Exploring the mathematical knowledge for teaching geometry and measurement through the design and use of rich assessment tasks

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DO NOT CITE OR REPRODUCE WITHOUT AUTHORS PERMISSION

GEOMETRY AND MEASUREMENT ASSESSMENT TASKS 1

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.


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,


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


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


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


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.

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.


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


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


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


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


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.



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.

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.



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


4 Tangram tiles, rulers, and grid paper were available to teachers during the assessment.

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.


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,


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)


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


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.


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.


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


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). .


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.


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.


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


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.


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.


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.


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.

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


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


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.


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


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.


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


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.


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


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, &


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


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.


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


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.


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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


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


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


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


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


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)


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.

!


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.


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.


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.


! 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.


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.


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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