

THE MEASURE OF SPACE

374

Mental Representation of Distance

375

Developing Conceptions of Unit

375

Developing Conceptions of Scale

376

Design Studies

376

Estimation and Error

378

STRUCTURING SPACE

380

Potential Affordances of Motion Geometries

380

Learning in Motion

380

MODELING PERSPECTIVES

382

Bridging Epistemologies

382

Cycles of Modeling

383

IMPLICATIONS

384

REFERENCES

385

MATHEMATICAL LEARNING

Does beauty have structure? How does a hinge work? What

happens if zero divides a number? Do the symmetries of a

triangle and the set of integers under addition have any struc-

ture in common? How many distinct patterns of wallpaper

design are possible? What are Nature’s numbers? How do

nurses determine the dosage of drugs (e.g., Pozzi, Noss, &

Hoyles, 1998) or entomologists quantify relations among ter-

mites (e.g., Hall, Stevens, & Torralba, in press)? What forms

of mathematical activity are found in automotive production

(Smith, 1999)? Questions like these suggest the enormous

imaginative scope and practical reach of mathematics and

demonstrate that mathematicians are jugglers not of num-

bers, but of concepts (e.g., Stewart, 1975). Mathematical

practice spans a universe of human endeavor, ranging from

art and craft to engineering design, and its products extend

over much of recorded history. Despite this long history of

mathematics, systematic study of mathematical learning oc-

cupies only a brief slice in time. Nevertheless, research in

mathematics education and in the psychology of mathemati-

cal learning continues to grow, so that any review of this

research is necessarily incomplete and highly selective.

Our choices for this review stem from a genetic view of

knowledge (Piaget, 1970), a “commitment that the structures,

forms, and possibly the content of knowledge is determined

in major respects by its developmental history” (diSessa,

1995, p. 23). Mathematics develops within a collective his-

tory of argument  and inscription (Davis & Hersh, 1981;

Devlin, 2000; Kline, 1980; Lakatos, 1976; Nunes, 1999;

Polya, 1945), so a genetic account of mathematical learning

describes potential origins and developmental landscapes of

these modes of thought. Accordingly, we first examine the

nature of mathematical argument, tracing a path between

everyday forms of argument and those that are widely recog-

nized as distinctly mathematical. In this first section we focus

on the epistemology (the grounds for knowing) and skills of

argument, rather than on the more familiar heuristics and

processes of mathematical reasoning (see, e.g., Haverty,

The authors appreciate the thoughtful comments and suggestions

of Leona Schauble, David Williamson Shaffer, Kathy Metz, Ellice

Forman, and the editors, William Reynolds and Gloria Miller.

This work was supported by the National Science Foundation

(REC 9903409). The opinions expressed do not necessarily reflect

the position, policy, or endorsement of the NSF.

358

Mathematical Learning

Koedinger, Klahr, & Alibali, 2000; Leinhardt & Schwarz,

1997; Schoenfeld, 1992). We suggest that developmental

roots of mathematical argument reside in the structure of nar-

rative and pretend play but note how these roots must be

nurtured to promote epistemic appreciation of proof and

related forms of mathematical argument.

We next turn to the role that inscriptions (e.g., markings in

a medium such as paper) and notations play in the growth and

development of mathematical ideas. Our intention once again

is to illuminate the developmental relationship between infor-

mal scratches on paper and the kinds of symbol systems em-

ployed in mathematical practice. In concert with the core role

assigned to argument, we suggest that mathematical thinking

emerges as refinement of everyday claims about pattern and

possibility yet departs from these everyday roots as these

claims are progressively inscribed and otherwise symbolized.

Inscription and mathematical thinking co-originate (Rotman,

1993), so that mathematics emerges as a distinct form of lit-

eracy, much in the manner in which writing distinguishes

itself from speech.

From these starting points we examine how these general

qualities of mathematical thinking play out in two realms:

geometry measurement and mathematical modeling. We

chose the former because spatial mathematics typically

receives short shrift in reviews of this kind, yet it encom-

passes a tradition that spans two millennia. Furthermore, spa-

tial visualization is increasingly relevant to scientific inquiry

and is undergoing a renaissance in contemporary computa-

tional mathematics. Modeling was selected as the second

strand because modeling emphasizes the need for a broad

mathematical education that includes several forms of math-

ematical inquiry. Moreover, modeling underscores the need

to develop accounts of mathematical learning at the bound-

aries of professional practices and conventionally recognized

mathematical activity (e.g., Moschkovich, 2002).

The studies selected for this review reflect both cognitive

(e.g., Anderson & Schunn, 2000) and sociocultural perspec-

tives (e.g., Forman, in press; Greeno, 1998) on learning. Stud-

ies of cognitive development typically shed light on

individual cognitive processes, for example, how young

students might think about units of measure and how their un-

derstandings might evolve. In contrast, sociocultural perspec-

tives typically underscore thinking as mediated activity (e.g.,

Mead, 1910; Wertsch, 1998). For example, one might con-

sider the history of cultural artifacts, such as rulers, in chil-

dren’s developing conceptions of units. We believe that both

forms of analysis are indispensable and that, in fact, these per-

spectives are interwoven for learners, regardless of re-

searchers’ proclivities to consider them as distinct enterprises.

Consider, for example, the idea of learning to construct a

geometric proof. On the one hand, a cognitive analysis char-

acterizes the kinds of skills required to develop a proof and

describes how those skills must be orchestrated (e.g.,

Koedinger & Anderson, 1990). These forms of char-

acterization seem indispensable to instructional design

(Anderson & Schunn, 2000). On the other hand, the need for

proof is cultural, arising from an epistemology that values

proof as explanation (Harel & Sowder, 1998; Hersh, 1993).

Accordingly, this perspective poses the challenge not just of

accounting for the understanding of proof, but also of how

one might inculcate a classroom culture that values proof. In

the sections that follow, we attempt to strike a balance be-

tween these two levels of explanation because both supply

important accounts of mathematical learning. Because we as-

sume that readers are familiar with the general nature of these

two kinds of analysis, we will not flesh out the assumptions of

each perspective in this chapter.