ZDM Mathematics Education (2011) 43:637–650
DOI 10.1007/s11858-011-0368-8
ORIGINAL ARTICLE
Children’s unit concepts in measurement: a teaching experiment
spanning grades 2 through 5
Jeffrey E. Barrett • Craig Cullen • Julie Sarama
Douglas H. Clements • David Klanderman •
Amanda L. Miller • Chepina Rumsey
•
Accepted: 6 September 2011 / Published online: 27 September 2011
Ó FIZ Karlsruhe 2011
Abstract We examined ways of improving students’ unit
concepts across spatial measurement situations. We report
data from our teaching experiment during a six-semester
longitudinal study from grade 2 through grade 5. Data
include instructional task sequences designed to help
children (a) integrate multiple representations of unit,
(b) coordinate and group units into higher-order units, and
(c) recognize the arbitrary nature of unit in comparison
contexts and student’s responses to tasks. Our results
suggest reflection on multiplicative relations among quantities prompted a more fully-developed unit concept. This
research extends prior work addressing the growth of unit
concepts in the contexts of length, area, and volume by
demonstrating the viability of level-specific instructional
actions as a means for promoting an informal theory of
measurement.
Keywords Measurement Length Area Volume Formative assessment Units Learning trajectory
J. E. Barrett (&) C. Cullen A. L. Miller C. Rumsey
Illinois State University, 4520 Math, 313 STV Hall,
Normal, IL 61790-4520, USA
e-mail: jbarrett@ilstu.edu
J. Sarama D. H. Clements
University at Buffalo, State University of New York,
505 Baldy Hall, Buffalo, NY 14260, USA
D. Klanderman
Trinity Christian College, 6601 West College Drive,
Palos Heights, IL 60463, USA
1 Introduction
We are interested in describing ways of helping elementary
children establish rich conceptual knowledge of units of
spatial measurement and use that knowledge as they
measure in complex situations. We set out to design specific instructional environments for teaching children about
spatial measurement and for promoting the development of
flexible and adaptive strategies that depend on the use of
mathematical structures (Verschaffel, Luwel, Torbeyns, &
Van Dooren, 2009). We believe such environments for
instruction will help students forge a conceptual, mathematical sense for measurement (Siegler, 2003). We share
the goals of learning-sciences researchers working to
describe ‘‘features of a learning environment intended to
support the longer term development of learners’ engagement in disciplinary practices, including reasoning about
evidence and explanation, representing and communicating
information, evaluating knowledge claims and building and
refining models and theories’’ (Ruthven, Laborde, Leach,
& Tiberghien, 2009, p. 330). Our particular investigation
describes features of a learning environment focused on
unit concepts of spatial measurement including researcherdesigned interventions in the context of clinical interviews.
Elementary students need a flexible, practical yet
theoretical knowledge of quantity and space as a foundation
for proportional reasoning, rational number knowledge, and
adequate understanding of algebraic variables for later
coursework in mathematics and science (Corcoran, Mosher,
& Rogat, 2009). Providing opportunities to establish such
knowledge requires thoughtful design of spatial measurement activities. We conjecture that an informal theory of
unit measures would inform students’ concept of units of
quantity across varying dimensions and tasks (Lehrer &
Schauble, 2007).
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Research on geometry and spatial measurement conducted by Piaget and his colleagues (1960) emphasized a
sequence of logical operations that depended on a succession of restructuring operations with spatial dimensions,
primarily length, then area, and also volume. More recent
work in child development and classroom studies on spatial
measurement have focused on conceptual foundations of
measure related to unit concepts: unit attributes, unit iteration, unit structuring and tiling, proportionality among
different units, additivity and the use of zero on measuring
scales, constituting an informal theory of measure (Lehrer,
2003, pp. 181–182). Researchers have hypothesized that
focusing on conceptual understanding of units is a productive way of anticipating and explaining changes in
conceptual understanding about measurement (Clarke,
Cheeseman, McDonough, & Clarke, 2003; Outhred,
Mitchelmore, McPhail, & Gould, 2003). Yet these more
recent studies have not addressed the relation between
instruction on unit concepts and strategy development in
complex task settings that engage various dimensions over
long developmental timeframes, nor have they focused on
comparison as a unifying operation relating measures and
units. Comparative reasoning underlies every act of measurement; one must relate each object to some unit object
and report its size in terms of those units.
Presently, most of the curriculum for US elementary
schools presupposes that spatial measurement of length,
area, and volume should be taught in an isolated sequence
instead of integrating these dimensions by focusing centrally on unit operations (Smith, et al., 2008). Few studies
have examined the possible concurrence of unit concept
development across dimensions. Such concurrent growth
of unit concepts for measuring continuous spatial quantity
might engender growth in students’ thinking about fractions and support their understanding of the continuous
nature of rational number (Steffe & Olive, 2010) by
establishing an integrated and abstract scheme for operating on units of measure in continuous spatial contexts.
There is virtue to working with all three measures together,
rather than treating them as separate topics.
Work in the Netherlands supports the potential value of
integrating instruction across length, area, weight and
volume in a sequence beginning with comparing and
ordering, then using a single unit to find quantity, and
lastly, using an instrument to read off a scaled value (van
den Heuvel-Panhuizen & Buys, 2008, p. 110). These
experiences begin with the quantity length, but include the
area and volume instruction in early elementary years. We
hypothesize that measurement is a conceptual domain of
learning that may be taught more efficiently if the ‘big
ideas’ can be developed around a core set of concepts for
measure units. Our purpose here is to describe tasks and
ways of designing tasks that focus on conceptual aspects of
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learning measurement units, especially by comparing
quantities and establishing units (Langrall, Mooney,
Nisbet, & Jones, 2008).
1.1 What is already known about student’s
understanding of unit concepts?
Many researchers have described the importance of the unit
concepts in measurement (e.g., Battista, 2006; Kamii,
2006; Mulligan, Mitchelmore, & Prescott, 2005), but rarely
have they analyzed students’ use of units across several
dimensions of space. Researchers concur about conceptually sophisticated ways of using units. These include the
following mathematical concepts of measurement: identifying units, abstracting, iterating, coordinating nested units
and using units to produce ratios (Lehrer, 2003; van den
Heuvel-Panhuizen & Buys, 2008). Students’ unit concepts
are rarely well formed. Although students may use unit
labels to name a quantity, they often do so without being
able to show the meaning of the relevant unit. Students
older than 10 years often fail on broken ruler or misaligned
ruler problems, indicating an inadequate unit concept (e.g.,
Hart, 1981).
Measurement units are closely related to rational number units and proportional thinking. Yet when asked about
measurement as a ratio, older elementary students often
mistakenly make additive comparisons rather than making
a multiplicative comparison for area units (e.g., Empson,
Junk, Dominguez, & Turner, 2006), or misrepresent ratios
of length measures along number lines (Moss & Case,
1999; Pettito, 1990). For example, Pettito asked students in
grades 1 through 3 to locate a number along an empty
number line marked with endpoints of 0 and 100. The
youngest students often marked values such as 16 too far to
the right, stepping right by intervals of one with over-sized
steps. In contrast, the older students used midpoints to
establish a ratio of ‘ and even  as referent values for 50
and 25. They also counted by intervals larger than one,
often counting by 10s to move to the right of zero. This
latter strategy illustrates reasoning by multiplicative ratios,
using ratios of 1/10 and 50/100 to help locate the quantities
16/100, or 44/100.
Other researchers have emphasized the importance of
unit operations as a basis for thinking about number and
algebraic variable, thereby supporting multiplicative operations (Davydov, 1991, pp. 42–43). Davydov suggests
multiplication is based on childrens’ experience of shifts
between units as they structure measures of continuous
quantity. He rehearses a classical problem of finding the
number of spoonfuls of water in a barrel. Rather than carry
out such a tedious exercise, the wise student finds a cup to
fill with spoonfuls, and then a jug to fill with cupfuls, and
finally fills the barrel with jugfuls. By translating from units
Children’s unit concepts in measurement
of spoons, to cups, to jugs to barrels, this student finds the
ratio between spoonfuls and barrel. This task and solution
requires the student to coordinate four different groups of
volume units in a systematic way because each unit represents a different quantity of water with respect to the
original spoon or barrel. Grouping units is an important
way of reasoning about quantity through unit measures that
yields efficient measures. Various researchers have shown
patterns in children’s developing ways of noticing and
using structure as they quantify space, such as the
increasingly organized sets of squares used as students
progress from disconnected square tiles, to uncoordinated
rows of tiles, to spiral sets, and then partially coordinated
sets of rows (Battista, Clements, Arnoff, Battista, & Van
Auken Borrow, 1998; Outhred & Mitchelmore, 2000).
Their work suggests that students benefit as they learn to
recognize, employ and coordinate collections of area units
in patterns and groups.
Science education researchers argue that students need
to gain a coherent, consistent, and ultimately theoretical
way of thinking about units across a range of dimensions
and attributes (Smith, Wiser, Anderson, & Krajcik, 2006).
Similarly, Langrall et al. (2008) indicate the generality and
importance of unit concepts in a theoretical perspective for
the mathematics curriculum. Students appeared to benefit
as they worked on measuring tasks emphasizing common
unit and structuring concepts across spatial dimensions of
length, perimeter, area and volume (Irwin, Vistro-Yu, &
Ell, 2004; Mulligan, et al., 2005). Focusing on unit concepts has apparent benefits, but little is known about the
kinds of decision making and task design considerations
involved as a researcher and her students investigate unit
concepts across a range of contexts from length, area and
volume measurement.
1.2 Theoretical framework and key terms
We employed hierarchic interactionalism (Clements &
Sarama, 2007, pp. 463–466) as our theoretical framework,
a blended perspective between nativist, socio-cultural
interactionalism and empiricism that posits development
as a coordination of both broad and local influences on
cognitive changes in the individual; interactions among
available knowledge gained from experience, established
acquisition schemes, and the influence of others prompt
development. Learning trajectories (LTs) serve as key
theoretical tools for advancing knowledge about development within this theoretical framework.
We find a complementary relation between learning
trajectories and formative assessment (Heritage, Kim, &
Vendlinksi, 2008). Formative assessment informs the
teacher of possibilities to adapt instruction to student
thinking. Learning trajectories describe students’ typical
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ways of thinking. Both use immediate, informal student
performance data and learning progressions to guide
instructional dialog with students. Learning progressions
are, ‘‘descriptions of successively more sophisticated
ways of reasoning within a content domain based on
research syntheses and conceptual analyses’’ (Smith,
et al., 2006, p. 1). Yet, the conceptual aspects of student
performance are difficult to discover, even with extensive
observation without specifying links to productive
instructional tasks for each progression level (Simon,
1995; Simon & Tzur, 2004). Thus, we prefer using
learning trajectories (LTs) rather than learning progressions (LPs).
LTs differ from LPs in that they are constructed by
linking claims about conceptual development with specific
instructional actions, tasks and support structures (Sarama
& Clements, 2011). As such, LTs are closely related to
what Ginsburg (2009) describes as mid-level theory, which
‘‘forms or informs instruction in a principled and effective
manner’’ (p. 111). He argues that the clinical interview
method provides insight into students’ underlying cognitive
competence. Researchers cannot merely pose tasks, but
must first situate each task, translate it for students and then
interpret responses to the task. One must press for students’
ways of thinking about their own actions as they work to
resolve tasks. The present report is a chronicle of the
efforts of a collaborative research group to implement
learning trajectories about length, area and volume to
engage students in conceptual knowledge of measurement
units.
1.3 Goals and research questions for teaching
and learning unit concepts
One goal for this study is to describe tasks and ways of
designing tasks to address conceptual aspects of unit concepts. A second goal is to characterize formative assessment on measurement unit concepts. We focus on student
thinking and strategies, using observations of student
thinking to inform our instructional design and to improve
our theoretical accounts of student learning.
We pursued three questions: (1) What kinds of unit
schemes and reasoning do students exhibit as they
encounter measuring tasks that require the recognition
of measurement units across various representations?
(2) What are critical aspects of students’ unit composition
schemes that constitute increasingly sophisticated ways of
understanding measurement? (3) What kinds of measurement tasks, questioning routines, and scaffolding help
students generalize unit concepts across spatial dimensions
of length, area and volume and eventually establish their
own theory of unit iteration and structuring? The study
offers partial answers to these questions.
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2 Methods
2.1 Using design research in a teaching experiment
and hypotheses for teaching unit concepts
We employed the design-research methodology of the
teaching experiment (Cobb & Gravemeijer, 2008; Steffe &
Thompson, 2000) to follow students’ growth, document
their ways of understanding measurement, and to examine
students’ ways of explaining and understanding their
measuring actions. Data were from a longitudinal study of
children’s measurement with students in upper elementary
school. The children in this study represent a stratified
range of abilities identified within a group of 45 students
from two classrooms in a midwestern US city. Some data
were collected during classroom sessions, but most were
collected during clinical interviews with small groups or
individual students. As a representative subset, we selected
two low-performing students, four middle and two higher
performing students, based on our initial assessment and
teacher report. We began interviewing these students in
clinical settings during their second grade year (students
were age 7–8 years) and continued interviewing them
through their grade 5 year. Seven of the students were
interviewed at least 28 times each across this time frame.
An eighth student left the school partway through the
study, so we replaced that student with another who had
participated in baseline interviews during the first semester
of the study. Each interview lasted from 15 to 25 min and
consisted of two to five tasks. In addition, we selected
another subset of eight students using similar criteria whom
we interviewed occasionally for further confirmation of
student behaviors in the main group.
We began each round of interviewing by choosing or
designing tasks that would extend prior tasks and interview
protocols. After administering the tasks for each interview,
each video was reviewed to classify strategies along the
learning trajectory. We report student progress along the
length measurement tasks elsewhere (Barrett, et al., 2011;
Sarama, Clements, Barrett, Van Dine, & McDonel, 2011).
In contrast, this report focuses on results for several tasks,
looking for patterns of strategy usage across measures
within students, not only for length, but also area and
volume. Task-level data included: researcher lesson plans
with predictions related to levels of measurement from the
conceptually aligned LTs for length,1 area and volume
measurement described elsewhere (Sarama & Clements,
2009); video recordings of the interview sessions; analytical summary notes regarding student responses; and our
subsequent framing of follow-up tasks. Unless noted
1
See Evaluation of hypothetical learning trajectory for length in the
early years (Sarama et al., 2011).
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otherwise, data from these tasks were obtained during
interviews with the primary subset of eight students.
We reviewed a database of journal notes about task
design. Journal entries included plans for assessment tasks,
supportive or explanatory discussion notes, along with
predictions about student performance for all 28 interviews
across six semesters for 8 students. We also searched our
summary reflective field notes on each session for each
child and our summary memos from monthly design-cycle
meetings spanning all six semesters to identify interviews
and tasks related to unit identity, representational issues,
and coordination across different dimensions. Thus, we
identified 11 different tasks that addressed three major
themes: (1) the integration of multiple representations of
units (see Fig. 1), (2) the coordination of related units (see
Fig. 2), and (3) theoretical perspectives on unit measures,
especially multiplicative relations (see Fig. 3).
Our design research assumes that the development of
unit and unit measurement concepts proceeds broadly as
children shift from actions on quantities to making records
of those actions and eventually abstract those actions
(Moss & Case, 1999; Piaget, et al., 1960). This research
tests that assumption. Furthermore, we purposely interspersed tasks from length, area and volume measurement
spanning four academic years of the children’s development. Although this approach was a productive way of
discovering student thinking about unit concepts, we do not
claim that interspersing tasks from these dimensions is
inherently better than the more standard sequence of
teaching first length, then area and lastly volume topics.
We hypothesized that children would construct meaningful and efficient ways of measuring as they forged ways
of connecting images of unit and unit iteration among
multiple figural and motor representations (Battista, 1999;
Steffe & Cobb, 1988). We also expected children to gain
efficiency in measuring and more clear knowledge about
equivalence and the meaning of variable in algebraic
contexts (cf. Knuth, Alibali, McNeil, Weinberg, & Stephens, 2005) as they integrated their knowledge and
strategies involving multiple representations of unit (e.g.,
Cannon, 1992), coordinated groups of units (e.g., Stephan,
Bowers, Cobb, & Gravemeijer, 2004) and implemented
generalized rules for making unit assignments within
various dimensions and across spatial dimensions (e.g.,
Carpenter & Lewis, 1976).
3 Findings: overview of design research phases
We identify three design phases in the teaching experiment, which we narrate as the findings of this study. Phase
One reports on students’ concept of a unit, multiple ways
students might represent units, and ways of coordinating
Children’s unit concepts in measurement
Fig. 1 Phase One tasks
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A Broken Ruler –Spring, Gr. 2
Ask: “Measure this strip with this ruler” and then,
“What are you counting?” If tick marks, align the
blue strip to a sequence of one-unit long strips and
relate back to A. broken ruler or B. unnumbered
ruler. Ask for explanations.
B Unnumbered Ruler – Spring, Gr. 2
C Partially-covered Ruler – Spring, Gr. 2 Give an eight-foot ruler with inch markings and a
ribbon obscuring the middle of the ruler. Ask, “Can
you measure this ribbon?” If students count their
own pointing motions, try the shorter section of
ribbon.
those representations to measure. Phase Two describes
students’ ways of grouping or subdividing units, anticipating ways they might construct flexible and dynamic
structures for quantity. Lastly, Phase Three incorporates
students’ work with arbitrary units and proportionality as
they built a theory of measurement. We now describe each
phase in detail by describing the tasks, the student work,
and our interpretation of the students’ work and strategies.
3.1 Findings and analysis, Phase One: abstracting
spatial units from various representations
In Phase One of our study, we focused on helping students
recognize and use length, area or volume units as they
compared objects by measuring them. Additionally, we
expected to help students measure consistently by having
them identify units represented with a variety of tools and
tasks. We wanted students to count units directly for each
measure. At the outset of the study, students were 7 or
8 years old (grade 2). Data from an initial clinical assessment2 indicated that seven of eight students (Arielle,
Owen, Drew, Abby, Ryan, Anselm and Danny) could find
length by counting linear units, and were able to iterate a
single unit to represent a collection of units with few
counting errors. However, they produced inexact measures
in special cases where the zero point was not available on
the ruler (refer to the level: unit relater and repeater in
Table 1. See also: Sarama & Clements, 2009). We wanted
2
Portions of the assessment were validated in other work.
students to develop a unit-based, consistent way of finding
length measures even where a ruler might be misaligned or
broken; we wanted them to use various images or tools and
their actions with these objects to identify the units as
conceptual entities. We wanted to extend this kind of unit
abstraction to area and volume measuring as well.
Thus, we wanted to motivate these seven students to
build more efficient ways to measure length, and to make
optimal use of number labels along rulers. To accomplish
this, we modified the display of units on tools to encourage
integration between tick marks, intervals between tick
marks, and number labels. We covered up a portion of the
ruler object, obscuring some of the unit intervals and tick
marks along the ruler so students would need to rely on the
higher number labels beyond the covered section to reason
backwards and deduce the length of an object along such
an obscured ruler. We expected students to identify a line
segment as a unit and learn to connect number labels or
tick marks with some set of unitary segments.
We designed three tasks that supported recognition of
unit intervals (See Fig. 1, Tasks A and B in Phase One,
along with a pilot task on wraps and sides in Phase Two).
These tasks were administered at two-week intervals over a
six-week span. The tasks supported shifts by five of the
students (Arielle, Owen, Drew, Abby, and Ryan) toward
reasoning at the next level in the LT, consistent length
measurer (See Table 1). These five students successfully
identified units from varying representations. However,
two students attributed the inconsistency of their measures
with different tools to a poor fit between these tools instead
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Fig. 2 Phase Two:
coordinating groups of units
with subordinate units or
superordinate units
J. E. Barrett et al.
D
Wraps and Sides Spring, Gr. 3
Give the pipe cleaner the length of the
perimeter of one square. Ask, “How
does the distance around this set of
tiles compare to the length of the pipe
cleaner?” If the student gives a
qualitative comparison, ask “How
much longer?”
E Volume comparison of inch cube
and box – Fall, Gr. 4
Show a pile of 30 cubes and ask the
student how many would fill the large
box. Arrange the 30 cubes as pictured
and asked, “Compare the volume of
this (inch cube) to the volume of that
(box). How much larger is the box
than one cube?”
F
Area of Blobs – Spring, Gr. 4
Provide two drawings of blobs on a
single transparency and several grids.
Ask, “How does the area of this small
blob compare to the area of the large
blob? How much bigger?”
G
Volume comparison of dm cube
and room – Fall, Gr. 5
Ask a pair of students to compare the
space inside a cubic decimeter to the
space inside a classroom, given meter
sticks, a decimeter cube, and a meter
cube. After 5 minutes, ask, “What
were you doing to solve this? How did
you make progress?”
of changing their own scheme in a way that would enable
them to identify the unit concepts across tools.
Although we saw progress in seven of the eight students’ ability to integrate units among representations,
three of the seven students still did not take advantage of
number labels; instead, they relied on counting visually
perceptible units. To determine if these students did not
effectively use the number labels because they did not
possess the computational skills or if they preferred to
identify and count available units, we constrained their
access to perceptual displays of units along tools. For
example, given a 96-in. ruler, students were asked to
measure a 26-inch ribbon stretched next to the ruler
between the 8th and 34th tick marks. There was a second
ribbon obscuring part of the ruler between the 15th and
29th tick marks (Task C, Fig. 1). The interviewer asked
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students to measure without uncovering that part of the
ruler (cf. Steffe & Cobb, 1988).
In response, two students adopted a ‘‘counting on’’ or
‘‘adding on’’ strategy to produce a value for the ribbon
length. For example, Arielle explained that she could get
from the initial value of 8 to the final value of 34 by adding
two 10s, one 2, and one 4. Thus, her answer of 26 was
composed of subunits of size 10, 2, and 4. Working on a
related task, Anselm explained that he would find the
length of an object that spanned from 3 to 27 on the ruler
because he could ‘‘back it up 3’’ and the end would be at
24. These strategies illustrate the use of numbers to reason
about units of length, exactly as we intended. All seven
students began to coordinate tick marks, intervals and
number labels successfully. However, incomplete knowledge of arithmetic operations on numbers greater than 20
Children’s unit concepts in measurement
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Fig. 3 Phase Three: building
an informal theory of units and
arbitrary unit assignments
A
0 1 2 3 4 5 6
impeded three students’ success at the computational
aspects of the task.
This task fits into a sequence of tasks that incorporates
numerical approaches and strategies with spatial strategies
for measuring by demanding systematic integration of tick
marks, interval spaces, and number labels beside the tick
marks on a ruler. It helped shift students’ attention to the
comparison of quantities, rather than an analysis of isolated
objects and their lengths. Students were prompted to notice
and coordinate units across varied representations. The
ribbon tasks were further revised to compare pairs of ribbons stretched along different portions of the same long
ruler, highlighting the comparative nature of measurement
activity. We asked them to explain how much longer one
ribbon is than the other, rather than simply identifying the
longer ribbon.
3.2 Findings and analysis, Phase Two: efficient
grouping of superordinate and subordinate units
During the second semester of the study we began Phase
Two, looking for ways to help students coordinate sub- and
superordinate units with units. Our review of our notes and
our summary analysis of student work from the prior
semester indicated that students struggled to relate different-sized units in a systematic way. We expected to help
students find ways of coordinating units within a measuring
system. For example, we know mm2, cm2 are smaller
denominations of units that can be grouped to fit exactly
into the larger denomination of a square meter. Similarly,
we wanted to find whether students might invent or adopt
ways of coordinating small units in composite units, for the
sake of efficiency.
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Table 1 Learning trajectory for length measurement
Trajectory level
Student thinking and actions
Instructional tasks
End-to-end length measurer Lays units endto-end and counts to report length. May not
recognize the need for equal-length units.
Needs a complete set of units to span along
an object
Student expects that length is quantifiable as a
composition of shorter lengths. Compares an
end-to-end train of countable objects to the
linear extent of an object. The scheme is
enhanced by the growing conception of
length measuring as sweeping through large
units coordinated with composing a length
with unit sticks
Provide student with fewer objects than
needed to span along an object and ask for
length
Length unit relater and repeater Measures by
repeated use of a unit, even if imprecisely.
Relates size and number of units explicitly.
Can add lengths to obtain the length of a
whole
Student can iterate a mental unit along an
object. Cardinal values are connected to
space units for small quantities but weaker
beyond these. Student recognizes that fewer
larger units will be required, if units are
visible
Instructor provides a tick mark tool with no
number labels and a band of connected unit
strips. Student is asked to coordinate these
tools to measure various lengths
Consistent length measurer Finds length on a
bent path as the sum of its parts. Measures
consistently, knowing the need for identical
units and a zero point. May coordinate units
and subunits
Can compose and partition length units. Can
think of the length of a bent path as the sum
of its parts. Mentally iterates a unit and
subunits (internalized ruler)
Student is asked to predict length with mental
iterations before checking with a tool
Conceptual ruler measurer Finds length by
imagining an end-to-end collection of units
Operates mentally with units and composite
units. Can mentally project a known length
along an object to measure or partition an
unknown length
Instructor asks students to find multiple shapes
given the constraint of a constant perimeter
We began with an exploratory task to coordinate the
side lengths and the entire perimeter of square tiles as two
different, but related linear units of measure in several
comparison situations (see Fig. 2, Task D). Researchers
administered the task to all 45 students in both classrooms.
Although all of the students related the longer units called a
wrap (the perimeter of one square, and the length of the
pipe cleaner) to just one part of the wrap called a side (a
side length of one square), they rarely integrated wraps and
sides together with a single unit. They reported them separately; students reported the mixed fraction number 2
wraps and 3 sides, but did not report it as 2 and ’ wraps,
nor did they describe it as 11 sides. By asking students to
switch from one unit to the other and characterize the same
unit, students were challenged to relate sides and wraps in
two different ways.
We wanted to demonstrate measuring situations that
would naturally involve the coordination of small, large
and larger units. Thus, we designed tasks like the Blobs
task to motivate translations within a system of units
(See Task F in Fig. 2). This type of task represented a
substantive design shift. Previously, we asked students to
compare quantities that were nearly the same. Here we
called for comparisons of quantities spanning a broad shift
of magnitude (e.g., compare the sizes of a small puddle
and a vast lake); we wanted students to engage in multiplicative relations and gain efficiency as they measure
area (Davydov, 1991). We used comparison problems,
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Instructor makes exaggerated mistakes by
gapping and overlapping sets of units with
the expectation that the student will identify
and correct these errors
Provide broken ruler for student to measure
objects
Instructor provides a physical unit and a ruler
to measure objects that require the student to
both iterate and subdivide the unit. Objects
should require quarter units and eighth units
emphasizing the coordination of various units to promote a
clear translation from small units to a bundled set of units
seen as a larger unit.
We reviewed eight students’ responses to area measure
tasks based on our learning trajectory on area (Sarama &
Clements, 2009, p. 302). We summarize the upper four
levels of the area trajectory to situate our subsequent
analysis. At the Primitive Coverer level, children represent
a covering of a region by drawing, but their alignment of
shapes is intuitive; therefore, the rows and columns are not
accurate. Not until the Partial Row Structurer level do they
structure the rectangle as a set of rows, understanding the
collinearity of rows and the constraint that each row must
have the same number of units. At the following level, Row
and Column Structurer, children have a mental construct of
a row as consisting of a composite of aligned, congruent
unit squares and determine the number of squares by iterating those rows. At the Array Structurer level, children
understand that the rectangle’s dimensions provide the
number of squares in rows and columns and thus can
meaningfully calculate the area from these dimensions
without perceptual support.
We expected five of the students (Arielle, Abby, Jessie,
Ryan, and Drew) to partially structure regions and to count
by groups of units (area LT level: Partial Row Structurer)
with Task F. We believed they could also align collections
of units in columns but had not yet coordinated rows and
columns at once; they often miscounted the corner unit.
Children’s unit concepts in measurement
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Table 2 Observed strategies of students and LT levels while working on Blobs task
Number of students
Observed strategies
Level within area LT
5 of 8 Jessie*, Ryan*,
Arielle*, Abby*, Owen
Created an upper bound to compare areas, counting individual
squares systematically in either rows or columns
Partial Row Structurer–if they drew; or if
not, then Row and Column Structurer
1 of 8 Danny
Created a lower bound to compare areas, counting individual squares
unsystematically in the interior
Partial Row Structurer
2 of 8 Anselm*, Drew
Used a decomposition and recomposition strategy to account for the
partial units, counting whole units per row or per column
Area conserver/Row and Column Structurer
1 of 8 Owen*
Counted around the border of the blob and then created a partial array
on the interior
Row and Column Structurer
In contrast, we expected the other three students (Owen,
Danny and Anselm) to count more systematically using
groups of rows or columns (Row and Column Structurer)
because they had demonstrated more advanced reasoning
using arrays.
We gave them drawings of two different blobs on a
single sheet of paper along with three different-sized
square grids on separate clear plastic sheets. These sheets
were provided as measuring tools. The larger blob contained an area 290 times larger than the area of the small
blob. The grids in the Blob task were constructed so that
each grid could be related to the others in a simple ratio
(they were commensurate area units in the ratio 1:16:64).
Hypothetically, if a student treated one square from the
larger grid as a 1 9 1 unit, then the same square region on
the middle grid would be a 4 9 4 unit of units, and the
same-sized square region on the smallest grid would be an
8 9 8 unit of units. If a student wanted to report an area
measure using both the smallest and middle-sized grids,
they would need to understand that four of the smallest grid
squares could fit within each square of the middle-sized
grid.
The interviewer negotiated students’ responses to the
task in the following ways: If the student offered a qualitative comparison (e.g., saying, ‘‘this blob is much larger!’’), then the interviewer asked, ‘‘How many times
larger is it?’’ By asking students to measure across a vast
change of magnitude, we expected them to attempt to
group units of units to coordinate quantities among the
three grids. We also expected them to make consistent
translations from smaller to larger units by referring to
ratios between groups of the two smaller units and each
large unit.
Based on previous research (Lehrer, Jenkins, & Osana,
1998) and earlier interviews, the research team expected
some students to use an upper bound strategy by counting
every square that had any part inside the blob shape or a
lower bound strategy by counting only squares completely
inside that blob shape. We also expected some students to
use a decomposition and re-composition strategy. During
the interviews, we observed the following strategies (see
Table 2). We noted six students (*) performed at the level
we predicted.
The task prompted six of eight students to form a ratio
and use it to coordinate units, but not necessarily the ratio
we intended. Two students, Arielle and Abby, used a visual
comparison to estimate that the small blob was  the area
of one of the squares in the smallest grid. Three students,
Abby, Anselm, and Jessie, correctly related the 64 small
square units to 1 large square. Danny attempted to relate
the small and large square area units but computed the ratio
as 1:67 instead of 1:64 due to counting errors. Owen related
16 medium square units to 1 large square unit. In all, six
students coordinated area units to compare regions as we
hoped. The grids may have prompted these students to see
a smaller 2D array within a large 2D array. Five students
correctly reported eight small squares across and eight
small squares down within a larger grid square. These
students apparently coordinated their unit of 1 square of 8
rows of 8 small tiles to form rows that constituted a superunit of 64 squares.
Abby, Danny, and Jessie reported multiplicative comparisons of the small to large blobs in terms of the small
squares: 112, 402, and 288, respectively (-61, 39, and
-1% error) through the use of an intermediate unit.
Although the percentage errors are high for two answers,
the students each demonstrated ability to coordinate multiple units through a set of multiplication operations to
relate the smallest unit to the largest unit. We take this
strategy as more sophisticated than directly counting how
many additional of the smallest squares it would take to be
as large as the largest region. In contrast, Owen and Jessie
(in a second response) gave additive comparisons. Owen
said the larger blob was 686 more by counting medium
squares and Jessie said it would be 2,870 more small
squares than the large blob. Nevertheless, all eight students
related different units from the available grids in coordinated patterns, suggesting an initial multiplicative scheme.
Based on the outcomes with Task F, we decided to
continue setting tasks that required a vast shift in magnitude to motivate and support students to coordinate
commensurate units, subunits and super-ordinate units.
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646
We shifted dimensions from area to volume to help students generalize the concept of unit coordination. We
eventually used the volume comparison of the inch cube
and box task (Task E in Fig. 2) and the volume comparison
of dm cubes (Task G in Fig. 2) to help students continue to
coordinate various units in a systematic and efficient way.
Surprisingly, while students coordinated units into portions
of layers, layers, and sets of layers to report the number of
inch cubes (reporting a range from 900 to 1,500 cubes),
students did not use that same measurement to answer:
‘‘how many times larger is the box than the one inch
cube?’’ For example, a student who claimed the larger box
could contain 900 cubes also said that it was 899 times
larger than the small cube (899 is the difference, rather than
the ratio). Apparently students do not always interpret a
measurement as a ratio between the unit and the object
being measured.
To summarize Phase Two, we found three of eight
students using multiplicative reasoning extensively and all
eight coordinating units on some aspects of the Blob task.
For the task on volume, students mistakenly discriminated
between our question about the volume measured in unit
cubes and how many times larger it would be than a unit
cube. Nevertheless, they did coordinate units by grouping
them and multiplying to find a measure in a comprehensive
unit value (i.e., they were purposely relating different-sized
units). Having addressed students’ integration of units
across representations, and having prompted them to
coordinate units within a system of related units, we now
report on our efforts to support students as they developed
a theory of measure to include the use of arbitrary units and
to emphasize multiplicative relations among quantities
involving units.
3.3 Findings and analysis, Phase Three: theoretical
analysis of measure units
We began Phase Three of our design research work by
synthesizing our design work from the prior two phases; we
still wanted students to connect representations and coordinate units in groups. Further, we wanted them to understand units as arbitrary assignments of value; the value of a
measure depends on the selected unit (e.g., see Task I,
Fig. 3, carried out with the secondary set of eight students).
As early as the third semester of the project (grade 3 spring
semester), we designed tasks to investigate students’
understanding of units of volume and their ability to cope
with inverse relations among unit size and quantity. We
illustrate the design of this group of tasks by describing our
design of Task H (see Fig. 3), the volume comparison of
prisms.
We used Task H to prompt students to reflect on the
process of selecting and identifying a unit of measure and
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J. E. Barrett et al.
to emphasize the inverse relation of unit and measures.
Based on the students’ performance on an assessment of
volume measurement knowledge during the previous
spring, the research team conjectured that all but one of the
students seemed to have at least a partial understanding of
cubes as filling a space in a countable and structured
arrangement (at least primitive 3D array counter: in Sarama
& Clements, 2009, pp. 306–308). Five students (Arielle,
Abby, Ryan, Drew, and Anselm) counted cubes one at a
time in a carefully structured context but were unable to
use more systematic and accurate counting strategies
(partial 3D structurer). Two students (Danny and Owen)
demonstrated ability to count more systematically than the
other six, accounting for internal and external cubes and
operating on composite columns of cubes (3D Row and
Column Structurer). Thus, each of the seven students who
exhibited a structuring strategy for identifying a set of unit
cubes was given this task (See Fig. 3, Task H).
Each student was shown two rectangular prisms whose
volumes were in a ratio of 1 to 2 and then was asked to
compare these two prisms by volume. Next, they were
given further pairs of prisms to compare. We expected the
students to correctly identify the volume of the larger prism
for any pairing using the smaller prism as unit. We thought
they would struggle to identify the volume of a smaller
prism if the unit was the larger prism; this would require
fractional quantities (halves or quarters). In order to relate
observed strategies (see Table 3) with the final solutions
(see Table 4), the names of students producing correct
volume comparisons have been bolded; the names of students producing incorrect volume comparisons have been
italicized.
Among these eight students, we expected Danny and
Owen to make correct comparisons and they did. Three
others, Arielle, Ryan, and Anselm (with a minor error) also
produced correct values despite our classification of their
strategy at a lower level that would predict difficulties on
such a task. Two of the three students who made inaccurate
Table 3 Students exhibited strategies for the multiple prism task
Number of students
Observed strategies
3 of 8 Owen, Drew,
Sara
Visually compared prisms without setting
them side-by-side, keeping them separate
1 of 8 Danny
Directly compared prisms by setting them
side-by-side or by moving the smaller prism
step-wise as if to traverse the space occupied
by the larger prism
Alternated between directly comparing prisms
side-by-side, and indirectly comparing them
while separate
3 of 8 Arielle,
Abby, Anselm
1 of 8 Ryan
Used transitive reasoning to compare the three
prisms at once without moving them
Children’s unit concepts in measurement
Table 4 Students’ responses to the multiple prism task
Number of students
Correctness
4 of 8 Owen, Arielle,
Danny, Ryan
Correctly identified the volume of each
prism when comparing all three prisms
without any errors, regardless of which
one was chosen as the unit
1 of 8 Anselm
Correctly identified the volume of each
prism when comparing all three prisms,
regardless of which one was chosen as the
unit, yet with one minor error
Made multiple inaccurate quantitative
comparisons
3 of 8 Abby, Drew,
Sara
comparisons, as predicted, used only visual comparisons
without moving or iterating a prism to arrange it next to
another. The third, Abby, used motions with the related
prisms, but appeared to guess at quantities without carefully iterating the smaller along the larger prism. However,
Ryan successfully compared all pairings without moving
any prisms.
The four students who correctly identified the volume,
given all combinations of unit and target prisms, were
posed a follow-up task. They were asked to compare the
volume of a composite solid made of one small and one
medium prism to the volume of the large prism. Ryan
analyzed the task successfully, identifying a quantity of 4/3
as the measure of the larger prism, taking the composite
solid as a unit. Ryan, Arielle, and Owen correctly identified
a quantity of 3/4 as the measure of the composite prism,
treating the larger prism as the unit, but did not find the
quantity if they were given composite prisms as a unit.
The following excerpt of an interview exemplifies the
structured process of questioning, including ways of
extending questions to probe an incorrect response:
I (interviewer): Let’s say again this (smallest prism,
S, 1 9 1 9 2 units) is our unit, how big is that
(pointing to largest prism, L, 2 9 2 9 2)?
Abby: Three of those (medium prism, M, 1 9 29 2)
and 9 of those (pointed to S).
[Actually, M measures 2S and L measures 4S]
I: Ok, so, this thing (L) is 3 of these (pointed to M)?
Abby: Yeah, and 9 of those (pointed to S, but pausing). No, wait it’s not 9 is it.
I: You can touch them. Would it help?
(Abby picked up S and iterated it four times on the
top face of the L. Then she picked up M and iterated
it twice above L.)
Abby: This one (L) is two of these (M). [correct]
Later in this interview, Abby identified the relations
among the prisms when the medium prism was assigned
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the value of 1 unit, correctly changing the measures of the
other two. In contrast, the other seven students were unable
to restate values and consistently find related quantities
based on the renaming of a unit object. We often allowed
students multiple trials if they were able to revise their
strategies like this because we wanted to know what they
could do given feedback and opportunities to verify claims
(Lesh & Hjalmarson, 2008).
We also developed a task (See Task J, Fig. 3) using
those prisms to find whether students would be able to set a
correspondence to a number line and map volume quantity
to a number line. We intended to prompt students to
coordinate volume units and length units as they interpreted the prism locations along the line. We wanted to find
whether students would recognize that the ratio between
measures of the prisms would depend on the unit location.
Here we present anecdotal evidence of responses from
one student in grade 4. The student correctly identified the
ratio between the unit and object being measured whenever
the objects’ volumes were in a ratio of 1:2 or 1:4 if he was
allowed to put the smaller prism inside the other (they were
hollow). In each case the student needed to handle both
prisms simultaneously and physically iterate the smaller
inside the larger. However, the student did not solve a task
where the middle prism was located at the position of one
on the number line, requiring a placement at one half for
the smaller prism. The student was only able to solve cases
where the smallest prism was placed at or beyond the value
of one.
A key idea in guiding students along the trajectories for
length, area and volume is to shift toward multiplicative
comparisons instead of additive comparisons (Sarama &
Clements, 2009, pp. 304, 308). The highest level of our trajectories for area and volume specify age 9 (grade 3 or 4)
based on multiplicative operations. Understanding that all
measures are in fact ratios between a unit quantity and other
quantities constitutes sophisticated and conceptual knowledge for measurement. During the third semester, our students were at least this age, yet they tended to compare by
additive operations. So we designed tasks to prompt multiplicative relations. We wanted students to integrate both
additive and multiplicative schemes for measuring. Based on
our work with the prism tasks in Phase Three and the Blobs
task in Phase Two, we tested and carried out a task on length
(Fig. 3, Task K) that required students to integrate arithmetic
and multiplicative comparisons as they searched for invariance. We asked for a multiplicative comparison by holding
the ratio of 1:3 constant across three pairs of line segments
and asking students what was the same in all three pairings.
Two of eight students given this task were able to make both
multiplicative and additive comparisons. Two more students
noticed multiplicative comparisons. The other four students
only noticed additive comparisons.
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We also attempted to direct students’ attention to multiplicative comparisons by modifying our questions. We
prompted students to compare prisms by volume and followed up by asking, ‘‘How many times larger is the box
compared to one unit cube?’’ (Task E in Fig. 2). Only two
of the eight students noticed that the measure of the box
and the comparison between the box and the unit cube as a
ratio should be the same. We plan to address this difficulty
by giving students more opportunities to describe measures
in words, including asking them to state a ratio between a
unit and the object measured with that unit.
4 Discussion and implications
Students in the study developed increasingly integrated
representations for units of length during Phase One. At
first, the students often relied on figural images alone to
measure length. Thus, we developed tasks to emphasize
arithmetic operations with number labels along rulers. As
the interviewer prompted repeated measures with different
tools and highlighted inconsistent cases, the students came
to integrate number labels along rulers, tick marks along
ruler objects, and the intervals between marks (Barrett,
Clements, Klanderman, Pennisi, & Polaki, 2006, p. 218).
Students learned to represent units in a variety of ways
including iterative motions, and the collection of segments
or ‘‘sticks’’; we found support for claims that students
benefit by encountering related measures carried out by
moving their finger along a path in a syncopated motion
and by pointing to successive midpoints of intervals
(Cullen & Barrett, 2010; Lakoff & Nunez, 2000). It
appeared to be critical to devise tasks like the Covered
Ruler task to integrate length units among representations.
Otherwise, students were counting directly, an inefficient
but reliable approach. Further development of similar tasks
would likely benefit students by prompting students to
integrate schemes for visual and motor counting operations
as they measure.
We also noticed that the students struggled to coordinate
sub- and superordinate units. This prompted Phase Two of
our work, focusing on the coordination of units. Although
the challenges students face in learning to coordinate units
are clear from previous research about structuring space
(Battista, et al., 1998; Mulligan, et al., 2005; Outhred &
Mitchelmore, 2004; Piaget, 1970), it is less clear how to
design tasks and administer tasks in formative assessment
so students develop more structured unit concepts. We
found that devising effective tasks required purposeful
shifts in phrasing from qualitative comparison to quantitative comparison (often additive in nature) and asking
specifically about multiplicative comparisons (Cullen,
et al., 2010). We also found that tasks requiring students to
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compare objects with extremely different areas or volumes
motivated students to coordinate a wide range of units; this
led to the growth of a coordination scheme for relating
counting at various scales within a single scale.
Students struggled in this study to transition from
making additive comparisons of measures to making
multiplicative comparisons, even when given several
opportunities, supporting prior research (Empson, et al.,
2006; Moss & Case, 1999). Although we were surprised
that measures were not often understood as ratios, we take
it as a challenge to prompt multiplicative relations with
more focused questions; this requires further research. We
conjecture that students have not had enough experience
reporting measures in verbal ways, and believe they should
be asked to explain ratios that are implicit with any measurement. That is, saying that a board is 1.3 m long is the
same thing as saying that the board is 1.3 times as long as a
meter, but the latter phrasing better supports unit concept
growth.
During the second phase, we expected students to
understand the multiplicative relations that constitute a
sophisticated level of theorizing about measures. This
prompted Phase Three of design in this study that included
both the ideas of multiple representations of unit and the
coordination of units, but also the arbitrary nature of unit
assignments, and the proportionality of related units within
a theoretical account of measurement. Our findings about
Task G illustrated one way of helping students shift units as
they compared; repeated shifts among units and measures
within one context may prompt students to recognize the
arbitrary nature of assignments of unit value (Seymour &
Lehrer, 2006).
The three phases described in this paper constitute a
pattern of increasingly sophisticated practices for establishing and using quantity to conduct scientific inquiry or
perhaps to test models in technological or engineering
settings (Ruthven, et al., 2009). Using the concept of a unit
as our analytic theme, we began by attending to students’
ways of coordinating representations. Next, we studied
student’s ways of grouping or subdividing units, allowing
flexible and dynamic structures for quantity. We also
included students’ work with arbitrary units and proportionality to establish a theory of measurement that built
upon the prior phases. Taken together, these phases
allowed a wide-angled view of student cognitive development rather than focusing narrowly on a single trajectory.
The phases served to integrate a broad collection of related
concepts. Other researchers may benefit by using a similar
pattern for coordinating across learning trajectories as we
have coordinated length, area and volume. One might
analyze multiple content strands at once by taking such a
lens on increasingly sophisticated mathematical practices
as we exemplified in these three phases. This account of
Children’s unit concepts in measurement
our design phases contributes to the broad understanding of
what constitutes a child’s theory of measurement (Lehrer &
Schauble, 2007, p. 156). Unit representations, unit coordination tasks, and arbitrary unit assignments that prompt
reflection on multiplicative relations are critical elements
for such a theory of measurement.
Acknowledgments The research reported here was supported by
the National Science Foundation through Grant No. DRL-0732217,
‘‘A Longitudinal Account of Children’s Knowledge of Measurement.’’ The opinions expressed are those of the authors and do not
represent views of the NSF.
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