Handbook of psychology volume 7 educational psychology

Inscriptions Transform Mathematical Thinking and Learning

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inscription is more accurately described as a fusion between

signifiers and signified. In their view, coming to interpret an

inscription mathematically often involves treating the signi-

fiers and the signified as undifferentiated, even though one

knows very well that they can be treated distinctly (the roots

of these capabilities are likely found in pretense and possibil-

ity, as we described previously). In their studies of students’

attempts to interpret graphical displays of physical motion,

they recounted an instance of teacher scaffolding by using

“these” to refer simultaneously to lines on a graph, objects

(toy bears), and a narrative in which the bears were nearing

the finish of a race. This referential ambiguity helped the

student create an interpretation of the inscription that was

more consistent with disciplinary practice as she sorted out

the relations among inscription, object, and the ongoing nar-

rative that anchored use of the inscription to a time course of

events.

According to Stevens and Hall (1998), mathematical

learning mediated by inscription is tantamount to disciplining

one’s perception: coming to see the inscription as a mathe-

matical marking consistent with disciplinary interpretations,

rather than as a material object consistent with everyday in-

terpretations. That such a specialized form of perception is

required is evident in the confusions that even older students

have about forms of notation like the graph of a linear func-

tion. For example, a student’s interpretation of slope in a case

study conducted by Schoenfeld, Smith, and Arcavi (1993)

included a conception of the line as varying with slope, y-

intercept, and x-intercept. The result was that the student’s

conception of slope was not stable across contexts of use.

Stevens and Hall (1998) traced the interventions of a tutor

who helped an eighth-grade student working on similar

problems of interpretation of graphical displays. Their analy-

sis focused on the tutoring moves that helped reduce the

student’s dependence on a literal grid representing Cartesian

coordinates. Some of the teacher’s assistance included literal

occlusion of grid, a move designed to promote disciplinary

understanding by literally short-circuiting the student’s

reliance on the grid in order to promote a disciplinary focus

on ratio of change to describe the line. Moschkovich (1996)

examined how pairs of ninth-grade students came to disci-

pline their own perceptions by coordinating talk, gestures,

and inscriptions of slope and intercept. Inscriptions helped

orient students toward a shared object of reference, and the

use of everyday metaphors such as hills and steepness

grounded this joint focus of conversation. Ultimately, how-

ever, the relative ambiguity of these everyday metaphors

instigated (for some pairs) a more disciplined interpretation

because meanings for these terms proved ambiguous in the

context of conversation. However, not all pairs of students

evolved toward disciplinary-centered interpretation, again

suggesting the need for instructional support.

Studies of Inscription in Classrooms Designed to Support

Invention and Appropriation

Some research provides glimpses of invention and use of

inscription in classrooms where the design of instruction

supports students’ invention and appropriation of varying

forms of mathematical inscription. These studies are oriented

toward a collective level of analysis (i.e., treating the class as a

unit of analysis) because the premise is that, following Latour

(1990), inscriptions mobilize arguments in particular commu-

nities. In these studies the community is the mathematics cul-

ture of the classroom. Moreover, “a focus on inscriptions

requires traditional learning environments to be redesigned in

such a way that students can appropriate inscription-related

practices and discourses” (Roth & McGinn, 1998, p. 52).

Cobb, Gravemeijer, Yackel, McClain, and Whitenack

(1997) traced children’s coordination of units of 10 and 1 in a

first grade class. Instruction was designed to situate investi-

gation of these units and unit collections in a context of pack-

aging candies. Arithmetic reasoning was constituted as a

“chain of signification” (Walkerdine, 1988) in which unifix

cubes first signified a quantity of candies packed in the shop

and then this sign (the unifix cubes–candies relation) was

incorporated as a signified of various partitions of candies

inscribed as pictured collections. At this point the structure of

the collection, rather than the original packaging of candy,

became the object of thinking. The structure of the collection,

in turn, served as the signified of yet another signifier, a

notational rendering of collections as, for instance, 3r13c

(3 rolls, 13 candies). Cobb et al. (1997) noted that this ren-

dering served as the vehicle by means of which the pictured

collections became models of arithmetic reasoning (also see

Gravemeijer, Cobb, Bowers, & Whitenack, 2000).

Kemeny (2001) examined the collective dialogic pro-

cesses during a lesson in which a third-grade teacher helped

students construct the mathematical object referred to by the

inscription of the Cartesian system. Her analysis underscores

the interplay between collective argument and inscription. It

also highlights the role of the teacher’s orchestration of con-

versation and inscription. First, the teacher introduced a new

signifier, drawing the axes of the coordinate system on the

blackboard, and invited students to consider whether it might

be a good tool for thinking about relationships between the

sides of similar rectangles. Because these students had a

prior history of investigating concepts of ratio via the study

of geometric similarity (Lehrer, Strom, & Confrey, in press),

the introduction of the signifier (the inscription) created an

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opportunity for students to create the signified—the Carte-

sian grid (see Sfard, 2000). Children’s first attempts to gen-

erate a signified were based on projecting metaphors of

measure. They decided, for example, that the lengths of the

axes should be subdivided into equal measures and that this

subdivision implied an origin labeled numerically as zero

because movement along the axis was a distance, not a

count. They debated where this origin should be placed and

generated several valid alternatives. At this point, the teacher

stepped in to introduce a convention, which students ac-

cepted as sensible.

Some students then transported a practice they had gener-

ated in previous investigations, superimposing paper models

of similar rectangles to observe their growth, to the axes on

the blackboard, drawing rectangles that mimicked the paper

material. This invited consideration of the axes as a literal

support (and raised questions about what to label them), but

it also inspired one student to notice a stunning possibility—

a rectangle might be represented by one of its vertices. Per-

haps there was no need to draw the whole thing! Their

teacher promptly seized upon this suggestion, and the stu-

dents went on to explore its implications. Eventually, they

concluded that there could be as many rectangles as they

liked, not just the cases initially considered, and that all sim-

ilar rectangles could be represented and generated as a line

through the origin.

Inscription (Cartesian coordinates) and argument (a gen-

eralization about similar figures) were co-originated. The

inscription did not spring out of thin air, but it became a tar-

get of metaphoric projection and extension and was ulti-

mately treated as an object in its own right. The construction

of this object invited a format for generalization, the line rep-

resenting all rectangles, and also an epistemology of pattern.

What was true for three or four cases was accepted as true for

infinitely many. Over the course of several lessons, students’

inscriptions of similarity as numeric ratio, as algebraic

pattern (e.g., the class of similar rectangles described by

LS

ϭ 3 ϫ SS, where LS and SS refer to “long side” and

“short side,” respectively), and as a line in the Cartesian sys-

tem introduced a resonance among inscriptional forms. For

example, the sense of pattern generalization could be ex-

pressed in three distinctive forms of inscription, yet the

equivalence of these forms invited construction of a signified

that spanned all three (Lehrer et al., in press).

The lesson analyzed by Kemeny (2001) was anchored in a

history of inscription in the classroom (Lehrer, Jacobson,

Kemeny, & Strom, 1999; Lehrer & Pritchard, in press). The

norms in the classroom included a stance toward adopting

inscriptions as tools for thinking and, further, toward assum-

ing that no inscription would be wasted; that is, if students de-

veloped a stable (and public) system of mathematical

inscription, they could reasonably expect to use it again. One

such opportunity was presented to students later in the year

when they conducted investigations about the growth of

plants. Lehrer, Schauble, Carpenter, and Penner (2000)

tracked students’ inscriptions of plant growth during succes-

sive phases of inquiry over the course of approximately three

months. The investigators found a reflexive relationship be-

tween children’s inscriptions of growth and their ideas about

growth. Over time, children either invented or appropriated

inscriptions that increasingly drew things together by increas-

ing the dimensionality of their models of growth. For exam-

ple, initial inscriptions were one-dimensional records of

height, but these were later supplanted by models of plant vol-

ume that incorporated variables of height, width, and depth

and that were sequenced chronologically to facilitate test of

the conjecture that plant growth was an analogue of geometric

growth (which it was not). Inscription and conception of

growth were co-originated in Rotman’s (1993) sense.

Notation: A Privileged Inscription

Developmental studies of children’s symbolization, microge-

netic studies of individuals’ efforts to appropriate inscription,

and collective studies of classrooms where inscriptions are

recruited to argument describe a complementary genetic path-

way for the development of mathematical reasoning: the in-

teractive constitution of inscription and mathematical objects.

These studies also reveal the cognitive and social virtues of

privileging notations among inscriptions.

Goodman (1976) suggested heuristic principles to dis-

tinguish notational systems from other systems of inscrip-

tion. The principles govern relations among inscriptions

(signifiers–literal markings), objects (signified), character

classes (equivalent inscriptions, such as different renderings

of the numeral 7), and compliance classes (equivalent ob-

jects, such as dense materials or emotional people). Two prin-

ciples govern qualities of inscriptions that qualify as notation:

(a) syntactic disjointedness, meaning that each inscription be-

longs to only one character class (e.g., the marking 7 is rec-

ognized as a member of a class of numeral 7s, but not numeral

1s), and (b) syntactic differentiation, meaning that one can

readily determine the intended referent of each mark (e.g., if

one marked quantity with length, then the differences in

length corresponding to differences in quantity should be per-

ceived readily).

Two other principles regulate mappings between charac-

ter classes and compliance classes. The first is that all inscrip-

tions of a character class should have the same compliance

class, which Goodman (1976) referred to as a principle of

unambiguity. For example, all numeral 7s should refer to the

same quantity, even though the quantity might be comprised

Geometry and Measurement

373

of seven dogs or seven cats. It follows, then, that character

classes should not have overlapping fields of compliance

classes—the principle of semantic disjointedness. For exam-

ple, the numeral 7 and the numeral 8 should refer to different

quantities. This requirement rules out natural language’s inter-

secting categories, such as whale and mammal. Finally, a prin-

ciple of semantic differentiation indicates that every object

represented in the notational scheme should be able to be clas-

sified discretely (assigned to a compliance class)—a principle

of digitalization of even analog qualities. For example, the

quantities 6.999 and 7.001 might be assigned to the quantity 7,

either as a matter of practicality or as a matter of necessity

before the advent of a decimal notation.

These features of notational systems afford the capacity

to treat symbolic expressions as things in themselves, and

thus to perform operations on the symbols without regarding

what they might refer to. This capacity for symbolically

mediated generalization creates a new faculty for mathemat-

ical reasoning and argument (Kaput, 1991, 1992; Kaput &

Schaffer, in press). For example, the well-formedness of no-

tations makes algorithms possible, transforming ideas into

computations (Berlinski, 2000). Notational systems simulta-

neously provide systematic opportunity for student expres-

sion of mathematical ideas, but the same systematicity

places fruitful constraints on that expression (Thompson,

1992).

We have seen, too, how notations transform mathematical

experiences genetically, both over the life span (from early

childhood to adulthood) and over the span of growing

expertise (from novices to professional practitioners of math-

ematics and science). Consider, for example, the van Oers

(2000, in press) account of parental scaffolding to notate

children’s counting. This marking objectifies counting activity

so that it becomes more visible and entity-like. The use of a

symbolic system for number foregrounds the quantity that

results from the activity of counting and backgrounds the

counting act itself. This separation of activity (counting) from

its product (quantity) sets the stage for making quantity a sub-

strate for further mathematical activities, such as counts of

quantities as exemplified in the Cobb et al. (1997) study of first

graders. Microgenetic studies like those of Hall (1990) and

Meira (1995) suggest that inscriptions tend to drift over time

and use toward notations that stabilize interactions among par-

ticipants. The classroom studies by Kemeny (2001) and

Lehrer et al. (2000) also suggest a press toward notation as a

means of fixing, selecting, and composing mathematical

objects as tools for argument. These studies, however, concen-

trate largely on the world on paper, so in the next section

we address the implications of electronic technologies for

bootstrapping the reflexive relation between conception and

inscription.

Dynamic Notations

The chief effect of electronic technologies is the correspond-

ing development of new kinds of notational systems, often

described as dynamic (Kaput, 1992). The manifestations of

electronically mediated notations are diverse, but what they

share in common is an expression of mathematics as compu-

tation (Noss & Hoyles, 1996). DiSessa (2000) suggested that

computation is a new form of mathematical literacy, conclud-

ing that computation, especially programming, “turns analy-

sis into experience and allows a connection between analytic

forms and their experiential implications” (p. 34). Moreover,

simulating experience is a pathway for building students’ un-

derstanding, yet it is also integral to the professional practices

of scientists and engineers.

Sherin (2001) explored the implications of replacing alge-

braic notation with programming for physics instruction.

Here again, notations did not simply describe experience for

students, but rather reflexively constituted it. Programming

expressions of motion afforded more ready expression of

time-varying situations. This instigated a corresponding shift

in conception from an algebraically guided physics of bal-

ance and equilibrium to a physics of process and cause.

Resnick (1994) pointed out that introducing students to

parallel programming (e.g., multiple screen “turtles”) pro-

vides an opportunity to develop mathematical descriptions at

multiple levels and to understand how levels interact. The

programming language provides an avenue for decentralized

thinking. Wilensky and Resnick (1999) noted the difficulties

that people have in comprehending levels of phenomena such

as traffic jams. At one level, traffic jams result from cars mov-

ing forward, but the interactions among cars create jams that

proliferate backward. This effect seems at first glance to

violate common sense, so it is hard for people to compre-

hend, but dynamic notations such as parallel programming

place new tools in the hands of students for thinking about re-

lations between local agents and aggregate levels of descrip-

tion. Our (much) abbreviated tour of dynamic notations

clearly indicates that this form of inscription affords new op-

portunities to coconstitute mathematical thought and writing.

In the sections that follow, we revisit this theme in the realms

of geometry measurement and mathematical modeling.

GEOMETRY AND MEASUREMENT

Geometry is a spatial mathematics that has its roots in antiq-

uity yet continues to evolve in the present, as witnessed by

continuing concern with computer-generated experiments in

visualization. Although common school experiences of

geometry emphasize the construction and proof schemes

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of the ancient Greeks, the scope of geometry is far wider,

ranging from consideration of fundamental qualities of space

such as shape and dimension (e.g., Banchoff, 1990; Senechal,

1990) to the very fabric of artistic design, commercial craft,

and models of natural processes (e.g., Stewart, 1998). Con-

sider, for example, the designs displayed in Figure 15.1. Both

were created from the same primary cell (unit) but with dif-

ferent symmetries (the left by a translation, the right by a ro-

tation). Systematic analyses of symmetries of design

stimulate both mathematical inquiry (e.g., Schattschneider,

1997; Washburn & Crowe, 1988) and the ongoing practice of

crafts such as quilting (e.g. Beyer, 1999).

Geometry’s versatility and scope have oriented us to sur-

vey a range of studies that demonstrate the potential role of

geometry in a general mathematics education (Goldenberg,

Cuoco, & Mark, 1998; Gravemeijer, 1998). Our chief em-

phasis is on studies of the growth and development of spatial

reasoning in contexts designed to support development (prin-

cipally, schools). We first consider studies of children’s

unfolding understanding of the measure of space. Although

measurement is (now) traditionally separated from geometry

education, we argue for its reinstatement on two grounds.

First, measuring a quality of a space invokes consideration of

its nature. For example, although measure of dimension

seems transparent, the dimension of fractal images in not ob-

vious, and consideration of their measure leads one toward

more fundamental ideas about their construction (e.g.,

Devaney, 1998). Second, measurement is inherently approx-

imate so that it constitutes a bridge to related forms of math-

ematics, such as distribution and reasoning about variation.

Third, practices of measurement span multiple realms of en-

deavor, especially the quantification of physical reality

(Crosby, 1997). Even apparently simple acts, such as match-

ing the color of a sample of dirt to an existing classification

scheme, are in fact embedded within systems of inscription

and practice, so that measurement is a window to the inter-

play between imagined qualities of the world and the practi-

cal grasp of these qualities (Goodwin, 2000). Consequently,

our review focuses on research that helps us understand the

kinds of thinking at the heart of the interplay between this

imaginative leap (i.e., an imagined quality of space) and

practical grasp (e.g., its measure).

After completing our review of measure, we consider how

inquiry about shape and form frames developing types of

arguments, especially proof and related “habits of mind”

(Goldenberg et al., 1998). Here we focus on the role of

dynamic notational systems, embodied (currently) as soft-

ware tools such as Logo (Papert, 1980) and the Geometer’s

Sketchpad (Jackiw, 1995), because these spotlight the role

of dynamic notation in the development of mathematical

reasoning and argument about space.

THE MEASURE OF SPACE

In the sections that follow, we review investigations of chil-

dren’s reasoning about measure. We focus primarily on stud-

ies of linear measure to illuminate the interactive roles of

inscription and developing conceptions of space because

these studies encapsulate many of the findings, issues, and

approaches that emerge in investigations of other dimensions

and qualities of space, such as area, volume, and angle (see

Lehrer, 2002; Lehrer, Jaslow & Curtis, in press, for more

extensive review of the latter). We include studies from mul-

tiple perspectives. Studies of cognitive development typically

compare children at different ages (cross-sectional) or follow

the same children for a period of time (longitudinal) to ob-

serve transitions in thinking, typically about units of mea-

sure. These studies provide glimpses of children’s thinking

under conditions of activity and learning that are typically

found in the culture. They follow from the tradition first es-

tablished by Piaget and his colleagues (e.g., Piaget, Inhelder,

& Szeminska, 1960). In contrast, design studies modify the

learning environment and then investigate the effects of

these modifications (Brown, 1992; Cobb, 2001). These stud-

ies are often conducted from sociocultural perspectives with

attendant attention to forms of inscription and notation and

to forms of classroom talk that seem important to help