Handbook of psychology volume 7 educational psychology

Figure 15.1

Symmetries of design produced by varying transformations.

The Measure of Space

375

learning to push development in the manner first articulated

by Vygotsky (1978).

Mental Representation of Distance

Piaget et al. (1960) proposed that to obtain a measure of

length, one must subdivide a distance and translate the subdi-

vision. Thus, n iterations of a unit represent a distance of n

units. Because distance is not a topological feature, Piaget

et al. (1960) proposed that children may fail to understand

that translation does not affect distance (i.e., that simple mo-

tion of a length does not change its measure), a symptom in

Piaget’s view of topological primacy in children’s represen-

tations of space. For example, preschool children often assert

that objects become closer together when they are occluded.

Piaget et al. (1960) believed that this assertion revealed chil-

dren’s use of a topological representation that would preserve

features such as continuity between points but not (necessar-

ily) distance because occlusion disrupts the topological prop-

erty of continuity.

A series of experiments conducted by Miller and

Baillargeon (1990) suggested instead that children’s asser-

tions reflected their relative perceptions of occluded and un-

occluded distances. Children from 3 to 6 years of age

proposed wooden lengths that would span a distance between

two endpoints of a bridge. The distance was then partially oc-

cluded. Although children often reported that the occluded

endpoints were closer together, they also asserted that the

length of the stick that “just fit” between them was unaf-

fected. This lack of correspondence between what children

said and what they did refuted the topological hypothesis, in-

dicating instead that children’s responses were guided by ap-

pearances, not mental representations of distance governed

by continuity of points. Research solidly refutes Piaget’s

equating of the historic structuring of geometries (e.g., pro-

gressing from Euclidean to topological) to changes over the

life span in ways of mentally representing space (e.g., Darke,

1982). For example, more contemporary research demon-

strates that infants (and rats) encode (Euclidean) metric in-

formation (see Newcombe & Huttenlocher, 2000). Although

it then seems reasonable to assume an implicit metric repre-

sentation of distance, Piaget’s core agenda of documenting

transitions in children’s constructions of invariants about

units of measure has proven fruitful.

Developing Conceptions of Unit

Children’s first understandings of length measure often in-

volve direct comparison of objects (Lindquist, 1989; Piaget

et al., 1960). Congruent objects have equal lengths, and

congruency is readily tested when objects can be superim-

posed or juxtaposed. Young children (first grade) also typi-

cally understand that the length of two objects can be

compared by representing them with a string or paper strip

(Hiebert, 1981a, 1981b). This use of representational means

likely draws on experiences of objects “standing for” others

in early childhood, as we described previously. First graders

(6- and 7-year-olds) can use given units to find the length of

different objects, and they associate higher counts with longer

objects (Hiebert, 1981a, 1981b; 1984). Most young children

(first and second graders) even understand that, given the

same length to measure, counts of smaller units will be larger

than counts of larger units (Carpenter & Lewis, 1976).

Lehrer, Jenkins, and Osana (1998) conducted a longitudi-

nal investigation of children’s conceptions of measurement in

the primary grades (a mixed age cohort of first-, second-, and

third-grade children were followed for three years). They

found that children in the primary grades (Grades 1–3, ages

6–8) may understand qualities of measure like the inverse

relation between counts and size of units yet fail to appreciate

other constituents of length measure, like the function of

identical units or the operation of iteration of unit. Children in

this longitudinal investigation often did not create units of

equal size for length measure (Miller, 1984), and even when

provided equal units, first and second graders typically did

not understand their purposes, so they freely mixed, for

example, inches and centimeters, counting all to measure a

length.

For these students, measure was not significantly differen-

tiated from counting (Hatano & Ito, 1965). Thus, younger

students in the Lehrer, Jenkins, et al. (1998) study often

imposed their thumbs, pencil erasers, or other invented units

on a length, counting each but failing to attend to inconsis-

tencies among these invented units (and often mixing their

inventions with other units). Even given identical units,

significant minorities of young children failed to iterate spon-

taneously units of measure when they ran out of units, despite

demonstrating procedural competence with rulers (Hatano &

Ito, 1965). For example, given 8 units and a 12-unit length,

some primary-grade children in the longitudinal study

sequenced all 8 units end to end and then decided that they

could not proceed further. They could not conceive of how

one could reuse any of the eight units, indicating that they

had not mentally subdivided the remaining space into unit

partitions.

Children often coordinate some of the components of

iteration (e.g., use of units of constant size, repeated applica-

tion) but not others, such as tiling (filling the distance with

units). Hence, children in the primary grades occasionally

leave spaces between identical units even as they repeatedly

376

Mathematical Learning

use a single unit to measure a length (Lehrer, 2002). The

components of unit iterations that children employ appear

highly idiosyncratic, most likely reflecting individual differ-

ences in histories of learning (Lehrer, Jenkins, et al., 1998).

Developing Conceptions of Scale

Measure of length involves not only the construction of unit

but also the coordination of these units into scales. Scales

reduce measurement to perception so that the measure of

length can be read as a point on that scale. However, only a

minority of young children understand that any point on a

scale of length can serve as the starting point, and even a sig-

nificant minority of older children (e.g., fifth graders) re-

spond to nonzero origins by simply reading off whatever

number on a ruler aligns with the end of the object (Lehrer,

Jenkins, et al., 1998).

Many children throughout schooling begin measuring

with one rather than with zero (Ellis, Siegler, & Van Voorhis,

2000). Starting a measure with one rather than zero may re-

flect what Lakoff and Nunez (2000) referred to as metaphoric

blend. One everyday metaphor for measure is that of the mea-

suring stick, where physical segments such as body parts

(e.g., hands) are iterated and the basic unit is one stick. An-

other everyday metaphor is that of motion along a path,

corresponding to children’s experiences of walking (Lakoff

& Nunez, 2000). Measure of a distance is then a blend of

motion and measuring-stick metaphors, which may lead to

mismappings between the 1 count of unit sticks and 0 as the

origin of the path distance (Lehrer et al., in press). The diffi-

culties entailed by this metaphoric blend are often most evi-

dent when children need to develop measures that involve

partitions of units. For example, Lehrer, Jacobson, Kemeny,

and Strom (1999) noted that some second-grade children

(7–8 years of age) measured a 2 1

͞2-unit strip of paper as

3 1

͞2 units by counting, “1, 2, [pause], 3 [pause], 3 1͞2.”

They explained that the 3 referred to the third unit counted,

but “there’s only a 1

͞2,” so in effect the last unit was repre-

sented twice, first as a count of unit and then as a partition of

a unit. Yet these same children could readily coordinate dif-

ferent starting and ending points for integers (e.g., starting at

3 and ending at 7 was understood to yield the same measure

as starting at 1 and ending at 5).

Design Studies

Design studies focus on establishing developmental trajecto-

ries for children’s conceptions of linear measure in contexts de-

signed to promote children’s use of inscription and tools. These

tools and inscriptions are typically objects of conversation in

classrooms, recruited to resolve contested claims about com-

parative lengths of objects or about reasonable estimates of

an object’s length. Hence, these studies are representative of

contexts in which conversation, inscription, and tool use are

typically interwoven.

Inscriptions and Tools Mediate Development

of Conceptions of Measure

Choices of tools often have consequences for children’s con-

ceptions of length (Nunes, Light, & Mason, 1993). Clements,

Battista, and Sarama (1998) reported that using computer

tools that mediated children’s experience of unit and iteration

helped children mentally restructure lengths into units. Third

graders (9-year-olds) created paths on a computer screen with

the Logo programming language. Many activities focused on

composing and decomposing lengths, which, in combination

with the tool, encouraged students to privilege some seg-

ments and their associated command (e.g., forward 10) as

units. Subsequently, children found unknown distances by it-

erations of these units. For example, one student found a

length of 40 turtle units by iterating 10 turtle units. Students

in this and related investigations apparently developed con-

ceptual rulers to project onto unmarked segments (Clements,

Battista, Sarama, Swaminathan, & McMillen, 1997). In an

investigation conducted by Watt (1998), fifth-grade students

employed a children’s computer-aided design tool, kidCAD,

to create blueprints of their classroom. At the outset of the in-

vestigation, students displayed many of the hallmarks of con-

ceptions of measure that one might expect from the studies of

cognitive development. That is, students evidenced tenuous

grasp of the zero point of the measurement scale and mixed

units of length measure. Here, students’ efforts to create con-

sistency between their kidCAD models and their classroom

helped make evident the rationale for measurement conven-

tions. These recognitions led to changes in measurement

practices and conceptions.

Other studies place a premium on children’s constructions

of tools and inscriptions for practical measurements. This

form of practical activity facilitates transition from embodied

activity of length measure, such as pacing, to symbolizing

these activities as “foot strips” and related measurement tools

(Lehrer et al., 1999; McClain, Cobb, Gravemeijer, & Estes,

1999). By constructing tools and inscribing units of measure,

children have the opportunity to discover, with guidance,

how scales are constructed. For example, children often puz-

zle about the meaning of the marks on rulers, and the func-

tions of these marks become evident to children as they

attempt to inscribe units and parts of units on their foot strips

(Lehrer et al., 1999). Moreover, when all students do not

The Measure of Space

377

employ the same unit of measure, the resulting mismatches in

the measure of any object’s length spurs the need for a

conventional unit (Lehrer et al., in press). These mismatches

highlight that measurement is not purely a cognitive act. It

also relies on perceiving the social utility of conventional

units and the communicative function served by common

methods of measure.

Tools Enhance the Visibility of Children’s Thinking

for Teachers

The construction of tools also makes children’s thinking

more visible to teachers, who can then transform instruction

as needed (Lehrer et al., in press). For example, Figure 15.2

displays a facsimile of a foot-strip tape measure designed by

a third-grade student, Ike, who indicated that the measure of

the ruler’s length was 4 because 4 footprint units fit on the

tape. Some components of iteration of unit are salient; the

units are all alike, and they are sequenced. On the other hand,

the process to be repeated appears to be a count, rather than a

measure, as indicated by the lack of tiling (space filling) of

the units. Construction of this tool mediated this student’s

understanding of unit, helping make salient some qualities of

unit. As we noted previously, these qualities of selection and

lifting away from the plane of activity are commonplace fea-

tures of notational systems. Other qualities that were evident

in this student’s paces (when he walked a distance, he placed

his feet heel to toe) remained submerged in activity. Hence, in

this classroom, creation of the tool provided a discursive

opening for the teacher and for other students who disagreed

with Ike’s production and who suggested that perhaps the

“spaces mattered.”

Splitting and Rational Number

Measurement can serve as a base metaphor for number.

Confrey (1995; Confrey & Smith, 1995) suggested an inter-

penetration between measure and conceptions of number via

splitting. Splitting refers to repeated partitions of a unit to

produce multiple similar forms in direct ratio to the splitting

factor. For example, halving produces ratios of 1 : 2. Rather

than simply split paper strips as an activity for its own sake,

measurement provides a rationale for splitting. Consequently,

in a classroom study Lehrer et al. (1999) observed second-

grade children repeatedly halving unit lengths as they

designed rulers. The need for these partitions of unit arose as

children attempted to measure lengths of objects that could

not be expressed as whole numbers.

Most children folded their unit (represented as a length of

paper strip) in half and then repeated this process to create

fourths, eighths, and even sixty fourths. These partitions were

then employed in children’s rulers, and children noticed that

they could increase the precision of measure. Eventually, these

actions helped children develop operator conceptions of ratio-

nal numbers, such as 1

͞2 ϫ 1͞2 ϫ 1͞2 ϭ 1͞8, and so on.

Similarly, division concepts of rational numbers were pro-

moted by classroom attention to problems involving

exchanges among units of measure for a fixed length. For

example, if one Stephanie (unit) is one-half of a Carmen

(unit) and a board is 4 Carmens long, what is its measure in

Stephanies? The visual relations among paper-strip models of

these units helped children differentiate between “one half

of” and “divided by one half.” Moss and Case (1999) also

featured splitting of linear measurement units as a means to

help students develop concepts of rational numbers. Their

work with fourth-grade students indicated that measure and

splitting, coupled with an emphasis on equivalence among

different notations of rational number, helped students

develop understanding of proportionality and, correlatively,

of rational numbers.

Measure and Modeling as a Gateway to Form

Classroom studies point to ways of melding linear measure

and the study of form in the elementary grades in ways that

recall their historical codevelopment. Children in Elizabeth

Penner’s first- and second-grade classes searched for forms

(e.g., lines, triangles, squares) that would model the configu-

ration of players in a fair game of tag (Penner & Lehrer,

2000). Attempts to inscribe the shape of fairness initiated

Figure 15.2

A foot-strip measure designed by a third grader.

378

Mathematical Learning

cycles of exploration involving length measure and proper-

ties related to length in each form (e.g., distances from the

sides of a square to the center). Eventually, children decided

that circles were the fairest of all forms because the locus of

points defining a circle was equidistant from its center. This

insight was achieved by emerging conceptions of units of lin-

ear measure (e.g., children created foot strips and other tools

to represent their paces) and by employing these understand-

ings to explore properties of shape and form. For example,

children were surprised to find that the distance between the

center of a square and a side varied with the path chosen.

Diagonal paths were longer than those that were perpendicu-

lar to a side, so they concluded that square configurations

were not fair, despite the congruence of their sides.

Children in Carmen Curtis’s third-grade class investigated

plant growth and modeled changes in their canopy as a series

of cylinders. Developing the model posed a new challenge in

mathematics, namely, grasping the correspondence between a

measure of “width” (the diameter of the base of the cylinder)

and its circumference. In other words, children could readily

measure the width but then had to figure out how diameter

could be used to find circumference. This challenge instigated

mathematical investigation, one that culminated in an approx-

imation of the relation between circumference and diameter

as “about 3 1

͞5.” So, in the course of modeling nature, chil-

dren developed a conjecture about the relationship between

properties of a circle. Of course, their investigations did not

end here, because having convinced themselves and others

about the validity of their model of the canopy of the plant,

they next had to concern themselves with how to measure its

volume (Lehrer et al., in press). In sum, tight couplings be-

tween space and measure in these modeling applications are

reminiscent of Piaget’s investigations but acknowledge that

these linkages are the object of instructional design, instead

of regarding them as preexisting qualities of mind.

Measure and Argument

In some classrooms measures are recruited in service of

argument. For example, in one of the second-grade class-

rooms referred to previously (Lehrer et al., 1999), children

saw paper models of three different rectangles and were

asked to consider which covered the most space on the black-

board. The rectangles all had the same area but were of dif-

ferent dimension (1

ϫ 12, 2 ϫ 6, 3 ϫ 4 units). The rectangles

were not marked in any way, nor were any tools provided.

Children’s initial claims were based on mere appearance.

Some thought that the “fat” rectangles (i.e., the 3

ϫ 4) must

cover the most space, others that the “long” (i.e., 1

ϫ 12) rec-

tangles must. These contested claims set the stage for the

teacher’s orchestration of argument: How could these claims

be resolved? Strom et al. (2001) analyzed the semantic struc-

ture of the resulting classroom conversation and rendered

its topology as a directed graph. The nodes of the graph

consisted of various senses of area as children conceived it

(e.g., as space covered, as composed of units), as enacted

(e.g., procedures to partition and reallot areas, procedures

that privileged certain partitions as units), and as historically

situated (e.g., children’s senses of this situation as related to

others that they had previously encountered). The analysis

highlighted the interplay among these forms of knowledge—

an interplay characteristic also of professional practice (e.g.,

Rotman, 1988)—and illustrated that the genetic trajectories

of conceptual, procedural, and historical knowledge were

firmly bound, not distinct. Moreover, a pivotal role was

played by notating the unit-of-area measure, a process that

afforded mobility and consequent widespread deployment of

unit in service of argument. That is, once the unit-of-area

measure assumed consensual status as a legitimate tool in the

classroom, it was used literally to mark off segments of area

on the three rectangles, eventually establishing that regard-

less of appearance, each covered 12 square units of space:

All three rectangles covered the same space. Of course, the

argument constructed by children was orchestrated by the

teacher, who animated certain students’ arguments, juxta-

posed temporally distant forms of reasoning, and reminded

students of norms of argument and justification throughout

the lesson.

Estimation and Error

Much of the research about measurement explores preci-

sion and error of measure in relation to mental estimation

(Hildreth, 1983; Joram, Subrahmanyam, & Gelman, 1998).

To estimate a length, students at all ages typically employ the

strategy of mentally iterating standard units (e.g., imagining

lining up a ruler with an object). In their review of a number

of instructional studies, Joram et al. (1998) suggested that

students often develop brittle strategies closely tied to the

original context of estimation. Joram et al. recommended that

instruction should focus on children’s development of refer-

ence points (e.g., landmarks) and on helping children estab-

lish reference points and units along a mental number line. It

is likely that mental estimation would also be improved with

more attention to the nature of unit, as suggested by many

of the classroom studies reviewed previously. However,

Forrester and Pike (1998) indicated that in some classrooms,

estimation is treated dialogically as distinct from measure-

ment. Employing conversation analysis, they examined

the discursive status of measurement and estimation in two