exploring the mathematical knowledge for teaching geometry and measurement through the design and...
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By Michael D. SteelePublished in Journal of Mathematics Teacher Education. Pre-press edition.TRANSCRIPT
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RUNNING HEAD: GEOMETRY AND MEASUREMENT ASSESSMENT TASKS
Exploring the Mathematical Knowledge for Teaching Geometry and Measurement through the
Design and Use of Rich Assessment Tasks
Michael D. Steele
Michigan State University
Accepted for publication in Journal of Mathematics Teacher Education
DRAFT
DO NOT CITE OR REPRODUCE WITHOUT AUTHORS PERMISSION
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Abstract
While recent national and international assessments have shown mathematical progress being
made by US students, little to no gains are evident in the areas of geometry and measurement.
These reports also suggest that practicing teachers have traditionally had few opportunities to
engage in content learning around topics in geometry and measurement. This article describes a
set of assessment tasks designed to measure teachers mathematical knowledge for teaching ge-
ometry and measurement in a nuanced way. The tasks, focused on relationships between measur-
able quantities of figures, adhere to three key design principles: Tasks are grounded in the con-
text of teaching, measure common and specialized content knowledge, and capture nuanced per-
formance beyond correct and incorrect answers. Six tasks are presented that reflect these design
principles, with teacher data illustrating the ways in which the tasks differentiate performance
and reveal important aspects of teacher knowledge.
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Understanding the nature and nuance of students mathematical performance has been a
key goal of national and international comparative studies over the past two decades (e.g.,
Gonzalez et al., 2004; Kloosterman, 2007; Provasnik, Gonzales, & Miller, 2009). In studies
which break down performance into content strands, two content areas consistently lag in
performance: geometry and measurement (National Center for Education Statistics, 2012), and
this performance gap is particularly visible in the United States. Standards documents have
highlighted geometry and measurement as as vital to positioning students well to enter the
workplace and science, technology, mathematics and engineering fields (Common Core State
Standards Initiative, 2010). Three reasons for poor performance in geometry and measurement
have been posited in the research literature: 1) weak treatment in K-12 curricula, 2) challenges
with the implementation of geometry and measurement in the classroom, and 3) limited teacher
knowledge related to geometry and measurement (e.g., Clements, 1999; Lehrer, 2003, Strom,
Kemeny, Lehrer, & Forman, 2001). Of these three areas, investigations of teacher knowledge
related to geometry and measurement have been nearly nonexistent in the research literature.
This study draws on the current body of research related to teachers mathematical knowledge in
designing a set of tools to assess teacher knowledge related to geometry and measurement. These
tasks as a set posit a particular way of thinking about teaching geometry and measurement that
rests on a broad and well-connected knowledge base, using the construct of mathematical
knowledge for teaching.
Over the past decade, scholars argued that teaching in ways that support meaningful
student learning involves a number of identifiable and differentiable knowledge bases,
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collectively referred to as mathematical knowledge for teaching (Ball, Thames, & Phelps, 2008).
Mathematical knowledge for teaching (MKT) includes knowledge that resides at the intersection
of pedagogy and content, incorporating what scholars have referred to as pedagogical content
knowledge as well as subject-matter knowledge. Ball, Thames, and Phelps (2008) differentiate
between two particular aspects of subject-matter knowledge: common content knowledge, the
ability to solve a mathematics problem that any educated user of mathematics might need; and
specialized content knowledge, the mathematical knowledge unique to the work of teaching.
Common content knowledge (CCK) is not specific to teaching, and consists of the body of
knowledge that any well-educated adult might need and use; or put another way, the knowledge
that a teacher might hope to instill in his or her students. Specialized content knowledge (SCK)
refers to knowledge that is inherently mathematical in nature, but unique to the work of teaching.
This knowledge base includes methods of presenting mathematical ideas, identifying the
mathematics contained in an instructional task, knowing and making sense of alternative
methods to solve a mathematical task, and being able to anticipate different ways to think about
the mathematics, including common misconceptions.
SCK is an important construct for two reasons. First, findings from studies of teacher
knowledge at the elementary level find that CCK and SCK are differentiable constructs, and that
higher levels of SCK correlate with stronger student learning outcomes (Ball & Hill, 2008; Hill
et al., 2008; Hill, Rowan, & Ball, 2005). Moreover, SCK tends to be underdeveloped in teachers
(Hill, Rowan, & Ball, 2005), and few specific learning experiences exist to develop specialized
content knowledge in teacher preparation and professional development. Developing robust
measures of SCK has been an important thrust of the current research agenda around
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mathematical knowledge for teaching; however, the majority of work in conceptualizing and
measuring SCK has taken place at the elementary level. Few studies have focused specifically on
secondary mathematics teachers mathematical knowledge for teaching, with geometry and
measurement almost entirely unexplored (see Swafford, Jones, & Thornton, 1997 for a notable
exception). There has recently been questions about the nature and differentiability of CCK and
SCK in secondary mathematics teachers, whom unlike most elementary certified teachers have a
mathematics major or significant mathematical training (Speer & King, 2009).
Given the urgent need to improve student performance in geometry and measurement and
the role of teacher knowledge in supporting or inhibiting that performance, there has been a
recent push to develop professional learning materials for teachers that support the development
of a richer understanding of geometry and measurement (Driscoll, DiMatteo, Nikula, & Egan,
2007; Driscoll & Seago, 2009). An important component of these development efforts is the
ability to measure changes in teachers CCK and SCK related to geometry. In this article, I
describe the development of a series of tasks designed to investigate and measure teachers
mathematical knowledge for teaching geometry and measurement, with a specific focus on CCK
and SCK. In the sections that follow, I use the CCK and SCK constructs to describe an important
aspect of MKT related to geometry and measurement: the relationships between length,
perimeter, and area. I then describe a set of tasks designed to measure teacher knowledge of this
content, focusing on the design features of the tasks. In doing so, the ways in which the tasks
differentiate student performance are illustrated using data from an administration of these tasks
at the start of a course for secondary teachers focused on building content knowledge for
teaching geometry and measurement. To begin, I examine the nature of geometry and
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measurement for students in the middle grades as a springboard for discussing the mathematical
knowledge teachers might need to support students learning.
A Slice of Geometry and Measurement: Relationships between Length, Perimeter, and Area
Measurable attributes of geometric figures length1, perimeter, and area; and length,
surface area, and volume are keystones of the measurement strand of school mathematics in the
elementary grades. Elementary students (grades K-5) learn how to find length, perimeter, and
area of one- and two-dimensional shapes, first working empirically, then progressing to the use
of a formula (Common Core State Standards, 2010; Kasten & Newton, in press; NCTM, 2000).
Work in the middle grades (grades 6-8) should link these empirical measurement and formula
experiences to geometric properties of shapes in order to develop understandings of the
relationships between these measurable attributes. This move sets the stage for work with
generalized figures and theorems related to congruence and similarity in high school geometry,
typically taken in grade 10 in the United States. Tasks in the middle grades should prompt
students to consider questions such as: what happens to the area of a rectangle if one doubles the
height? What happens to its perimeter? Are the results the same if we consider a triangle? A
trapezoid? A parallelogram? What properties of a shape are variant when perimeter or area
changes, and what properties are invariant? How is this variation linked to general formulas and
algorithms for determining these measurable quantities? What are the characteristics of these
shapes that make these relationships generalizable?
In general, middle grades students are successful in determining the perimeter and area of
common polygons given length measures (Blume, Galindo, & Wolcott, 2007), and it is
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1 In this work, I use the term length to denote one-dimensional measurable attributes of two- and three-dimensional figures. This might include the sides or edges of polygons or polyhedra or non-side distances such as the height of a non-rectangular parallelogram or an oblique prism.
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reasonable to expect secondary mathematics teachers to be similarly adept. More problematic is
the move to articulating the more general relationships between measurable quantities such as
length, perimeter, and area. Most tasks involving measurable quantities on the 2003 National
Assessment of Educational Progress (an assessment administered to a nationally-representative
sample of US students) did not show significant student performance gains (Blume, Galindo, &
Wolcott, 2007), echoing the results of previous research (e.g., Bright & Hoeffner, 1993; Chappell
& Thompson, 1999; Clements & Battista, 1989; Hoffer, 1983; Martin & Strutchens, 2000;
Sarama, Clements, Swaminathan, McMillen, & Gmez, 2003). Performance difficulties were
characterized by confusion between formulas for perimeter and area and suggested that students
did not have a conceptual understanding that allowed them to untangle the confusion. This
difficulty in distinguishing between area and perimeter suggests that the concepts are not well
connected to properties of the shapes in question. Whether this same confusion is prevalent in
secondary mathematics teachers is an open question. Early studies of secondary teachers suggest
that their conception of geometry was similar to that of typical middle school students
(Hershkowitz & Vinner, 1984; Mayberry, 1983). However, more recent work has suggested that
interventions focused on developing teachers subject-matter knowledge related to geometry and
measurement can positively impact both teacher knowledge and the quality of tasks they enact
with secondary students (Swafford, Jones, & Thornton, 1997). While the Swafford, Jones, &
Thornton (1997) study showed significant changes in teachers subject-matter knowledge, the
authors note that the assessments used to measure teacher knowledge had limitations (limited
content and form, ceiling effects), suggesting a more fine-grained set of assessment tasks might
yield more nuanced information about teacher knowledge of geometry and measurement.
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Designing Tasks that Measure Teacher Knowledge of Length, Perimeter, and Area
Early efforts to measure teacher knowledge focused on two primary vectors: teacher
certification and teacher course-taking. Studies of certification have shown inconclusive and
sometimes conflicting results, exacerbated by the issue of generalizing across unique state-based
certification requirements and assessments (Ball & Hill, 2008). Mathematics course-taking
studies also showed slightly more consistent results while adding several complications; namely,
that the taking of a course does not directly measure what teachers may have learned from the
course, and that content may have little or no bearing on the content teachers are teaching (Ball
& Hill, 2008). Direct assessments that measure content that teachers teach are much more
promising; large-scale multiple choice assessments of this style have been used to differentiate
CCK and SCK and to link teacher knowledge to student achievement (Hill, Rowan, & Ball,
2005). Two issues arise with these large-scale multiple choice measures. First, they provide little
detail on the teachers thinking in arriving at their solution. Second, they do not necessarily
measure the interactions between common and specialized content knowledge. For example, the
failure to decide which of two formulas for the volume of a rectangular prism would be most
appropriate for a particular mathematical task might indicate that a teacher does not necessarily
see the difference in the two formulas (implicating SCK), or that the teacher has a flawed
conception of how one or both of the formulas calculates volume (implicating CCK).
In order to address some of the limitations of previous work developing CCK and SCK
instruments, the tasks designed for this study had to fulfill three core design principles, listed
here and elaborated below:
Tasks are grounded in the context of teaching
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Tasks, as a set, measure aspects of and relationships between common and specialized
content knowledge related to geometry and measurement
Tasks capture nuances of teacher knowledge beyond correct and incorrect answers,
including specific misconceptions and the identification of vectors for change
Context of teaching. Teacher learning experiences that have the potential to transform
teachers knowledge and practices are most effective when they are grounded in the work and
context of teaching (Ball & Cohen, 1999; Thompson & Zeuli, 1999). This same premise holds
true for assessing teacher knowledge tasks in which teachers are asked to explore authentic
mathematical tasks that students would do, consider issues of lesson planning, and analyze
student work are more likely to measure knowledge that can be and is used in teaching practice.
Measuring CCK and SCK. The second principle stipulates that tasks individually must
measure specific aspects of subject-matter knowledge. That is, each task should illuminate
aspects of teachers understandings related to the content, not necessarily related to issues of
pedagogy or more content-general teaching skills. In addition, the tasks as a set should measure
both common and specialized content knowledge related to geometry and measurement. This
aspect of the principle is important for articulating the relationships between CCK and SCK; if
an assessment item reveals issues with teachers specialized content knowledge, these
deficiencies may relate to the underlying common content knowledge.
Capturing nuanced performance. Finally, tasks should provide rich, nuanced data on
teacher knowledge that provide insight into ways that teachers are making sense of the content as
well as possible vectors for change in teacher knowledge. A key aspect of mathematical
knowledge for teaching across content areas is the ability to produce multiple representations of
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a mathematical idea, make connections between those representations, and make decisions about
which representations are best for work with a particular set of students (Ball, Thames, & Phelps,
2008). Assessing each of these three ideas separately (producing representations, connecting
representations, and selecting representations for use) provides important data on teacher
knowledge. However, items that assess multiple aspects of this knowledge base at once have the
potential to demonstrate important features of teacher knowledge and performance not otherwise
captured, as well as providing information about improving teacher performance.
Capturing nuanced performance relates not only to the task design but to the lens used to
assess performance on the task. As such, task development also should include rubrics that focus
attention on the knowledge teachers display in solving the task and the generalizability of the
solution. These rubrics can serve both a diagnostic and prognostic purpose. For example, a
teacher might be able to represent a mathematical pattern related to a fixed area/changing
perimeter situation as a table, graph, and equation, but may not be able to make connections
between those specific representations. This result would suggest a particular sort of intervention
that would work from the ability to produce the representations towards the ability to make
connections between them that individual items may not have suggested independently.
To develop a set of tasks around the relationships between length, perimeter, and area, the
constructs of common content knowledge (CCK) and specialized content knowledge (SCK) were
used in conjunction with the National Council of Teachers of Mathematics standards documents
(NCTM, 2000) and previous research to identify key mathematical ideas around which tasks
could be built. The articulation of this knowledge base is shown in Table 1. While the
framework draws across a diverse set of research literature, it is important to note that is only one
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conceptualization and is not intended as a canonical representation of the complete set of
knowledge needed to teach relationships between length, perimeter, and area.
INSERT TABLE 1 ABOUT HERE
The core ideas represented in the CCK section of Table 1 involve being able to calculate
area and perimeter, and describe and distinguish the ways in which one-dimensional
measurements relate to area and perimeter. Evidence of these understandings could be
represented by the ability to find area and perimeter and to describe the ways in which the
quantities are mediated by length in a number of ways, ranging from the use of specific examples
to the application of general geometric principles. The aspects of SCK in Table 1 describe the
teachers representational fluency with respect to the ideas of length, perimeter and area, their
abilities to identify affordances and constraints of different formulas for calculating area and
perimeter, and their abilities to identify important ideas in this space for students to learn and
tasks that have the potential to support that learning. In considering these aspects of CCK and
SCK, one might expect most secondary mathematics teachers to perform well on tasks related to
CCK, such as finding perimeter and area of basic shapes and understanding that shapes with the
same perimeter can have different areas and vice-versa. Results of previous studies of teachers
suggest that tasks related to the SCK in Table 1 might prove more challenging for secondary
teachers (Fuys, Geddes, & Tischler, 1988; Hershkowitz & Vinner, 1984; Mayberry, 1983;
Swafford, Jones, & Thornton, 1997).
These tasks were created as a part of the process of designing a content-focused methods
course (Markovits & Smith, 2008; Steele & Hillen, 2012), designed as an opportunity for
teachers to enhance their mathematical knowledge for teaching geometry and measurement in
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the middle grades. Teachers performance on the tasks at the start of the course is used to
illustrate the ways in which the tasks met the three design criteria, focusing on the set of six tasks
related to two-dimensional geometry and measurement. A brief description of the course
follows.2.
Research Context and Process
The six tasks described below were created as a part of the design of a Masters-level
content-focused methods course taught by the author at a mid-sized public university in the
eastern United States. The instructional intervention consisted of a 6-week course related to
mathematical knowledge for teaching geometry and measurement in the middle grades,
including work on solving mathematical tasks, analyzing student work, and planning lessons
around geometry and measurement content. The first three weeks focused on two-dimensional
geometry and measurement, and the tasks were used with teachers during the first course
meeting to assess the groups MKT related to course content3. Table 2 shows demographic
information on the 25 teachers who participated in the course. All 25 teachers completed the
written assessment, which contained all tasks except the Minimizing Perimeter Lesson Plan,
which was completed in an interview setting by a subset of 20 teachers.
INSERT TABLE 2 ABOUT HERE
GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 12
2 While the tasks were designed in part to measure teacher learning, a detailed reporting of what teachers learned is beyond the scope of this article (see Steele, 2006 for more detail).
3 In addition to the secondary-certified teachers, two additional populations are included: elementary-certified teach-ers and teacher leaders. At the time of the study, elementary certified teachers were able to teach middle grades mathematics, and the elementary teachers in the course either were or had an interest in teaching middle school. Similarly, the teacher leaders in the course were all secondary-certified teachers who had recently left the classroom, tasked with supporting secondary teachers throughout the region. As such, including these teachers in the population is representative of the range of teachers likely to be teaching middle grades mathematics.
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For teachers written assessment tasks, categorical rubrics were developed that sought to
characterize performance with respect to the aspects of knowledge in Table 2. Specifically, these
rubrics were designed to assess correctness of responses to mathematical tasks and particular
features of responses (number and types of representations, rationale for response) for
mathematical tasks. For open response tasks, emergent categories were identified using a
Straussian grounded theoretical approach (Corbin & Strauss, 1990). Teacher performance on the
Minimizing Perimeter Lesson Plan interview task was audiotaped, transcribed and coded for the
number and type of mathematical goals identified by teachers. Teachers were invited to modify
the task as a part of the interview protocol; the nature of these modifications were also coded
using emergent categories. The author and a second researcher double-coded 25% of the data
(randomly selected) to establish reliability. Inter-rater reliability for rubrics ranged from 88% to
100%, with all disagreements resolved through discussion.
The Tasks and Findings
Six tasks were designed to assess aspects of teachers MKT related to length, perimeter,
and area. Table 3 maps the six tasks to the aspects of CCK and SCK related to length, perimeter,
and area that they are designed to assess. In each of the sections that follow, the tasks are
described in turn. For each task, I detail the ways in which the tasks met the three critical design
criteria and in particular, what teacher work on the tasks showed with respect to their common
and specialized content knowledge. The description of the task establishes the ways in which the
tasks meet the context of teaching. Sections describing the ways in which each task measures
CCK and SCK and captures nuanced performance follow for each.
INSERT TABLE 3 ABOUT HERE
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Tangrams Task
The Tangrams task, shown in Figure 1, uses the familiar set of tangram tiles to create a
fixed area-changing perimeter situation. The task presents the set of tangram tiles, commonly
used in elementary and middle grades, in a square configuration and two other configurations
and asked teachers to determine which of the two rearrangements, if any, had the greater area and
greater perimeter.
INSERT FIGURE 1 ABOUT HERE
Measuring CCK and SCK. The Tangrams task assessed the relationships between length,
perimeter, and area in the context of irregular polygons. In particular, the task measured
teachers CCK in knowing that partitioning a figure into pieces and rearranging them can change
the perimeter but not the area. To respond correctly, teachers needed to articulate the non-
constant relationship between measurable attributes and perimeter and explain why neither of the
two rearrangements (B and C) changed the area, which of the two had the greater perimeter, and
why. Teachers were asked to justify their responses, giving them the opportunity to use general
mathematical principles to compare the two arrangements.
Capturing nuanced performance. Responses to this task assess whether or not teachers
understand that rearranging the tiles maintains area and potentially changes perimeter. The task
also captures the nature of the argument teachers can make to justify that conclusion. Teachers
could respond correctly to the task in three ways. They could make a visual estimation of area
and perimeter and base their responses on this estimate. Teachers could use a formal or informal
measuring device4 to determine the area and perimeter of each shape and make a quantitative
GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 14
4 Tangram tiles, rulers, and grid paper were available to teachers during the assessment.
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comparison. A general argument could also be made regarding the relationships between the
tiles (their area covered and lengths of sides) and the perimeter and area of Figures B and C.
Table 4 shows a rubric designed to evaluate these three types of responses, with examples of
teacher responses included.
INSERT TABLE 4 ABOUT HERE
Teacher performance on these tasks was able to differentiate the extent to which teachers
relied on empirical arguments as compared to general principles in making the perimeter and
area comparisons. For the area question, 21 of 25 teachers correctly noted that both areas were
the same, with all but two of the correct responses scoring at Level 3 on the rubric. On the
perimeter question, nearly all teachers (23 of 25) successfully selected arrangement C as having
the greater perimeter, but the reasoning was more varied. About half justified the change
empirically or qualitatively (Level 1 or 2). These results show that while the majority of teachers
displayed an important aspect of their common content knowledge, in knowing that figures with
the same area can have different perimeters, there was significant variation in the extent to which
teachers used general mathematical principles to justify this conclusion. Such general
mathematical principles, while not needed to generate a correct answer to the task, are an
important aspect of specialized content knowledge. Being able to describe relationships between
length, area, and perimeter across shapes using general principles reflects a deeper understanding
of the geometric principles underlying plane geometry. .
Area of a Parallelogram Task
The Area of a Parallelogram task (Figure 2) presents a common geometric figure in the
elementary and middle grades - the parallelogram - and fixes two attributes of the parallelogram,
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height and area. The task asks whether or not fixing these two attributes implies a fixed
perimeter, and prompts for an explanation of why this is or is not the case. The parallelogram is
an important transitional plane figure in the middle school, as along with the triangle, it is one of
the first times that students encounter an important measurable attribute (the height) that is not a
side of the figure. This task uses this familiar teaching context of the parallelogram and prompts
teachers to identify the quantities that are variant and invariant with respect to perimeter and
area. This notion of variant and invariant properties is critical to a generalized understanding of
geometry and has significant implications for the use of dynamic geometry software in the
classroom (Ruthven, Hennessy, & Deaney, 2008)
INSERT FIGURE 2 ABOUT HERE
Measuring CCK and SCK. The Area of a Parallelogram task assessed teachers CCK
regarding the common misconception that a fixed area implies a fixed perimeter (and vice versa),
and that a changing area implies a changing perimeter. This concept is not likely to be
challenging for secondary mathematics teachers. The interesting features of the task include the
ways in which teachers might justify their conclusion and the representations used in explaining
their response, a measure of SCK.
Capturing nuanced performance. The task allows teachers to make their argument either
using empirical examples (specifically, a counterexample) or reasoning using general geometric
measurement properties. Responses were coded using the rubric shown in Table 5. The coding
scheme distinguishes correct and incorrect responses to the question; answers coded Correct-1
represent a mathematical explanation for why the perimeter is not fixed, indicating a conceptual
understanding of the relationship between length, perimeter, and area. Answers coded
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Incorrect-1 represented evidence of the misconception that fixed area implies fixed perimeter.
The number and type of representation used to respond to the question was also tracked.
INSERT TABLE 5 ABOUT HERE
Similarly to the Tangrams item, the Parallelogram task measures an aspect of teachers
abilities to describe the relationships between dimension, perimeter, and area. Performance on
this item as compared to the Tangrams item suggests that it captures a different aspect of
mathematical knowledge for teaching. Performance on the item was mixed, with only 12 of 25
teachers generating correct responses with an explanation. Incorrect answers were frequently
accompanied by a figure that correctly represented the situation, but arrived at an incorrect
conclusion. Figure 3 shows two such responses.
INSERT FIGURE 3 ABOUT HERE
The incorrect responses shown in Figure 3 reveal that while teachers might be able to
articulate the relationships between base, height and area both symbolically and in a diagram,
this does not guarantee that they understand that fixed height and area implicate a unique
perimeter. In both cases, responses show that teachers may conflate the height of a parallelogram
as contributing to its perimeter even in non-rectangular parallelograms. Measures of
representational use showed that all teachers administered the item used a diagram
representation, often to provide the requested example as shown in both responses in Figure 3.
Most teachers used at least one other representation (mean of 1.72), with the most common
secondary representations written explanations, as in Response A, and symbolic, as in Response
B. The symbolic response, while it may be comfortable for teachers, often required
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interpretation to determine how the symbolic representation justified the conclusion, as in
Response B. In fact, 75% of teachers using a symbolic representation as the only other
representation produced an incorrect response. The use of a written explanation did allow more
responses to be coded as holding the misconception about the relationship between height,
perimeter, and area of the parallelogram (Incorrect-1). These results suggest that while
representational fluency is clearly an important aspect of teacher knowledge, it does not always
guarantee that the teacher understands the underlying mathematical principles at play.
Fence in the Yard Task
The Fence in the Yard task (Figure 4) presents a situation in which a fixed length of fence
needs to be used to create a rectangular pen of maximum area. This task was administered to 8th
graders on the 1992 NAEP. Results were poor, with less than 1% of students scoring extended
or satisfactory on the item (Kenney & Lindquist, 2000). Poor performance was attributed in
part to lack of familiarity with a complex fixed perimeter-changing area situation. Previous
research has shown that students often struggle to make sense of changing x-changing y
situations such as the relationships in this task (e.g., Stavy & Tirosh, 1996). .
INSERT FIGURE 4 ABOUT HERE
Measuring CCK and SCK. With respect to CCK, the task requires teachers to generate
length/width pairs and calculate perimeter and area and to recognize that a constant perimeter
does not imply a constant area. With respect to SCK, the task provides opportunities to use a
variety of representations (tables, graphs, symbols, pictures, verbal explanations) in the service
of explaining the relationships between length, perimeter, and area.
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Capturing nuanced performance. Similar to the Tangrams and Parallelograms tasks,
Fence in the Yard is designed to capture aspects of teachers abilities to describe the relationships
between length, perimeter, and area. Responses to the task were coded for the ability to calculate
perimeter and area correctly, to recognize that a constant perimeter does not imply a constant
area, both important aspects of CCK. The representations used to explain the relationship
between length, perimeter, and area and the extent to which responses made that relationship
explicit (aspects of SCK) were coded in order to differentiate teacher performance. Examples of
responses that do and do not make this relationship salient are shown in Figure 5. Response A is
in the form of a written explanation, and makes explicit the ways in which length/width pairs,
perimeter, and area are related in the context of the task, suggesting that the teacher not only can
calculate different areas for a fixed perimeter, but can also articulate the generalized principle
behind the empirical examples. Response B is in the form of a table with explanatory text; while
this response identifies the values that maximize the area correctly, there is no explicit
explanation of how the length/width pairs, perimeter, and area are related in the context of the
task. In this case, it is unclear the extent to which the teacher knows the general underlying
principle.
INSERT FIGURE 5 ABOUT HERE
Teachers who were administered the Fence in the Yard task were able to identify the fixed
perimeter-changing area situation and calculate perimeter and area of a variety of fences.
Interesting patterns were observed in the representations and rationale used to describe the
impact of changing the dimensions of the rectangle on area while keeping the perimeter constant.
This involves both CCK in being able to coordinate and calculate the measurable quantities, and
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SCK in the ability to relate the perimeter and areas to the geometric properties of the figures (see
response A in Figure 5). Teachers responses were able to differentiate these performances, with
about half of teachers showing empirical examples and half making the general principles related
to the geometric properties explicit in addition to finding the maximum area.
Similar to the Parallelogram task, teachers SCK in the form of representational use was
examined and compared to their abilities to articulate the general principles underlying the task.
The task revealed interesting nuances with respect to representational fluency. On this task,
representations were categorized (graph, table, written explanation, and symbolic) and cross-
coded with respect to the extent to which the response illuminated the relationships between the
length/width pairs, perimeter, and area. Most teachers solving the Fence in the Yard task in
general used multiple representations in their responses, with the average response including
1.96 representations. There was also an interesting relationship between the representations used
and the extent in which the response made salient the relationships between the length/width
pairs, perimeter, and area. Responses that made explicit the ways in which changes to length/
width impacted perimeter and area were more likely to use a table (2(1, 50) = 3.84, p < 0.01).
The table is a notable representation because it shows specific values of length, width, perimeter,
and area, and allows for pattern recognition that can facilitate a description of the general
principles that explain how changes to one quantity impact changes in another. This result
suggests that a table can be a particularly rich reasoning tool in examining relationships between
length, perimeter, and area.
GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 20
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Minimizing Perimeter Lesson Planning Task
The Minimizing Perimeter Lesson Planning task was implemented in an interview setting
and assessed several aspects of teachers CCK and SCK. The task, shown in Figure 6, was
adapted from a task in NCTMs Navigating through Geometry Grades 6-8 (Pugalee et al., 2002).
The planning task was designed to approximate the authentic work of solving a mathematical
task that one would design a lesson around, planning the lesson, and describing the potential
implementation with a middle school class. Teachers were asked to examine the task prior to the
interview, which involved a fixed area-changing perimeter scenario. In solving the task
themselves, teachers had to calculate perimeter and area, explain the non-constant relationship
between perimeter and area, articulate the impact of changes to length on perimeter and area, and
use a variety of representations to explain these relationships. In planning the lesson, teachers
had to decide what aspects of the problem (if any) to modify, articulate their specific goals for
the lesson, and describe how the lesson would unfold.
INSERT FIGURE 6 ABOUT HERE
Measuring CCK and SCK. This task assesses several aspects of teachers mathematical
knowledge for teaching related to dimension, perimeter, and area. The task asks teachers to
engage in solving the task themselves, and to consider the ways in which they might plan a
lesson for a middle school class using this task. As such, the task measures teachers CCK in
their ability to solve the problem correctly, and their SCK5 with respect to the choices they made
about representational use and modifications to the problem for implementation in the classroom.
GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 21
5 While this task does ask teachers to think about classroom practice, the fact that they are considering in general how this task might be implemented, and not with respect to a specific group of students or in the context of a class, distinguishes the measurement of SCK from knowledge of content and students or knowledge of content and teach-ing.
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Teachers goals also evaluate an aspect of their specialized content knowledge related to their
task selection.
Capturing nuanced performance. The Minimizing Perimeter task assesses yet another
aspect of teachers CCK related to the relationships between dimension, perimeter, and area.
Responses that correctly determine area and perimeter, free of evidence of the misconception that
a fixed area implied a fixed perimeter, represent evidence of this knowledge. Similar to Fence in
the Yard, SCK can be measured by representational use. In addition, the extent to which teachers
might modify the task and articulate their goals for the task represent SCK. Rubrics were used
for correctness and representational use, with modifications and goals analyzed using open
coding. Modifications can have one of two general effects - to maintain the cognitive demands
of a task, keeping the focus on conceptual understanding; or to diminish the demands, often
through proceduralizing the task (Stein, Smith, Henningsen, & Silver, 2009). Responses were
grouped into themes, with modifications and goals assessed for whether or not they supported a
procedural or conceptual understanding of length, perimeter, and area (or both).
With respect to CCK, teachers in the sample performed well - every teacher was able to
create a variety of rectangular length/width pairs and calculate perimeter and area, keeping area
constant while varying perimeter. This mirrors performance on the Fence in the Yard task. How-
ever, when examined in concert with representational uses, interesting patterns emerged. Repre-
sentations used included tables, written explanations, and symbols, as well as the graph that is
specifically requested. The graphical representation of the non-linear relationship between
length and perimeter was one site that differentiated performance. Many teachers either assumed
that the relationship would be a symmetric parabola or expressed uncertainty about the shape of
GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 22
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the graph. (This relationship can be represented by the rational function ). This dis-
crepancy underscores the complex relationship between CCK and SCK. While teachers might
demonstrate the ability to generate examples of rectangles that satisfy the task conditions, repre-
senting the generalized relationship accurately can pose a challenge, particularly in the context of
a graphical representation.
The task also sought to capture the ways in which teachers select and modify
mathematical tasks related to length, perimeter, and area, and the student learning goals they
identify for those tasks. The initial Minimizing Perimeter task was relatively strong in terms of
the cognitive demands and had the potential to illuminate key aspects of the relationships
between dimension, perimeter, and area. Results showed that most teachers engaged in a
modification of some sort: of the 20 responses collected, 15 made modifications that fell into
these two categories. Excerpts that exemplify each type of modification are shown below.
So and then in the second [part], since my mathematical goal is to get them to understand the relationship between perimeter and area, [question 3] restricts our area and wants to find the minimum amount of fencing. Ok so what I would do with [question 4] is alter it and say ok well the principal gives them a certain amount of fence, instead of giving them a certain amount of area he says, ok youre gonna use the leftover fence from the, 10th grade class garden I think that by forcing them to look at keeping area constant and keeping perimeter constant 2 different situations it, will allow them to see the relationship on both sides of the coin.
Lana, Getting at the conceptual ideas.
Since this is a 7th grade class, my first approaches [with the class] would probably be to have them just throw out a set of numbers. Such that the area- know that area is length times width[and] say, ok what would the perimeter be. And determine that the area is in fact 36 square feet. And I guess the perimeter would be 26 feet... And then, have them begin to list in the chart length vs. width the idea that if we have 1 for the length, the width is 36. Two for the length, the width is 18. And so forth. List every possibility.
Noah, Proceduralizing the task
GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 23
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Interesting correlations also emerged when teachers modifications were considered with
respect to their knowledge about the underlying relationship. Of the ten teachers that exhibited a
misconception or uncertainty about the nature of the relationship between length and perimeter,
only one made modifications that targeted conceptual understanding. These data suggest a
critical correlation - the nature of teachers CCK and SCK directly related to the mathematical
ideas influences the ways in which teachers might choose to plan for a lesson around that task.
Another influence on the ways in which teachers might implement tasks are the extent to
which they are able to determine a tasks mathematical goals, noted in previous research as a
critical influence on teachers planning practices (Morris, Hiebert, & Spitzer, 2009). The teachers
who piloted this item identified between two and three mathematical goals for the task (a mean
of 2.15). What was notable about these goals was the extent to which they targeted the
generalized principles as compared to specifics. For example, while thirteen teachers identified a
goal related to constant area and changing perimeter, five of these teachers limited their
discussion of that goal to the ability to identify the square as the minimum perimeter condition.
The articulation of specific goals that relate to the general principles have implications for
students opportunities to learn. For example, if students know that rectangles can have the same
area and different perimeters, this does not necessarily ensure that they will be able to describe
the impact of changing length and width on that perimeter while keeping area constant. In
contrast to the results related to task modification, the number of goals specified and the nature
of the goals did not show any specific correlation to teachers mathematical responses to the task,
although this is an important area for further inquiry.
GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 24
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Together, these results show important connections between CCK and SCK. Given a
relatively robust mathematical task, the depth of teachers understanding of the mathematics in
the task has implications for teaching decisions that either maintain the cognitive demand or
proceduralize the task. This also suggests the importance of a teacher working through a
mathematical task as a part of the planning process and as a vector for developing SCK. In
addition, having teachers specify their goals for a rich mathematical task can produce goals of
varying grain sizes. While no direct correlation was found in this study between goals and
planning practices, previous research suggests that this is a fruitful area for further inquiry.
Big Ideas Task
In order to make informed choices about mathematical tasks and topics, teachers need to
determine what the important content related to length, perimeter, and area might be for students
to learn. The Big Ideas task asked teachers identify the ideas that they felt middle grades students
should learn related to two-dimensional geometry, including length, perimeter, and area.
Measuring CCK and SCK. This task measures SCK in a similar way to the Minimizing
Perimeter task, in which teachers were asked to specify goals for a mathematical task. This task
zooms out to evaluate the extent to which teachers are able to articulate broader goals that cut
across grades for geometry and measurement learning. Teachers written responses were
examined for themes using open coding, and distilled to a set of general categories. These
general categories were identified as involving understanding of a concept, the learning of a
specific procedure or set of procedures, or a combination of conceptual and procedural. Grouws,
Smith, and Sztajns (2004) analysis of a national sample of US middle school teachers suggests
that the bulk of middle school geometry and measurement instruction focuses on procedural
GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 25
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work, such as using formulas to calculate perimeter and area of basic shapes. This task was
designed to assess what teachers value with respect to geometry and measurement instruction.
Capturing nuanced performance. The open nature of this task allowed teachers to express
a wide range of topics related to geometry and measurement that they felt were important.
Responses were coded as to whether the goals were primarily procedural in nature or conceptual
in nature. While a rich understanding of geometry and measurement would likely contain both
sorts of goals, this task provides the opportunity to measure both the mathematical content of the
goals and the extent to which conceptual and procedural goals are balanced.
INSERT TABLE 6 ABOUT HERE
Data from this task was coded thematically, with goal themes classified as primarily
procedural or conceptual. The results are shown in Table 6, with the number of teachers naming
each type of goal in parentheses. These data show that in general, teachers identified both
procedural and conceptual goals, with a stronger emphasis on procedural as measured by the
number of teachers identifying each type of goal. Moreover, there is little consensus on the
nature of geometry and measurement in the middle grades amongst teachers, as the most popular
goal theme only encompassed slightly more than half of the teachers responding.
Considering Formula Use Task
In addition to making decisions about the topics students will explore related to geometry
and measurement, teachers also must make decisions about the the methods and tools that will be
promoted in the learning of these topics. In the course of teaching lessons on perimeter and area,
teachers need to make decisions about the formulas they will introduce to students to facilitate
calculation. These decisions are often influenced by the textbook in use, teachers own
GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 26
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mathematical experiences, standardized tests, and the prior knowledge that students bring to the
work in the classroom. The Considering Formula Use task (Figure 7) is designed to assess the
reasons why teachers might select one formula for calculating the area of a rectangle over
another.
INSERT FIGURE 7 ABOUT HERE
Measuring CCK and SCK. Formulas are critical in facilitating efficient calculation of
measurable quantities such as perimeter and area. They also afford the opportunity to make
connections between the specific measures and geometric properties of figures, as represented by
the attributes used in the calculation (e.g., base, height, etc.)6. Similar to representational use,
understanding the affordances and constraints of different formulas for the same measure is
prerequisite to teachers making informed decisions about formula use with their students. The
task asked teachers to consider which formula for area of a rectangle, A=bh or A=lw, they might
use with a group of students and why.
Capturing nuanced performance. Responses were coded based on the formula chosen
(bh, lw, both, or no preference) and the rationale for why the formula was chosen, aggregated
into thematic categories. The results from this item show interesting links between formula
choice and the rationale. All teachers choosing the A=lw formula and those justified their choice
by stating that it was easier for students to use or that measurements were easier to find when
referring to length and width. Teachers preferring the A=bh justified their choices by citing the
generality of the formula to figures beyond the rectangle and the connection to the formula for
GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 27
6 The nature of these formulas and their typical use in schools can obscure the distinctions between geometric and measurement quantities. However, the formulas are ubiquitous in their use and a discussion of these geometry and measurement distinctions was beyond the scope of both the task and this article.
-
the volume of a rectangular prism or cylinder. When considered alongside their rationales,
teachers pedagogical choices of a formula to use with students suggest important relationships to
their own content knowledge. Teachers attuned to the conceptual relationships between different
two-dimensional figures and the relationships between area and volume formulas may be more
likely to choose one area formula over the other in work with their students. Additionally,
teachers focused on procedural issues for their students, such as the ability to easily identify
measurements and make calculations, are likely to choose a different formula. This contrast
provides important insight into the extent to which teachers might draw on their conceptual
knowledge, as compared to procedural issues such as ease of student calculation, in making
decisions about how to teach mathematics.
DISCUSSION
In this article, I have presented three design features for rich, open-response items that
assess mathematical knowledge for teaching. The set of six two-dimensional geometry and
measurement tasks presented here embody these design features and illustrate the ways in which
the tasks are grounded in the context of teaching, capture nuanced teacher performance, and
measure common and specialized content knowledge. The examples of teacher performance on
the tasks presented here illustrates the ways in which the tasks can differentiate teacher
performance. Moreover, teacher work on the tasks provides important windows into the
connections between common and specialized content knowledge in teaching.
Previous research has underscored the importance of specialized content knowledge as a
causal influence on student achievement (Hill, Rowan, & Ball, 2005) and as an important factor
in teachers abilities to unpack mathematical goals in lesson planning (Morris, Hiebert, &
GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 28
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Spitzer, 2009), and has noted that geometry and measurement is a content area in which teachers
often have less expertise than they might like in thinking about how to enact rich lessons with
students (Swafford, Jones, & Thornton, 1997). The tasks described here illustrate the important
connections between common and specialized content knowledge and the ways in which CCK
can influence how teachers make use of SCK. For example, teachers with stronger abilities to
describe the relationships between length, perimeter, and area clearly on the Fence in the Yard
Task were more likely to use multiple representations in their response. The ability to use
multiple representations and to understand the ways in which those representations make aspects
of the mathematics salient implicates a wider range of pedagogical choices they might make in
their classrooms. Moreover, these teachers selected a particular representation - the table - that
has purchase in making those relationships visible. This strong SCK is more likely to provide
teachers with observable pedagogical tools and practices to support the development of students
understanding of this relationship in their own classroom. These results resonate with Hill et al.s
(2008) analysis of teacher practice, in which they found correlations between specific
performances on SCK items and the quality and richness of teachers mathematics instruction,
particularly with respect to mathematical language.
These items also illustrate important connections between CCK and SCK, a relationship
that previous work on mathematical knowledge for teaching has not fully explored. Given that
these tasks are designed for use with secondary teachers, it is anticipated that teachers will
approach these items with relatively strong CCK. Teachers are likely to be able to calculate
perimeter and area and make basic connections between length, perimeter, and area; the data
from the items bore that hypothesis out at some level. However, items that measured those
GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 29
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aspects of CCK from multiple angles (e.g., using both rectangles and parallelograms, reasoning
from diagrammatic and contextual starting situations) and that provided opportunities to
mobilize both CCK and SCK in the service of the same task, revealed important nuances and
connections. One such example is with respect to the relationship between dimension, perimeter,
and area. Tasks using squares and rectangles embedded in a context, in which empirical
examples could be used to describe the relationship, showed strong CCK performance. The
parallelogram task, which used a more complex shape with respect to the relationships between
dimension, perimeter, and area and provided opportunities to make use of general principles,
illustrated that the connections between CCK and SCK were not robust in some cases.
The Minimizing Perimeter task also revealed an important relationship between CCK and
SCK and teachers ability to write goals for a mathematical lesson. Teachers with stronger
mathematical performances on the task were better able to write more specific goals for the use
of the Minimizing Perimeter task with students. Unpacking mathematical goals is a critical
factor in planning for and enacting meaningful mathematics instruction, and learning to write
such goals can be challenging (Morris, Hiebert, & Spitzer, 2009). Morris and colleagues in their
study posit that teachers SCK influences their ability to unpack mathematical goals; the results
of the Minimizing Perimeter task support this hypothesis. Given that teachers were also required
to solve the task prior to the interview, the findings support the assertion that engaging in a
mathematical task prior to teaching it is a critical skill that supports high-quality mathematics
teaching (Smith & Stein, 2011; Steele, 2008). This finding has particularly important
implications for supporting prospective teachers in building capacity to plan for, teach, and
reflect on strong, conceptually-based mathematics lessons.
GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 30
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The three design features exemplified by these tasks - grounded in the context of
teaching, measuring CCK and SCK, and capturing nuanced performance - are intended to be of
use to researchers interested in investigating other aspects of mathematical knowledge for
teaching. An important first step is using the research literature on student and teacher learning in
the development of an MKT framework for the content in question. Items can then be adapted or
designed that measure these mathematical ideas across multiple aspects of MKT. An important
limitation of this study was the development of items that only measured common and
specialized content knowledge; researchers interested in interactions across the other MKT
categories, such as knowledge of content and students or curricular knowledge, might design
tasks that assess both subject-matter knowledge and pedagogical content knowledge.
Understanding these interactions through teacher knowledge assessments could significantly
advance the fields understanding of the nature and use of mathematical knowledge for teaching.
The construction of such tasks could aid the work of assessing the impact of mathematics teacher
education and professional development efforts. The tasks themselves could also be positioned
as objects of inquiry for teachers to support them in deepening their own understandings.
While there has been significant research effort in developing measures of teachers
mathematical knowledge for teaching, the bulk of this work has focused on elementary content
and multiple-choice items that afford the collection of large data sets. This research advances the
field and supports continued theory-building related to MKT. The items in this article and the
design features upon which they are built provide a model for a finer-grained examination of the
relationships between dimensions of teachers mathematical knowledge. They also provide us
with tools to assess teacher knowledge in geometry and measurement at the secondary level. The
GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 31
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limitations of these items is that they do take significant time and energy to both create and
assess; however, they provide lenses into mathematical knowledge for teaching that other
assessments do not. In addition, the articulation of the three design features constitutes a
framework for future work creating items in other content areas, While these items focused on
common and specialized content knowledge, the design features are likely extensible to other
subdomains of MKT. These design features and prototype items can serve as a foundation for
future research to build a cohesive set of measures, and ultimately a robust knowledge base,
about secondary mathematical knowledge for teaching.
GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 32
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Table 1
Mathematical Knowledge for Teaching Length, Perimeter, and Area.
Common Content Knowledge Specialized Content Knowledge
Calculate the perimeter and area of shapes given length measurements
Know the affordances and constraints of dif-ferent formulas related to length, perimeter, and area
Demonstrate a conceptual understanding of the relationships between lengths, perimeter, and area, including: the non-constant relationship between pe-
rimeter and area the impact of changes to one-dimensional
attributes on perimeter and area
Demonstrate representational fluency (sym-bolic, tabular, pictoral/graphical and moves between them) in describing the relationships between length, perimeter, and area
Identifying aspects of the relationship be-tween length, perimeter, and area that are im-portant for students to learn
Identify mathematical tasks that support stu-dents understandings of length, perimeter, and area
GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 39
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Table 2
Teachers providing data on the geometry and measurement items.
Subgroup Teachers providing written data Teachers providing interview data
Secondary prospective 9 8
Elementary prospective 3 3
Secondary practicing 10 7
Teacher leaders 3 2
GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 40
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Table 3
Mapping Mathematical Knowledge for Teaching Length, Perimeter, and Area to the Tasks
Common Content Knowledge Data Sources
Calculate the perimeter and area of shapes given length measurements
Fence in the Yard taskMinimizing Perimeter Lesson Plan interview task
Demonstrate a conceptual understanding of the relation-ships between length, perimeter, and area, including: the non-constant relationship between perimeter and
area the impact of changes to one-dimensional attributes on
perimeter and area
Fence in the Yard taskArea of a Parallelogram taskTangrams taskMinimizing Perimeter Lesson Plan interview task
Specialized Content Knowledge Data Sources
Know the affordances and constraints of different formu-las related to length, perimeter, and area Considering Formula Use task
Demonstrate representational fluency (symbolic, tabular, pictoral/graphical and moves between them) in describing the relationships between length, perimeter, and area
Fence in the Yard taskArea of a Parallelogram task
Identifying aspects of the relationship between length, perimeter, and area that are important for students to learn Big Ideas task
Identify mathematical tasks that support students under-standings of length, perimeter, and area
Minimizing Perimeter Lesson Plan interview task
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Table 4
Coding rubric for Tangrams Task.
Question 1 (Area) Question 2 (Perimeter)
Score Point 3: Response is correct Justification uses the concept of area to
make a general argumentExample: They both have the same area be-cause each tile has a fixed area, they are just in a different arrangement
Score Point 3: Response is correct Justification uses the concept of perimeter
to make a general argumentExample: Figure 3 spreads the figure out and exposes more edges of the seven tiles, giving a greater perimeter
Score Point 2: Response is correct Justification uses a form of empirical meas-
urementExample: I broke each figure into triangles and rectangles and calculated the areas
Score Point 2: Response is correct Justification uses a form of empirical meas-
urementExample: I measured the perimeter of both and Figure 3 has the greater perimeter
Score Point 1: Response is correct Justification is based on qualitative observa-
tion or no justification is providedExample: They look like they cover the same space
Score Point 1: Response is correct Justification is based on qualitative observa-
tion or no justification is providedExample: Figure 3 looks like it has more length around the outside
Score Point 0: Incorrect, missing, or vague response
Score Point 0: Incorrect, missing, or vague response
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Table 5
Coding rubric for Area of a Parallelogram Task.
Category Description
Correct-1Correct response (statement is false) Shows at least 2 examples that demonstrate why the statement is false, or a generalization that explains why
Correct-2Correct response (statement is false) Does not provide examples or a generalization that explains why
Incorrect-1Incorrect response (statement is true) Evidence that the teacher thinks there is only one possible parallelogram with the specified base and area
Incorrect-2
Incorrect response (statement is true) No evidence that the teacher thinks there is only one possible parallelo-gram with the specified base and areaOR Correct response, erroneous reason
Vague/Inconclusive Cannot be classified or response is incomplete
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Table 6
Goal categories identifiedBig Ideas task
Procedural Goals Conceptual Goals
Calculate or find area and perimeter (13)Understand perimeter and area conceptually, in-cluding perimeter as distance around, area as cov-ering (8)
Identify the names and characteristics of 2-D shapes (12)
Describe the relationships between length, pe-rimeter, and area (7)
Memorize and use formulas (3) Generate and explain formulas (6)
Find missing sides of a shape given pe-rimeter and/or area (3)
Describe the difference between linear and square units (4)
Differentiate between perimeter and area on a shape (3)
Use or apply perimeter and area (4)
Decompose and manipulate shapes into other shapes (2)
Develop spatial sense (2)
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Tangrams are a special set of 7 geometric tiles shown below in Figure 1. The shapes in Figures 2 and 3 were formed using all the tangram tiles.
a. Which figure, 2 or 3, has the greater area? Justify your answer.
b. Which figure, 2 or 3, has the greater perimeter? Justify your answer.
Figure 1. The Tangrams Task.
!
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True or false: A parallelogram with a base of 6 cm and an area of 24 cm2 will always have the same perimeter. Provide at least one example to support your answer.
Figure 2. Area of a Parallelogram task.
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Response A:
True there is only one such parallelogram that can be formed with these dimensions, so the perimeter will always be the same.
Response B: True
Figure 3. Incorrect solutions to the Area of a Parallelogram task.
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Julie wants to fence in an area in her yard for her dog. After paying for the materials to build her doghouse, she can afford to buy only 36 feet of fencing. She is considering various different shapes for the enclosed area. However, she wants all of her shapes to have 4 sides that are whole number lengths and contain 4 right angles. All 4 sides are to have fencing. What is the largest area that Julie can enclose with 36 feet of fencing? Support your answer by showing the work that would convince Julie that your area is the largest.
(From 1996 NAEP, as cited in Kenney & Lindquist, 2000)
Figure 4. Fence in the Yard task.
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! Response A! Response B
As the numbers/length of fencing gets fur-ther from one another the area decreases. This is the greatest area one can have w/36 ft of fencing. All the sides are the same, thus making it a square. As we discussed in class a square maximizes the area.
...pattern will continue... (posttest response)
l w Perimeter (ft.) Area (ft2)
1 17 36 17
2 16 36 32
3 15 36 45
4 14 36 56
5 13 36 65
6 12 36 72
7 11 36 77
8 10 36 80
9 9 36 81
10 8 36 80
Specifically describes how dimension, perimeter, and area are related in the context of the problem
Does not specifically describe relationship be-tween dimension, perimeter, and area
Figure 5. Responses to Fence in the Yard.
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Your final task is to plan a lesson around this problem. Im going to give you 5 to 8 minutes to write down your ideas about how you might implement a lesson with this problem. Your target mathematical goal will be to get students to understand the relationships between area and perimeter. You are free to modify the problem in any way.
When the teacher indicates they are finished planning the lesson ask, Could you walk me through the lesson you have planned around this task? What do you hope students will learn through engaging in this lesson? Is there anything else you would like to say about this lesson?
Minimizing Perimeter Task (Adapted from Pugalee et al., 2002)
The 7th grade class wants to start a small organic school garden to grow vegetables for the cafeteria. The principal has told the class that they can have a 36 ft2 rectangular area behind the school. The rectangle can be any shape they choose, so long as it is 36 square feet in area.
1. Find the least amount of fencing for a rectangular garden plot that is 36 square feet in area. Organize the information using a table like the one below.
Length (feet) Width (feet) Perimeter (feet) Area (square feet)
2. Use the data in your table to create a graph of perimeter vs. length.
3. The 6th grade decides they also want to start a small garden. The principal gives them 24 ft2 to create their garden in any rectangular shape they choose. Find the least amount of fencing for a rectangular garden plot that is 24 square feet in area. Make a table and graph similar to the ones you created above.
4. When they hear of the success of the middle school gardens, the local high school wants to create a garden of their own. Their principal allows the high school to have 100 ft2. Make a conjecture about the minimum fencing needed for an area of 100 square feet and write a paragraph defending your conjecture.
Figure 6. The Minimizing Perimeter Interview Prompt & Instructional Task.
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There are two common forms that textbooks use for the area of a rectangle:
Area = length width and Area = base height
Is there a difference between the two formulas? If so, describe the difference. Which would you choose to use with students, and why?
Figure 7. The Considering Formula Use Task.
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