Handbook of psychology volume 7 educational psychology

Inscriptions Transform Mathematical Thinking and Learning

367

teachers who were schooled traditionally. At first, students

were satisfied by induction over several cases to “prove” that

the area of a rectangle could be constituted by multiplication

of its width and length. Rather than challenging something

that the students all knew to be true, the teacher (Simon)

directed the conversation toward explanation, subtly reori-

enting the grounds of argument from the particular to the

general (e.g., whether this would work all the time). Simon’s

emphasis on the general was further illustrated in another

episode in which students attempted to determine the area of

an irregular blob by transforming its contour to a more famil-

iar form. Although students could see in a case that their strat-

egy in fact also transformed the area, they were not bothered

by this refutation (see also Schauble, 1996), a manifestation

of an everyday sense of the general, rather than a mathemati-

cal sense. Simon and Blume (1996) also encountered the lim-

its of persuasion when students considered justifications of

their predictions about the taste of mixtures that were in

different ratios. Here students talked past one another, appar-

ently because some thought of the situation as additive and

others as multiplicative. Such studies of teaching and learn-

ing again emphasize both the role of the teacher in establish-

ing formats of argument consistent with the discipline and the

need for enculturation so that students can see the functions

of proof, not simple exposure to proof practices.

Reprise of Mathematical Argument

Mathematical argument emphasizes generality and certainty

about patterns and is supported by cognitive capacities to

represent possibility and to reason counterfactually about pos-

sible patterns. These capacities seem to be robustly supported

by cultural practices such as pretense and storytelling. Never-

theless, dispositions to construct mathematically sound argu-

ments apparently do not arise spontaneously in traditional

schooling or in everyday cultural practices. Mathematical

forms such as proof have their genesis in mathematics

classrooms that emphasize conjecture, justification, and

explanation. These forms of thinking demand high standards

of teaching practice because the evidence suggests that

although students may be the primary authors of these

arguments, it is the teachers who orchestrate them. Classroom

dialogue can spawn overlapping epistemologies, as students

are oriented toward mathematics as structure and pattern

while they simultaneously examine the grounds of knowl-

edge. Ideally, pattern and proof epistemologies co-originate in

classrooms because pattern provides the grounds for proof and

proof the rationale for pattern. Thus, classroom conversation

and dialogue constitute one possible genetic pathway toward

the development of proof reasoning skills and an appreciation

of the epistemology of generalization. Yet even as we empha-

size proposition and language, we are struck with the role

played by symbolization and tools in the development of argu-

ments in classrooms and in various guises of mathematical

practice. This is not surprising when one considers the central

historical role of such symbolizations in the development of

mathematics. We turn next to considering a complementary

genetic pathway to mathematical knowledge, that of students

as writers of mathematics.

INSCRIPTIONS TRANSFORM MATHEMATICAL

THINKING AND LEARNING

In this section we explore the invention and appropriation of

inscriptions (literal marks on paper or other media, following

Latour, 1990) as mediational tools that can transform mathe-

matical activity. This view follows from our emphasis in the

previous section on mathematics as a discursive practice in

which everyday resources, such as conversation and pre-

tense, provide a genetic pathway for the development of an

epistemology of mathematical argument, of literally talking

mathematics into being (Sfard, 2000; Sfard & Kieran, 2001).

Here we focus on the flip side of the coin, portraying mathe-

matics as a particular kind of written discourse— “a business

of making and remaking permanent inscriptions . . . operated

upon, transformed, indexed, amalgamated” (Rotman, 1993,

p. 25). Rotman distinguished this view from a dualist view of

symbol and referent as having independent existence,

proposing instead that signifier (inscription) and signified

(mathematical idea) are “co-creative and mutually origina-

tive” (p. 33). Accordingly, we first describe perspectives that

frame inscriptions as mediators of mathematical and scien-

tific activity, with attention to sociocultural accounts of

inscription and argument. These accounts of inscription but-

tress the semiotic approach taken by Rotman (1988, 1993)

and set the stage for cognitive studies of inscription. We go

on to describe children’s efforts to invent or appropriate in-

scriptions in everyday contexts such as drawing or problem

solving. Collectively, these studies suggest that the growth of

representational competence, as reviewed in the previous

section, is mirrored by a corresponding competence in the

uses of inscription and notation. In other words, the having of

ideas and the inscribing of ideas coevolve. Studies of inscrip-

tionally mediated thinking in mathematics indicate that math-

ematical objects are created as they are inscribed. This

perspective calls into question typical accounts in cognitive

science, where inscriptions are regarded as simply referring

to mathematical objects, rather than constituting them. We

conclude this section with the implications of these findings

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Mathematical Learning

for an emerging arena of dynamic inscriptions, namely, com-

putational media.

Disciplinary Practices of Inscription and Notation

Studies in the sociology of science demonstrate that scientists

invent and appropriate inscriptions as part of their everyday

practice (Latour, 1987, 1990; Lynch, 1990). Historically, sys-

tems of inscription and notation have played important roles

in the quantification of natural reality (Crosby, 1997) and are

tools for modeling the world on paper (Olson, 1994). DiSessa

(2000, p. 19) noted,

Not only can new inscription systems and literacies ease learn-

ing, as algebra simplified the proofs of Galileo’s theorems, but

they may also rearrange the entire terrain. New principles

become fundamental and old ones become obvious. Entirely

new terrain becomes accessible, and some old terrain becomes

boring.

Visualizing Nature

One implication of this view of scientific practice as the in-

vention and manipulation of the world on paper (or electronic

screen) is that even apparently individual acts of perceiving

the world, such as classifying colors or trees, are mediated by

layers of inscription and anchored to the practices of discipli-

nary communities (Goodwin, 1994, 1996; Latour, 1986).

Goodwin (1994) suggested that inscriptions do not mirror

discourse in a discipline but complement it, so that profes-

sional practices in mathematics and science use “the distinc-

tive characteristics of the material world to organize

phenomena in ways that spoken language cannot—for exam-

ple, by collecting records of a range of disparate events onto

a single visible surface” (p. 611). For example, archaeolo-

gists classify a soil sample by layering inscriptions, field

practices, and particular forms of talk to render a professional

judgment (Goodwin, 2000). Instead of merely looking,

archaeologists juxtapose the soil sample with an inscription

(the Munsell color chart) that arranges color gradations into

an ordered grid, and they spray water on the soil to create a

consistent viewing environment. These practices format dis-

cussion of the appropriate classification and illustrate the

moment-to-moment embedding of inscription within particu-

lar practices.

Repurposing Inscription

Inscriptions in scientific practice are not necessarily stable.

Kaiser (2000) examined the long-term history of physicists’

use of Feynman diagrams. Initially, these diagrams were

invented to streamline, and make visible, computationally

intensive components of quantum field theory. They drew

heavily on a previous inscription, Minkowski’s space-time

diagrams, which lent an interpretation of Feynman diagrams

as literal trajectories of particles through space and time. Of

course, physicists knew perfectly well that the trajectories so

described did not correspond to reality, but that interpretation

was a convenient fiction, much in the manner in which physi-

cists often talk about subatomic particles as if they were

macroscopic objects (e.g., Ochs, Jacoby, & Gonzales, 1994;

Ochs, Gonzales, & Jacoby, 1996). Over time, the theory for

which Feynman developed his diagrams was displaced, and a

competing inscription tuned to the new theory, dual dia-

grams, was introduced. Yet despite its computational advan-

tages, the new inscription (dual diagrams) never replaced the

Feynman diagram. Kaiser (2000) suggested that the reason

was that the Fenyman diagrams had visual elements in com-

mon with the inscriptions of paths in bubble chambers, and

this correspondence again had an appeal to realism:

Unlike the dual diagrams, Feynman diagrams could evoke, in an

unspoken way, the scatterings and propagation of real particles,

with “realist” associations for those physicists already awash

in a steady stream of bubble chamber photographs, in ways that

the dual diagrams simply did not encourage. (Kaiser, 2000,

pp. 76–77) 

Hence, scientific practices of inscription are saturated in

some ways with epistemic stances toward the world and thus

cannot be understood outside of these views. 

Inscription and Argument

Nevertheless, Latour (1990) suggested that systems of

inscription, whether they are about archaeology or particle

physics, share properties that make them especially well

suited for mobilizing cognitive and social resources in

service of argument. His candidates include (a) the literal

mobility and immutability of inscriptions, which tend

to obliterate barriers of space and time and fix change,

effectively freezing and preserving it so that it can serve as

the object of reflection; (b) the scalability and reproducibility

of inscriptions, which guarantee economy even as they

preserve the configuration of relations among elements of the

system represented by the inscription; and (c) the potential

for recombination and superimposition of inscriptions, which

generate structures and patterns that might not otherwise be

visible or even conceivable. Lynch (1990) reminded us, too,

that inscriptions not only preserve change, but edit it as well:

Inscriptions reduce and enhance information. In the next

section we turn toward studies of the development of children

Inscriptions Transform Mathematical Thinking and Learning

369

as inscribers, with an eye toward continuities (and some

discontinuities) between inscriptions in scientific and every-

day activity.

The Development of Inscriptions as Tools for Thought

Children’s inscriptions range from commonplace drawings

(e.g., Goodnow, 1977) to symbolic relations among maps,

scale models, and pictures and their referents (e.g., DeLoache,

1987) to notational systems for music (e.g., Cohen, 1985),

number (e.g., Munn, 1998), and the shape of space (Newcombe

& Huttenlocher, 2000). These inscriptional skills influence

each other so that collectively children develop an ensemble of

inscriptional forms (Lee & Karmiloff-Smith, 1996). As a con-

sequence, by the age of 4 years children typically appreciate

distinctions among alphabetical, numerical, and other forms of

inscription (Karmiloff-Smith, 1992).

Somewhat surprisingly, children invent inscriptions as

tools for a comparatively wide range of circumstances and

goals. Cohen (1985) examined how children ranging in age

from 5 to 11 years created inscriptions of musical tunes they

first heard, and then attempted to play with their invented

scores. She found that children produced a remarkable diver-

sity of inscriptions that did the job. Moreover, a substantial

majority of the 8- to 11-year-olds created the same inscrip-

tions for encoding and decoding. Their inscriptions adhered

to one-to-one mapping rules so that, for example, symbols

consistently had one meaning (e.g., a triangle might denote a

brief duration) and each meaning (e.g., a particular note) was

represented by only one symbol. Both of these properties are

hallmarks of conventional systems of notation (e.g., Good-

man, 1976). Other studies of cognitive development focus on

children’s developing understandings and uses of inscription

for solving puzzle-like problems.

Karmiloff-Smith (1979) had children (7–12 years) create

an inscriptional system that could be used as an external

memory for driving (with a toy ambulance) a route with a

series of bifurcations. Children invented a wide range of ade-

quate mnemonic marks, including maps, routes (e.g., R and

L to indicate directions), arrows, weighted lines, and the like.

Often, children changed their inscriptions during the course

of the task, suggesting that children transform inscriptions in

response to local variation in problem solving. All of their re-

visions in this task involved making information that was im-

plicit, albeit economically rendered, explicit (e.g., adding an

additional mark to indicate an acceptable or unacceptable

branch), even though the less redundant systems appeared

adequate to the task. Karmiloff-Smith (1992) suggested that

these inscriptional changes reflected change in internal repre-

sentations of the task. An alternative interpretation is that

children became increasingly aware of the functions of

inscription, so that in this task with large memory demands,

changes to a more redundant system of encoding provided

multiple cues and so lightened the burden of decoding—a

tradeoff between encoding and decoding demands.

Communicative considerations are paramount in other

studies of children’s revisions of inscriptions. For example,

both younger (8–9 years) and older (10–11 years) children

adjusted inscriptions designed as aides for others (a peer or

a younger child) to solve a puzzle problem in light of the age

of the addressee (Lee, Karmiloff-Smith, Cameron, &

Dodsworth, 1998). Compared with adults, younger children

were more likely to choose minimal over redundant inscrip-

tions for the younger addressee, whereas the older children

were equally likely to chose either inscription. Overall, there

was a trend for older children to assume that younger

addressees might benefit from redundancy.

In a series of studies with older children (sixth grade

through high school), diSessa and his colleagues (diSessa, in

press; diSessa, Hammer, Sherin, & Kolpakowski, 1991) in-

vestigated what students know about inscriptions in a general

sense. They found that like younger children, older children

and adolescents invented rich arrays of inscriptions tuned

to particular goals and purposes. Furthermore, participants’

inventions were guided by criteria such as parsimony, econ-

omy, compactness (spatially compact inscriptions were pre-

ferred), and objectivity (inscriptions sensitive to audience,

so that personal and idiosyncratic features were often

suppressed).

Collectively, studies of children’s development suggest an

emerging sense of the uses and skills of inscription across a

comparatively wide range of phenomena. Invented inscrip-

tions are generative and responsive to aspects of situation.

They are also effective: They work to achieve the goal at

hand. Both younger and older children adapt features of in-

scriptions in light of the intended audience, suggesting an

early distinction between idiosyncratic and public functions

of inscription. Children’s invention and use of inscriptions

are increasingly governed by an emerging meta-knowledge

about inscriptions, which diSessa et al. (1991) termed

metarepresentational competence. Such capacities ground

the deployment of inscriptions for mathematical activity,

although we shall suggest (much as we did for argument) that

if mathematics and inscription are to emerge in coordination,

careful attention must be paid to the design of mathematics

education.

Inscriptions as Mediators of Mathematical Activity

and Reasoning

Mathematical inscriptions mediate mathematical activity and

reasoning. This position contrasts with inscriptions as mere

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Mathematical Learning

records of previous thought or as simple conveniences for syn-

tactic manipulation. In this section we trace the ontogenesis

of this form of mediated activity, beginning with children’s

early experiences with parents and culminating with class-

rooms where inscriptions are recruited to create and sustain

mathematical arguments.

Early Development

Van Oers (2000, in press) claimed that early parent-child in-

teractions and play in preschool with counting games set the

stage for fixing and selecting portions of counting via in-

scription. In his account, when a child counts, parents have

the opportunity to interpret that activity as referring to cardi-

nality instead of mere succession. For example, as a child

completes his or her count, perhaps a parent holds up fingers

to signify the quantity and repeats the last word in the count-

ing sequence (e.g., 3 of 1, 2, 3). This act of inscription,

although perhaps crudely expressed as finger tallies, curtails

the activity of counting and signifies its cardinality. As

suggested by Latour (1990), the word or tally (or numeral)

can be transported across different situations, such as three

candies or three cars, so number becomes mobile as it is

recruited to situations of “how many.”

Pursuing the role of inscription in developing early num-

ber sense, Munn (1998) investigated how preschool

children’s use of numeric notation might transform their

understanding of number. She asked young children to par-

ticipate in a “secret addition” task. First children saw blocks

in containers, and then they wrote a label for the quantity

(e.g., with tallies) on the cover of each of four containers. The

quantity in one container was covertly increased, and

children were asked to discover which of the containers had

been incremented. The critical behavior was the child’s

search strategy. Some children guessed, and others thought

that they had to look in each container and try to recall its

previous state. However, many used the numerical labels

they had written to check the quantity of a container against

its previous state. Munn found that over time, preschoolers

were more likely to use their numeric inscriptions in their

search for the added block, using inscriptions of quantity to

compare past and current quantities. In her view, children’s

notations transformed the nature of their activity, signaling an

early integration of inscriptions and conceptions of number.

Coconstitution of conceptions of number and inscription

may also rely on children’s capacity for analogy. Brizuela

(1997) described how a child in kindergarten came to under-

stand positional notation of number by analogy to the use of

capital letters in writing. For this child, the 3 in 34 was a

“capital number,” signifying by position in a manner

reminiscent of signaling the beginning of a sentence with a

capital letter. 

Microgenetic Studies of Appropriation of Inscription

The cocreation of mathematical thought and inscription is

elaborated by microgenetic examination of mathematical ac-

tivity of individuals in a diverse range of settings. Hall (1990,

1996) investigated the inscriptions generated by algebra prob-

lem solvers (ranging from middle school to adult participants,

including teachers) during the course of solution. He sug-

gested that the quantitative inferences made by solvers were

obtained within representational niches defined by interaction

among varied forms of inscription (e.g., algebraic expres-

sions, diagrams, tables) and narratives, not as a simple result

of parsing strings of expressions. These niches or material de-

signs helped participants visualize relations among quantities

and stabilized otherwise shifting frames of reference.

Coevolution of inscription and thinking was also promi-

nent in Meira’s (1995, in press) investigations of (middle

school) student thinking about linear functions that describe

physical devices, such as winches or springs. His analysis

focused on student construction and use of a table of values

to describe relations among variables such as the turns of a

winch and the distance an object travels. As pairs of students

solved problems, Meira (1995) noted shifting signification,

reminiscent of the role of the Feynman diagrams, in that

marks initially representing weight shifted to represent

distance. He also observed several different representational

niches (e.g., transforming a group of inscriptions into a single

unit and then using that unit in subsequent calculation), a

clear dependence of problem-solving strategies on qualities

of the number tables, and a lifting away from the physical

devices to operations in the world of the inscriptions—a way

of learning to see the world through inscriptions.

Izsak (2000) found that pairs of eighth-grade students

experimented with different possibilities for algebraic

expressions as they explored the alignment between computa-

tions on paper and the behavior of the winch featured in the

Meira (1995) study. Pairs also negotiated shifting signification

between symbols and aspects of device behavior, suggesting

that interplay between mathematical expression and qualities

of the world may constitute one genetic pathway for mediat-

ing mathematical thinking via inscriptions. (We pick this

theme up again in the section on mathematical modeling.)

In their studies of student appropriation of graphical

displays, Nemirovsky and his colleagues (Nemirovsky &

Monk, 2000; Nemirovsky, Tierney, & Wright, 1998) sug-

gested that learning to see the world through systems of