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Area Measurement: Nonsquare Units and New Connections
Amanda L. Miller Illinois State University
Abstract
In this paper, I report the results of a dissertation study on area measurement. The study
utilized structured, task-based interviews conducted to explore how students think about area
measurement when using not only square units but also nonsquare, rectangular units and
triangular units The major design was that of a cross-sectional study, with five students from
each of four different grade groups: Grades 1, 3, 5, and 7 (ages 7, 9, 11, and 13). Overall, the
data in this study extends the literature on area measurement development to indicate
progressions in students’ covering, enumerating, subdividing, and spatial structuring
schemes, regardless of unit shape. Implications for instruction and research are discussed.
Although area measurement is one of the most commonly taught (and used) forms of
geometric measurement (Curry, Mitchelmore, & Outhred, 2006), research indicates that
measuring area is more than a procedure or skill quickly learned and retained. Instead,
students can perform rote procedures without understanding (Sarama & Clements, 2009). In
order to investigate why students struggle with area measurement, researchers have analyzed
textbooks and manipulative materials and attempted to identify common errors and
misconceptions. One of these common misconceptions stems from measuring a continuous
quantity (area) with discrete objects (e.g., foam squares). Cavanaugh (2008) found that,
“Many textbooks present regions which are already partitioned so that students need only
count the squares one by one to discover the area of a shape” (p. 55). Other researchers have
exhibited the prevalence of textbook exercises that present rectangular regions with
dimensions displayed with number labels on a pair of orthogonal sides, prompting students to
consider and operate on the displayed number labels only (e.g., Miller, Kara, & Eames,
2012). Such representations had a tendency to encourage students to limit the learning of area
conceptions to rote counting of units or formula usage. These narrowed approaches cause
some students to develop an incomplete conceptual understanding of area concepts and
principles, including identifying a unit, relating unit size to the dimensions, constructing of
arrays, and tessellating the plane (Battista, 2003; Cavanagh, 2008; Clements & Sarama,
2007).
Similar results were reported when area measurement experiences were focused on
covering with physical tiles and then counting those tiles. In their review of related literature,
Outhred and Mitchelmore (2000) argued that using concrete materials to tile a rectangular
figure “may conceal the very relations they are intended to illustrate” (p. 146). Some
manipulatives, such as foam squares or grid-overlays, pre-structure an array, allowing
students to determine correctly the area of the region without attending to the structure
(Lehrer, 2003; Outhred & Mitchelmore, 2000). In other words, students are often able to
create an array of square tiles or drawn units without conceptually understanding that a
square within an array is an individual unit, a component of a row, and a component of a
column (Sarama & Clements, 2009). As Kamii and Kysh (2006) argued, “Empirically
covering a surface with squares or a grid and counting them is one thing, and being able to
think about a square as a unit for area is quite another thing” (p. 113).
Other researchers have reported that students lack a conceptual understanding of the
rectangular area formula, and thus, are unable to recognize the unit of measurement for area.
Battista, Clements, Arnoff, Battista, and Borrow (1998) found that students had difficulties
visualizing the spatial structure of rectilinear regions or the row and column array of squares.
When shown an incomplete array (Figure 1), only 19% of second graders, 31% of fourth
graders, and 78% of fifth graders were able to determine correctly the number of squares in
the completed array.
Figure 1. Example of an incomplete array
While continuing to study students’ development of area measurement, Battista
(2003) determined, “many children simply learn to multiply lengths to generate areas without
understanding that these products are generating arrays” (p. 110).
Rationale for Studying Area Measurement with Nonsquare Area Units
Although researchers have identified several student misconceptions related to area
units, the field has yet to determine how to address them. Since the 1970s, researchers have
analyzed how children learn area unit concepts, such as selecting units (Heraud, 1987; Maher
& Beattys, 1986), constructing units (Reynolds & Wheatley, 1996), identifying units (Kamii
& Kysh, 2006), and relating area units (e.g., Carpenter & Lewis, 1976; Hiebert, 1981; Barrett
et al., 2011). However, children’s conceptions of area are usually investigated solely in terms
of square (standard) units (e.g., Battista, Clements, Arnoff, Battista, & Borrow, 1998; Kamii
& Kysh, 2006). From the literature, it is clear that providing students with area tasks
involving only square (standard) units is insufficient.
More attention needs to be paid to students’ development of area conceptions in terms
of nonsquare (nonstandard) units in order to provide researchers and teachers with a more
generalized characterization of the ways elementary and middle school students think about
area measurement. Such a characterization would integrate students’ behaviors and reasoning
processes with different types of area units, which would include but not be limited to square
units.
Multiple researchers have called for such a de-limitation. During the 1980s, Hiebert
(1981) argued, “Children can also benefit from working with nonsquare units to cover a
region” (p. 42), not just square units. During the 1990s, Simon (1995) and Simon and Blume
(1994) made similar assertions but in reference to pre-service teachers. Specifically, Simon
(1995) claimed,
Measuring with a nonsquare rectangle to determine the area encourages a level of
visualization that is not required when one uses a ruler to determine square units, that
is, they will have to take into account what they are counting, the unit of measure,
which is based on how they are laying the tiles on the table. (p. 133)
Despite the history of these supplications, such a generalized characterization of the ways
elementary and middle school students think about area measurement has yet to be described.
To initiate such a characterization, this descriptive study was conducted to explore the ways
elementary and middle school students think about area measurement when using not only
square units but also nonsquare, rectangular units and triangular units.
Prior to this study, it was my conjecture that (a) the use of nonsquare, rectangular
units would help elementary and middle school students develop an appreciation for the use
of square units as standard area units and (b) doing so would clarify conceptual, procedural,
and intuitive aspects of area measurement.
Research Questions
The following questions guided this study:
1. In what ways do students enumerate and structure two-dimensional space with a
variety of area units?
2. What are conceptual, procedural, and intuitive aspects of area measurement that relate
to students’ enumeration and structuring techniques when working with area units?
Theoretical Considerations
This study was informed by intuitionism and constructivism. As asserted by Fischbein
(1990), “mathematical activity is essentially a constructive process;” hence, researchers have
much to learn from “observing the child’s spontaneous behaviour when coping with
mathematical problems” (p. 7). He argued that such spontaneous behaviour cannot be
categorized as only conceptual or procedural in nature. He asserted that there are conceptual,
procedural, and intuitive aspects at every level of mathematical activity (Fischbein, 1990;
2002). These components can act in harmony or disharmony, creating a dynamic that should
be investigated. Although intuitivism and constructivism are not, generally, integrated, this
study required a theoretical approach that accounted for conceptual, procedural, and intuitive
aspects of mathematical thinking and learning related to the domain of area measurement.
Context and Methods
This research study incorporated qualitative methods to explore the ways elementary
and middle school students resolved area measurement tasks with square and nonsquare
units. The major design was that of a cross-sectional study, with students from four different
grade groups: Grades 1, 3, 5, and 7 (ages 7, 9, 11, and 13). In each age group, five students
were selected from two classes per grade within one school, providing a total of 20
participants.
Research Design
This study employed structured, task-based interviews (Goldin, 2000) aimed at
understanding the students’ current mathematical knowledge at a defined time. Individual,
structured, task-based interview design was selected due to its use in prior studies attempting
to model the relationship between internal and external representational systems (Goldin,
2000). This design has four components: a subject, an interviewer, a set of preplanned tasks,
and a carefully described theoretical framework. Individual, structured, task-based interviews
require that all participants receive the same structured mathematical tasks, yet allow for
further probing. At times, I chose to gather more information and pose unscripted follow up
questions, such as, “Tell me more,” or “How did you come up with your answer?”
Participants
I chose to work with elementary and middle school students from a Midwestern
public school district in the United States, hereafter referred to as Prairie School. The school
is located in a sub-urban community with a population of approximately 125,000 people in
2012. According to Prairie School documents, the school’s student population is reflective of
the community’s demographics. During the 2010 – 2011 school year (the year before this
study), students were 67.7% White/ nonHispanic, 9.3% Hispanic, 8.8% Black/ nonHispanic,
8.3% Asian, and 5.9% multi-ethnic. Prairie School uses the same curricular materials and
texts as nearby districts, Everyday Mathematics (e.g. Bell et al., 2004).
Data Sources
Participant selection was based on a seven item initial survey administered verbally
for first graders and in written form for third, fifth, and seventh graders. After five students
per grade were selected, 10 tasks were posed within two one-on-one interviews. For each
interview task, a child was presented with a rectilinear region and rectilinear unit. Linear
dimensions were displayed in a variety of ways (e.g., tickmarks, dots, grids), but not with
number labels (see Figure 2). The participant was then presented with tasks that required him
or her to operate on conceptual, figural, and physical area units (Steffe, von Glasersfeld,
Richards, & Cobb, 1983).
Figure 2. Ten interview tasks
The participants were posed the same tasks in the same order. Each study interview was
between 11 and 30 minutes in length and occurred during the normal school day. The total
interview time per child was between 31 and 51 minutes in length. All interviews were video
recorded and transcribed. Thus, the data collected as part of this study included transcripts
from the initial surveys and the study interviews, interviews scripts, video recordings, my
reflection after each interview, and copies of student work.
Analysis
After the 40 study interviews had been administered, I performed three phases of
analysis. In the first phase, I started naming and describing the strategies I observed students
using. In the second phase, I performed a comparative analysis (Corbin & Strauss, 2008) to
create initial concepts, or free nodes, to account for and make sense of the strategies I
observed. This comparative analysis consisted of constant comparisons, a process of first
sorting data bits, next organizing data bits, then creating categories of data bits, and finally
constructing, revising, and refining categories or themes in the data while repeatedly and
systematically searching the data (Corbin & Strauss, 2008). By constantly comparing units or
bits of data, I developed an analytical tool that was rooted in my own data. This is in contrast
to using an analytical tool developed by someone else based on a different sample, different
set of tasks, and different research questions.
With the constant comparative method, I interpreted my data to investigate the ways
students resolved area measurement tasks with a variety of area units. To make these constant
comparisons, I rewatched a subtask and attributed a code, or node, for a sentence, action, or
strategy. When I finished a subtask, I went on to the next subtask. When I finished an
interview, I rewatched the interview to see if created any new nodes near the end of the
interview that I would want to use on an earlier subtask. If a meaningful bit of data was coded
with a free node, it was also marked with additional nodes to mark the participant, the task,
and the sub-task to identify the data bit source. I repeated this process per subtask, task,
interview, child, and grade.
In the third phase, I made theoretical comparisons per child, per grade, per task, and
per unit type. According to Corbin and Strauss (2008), “theoretical comparisons are tools
designed to assist the analysis with arriving at a definition or understanding of some
phenomenon by looking at the property and dimensional level” (p. 75). To make theoretical
comparisons, I uploaded all of my transcripts and created nodes for all of my codes in the
qualitative research software Nvivo 8 developed by QSR International (2010). I ran
matrix-coding queries on my nodes to compare codes and groups of codes. In a
matrix-coding query, one free node, tree node, or attribute can be selected to be displayed in a
column (such as structuring with individual units one by one) and a different free node, tree
node, or attribute can be selected to be displayed in a row (such as grade).
After completing the three-phase process, I recruited a check coder. This individual
independently coded 10% of my data, at least one subtask per participant. Between one and
six codes were utilized per subtask. Interrater reliability among Rater 1 and Rater 2 was
computed to be R = 0.73, which was above Marques and McCalls’s (2005) recommended R
for interraters and dissertation researchers. During the post-coding discussion, we compared
codes and came to a consensus for each disagreement. Definitions and use provisions were
elaborated upon based on this conversation. A revised R after coding incongruities were
resolved was not computed.
Summary of Findings
In the section below, I share the results and analyses that support the answering of
Research Questions 1 and 2: In what ways do students enumerate and structure
two-dimensional space with a variety of area units; and what are conceptual, procedural, and
intuitive aspects of area measurement that relate to students’ enumeration and structuring of
two-dimensional space? To do so, I present descriptive accounts of the ways four groups of
students performed on the individual interview tasks. These four groups, or tiers, were
created based on student overall success indices. Student overall success indices were
determined by scaling raw counts for correct responses per sub-task, incorrect responses per
sub-task, and corrected incorrect responses per sub-task. Note that an uncorrected correct
responses was classified as an incorrect response. These raw counts were then scaled. Correct
responses were multiplied by three points, corrected incorrect responses by two points, and
incorrect responses by one point. Next, these scaled counts were summed and then divided by
the total number of subtasks that the individual participant was posed. Table 1 lists each
participant’s tier group number, overall success index, grade, and gender.
Table 1. Overall Success Index per Student
Fictional Name Tier Group Overall Success Index Grade Gender Charlie 1 92.47% 5 Male Jacob 1 89.66% 5 Male
Stephanie 1 89.58% 7 Female Irene 1 88.17 5 Female
Matthew 1 86.67% 7 Male Chris 2 86.46% 5 Male Abel 2 86.46% 7 Male
Dameon 2 83.87% 7 Male Nora 2 82.80% 7 Female Cathy 2 81.48% 1 Female
Eli 3 78.79% 3 Male Octavia 3 75.00% 3 Female Salena 3 73.81% 5 Female Jerome 3 65.69% 3 Male Rebecca 3 63.81% 3 Female
Sally 4 60.19% 1 Female Carter 4 60.19% 3 Male Olive 4 52.69% 1 Female Orrin 4 51.96% 1 Male
Brandon 4 48.48% 1 Male
Although trends per grade, gender, and handedness were also considered, more similarities
were found per tier group than the other categorizations. Individual profiles from Phase 1 of
analysis and matrix queries from Phase 3 of analysis informed the writing of tier group
synopses.
Tier Group 1 Results
The five students categorized into tier group 1 (TG1) had overall success indices
between 86.67% and 92.47% (see Table 1). TG1 students (spanning grades 5 and 7) exhibited
the least variability of the four tier groups in their enumeration and structuring techniques.
They exhibited a capacity to operate mentally and figurally on a unit of units (row and
nonrow), use perceptual support across subtasks, determine the number of units in a row or
column, a unit of units and iterate this unit of units repeatedly and exhaustively in one-to-one
correspondence with the elements of the orthogonal column to structure an array, and apply
the rectangular area formula with understanding. However, they not only exhibited this
success with square units but also nonsquare units (see Figure 3). In fact, TG1 students were
more successful on triangular unit tasks than square unit tasks.
Figure 3. Matthew’s drawing on Task 9
These results indicate that TG1 students had developed a set of complete and integrated
covering, enumerating, and spatial structuring schemes. Their abstraction of spatial
structuring techniques with square units allowed for transfer on novel tasks that included
nonsquare units and a variety of dimension demarcation displays.
Tier Group 2 Results
The five students categorized into tier group 2 (TG2) had overall success indices
between 81.48% and 86.46% (see Table 1). TG2 students (spanning grades 1 to 7) exhibited
more variability in their enumeration and structuring techniques than TG1 students. They
exhibited a capacity to operate mentally and figurally on a combination of a unit of units (row
and nonrow) and individual units, use perceptual support on some sub-tasks but not all, use
mixed drawing and counting actions by recognizing, drawing, or counting at least some rows
(or columns) as a unit of units, and apply arithmetic approaches with and without
understanding. TG2 students were more successful on square unit tasks than triangular or
rectangular unit tasks (see Figure 4).
Figure 4. Abel’s enumeration of physical tiles on Task 5
This indicates that TG2 students had developed a set of mental construct of unit of units that
had not yet been abstracted and thus integrated into a general concept of area unit, as it had
for TG1 students. That is, TG2 students may have integrated enumerating and covering
schemes without a complete set of spatial structuring schemes. Although TG2 students had
developed an expectation of congruity, they had not yet determined how to articulate or enact
that congruity across unit types and dimension demarcation displays.
Tier Group 3 Results
The five students categorized into tier group 3 (TG3) had overall success indices
between 63.81% and 78.79% (see Table 1). TG3 students (spanning grades 3 and 5) exhibited
the most variability in enumeration and structuring techniques of the four tier groups (see
Figure 5).
a. b. c. Figure 5. Octavia's a) Structuring with Individual Units on Task 1, b) Use of a Mixed
Drawing Strategy on Task 2, and c) Use of Parallel Row and Column Segments on Task 6
Although they attempted to structure with a unit of units (row and nonrow), TG3 students
were more successful when structuring with individual units. In addition, TG3 students were
more likely to use perceptual support incorrectly than to use it correctly. They exhibited a
propensity to count individual units one by one systematically, rather than use more advanced
enumeration strategies. These results indicate that TG3 students had made progress with
mixed drawing and counting actions by recognizing, drawing, or counting at least some rows
(or columns) as a unit of units but had not yet developed a mental construct of a unit of units,
as the TG2 students had. That is, TG3 students may have partially integrated enumerating
and covering schemes without a complete set of spatial structuring schemes. Furthermore,
TG3 students had also not yet developed an expectation of congruity among strategies, as
evidenced by their lack of consistency in strategies among sub-tasks, unit types, and
demarcation of dimension types.
Tier Group 4 Results
The five students categorized into tier group 4 (TG4) had overall success indices
between 48.48% and 60.19% (see Table 1). TG4 students (spanning grades 1 and 3) were
comparably invariant in their strategies. They only exhibited more variability in enumeration
and structuring techniques than TG1 students. They attempted to cover with individual
shapes that were sometimes approximations of squares, rectangles, or triangles that were not
constrained in size, shape, or location. Other times, TG4 students covered with individual
units that were somewhat constrained in size, shape, and/ or location. Due to their struggles
with unit concepts, TG4 students rarely operated on a unit of units, and when they did, the
units of units were usually row-based. Furthermore, TG4 students were less likely use
perceptual support than use it (correctly or incorrectly). These results indicate the TG4
students had not yet developed a mental construct of unit. Instead, they struggled to make
sense of experiences not only when they were asked to structure mentally or figurally with a
variety of area units, but also when they were asked to tile with physical manipulatives. That
is, TG4 students exhibited an inability to organize or coordinate the space with physical,
figural, and mental units (see Figure 6). They had neither integrated their enumerating and
covering schemes, nor developed spatial structuring schemes.
a. b. Figure 6. a) Carter’s tiling and b) Olive’s drawing on Task 9
Nontier Group Based Findings
By comparing strategies across tiers, I identified three collections of trends. These
three collections are conceptual aspects, procedural aspects, and intuitive aspects of area
measurement. Because these aspects influence one another, it is important to examine the role
each of them plays in area measurement.
Conceptual Aspects. The first collection of trends indicated conceptual aspects of
area measurement. Conceptual aspects are defined to be implicit or explicit principles or
features of a concept. These aspects are relational, flexible, and generalizable (Rittle-Johnson,
Sieger, & Alibali, 2001; Schneider & Stern, 2005). A student’s mental image of an area unit
and how he operates on it is a conceptual aspect of area measurement. This is because a unit
is a conceptual entity (cf. Barrett et al., 2011). I interpreted the strategy codes no anchoring,
anchoring, structuring with individual units, structuring with units of units as rows or
columns, structuring with units of units not as rows or columns, and structuring with parallel
row and column segments to be attending to conceptual aspects of area measurement.
Procedural Aspects. The second collection of trends indicated procedural aspects of
area measurement. Procedural aspects are defined to be conventional and routine sequences
of actions such that the objects operated on are symbols (cf., Hiebert & Lefevre, 1986).
Algorithms and other arithmetic-based computation strategies are examples of procedures.
Hence, I classified many of the enumeration strategy codes as procedural aspects of area
measurement.
Intuitive Aspects. Intuitive aspects of area measurement are defined to be
self-evident, immediate, and natural ideas or notions about area measurement concepts and
operations. Some intuitive aspects are based on everyday experiences, and others are based
on educational experiences (cf., Fischbein, 2002).
Two of the emergent strategies considered to be related to intuitive aspects of area
measurement are the one row, one column strategy and the corner strategy. The one row, one
column strategy was used by five students (all from TG2 and TG3 but spanning grades 3, 5,
and 7) when drawing on five tasks across unit types. Sometimes the student using he strategy
first drew individual units in one row and then one column before drawing individual units to
cover the interior of the region. Other times they used the strategy before curtailing to draw
row and column segments, indicating a mixed drawing strategy.
In contrast to the one row, one column strategy, the corner strategy is based on
(Western) everyday experiences. Young children, such as Orrin, placed four units in the four
corners and then considered the region covered (see Figure 7). Some of the participants first
covered the corners, then the remaining border, and finally the interior of the region with
figural or physical units. Others covered the corners before filling in the rest of the region
without focusing on the remaining border next. The corner strategy was used by five children
(from TG2 and TG3 but spanning all four grades) on seven tasks, spanning the unit types.
Figure 7. Orrin’s tiling on Task 9
Only one student gave an explanation for her strategy. Rebecca started in the four
corners six times on five different tasks. When asked about her strategy, Rebecca asserted, “I
like to do it because there’s a game named four corners.” Although I asked the students if
they liked puzzles, the corner strategy users did not explicitly make the connection.
Discussion of Findings
Overall, the data in this study support previous findings that area measurement is
developmental in nature (e.g., Battista et al., 1998; Sarama & Clements, 2009). I identified an
increasing pattern, generally based on grade, in children’s use of more sophisticated
strategies and a decreasing pattern in their use of less sophisticated strategies. This was
especially pronounced with transitions from structuring with individual units to structuring
with groups of units (as rows and nonrows) and then to structuring with parallel row and
column segments. These results extend the existing literature on area measurement
development to indicate that the construction of individual units, groups of units, and arrays
are developmental in nature, regardless of unit shape.
There were two notable emergent strategies: one row, one column strategy and corner
strategy. I consider the one row, one column strategy to be based on educational experiences
because of its potential connection to textbook displays of regions. Figure 8 is an area task
that imitates the format of area measurement tasks often printed in elementary and middle
school mathematics textbooks.
Figure 8. Textbook area task
In such a task, a child may be asked to determine the area of the rectangle, prompting the
child to multiply the length and the width of the rectangle. Hence, the child may be prompted
to acknowledge the importance of a pair of orthogonal sides. The one row, one column first
strategy is an extension of this acknowledgement that may or may not be used to quantify a
two-dimensional region correctly. In some cases, children that used this strategy then
multiplied the number of units in a row times the number of units in a column to quantify the
number of units covering the region. Within an array of squares, the number in a row times
the number in a column does produce a correct number but not a correct quantity. Referring
to the operation of n rows times m columns produces mn row-columns, which is an erroneous
abstraction of the rectangular area formula. Rather, one correct arithmetic approach would be
n rows times m square units per row, producing nm square units. Within an array of
nonsquare units, the number in a row times the number in a column does not necessarily
produce a correct number or a correct quantity.
The second notable strategy was the corner strategy. There were many variations of
this strategy. I consider the corner strategy to be based on everyday experiences because of
its connection to jigsaw puzzles. A common strategy for assembling a jigsaw puzzle is first
finding the corner pieces (pieces with two straight edges), then finding the border pieces
(pieces with one straight edge), and finally assembling the interior pieces (pieces with no
straight edges), assembling adjacent pieces by fit.
There are many parallels to the ways young children cover, figurally or physically, a
rectilinear region with area units and the ways individuals assemble a jigsaw puzzle. In both
situations individuals have a tendency to use similar strategies, including corner strategies,
border strategies, next to existing strategies, and trial and error strategies. Furthermore, when
covering an area and assembling a puzzle, it is important to eliminate gaps (large and small)
and overlaps in order to create a planar region completely covered (although in the assembly
of a puzzle, the "units" are not congruent). These connections have not been investigated in
the literature, neither the strategy correspondences nor the space-covering comparisons. Yet,
they have direct instructional implications. Such a comparison may help children recognize
and then eliminate their space-covering errors.
Educational Importance of the Research
These results indicate that the account of ways children develop spatial structuring
schemes is too narrow. Researchers have described how children progress from iterating
individual units to cover, to row-based, two-dimensional structuring schemes, and then to
array-based, two-dimensional structuring schemes (Battista et al., 1998; Sarama & Clements,
2009). However, such a progression does not account for nonrow, two-dimensional
structuring. The results from this study, especially from the triangular unit tasks, indicate that
nonrow, two-dimensional structuring should be considered. It is important for future research
to account for such a delimited notion of nonrow, two-dimensional structuring schemes and
the ways they support meaningful use of arrays, grids, and coordinate systems (Miller, 2013).
These results not only extend the existing literature on how children think about area
measurement, but also suggest multiple instructional implications. The emergent strategies of
one row, one column first and corners are worthy of further investigation. However, before
promoting the use of the one row, one column first strategy or discussing the use of jigsaw
puzzles in math lessons, more study is needed. Is the one row, one column first strategy a
transient strategy supporting the shift from around the border to array structuring? Does the
one row, one column first strategy propagate an erroneous abstraction of the rectangular area
formula? Does the connection of covering and area to assembling a jigsaw puzzle help
children recognize and then eliminate their space-covering errors? Are there other strategy
correspondences? Does the lack of congruence among jigsaw puzzle pieces detract from
these correspondences? These and other questions need answering before recommending that
jigsaw puzzles be used to support area measurement development.
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