Handbook of psychology volume 7 educational psychology

The Measure of Space

379

fifth-grade classrooms. Teachers formatted estimation as an

activity that preceded measure and as one characterized by a

lack of precision. In contrast, measurement was associated

with real length (i.e., perimeter) and the use of a ruler. The

consequence of this formatting was that students who em-

ployed nonstandard units to estimate, such as their fingers,

could not conceive of any way in which the use of such units

might be considered as measure. In short, treating estimation

and measurement as discursively distinct resulted in a corre-

sponding conceptual division between them.

In contrast, Kerr and Lester (1986) underscored a fusion

between measurement and estimation. They suggested that

instruction in measure should routinely encompass consider-

ations of sources of error, especially (a) the assumptions

(e.g., the model) about the object to be measured, (b) choice

of measuring instrument, and (c) how the instrument is used

(e.g., method variation). Historically, the recognition of error

was troubling to scientists. For example, Porter (1986)

documented the struggles in astronomy to come to grips with

variability in the measures of interstellar distances. Varelas

(1997) examined how third- and fourth-grade students made

sense of the variability of repeated trials. Many children

apparently did not conceptualize the differences among

repeated observations as error and often suggested that fewer

trials might be preferable to more. In other words, their solu-

tion was to sidestep the problem by avoiding the production

of troubling variability. Their conceptions seemed bound

with relatively diffuse conceptions of representative values

of a set of repeated trials. In a related study, Lehrer et al.

(2000) found that with explicit attention to ways of ordering

and structuring trial-to-trial variability, second-grade chil-

dren made sense of trial-to-trial variation by suggesting rep-

resentative (“typical”) values of sets of trials. Choices of

typical values included “middle numbers” (i.e., medians) and

modes, with a distinct preference for the latter. In contexts

where the distinction between signal and noise was more

evident, as in repeated measures of mass and volume of

objects, fifth-grade students readily proposed variations of

trimmed means as estimates of “real” weights and volumes

(Lehrer, Schauble, Strom, & Pligge, 2001).

Petrosino, Lehrer, and Schauble (in press) further investi-

gated children’s ideas about sources and representations of

measurement error. In a classroom study with fourth graders,

children’s conceptions of error were mediated by the intro-

duction of concepts of distribution. Students readily concep-

tualized the center of a distribution of measures of the height

of the school’s flagpole as an estimate of its real height. Fur-

thermore, indicators of variability were related to sources of

error, such as individual differences and differences in tools

used to measure height. Hence, students in this fourth-grade

classroom came to understand that errors in measure might

be random, yet still evidence a structure that could be pre-

dicted by information about sources of error, such as instru-

mentation. Konold and Pollatsek (in press) suggested that

contexts of repeated measures like those just described offer

significant advantages for assisting students to come to see

samples of measures as outcomes of processes, and statistics

like center and spread as indicators of signal and noise in

these processes, respectively.

Collectively, design studies and research in cognitive

development suggest several trends. First, children’s initial

understandings of the measurement of length are likely

grounded in commonplace experiences like walking and

commonplace artifacts, like measuring sticks (e.g., rulers).

Accordingly, engaging students in inscribing motion and

designing tools leads to significant transitions in conceptual

development. These transitions exceed those that one might

expect from everyday activity and suggest some of the ways

in which instructional design and learning can lead children’s

development. Second, understanding of length measure

emerges as children coordinate conceptual constituents of the

underpinnings of unit, such as subdivision of a length and

iteration of these subdivisions, with the underpinnings of

scale, such as origin and its numeric representation as zero.

These coordinations appear to emerge in pieces, with proce-

dural manipulation of given units to measure a length often

preceding fuller understanding of the entailments of these

procedures. Constructs of unit are intertwined with those of

scale, so that, for example, the correspondence of zero and

the origin of a scale likely undergo several transitions.

Third, length measure can serve as a springboard to

related forms of mathematics. The continuity of linear mea-

sure, coupled with procedures of splitting, appears to offer

important resources for the development of rational number

concepts. Measure and modeling can also serve as a founda-

tion for children’s conceptions of shape, especially properties

of shape. Fourth, measure can be recruited in service of math-

ematical argument. Such recruitment leads to conceptual

change as students grapple with ways of resolving contesting

claims by developing and refining their conceptions of unit.

Fifth, considering measure as inherently imprecise provides a

lead-in to the mathematics of distribution, especially when

students are asked to develop accounts (and measures) of

the contributions of different sources of error. Measurement

processes are a good entry point for distribution because they

clarify the contributions both of signal and error to the result-

ing shape of distribution. Consequently, children can come to

see the structure inherent in a random process.

Finally, the need to promote conceptual development

about measurement explicitly is acute when one considers

380

Mathematical Learning

that typical beginning university students often exhibit a

relatively tenuous grasp of the measure of space. For exam-

ple, Baturo and Nason (1996) noted that for the majority of a

sample of preservice teachers, area measure was tightly

bound to recall of formulas, like that used to find the area of

a rectangle. Yet none had any idea about the basis of any for-

mula. Most asserted that 128 cm

2

were larger than 1 m

2

be-

cause there were 100 cm in a meter. Many thought that area

measure applied only to polygons and confused area with

volume when presented with three-dimensional shapes.

These fragile conceptions of measure appear similar to those

of other preservice teachers as well (e.g., Simon & Blume,

1996, as we described earlier).

STRUCTURING SPACE

In the preceding section we described how children come

to structure space through its measure, assisted by efforts

to model and inscribe length. We reprise these themes by

turning to studies that describe how children come to struc-

ture space through its construction. We focus on dynamic

notations afforded by electronic technologies. These elec-

tronic technologies loosen the tether of geometry to its

euclidean foundation by introducing motion to form, in

contrast to the static geometry of the Greeks (Chazan &

Yerushalmy, 1998). Motion is inscribed from either local or

global perspectives. The former is represented by tools like

Logo (Papert, 1980), which approach the tracing of a locus of

points through the action of an agent. The agent’s perspective

is local because a pattern like a circle or square emerges from

a series of movements of the agent, often called a turtle, such

as the line segment that results from FD 40 (which traces a

path 40 units from the current orientation of the turtle). In

contrast, tools like the Geometer’s Sketchpad (Jackiw, 1995)

introduce motion from the perspective of the plane so that

movement is defined globally by stretching line segments

(or entire figures). For example, a construction of a square

can be resized by dragging one of its vertices or sides. The

resulting dynamic geometry is a new mathematical entity

(Goldenberg, Cuoco, & Mark, 1998). So, too, is the geometry

afforded by Logo, albeit in a different voice (Abelson &

diSessa, 1980).

Potential Affordances of Motion Geometries

Like other innovations in notational systems, agent-based

and dynamic geometries afford new ways of thinking about

shape and form. Logo (representing agent-based geometries)

affords a path perspective to shape and form—one comes to

see a figure as a trace of an agent’s (e.g., the turtle’s) motions.

It allows procedural specification of figures, thus creating

grounds for linking properties of a figure with operations

necessary to generate those properties. For example, the three

sides and three angles of a triangle correspond to three linear

motions (e.g., Forward 70) and three turns (e.g., Right 120)

of a turtle. Procedural specification, in turn, affords a distinc-

tion between the particular and the general. For instance, any

polygon can be defined by the same procedure simply by

varying the inputs to that procedure (e.g., the number of

sides). Thus, a procedure can simultaneously represent a spe-

cific drawn polygon or any polygon. Dynamic geometries

(e.g., the Geometer’s Sketchpad) create a clear distinction

between the particular and the general in a different way.

Drawing allows the creation of particular figures, but con-

struction allows the creation of general figures. The distinc-

tion between the two has a practical consequence in dynamic

geometry. When dragged (e.g., continuously deforming a

shape by pulling on a vertex), the relationships among

constituents of drawings change, but the relationships among

constituents of constructions do not. The result is that “the

diagrams created with geometry construction programs seem

poised between the particular and the general. They appear in

front of us in all their particularity, but, at the same time, they

can be manipulated in ways that indicate the generalities

lurking behind the particular” (Chazan & Yerushalmy, 1998,

p. 82). So, like Logo, geometry construction environments

relax the notational constraint of semantic disjointedness,

moving notation in the direction of natural language. The

drag mode of dynamic geometries creates multiple examples,

and the measurement capabilities of dynamic geometry tools

provide a fertile ground for conjecture and experiment. Both

Logo and dynamic geometry tools also provide means for

individual expression—especially when they are harnessed

to design (Harel & Papert, 1991; Lehrer, Guckenberg & Lee,

1988; Shaffer, 1998).

Learning in Motion

What, then, of learning? Do motion geometries create conse-

quential opportunities for pedagogical improvement, or are

they simply different? Such questions are fraught with diffi-

culty because media are not neutral, yet their effects are

usually bound with the kinds of pedagogical practices that

they afford. When these tools for dynamic notation are used

in ways that preserve the forms of teaching practice articu-

lated by Schoenfeld (1988; e.g., separating construction and

deduction), there seems to be little evidence of any substan-

tive change in student conceptions or epistemologies

(Chazan & Yerushalmy, 1998). However, when these tools

Structuring Space

381

are coupled with forms of instruction that emphasize conjec-

ture, explanation, and individual expression, the research

clearly indicates substantive conceptual change.

Logo Geometry

Perhaps because Logo and its descendants have a longer his-

tory, the evidence for learning with Logo spans multiple

decades and forms of inquiry. Early studies of learning with

Logo were conducted by its founders and featured carefully

articulated cases of student investigation of, among other

things, conjectures about the invariant sum of the turns (i.e.,

360) in the paths of polygons and explorations of the rela-

tionships among constituents of shape, such as sides and

angles (e.g., Papert, Watt, diSessa, & Weir, 1979). Follow-up

studies attempted to articulate relations between teaching and

learning with and without Logo tools, and again a subset of

this work focused on children’s learning about shape and

form.

When students use Logo in environments crafted to invite

student investigation and reflection, students (most research

was conducted with elementary students) tended to analyze

properties of shape and form, such as angle and side, and to

develop concepts of definition of classes of forms, as well

as relations among classes, such as squares and rectangles

(e.g., Clements & Battista, 1989, 1990; Lehrer et al., 1988a;

Lehrer, Guckenberg, & Sancilio, 1988b; Noss, 1987; Olive,

1991). Collectively, these studies painted portraits of chil-

dren’s learning of shape and form that (at the time) appeared

unobtainable with conventional tools and instruction. More-

over, children’s responses suggested that their learning fol-

lowed from their use of Logo tools. For instance, third-grade

children often compared forms such as triangles and squares

by considering the programs they used to make them: “Well,

it’s . . . 3 times 120 here and 4 times 90 here equal 360 and

that’s once around” (Lehrer et al., 1988a, p. 548). Moreover,

in the Lehrer et al. (1988a) study, independent measures of

children’s knowledge of Logo’s turn and move commands

and their ability to implement variables (tools for generaliza-

tion in Logo) correlated substantially with measures of chil-

dren’s knowledge of angles and of relations among

polygons, respectively. Not surprisingly, these effects were

stronger when instruction was designed to help students de-

velop knowledge of geometry, rather than simply good pro-

gramming skills. Lehrer, Randle, and Sancilio (1989)

suggested that some of what children were learning with

Logo could be attributed to formats of instruction and argu-

ment because researchers were often serving as teachers, and

most tended to promote conjecture and explanation in their

teaching.

Lehrer et al. (1989) worked with groups of fourth-grade

children with similar instructional goals and similar em-

phases on conjecture and explanation, but only some of the

students used Logo as a tool. They found no differences be-

tween the groups on measures of simple attributes of shape

and form, like angle measure or identification of properties

like parallelism. However, students using Logo tools learned

more about class inclusion relationships among quadrilater-

als and were far better at distinguishing necessary and suffi-

cient conditions in the definition of polygons. Moreover,

these differences between groups endured beyond the cycle

of instruction. Protocol analysis suggested that one likely

source of these differences was children’s use of variables to

define shapes in ways that allowed them to coconstitute the

general (the procedure defined with one or more variables)

and the particular (the figure drawn on the screen). Related

research with Logo-based microworlds expanded the scope

of geometry to transformation and symmetry and to ratio

and proportion (Edwards, 1991; see Edwards, 1998; Miller,

Lehman, & Koedinger, 1999, for general perspectives on

microworlds and learning).

A contemporary cycle of research featuring Logo as a tool

for teaching and learning geometry significantly extends its

reach and is best exemplified by the work of Clements,

Battista, and Samara (2001), who documented a program of

research conducted over the last decade. Teachers in Grades

1 through 6 used a Logo-based curriculum of ambitious

scope in which study of shape and form featured cycles of

conjecture and explanation. Their results replicated the major

findings of previous research but also significantly expanded

them to include broader portraits of student learning and

development with diverse samples of students (See also

Clements, Sarama, Yelland, & Glass, in press). In summary,

although the path of research with Logo has hit its share of

snags and setbacks, investigations of Logo as a tool for teach-

ing and learning geometry in carefully crafted environments

suggest clear support for the claim that it provides a new

form of mathematical literacy.

Dynamic Geometries

Research with dynamic geometries, again conducted in envi-

ronments crafted to support learning, also suggested produc-

tive means by which these tools can be harnessed to inform

conceptual change. However, our tour of this literature is

abbreviated due both to its relative novelty and to the practi-

cal limitations of space. Several studies indicate that the dis-

tinction between drawings and constructed diagrams

exemplified in dynamic geometry tools constitutes a form of

instructional capital. Constructions that can be subjected to

382

Mathematical Learning

motion afford systematic experimentation, and this capacity

for experimentation can be instructionally focused to a search

for an explanation of the invariants observed (Arcavi &

Hadas, 2000; de Villiers, 1998; Olive, 1998). Koedinger

(1998) proposed an explicit model of instructional support

for encouraging generation and refinement of student conjec-

tures, thus changing the grounds of deduction. For example,

his model develops a tutoring architecture that supports stu-

dents’ constructions of diagrams and associated experiments.

Arcavi and Hadas (2000) described instructional support for

use of dynamic geometry tools to model situations, with par-

ticular attention to how symbolic expression of function is in-

formed by systematic experimentation. Chazan (1993) found

that the use of construction-geometry tools in concert with in-

struction that supported student conjecturing helped high

school students become more aware of distinctions between

empirical and deductive forms of argument.

Technologically Assisted Design Tools

Although dynamic geometry tools are most often employed

to solve mathematical problems posed by teachers, Shaffer

(1997) designed a dynamic geometry construction micro-

world, Escher’s World, that high school students used for cre-

ating artistic designs by generating systems of mathematical

constraints and searching for solutions to mathematical prob-

lems with particular design properties and, consequently,

aesthetic appeal. Shaffer’s instructional design deliberately in-

corporated practices of architectural design studios so that

student design practices also included public displays (e.g.,

pinups) and conversations with critics about their evolving de-

signs. This coupling of mathematics and design resulted in in-

creased knowledge about transformational design as well as an

appreciation of mathematics as a vehicle for expressive intent.

Studies with younger designers and related electronic

technologies also indicate the fruitfulness of design contexts

that intersect worlds of artistic expression and mathematical

intent. Watt and Shanahan (1994) developed a computer

microworld and curriculum materials to support design of

quilts via transformational geometry. Research conducted

with these tools and materials, together with professional

development efforts to help teachers understand children’s

thinking, promoted primary grade students’ understanding

of transformational geometry, as well as their exploration of

algebraic structure, qualities of symmetry, and the limits of

induction (Jacobson & Lehrer, 2000; Kaput, 1999; Lehrer,

Jacobson, et al., 1998). As with the designers described by

Shaffer (1997), children’s conversations often reflected their

appreciation of an interaction between mathematics and ex-

pressive intent. For example, students debated the qualities of

“interesting” design; one student, for example, suggested that

some units would be “boring” no matter what transforma-

tions or sequences of transformations were applied to make a

quilt. He argued that multiple lines of symmetry would re-

strict the quilt design to simple translation of units (Hartmann

& Lehrer, 2000). That is, units with four lines of symmetry

restricted the space of possible design. In contrast, asymmet-

ric units allowed for the greatest number of potential designs.

Zech et al. (1998) developed dynamic design tools for chil-

dren’s (Grade 5) expression of architectural designs, such as

those of swing sets on playground. Designing blueprints for

these architectural challenges served as a forum for explo-

ration of measure, shape, and their relations.

In summary, the development of motion geometry tools

and related technologies affords new forms of mathematical

expression. The dual expression of the particular and the gen-

eral, together with experimentation about their relation, cre-

ates pedagogical opportunities to orient students toward

mathematical argument as explanation, not just verification.

Moreover, because these tools create conditions for construc-

tion and experimentation about shape and form, students at

all ages tend to develop analytic capabilities that have long

proven difficult to achieve. Perhaps most exciting is the po-

tential for pedagogy at the boundaries of mathematics and

design that capitalizes on the expression of mathematical in-

tent. Of course, mathematical intent, in turn, is supported and

shaped by these tools.