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RUNNING HEAD: MAKING SENSE ANGLES
1
Making Sense by Measuring Arcs: A Teaching Experiment in Angle Measure
Kevin C. Moore
University of Georgia
Moore, K. C. (2013). Making sense by measuring arcs: A teaching experiment in
angle measure. Educational Studies in Mathematics, 83(2), 225-245.
Available at: http://link.springer.com/article/10.1007%2Fs10649-012-9450-6
DOI 10.1007/s10649-012-9450-6
© 2012 Springer Science+Business Media Dordrecht 2012
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
2
Making sense by measuring arcs: A teaching
experiment in angle measure
Kevin C. Moore
Department of Mathematics and Science Education, University of Georgia, 105
Aderhold Hall, Athens, GA 30602
Phone: 706.542.3211
Fax: 706.542.4551
kvcmoore@uga.edu
Abstract: I discuss a teaching experiment that sought to characterize precalculus
students’ angle measure understandings. The study’s findings indicate that the
students initially conceived angle measures in terms of geometric objects. As the
study progressed, the students formed more robust understandings of degree and
radian measures by constructing an arc length image of angle measures; the
students’ quantification of angle measure entailed measuring arcs and conceiving
multiplicative relationships between a subtended arc, a circle’s circumference, and
a circle’s radius. The students leveraged these quantitative relationships to
transition between units with a fixed magnitude (e.g., an arc length’s measure in
feet) and various angle measure units, while maintaining invariant meanings for
angle measures in different units. These results suggest that quantifying angle
measure, regardless of unit, through processes that involve measuring arc lengths
can support coherent angle measure understandings.
Keywords: Angle Measure; Teaching Experiment; Quantitative Reasoning;
Student Thinking; Trigonometry; Multiplicative Reasoning
1. Introduction
Angle measure and trigonometric functions have been high school and
undergraduate mathematics topics for well over a century. In addition to their
deep-rooted place in mathematics, both topics are frequently used in physics,
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
3
engineering, and applied mathematics. The results of several research studies have
indicated, however, that students and teachers have difficulty understanding and
reasoning about trigonometric functions (Brown, 2005; Fi, 2003; Thompson,
Carlson, & Silverman, 2007; Weber, 2005). Other studies have reported that
teachers’ and students’ fragmented understandings of angle measure lead to
disconnected understandings of trigonometric functions (Akkoc, 2008; Topçu,
Kertil, Akkoc, Kamil, & Osman, 2006). Complicating the issue, few research
studies have addressed how students come to understand angle measure in the
particular ways in which they do.
The present study examines two undergraduate precalculus students’ evolving
angle measure understandings
1
as they participated in a teaching experiment
(Steffe & Thompson, 2000). Thompson’s (1990) theory of quantitative reasoning,
research on the teaching and learning of trigonometry and angle measure, and the
historical roots of angle measure inform the study. In the present work, I develop
hypotheses of the students’ angle measure understandings with a focus on the
students’ quantification of angle measure. I also discuss the students’ angle
measure understandings in terms of several ideas of quantitative reasoning.
2. Research Literature on Angle Measure
Previous research on students’ angle measure understandings has mainly
focused on elementary students’ angle conceptions (Clements & Burns, 2000;
Kieran, 1986; Mitchelmore & White, 2000). These studies have identified the
important role of physical experience in students’ construction of angle
(Mitchelmore & White, 2000), including their ability to coordinate turn as a
physical action and turn as a number (Clements & Burns, 2000; Kieran, 1986).
Such findings offer valuable insights into elementary students’ development of
angle conceptions, but leave much to uncover about students’ angle measure
concept past the elementary level.
Research literature on trigonometric functions (Akkoc, 2008; Brown, 2005; Fi,
2003; Thompson et al., 2007; Topçu et al., 2006; Weber, 2005) also generates a
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
4
need to better understand students’ angle measure conceptions. Two studies
(Akkoc, 2008; Topçu et al., 2006) characterized Turkey pre-service and in-service
mathematics teachers as holding understandings of radian angle measures
dominated by degree measure; when given radian measures, the teachers
converted these measures into a number of degrees in order to attribute a meaning
to the measures.
2
Not one of the four teachers interviewed by Topçu and
colleagues defined radian measures as a ratio of lengths. As Akkoc (2008)
reported, and compatible with previous findings (Fi, 2003), pre-service teachers
also claimed that radian measures are only given in terms of π, leading teachers to
interpret 30 as a number of degrees in expressions such as sin(30). In light of his
findings, Akkoc suggested that impoverished radian angle measure
understandings likely contribute to teacher and student difficulties in
trigonometry.
Collectively, the aforementioned research has revealed that students and
teachers construct shallow and fragmented angle measure understandings that
inhibit their ability to construct flexible trigonometric function understandings. Of
particular relevance to the present study, the research suggests that teachers’ and
students’ angle measure understanding often lack meaningful connections to arcs.
These observations illustrate a need to gain better insight into students’ ways of
thinking about angle measure that support connected
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angle measure
understandings.
3. An Arc Approach to Angle Measure
United States (US) and international textbooks provide many different
characterizations of angle measure. In a survey of thirty elementary and secondary
textbooks, I found angle measures defined as an amount of a rotation, a number
1
I use the term understanding to refer to a students’ system of schemes and
conceptual operations.
2
These studies did not investigate the teachers’ degree measure understandings.
3
For the purpose of this study I define connected angle measure understandings
as meanings that can be used to interpret angle measures in consistent ways
regardless of the angle measure unit. Connected understandings of angle measure
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
5
obtained by using a protractor, and a measurement of an arc length, to name a
few. The roots of each definition can be linked to various points in the historical
development of angle measure. Matos (1990) and Bressoud (2010) have provided
summaries of the history of trigonometry and the history of angle, respectively,
while highlighting critical angle measure developments.
Of relevance to the present article, Matos’s (1990) and Bressoud’s (2010)
historical accounts illustrate that the majority of angle measure developments
occurred in the context of measuring arcs. For instance, the Babylonians measured
the circular movement of celestial bodies using 360 whole parts for the sky
(Matos, 1990). Despite the rich historical focus on measuring arcs, an educator
can be, depending on her or his country, hard pressed to find curricula that relate
degree measure to the length of an arc. A textbook might use an arc to label an
angle and its measure, but it is not necessarily the case that degree angle measure
is presented as a process of partitioning an arc using a specified unit length.
Consider an approach to angle measure that defines degrees as the standard
unit of angle measure and asks students to use a protractor to assign degree
measures to angles. The approach then defines right angles, supplementary angles,
obtuse angles, etc., and students execute calculations using angle measures and
these definitions. Such an approach, which is customary in the US, might
facilitate relating angle measures through calculations, but it fails to address the
quantitative structure behind the process of determining an angle’s measure. In
contrast to such an approach, radian angle measures are typically introduced in
terms of measuring an intersected or subtended arc using a circle’s radius as a
unit; the measure is defined in terms of a quantitative relationship with an explicit
connection to a subtended arc. Such a definition for radian measures
fundamentally differs from more common approaches to degree angle measure.
One approach that avoids a divide in angle measure meanings is to develop
angle measures, regardless of unit, as representative of the same quantitative
relationship. This can be accomplished by conceiving angle measure as the
process of determining the fractional amount of a circle’s circumference
also enable students to flexibly convert between units of angle measure while
maintaining these common meanings.
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
6
subtended by an angle, provided that the circle is centered at the vertex of the
angle (Thompson, 2008). An angle that measures one degree subtends 1/360 of
the circumference of any circle centered at the vertex of the angle and an angle
that measures one radian subtends 1/2π of the circumference of any circle
centered at the vertex of the angle; radians and degrees measure the same quantity
and are thus scaled versions of one another (Thompson, 2008).
With angle measure understandings that foreground relationships between
quantities, students should interpret an arc used to denote an angle measure as
more than a label; they should understand the arc as denoting an equivalence class
involving arc lengths (including the given arc length) and circles. For instance, an
angle measure of 2 radians conveys that the angle subtends 2 radii on all circles
centered at the vertex, and that same angle subtends 2/2π of each circle’s
circumference (Figure 1). A benefit of radian angle measures is that once a circle
is specified, the unit magnitude is more explicit than the magnitude associated
with one degree. Also, by understanding radian measures as equivalence classes
grounded in measuring in radii, a foundation is in place for students to understand
the values associated with the unit circle in terms of equivalence classes. If
students are to connect the unit circle–an arbitrary circle of radius 1–to all circles
and radian angle measures, it is important that they connect the unit circle to
measuring in radii (Moore, LaForest, & Kim, 2012).
Fig. 1 An arc length image of angle measure that involves equivalence of arcs
4. Theoretical Perspective
Principles of quantitative reasoning (Smith III & Thompson, 2008; Thompson,
1990), which call attention to the importance of students constructing quantities
and relationships between quantities when learning mathematics, provide a
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
7
foundation for the present investigation into student thinking. Quantitative
reasoning involves an individual’s mental actions when conceiving of a situation,
constructing measurable attributes of the situation (which are called quantities)
and constructing about relationships between conceived quantities. These mental
actions create a (quantitative) structure that forms a foundation for constructing
mathematical understandings.
A quantity is defined as a conceived attribute of something that admits a
measurement process (Thompson, 1990). Relative to the focus of this research,
two intersecting rays in relation to each other form an object (the something) that
has a measurable attribute of openness. Quantifying the openness of an angle
entails conceiving a quantitative relationship–specifically, a multiplicative
relationship–between a subtended arc and a unit length (e.g., the radius). Also, as
described above, quantifying angle measure involves coming to understand a unit
in terms of a multiplicative relationship between a class of subtended arcs and the
corresponding circles’ circumferences.
I adopt Thompson’s (1990) definition of value to refer to a number that
reflects the result of a measurement process. I use number to refer to a number
that does not reflect the result of a measurement process.
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As an example of the
difference between a number and a value, consider a student solving a problem in
which they use a ratio between an arc length that subtends an angle and the
circumference of the corresponding circle. When calculating the ratio, the student
might conceive the ratio only as a call to execute a calculation (e.g., ratio as a
number), or the student might interpret the ratio as representing a measure of the
fractional amount of the circle’s circumference that subtends the angle (e.g., ratio
as a value). In the case of ratio as a value, the student’s subsequent actions might
stem from reasoning about the ratio as a value (e.g., I multiply the ratio by the
total number of angle measure units in a circle to determine the angle measure that
is the equivalent fractional amount of the total angle measure units).
4
Values and numbers can be specified or unspecified. For instance, an individual
can anticipate making or determining a measure and consider the meaning of this
anticipated result independent of a specified value.
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
8
It is important to note that a quantity is a cognitive object. Quantities can and
will differ from individual to individual; an individual’s quantification of an
angle’s openness may or may not include a subtended arc as inherent to the
measurement process. As Thompson described, quantifying a quantity “is a
process of settling what it means to measure a quantity, what one measures to do
so, and what a measure means after getting one” (2011, p. 38).
5. Methodology of the Study
One purpose of education research is building models of students’
mathematics that provide a viable explanation of the schemes and operations
driving the students’ behaviors (Steffe & Thompson, 2000). I conducted the
present study in order to develop such models in the context of the students’
quantification of angle measure over the course of a teaching experiment (Steffe
& Thompson, 2000).
5.1 Subjects and Setting
Three full-time students from an undergraduate precalculus course at a large
public university in the southwest US participated in the teaching experiment. The
three students were chosen out of necessity, as they were the only volunteers from
the class who had schedules that aligned with my schedule. Two students (Judy
and Zac) are the focus of the present study.
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Judy was a female in her mid-
twenties and a biochemistry student. Zac was a male in his early twenties and an
ethnomusicology/audio technology major. Judy received an ‘A’ for her final
course grade and Zac received a ‘B’ for his final course grade. The students were
monetarily compensated for their time. I chose precalculus students because
precalculus is typically the course that introduces trigonometric functions and
radian angle measure.
The aforementioned precalculus classroom was part of a design research study
informed by literature on covariational reasoning (Carlson, Jacobs, Coe, Larsen,
& Hsu, 2002) and quantitative reasoning. An observer to the teaching experiment
5
Judy and Zac are pseudonyms.
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
9
and I served as teaching assistants in the course. Course topics included quantity,
rate of change, linear functions, function, exponential functions, and rational
functions. Angle measure and trigonometric functions formed the last module of
the course. The participants did not attend the angle measure and trigonometry
class sessions, instead participating in the teaching experiment.
Students who take precalculus at the students’ university (and peer
institutions) represent a wide range of majors, and a majority of these students are
not mathematics majors. Research with undergraduate students in precalculus
suggests that they encounter similar difficulties as their secondary level
counterparts when reasoning about quantities and relationships between quantities
(Moore & Carlson, 2012). Also, undergraduate precalculus students’ performance
on a research-based precalculus assessment (Carlson, Oehrtman, & Engelke,
2010) is comparable to that of their secondary school counterparts. Thus, although
the participants in the present study represent a convenience sample,
undergraduate and secondary precalculus populations are not dissimilar and the
students were not atypical relative to their peers.
5.2 Data Collection and Analysis Methods
The angle measure portion of the teaching experiment involved two 75-minute
teaching sessions occurring within a span of four days. Each teaching session
included Judy, Zac, an observer, a third student, and myself. In an attempt to gain
insights into the students’ angle measure conceptions upon entering the study, I
conducted pre-interviews with each student that followed the design of a clinical
interview (Clement, 2000) and Goldin’s (2000) principles of structured, task-
based interviews. The pre-interviews occurred one week prior to the first teaching
session.
Following two teaching sessions on trigonometric functions
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, the teaching
experiment included a two-hour researcher-student teaching session with each
student. The one-on-one sessions occurred two weeks after the second teaching
session (Figure 2) and followed the teaching experiment principles. I use teaching
sessions and interview sessions to distinguish between the sessions that included
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
10
more than one student and the sessions that included only one student,
respectively.
Fig. 2 The progression of the teaching experiment, with solid frame boxes
indicating the focus of the present work
Teaching experiments (Steffe & Thompson, 2000) adhere to the stance that
students are in a constant mode of construction, a stance that is in line with the
principles of quantitative reasoning. During a teaching experiment, a researcher
aims to build viable models of students’ mathematical understandings and
document shifts in these understandings. These models may become more precise
over time, but the models are never to be interpreted as one-to-one representations
of the students’ thinking. The researcher’s mathematical understandings, the
perspective that the researcher uses during the study (e.g., quantitative reasoning),
and the researcher’s learning goals for the students shape his models (Steffe &
Thompson, 2000). Relative to the present study, I was involved (both in teaching
and design) in the aforementioned precalculus design research study. Hence, I was
aware the materials addressed research-established student difficulties and needs
for calculus. My previous teaching experiences and understanding of the project
goals thus influenced the design of the study and my models. For instance, I was
aware that students often face difficulty conceiving calculations in terms of
quantitative relationships, and many of my decisions during the study revolved
around determining students’ meanings for calculations.
All sessions during the teaching experiment were videotaped and I digitized
all student work. The sessions included two cameras, one for the students’ table
and one for work produced on a whiteboard, and a computer feed capture.
6
The results of these teaching sessions are part of a manuscript in preparation.
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
11
Following the teaching experiment methodology (Steffe & Thompson, 2000), data
analysis included both ongoing and retrospective elements that included
documenting decisions during the teaching experiment and revisiting these
decisions during a retrospective analysis of the data. Upon the completion of data
collection, a retrospective analysis of the data occurred. After transcribing all
sessions, I conducted an open coding (Strauss & Corbin, 1998) of the data that
involved identifying episodes that offered insights into the students’ thinking.
Consistent with Thompson’s (2008) description of a conceptual analysis, I
analyzed the identified instances in an attempt to create hypotheses of the
students’ schemes and operations of thought. I tested these tentative models by
searching and analyzing the data for evidence that either contradicted or supported
the generated models. Such analyses led to modifications of these hypothesized
models of each student’s mathematics, as well as documenting shifts in the
students’ thinking. Additionally, the students’ ways of thinking were compared
and contrasted.
6. Results
I first provide an overview of the pre-interview findings. I then characterize
the students’ quantification of angle measure during the teaching sessions. To
further illustrate the outcomes of the students’ quantification of angle measure, I
summarize their activity during the post-teaching interview sessions.
6.1 Pre-interviews
Initially, I asked the students to describe their meanings for various angle
measures (e.g., 1, 34, and 90 degrees). The students predominantly described
angle measures as properties of familiar geometric objects (e.g., a line has 180
degrees, two perpendicular segments have 90 degrees, and a circle has 360
degrees) and had difficulty describing angle measures that did not correspond to
such geometric objects (e.g., angle measures of 1 degree). The students often drew
a circle or an arc when measures did not relate to a familiar geometric object, but
the students treated arcs as labels and not as measurable attributes (e.g., a
subtended arc and a circumference). To illustrate, when tasked with describing
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
12
one degree, Judy claimed, “Wow…I really don’t know what an angle is outside of
formulas,” and explained that the angle is supplementary to an angle of measure
179 degrees. When asked to provide another meaning for one degree, she stated
that she could only think about it in terms of a line having a measure of 180
degrees.
I also asked the students to measure an angle using a compass, waxed string,
and a ruler (Figure 3). Both students constructed a circle centered at the vertex of
the angle, but neither student completed the task. After drawing a circle, Zac
stated, “they have…a [protractor] that’s already designed out, shows you where
all the angles are.” Yet, he did not relate the protractor to the circle. Judy
determined the circumference of the circle, the arc length subtended by the angle,
and divided the arc length by the circumference (obtaining 0.087). Yet, she was
unable to provide a meaning for her calculation or solve the problem. To Judy,
0.087 was a number; the calculation did not reflect a multiplicative comparison
between the arc length and circumference.
Fig. 3 Measuring an angle with tools
To conclude the interview, I tasked the students with The Traversed Arc
Problem (Figure 4), which provided specified values for an arc length and radius.
Despite their difficulties during the previous problems, both students obtained a
correct solution by using an equation with two ratios (e.g.,
32 / (102
π
)
( )
= x / 360
( )
or
90 / 80.1
( )
= x / 32
( )
).
Fig. 4 The Traversed Arc Problem
Measure the openness of the following angle. You may use a compass, a
waxed string, a ruler, and a calculator.
An individual is riding a Ferris wheel that has a radius of 51 feet. On part of a
trip around the Ferris wheel, the individual covers an arc-length of 32 feet. How
many degrees did the individual rotate?
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
13
The students justified their solutions by comparing the units of each value and
using the calculational
7
strategy of cross-multiplication. As an example, after Zac
determined the arc length (80.1 feet) that corresponded to 90 degrees, he
described that the equation
90 / 80.1
( )
= x / 32
( )
is appropriate for solving a
problem in which there are three known numbers and an unknown number
(Excerpt 1).
Excerpt 1
1
2
3
4
5
6
7
8
9
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Zac: Well it's just, if you're given three variables and you just need one
more. Well, you, uh, 'cause you're given degrees and feet and
degrees and feet…And it just, it gives you three of the four you
need. It's a very easy equation to find a fourth…
Int: OK, and so, and how do you know how, which way to set up the
proportion?
Zac: Well you could do it either way. I could do eighty point one over
ninety is thirty two, as long as the top's are both the same unit, their
both degrees, and these are both feet (writing units by the
measurements).
I asked Zac to provide a meaning of the ratio 90/80.1 independent of the
calculations he used to solve the problem, but he only focused on the calculations
he performed. Zac conceived 90/80.1 as a number. As an alternative conception,
90/80.1 can be thought of as representing the number of degrees per foot of arc
length for that circle.
I took the students’ actions during the pre-interview to imply that they had not
quantified angle measure in a way that entailed quantitative relationships. Instead,
they predominantly conceived angle measures as intrinsic properties of geometric
objects or in terms of calculations with corresponding angle measures (e.g.,
7
My use of the term calculational is consistent with the notion of a calculational
orientation (Thompson, Philipp, Thompson, & Boyd, 1994), which describes an
orientation towards identifying procedures, executing calculations, and working
with numbers/expressions devoid of contextual reference.
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
14
supplementary angles).
8
The students did draw arcs, but their arcs were not a
component of a measurement scheme. I also note that neither student mentioned
the radian (or radius) as a unit of angle measure during the pre-interview. This
observation aligns with research (Akkoc, 2008; Brown, 2005; Fi, 2003; Topçu et
al., 2006) that has identified degree angle measure as dominating individuals’
angle measure conceptions.
6.2 Connecting Angle Measure and Arc Length
I designed the first teaching session to challenge the students to determine a
process to create a protractor (The Protractor Problem, Figure 5), with each
student using a different sized blank protractor. I expected the students’ angle
measure understandings, as inferred from the pre-interview, would not support
solving the problem. Thus, I hoped the students would face a perturbation that
caused them to consider what one quantifies when measuring an angle.
Task 1: Using the supplies of a waxed string, a ruler, a blank protractor and a
calculator, create a protractor that measures an angle in a number of gips, where
8 gips rotate a circle
9
.
Task 2: Using the supplies of a waxed string, a ruler, a blank protractor, and a
calculator, create a protractor that measures an angle in a number of quips,
where 15 quips rotate a circle.
Fig. 5 The Protractor Problem
When working on Task 1, Judy first used the waxed string to measure the
circumference of the protractor.
10
The students quickly discarded the measure and
relied on folding the protractor to create the equal areas (Excerpt 2).
Excerpt 2
8
I note that Zac did refer to a vague area or space between two rays during the
interview, but this only occurred once.
9
I chose the word circle to be intentionally vague. I did not wish to point the
students directly to a measurable attribute of the circle (e.g., area or
circumference) in the hopes of determining which imagery was more natural for
the students.
10
To improve readability, I refer to the perimeter of the curved part of a protractor
– a half circle – as the circumference of the protractor.
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
15
1
2
3
4
5
6
7
8
9
10
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Zac: Well, I already figured out what two gips is by just dividing it in
half.
Int: So dividing what in half?
Zac: The protractor. I just drew a line down the middle (waving hand
over the protractor) and that gives me two gips. And then I just need
to figure out how to find…(pause)
Int: So how’d you know how to draw the line?
Zac: Uh, I figure out that the diameter is four inches, and just found out
where two inches is, marked it (referring to the midpoint of the
diameter), and found my best two inches this way (waving pen tip
from the bottom to the top of the protractor) and drew it up.
Despite Judy’s initial act of determining the circumference, Zac did not
attempt to measure or discuss the circumference. Instead, Zac reasoned about
dividing the entire protractor into equal pieces (e.g., areas). After creating the two-
gip mark, both students divided the protractor into four equal areas by folding the
protractor. Believing the students were reasoning about areas, I modified Task 1
so that a folding method does not produce integer measures (Task 2). The students
immediately claimed they needed to determine 7.5 equal areas and started folding
the protractor. They then realized that folding the protractor into successive halves
did not yield 7.5 equal areas and explained that they could only approximate the
partitioning of equal areas.
To support a shift in the students’ thinking, I asked them for a second method
to decide, as precisely as possible, the placement of the quip marks with the
stipulation that they could not fold the protractor. Zac grabbed the waxed string
and claimed, “Measure out the whole thing…measure out the perimeter.” Zac and
Judy used the waxed string to measure the circumferences of their protractors, and
then divided their circumferences by the total number of quips (e.g., 7.5 for the
half-circle or 15 for the entire circle). They explained that the value represented
the arc length corresponding to one quip on their circle (approximately 1.885
cm/quip and 1.6336 cm/quip); their calculations emerged from reasoning about
partitioning the circumference of the protractor into equal arc lengths (e.g., ratio
as a value). This moment marked the first instance of the students generating and
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
16
relating measurable arc lengths to angle measures, which appeared to be fostered
by facing a situation in which their area image of angle measure did not support
solving the task.
To continue exploring relationships between angle measure and arc lengths, I
drew the students’ attention to a diagram (Figure 6) of their two protractors. Just
prior to drawing the diagram, the students observed that the arc length per unit of
angle measure varied between their protractors, yet there was something “the
same” about the arc lengths. In an attempt build on their observation, I posed
ratios comparing the unit arc lengths to the total circumference of the
corresponding circle (1.885/28.2744 and 1.63/24.5044), as well as the ratio of
1/15.
Fig. 6 Two protractors of different size (figure not to scale)
After the students calculated each ratio (approximately 0.067), I emphasized
that I wanted a meaning for the ratios in terms of the quantities of the situation
and not just the numerical value. Zac claimed, “It’s just taking the full
circumference and then a fifteenth of a full circumference…It’s the exact same.
You’re taking one-fifteenth of the full circumference and dividing it by the full
circumference.” Zac conceived each ratio as a value representing an arc length’s
fraction of a circle’s circumference, which supported him in identifying that the
angle cut off an equivalent fraction of each circle’s circumference. The students
then generalized this relationship to conclude that quip measures convey a
fractional amount of a circle’s circumference cut off by the angle that holds for all
circles centered at the vertex of the angle.
3.9 inches
4.5 inches
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6.3 Extending to Degree Measure
The Protractor Problem occurred over 45 minutes, a period during which the
students conceived of an arc length as a measurable attribute related to angle
measure in two ways. First, they reasoned about an arc length per unit of angle
measure that can be iterated along the circumference a circle. Second, they
conceived angle measures in quips (and gips) in terms of the fractional amount of
any circle’s circumference cut off by the corresponding angle.
I hypothesized that their meanings for degree angle measure shift accordingly,
and thus asked the students to respond to a question from the pre-interview:
“What does it mean for an angle to have an openness of one degree?” Zac
claimed, “[The angle cuts off] one three-hundred and sixtieth of a circle’s
circumference.” Judy further described that an angle of 10 degrees subtends
10/360
ths
of a circle’s circumference and she used a calculator to determine that
the subtended arc is approximately 2.7% of the circumference for any circle
centered at the vertex of the angle.
I took the students’ actions to imply they had quantified angle measure such
that they had an invariant meaning for different units. Specifically, they
interpreted angle measures, regardless of unit, as a multiplicative comparison
between an arc length and a circle’s circumference that is not dependent on the
size of the circle used to make this comparison. I hypothesized that a method for
converting between angle measures might emerge as a consequence of giving an
invariant meaning to angle measures of different units. In an attempt to test this
conjecture, I asked them to determine the angle measure in quips that is equivalent
to 10 degrees.
Judy suggested using the equation
10 / 360
( )
= x / 15
( )
to solve for the number
of quips that are equivalent to 10 degrees, and the students claimed that each ratio
represented an equivalent percentage (2.7%) of the total number of angle measure
units. Recall that Judy and Zac used a similar equation and calculations to
determine an angle measure during the pre-interview, but their explanations
conveyed that their solution stemmed from a calculational strategy (Excerpt 1);
the ratios did not represent the measure of a quantity formed by making a
multiplicative comparison between two other quantities. In the present case, their
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ratios represented an equivalent percentage (e.g., a multiplicative relationship) of
the total number of angle measure units (e.g., ratio as value).
To conclude the first teaching session, I implemented The Protractor Applet
(Figure 7). I designed the applet so that the user can increase/decrease the radius
of the circle or the angle’s openness, with the displayed measures varying
correspondingly. I used the applet to formalize the prior outcomes and continue
promoting a focus on expressions and calculations as representative of quantities’
values.
Fig. 7 The Protractor Applet
As I posed either a varying radius or a varying openness of the angle, both
students correctly predicted how the measures change or stay constant. For
instance, Zac predicted that for a decreasing radius, “The arc length of A B and the
circumference will get smaller, but the angle [measure] and percent will stay the
same.” It is important to note that the students’ ability to correctly predict how the
measures change reveal a capacity to reason about indeterminate values. That is,
they were able to imagine the measures of the various quantities varying and
justify these variations using their angle measure understandings without
reasoning about specified values and performing calculations to determine these
comparisons. The students also spoke with extreme precision when describing the
values on the applet. As opposed to referencing “the circle” or using ambiguous
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referents (e.g., it), the students explicitly mentioned “the circumference” and “the
arc length” and discussed the beginning and ending points for all identified
lengths. Although using such specific terms might seem of little consequence,
previous research (Moore & Carlson, 2012) has revealed that precalculus students
frequently reason about lengths in ambiguous ways and thus don’t construct
precise quantitative structures of problem situations that support products (e.g.,
calculations, formulas, and graphs) correctly relating quantities’ values.
6.4 Creating Circles and Measuring in Radii
During the second teaching session, I transitioned the instruction to radian
angle measure. The Circumference Problem (Figure 8) prompted the students to
create a circle with a radius having the length of a piece of waxed string and then
measure various string (radii) lengths along the circle’s circumference. During the
task, and in line with their actions during The Protractor Problem, I intended that
the students develop a scheme for radian angle measure involving partitioning an
arc length into a number of unit lengths (e.g., radii).
Create a circle that has a radius the length of your waxed string. Then,
approximate how many string lengths mark off the circumference of your
circle. Create an angle that cuts off an arc of approximately one string length
and 1.5 string lengths. Compare your results with those of your classmates.
Fig. 8 The Circumference Problem
I intended that The Circumference Problem support the students in leveraging
several facets of their reasoning from the first teaching session. First, I hoped to
engage the students in measuring along a circle’s circumference to partition the
circumference into equal lengths. Second, a major outcome of the first teaching
session was that an angle measure conveys the fractional amount of a circle’s
circumference cut off by the angle that is not dependent on the size of the circle.
Building off of this outcome, Judy’s and Zac’s strings were of different lengths
with the hope that the students realize the size of the circle used to create an angle
subtending a particular number of radii is inconsequential.
At the outset of The Circumference Problem, Zac recalled the formula relating
the circumference and radius of a circle (C = 2πr), yet neither student was able to
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clarify how the task relates to this formula. After creating circles and using the
waxed strings to determine that approximately the same number of string lengths
(2π) marked off the circles’ circumferences, they conjectured that if an individual
measures an equivalent number of string lengths along her circumference, then the
resulting arc lengths correspond to angles with an equivalent amount of openness.
Following their observation, I asked the students to discuss using a unit of angle
measure based on a circle’s radius (Excerpt 3).
Excerpt 3
1
2
3
4
5
6
7
8
9
10
Zac: Is it a useful unit of measure, well yes or no, what do you think? We
could use quips, but is it useful?
Judy: I don’t know.
Zac: It simplifies the circle, you know, the circumference of the circle is
equal to two pi r. The radius is the unit, not inches or centimeters or
anything like that.
Judy: Oh, you mean, oh! OK. So you mean, OK. I think I’m thinking in
terms of the length of the radius.
Zac: We’re using it as an actual unit… One radius, and then six point two
eight radius, or radians.
While Zac appeared to conceive the radius as a unit that “simplified a circle"
to a radius of “one radius” and a circumference of “six point two eight radius,”
Judy’s response to Zac suggests that she was initially considering measuring the
length of the radius (lines 7-8) and not considering the length of the radius as a
unit to measure other lengths. As the discussion continued, Judy claimed, “OK,
yeah, in that sense, you don’t have to bother with length numbers,” using “length
numbers” to refer to measures of the radius in a unit other than radii.
I returned the discussion to the number of 6.28. The students’ responses
suggest that both were considering the radius as a unit of measure that could be
used for any circle (Excerpt 4).
Excerpt 4
1
2
Int.: So what measurement does that represent, six point two eight?
Zac: That’s always the circumference on every circle.
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3
4
Int.: That’s always the circumference measured in what units?
Judy: Radiuses.
At this point in the lesson, we defined the term radian
11
as a measure that
stems from the process of measuring in radii and subsequently asked the students
to describe the meaning of various radian angle measures, including 1.5, 3.5, and
π radians. In each case, the students described the measures in terms of an angle
subtending an arc of that number of radii. Zac explained that an angle with a
measure of 1.5 radians subtends “one and a half radius-es,” regardless of the circle
used to measure the angle. As another example, Judy stated that multiplying π
radians by the length of a circle’s radius determines the arc length subtended by
an angle that measures π radians, “Because π is just an amount of, um, radiuses
along the circumference. So you would just need to multiply by the radius
length.”
6.5 Post-interviews
By the completion of the second teaching session, the students had exhibited
multiple ways of thinking about angle measure that stemmed from measuring
(e.g., an act of partitioning) along arc lengths in various units (e.g., radii, degrees,
and conventional length units) and coordinating these measures. Of particular
importance, the students reasoned about (a) multiplicative comparisons between
arc lengths and circles’ circumferences and (b) multiplicative comparisons
between arc lengths and circles’ radii. The teaching sessions did not fully explore
connections between these ways of thinking and I questioned if the students
would continue to use such reasoning after the teaching sessions. I conducted an
interview session with each student two weeks after the second teaching session to
gain deeper insights into their thinking.
I first tasked Zac with The Arc Length Problem (Figure 9). Zac explained his
solution previous to executing calculations (Excerpt 5).
11
The students used the terms radians and radii (or radius) interchangeably when
referencing any measure that involved measuring relative to a circle’s radius, even
in the arbitrary case.
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Given that the following angle measurement
θ
is 35 degrees, determine the
length of each arc cut off by the angle. Consider the circles to have radius
lengths of 2 inches, 2.4 inches, and 2.9 inches (figure not to scale).
Fig. 9 The Arc Length Problem
Excerpt 5
1
2
3
4
5
6
7
8
Zac: So what I plan on doing for this one is converting thirty-five degrees
into radians. And a very easy way of doing that is putting thirty-five
over three sixty is equal to x over two pi (writing corresponding
equation)…And then with that all I have do is just multiply the
answer (pointing to x) by two inches, two point four inches, and two
point nine inches (pointing to each value in the problem statement)
to get the different arc lengths (identifying each arc length with his
pen tip) right there, because radians is just a percentage of a radius.
Stemming from understanding radian measures as a multiplicative relationship
between a set of arcs and corresponding radius lengths, Zac anticipated using the
radian measure to determine each arc length (lines 4-8). That is, his understanding
of radian angle measures was such that he was able to give meaning to
calculations involving these measures without executing the calculations.
Likewise, Zac anticipated converting the angle measure by conceiving angle
measures as a fractional amount of a circle’s circumference. Zac explained, “Well
what you're doing is just technically finding a percentage. Like thirty-five over
three sixty is (using calculator), is nine point seven percent of the full
circumference.”
θ
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Differing from Zac, Judy did not first convert the angle measures to a number
of radians. Instead, Judy used the equation
35 / 360
( )
= x / (2
π
r)
( )
for each radius.
Judy claimed, “if you have your first circle and you increase the radius then even
though the percentage of the entire circle is the same, you have to compensate
with a larger arc length in inches.” Compatible with Zac’s conversion method,
Judy’s solution came as a consequence of understanding angle measures as
conveying a fractional amount of a circle’s circumference.
At various points in the interviews, I also asked the students to describe radian
angle measures. When explaining the meaning of an angle measure of 0.61
radians, Zac fluently transitioned between reasoning about a radian angle measure
as a multiplicative relationship between a subtended arc and a radius length and as
a multiplicative relationship between a subtended arc and the circumference of a
circle. Moreover, understanding angle measures as a fraction of a circle’s
circumference subtended by the angle enabled Zac to give an invariant meaning to
degree and radian angle measures (Excerpt 6).
Excerpt 6
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Zac: That means that this arc length right here (tracing the arc length on
the smallest circle) is point six one, or sixty one percent of the
radius.
Int: OK. Now (pause), and then so why times…with the point six one
(pointing to 0.61(x), the calculation Zac used to solve the problem).
Zac: Well because I couldn't just leave it as radians, so I have to get it in
inches, or you know, an actual distance measure. So knowing that
it's sixty one percent, or it's equal to sixty one percent of the radius,
all I have to do is just multiply it by the radius and I know what it
is.
Int: OK, so one last question. So we have all these different lengthed
arcs right (tracing the three arcs)…but yet that angle measurement
doesn't change?
Zac: Well, because it's always the same percent of the circle it's cutting
out for each different circle…
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16
17
18
19
20
21
22
23
Int: OK…so what's the same percent there you said?
Zac: The, well, the degrees or radians is showing a chunk of the circle
being cut out, and that's a certain percent of the circle being cut out.
It's always the same no matter what, as long as you're using that
same degree or radian length, then you're always going to have that
same amount, or same percent of the circle, or circumference being
cutout no matter what size the radius is, or the circumference of the
circle is (making circular motion with hand).
Like Zac, Judy also reasoned about measures in radians as conveying a
multiplicative relationship between a subtended arc and the radius. Specifically,
she described that she thought of radian measures as a “function,” meaning that
the measures described a relationship between two quantities that holds for all
circles. In Excerpt 7, she describes measures of 5.27 and 1.2π radians in the
context of a circle with a radius of 3.5 inches (Figure 10).
Excerpt 7
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Judy: It means that if you travel from your starting point here, and you
travel counterclockwise (tracing arc length), you travel five point
two seven times your radius length along the circumference
(moving her pen tip in the shape of a circle).
Int: OK, what about the one point two pi radians? What's that mean…
Judy: Um, I guess one point two pi times the radius length, which is,
(using calculator) um, about three point seven seven radius lengths.
Int: OK, so what role does pi play in that?
Judy: Just a number. Oh, and then the arc length is, um, I just multiplied
the radians times the radius length (3.5 inches) to get the arc length,
which is eighteen point four four five.
Int: So why does that operation work, why does that give you the arc
length?
Judy: Um, because, uh, I kind of look at the radians like a function
almost, so I always look at it as five point two seven times whatever
the radius length is, is the linear arc length 'cause um, that's how it
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17
18
translates to any other circle you use. So if it was a larger one where
the radius was five, then I'd multiply it by five instead.
I note that Judy’s description suggests that she imagined traveling along an
arc. Zac also exhibited behaviors throughout his interview suggestive of this
imagery. I conjecture that their image of traveling along an arc stemmed from
repeatedly measuring along a subtended arc length during previous tasks.
Fig. 10 Judy’s diagram for angle measures
As the study progressed there was a noticeable shift in the students’ thinking
from reasoning about numbers and calculations to reasoning about quantitative
relationships independent of specified values. During the teaching sessions, we
had not a focused on developing formulas, but I expected that their thinking about
quantitative relationships involving indeterminate values would support the use of
formulas to represent relationships. To test this conjecture, I gave each student
The Arc Problem (Figure 11). Both students’ formulas emerged as representative
of a multiplicative relationship between a subtended arc and corresponding radius
(Excerpt 8, Excerpt 9).
Using the following diagram, determine a formula between the measurements r,
θ
, and s.
Fig. 11 The Arc Problem
Excerpt 8
1
2
3
Judy: Oh, um, then, theta is equal to s over r…because radians is, it
calculates the number of radius lengths that have passed along the
circumference.
s
r
r
θ
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Excerpt 9
1
2
3
4
5
Zac: We'll say theta equals radians (writing
θ
= rad), very very simple
then. r theta is equal to s (writing r
θ
= s). 'Cause theta is in radians,
that means a percentage of the radius. Which would then be equal
to this length (tracing arc length). So you multiply the percentage
of the radius by the radius, you'll get the arc length.
7. Discussion
Returning to Thompson’s characterization of quantification, quantifying a
quantity involves “a process of settling what it means to measure a quantity, what
one measures to do so, and what a measure means after getting one” (2011, p. 38).
In other words, one’s understanding of a quantity inherently involves her
quantifying of that quantity. I discuss the students’ quantification of angle
measure in two parts. First, I discuss critical operations involved in the students’
quantification of angle measure. I then highlight other central aspects of
quantitative reasoning that emerged as critical for their quantification process,
such as their capacity to reason about indeterminate values.
7.1 The Students’ Quantification of Angle Measure
Upon entering the study, Judy and Zac’s thinking about angle measure was
constrained to situations in which they were given angle measures or numbers
with which to execute calculational strategies to obtain angle measures. Angle
measures, to Judy and Zac, were intrinsic properties of geometric objects and
relatable to other angle measures through calculations (e.g., supplementary or
complementary angles). They appeared to have no discernable scheme of what
one measures when measuring an angle, nor did their angle measure
understandings include a quantitative meaning for the units used to measure the
openness of an angle. Without a robust understanding of the process of measuring
an angle and how the structure of the unit relates to this process, it made little
sense for them to discuss the meaning of angle measures independent of other
angle measures and geometric objects.
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As the study progressed, the students conceived situations in ways that
supported their constructing specific quantitative meanings for angle measure.
More pointedly, by encountering situations in which they engaged in the
partitioning of subtended arc lengths and reflecting on these actions, the students
conceived degree and radian angle measures (as well as other informal units like
quips) as one in the same: both units convey the fractional amount of a circle’s
circumference that is subtended by an angle with that measure. To quantify angle
measure as such required that the students conceive a subtended arc length as a
property of an angle’s openness to be partitioned into unit lengths, where each
unit length is a fractional amount of a circle’s circumference. Encountering
multiple situations in which these actions were purposeful likely supported such
quantification acts, leading to their abstracting angle measure as representative of
quantitative relationships. For instance, their connecting angle measure to the
subtended fractional amount of any circle’s circumference stemmed from
identifying and comparing arc lengths per unit of angle measure on multiple
circles. Similarly, the students came to understand radian angle measures as a
multiplicative relationship that holds for any circle by comparing radii measures
on multiple circles.
Judy and Zac’s capacity to reason about multiplicative relationships, which
included interpreting expressions as representing multiplicative relationships,
played a critical role in their quantification of angle measure. During the pre-
interview, the students used ratios during their solutions, but these ratios did not
represent multiplicative relationships. A critical transition in their thinking
occurred when they conceived several ratios between arc lengths and
circumferences as representing the same multiplicative relationships. As the
students reflected on different arc lengths on different circles corresponding to the
same angle measure, the students reasoned that the ratio a/b conveys that a is a/b
times as large as b. In turn, they concluded that an angle measure conveys the
fractional amount of any circle’s circumference that is subtended by the angle.
Likewise, Judy and Zac concluded that a measure in radians conveys that the
subtended arc is so many times as large the corresponding circle’s radius. Without
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such multiplicative reasoning schemes available, the students likely would have
quantified angle measures is an entirely different manner.
7.2 Quantitative Reasoning and Indeterminate Values
The students came to understand angle measure as the process of: (a)
constructing a circle centered at the vertex of the angle, (b) measuring the circle’s
circumference and subtended arc length, (c) determining the fractional amount of
the circumference subtended by the angle, and (d) determining the number of
angle measure units that correspond to that fractional amount.
12
It is important to
note that the end product of the students’ quantification of angle measure was
such that they were not constrained to carrying out these actions. That is, they
reasoned about such actions and relationships without having to physically carry
them out, execute calculations, or determine specified values.
In principle, quantitative reasoning is based on constructing relationships
between quantities such that specified values are inconsequential to the conceived
relationships (Smith & Thompson, 2008). Specified values come into play when
attempting to determine particular quantities’ values, but they are not required to
reason about quantities and relationships between these quantities. Thus,
quantitative reasoning rests on the capacity to reason about indeterminate values.
As evidence of reasoning based in quantitative relationships, during the latter
parts of the study the students anticipated calculations during their solutions by
reasoning about indeterminate values. The students also produced formulas and
equations that stemmed from quantitative relationships between indeterminate
values. The students’ tendency to reason about indeterminate values and
quantitative relationships, as opposed to numbers and calculations, was most
apparent during the post-interview sessions (Excerpts 5-9). For instance, both
students determined the standard formulas of s = r
θ
and
θ
= s/r by reasoning
about unspecified radian angle measures as measures of arcs in radii. The fluency
by which they produced these formulas, and the reasoning that supported this
12
As an alternative to (c) and (d), the students also reasoned about
multiplicatively comparing the arc length with the radius to determine the arc
length in radii.
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fluency, stands in stark contrast with previous research on students’ and teachers’
understandings of radian angle measure (Akkoc, 2008; Topçu et al., 2006).
In their description of quantitative reasoning, Smith and Thompson (2008)
argued, “If students are eventually to use algebraic notation and techniques to
express their ideas and reasoning productively, then their ideas and reasoning
must become sufficiently sophisticated to warrant such tools” (pg. 98). In the
present study, the students’ reasoning indicates that paying particular attention to
students’ quantification of a quantity can support the productive use of algebraic
notation and techniques. By developing understandings rooted in quantities and
relationships between these quantities, the students’ understandings afforded the
use of formulas, equations, and calculations to represent relationships between
quantities and indeterminate values.
8. Concluding Remarks
The results of this study contribute to the growing body of research (e.g.,
Castillo-Garsow, 2010; Ellis, 2007; Johnson, 2012; Thompson, 1994) that reveals
the critical role of quantitative reasoning in learning mathematics. Specifically,
the students’ progress was contingent on ideas of measurement, multiplicative
reasoning, and reasoning about indeterminate values, all central strands of
quantitative reasoning. How students who lack critical measurement and
multiplicative reasoning schemes would progress during a similar instructional
sequence remains a question open for investigation. Studies that pursue students’
progress under these circumstances have the potential to contribute to the body of
knowledge on students’ angle measure understandings, including how various
reasoning abilities, or the lack thereof, influence their angle measure
understandings.
In addition to being of interest to teachers and researchers of undergraduate
precalculus students, the findings are also pertinent to those involved in the
teaching and learning of secondary students. Research has identified that
quantitative reasoning can support secondary students’ learning of precalculus
ideas, including exponential (Castillo-Garsow, 2010) and quadratic (Ellis, 2007)
functions. The students of the present study also showed performance in their
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precalculus course compatible to that of secondary precalculus students. Thus,
secondary students might also benefit from an approach to angle measure that is
based on principles of quantitative reasoning. Future studies that investigate
secondary students’ quantification of angle measure are needed, the results of
which will likely produce additional insights into the role of quantitative
reasoning in secondary students’ learning.
While the study’s findings suggest that an arc approach to angle measure can
foster coherent experiences for students, I caution the reader to conclude that all
arc approaches to angle measure will produce such experiences. Instead, the
students’ actions emphasize that an arc approach to angle measure is beneficial
when the approach foregrounds quantitative reasoning, and namely the
quantification of angle measure. Quantification is not a passive activity; it is a
cognitive process. As Thompson (2011) described, the quantification process is a
critical and complex part of conceiving a quantity and this process takes time
(sometimes years). If we, as educators, expect students to understand angle
measure (and other quantities), then we must take seriously their quantification of
angle measure, which includes creating a need and giving time for students’
quantification acts. Introducing angle measures (and using arcs) without taking
seriously the quantification of angle measure likely sends students the message:
use these numbers to perform calculations and find other numbers, but do not
worry about what the numbers, arcs, and calculations mean. Likewise, giving
students a protractor without addressing how to measure an angle in the absence
of protractor likely sends the message: use this protractor to measure an angle, but
do not worry about the structure behind the protractor. In contrast, placing
quantification at the forefront of the teaching of angle measure positions students
to conceive quantities and relationships between quantities as inherent to the
process of measuring an angle.
Acknowledgements: National Science Foundation (NSF) grant number EHR-
0412537, led by Principal Investigator Marilyn Carlson, supported this work. All
opinions expressed are solely those of the author and do not necessarily reflect the
views of the NSF. I thank Marilyn Carlson, Andrew Izsák, and the reviewer team
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for providing feedback on previous versions of this article. I also thank Pat
Thompson for his substantive conversations about angle measure and quantitative
reasoning. These conversations have undoubtedly influenced my thinking on these
topics and shaped the study.
References
Akkoc, H. (2008). Pre-service mathematics teachers' concept image of radian.
International Journal of Mathematical Education in Science and
Technology, 39(7), 857-878.
Bressoud, D. M. (2010). Historical reflections on teaching trigonometry.
Mathematics Teacher, 104(2), 106-112.
Brown, S. A. (2005). The trigonometric connection: Students' understanding of
sine and cosine. Ph.D. Dissertation. Illinois State University: USA.
Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying
covariational reasoning while modeling dynamic events: A framework and
a study. Journal for Research in Mathematics Education, 33(5), 352-378.
Carlson, M., Oehrtman, M., & Engelke, N. (2010). The precalculus concept
assessment (PCA) Instrument: A tool for assessing reasoning patterns,
understandings, and knowledge of precalculus level students. Cognition
and Instruction, 28(2).
Castillo-Garsow, C. C. (2010). Teaching the Verhulst model: A teaching
experiment in covariational reasoning and exponential growth. Ph.D.
Dissertation. Arizona State University, Tempe, AZ.
Clement, J. (2000). Analysis of clinical interviews: Foundations and model
viability. In A. E. Kelly & R. A. Lesh (Eds.), Research design in
mathematics and science education (pp. 547-589). Mahwah, NJ: Lawrence
Erlbaum Associates, Inc.
Clements, D. H., & Burns, B. A. (2000). Students' development of strategies for
turn and angle measure. Educational Studies in Mathematics, 41(1), 31-45.
Ellis, A. B. (2007). The influence of reasoning with emergent quantities on
students' generalizations. Cognition and Instruction, 25(4), 439-478.
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
32
Fi, C. (2003). Preservice secondary school mathematics teachers' knowledge of
trigonometry: Subject matter content knowledge, pedagogical content
knowledge and envisioned pedagogy. Ph.D. Dissertation. University of
Iowa: USA.
Goldin, G. A. (2000). A scientific perspective on structured, task-based interviews
in mathematics education research. In A. E. Kelly & R. A. Lesh (Eds.),
Research design in mathematics and science education (pp. 517-545).
Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
Johnson, H. L. (2012). Reasoning about variation in the intensity of change in
covarying quantities involved in rate of change. Journal of Mathematical
Behavior, 31(3), 313-330.
Kieran, C. (1986). 'LOGO and the notion of angle among fourth and sixth grade
children'. In L. Burton & C. Hoyles (Eds.), Proceedings of the 10th
International Conference on the Psychology of Mathematics Education
(pp. 99-104). London.
Matos, J. (1990). The historical development of the concept of angle. The
Mathematics Educator, 1(1), 4-11.
Mitchelmore, M. C., & White, P. (2000). Development of angle concepts by
progressive abstraction and generalization. Educational Studies in
Mathematics, 41(3), 209-238.
Moore, K. C., & Carlson, M. P. (2012). Students’ images of problem contexts
when solving applied problems. The Journal of Mathematical Behavior,
31(1), 48-59.
Moore, K. C., LaForest, K., & Kim, H. J. (2012). The unit circle and unit
conversions. Proceedings of the Fifteenth Annual Conference on Research
in Undergraduate Mathematics Education (pp. 16-31). Portland, OR:
Portland State University.
Smith III, J., & Thompson, P. W. (2008). Quantitative reasoning and the
development of algebraic reasoning. In J. J. Kaput, D. W. Carraher & M.
L. Blanton (Eds.), Algebra in the Early Grades (pp. 95-132). New York,
NY: Lawrence Erlbaum Associates.
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
33
Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology:
Underlying principles and essential elements. In A. E. Kelly & R. A. Lesh
(Eds.), Research design in mathematics and science education (pp. 267-
307). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
Strauss, A. L., & Corbin, J. M. (1998). Basics of qualitative research : techniques
and procedures for developing grounded theory (2nd ed.). Thousand
Oaks: Sage Publications.
Thompson, A. G., Philipp, R. A., Thompson, P. W., & Boyd, B. A. (1994).
Calculational and conceptual orientations in teaching mathematics. In A.
Coxford (Ed.), 1994 Yearbook of the NCTM (pp. 79-92). Reston, VA:
NCTM.
Thompson, P. W. (1990). A theoretical model of quantity-based reasoning in
arithmetic and algebraic. Center for Research in Mathematics & Science
Education: San Diego State University.
Thompson, P. W. (1994). Images of rate and operational understanding of the
fundamental theorem of calculus. Educational Studies in Mathematics,
26(2-3), 229-274.
Thompson, P. W. (2008). Conceptual analysis of mathematical ideas: Some
spadework at the foundations of mathematics education. In O. Figueras, J.
L. Cortina, S. Alatorre, T. Rojano & A. Sépulveda (Eds.), Proceedings of
the Annual Meeting of the International Group for the Psychology of
Mathematics Education (Vol. 1, pp. 45-64). Morélia, Mexico: PME.
Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In
S. Chamberlin, L. L. Hatfield & S. Belbase (Eds.), New perspectives and
directions for collaborative research in mathematics education: Papers
from a planning conference for WISDOM^e. Laramie, WY: University of
Wyoming.
Thompson, P. W., Carlson, M. P., & Silverman, J. (2007). The design of tasks in
support of teachers' development of coherent mathematical meanings.
Journal of Mathematics Teacher Education, 10, 415-432.
Topçu, T., Kertil, M., Akkoc, H., Kamil, Y., & Osman, Ö. (2006). Pre-service and
in-service mathematics teachers' concept images of radian. In J. Novotná,
MAKING SENSE ANGLES
Pre-proof version of: Moore, K. C. (2013). Making sense by measuring arcs: A
teaching experiment in angle measure. Educational Studies in
Mathematics, 83(2), 225-245.
34
H. Moraová, M. Krátká & N. Stehlíková (Eds.), Proceedings of the 30th
Conference of the International Group for the Psychology of Mathematics
Education (Vol. 5, pp. 281-288). Prague: PME.
Weber, K. (2005). Students' understanding of trigonometric functions.
Mathematics Education Research Journal, 17(3), 91-112.
